text stringlengths 8 1.01M |
|---|
Everything you get from a traditional test prep book PLUS one-on-one instruction covering all the math on the ACT.**"I just love YourTeacher and the way you explain things. I felt like I was in a classroom instead of just looking at examples."Diane"I am needing prep work for my daughter's ACT. It's been very helpful and I do like how it's set up. The lessons are very clear and understandable - I wish I had this when I was in school!!"Ann"My daughter, who's a junior, is using Your Teacher to prepare for the ACT in the spring. She loves the interactive, go at your on pace style of Your Teacher. I absolutely love the program and certainly would and have recommended it."Sonya**Need more than practice problems to get ready for the ACT…YourTeacher's ACT Math Prep Course covers the exact math you need to know to ace the ACT. Our app includes everything you would get in a traditional ACT test prep course (i.e. practice problems), but ALSO includes the one-on-one instruction you need to truly learn math.Our lessons include:-Multiple video example problems(similar to how a teacher starts class at the board by explaining the examples from the textbook)-Interactive practice problems with built-in support(similar to how a teacher assigns practice and walks around the class providing help)-A Challenge Problem(similar to how a teacher assigns a higher level problem which students must work on their own to prove mastery)-Extra problem worksheets(similar to how a teacher assigns additional problems for homework)-Review notes(similar to how a teacher provides summary handouts or refers you to your textbook)******ACT Math Prep App - Scope and SequenceYourTeacher's ACT Math Prep App covers the exact math you need to ace the ACT. CHAPTER 1: INTEGERSAddition and SubtractionMultiplication and DivisionOrder of OperationsEvaluationAbsolute ValueCHAPTER 2: FRACTIONSDivisibility RulesFactors and PrimesPrime FactorizationMultiples and Least Common MultiplesGreatest Common FactorIntroduction to FractionsEquivalent Fractions (Part I)Lowest TermsEquivalent Fractions (Part II)Improper Fractions and Mixed NumbersComparing Proper FractionsComparing Mixed Numbers and Improper FractionsLeast Common MultipleAddition and SubtractionMultiplication and DivisionOrder of OperationsEvaluationAbsolute ValueCHAPTER 3: DECIMALSUnderstanding DecimalsComparing DecimalsRounding DecimalsLong DivisionDividing Decimals by Whole NumbersDividing Decimals by DecimalsPowers of TenConverting from Scientific to Standard NotationConverting from Standard to Scientific NotationCHAPTER 4: EXPRESSIONS AND EQUATIONSCombining Like TermsDistributive PropertyDistributive / Like TermsOne-Step EquationsTwo-Step EquationsEquations with FractionsEquations Involving DistributiveVariable on Both SidesVariable on Both Sides / FractionsVariable on Both Sides / DistributiveEquations with DecimalsEquations with Fractional SolutionsBeginning FormulasAdvanced FormulasCHAPTER 5: WORD PROBLEMSModeling Two-Step EquationsNumber ProblemsConsecutive Integer ProblemsGeometry ProblemsValue ProblemsIntroductory Motion ProblemsAdvanced Motion ProblemsCHAPTER 6: INEQUALITIES, ABSOLUTE VALUE, FUNCTIONS, & GRAPHINGCHAPTER 7: LINEAR EQUATIONSCHAPTER 8: SYSTEMS OF EQUATIONSCHAPTER 9: EXPONENTS & POLYNOMIALSCHAPTER 10: FACTORINGCHAPTER 11: RATIONAL EXPRESSIONS & EQUATIONSCHAPTER 12: RADICALS & QUADRATICSCHAPTER 13: PROBABILITY & STATISTICSCHAPTER 14: INTRODUCTION TO GEOMETRYCHAPTER 15: PARALLEL LINES & POLYGONSCHAPTER 16: RATIO, PROPORTION, & PERCENTCHAPTER 17: RIGHT TRIANGLESCHAPTER 18: CIRCLESCHAPTER 19: MEASUREMENTCHAPTER 20: ADVANCED AREA & VOLUME(Wifi or 3G connection required)ACT is the registered trademark of ACT, Inc. YourTeacher has no affiliation with ACT, Inc., and the YourTeacher ACT app is not approved or endorsed by ACT |
17
3 3
2 6
1 2Similar
Learn computer science in a fun way. There are lot of subjects to explore and learn. You don't need internet and need not worry about ads. Its completely educational with the hope of making you learn somethingDownload Computer Science Dictionary today to know more about the latest computer terminology and the actual meaning.
Main Features: 1. Equipped with quick dynamic search function – The dictionary will start searching for the words while you type. 2. Bookmark – you are able to bookmark the Computer Science Computer limited review of significant people in computing history. The intended audience is High School to Computer Science Undergraduate students, professionals refreshing their knowledge, and people that like triviaAP Math and Computer science can stand for many distinct Advanced Placement Math & Computer science high school students, provided by the College Board: - AP Calculus AB is traditionally taken after precalculus and is the first calculus course offered at most schools except for the regular calculus class. The Pre-Advanced Placement pathway for math will help prepare students for further Advanced Placement classes and exams. - Calculus BC is a full-year course in the calculus of functions of a single variable. It includes all topics covered in Calculus AB plus additional topics...Students who take an AP Calculus course should do so with the intention of placing out of a comparable college calculus course - Advanced Placement Computer Science is meant to be the equivalent of a first-semester course in computer science The AP exam currently tests students on their knowledge of Java. - Advanced Placement Statistics (AP Statistics, AP Stat or AP Stats) is equivalent to a one semester, non-calculus-based introductory college statistics course and is normally offered to juniors and seniors in high school.
In this app, we provide many Questions and Flashcards, They may help you to prepare for all types of AP Math & Computer science. Practice them everyday to get high score, Good luck :) applicationsThis application is an excellent way to explore the innovative Business, Computer Science, and Technology courses in Ontario high schools. The application includes information about each course along with extra information such as business competitions and an interactive game. The game involves identifying common Myths about Computer Science. Explore and enjoy. Updates to follow. |
You are here
Mathematical Modeling
MATH
340
MATH 340 - Mathematical Modeling3 hours An introduction to the development and analysis of deterministic and probabilistic models. Includes curve fitting, simulations, difference and differential equations. Applications from ecology, environmental science, economy and other fields.
Pre-Requisites:
Prerequisite(s): MATH 262 with C or better, and sophomore standing or higher. |
In his 1957 book, How to solve it, Polya describes a four-stage approach to mathematical problem-solving. He bases his approach on common-sense questions that would naturally occur to an experienced problem-solver. Polya claims teachers should pose these questions to students in as natural and unobtrusive a way as possible, the goal being to encourage independence and internalization of this framework. A geometry.pre-college newsgroup discussion. |
More About
This Textbook
Overview
Mathematics is a universal language. Differential equations, mathematical modeling, numerical methods and computation form the underlying infrastructure of engineering and the sciences. In this context mathematical modeling is a very powerful tool for studying engineering problems, natural systems and human society. This interdisciplinary book contains a comprehensive overview, including practical examples, of the progress achieved to date in the modeling of coupled phenomena, computational mathematics and mechanics, heat transfer, fluid-structure interactions, biomechanics, and the flow of mass and energy in porous media. Numerical subjects such as grid generation, optimization, finite elements, finite differences, spectral methods, boundary elements, finite volumes and meshless methods are also discussed in detail using real examples.
The book provides a thorough presentation of the existing numerical techniques with specific applications to concrete, practical topics. The models and solutions presented here describe various systems: mechanical, biological, geophysical, technical, ecological, etc. The book is organized in thirty six chapters, each written by distinguished experts in their respective fields. The topics presented cover the current state of knowledge in numerical engineering practice including recent and ongoing developments and the presentation of new ideas for future research on applied computational engineering mathematics.
The book will be of interest to scientists working in engineering (structural, civil, mechanical), geology, geophysics, aquifer research, petroleum engineering, applied mathematics, and physics, as well as students in any of these |
Description:
This first semester of seventh-grade math teaches mathematics as a step-by-step process. The course includes adding, subtracting, multiplying, and dividing decimals and fractions. Pre-algebraic expressions and equations, number theory, and geometry are also covered 7A / Online
Schedule Number:
10004
Instructor(s):
Robin Cottrell
Location:
Dates:
Units:
0.5 Academic Credits
Lessons/Exams:
7 |
Note: Offsite links will open in a new browser window if your browser is capable of doing that. Inclusion of offsite links should not be considered an endorsement of any product offers nor any guarantee of accuracy of information presented. If you find any broken links, or have other suggestions, please e-mail the webmaster using the link at the bottom of the page.
Math
Help for Math Anxiety --link to a page at Middle Tennessee State University. Gives study tips, how to be successful in math classes, how to deal with math anxiety, and more!
Coping with Math Anxiety --a mini-text with a definition, information about the roots of math anxiety, and how to overcome it.
Mathnotes.com --link to a site with tutorials for slightly different prealgebra, elementary algebra, and intermediate algebra books from the ones we used to use, but by the same authors. You'll need to match book names and chapter titles with the ones in our books.
"Math for Morons Like Us" -- Despite the title, this is a site developed by high school students to teach each other prealgebra, algebra, and other math topics. This could be helpful for both elementary and intermediate algebra. You may need to browse through both Algebra I and Algebra II to find the topic you need. Also has information helpful for studying prealgebra as well as precalculus and calculus.
Professor Freedman's Math Help has tutorials in developmental math written by students in those courses. It also has information on math anxiety, learning styles and study skills.
Math Power--Also known as Professor Freedman's Math Help. Written by a developmental math professor at another college, this site includes algebra tutorials she wrote as well as some written by students.
SOS Math is not quite as user friendly as some of the other sites, but does cover a wide variety of topics and goes way beyond basic algebra and into calculus and higher courses.
Math Expression is another math tutorial site. It has the disadvantage of having a fair number of advertisements all over each page.
Math TV is a website that has videos that cover many of the same types of problems that you encounter in your course. On the left side, look under algebra (you may have to click on the word to expand that topic) and then click on a larger topic (which are similar to the titles of our chapters). Below that you'll see a number of sample problems, and each problem is usually worked out by several different students on the video screen on the right. Click on the student's name or picture to see the video. There ads at the bottom of the video screen, but you can click on the X in the upper right hand corner of the ad to get rid of them.
Practical Study Strategies and Tips --Link to a page at the University of Illinois at Chicago with practical tips on note taking, concentration, studying for exams, studying with a group, etc. Don't use their explanation of how to calculate GPA, though--we calculate ours with a different scale. |
COURSE NUMBER AND TITLE: MATH 1101 Introduction to Mathematical Modeling
CREDIT HOURS: 3
CATALOG DESCRIPTION: An introduction to mathematical modeling using graphical, numerical, symbolic, and verbal techniques to describe and explore real-world data and phenomena. Emphasis is on the use of elementary functions to investigate and analyze applied problems and questions, supported by the use of appropriate technology, and on effective communication of quantitative concepts and results. (Credit will not be awarded for both MATH 1101 and MATH 1111. Not recommended for students required to take MATH 1113 or MATH 1220.) |
Survey of Mathematics with Applications, A (9th Edition)
9780321759665
ISBN:
0321759664
Edition: 9 Pub Date: 2012 Publisher: Addison Wesley
Summary: This textbook serves as a broad introduction to students who are looking for an overview of mathematics. It is designed in such a way that students will actually find the text accessible and be able to easily understand and most importantly enjoy the subject matter. Students will learn what purpose math has in our lives and how it affects how we live and how we relate to it. It is not heavy on pure math; its purpose ...is as an overview of mathematics that will enlighten students without an intense background in math. If you want to obtain this and other cheap math textbooks we have many available to buy or rent in great condition online.
Allen R. Angel is the author of Survey of Mathematics with Applications, A (9th Edition), published 2012 under ISBN 9780321759665 and 0321759664. Nine hundred twenty seven Survey of Mathematics with Applications, A (9th Edition) textbooks are available for sale on ValoreBooks.com, three hundred twenty seven used from the cheapest price of $70.05, or buy new starting at $132 Instructor Edition: Same as student edition with additional notes or answers. May include moderately worn cover, writing, markings or slight discoloration. SKU:9780321639288 ***Warning***Text Only. Still in Shrink Wrap Annotated Instructor's Copy, 9th edition but No Supplementary Materials otherwise same as student with help added tips,and answers.Shipping from California.[less] |
Download
Chapter 9: Polynomials and Factoring; More on Probability
Chapter Outline
Loading Contents
Chapter Summary
Image Attributions
Description
This chapter introduces students to polynomials and their basic operations as well as the process of factoring polynomials, quadratic expressions, and special products. Also considered is probability through compound events. |
Latihan Matematik Tingkatan 4 |
Wolfram|Alpha as a self-verification tool
Last week, I wrote about structuring class time to get students to self-verify their work. This means using tools, experiences, other people, and their own intelligence to gauge the validity of a solution or answer without uncritical reference an external authority — and being deliberate about it while teaching, resisting the urge to answer the many "Is this right?" questions that students will ask.
Among the many tools available to students for this purpose is Wolfram|Alpha, which has been blogged about extensively. (See also my YouTube video, "Wolfram|Alpha for Calculus Students".) W|A's ability to accept natural-language queries for calculations and other information and produce multiple representations of all information it has that is related to the query — and the fact that it's free and readily accessible on the web — makes it perhaps the most powerful self-verification tool ever invented.
For example, suppose a student were trying to calculate the derivative of \(y = \frac{e^x}{x^2 + 1}\). Students might forget the Quotient Rule and instead try to take the derivative of both top and bottom of the fraction, giving:
\(y' = \frac{e^x}{2x}\)
Then, if they're conscientious students, they'll ask "Is this right?" What I suggest is: What does Wolfram|Alpha say? If we type in derivative of e^x/(x^2+1) into W|A, we get:
The derivative W|A gets is clearly nowhere near the derivative we got, so one of us is wrong… and it's probably not W|A. Even if we got the initial derivative right in an unsimplified form, the probability of a simplification error is pretty high here thanks to all the algebra; we can check our work in different ways by looking at the alternate form and at the graphs. (Is my derivative always nonnegative? Does it have a root at 0? If I graph my result on a calculator or Winplot, does it look like the plot W|A is giving me? And so on.)
But how is this better than just having a very sophisticated "back of the book", another authority figure whose correctness we don't question and whose answers we use as the norm? The answer lies in the "Show steps" link at the right corner of the result. Click on it, and we get the sort of disclosure that oracles, including backs of books, don't usually provide:
Every step is generated in complete detail. Some of the details have to be parsed out (especially the first line about using the Quotient Rule), but nothing is hidden. This makes W|A much more like an interactive solutions manual than just the back of the book, and the ability given to the student to verify the correctness of the computer-generated solution is what makes W|A much more than just an oracle whose results we take on faith.
Using W|A as a self-checking tool also trains students to think in the right sort of way about reading — and preparing — mathematical solutions. Namely, the solution consists of a chain of steps, each of which is verifiable and, above all, simple. "Differentiate the sum term by term"; "The derivative of 1 is zero". When students use W|A to check a solution, they can sit down with that solution and then go line by line, asking themselves (or having me ask them) "Do you understand THIS step? Do you understand THE NEXT step?" and so on. They begin to see that mathematical solutions may be complex when taken in totality but are ultimately made of simple things when taken down to the atomic level.
The very fact that solutions even have an "atomic level" and consist of irreducible simple steps chained together in a logical flow is a profound idea for a lot of students, and if they learn this and forget all their calculus, I'll still feel like they had a successful experience in my class. For this reason alone teachers everywhere — particularly at the high school level, where mechanical fluency is perhaps more prominent than at the college level — ought to be making W|A a fixture of their instructional strategies |
I just finished self-studying Calculus using this book. During the last six months that it took me to finish, never did I find the need for a teacher.
This book covers the topics taught in the first three semesters of an undergraduate math-related course. There are 14 chapters in all: 1) Functions, Graphs, and Models 2) Prelude to Calculus 3) The Derivative 4) Additional Applications of the Derivative 5) The Integral 6) Applications of the Integral 7) Techniques of Integration 8) Differential Equations 9) Polar Coordinates and Parametric Curves 10) Infinite Series 11) Vectors, Curves, and Surfaces in Space 12) Partial Differentiation 13) Multiple Integrals 14) Vector Calculus.
Each chapter begins with a page of related historical details in order to engage the reader. This is followed by around 4-10 sections, each of which has an exercise with 30 problems (on an average) and 10 True/False questions. After reading the theory for a section, solve 10 problems from the exercise. [Make sure you use MATLAB (or any similar software) and a Graphing Calculator whenever a problem requires it. Wolfram Mathematica Online Integrator is another useful tool.] Then do the True/False questions and move on to the next section. |
Summary: The authors help students ''see the math'' through their focus on functions; visual emphasis; side-by-side algebraic and graphical solutions; real-data applications; and examples and exercises. By remaining focused on today's students and their needs, the authors lead students to mathematical understanding and, ultimately, success in class.26.39 +$3.99 s/h
Good
DOLLHOUSE BOOKS CALUMET CITY, IL
Good INCLUDES GRAPHING CALCULATOR MANUAL. BOK IS IN GOOD CONDITION BUT HAS VERY SLIGHT RAIN DAMAGE ON THE TOP EDGE OF SOME PAGES. DOES NOT INTERFERE WITH THE READING OF THE PAGES. BOOK IS CLEAN INSI...show moreDE. COVER MAY HAVE SLIGHT WEAR ON CORNERS. WILL SHIP BEST AVAILABLE COPY. ...show less
$134 |
Have additional students using Teaching Textbooks Math 6? This additional student workbook and answer booklet will allow extra students to complete the course in their own book. Perfect for co-ops or siblings! Workbook is 623 pages, softcover, spiral-bound. Answer booklet is 46 pages, softcover. This set does NOT include CD-ROMs; this book is not designed to be used without the Math 6 CD-ROMs. Teaching Textbooks Grade 6 Math 6: Teaching Textbooks, Extra Workbook
Review 1 for Math 6: Teaching Textbooks, Extra Workbook
Overall Rating:
5out of5
Excellent curriculum for independent math learning
Date:April 4, 2011
Jimmie
Age:35-44
Quality:
5out of5
Value:
5out of5
Meets Expectations:
5out of5
My daughter completed the Teaching Textbooks 5 and wanted to continue on with level 6. For 5 we used both CDs and the workbook, but she often didn't need the book. So for 6 I bought the CDs only. After a few lessons, she requested a workbook. She realized that copying the problems onto paper introduced more possibility for error and took longer. She also told me that the book had information that she liked to turn back and reference. You certainly can use TT curriculum without the workbook, but having the book makes it much easier.
Share this review:
0points
0of0voted this as helpful.
Review 2 for Math 6: Teaching Textbooks, Extra Workbook
Overall Rating:
5out of5
These math books are wonderful. Easy to learn!!
Date:March 26, 2011
Chris
Location:Newton, NC
Age:35-44
Gender:female
Quality:
5out of5
Value:
5out of5
Meets Expectations:
5out of5
I am so glad that we found the Teaching Textbook. They make learning math so much easier. We will absolutely continue to use them each year. Each lesson teaches something new, but reviews the past lessons. That way you continue to keep reviewing what you have learned. We did not get the cd's this year, but we will for next year.
Share this review:
0points
0of0voted this as helpful.
Review 3 for Math 6: Teaching Textbooks, Extra Workbook
Overall Rating:
5out of5
The content is wonderful!
Date:November 3, 2010
Renee
Location:Charlotte, NC
Age:35-44
Quality:
5out of5
Value:
5out of5
Meets Expectations:
5out of5
This is the first time I have changed our math curriculum in seven yrs. of homeschooling. I love the teaching textbooks. The workbook is great practice to reinforce what is on the CDs. I would highly recommend this program. |
Mathematical Physics The Basics
9781420053029
ISBN:
1420053027
Pub Date: 2007 Publisher: C R C Press LLC
Summary: The first of a two-volume set, Mathematical Physics: The Basics provides a masterful introduction to the mathematical methods encountered by undergraduate students in physics, chemistry, and engineering. Topics include vectors and Cartesian tensors, vector calculus, Lorentz tensors, curvilinear coordinates, linear vector spaces and linear operators, eigenvalues and eigenvectors, matrix representations and diagonaliza...tion of matrices, complex variables, analytic functions, Taylor and Laurent series, contour integrals, residue theorem and applications, and method of steepest descent. Numerous worked examples and problems are featured to demonstrate principles and facilitate learning.
Joglekar, S D is the author of Mathematical Physics The Basics, published 2007 under ISBN 9781420053029 and 1420053027. One Mathematical Physics The Basics textbook is available for sale on ValoreBooks.com, or buy new starting at $168.75.[read more] |
Share
WIRIS cas
WIRIS cas is an online platform for mathematical calculations designed for education. You can access a powerful calculation toolbar through an HTML page that includes integrals and limits calculation, function graphing in 2D or 3D and symbolic matrices manipulation, among others.
WIRIS cas covers all mathematical topics from primary school to university level (Calculus, Algebra, Geometry, Differential Equations...).
Sort
You're browsing Popular Alternatives to WIRIS casIRIS casage is a free open-source mathematics software system licensed under the GPL. It combines the power of many existing open-source packages into a common Python-based interface. Mission: Creating a viable...
Microsoft Mathematics is a desktop graphing calculator that can help you visualize and see mathematical concepts as you've never seen them before. Microsoft Mathematics includes ink handwriting support toSMath Studio - is a math program with 'paper'-like interface and numerous computing features. It has an ability to work with systems, matrices, vectors, complex numbers, infinities and fractions. SMath |
More
2,500 fully worked problems of varying difficulty
Clear, concise explanations of arithmetic, algebra, and geometry
Outline format supplies a concise guide to the standard college courses in elementary mathematics
Appropriate for the following courses: Basic Mathematics, Elementary Mathematics, Introduction to Mathematics, Review of Arithmetic, Elementary Algebra, Review of Algebra, Business Mathematics I, Math for the GED
Detailed explanations and practice problems in arithmetic, algebra, and geometry
Description:
Study faster, learn better, and get top gradesA clear review
of standard college course of Mathematics for Elementary School Teachers, this book will be designed to improve your basic knowledge of math content required for this level, engage you ... |
Introduction to Algebra 1 Jacobs, Russell F. Jacobs Holt Rinehart and Winston HardBack 1993 602 pages *****there is a small crease in upper right hand corner of cover*****PLEASE
Anthology of Science Fiction by Rinehart and Winston Staff Holt ISBN 0-03-052947-6 Condition :Pre-owned Like new condition. Student/school name blacked out. No writing/highlight ing in text. Binding Tight. Description:Part I Life Science Part II Earth Science Part III Physical Science Answer Keys for the think about it questions at the end of each segment. Shipping Policy All orders are processed within 2 business days. Estimated shipping time is 3-5 days. If you have not received your items with
WORLD HISTORY Continuity and Change Holt, Rinehart and Winston Edited by William Travis hanes III Brand new Never read I ship everyday except Sundays and Holidays and always provide tracking Retail Price is $106.00 eaglepri2 Store
Buy With 100% Confidence Fast Shipping !! Brand New!! Check Other Great Sales | Read our Feedbacks | Add us to Favorite Sellers Item Description: Take students a step further in learning algebra Specially written for low-level learners, Algebra 2 covers several methods for solving quadratic equations, such as factoring, completing the square, and graphing. The text also introduces trigonometry and exponential functions—vital concepts for real world applications. Filled with full-color illust |
"Advanced Mathematical Concepts, (c)2006 provides comprehensive coverage of all the topics covered in a full-year Precalculus course. Its unique unit organization readily allows for semester courses in Trigonometry, Discrete Mathematics, Analytic Geometry, and Algebra and Elementary Functions. Pacing and Chapter Charts for Semester Courses are conveniently located on page T4 of the Teacher Wraparound Edition. <BR>"Advanced Mathematical Concepts lessons develop mathematics using numerous examples, real-world applications, and an engaging narrative. Graphs, diagrams, and illustrations are used throughout to help students visualize concepts. Directions clearly indicate which problems may require the use of a graphing calculator |
The consequence of multiplication
Life is all arithmetic. We try to add to our incomes, subtract from our waistlines, divide our time, and for a while, we avoid multiplying. (Lest we not secure a career that adds to our income!) But a time comes when we want to be fruitful and multiply. We want to add to the legion of geeks.
Then we become the parent that says, "If I've told you n times, I've told you n + 1 times... clean your room or you won't get to eat any of the first derivative of a cow tonight!"* This book is for the parents of future mathletes. It's the storybook adventure of two friends as they discover the wonders of calculus. Who knows, you might even learn something yourself, parental unit!
*Prime Rib
Author: Omi M. Inouye
Publisher: omionline.ca
ISBN: 978-0987823915
Pages: 54
Cover: Softcover
Year: 2011
Edition: 1st
Genre: Reference
Wanna chat about Introductory Calculus For Infants?
Have questions about Introductory Calculus For Infants or your order? We monitor these comments daily, but it may be faster to email us directly or call us at 1-888-GEEKSTUFF. |
The nation's first choice for an NSF reform high school mathematics series This new 2nd edition features a colorful lesson design; earlier development of algebraic topics; expanded use of technology; pre-requisite skills review in every lesson; Unit... (read more)
Tessellations--shapes repeated over and over to fill a plane without overlapping--have inspired beautiful art, from intricate tile work to M.C. Escher's playful graphics. Now, master origami artist Eric Gjerde has produced the same stunning kaleidoscopic... (read more)
The two volumes contain 65 chapters, which are based on talks presented by reputable researchers in the field at the Tenth International Conference on Integral Methods in Science and Engineering. The chapters address a wide variety of methodologies, |
Algebra IA
Specific subject areas of study include a review of elementary algebra, number systems and set notation, linear and absolute value functions, equations and inequalities, systems of linear equations, inequalities, nonlinear systems, radicals and complex numbers, quadratic functions, equations and inequalities, rational expressions and equations, and function notation. The graphing calculator is introduced and used frequently throughout the course.
Algebra
Specific subject areas of study include a review of elementary algebra, number systems and set notation, linear and absolute value functions, equations and inequalities, systems of linear equations, inequalities, nonlinear systems, radicals, quadratic functions, equations, rational expressions and equations, and function notation. Time is allowed for individualized instruction.
MATH IIA: GEOMETRY
This course develops geometry as a system of logical reasoning using clear definitions, postulates, theorems and corollaries to foster creative and critical thinking. Geometric elements, structure and relationships are presented and integrated with algebra, trigonometry, logic and arithmetic. Deductive reasoning is introduced and applied to geometry to develop formal mathematical proofs. The topics of congruence, parallelism, quadrilaterals, circles, similarity, areas, regular polygons, coordinate geometry and locus are studied from geometric, algebraic and graphic perspectives for the improvement of reasoning. This course amply covers all the geometry necessary for achievement on the SAT I.
Permutations and Combinations
Students learn how to compute the number of possible arrangements we can make within a particular group of items with and without regard to the order in which the items appear in each arrangement. Students also calculate the probabilities of particular arrangements occurring. Students will work with simple arrangements such as rolling dice and drawing cards as well as with more complex arrangements such as license plates and telephone numbers. |
This course is an introduction to algebraic and enumerative combinatorics. You will discover the beautiful interplay
between algebra and combinatorics, learning how to
apply algebraic techniques to solve enumeration problems, and how to use
combinatorial methods to solve questions arising in other areas of mathematics.
No prior knowledge of combinatorics is expected, but
some familiarity with linear algebra and group theory is preferable.
The homework will consist of a problem set roughly every two weeks.
Collaboration is permitted, but you are not allowed to copy someone else's
work. The solutions must be written individually. You have to mention on your
problem set the names of the students that you worked with, and also which
books or articles you used.
The final will be a take-home exam. You must work on the problems on your own. No
collaboration permitted in the exam.
The final project will consist of preparing a topic and presenting it in
class. Students can work in pairs. Here are possible topics
for the project.
Students with disabilities: Students with disabilities enrolled in this
course that may need disability-related classroom accommodations are encouraged
to make an office appointment to see me before the end of the second week of
the term. All discussions will remain confidential, although the Student
Accessibility Services office may be consulted to discuss appropriate implementation of any
accommodation requested. |
req... read moreMathematical Tools for Physics by James Nearing Encouraging students' development of intuition, this original work begins with a review of basic mathematics and advances to infinite series, complex algebra, differential equations, Fourier series, and more. 2010 edition.
Mathematical Theory of Compressible Fluid Flow by Richard von Mises Suitable for advanced undergraduate and graduate students, this text covers general theorems, conservation equations, waves, shocks, and nonisentropic flows, with emphasis on the basics, both conceptual and mathematical. 1958 edition.
Dynamics of Fluids in Porous Media by Jacob Bear This is the definitive work on the subject by one of the world's foremost hydrologists, designed primarily for advanced undergraduate and graduate students. 335 black-and-white illustrations. Exercises, with answers.
Hydrodynamics by Sir Horace Lamb A classic presentation of the subject and a standard reference, unexcelled for its exposition of fundamental theorems, equations, and detailed methods of solution. 1932Mathematical Modelling Techniques by Rutherford Aris "Engaging." — Applied Mathematical Modelling. A theoretical chemist and engineer discusses the types of models — finite, statistical, stochastic, and more — as well as how to formulate and manipulate them for best results.
Product Description: |
Transformation Geometry : An Introduction to Symmetry - 82 edition
Summary: Transformation Geometry: An Introduction to Symmetryis a modern approach to Euclidean Geometry. This study of the automorphism groups of the plane and space gives the classical concrete examples that serve as a meaningful preparation for the standard undergraduate course in abstract algebra. The detailed development of the isometries of the plane is based on only the most elementary geometry and is appropriate for graduate courses for secondary teachers.
Reading copy. May have notes, underlining or highlighting. Dust jacket may be missing.
$113725 +$3.99 s/h
Good
Nettextstore Lincoln, NE
199754.0563 +$3.99 s/h
New
indoo Avenel, NJ
BRAND NEW
$63.00 +$3.99 s/h
New
surplus computer books fallbrook, CA
0387906363 BRAND NEW NEVER USED IN STOCK 125,000+ HAPPY CUSTOMERS SHIP EVERY DAY WITH FREE TRACKING NUMBER |
Mcdougal Littell Middle School Math Indiana Students Edition Course 1
Mcdougal Littell Middle School Math strengthens the students understanding and provides the tools that the students need to excel in mathematics.
Long Synopsis:
Mcdougal Littell Middle School Math strengthens the students understanding and provides the tools that the students need to excel in mathematics. It has written lessons with frequent step-by-step examples which make even difficult math concepts and methods easier to understand. |
Barron's Dictionary of Mathematics Terms - 3rd edition
Summary: This quick-reference dictionary for math students, teachers, engineers, and statisticians defines more than 700 terms related to algebra, geometry, analytic geometry, trigonometry, probability, statistics, logic, and calculus. It also lists and defines mathematical symbols, includes a brief table of integrals, and describes how to derive key theorems. Filled with illustrative diagrams and equations |
Synopses & Reviews
Publisher Comments:
The true power of vectors has never been exploited, for over a century, mathematicians, engineers, scientists, and more recently programmers, have been using vectors to solve an extraordinary range of problems. However, today, we can discover the true potential of oriented, lines, planes and volumes in the form of geometric algebra. As such geometric elements are central to the world of computer games and computer animation, geometric algebra offers programmers new ways of solving old problems. John Vince (best-selling author of a number of books including Geometry for Computer Graphics, Vector Analysis for Computer Graphics and Geometric Algebra for Computer Graphics) provides new insights into geometric algebra and its application to computer games and animation. The first two chapters review the products for real, complex and quaternion structures, and any non-commutative qualities that they possess. Chapter three reviews the familiar scalar and vector products and introduces the idea of 'dyadics', which provide a useful mechanism for describing the features of geometric algebra. Chapter four introduces the geometric product and defines the inner and outer products, which are employed in the following chapter on geometric algebra. Chapters six and seven cover all the 2D and 3D products between scalars, vectors, bivectors and trivectors. Chapter eight shows how geometric algebra brings new insights into reflections and rotations, especially in 3D. Finally, chapter nine explores a wide range of 2D and 3D geometric problems followed by a concluding tenth chapter. Filled with lots of clear examples, full-colour illustrations and tables, this compact book provides an excellent introduction to geometric algebra for practitioners in computer games and animation.
Synopsis:
This book uses 3D colour drawings and tabulations of algebraic expansions to provide new insights into geometric algebra and its application to computer games and animation. It is filled with many worked examples and full-colour illustrations and tables.
Synopsis:
"Synopsis"
by Springer,
This book uses 3D colour drawings and tabulations of algebraic expansions to provide new insights into geometric algebra and its application to computer games and animation. It is filled with many worked examples and full-colour illustrations and tables.
"Synopsis"
by Springer, |
TI 84/84+GuidebookDocument Transcript
TI-84 Plus and TI-84 Plus Silver Edition GuidebookNote: This guidebook for the TI-84 Plus or TI-84 Plus Silver Edition with operating system (OS)version 2.53MP. If your calculator has a previous OS version, your screens may look differentand some features may not be available. You can download the latest OS at education.ti.com.
Chapter 1:Operating the TI-84 Plus Silver EditionDocumentation ConventionsIn the body of this guidebook, TI-84 Plus refers to the TI-84 Plus Silver Edition. Sometimes, as inChapter 19, the full name TI-84 Plus Silver Edition is used to distinguish it from the TI-84 Plus.All the instructions and examples in this guidebook also work for the TI-84 Plus. All the functions ofthe TI-84 Plus Silver Edition and the TI-84 Plus are the same. The two graphing calculators differonly in available RAM memory, interchangeable faceplates, and Flash application ROM memory.Screen shots were taken using OS version 2.53MP in either MathPrint™ or Classic mode. Allfeatures are available in both modes; however, screens make look slightly different depending onthe mode setting. Many examples highlight features that are not available in previous OS versions.If your calculator does not have the latest OS, features may not be available and your screens maylook different. You can download the latest OS from education.ti.com.TI-84 Plus KeyboardGenerally, the keyboard is divided into these zones: graphing keys, editing keys, advancedfunction keys, and scientific calculator keys.Keyboard ZonesGraphing — Graphing keys access the interactive graphing features. The third function of thesekeys (t ^-a) displays the shortcut menus, which include templates for fractions, n/d,quick matrix entry, and some of the functions found on the MATH and VARS menus.Editing — Editing keys allow you to edit expressions and values.Advanced — Advanced function keys display menus that access the advanced functions.Scientific — Scientific calculator keys access the capabilities of a standard scientific calculator. Chapter 1: Operating the TI-84 Plus Silver Edition 1
TI-84 Plus Silver EditionGraphing KeysEditing KeysAdvancedFunction KeysScientificCalculator KeysUsing the Color.Coded KeyboardThe keys on the TI-84 Plus are color-coded to help you easily locate the key you need.The light colored keys are the number keys. The keys along the right side of the keyboard are thecommon math functions. The keys across the top set up and display graphs. The Œ key providesaccess to applications such as the Inequality Graphing, Transformation Graphing, Conic Graphing,Polynomial Root Finder and Simultaneous Equation Solver, and Catalog Help.The primary function of each key is printed on the keys. For example, when you press , theMATH menu is displayed.Using the y and ƒ KeysThe secondary function of each key is printed above the key. When you press the y key, thecharacter, abbreviation, or word printed above the other keys becomes active for the nextkeystroke. For example, when you press y and then , the TEST menu is displayed. Thisguidebook describes this keystroke combination as y :.Many keys also have a third function. These functions are printed above the keys in the samecolor as the ƒ key. The third functions enter alphabetic characters and special symbols aswell as access SOLVE and shortcut menus. For example, when you press ƒ and then ,the letter A is entered. This guidebook describes this keystroke combination as ƒ [A]. Chapter 1: Operating the TI-84 Plus Silver Edition 2
If you want to enter several alphabetic characters in a row, you can press y 7 to lock thealpha key in the On position and avoid having to press ƒ multiple times. Press ƒ asecond time to unlock it.Note: The flashing cursor changes to Ø when you press ƒ, even if you are accessing afunction or a menu. ƒ^-a Access shortcut y menus forAccesses the functionalitysecond function including templatesprinted above each for fractions, n/d,key. and other functions.ƒAccesses the thirdfunction printedabove each key.Turning On and Turning Off the TI-84 PlusTurning On the Graphing CalculatorTo turn on the TI-84 Plus, press É. An information screen displays reminding you that you canpress t ^ - a to display the shortcut menus. This message also displays when you resetRAM. To continue but not see this information screen again, press 1. To continue and see this information screen again the next time you turn on the TI-84 Plus, press 2.• If you previously had turned off the graphing calculator by pressing y M, the TI-84 Plus displays the home screen as it was when you last used it and clears any error. (The information screen displays first, unless you chose not to see it again.) If the home screen is blank, press } to scroll through the history of previous calculations.• If Automatic Power Down™ (APD™) had previously turned off the graphing calculator, the TI-84 Plus will return exactly as you left it, including the display, cursor, and any error. Chapter 1: Operating the TI-84 Plus Silver Edition 3
• If the TI-84 Plus is turned off and connected to another graphing calculator or personal computer, any communication activity will "wake up" the TI-84 Plus.To prolong the life of the batteries, APD™ turns off the TI-84 Plus automatically after about fiveminutes without any activity.Turning Off the Graphing CalculatorTo turn off the TI-84 Plus manually, press y M.• All settings and memory contents are retained by the Constant Memory™ function.• Any error condition is cleared.BatteriesThe TI-84 Plus uses five batteries: four AAA alkaline batteries and one button cell backup battery.The backup battery provides auxiliary power to retain memory while you replace the AAAbatteries. To replace batteries without losing any information stored in memory, follow the steps inAppendix C.Setting the Display ContrastAdjusting the Display ContrastYou can adjust the display contrast to suit your viewing angle and lighting conditions. As you changethe contrast setting, a number from 0 (lightest) to 9 (darkest) in the top-right corner indicates thecurrent level. You may not be able to see the number if contrast is too light or too dark.Note: The TI-84 Plus has 40 contrast settings, so each number 0 through 9 represents foursettings.The TI-84 Plus retains the contrast setting in memory when it is turned off.To adjust the contrast, follow these steps. Press y } to darken the screen one level at a time. Press y † to lighten the screen one level at a time.Note: If you adjust the contrast setting to 0, the display may become completely blank. To restorethe screen, press y } until the display reappears.When to Replace BatteriesWhen the batteries are low, a low-battery message is displayed when you turn on the graphingcalculator.To replace the batteries without losing any information in memory, follow the steps in Appendix C. Chapter 1: Operating the TI-84 Plus Silver Edition 4
Generally, the graphing calculator will continue to operate for one or two weeks after the low-battery message is first displayed. After this period, the TI-84 Plus will turn off automatically andthe unit will not operate. Batteries must be replaced. All memory should be retained.Note:• The operating period following the first low-battery message could be longer than two weeks if you use the graphing calculator infrequently.• Always replace batteries before attempting to install a new operating system.The DisplayTypes of DisplaysThe TI-84 Plus displays both text and graphs. Chapter 3 describes graphs. Chapter 9 describeshow the TI-84 Plus can display a horizontally or vertically split screen to show graphs and textsimultaneously.Home ScreenThe home screen is the primary screen of the TI-84 Plus. On this screen, enter instructions toexecute and expressions to evaluate. The answers are displayed on the same screen. Mostcalculations are stored in the history on the home screen. You can press } and † to scroll throughthe history of entries on the home screen and you can paste the entries or answers to the currententry line.Displaying Entries and Answers• When text is displayed, the TI-84 Plus screen can display a maximum of 8 lines with a maximum of 16 characters per line in Classic mode. In MathPrint™ mode, fewer lines and fewer characters per line may be displayed.• If all lines of the display are full, text scrolls off the top of the display. - To view previous entries and answers, press }. - To copy a previous entry or answer and paste it to the current entry line, move the cursor to the entry or answer you want to copy and press Í. Note: List and matrix outputs cannot be copied. If you try to copy and paste a list or matrix output, the cursor returns to the input line.• If an expression on the home screen, the Y= editor (Chapter 3), or the program editor (Chapter 16) is longer than one line, it wraps to the beginning of the next line in Classic mode. In MathPrint™ mode, an expression on the home screen or Y= editor that is longer than one line scrolls off the screen to the right. An arrow on the right side of the screen indicates that you can scroll right to see more of the expression. In numeric editors such as the window screen (Chapter 3), a long expression scrolls to the right and left in both Classic and MathPrint™ modes. Press y ~ to move the cursor to the end of the line. Press y | to move the cursor to the beginning of the line. Chapter 1: Operating the TI-84 Plus Silver Edition 5
When an entry is executed on the home screen, the answer is displayed on the right side of thenext line. Entry AnswerThe mode settings control the way the TI-84 Plus interprets expressions and displays answers.If an answer, such as a list or matrix, is too long to display entirely on one line, an arrow(MathPrint™) or an ellipsis (Classic) is displayed to the right or left. Press ~ and | to display theanswer. MathPrint™ Classic Entry Entry Answer Answer Entry Entry Answer AnswerUsing Shortcut Menus t^ t_ t` ta Opens FRAC Opens FUNC Opens MTRX Opens YVAR menu. menu. menu. menu.Shortcut menus allow quick access to the following:• Templates to enter fractions and selected functions from the MATH MATH and MATH NUM menus as you would see them in a textbook. Functions include absolute value, summation, numeric differentiation, numeric integration, and log base n. Chapter 1: Operating the TI-84 Plus Silver Edition 6
• Matrix entry.• Names of function variables from the VARS Y-VARS menu.Initially, the menus are hidden. To open a menu, press t plus the F-key that corresponds tothe menu, that is, ^ for FRAC, _ for FUNC, ` for MTRX, or a for YVAR. To select a menuitem, either press the number corresponding to the item, or use the arrow keys to move the cursorto the appropriate line and then press Í.All shortcut menu items except matrix templates can also be selected using standard menus. Forexample, you can choose the summation template from three places:FUNC shortcut menuMATH MATH menuCatalogThe shortcut menus are available to use where input is allowed. If the calculator is in Classicmode, or if a screen is displayed that does not support MathPrint™ display, entries will bedisplayed in Classic display. The MTRX menu is only available in MathPrint™ mode on the homescreen and in the Y= editor.Note: Shortcut menus may not be available if t plus F-key combinations are used by anapplication that is running, such as Inequality Graphing or Transformation Graphing.Returning to the Home ScreenTo return to the home screen from any other screen, press y 5.Busy IndicatorWhen the TI-84 Plus is calculating or graphing, a vertical moving line is displayed as a busyindicator in the top-right corner of the screen. When you pause a graph or a program, the busyindicator becomes a vertical moving dotted line. Chapter 1: Operating the TI-84 Plus Silver Edition 7
Display CursorsIn most cases, the appearance of the cursor indicates what will happen when you press the nextkey or select the next menu item to be pasted as a character.Cursor Appearance Effect of Next KeystrokeEntry Solid rectangle A character is entered at the cursor; any existing $ character is overwrittenInsert Underline A character is inserted in front of the cursor __ locationSecond Reverse arrow A 2nd character is entered or a 2nd operation is Þ executedAlpha Reverse A An alpha character is entered, SOLVE is Ø executed, or shortcut menus are displayed.Full Checkerboard rectangle No entry; the maximum characters are entered at # a prompt or memory is fullMathPrint™ Right arrow The cursor moves to either the next part of the template or out of the template.If you press ƒ during an insertion, the cursor becomes an underlined A (A). If you press yduring an insertion, the underlined cursoSr becomes an underlined # (#).Note: If you highlight a small character such as a colon or a comma and then press ƒ or y,the cursor does not change because the cursor width is too narrow.Graphs and editors sometimes display additional cursors, which are described in other chapters.Interchangeable FaceplatesThe TI-84 Plus Silver Edition has interchangeable faceplates that let you customize theappearance of your unit. To purchase additional faceplates, refer to the TI Online Store ateducation.ti.com.Removing a Faceplate1. Lift the tab at the bottom edge of the faceplate away from the TI-84 Plus Silver Edition case.2. Carefully lift the faceplate away from the unit until it releases. Be careful not to damage the faceplate or the keyboard. Chapter 1: Operating the TI-84 Plus Silver Edition 8
Installing New Faceplates1. Align the top of the faceplate in the corresponding grooves of the TI-84 Plus Silver Edition case.2. Gently click the faceplate into place. Do not force.3. Make sure you gently press each of the grooves to ensure the faceplate is installed properly. See the diagram for proper groove placement.Using the ClockUse the clock to set the time and date, select the clock display format, and turn the clock on andoff. The clock is turned on by default and is accessed from the mode screen.Displaying the Clock Settings1. Press z.2. Press the † to move the cursor to SET CLOCK.3. Press Í.Changing the Clock Settings1. Press the ~ or | to highlight the date format you want. Press Í.2. Press † to highlight YEAR. Press ' and type the year.3. Press † to highlight MONTH. Press ' and type the number of the month (1-12).4. Press † to highlight DAY. Press ' and type the date.5. Press † to highlight TIME. Press ~ or | to highlight the time format you want. Press Í. Chapter 1: Operating the TI-84 Plus Silver Edition 9
6. Press † to highlight HOUR. Press ' and type the hour (a number from 1-12 or 0-23).7. Press † to highlight MINUTE. Press ' and type the minutes (a number from 0-59).8. Press † to highlight AM/PM. Press ~ or | to highlight the format. Press Í.9. To save changes, press † to highlight SAVE. Press Í.Error MessagesIf you type the wrong date for the month, for example,June 31 (June does not have 31 days), you willreceive an error message with two choices:• To quit the clock application and return to the home screen, select 1: Quit. — or —• To return to the clock application and correct the error, select 2: Goto.Turning the Clock OnThere are two options to turn the clock on. One option is through the MODE screen, the other isthrough the Catalog. Chapter 1: Operating the TI-84 Plus Silver Edition 10
Using the Mode Screen to turn the clock on1. If the clock is turned off, Press † to highlight TURN CLOCK ON.2. Press Í Í.Using the Catalog to turn the clock on1. If the clock is turned off, Press y N2. Press † or } to scroll the CATALOG until the selection cursor points to ClockOn.3. Press Í Í.Turning the Clock Off1. Press y N.2. Press † or } to scroll the CATALOG until the selection cursor points to ClockOff.3. Press Í Í.Entering Expressions and InstructionsWhat Is an Expression?An expression is a group of numbers, variables, functions and their arguments, or a combination ofthese elements. An expression evaluates to a single answer. On the TI-84 Plus, you enter anexpression in the same order as you would write it on paper. For example, pR2 is an expression.You can use an expression on the home screen to calculate an answer. In most places where avalue is required, you can use an expression to enter a value. Chapter 1: Operating the TI-84 Plus Silver Edition 11
Entering an ExpressionTo create an expression, you enter numbers, variables, and functions using the keyboard andmenus. An expression is completed when you press Í, regardless of the cursor location. Theentire expression is evaluated according to Equation Operating System (EOS™) rules, and theanswer is displayed according to the mode setting for Answer.Most TI-84 Plus functions and operations are symbols comprising several characters. You mustenter the symbol from the keyboard or a menu; do not spell it out. For example, to calculate the logof 45, you must press « 45. Do not enter the letters L, O, and G. If you enter LOG, the TI-84 Plusinterprets the entry as implied multiplication of the variables L, O, and G.Calculate 3.76 P (L7.9 + ‡5) + 2 log 45.3 Ë 76 ¥ £ Ì 7 Ë 9 Ãy C 5 ¤ ¤ à 2 « 45 ¤Í MathPrint™ ClassicMultiple Entries on a LineTo enter two or more expressions or instructions on a line, separate them with colons (ƒ [:]).All instructions are stored together in last entry (ENTRY).Entering a Number in Scientific Notation1. Enter the part of the number that precedes the exponent. This value can be an expression.2. Press y D. â is pasted to the cursor location.3. Enter the exponent, which can be one or two digits. Note: If the exponent is negative, press Ì, and then enter the exponent.When you enter a number in scientific notation, the TI-84 Plus does not automatically displayanswers in scientific or engineering notation. The mode settings and the size of the numberdetermine the display format.FunctionsA function returns a value. For example, ÷, L, +, ‡, and log( are the functions in the example on theprevious page. In general, the first letter of each function is lowercase on the TI-84 Plus. Mostfunctions take at least one argument, as indicated by an open parenthesis following the name. Forexample, sin( requires one argument, sin(value). Chapter 1: Operating the TI-84 Plus Silver Edition 12
Note: The Catalog Help App contains syntax information for most of the functions in the catalog.InstructionsAn instruction initiates an action. For example, ClrDraw is an instruction that clears any drawnelements from a graph. Instructions cannot be used in expressions. In general, the first letter ofeach instruction name is uppercase. Some instructions take more than one argument, as indicatedby an open parenthesis at the end of the name. For example, Circle( requires three arguments,Circle(X,Y,radius).Interrupting a CalculationTo interrupt a calculation or graph in progress, which is indicated by the busy indicator, press É.When you interrupt a calculation, a menu is displayed.• To return to the home screen, select 1:Quit.• To go to the location of the interruption, select 2:Goto.When you interrupt a graph, a partial graph is displayed.• To return to the home screen, press ' or any nongraphing key.• To restart graphing, press a graphing key or select a graphing instruction.TI-84 Plus Edit KeysKeystrokes Result~ or | Moves the cursor within an expression; these keys repeat.} or † Moves the cursor from line to line within an expression that occupies more than one line; these keys repeat. Moves the cursor from term to term within an expression in MathPrint™ mode; these keys repeat. On the home screen, scrolls through the history of entries and answers.y| Moves the cursor to the beginning of an expression.y~ Moves the cursor to the end of an expression.y} On the home screen, moves the cursor out of a MathPrint™ expression. In the Y=editor, moves the cursor from a MathPrint™ expression to the previous Y-var.y† In the Y=editor, moves the cursor from a MathPrint ™ expression to the next Y-var.Í Evaluates an expression or executes an instruction.' On a line with text on the home screen, clears the current line. On a blank line on the home screen, clears everything on the home screen. In an editor, clears the expression or value where the cursor is located; it does not store a zero. Chapter 1: Operating the TI-84 Plus Silver Edition 13
Keystrokes Result{ Deletes a character at the cursor; this key repeats.y6 Changes the cursor to an underline (__); inserts characters in front of the underline cursor; to end insertion, press y 6 or press |, }, ~, or †.y Changes the cursor to Þ; the next keystroke performs a 2nd function (displayed above a key and to the left); to cancel 2nd, press y again.ƒ Changes the cursor to Ø; the next keystroke performs a third function of that key (displayed above a key and to the right), executes SOLVE (Chapters 10 and 11), or accesses a shortcut menu; to cancel ƒ, press ƒ or press |, }, ~, or †.y7 Changes the cursor to Ø; sets alpha-lock; subsequent keystrokes access the third functions of the keys pressed; to cancel alpha-lock, press ƒ. If you are prompted to enter a name such as for a group or a program, alpha-lock is set automatically." Pastes an X in Func mode, a T in Par mode, a q in Pol mode, or an n in Seq mode with one keystroke.Setting ModesChecking Mode SettingsMode settings control how the TI-84 Plus displays and interprets numbers and graphs. Modesettings are retained by the Constant 'Memory™ feature when the TI-84 Plus is turned off. Allnumbers, including elements of matrices and lists, are displayed according to the current modesettings.To display the mode settings, press z. The current settings are highlighted. Defaults arehighlighted below. The following pages describe the mode settings in detail.Normal Sci Eng Numeric notationFloat 0123456789 Number of decimal places in answersRadian Degree Unit of angle measureFunc Par Pol Seq Type of graphingConnected Dot Whether to connect graph pointsSequential Simul Whether to plot simultaneouslyReal a+bi re^qi Real, rectangular complex, or polar complexFull Horiz G-T Full screen, two split-screen modesMathPrint Classic Controls whether inputs and outputs on the home screen and in the Y= editor are displayed as they are in textbooksn/d Un/d Displays results as simple fractions or mixed fractionsAnswers: Auto Dec Frac Controls the format of the answers Chapter 1: Operating the TI-84 Plus Silver Edition 14
GoTo Format Graph: No Yes Shortcut to the Format Graph screen (y .)StatDiagnostics: Off On Determines which information is displayed in a statistical regression calculationSet Clock Sets the time and dateChanging Mode SettingsTo change mode settings, follow these steps.1. Press † or } to move the cursor to the line of the setting that you want to change.2. Press ~ or | to move the cursor to the setting you want.3. Press Í.Setting a Mode from a ProgramYou can set a mode from a program by entering the name of the mode as an instruction; forexample, Func or Float. From a blank program command line, select the mode setting from themode screen; the instruction is pasted to the cursor location.Normal, Sci, EngNotation modes only affect the way an answer is displayed on the home screen. Numeric answerscan be displayed with up to 10 digits and a two-digit exponent and as fractions. You can enter anumber in any format.Normal notation mode is the usual way we express numbers, with digits to the left and right of thedecimal, as in 12345.67.Sci (scientific) notation mode expresses numbers in two parts. The significant digits display withone digit to the left of the decimal. The appropriate power of 10 displays to the right of å, as in1.234567â4.Eng (engineering) notation mode is similar to scientific notation. However, the number can haveone, two, or three digits before the decimal; and the power-of-10 exponent is a multiple of three, asin 12.34567â3.Note: If you select Normal notation, but the answer cannot display in 10 digits (or the absolutevalue is less than .001), the TI-84 Plus expresses the answer in scientific notation.Float, 0123456789Float (floating) decimal mode displays up to 10 digits, plus the sign and decimal. Chapter 1: Operating the TI-84 Plus Silver Edition 15
0123456789 (fixed) decimal mode specifies the number of digits (0 through 9) to display to the rightof the decimal for decimal answers.The decimal setting applies to Normal, Sci, and Eng notation modes.The decimal setting applies to these numbers, with respect to the Answer mode setting:• An answer displayed on the home screen• Coordinates on a graph (Chapters 3, 4, 5, and 6)• The Tangent( DRAW instruction equation of the line, x, and dy/dx values (Chapter 8)• Results of CALCULATE operations (Chapters 3, 4, 5, and 6)• The regression equation stored after the execution of a regression model (Chapter 12)Radian, DegreeAngle modes control how the TI-84 Plus interprets angle values in trigonometric functions andpolar/rectangular conversions.Radian mode interprets angle values as radians. Answers display in radians.Degree mode interprets angle values as degrees. Answers display in degrees.Func, Par, Pol, SeqGraphing modes define the graphing parameters. Chapters 3, 4, 5, and 6 describe these modes indetail.Func (function) graphing mode plots functions, where Y is a function of X (Chapter 3).Par (parametric) graphing mode plots relations, where X and Y are functions of T (Chapter 4).Pol (polar) graphing mode plots functions, where r is a function of q (Chapter 5).Seq (sequence) graphing mode plots sequences (Chapter 6).Connected, DotConnected plotting mode draws a line connecting each point calculated for the selected functions.Dot plotting mode plots only the calculated points of the selected functions.Sequential, SimulSequential graphing-order mode evaluates and plots one function completely before the nextfunction is evaluated and plotted.Simul (simultaneous) graphing-order mode evaluates and plots all selected functions for a singlevalue of X and then evaluates and plots them for the next value of X. Chapter 1: Operating the TI-84 Plus Silver Edition 16
Note: Regardless of which graphing mode is selected, the TI-84 Plus will sequentially graph all statplots before it graphs any functions.Real, a+bi, re^qiReal mode does not display complex results unless complex numbers are entered as input.Two complex modes display complex results.• a+bi (rectangular complex mode) displays complex numbers in the form a+bi.• re^qi (polar complex mode) displays complex numbers in the form re^qi.Note: When you use the n/d template, both n and d must be real numbers. For example, you canenter (the answer is displayed as a decimal value) but if you enter , a data type errordisplays. To perform division with a complex number in the numerator or denominator, use regulardivision instead of the n/d template.Full, Horiz, G-TFull screen mode uses the entire screen to display a graph or edit screen.Each split-screen mode displays two screens simultaneously.• Horiz (horizontal) mode displays the current graph on the top half of the screen; it displays the home screen or an editor on the bottom half (Chapter 9).• G-T (graph-table) mode displays the current graph on the left half of the screen; it displays the table screen on the right half (Chapter 9).MathPrint™, ClassicMathPrint™ mode displays most inputs and outputs the way they are shown in textbooks, such as 21 3-- + -- and x 2 dx . - -2 4 1Classic mode displays expressions and answers as if written on one line, such as 1/2 + 3/4.Note: If you switch between these modes, most entries will be preserved; however matrixcalculations will not be preserved. Chapter 1: Operating the TI-84 Plus Silver Edition 17
n/d, Un/dn/d displays results as a simple fraction. Fractions may contain a maximum of six digits in thenumerator; the value of the denominator may not exceed 9999.Un/d displays results as a mixed number, if applicable. U, n, and d must be all be integers. If U is anon-integer, the result may be converted U … n/d. If n or d is a non-integer, a syntax error isdisplayed. The whole number, numerator, and denominator may each contain a maximum of threedigits.Answers: Auto, Dec, FracAuto displays answers in a similar format as the input. For example, if a fraction is entered in anexpression, the answer will be in fraction form, if possible. If a decimal appears in the expression,the output will be a decimal number.Dec displays answers as integers or decimal numbers.Frac displays answers as fractions, if possible.Note: The Answers mode setting also affects how values in sequences, lists, and tables aredisplayed. Choose Dec or Frac to ensure that values are displayed in either decimal or fractionform. You can also convert values from decimal to fraction or fraction to decimal using the FRACshortcut menu or the MATH menu.GoTo Format Graph: No, YesNo does not display the FORMAT graph screen, but can always be accessed by pressingy ..Yes leaves the mode screen and displays the FORMAT graph screen when you press Í sothat you can change the graph format settings. To return to the mode screen, press z.Stat Diagnostics: Off, OnOff displays a statistical regression calculation without the correlation coefficient (r) or thecoefficient of determination (r2).On displays a statistical regression calculation with the correlation coefficient (r), and thecoefficient of determination (r2), as appropriate.Set ClockUse the clock to set the time, date, and clock display formats. Chapter 1: Operating the TI-84 Plus Silver Edition 18
• Although most variables can be archived, system variables including r, T, X, Y, and q cannot be archived (Chapter 18)• Apps are independent applications.which are stored in Flash ROM. AppVars is a variable holder used to store variables created by independent applications. You cannot edit or change variables in AppVars unless you do so through the application which created them.Storing Variable ValuesStoring Values in a VariableValues are stored to and recalled from memory using variable names. When an expressioncontaining the name of a variable is evaluated, the value of the variable at that time is used.To store a value to a variable from the home screen or a program using the ¿ key, begin on ablank line and follow these steps.1. Enter the value you want to store. The value can be an expression.2. Press ¿. ! is copied to the cursor location.3. Press ƒ and then the letter of the variable to which you want to store the value.4. Press Í. If you entered an expression, it is evaluated. The value is stored to the variable.Displaying a Variable ValueTo display the value of a variable, enter the name on a blank line on the home screen, and thenpress Í.Archiving Variables (Archive, Unarchive)You can archive data, programs, or other variables in a section of memory called user data archivewhere they cannot be edited or deleted inadvertently. Archived variables are indicated by asterisks(ä) to the left of the variable names. Archived variables cannot be edited or executed. They canonly be seen and unarchived. For example, if you archive list L1, you will see that L1 exists inmemory but if you select it and paste the name L1 to the home screen, you won't be able to see itscontents or edit it until it is unarchived. Chapter 1: Operating the TI-84 Plus Silver Edition 20
Recalling Variable ValuesUsing Recall (RCL)To recall and copy variable contents to the current cursor location, follow these steps. To leaveRCL, press '.1. Press y K. RCL and the edit cursor are displayed on the bottom line of the screen.2. Enter the name of the variable in one of five ways. • Press ƒ and then the letter of the variable. • Press y 9, and then select the name of the list, or press y [Ln]. • Press y >, and then select the name of the matrix. • Press to display the VARS menu or ~ to display the VARS Y-VARS menu; then select the type and then the name of the variable or function. • Press t a to display the YVAR shortcut menu, then select the name of the function. • Press |, and then select the name of the program (in the program editor only). The variable name you selected is displayed on the bottom line and the cursor disappears.3. Press Í. The variable contents are inserted where the cursor was located before you began these steps. Note: You can edit the characters pasted to the expression without affecting the value in memory.Scrolling Through Previous Entries on the Home ScreenYou can scroll up through previous entries and answers on the home screen, even if you havecleared the screen. When you find an entry or answer that you want to use, you can select it andpaste it on the current entry line.Note: List and matrix answers cannot be copied and pasted to the new entry line. However, youcan copy the list or matrix command to the new entry line and execute the command again todisplay the answer. Chapter 1: Operating the TI-84 Plus Silver Edition 21
Press } or † to move the cursor to the entry or answer you want to copy and then press Í. TThe entry or answer that you copied is automatically pasted on the current input line at the cursor location. Note: If the cursor is in a MathPrint™ expression, press y } to move the cursor out of the expression and then move the cursor to the entry or answer you want to copy. Press u or { to delete an entry/answer pair. After an entry/answer pair has been deleted, it cannot be displayed or recalled again.ENTRY (Last Entry) Storage AreaUsing ENTRY (Last Entry)When you press Í on the home screen to evaluate an expression or execute an instruction,the expression or instruction is placed in a storage area called ENTRY (last entry). When you turnoff the TI-84 Plus, ENTRY is retained in memory.To recall ENTRY, press y [. The last entry is pasted to the current cursor location, whereyou can edit and execute it. On the home screen or in an editor, the current line is cleared and thelast entry is pasted to the line.Because the TI-84 Plus updates ENTRY only when you press Í, you can recall the previousentry even if you have begun to enter the next expression.5Ã7Íy[Accessing a Previous EntryThe TI-84 Plus retains as many previous entries as possible in ENTRY, up to a capacity of 128bytes. To scroll those entries, press y [ repeatedly. If a single entry is more than 128 bytes,it is retained for ENTRY, but it cannot be placed in the ENTRY storage area.1 ¿ƒ AÍ2¿ƒ BÍy[If you press y [ after displaying the oldest stored entry, the newest stored entry is displayedagain, then the next-newest entry, and so on.y[ Chapter 1: Operating the TI-84 Plus Silver Edition 22
Executing the Previous Entry AgainAfter you have pasted the last entry to the home screen and edited it (if you chose to edit it), youcan execute the entry. To execute the last entry, press Í.To execute the displayed entry again, press Í again. Each subsequent execution displays theentry and the new answer.0 ¿ƒ N̓ N à 1 ¿ƒ Nƒ ã:䊃ÄN ¡ ÍÍÍMultiple Entry Values on a LineTo store to ENTRY two or more expressions or instructions, separate each expression orinstruction with a colon, then press Í. All expressions and instructions separated by colonsare stored in ENTRY.When you press y [, all the expressions and instructions separated by colons are pasted tothe current cursor location. You can edit any of the entries, and then execute all of them when youpress Í.Example: For the equation A=pr 2, use trial and error to find the radius of a circle that covers 200square centimeters. Use 8 as your first guess.8 ¿ ƒ R ƒ ã :äyB ƒ R ¡Íy[y | 7 y 6 Ë 95ÍContinue until the answer is as accurate as you want.Clearing ENTRYClear Entries (Chapter 18) clears all data that the TI-84 Plus is holding in the ENTRY storage area.Using Ans in an ExpressionWhen an expression is evaluated successfully from the home screen or from a program, the TI-84Plus stores the answer to a storage area called Ans (last answer). Ans may be a real or complexnumber, a list, a matrix, or a string. When you turn off the TI-84 Plus, the value in Ans is retained inmemory. Chapter 1: Operating the TI-84 Plus Silver Edition 23
You can use the variable Ans to represent the last answer in most places. Press y Z to copy thevariable name Ans to the cursor location. When the expression is evaluated, the TI-84 Plus uses thevalue of Ans in the calculation.Calculate the area of a garden plot 1.7 meters by 4.2 meters. Then calculate the yield per squaremeter if the plot produces a total of 147 tomatoes.1Ë7¯4Ë2Í147 ¥ y ZÍContinuing an ExpressionYou can use Ans as the first entry in the next expression without entering the value again orpressing y Z. On a blank line on the home screen, enter the function. The TI-84 Plus pastesthe variable name Ans to the screen, then the function.5¥2ͯ9Ë9ÍStoring AnswersTo store an answer, store Ans to a variable before you evaluate another expression.Calculate the area of a circle of radius 5 meters. Next, calculate the volume of a cylinder of radius5 meters and height 3.3 meters, and then store the result in the variable V.yB 5 ¡Í¯3Ë3Í¿ƒ VÍTI-84 Plus MenusUsing a TI-84 Plus MenuYou can access most TI-84 Plus operations using menus. When you press a key or keycombination to display a menu, one or more menu names appear on the top line of the screen.• The menu name on the left side of the top line is highlighted. Up to seven items in that menu are displayed, beginning with item 1, which also is highlighted. Chapter 1: Operating the TI-84 Plus Silver Edition 24
• A number or letter identifies each menu item's place in the menu. The order is 1 through 9, then 0, then A, B, C, and so on. The LIST NAMES, PRGM EXEC, and PRGM EDIT menus only label items 1 through 9 and 0.• When the menu continues beyond the displayed items, a down arrow ($) replaces the colon next to the last displayed item.• When a menu item ends in an ellipsis (...), the item displays a secondary menu or editor when you select it.• When an asterisk (ä) appears to the left of a menu item, that item is stored in user data archive (Chapter 18).Displaying a MenuWhile using your TI-84 Plus, you often will need toaccess items from its menus.When you press a key that displays a menu, thatmenu temporarily replaces the screen where you areworking. For example, when you press , theMATH menu is displayed as a full screen.After you select an item from a menu, the screenwhere you are working usually is displayed again.Moving from One Menu to AnotherSome keys access more than one menu. When youpress such a key, the names of all accessible menusare displayed on the top line. When you highlight amenu name, the items in that menu are displayed.Press ~ and | to highlight each menu name.Note: FRAC shortcut menu items are also found on theMATH NUM menu. FUNC shortcut menu items arealso found on the MATH MATH menu.Scrolling a MenuTo scroll down the menu items, press †. To scroll up the menu items, press }. Chapter 1: Operating the TI-84 Plus Silver Edition 25
To page down six menu items at a time, press ƒ †. To page up six menu items at a time,press ƒ }.To go to the last menu item directly from the first menu item, press }. To go to the first menu itemdirectly from the last menu item, press †.Selecting an Item from a MenuYou can select an item from a menu in either of two ways.• Press the number or letter of the item you want to select. The cursor can be anywhere on the menu, and the item you select need not be displayed on the screen.• Press † or } to move the cursor to the item you want, and then press Í.After you select an item from a menu, the TI-84 Plustypically displays the previous screen.Note: On the LIST NAMES, PRGM EXEC, and PRGM EDIT menus, only items 1 through 9 and 0 arelabeled in such a way that you can select them by pressing the appropriate number key. To movethe cursor to the first item beginning with any alpha character or q, press the key combination forthat alpha character or q. If no items begin with that character, the cursor moves beyond it to thenext item.Example: Calculate 3‡27.†††Í27 ÍLeaving a Menu without Making a SelectionYou can leave a menu without making a selection in any of four ways.• Press y 5 to return to the home screen.• Press ' to return to the previous screen.• Press a key or key combination for a different menu, such as or y 9.• Press a key or key combination for a different screen, such as o or y 0. Chapter 1: Operating the TI-84 Plus Silver Edition 26
2. Select the type of variable, such as 2:Zoom from the VARS menu or 3:Polar from the VARS Y-VARS menu. A secondary menu is displayed.3. If you selected 1:Window, 2:Zoom, or 5:Statistics from the VARS menu, you can press ~ or | to display other secondary menus.4. Select a variable name from the menu. It is pasted to the cursor location.Equation Operating System (EOS™)Order of EvaluationThe Equation Operating System (EOS™) defines the order in which functions in expressions areentered and evaluated on the TI-84 Plus. EOS™ lets you enter numbers and functions in a simple,straightforward sequence.EOS evaluates the functions in an expression in this order.Order Number Function 1 Functions that precede the argument, such as ‡, sin(, or log( 2 Functions that are entered after the argument, such as 2, M1, !, ¡, r, and conversions 3 x Powers and roots, such as 25 or 5 32 4 Permutations (nPr) and combinations (nCr) 5 Multiplication, implied multiplication, and division 6 Addition and subtraction 7 Relational functions, such as > or 8 Logic operator and 9 Logic operators or and xorNote: Within a priority level, EOS™ evaluates functions from left to right. Calculations withinparentheses are evaluated first.Implied MultiplicationThe TI-84 Plus recognizes implied multiplication, so you need not press ¯ to expressmultiplication in all cases. For example, the TI-84 Plus interprets 2p, 4sin(46), 5(1+2), and (2…5)7 asimplied multiplication.Note: TI-84 Plus implied multiplication rules, although like the TI-83, differ from those of the TI-82.For example, the TI-84 Plus evaluates 1à2X as (1à2)…X, while the TI-82 evaluates 1à2X as 1à(2…X)(Chapter 2). Chapter 1: Operating the TI-84 Plus Silver Edition 28
ParenthesesAll calculations inside a pair of parentheses are completed first. For example, in the expression4(1+2), EOS first evaluates the portion inside the parentheses, 1+2, and then multiplies the answer,3, by 4.NegationTo enter a negative number, use the negation key. Press Ì and then enter the number. On theTI-84 Plus, negation is in the third level in the EOS™ hierarchy. Functions in the first level, such assquaring, are evaluated before negation.Example: MX2, evaluates to a negative number (or 0). Use parentheses to square a negativenumber.Note: Use the ¹ key for subtraction and the Ì key for negation. If you press ¹ to enter a negativenumber, as in 9 ¯ ¹ 7, or if you press Ì to indicate subtraction, as in 9 Ì 7, an error occurs. Ifyou press ƒ A Ì ƒ B, it is interpreted as implied multiplication (A…MB).Special Features of the TI-84 PlusFlash – Electronic UpgradabilityThe TI-84 Plus uses Flash technology, which lets you upgrade to future software versions withoutbuying a new graphing calculator.As new functionality becomes available, you can electronically upgrade your TI-84 Plus from theInternet. Future software versions include maintenance upgrades that will be released free ofcharge, as well as new applications and major software upgrades that will be available forpurchase from the TI Web site: education.ti.com. For details, refer to Chapter 19.1.5 Megabytes of Available Memory1.5 MB of available memory are built into the TI-84 Plus Silver Edition, and 0.5 MB for theTI-84 Plus. About 24 kilobytes (K) of RAM (random access memory) are available for you tocompute and store functions, programs, and data. Chapter 1: Operating the TI-84 Plus Silver Edition 29
About 1.5 M of user data archive allow you to store data, programs, applications, or any othervariables to a safe location where they cannot be edited or deleted inadvertently. You can also freeup RAM by archiving variables to user data. For details, refer to Chapter 18.ApplicationsMany applications are preloaded on your TI-84 Plus and others can be installed to customize theTI-84 Plus to your needs. The 1.5 MB archive space lets you store up to 94 applications at onetime on the TI-84 Plus Silver Edition. Applications can also be stored on a computer for later use orlinked unit-to-unit. There are 30 App slots for the TI-84 Plus. For details, refer to Chapter 18.ArchivingYou can store variables in the TI-84 Plus user data archive, a protected area of memory separatefrom RAM. The user data archive lets you:• Store data, programs, applications or any other variables to a safe location where they cannot be edited or deleted inadvertently.• Create additional free RAM by archiving variables.By archiving variables that do not need to be edited frequently, you can free up RAM forapplications that may require additional memory. For details, refer to:Chapter 18.Other TI-84 Plus FeaturesThe TI-84 Plus guidebook that is included with your graphing calculator has introduced you tobasic TI-84 Plus operations. This guidebook covers the other features and capabilities of the TI-84Plus in greater detail.GraphingYou can store, graph, and analyze up to 10 functions, up to six parametric functions, up to six polarfunctions, and up to three sequences. You can use DRAW instructions to annotate graphs.The graphing chapters appear in this order: Function, Parametric, Polar, Sequence, and DRAW.For graphing details, refer to Chapters 3, 4, 5, 6, 8.SequencesYou can generate sequences and graph them over time. Or, you can graph them as web plots oras phase plots. For details, refer to Chapter 6.TablesYou can create function evaluation tables to analyze many functions simultaneously. For details,refer to Chapter 7. Chapter 1: Operating the TI-84 Plus Silver Edition 30
Split ScreenYou can split the screen horizontally to display both a graph and a related editor (such as the Y=editor), the table, the stat list editor, or the home screen. Also, you can split the screen vertically todisplay a graph and its table simultaneously. For details, refer to Chapter 9.MatricesYou can enter and save up to 10 matrices and perform standard matrix operations on them. Fordetails, refer to Chapter 10.ListsYou can enter and save as many lists as memory allows for use in statistical analyses. You canattach formulas to lists for automatic computation. You can use lists to evaluate expressions atmultiple values simultaneously and to graph a family of curves. For details, refer to:Chapter 11.StatisticsYou can perform one- and two-variable, list-based statistical analyses, including logistic and sineregression analysis. You can plot the data as a histogram, xyLine, scatter plot, modified or regularbox-and-whisker plot, or normal probability plot. You can define and store up to three stat plotdefinitions. For details, refer to Chapter 12.Inferential StatisticsYou can perform 16 hypothesis tests and confidence intervals and 15 distribution functions. Youcan display hypothesis test results graphically or numerically. For details, refer to Chapter 13.ApplicationsPress Πto see the complete list of applications that came with your graphing calculator.Documentation for TI Flash applications are on the product CD. Visit education.ti.com/guides foradditional Flash application guidebooks. For details, refer to Chapter 14.CATALOGThe CATALOG is a convenient, alphabetical list of all functions and instructions on the TI-84 Plus.You can paste any function or instruction from the CATALOG to the current cursor location. Fordetails, refer to Chapter 15.ProgrammingYou can enter and store programs that include extensive control and input/output instructions. Fordetails, refer to Chapter 16. Chapter 1: Operating the TI-84 Plus Silver Edition 31
ArchivingArchiving allows you to store data, programs, or other variables to user data archive where theycannot be edited or deleted inadvertently. Archiving also allows you to free up RAM for variablesthat may require additional memory.Archived variables are indicated by asterisks (ä) to theleft of the variable names.For details, refer to Chapter 16.Communication LinkThe TI-84 Plus has a USB port using a USB unit-to-unit cable to connect and communicate withanother TI-84 Plus or TI-84 Plus Silver Edition. The TI-84 Plus also has an I/O port using an I/Ounit-to-unit cable to communicate with a TI-84 Plus Silver Edition, a TI-84 Plus, a TI-83 Plus SilverEdition, a TI-83 Plus, a TI-83, a TI-82, a TI-73, CBL 2™, or a CBR™ System.With TI Connect™ software and a USB computer cable, you can also link the TI-84 Plus to apersonal computer.As future software upgrades become available on the TI Web site, you can download the softwareto your PC and then use the TI Connect™ software and a USB computer cable to upgrade yourTI-84 Plus.For details, refer to: Chapter 19Error ConditionsDiagnosing an ErrorThe TI-84 Plus detects errors while performing these tasks.• Evaluating an expression• Executing an instruction• Plotting a graph• Storing a valueWhen the TI-84 Plus detects an error, it returns an error message as a menu title, such asERR:SYNTAX or ERR:DOMAIN. Appendix B describes each error type and possible reasons for theerror. Chapter 1: Operating the TI-84 Plus Silver Edition 32
• If you select 1:Quit (or press y 5 or '), then the home screen is displayed.• If you select 2:Goto, then the previous screen is displayed with the cursor at or near the error location.Note: If a syntax error occurs in the contents of a Y= function during program execution, then theGoto option returns to the Y= editor, not to the program.Correcting an ErrorTo correct an error, follow these steps.1. Note the error type (ERR:error type).2. Select 2:Goto, if it is available. The previous screen is displayed with the cursor at or near the error location.3. Determine the error. If you cannot recognize the error, refer to Appendix B.4. Correct the expression. Chapter 1: Operating the TI-84 Plus Silver Edition 33
Chapter 2:Math, Angle, and Test OperationsGetting Started: Coin FlipGetting Started is a fast-paced introduction. Read the chapter for details. For more probabilitysimulations, try the Probability Simulations App for the TI-84 Plus. You can download this App fromeducation.ti.com.Suppose you want to model flipping a fair coin 10 times. You want to track how many of those 10coin flips result in heads. You want to perform this simulation 40 times. With a fair coin, theprobability of a coin flip resulting in heads is 0.5 and the probability of a coin flip resulting in tails is0.5.1. Begin on the home screen. Press | to display the MATH PRB menu. Press 7 to select 7:randBin( (random Binomial). randBin( is pasted to the home screen. Press 10 to enter the number of coin flips. Press ¢. Press Ë 5 to enter the probability of heads. Press ¢. Press 40 to enter the number of simulations. Press ¤.2. Press Í to evaluate the expression. A list of 40 elements is generated with the first 7 displayed. The list contains the count of heads resulting from each set of 10 coin flips. The list has 40 elements because this simulation was performed 40 times. In this example, the coin came up heads five times in the first set of 10 coin flips, five times in the second set of 10 coin flips, and so on.3. Press ~ or | to view the additional counts in the list. An arrow (MathPrint™ mode) or an ellipses (Classic mode) indicate that the list continues beyond the screen.4. Press ¿ y d Í to store the data to the list name L1. You then can use the data for another activity, such as plotting a histogram MathPrint™ (Chapter 12).Note: Since randBin( generates random numbers, yourlist elements may differ from those in the example. Classic Chapter 2: Math, Angle, and Test Operations 34
Keyboard Math OperationsUsing Lists with Math OperationsMath operations that are valid for lists return a list calculated element by element. If you use twolists in the same expression, they must be the same length.Addition, Subtraction, Multiplication, DivisionYou can use + (addition, Ã), N (subtraction, ¹), … (multiplication, ¯), and à (division, ¥) with realand complex numbers, expressions, lists, and matrices. You cannot use à with matrices. If youneed to input A/2, enter this as A †1/2 or A †.5.valueA+valueB valueA N valueBvalueA…valueB valueA à valueBTrigonometric FunctionsYou can use the trigonometric (trig) functions (sine, ˜; cosine, ™; and tangent, š) with realnumbers, expressions, and lists. The current angle mode setting affects interpretation. Forexample, sin(30) in radian mode returns L.9880316241; in degree mode it returns .5.sin(value) cos(value) tan(value)You can use the inverse trig functions (arcsine, y ?; arccosine, y @; and arctangent,y A) with real numbers, expressions, and lists. The current angle mode setting affectsinterpretation.sinL1(value) cosL1(value) tanL1(value)Note: The trig functions do not operate on complex numbers.Power, Square, Square RootYou can use ^ (power, ›), 2 (square, ¡), and ‡( (square root, y C) with real and complexnumbers, expressions, lists, and matrices. You cannot use ‡( with matrices.MathPrint™: valuepower value2 ‡(value) ÈClassic: value^power È Chapter 2: Math, Angle, and Test Operations 35
InverseYou can use L1 (inverse, œ) with real and complex numbers, expressions, lists, and matrices. Themultiplicative inverse is equivalent to the reciprocal, 1àx.value-1log(, 10^(, ln(You can use log( (logarithm, «), 10^( (power of 10, y G), and ln( (natural log, μ) with real orcomplex numbers, expressions, and lists.log(value) MathPrint™: 10power ln(value) Classic: 10^(power)Exponentiale^( (exponential, y J) returns the constant e raised to a power. You can use e^( with real orcomplex numbers, expressions, and lists.MathPrint™: epowerClassic: e^(power)Constante (constant, y [e]) is stored as a constant on the TI-84 Plus. Press y [e] to copy e to the cursorlocation. In calculations, the TI-84 Plus uses 2.718281828459 for e.NegationM (negation, Ì) returns the negative of value. You can use M with real or complex numbers,expressions, lists, and matrices. Chapter 2: Math, Angle, and Test Operations 36
MvalueEOS™ rules (Chapter 1) determine when negation is evaluated. For example, L42 returns anegative number, because squaring is evaluated before negation. Use parentheses to square anegated number, as in (L4)2.Note: On the TI-84 Plus, the negation symbol (M) is shorter and higher than the subtraction sign (N),which is displayed when you press ¹.Pip (Pi, y B) is stored as a constant in the TI-84 Plus. In calculations, the TI-84 Plus uses3.1415926535898 for p.MATH OperationsMATH MenuTo display the MATH menu, press .MATH NUM CPX PRB1: 4Frac Displays the answer as a fraction.2: 4Dec Displays the answer as a decimal.3: 3 Calculates the cube.4: 3 ‡( Calculates the cube root.5: x ‡ Calculates the xth root.6: fMin( Finds the minimum of a function.7: fMax( Finds the maximum of a function.8: nDeriv( Computes the numerical derivative. Chapter 2: Math, Angle, and Test Operations 37
MATH NUM CPX PRB9: fnInt( Computes the function integral.0: summation )( Returns the sum of elements of list from start to end, where start <= end.A: logBASE( Returns the logarithm of a specifed value determined from a specified base: logBASE(value, base).B: Solver... Displays the equation solver.4Frac, 4Dec4Frac (display as a fraction) displays an answer as its rational equivalent. You can use 4Frac withreal or complex numbers, expressions, lists, and matrices. If the answer cannot be simplified orthe resulting denominator is more than three digits, the decimal equivalent is returned. You canonly use 4Frac following value.value 4Frac4Dec (display as a decimal) displays an answer in decimal form. You can use 4Dec with real orcomplex numbers, expressions, lists, and matrices. You can only use 4Dec following value.value 4DecNote: You can quickly convert from one number type to the other by using the FRAC shortcutmenu. Press t ^ 4:4F3 4D to convert a value.Cube, Cube Root3 (cube) returns the cube of value. You can use 3 with real or complex numbers, expressions, lists,and square matrices.value3 (cube root) returns the cube root of value. You can use 3‡( with real or complex numbers,3‡ (expressions, and lists.3 ‡(value) Chapter 2: Math, Angle, and Test Operations 38
x‡ (Root)x ‡ (xth root) returns the xth root of value. You can use x‡ with real or complex numbers, expressions,and lists.xthrootx‡valuefMin(, fMax(fMin( (function minimum) and fMax( (function maximum) return the value at which the localminimum or local maximum value of expression with respect to variable occurs, between lower andupper values for variable. fMin( and fMax( are not valid in expression. The accuracy is controlled bytolerance (if not specified, the default is 1âL5).fMin(expression,variable,lower,upper[,tolerance])fMax(expression,variable,lower,upper[,tolerance])Note: In this guidebook, optional arguments and the commas that accompany them are enclosedin brackets ([ ]).MathPrint™ClassicnDeriv(nDeriv( (numerical derivative) returns an approximate derivative of expression with respect to variable,given the value at which to calculate the derivative and H (if not specified, the default is 1âL3).nDeriv( is valid only for real numbers. Chapter 2: Math, Angle, and Test Operations 39
MathPrint™:Classic: nDeriv(expression,variable,value[,H])nDeriv( uses the symmetric difference quotient method, which approximates the numericalderivative value as the slope of the secant line through these points. f x + – f x – f x = ----------------------------------------- - 2As H becomes smaller, the approximation usually becomes more accurate. In MathPrint™ mode,the default H is 1EM3. You can switch to Classic mode to change H for investigations.MathPrint™ClassicYou can use nDeriv( once in expression. Because of the method used to calculate nDeriv(, the TI-84Plus can return a false derivative value at a nondifferentiable point.fnInt(fnInt( (function integral) returns the numerical integral (Gauss-Kronrod method) of expression withrespect to variable, given lower limit, upper limit, and a tolerance (if not specified, the default is 1âL5).fnInt( is valid only for real numbers.MathPrint™:Classic: fnInt(expression,variable,lower,upper[,tolerance])In MathPrint™ mode, the default H is 1EM3. You can switch to Classic mode to change H forinvestigations. Chapter 2: Math, Angle, and Test Operations 40
Note: To speed the drawing of integration graphs (when fnInt( is used in a Y= equation), increasethe value of the Xres window variable before you press s.Using the Equation SolverSolverSolver displays the equation solver, in which you can solve for any variable in an equation. Theequation is assumed to be equal to zero. Solver is valid only for real numbers.When you select Solver, one of two screens is displayed.• The equation editor (see step 1 picture below) is displayed when the equation variable eqn is empty.• The interactive solver editor is displayed when an equation is stored in eqn.Entering an Expression in the Equation SolverTo enter an expression in the equation solver, assuming that the variable eqn is empty, followthese steps.1. Select B:Solver from the MATH menu to display the equation editor.2. Enter the expression in any of three ways. • Enter the expression directly into the equation solver. • Paste a Y= variable name from the YVARS shortcut menu (t a) to the equation solver. • Press y K, paste a Y= variable name from the YVARS shortcut menu, and press Í. The expression is pasted to the equation solver. The expression is stored to the variable eqn as you enter it.3. Press Í or †. The interactive solver editor is displayed. • The equation stored in eqn is set equal to zero and displayed on the top line. • Variables in the equation are listed in the order in which they appear in the equation. Any values stored to the listed variables also are displayed. Chapter 2: Math, Angle, and Test Operations 41
• The default lower and upper bounds appear in the last line of the editor (bound={L1â99,1â99}). • A $ is displayed in the first column of the bottom line if the editor continues beyond the screen.Note: To use the solver to solve an equation such as K=.5MV2, enter eqn:0=KN.5MV2 in theequation editor.Entering and Editing Variable ValuesWhen you enter or edit a value for a variable in the interactive solver editor, the new value is storedin memory to that variable.You can enter an expression for a variable value. It is evaluated when you move to the nextvariable. Expressions must resolve to real numbers at each step during the iteration.You can store equations to any VARS Y-VARS variables, such as Y1 or r6, and then reference thevariables in the equation. The interactive solver editor displays all variables of all Y= functionsrecalled in the equation.Solving for a Variable in the Equation SolverTo solve for a variable using the equation solver after an equation has been stored to eqn, followthese steps.1. Select B:Solver from the MATH menu to display the interactive solver editor, if not already displayed.2. Enter or edit the value of each known variable. All variables, except the unknown variable, must contain a value. To move the cursor to the next variable, press Í or †. Chapter 2: Math, Angle, and Test Operations 42
3. Enter an initial guess for the variable for which you are solving. This is optional, but it may help find the solution more quickly. Also, for equations with multiple roots, the TI-84 Plus will attempt to display the solution that is closest to your guess. upper + lower The default guess is calculated as ---------------------------------------- . - 24. Edit bound={lower,upper}. lower and upper are the bounds between which the TI-84 Plus searches for a solution. This is optional, but it may help find the solution more quickly. The default is bound={L1â99,1â99}.5. Move the cursor to the variable for which you want to solve and press ƒ . • The solution is displayed next to the variable for which you solved. A solid square in the first column marks the variable for which you solved and indicates that the equation is balanced. An ellipsis shows that the value continues beyond the screen. Note: When a number continues beyond the screen, be sure to press ~ to scroll to the end of the number to see whether it ends with a negative or positive exponent. A very small number may appear to be a large number until you scroll right to see the exponent. • The values of the variables are updated in memory. • leftNrt=diff is displayed in the last line of the editor. diff is the difference between the left and right sides of the equation when evaluated at the calculated solution. A solid square in the first column next to leftNrt indicates that the equation has been evaluated at the new value of the variable for which you solved.Editing an Equation Stored to eqnTo edit or replace an equation stored to eqn when the interactive equation solver is displayed,press } until the equation editor is displayed. Then edit the equation.Equations with Multiple RootsSome equations have more than one solution. You can enter a new initial guess or new bounds tolook for additional solutions.Further SolutionsAfter you solve for a variable, you can continue to explore solutions from the interactive solvereditor. Edit the values of one or more variables. When you edit any variable value, the solid Chapter 2: Math, Angle, and Test Operations 43
squares next to the previous solution and leftNrt=diff disappear. Move the cursor to the variable forwhich you now want to solve and press ƒ .Controlling the Solution for Solver or solve(The TI-84 Plus solves equations through an iterative process. To control that process, enterbounds that are relatively close to the solution and enter an initial guess within those bounds. Thiswill help to find a solution more quickly. Also, it will define which solution you want for equationswith multiple solutions.Using solve( on the Home Screen or from a ProgramThe function solve( is available only from CATALOG or from within a program. It returns a solution(root) of expression for variable, given an initial guess, and lower and upper bounds within which thesolution is sought. The default for lower is L1â99. The default for upper is L1â99. solve( is valid onlyfor real numbers.solve(expression,variable,guess[,{lower,upper}])expression is assumed equal to zero. The value of variable will not be updated in memory. guess maybe a value or a list of two values. Values must be stored for every variable in expression, exceptvariable, before expression is evaluated. lower and upper must be entered in list format.MathPrint™ClassicMATH NUM (Number) OperationsMATH NUM MenuTo display the MATH NUM menu, press ~.MATH NUM CPX PRB1: abs( Absolute value2: round( Round3: iPart( Integer part Chapter 2: Math, Angle, and Test Operations 44
MATH NUM CPX PRB4: fPart( Fractional part5: int( Greatest integer6: min( Minimum value7: max( Maximum value8: lcm( Least common multiple9: gcd( Greatest common divisor0: remainder( Reports the remainder as a whole number from a division of two whole numbers where the divisor is not zero.A: 4n/d3 4Un/d Converts an improper fraction to a mixed number or a mixed number to an improper fraction.B: 4F3 4D Converts a decimal to a fraction or a fraction to a decimal.C: Un/d Displays the mixed number template in MathPrint™ mode. In Classic mode, displays a small u between the whole number and fraction.D: n/d Displays the fraction template in MathPrint™ mode. In Classic mode, displays a thick fraction bar between the numerator and the denominator.abs(abs( (absolute value) returns the absolute value of real or complex (modulus) numbers,expressions, lists, and matrices.Note: abs( is also found on the FUNC shortcut menu (t _ 1).abs(value)MathPrint™ClassicNote: abs( is also available on the MATH CPX menu. Chapter 2: Math, Angle, and Test Operations 45
round(round( returns a number, expression, list, or matrix rounded to #decimals (9). If #decimals is omitted,value is rounded to the digits that are displayed, up to 10 digits.round(value[,#decimals])iPart(, fPart(iPart( (integer part) returns the integer part or parts of real or complex numbers, expressions, lists,and matrices.iPart(value)fPart( (fractional part) returns the fractional part or parts of real or complex numbers, expressions,lists, and matrices.fPart(value)Note: The way the fractional result is displayed depends on the Answers mode setting. To convert from oneformat to another, use 4F3 4D on the FRAC shortcut menu (t ^ 4).int(int( (greatest integer) returns the largest integer real or complex numbers, expressions, lists, andmatrices.int(value) Chapter 2: Math, Angle, and Test Operations 46
Note: For a given value, the result of int( is the same as the result of iPart( for nonnegative numbersand negative integers, but one integer less than the result of iPart( for negative nonintegernumbers.min(, max(min( (minimum value) returns the smaller of valueA and valueB or the smallest element in list. If listAand listB are compared, min( returns a list of the smaller of each pair of elements. If list and valueare compared, min( compares each element in list with value.max( (maximum value) returns the larger of valueA and valueB or the largest element in list. If listAand listB are compared, max( returns a list of the larger of each pair of elements. If list and value arecompared, max( compares each element in list with value.min(valueA,valueB) max(valueA,valueB)min(list) max(list)min(listA,listB) max(listA,listB)min(list,value) max(list,value)Note: min( and max( also are available on the LIST MATH menu.lcm(, gcd(lcm( returns the least common multiple of valueA and valueB, both of which must be nonnegativeintegers. When listA and listB are specified, lcm( returns a list of the least common multiple of eachpair of elements. If list and value are specified, lcm( finds the least common multiple of eachelement in list and value.gcd( returns the greatest common divisor of valueA and valueB, both of which must be nonnegativeintegers. When listA and listB are specified, gcd( returns a list of the greatest common divisor ofeach pair of elements. If list and value are specified, gcd( finds the greatest common divisor of eachelement in list and value.lcm(valueA,valueB) gcd(valueA,valueB)lcm(listA,listB) gcd(listA,listB)lcm(list,value) gcd(list,value) Chapter 2: Math, Angle, and Test Operations 47
remainder(remainder( returns the remainder resulting from the division of two positive whole numbers, dividendand divisor, each of which can be a list. The divisor cannot be zero. If both arguments are lists, theymust have the same number of elements. If one argument is a list and the other a non-list, the non-list is paired with each element of the list, and a list is returned.remainder(dividend, divisor)remainder(list, divisor)remainder(dividend, list)remainder(list, list)4n/d3 4Un/d4n/d3 4Un/d converts an improper fraction to a mixed number or a mixed number to an improperfraction. You can also access 4n/d3 4Un/d from the FRAC shortcut menu (t ^ 3). Chapter 2: Math, Angle, and Test Operations 48
4F3 4D4F3 4D converts a fraction to a decimal or a decimal to a fraction. You can also access 4F3 4D fromthe FRAC shortcut menu (t ^ 4).Un/dUn/d displays the mixed number template. You can also access Un/d from the FRAC shortcutmenu (t ^ 2). In the fraction, n and d must be non-negative integers.MathPrint™ "Classicn/dn/d displays the mixed number template. You can also access n/d from the FRAC shortcut menu(t ^ 1). n and d can be real numbers or expressions but may not contain complex numbers.MathPrint™ "ClassicEntering and Using Complex NumbersComplex-Number ModesThe TI-84 Plus displays complex numbers in rectangular form and polar form. To select a complex-number mode, press z, and then select either of the two modes.• a+bi (rectangular-complex mode)• re^qi (polar-complex mode) Entering and Using Complex Numbers 49
On the TI-84 Plus, complex numbers can be stored to variables. Also, complex numbers are validlist elements.In Real mode, complex-number results return an error, unless you entered a complex number asinput. For example, in Real mode ln(L1) returns an error; in a+bi mode ln(L1) returns an answer.Real mode a+bi mode $ $Entering Complex NumbersComplex numbers are stored in rectangular form, but you can enter a complex number inrectangular form or polar form, regardless of the mode setting. The components of complexnumbers can be real numbers or expressions that evaluate to real numbers; expressions areevaluated when the command is executed.You can enter fractions in complex numbers, but the output will always be a decimal value.When you use the n/d template, a fraction cannot contain a complex number. "You can use division to compute the answer: Entering and Using Complex Numbers 50
Note about Radian Versus Degree ModeRadian mode is recommended for complex number calculations. Internally, the TI-84 Plusconverts all entered trigonometric values to radians, but it does not convert values for exponential,logarithmic, or hyperbolic functions.In degree mode, complex identities such as e^(iq) = cos(q) + i sin(q) are not generally truebecause the values for cos and sin are converted to radians, while those for e^() are not. Forexample, e^(i45) = cos(45) + i sin(45) is treated internally as e^(i45) = cos(p/4) + i sin(p/4).Complex identities are always true in radian mode.Interpreting Complex ResultsComplex numbers in results, including list elements, are displayed in either rectangular or polarform, as specified by the mode setting or by a display conversion instruction. In the examplebelow, polar-complex (re^qi) and Radian modes are set.MathPrint™:Classic:Rectangular-Complex ModeRectangular-complex mode recognizes and displays a complex number in the form a+bi, where a isthe real component, b is the imaginary component, and i is a constant equal to –1 .To enter a complex number in rectangular form, enter the value of a (real component), press à or ¹,enter the value of b (imaginary component), and press y V (constant).real component(+ or N)imaginary component iPolar-Complex ModePolar-complex mode recognizes and displays a complex number in the form re^qi, where r is themagnitude, e is the base of the natural log, q is the angle, and i is a constant equal to –1 . Entering and Using Complex Numbers 51
To enter a complex number in polar form, enter the value of r (magnitude), press y J(exponential function), enter the value of q (angle), press y V (constant), and then press ¤.magnitudee^(anglei)MathPrint™Classic Entering and Using Complex Numbers 52
imag(imag( (imaginary part) returns the imaginary (nonreal) part of a complex number or list of complexnumbers.imag(a+bi) returns b.imag(re^(qi)) returns r†sin(q).MathPrint™ Classicangle(angle( returns the polar angle of a complex number or list of complex numbers, calculated as tanL1(b/a), where b is the imaginary part and a is the real part. The calculation is adjusted by +p in thesecond quadrant or Np in the third quadrant.angle(a+bi) returns tanL1(b/a).angle(re^(qi)) returns q, where Lp<q<p.MathPrint™ Classicabs(abs( (absolute value) returns the magnitude (modulus), , of a complex number or listof complex numbers. You can also access abs( from the FUNC shortcut menu (t _ 1).abs(a+bi) returns .abs(re^(qi)) returns r (magnitude). Entering and Using Complex Numbers 54
4Rect4Rect (display as rectangular) displays a complex result in rectangular form. It is valid only at theend of an expression. It is not valid if the result is real.complex result8Rect returns a+bi.4Polar4Polar (display as polar) displays a complex result in polar form. It is valid only at the end of anexpression. It is not valid if the result is real.complex result8Polar returns re^(qi).MATH PRB (Probability) OperationsMATH PRB MenuTo display the MATH PRB menu, press |.MATH NUM CPX PRB1: rand Random-number generator2: nPr Number of permutations3: nCr Number of combinations4: ! Factorial Entering and Using Complex Numbers 55
MATH NUM CPX PRB5: randInt( Random-integer generator6: randNorm( Random # from Normal distribution7: randBin( Random # from Binomial distribution8: randIntNoRep( Random ordered list of integers in a rangerandrand (random number) generates and returns one or more random numbers > 0 and < 1. Togenerate a list of random-numbers, specify an integer > 1 for numtrials (number of trials). Thedefault for numtrials is 1.rand[(numtrials)]Note: To generate random numbers beyond the range of 0 to 1, you can include rand in anexpression. For example, rand5 generates a random number > 0 and < 5.With each rand execution, the TI-84 Plus generates the same random-number sequence for agiven seed value. The TI-84 Plus factory-set seed value for rand is 0. To generate a differentrandom-number sequence, store any nonzero seed value to rand. To restore the factory-set seedvalue, store 0 to rand or reset the defaults (Chapter 18).Note: The seed value also affects randInt(, randNorm(, and randBin( instructions.nPr, nCrnPr (number of permutations) returns the number of permutations of items taken number at a time.items and number must be nonnegative integers. Both items and number can be lists.items nPr numbernCr (number of combinations) returns the number of combinations of items taken number at a time.items and number must be nonnegative integers. Both items and number can be lists.items nCr number Entering and Using Complex Numbers 56
Factorial! (factorial) returns the factorial of either an integer or a multiple of .5. For a list, it returns factorialsfor each integer or multiple of .5. value must be ' L.5 and 69.value!Note: The factorial is computed recursively using the relationship (n+1)! = n…n!, until n is reducedto either 0 or L1/2. At that point, the definition 0!=1 or the definition (L1à2)!=‡p is used to completethe calculation. Hence:n!=n…(nN1)…(nN2)… ... …2…1, if n is an integer ' 0n!= n…(nN1)…(nN2)… ... …1à2…‡p, if n+1à2 is an integer ' 0n! is an error, if neither n nor n+1à2 is an integer ' 0.(The variable n equals value in the syntax description above.)randInt(randInt( (random integer) generates and displays a random integer within a range specified bylower and upper integer bounds. To generate a list of random numbers, specify an integer > 1 fornumtrials (number of trials); if not specified, the default is 1.randInt(lower,upper[,numtrials])randNorm(randNorm( (random Normal) generates and displays a random real number from a specifiedNormal distribution. Each generated value could be any real number, but most will be within theinterval [mN3(s), m+3(s)]. To generate a list of random numbers, specify an integer > 1 for numtrials(number of trials); if not specified, the default is 1.randNorm(m,s[,numtrials]) Entering and Using Complex Numbers 57
ANGLE7: P8Rx( Returns x, given R and q8: P8Ry( Returns y, given R and qEntry NotationDMS (degrees/minutes/seconds) entry notation comprises the degree symbol (¡), the minutesymbol (), and the second symbol ("). degrees must be a real number; minutes and seconds must bereal numbers ' 0.Note: DMS entry notation does not support fractions in minutes or seconds.degrees¡minutesseconds"For example, we know that 30 degrees is the same as p/6 radians, and we can verify that bylooking at the values in degree and radian modes. If the angle mode is not set to Degree, you mustuse ¡ so that the TI-84 Plus can interpret the argument as degrees, minutes, and seconds.Degree mode Radian modeDegree¡ (degree) designates an angle or list of angles as degrees, regardless of the current angle modesetting. In Radian mode, you can use ¡ to convert degrees to radians.value¡{value1,value2,value3,value4,...,value n}¡¡ also designates degrees (D) in DMS format. (minutes) designates minutes (M) in DMS format." (seconds) designates seconds (S) in DMS format.Note: " is not on the ANGLE menu. To enter ", press ƒ [ã].Radiansr (radians) designates an angle or list of angles as radians, regardless of the current angle modesetting. In Degree mode, you can use r to convert radians to degrees.valuer Entering and Using Complex Numbers 59
Degree mode8DMS8DMS (degree/minute/second) displays answer in DMS format. The mode setting must be Degreefor answer to be interpreted as degrees, minutes, and seconds. 8DMS is valid only at the end of aline.answer8DMSR8Pr (, R8Pq(, P8Rx(, P8Ry(R8Pr( converts rectangular coordinates to polar coordinates and returns r. R8Pq( convertsrectangular coordinates to polar coordinates and returns q. x and y can be lists.R8Pr(x,y), R8Pq(x,y) Note: Radian mode is set.P8Rx( converts polar coordinates to rectangular coordinates and returns x. P8Ry( converts polarcoordinates to rectangular coordinates and returns y. r and q can be lists.P8Rx(r,q), P8Ry(r,q) Note: Radian mode is set. Entering and Using Complex Numbers 60
TEST (Relational) OperationsTEST MenuTo display the TEST menu, press y :.This operator... Returns 1 (true) if...TEST LOGIC1: = Equal2: ƒ Not equal to3: > Greater than4: ' Greater than or equal to5: < Less than6: Less than or equal toÄ=, ƒ, >, ', <, Relational operators compare valueA and valueB and return 1 if the test is true or 0 if the test is false.valueA and valueB can be real numbers, expressions, or lists. For = and ƒ only, valueA and valueB alsocan be matrices or complex numbers. If valueA and valueB are matrices, both must have the samedimensions.Relational operators are often used in programs to control program flow and in graphing to controlthe graph of a function over specific values.valueA=valueB valueAƒvalueBvalueA>valueB valueA'valueBvalueA<valueB valueAvalueBUsing TestsRelational operators are evaluated after mathematical functions according to EOS rules(Chapter 1).• The expression 2+2=2+3 returns 0. The TI-84 Plus performs the addition first because of EOS rules, and then it compares 4 to 5.• The expression 2+(2=2)+3 returns 6. The TI-84 Plus performs the relational test first because it is in parentheses, and then it adds 2, 1, and 3. Entering and Using Complex Numbers 61
TEST LOGIC (Boolean) OperationsTEST LOGIC MenuTo display the TEST LOGIC menu, press y : ~.This operator... Returns a 1 (true) if...TEST LOGIC1: and Both values are nonzero (true).2: or At least one value is nonzero (true).3: xor Only one value is zero (false).4: not( The value is zero (false).Boolean OperatorsBoolean operators are often used in programs to control program flow and in graphing to controlthe graph of the function over specific values. Values are interpreted as zero (false) or nonzero(true).and, or, xorand, or, and xor (exclusive or) return a value of 1 if an expression is true or 0 if an expression isfalse, according to the table below. valueA and valueB can be real numbers, expressions, or lists.valueA and valueBvalueA or valueBvalueA xor valueB valueA valueB and or xor ƒ0 ƒ0 returns 1 1 0 ƒ0 0 returns 0 1 1 0 ƒ0 returns 0 1 1 0 0 returns 0 0 0not(not( returns 1 if value (which can be an expression) is 0.not(value)Using Boolean Operations Entering and Using Complex Numbers 62
Boolean logic is often used with relational tests. In the following program, the instructions store 4into C. Entering and Using Complex Numbers 63
Chapter 3:Function GraphingGetting Started: Graphing a CircleGetting Started is a fast-paced introduction. Read the chapter for details.Graph a circle of radius 10, centered on the origin in the standard viewing window. To graph thiscircle, you must enter separate formulas for the upper and lower portions of the circle. Then useZSquare (zoom square) to adjust the display and make the functions appear as a circle.1. In Func mode, press o to display the Y= editor. Press y C £ 100 ¹ " ¡ ¤ Í to enter the expression Y=‡(100NX 2), which defines the top half of the circle. The expression Y=L‡(100NX 2) defines the bottom half of the circle. On the TI-84 Plus, you can define one function in terms of another. To define Y2=LY1, press Ì to enter the negation sign. Press t a to display the Y-VARS shortcut menu, and then press Í to select Y1.2. Press q 6 to select 6:ZStandard. This is a quick way to reset the window variables to the standard values. It also graphs the functions; you do not need to press s. Notice that the functions appear as an ellipse in the standard viewing window. This is due to the range of values that ZStandard defines for the X-axis and Y-axis.3. To adjust the display so that each pixel represents an equal width and height, press q 5 to select 5:ZSquare. The functions are replotted and now appear as a circle on the display. Chapter 3: Function Graphing 64
4. To see the ZSquare window variables, press p and notice the new values for Xmin, Xmax, Ymin, and Ymax.Defining GraphsTI-84 Plus—Graphing Mode SimilaritiesChapter 3 specifically describes function graphing, but the steps shown here are similar for eachTI-84 Plus graphing mode. Chapters 4, 5, and 6 describe aspects that are unique to parametricgraphing, polar graphing, and sequence graphing.Defining a GraphTo define a graph in any graphing mode, follow these steps. Some steps are not alwaysnecessary.1. Press z and set the appropriate graph mode.2. Press o and enter, edit, or select one or more functions in the Y= editor.3. Deselect stat plots, if necessary.4. Set the graph style for each function.5. Press p and define the viewing window variables.6. Press y . and select the graph format settings.Displaying and Exploring a GraphAfter you have defined a graph, press s to display it. Explore the behavior of the function orfunctions using the TI-84 Plus tools described in this chapter.Saving a Graph for Later UseYou can store the elements that define the current graph to any of 10 graph database variables(GDB1 through GDB9, and GDB0; Chapter 8). To recreate the current graph later, simply recall thegraph database to which you stored the original graph.These types of information are stored in a GDB.• Y= functions• Graph style settings• Window settings• Format settings Chapter 3: Function Graphing 65
You can store a picture of the current graph display to any of 10 graph picture variables (Pic1through Pic9, and Pic0; Chapter 8). Then you can superimpose one or more stored pictures ontothe current graph.Setting the Graph ModesChecking and Changing the Graphing ModeTo display the mode screen, press z. The default settings are highlighted below. To graphfunctions, you must select Func mode before you enter values for the window variables and beforeyou enter the functions.The TI-84 Plus has four graphing modes.• Func (function graphing)• Par (parametric graphing; Chapter 4)• Pol (polar graphing; Chapter 5)• Seq (sequence graphing; Chapter 6)Other mode settings affect graphing results. Chapter 1 describes each mode setting.• Float or 0123456789 (fixed) decimal mode affects displayed graph coordinates.• Radian or Degree angle mode affects interpretation of some functions.• Connected or Dot plotting mode affects plotting of selected functions.• Sequential or Simul graphing-order mode affects function plotting when more than one function is selected.Setting Modes from a ProgramTo set the graphing mode and other modes from a program, begin on a blank line in the programeditor and follow these steps.1. Press z to display the mode settings.2. Press †, ~, |, and } to place the cursor on the mode that you want to select.3. Press Í to paste the mode name to the cursor location.The mode is changed when the program is executed. Chapter 3: Function Graphing 66
Defining FunctionsDisplaying Functions in the Y= EditorTo display the Y= editor, press o. You can store up to 10 functions to the function variables Y1through Y9, and Y0. You can graph one or more defined functions at once. In this example,functions Y1 and Y2 are defined and selected.Defining or Editing a FunctionTo define or edit a function, follow these steps.1. Press o to display the Y= editor.2. Press † to move the cursor to the function you want to define or edit. To erase a function, press '.3. Enter or edit the expression to define the function. • You may use functions and variables (including matrices and lists) in the expression. When the expression evaluates to a nonreal number, the value is not plotted; no error is returned. • You can access the shortcut menus by pressing ƒ ^ - a. • The independent variable in the function is X. Func mode defines " as X. To enter X, press " or press ƒ [X]. • When you enter the first character, the = is highlighted, indicating that the function is selected. As you enter the expression, it is stored to the variable Yn as a user-defined function in the Y= editor.4. Press Í or † to move the cursor to the next function.Defining a Function from the Home Screen or a ProgramTo define a function from the home screen or a program, begin on a blank line and follow thesesteps.1. Press ƒ [ã], enter the expression, and then press ƒ [ã] again.2. Press ¿. Chapter 3: Function Graphing 67
3. Press ƒ a to display the YVAR shortcut menu, move the cursor to the function name, and then press Í. "expression"!YnWhen the instruction is executed, the TI-84 Plus stores the expression to the designated variableYn, selects the function, and displays the message Done.Evaluating Y= Functions in ExpressionsYou can calculate the value of a Y= function Yn at a specified value of X. A list of values returns a list.Yn(value)Yn({value1,value2,value3, . . .,value n})Selecting and Deselecting FunctionsSelecting and Deselecting a FunctionYou can select and deselect (turn on and turn off) a function in the Y= editor. A function is selectedwhen the = sign is highlighted. The TI-84 Plus graphs only the selected functions. You can selectany or all functions Y1 through Y9, and Y0.To select or deselect a function in the Y= editor, follow these steps.1. Press o to display the Y= editor.2. Move the cursor to the function you want to select or deselect.3. Press | to place the cursor on the function's = sign.4. Press Í to change the selection status.When you enter or edit a function, it is selected automatically. When you clear a function, it isdeselected. Chapter 3: Function Graphing 68
Turning On or Turning Off a Stat Plot in the Y= EditorTo view and change the on/off status of a stat plot in the Y= editor, use Plot1 Plot2 Plot3 (the topline of the Y= editor). When a plot is on, its name is highlighted on this line.To change the on/off status of a stat plot from the Y= editor, press } and ~ to place the cursor onPlot1, Plot2, or Plot3, and then press Í. Plot1 is turned on. Plot2 and Plot3 are turned off.Selecting and Deselecting Functions from the Home Screen or a ProgramTo select or deselect a function from the home screen or a program, begin on a blank line andfollow these steps.1. Press ~ to display the VARS Y-VARS menu.2. Select 4:On/Off to display the ON/OFF secondary menu.3. Select 1:FnOn to turn on one or more functions or 2:FnOff to turn off one or more functions. The instruction you select is copied to the cursor location.4. Enter the number (1 through 9, or 0; not the variable Yn) of each function you want to turn on or turn off. • If you enter two or more numbers, separate them with commas. • To turn on or turn off all functions, do not enter a number after FnOn or FnOff. FnOn[function#,function#, . . .,function n] FnOff[function#,function#, . . .,function n]5. Press Í. When the instruction is executed, the status of each function in the current mode is set and Done is displayed.For example, in Func mode, FnOff :FnOn 1,3 turns off all functions in the Y= editor, and then turnson Y1 and Y3. Chapter 3: Function Graphing 69
Setting Graph Styles for FunctionsMATH Graph Style Icons in the Y= EditorThis table describes the graph styles available for function graphing. Use the styles to visuallydifferentiate functions to be graphed together. For example, you can set Y1 as a solid line, Y2 as adotted line, and Y3 as a thick line. Icon Style Description ç Line A solid line connects plotted points; this is the default in Connected mode è Thick A thick solid line connects plotted points é Above Shading covers the area above the graph ê Below Shading covers the area below the graph ë Path A circular cursor traces the leading edge of the graph and draws a path ì Animate A circular cursor traces the leading edge of the graph without drawing a path í Dot A small dot represents each plotted point; this is the default in Dot modeNote: Some graph styles are not available in all graphing modes. Chapters 4, 5, and 6 list thestyles for Par, Pol, and Seq modes.Setting the Graph StyleTo set the graph style for a function, follow these steps.1. Press o to display the Y= editor.2. Press † and } to move the cursor to the function.3. Press | | to move the cursor left, past the = sign, to the graph style icon in the first column. The insert cursor is displayed. (Steps 2 and 3 are interchangeable.)4. Press Í repeatedly to rotate through the graph styles. The seven styles rotate in the same order in which they are listed in the table above.5. Press ~, }, or † when you have selected a style. Chapter 3: Function Graphing 70
Shading Above and BelowWhen you select é or ê for two or more functions, the TI-84 Plus rotates through four shadingpatterns.• Vertical lines shade the first function with a é or ê graph style.• Horizontal lines shade the second.• Negatively sloping diagonal lines shade the third.• Positively sloping diagonal lines shade the fourth.• The rotation returns to vertical lines for the fifth é or ê function, repeating the order described above.When shaded areas intersect, the patterns overlap.Note: When é or ê is selected for a Y= function that graphs a family of curves, such as Y1={1,2,3}X,the four shading patterns rotate for each member of the family of curves.Setting a Graph Style from a ProgramTo set the graph style from a program, select H:GraphStyle( from the PRGM CTL menu. To displaythis menu, press while in the program editor. function# is the number of the Y= function namein the current graphing mode. graphstyle# is an integer from 1 to 7 that corresponds to the graphstyle, as shown below.1 = ç (line) 5 = ë (path)2 = è (thick) 6 = ì (animate)3 = é (above) 7 = í (dot)4 = ê (below)GraphStyle(function#,graphstyle#)For example, when this program is executed in Func mode, GraphStyle(1,3) sets Y1 to é (above). Chapter 3: Function Graphing 71
Setting the Viewing Window VariablesThe TI-84 Plus Viewing WindowThe viewing window is the portion of the coordinate plane defined by Xmin, Xmax, Ymin, and Ymax.Xscl (X scale) defines the distance between tick marks on the x-axis. Yscl (Y scale) defines thedistance between tick marks on the y-axis. To turn off tick marks, set Xscl=0 and Yscl=0.Displaying the Window VariablesTo display the current window variable values, press p. The window editor above and to theright shows the default values in Func graphing mode and Radian angle mode. The windowvariables differ from one graphing mode to another.Xres sets pixel resolution (1 through 8) for function graphs only. The default is 1.• At Xres=1, functions are evaluated and graphed at each pixel on the x-axis.• At Xres=8, functions are evaluated and graphed at every eighth pixel along the x-axis.Note: Small Xres values improve graph resolution but may cause the TI-84 Plus to draw graphsmore slowly.Changing a Window Variable ValueTo change a window variable value from the window editor, follow these steps.1. Press † or } to move the cursor to the window variable you want to change.2. Edit the value, which can be an expression. • Enter a new value, which clears the original value. • Move the cursor to a specific digit, and then edit it.3. Press Í, †, or }. If you entered an expression, the TI-84 Plus evaluates it. The new value is stored.Note: Xmin<Xmax and Ymin<Ymax must be true in order to graph.Storing to a Window Variable from the Home Screen or a ProgramTo store a value, which can be an expression, to a window variable, begin on a blank line andfollow these steps. Chapter 3: Function Graphing 72
1. Enter the value you want to store.2. Press ¿.3. Press to display the VARS menu.4. Select 1:Window to display the Func window variables (X/Y secondary menu). • Press ~ to display the Par and Pol window variables (T/q secondary menu). • Press ~ ~ to display the Seq window variables (U/V/W secondary menu).5. Select the window variable to which you want to store a value. The name of the variable is pasted to the current cursor location.6. Press Í to complete the instruction.When the instruction is executed, the TI-84 Plus stores the value to the window variable anddisplays the value.@X and @YThe variables @X and @Y (items 8 and 9 on the VARS (1:Window) X/Y secondary menu; @X is alsoon the Window screen) define the distance from the center of one pixel to the center of anyadjacent pixel on a graph (graphing accuracy). @X and @Y are calculated from Xmin, Xmax, Ymin,and Ymax when you display a graph. Xmax – Xmin Ymax – Ymin X = -------------------------------------- - Y = -------------------------------------- - 94 62You can store values to @X and @Y. If you do, Xmax and Ymax are calculated from @X, Xmin, @Y,and Ymin.Note: The ZFrac ZOOM settings (Zfrac1/2, ZFrac1/3, ZFrac1/4, ZFrac1/5, ZFrac1/8, ZFrac1/10)change @X and @Y to fractional values. If fractions are not needed for your problem, you can adjust@X and @Y to suit your needs.Setting the Graph FormatDisplaying the Format SettingsTo display the format settings, press y .. The default settings are highlighted below.Note: You can also go to the Format Graph screen from the Mode screen by selecting YES at theGoTo Format Graph prompt. After you make changes, press zto return to the Mode screen.RectGC PolarGC Sets cursor coordinates.CoordOn CoordOff Sets coordinates display on or off.GridOff GridOn Sets grid off or on. Chapter 3: Function Graphing 73
AxesOn AxesOff Sets axes on or off.LabelOff LabelOn Sets axes label off or on.ExprOn ExprOff Sets expression display on or off.Format settings define a graph's appearance on the display. Format settings apply to all graphingmodes. Seq graphing mode has an additional mode setting (Chapter 6).Changing a Format SettingTo change a format setting, follow these steps.1. Press †, ~, }, and | as necessary to move the cursor to the setting you want to select.2. Press Í to select the highlighted setting.RectGC, PolarGCRectGC (rectangular graphing coordinates) displays the cursor location as rectangular coordinatesX and Y.PolarGC (polar graphing coordinates) displays the cursor location as polar coordinates R and q.The RectGC/PolarGC setting determines which variables are updated when you plot the graph,move the free-moving cursor, or trace.• RectGC updates X and Y; if CoordOn format is selected, X and Y are displayed.• PolarGC updates X, Y, R, and q; if CoordOn format is selected, R and q are displayed.CoordOn, CoordOffCoordOn (coordinates on) displays the cursor coordinates at the bottom of the graph. If ExprOffformat is selected, the function number is displayed in the top-right corner.CoordOff (coordinates off) does not display the function number or coordinates.GridOff, GridOnGrid points cover the viewing window in rows that correspond to the tick marks on each axis.GridOff does not display grid points.GridOn displays grid points.AxesOn, AxesOffAxesOn displays the axes. Chapter 3: Function Graphing 74
AxesOff does not display the axes.This overrides the LabelOff/ LabelOn format setting.LabelOff, LabelOnLabelOff and LabelOn determine whether to display labels for the axes (X and Y), if AxesOn formatis also selected.ExprOn, ExprOffExprOn and ExprOff determine whether to display the Y= expression when the trace cursor isactive. This format setting also applies to stat plots.When ExprOn is selected, the expression is displayed in the top-left corner of the graph screen.When ExprOff and CoordOn both are selected, the number in the top-right corner specifies whichfunction is being traced.Displaying GraphsDisplaying a New GraphTo display the graph of the selected function or functions, press s. TRACE, ZOOMinstructions, and CALC operations display the graph automatically. As the TI-84 Plus plots thegraph, the busy indicator is on. As the graph is plotted, X and Y are updated.Pausing or Stopping a GraphWhile plotting a graph, you can pause or stop graphing.• Press Í to pause; then press Í to resume.• Press É to stop; then press s to redraw.Smart GraphSmart Graph is a TI-84 Plus feature that redisplays the last graph immediately when you presss, but only if all graphing factors that would cause replotting have remained the same sincethe graph was last displayed.If you performed any of the following actions since the graph was last displayed, the TI-84 Plus willreplot the graph based on new values when you press s.• Changed a mode setting that affects graphs• Changed a function in the current picture• Selected or deselected a function or stat plot Chapter 3: Function Graphing 75
• Changed the value of a variable in a selected function• Changed a window variable or graph format setting• Cleared drawings by selecting ClrDraw• Changed a stat plot definitionOverlaying Functions on a GraphOn the TI-84 Plus, you can graph one or more new functions without replotting existing functions.For example, store sin(X) to Y1 in the Y= editor and press s. Then store cos(X) to Y2 andpress s again. The function Y2 is graphed on top of Y1, the original function.Graphing a Family of CurvesIf you enter a list (Chapter 11) as an element in an expression, the TI-84 Plus plots the function foreach value in the list, thereby graphing a family of curves. In Simul graphing-order mode, it graphsall functions sequentially for the first element in each list, and then for the second, and so on.{2,4,6}sin(X) graphs three functions: 2 sin(X), 4 sin(X), and 6 sin(X).{2,4,6}sin({1,2,3}X) graphs 2 sin(X), 4 sin(2X), and 6 sin(3X) .Note: When using more than one list, the lists must have the same dimensions. Chapter 3: Function Graphing 76
Exploring Graphs with the Free-Moving CursorFree-Moving CursorWhen a graph is displayed, press |, ~, }, or † to move the cursor around the graph. When youfirst display the graph, no cursor is visible. When you press |, ~, }, or †, the cursor moves fromthe center of the viewing window.As you move the cursor around the graph, the coordinate values of the cursor location aredisplayed at the bottom of the screen if CoordOn format is selected. The Float/Fix decimal modesetting determines the number of decimal digits displayed for the coordinate values.To display the graph with no cursor and no coordinate values, press ' or Í. When youpress |, ~, }, or †, the cursor moves from the same position.Graphing AccuracyThe free-moving cursor moves from pixel to pixel on the screen. When you move the cursor to apixel that appears to be on the function, the cursor may be near, but not actually on, the function.The coordinate value displayed at the bottom of the screen actually may not be a point on thefunction. To move the cursor along a function, use r.The coordinate values displayed as you move the cursor approximate actual math coordinates,accurate to within the width and height of the pixel. As Xmin, Xmax, Ymin, and Ymax get closertogether (as in a Zoom In) graphing accuracy increases, and the coordinate values more closelyapproximate the math coordinates. Free- moving cursor appears to be on the curveExploring Graphs with TRACEBeginning a TraceUse TRACE to move the cursor from one plotted point to the next along a function. To begin atrace, press r. If the graph is not displayed already, press r to display it. The trace cursoris on the first selected function in the Y= editor, at the middle X value on the screen. The cursorcoordinates are displayed at the bottom of the screen if CoordOn format is selected. TheY= expression is displayed in the top-left corner of the screen, if ExprOn format is selected. Chapter 3: Function Graphing 77
Moving the Trace CursorTo move the TRACE cursor do this:To the previous or next plotted point, press | or ~.Five plotted points on a function (Xres press y | or y ~.affects this),To any valid X value on a function, enter a value, and then press Í.From one function to another, press } or †.When the trace cursor moves along a function, the Y value is calculated from the X value; that is,Y=Yn(X). If the function is undefined at an X value, the Y value is blank. Trace cursor on the curveIf you move the trace cursor beyond the top or bottom of the screen, the coordinate values at thebottom of the screen continue to change appropriately.Moving the Trace Cursor from Function to FunctionTo move the trace cursor from function to function, press † and }. The cursor follows the order ofthe selected functions in the Y= editor. The trace cursor moves to each function at the same Xvalue. If ExprOn format is selected, the expression is updated.Moving the Trace Cursor to Any Valid X ValueTo move the trace cursor to any valid X value on the current function, enter the value. When youenter the first digit, an X= prompt and the number you entered are displayed in the bottom-leftcorner of the screen. You can enter an expression at the X= prompt. The value must be valid forthe current viewing window. When you have completed the entry, press Í to move the cursor.Note: This feature does not apply to stat plots. Chapter 3: Function Graphing 78
Panning to the Left or RightIf you trace a function beyond the left or right side of the screen, the viewing window automaticallypans to the left or right. Xmin and Xmax are updated to correspond to the new viewing window.Quick ZoomWhile tracing, you can press Í to adjust the viewing window so that the cursor locationbecomes the center of the new viewing window, even if the cursor is above or below the display.This allows panning up and down. After Quick Zoom, the cursor remains in TRACE.Leaving and Returning to TRACEWhen you leave and return to TRACE, the trace cursor is displayed in the same location it was inwhen you left TRACE, unless Smart Graph has replotted the graph.Using TRACE in a ProgramOn a blank line in the program editor, press r. The instruction Trace is pasted to the cursorlocation. When the instruction is encountered during program execution, the graph is displayedwith the trace cursor on the first selected function. As you trace, the cursor coordinate values areupdated. When you finish tracing the functions, press Í to resume program execution.Exploring Graphs with the ZOOM InstructionsZOOM MenuTo display the ZOOM menu, press q. You can adjust the viewing window of the graph quickly inseveral ways. All ZOOM instructions are accessible from programs.ZOOM MEMORY1: ZBox Draws a box to define the viewing window.2: Zoom In Magnifies the graph around the cursor.3: Zoom Out Views more of a graph around the cursor.4: ZDecimal Sets @X and @Y to 0.1.5: ZSquare Sets equal-size pixels on the X and Y axes.6: ZStandard Sets the standard window variables.7: ZTrig Sets the built-in trig window variables.8: ZInteger Sets integer values on the X and Y axes.9: ZoomStat Sets the values for current stat lists.0: ZoomFit Fits YMin and YMax between XMin and XMax.A: ZQuadrant1 Displays the portion of the graph that is in quadrant 1 Chapter 3: Function Graphing 79
ZOOM MEMORYB: ZFrac1/2 Sets the window variables so that you can trace in increments of , if possible. Sets @X and @Y to .C: ZFrac1/3 Sets the window variables so that you can trace in increments of , if possible. Sets @X and @Y to .D: ZFrac1/4 Sets the window variables so that you can trace in increments of , if possible. Sets @X and @Y to .E: ZFrac1/5 Sets the window variables so that you can trace in increments of , if possible. Sets @X and @Y to .F: ZFrac1/8 Sets the window variables so that you can trace in increments of , if possible. Sets @X and @Y to .G: ZFrac1/10 Sets the window variables so that you can trace in increments of , if possible. Sets @X and @Y to .Note: You can adjust all window variables from the VARS menu by pressing 1:Window andthen selecting the variable from the X/Y, T/q, or U/V/W menu.Zoom CursorWhen you select 1:ZBox, 2:Zoom In, or 3:Zoom Out, the cursor on the graph becomes the zoomcursor (+), a smaller version of the free-moving cursor (+).ZBoxTo define a new viewing window using ZBox, follow these steps.1. Select 1:ZBox from the ZOOM menu. The zoom cursor is displayed at the center of the screen.2. Move the zoom cursor to any spot you want to define as a corner of the box, and then press Í. When you move the cursor away from the first defined corner, a small, square dot indicates the spot.3. Press |, }, ~, or †. As you move the cursor, the sides of the box lengthen or shorten proportionately on the screen. Note: To cancel ZBox before you press Í, press '.4. When you have defined the box, press Í to replot the graph. Chapter 3: Function Graphing 80
To use ZBox to define another box within the new graph, repeat steps 2 through 4. To cancel ZBox,press '.Zoom In, Zoom OutZoom In magnifies the part of the graph that surrounds the cursor location. Zoom Out displays agreater portion of the graph, centered on the cursor location. The XFact and YFact settingsdetermine the extent of the zoom.To zoom in on a graph, follow these steps.1. Check XFact and YFact; change as needed.2. Select 2:Zoom In from the ZOOM menu. The zoom cursor is displayed.3. Move the zoom cursor to the point that is to be the center of the new viewing window.4. Press Í. The TI-83 Plus adjusts the viewing window by XFact and YFact; updates the window variables; and replots the selected functions, centered on the cursor location.5. Zoom in on the graph again in either of two ways. • To zoom in at the same point, press Í. • To zoom in at a new point, move the cursor to the point that you want as the center of the new viewing window, and then press Í.To zoom out on a graph, select 3:Zoom Out and repeat steps 3 through 5.To cancel Zoom In or Zoom Out, press '.ZDecimalZDecimal replots the functions immediately. It updates the window variables to preset values, asshown below. These values set @X and @Y equal to 0.1 and set the X and Y value of each pixel toone decimal place.Xmin=L4.7 Ymin=L3.1Xmax=4.7 Ymax=3.1Xscl=1 Yscl=1ZSquareZSquare replots the functions immediately. It redefines the viewing window based on the currentvalues of the window variables. It adjusts in only one direction so that @X=@Y, which makes thegraph of a circle look like a circle. Xscl and Yscl remain unchanged. The midpoint of the currentgraph (not the intersection of the axes) becomes the midpoint of the new graph. Chapter 3: Function Graphing 81
ZStandardZStandard replots the functions immediately. It updates the window variables to the standardvalues shown below.Xmin=L10 Ymin=L10 Xres=1Xmax=10 Ymax=10Xscl=1 Yscl=1ZTrigZTrig replots the functions immediately. It updates the window variables to preset values that areappropriate for plotting trig functions. Those preset values in Radian mode are shown below.Xmin=L(47à24)p (decimal equivalent) Ymin=L4Xmax=(47à24)p (decimal equivalent) Ymax=4Xscl=p/2 (decimal equivalent) Yscl=1ZIntegerZInteger redefines the viewing window to the dimensions shown below. To use ZInteger, move thecursor to the point that you want to be the center of the new window, and then press Í;ZInteger replots the functions.@X=1 Xscl=10@Y=1 Yscl=10ZoomStatZoomStat redefines the viewing window so that all statistical data points are displayed. For regularand modified box plots, only Xmin and Xmax are adjusted.ZoomFitZoomFit replots the functions immediately. ZoomFit recalculates YMin and YMax to include theminimum and maximum Y values of the selected functions between the current XMin and XMax.XMin and XMax are not changed.ZQuadrant1ZQuandrant1 replots the function immediately. It redefines the window settings so that onlyquadrant 1 is displayed. Chapter 3: Function Graphing 82
ZFrac1/2ZFrac1/2 replots the functions immediately. It updates the window variables to preset values, asshown below. These values set @X and @Y equal to 1/2 and set the X and Y value of each pixel toone decimal place.Xmin=L47/2 Ymin=L31/2Xmax=47/2 Ymax=31/2Xscl=1 Yscl=1ZFrac1/3ZFrac1/3 replots the functions immediately. It updates the window variables to preset values, asshown below. These values set @X and @Y equal to 1/3 and set the X and Y value of each pixel toone decimal place.Xmin=L47/3 Ymin=L31/3Xmax=47/3 Ymax=31/3Xscl=1 Yscl=1ZFrac1/4ZFrac1/4 replots the functions immediately. It updates the window variables to preset values, asshown below. These values set @X and @Y equal to 1/4 and set the X and Y value of each pixel toone decimal place.Xmin=L47/4 Ymin=L31/4Xmax=47/4 Ymax=31/4Xscl=1 Yscl=1ZFrac1/5ZFrac1/5 replots the functions immediately. It updates the window variables to preset values, asshown below. These values set @X and @Y equal to 1/5 and set the X and Y value of each pixel toone decimal place.Xmin=L47/5 Ymin=L31/5Xmax=47/5 Ymax=31/5Xscl=1 Yscl=1 Chapter 3: Function Graphing 83
ZFrac1/8ZDecimal replots the functions immediately. It updates the window variables to preset values, asshown below. These values set @X and @Y equal to 1/8 and set the X and Y value of each pixel toone decimal place.Xmin=L47/8 Ymin=L31/8Xmax=47/8 Ymax=31/8Xscl=1 Yscl=1ZFrac1/10ZFrac1/10 replots the functions immediately. It updates the window variables to preset values, asshown below. These values set @X and @Y equal to 1/10 and set the X and Y value of each pixel toone decimal place.Xmin=L47/10 Ymin=L31/10Xmax=47/10 Ymax=31/10Xscl=1 Yscl=1Using ZOOM MEMORYZOOM MEMORY MenuTo display the ZOOM MEMORY menu, press q ~.ZOOM MEMORY1: ZPrevious Uses the previous viewing window.2: ZoomSto Stores the user-defined window.3: ZoomRcl Recalls the user-defined window.4: SetFactors... Changes Zoom In and Zoom Out factors.ZPreviousZPrevious replots the graph using the window variables of the graph that was displayed before youexecuted the last ZOOM instruction.ZoomStoZoomSto immediately stores the current viewing window. The graph is displayed, and the values ofthe current window variables are stored in the user-defined ZOOM variables ZXmin, ZXmax, ZXscl,ZYmin, ZYmax, ZYscl, and ZXres.These variables apply to all graphing modes. For example, changing the value of ZXmin in Funcmode also changes it in Par mode. Chapter 3: Function Graphing 84
ZoomRclZoomRcl graphs the selected functions in a user-defined viewing window. The user-definedviewing window is determined by the values stored with the ZoomSto instruction. The windowvariables are updated with the user-defined values, and the graph is plotted.ZOOM FACTORSThe zoom factors, XFact and YFact, are positive numbers (not necessarily integers) greater than orequal to 1. They define the magnification or reduction factor used to Zoom In or Zoom Out around apoint.Checking XFact and YFactTo display the ZOOM FACTORS screen, where you can review the current values for XFact andYFact, select 4:SetFactors from the ZOOM MEMORY menu. The values shown are the defaults.Changing XFact and YFactYou can change XFact and YFact in either of two ways.• Enter a new value. The original value is cleared automatically when you enter the first digit.• Place the cursor on the digit you want to change, and then enter a value or press { to delete it.Using ZOOM MEMORY Menu Items from the Home Screen or a ProgramFrom the home screen or a program, you can store directly to any of the user-defined ZOOMvariables.From a program, you can select the ZoomSto and ZoomRcl instructions from the ZOOM MEMORYmenu. Chapter 3: Function Graphing 85
Using the CALC (Calculate) OperationsCALCULATE MenuTo display the CALCULATE menu, press y /. Use the items on this menu to analyze thecurrent graph functions.CALCULATE1: value Calculates a function Y value for a given X.2: zero Finds a zero (x-intercept) of a function.3: minimum Finds a minimum of a function.4: maximum Finds a maximum of a function.5: intersect Finds an intersection of two functions.6: dy/dx Finds a numeric derivative of a function.7: ‰f(x)dx Finds a numeric integral of a function.valuevalue evaluates one or more currently selected functions for a specified value of X.Note: When a value is displayed for X, press ' to clear the value. When no value is displayed,press ' to cancel the value operation.To evaluate a selected function at X, follow these steps.1. Select 1:value from the CALCULATE menu. The graph is displayed with X= in the bottom-left corner.2. Enter a real value, which can be an expression, for X between Xmin and Xmax.3. Press Í.The cursor is on the first selected function in the Y= editor at the X value you entered, and thecoordinates are displayed, even if CoordOff format is selected.To move the cursor from function to function at the entered X value, press } or †. To restore thefree-moving cursor, press | or ~. Chapter 3: Function Graphing 86
zerozero finds a zero (x-intercept or root) of a function using solve(. Functions can have more than onex-intercept value; zero finds the zero closest to your guess.The time zero spends to find the correct zero value depends on the accuracy of the values youspecify for the left and right bounds and the accuracy of your guess.To find a zero of a function, follow these steps.1. Select 2:zero from the CALCULATE menu. The current graph is displayed with Left Bound? in the bottom-left corner.2. Press } or † to move the cursor onto the function for which you want to find a zero.3. Press | or ~ (or enter a value) to select the x-value for the left bound of the interval, and then press Í. A 4 indicator on the graph screen shows the left bound. Right Bound? is displayed in the bottom-left corner. Press | or ~ (or enter a value) to select the x-value for the right bound, and then press Í. A 3 indicator on the graph screen shows the right bound. Guess? is then displayed in the bottom-left corner.4. Press | or ~ (or enter a value) to select a point near the zero of the function, between the bounds, and then press Í.The cursor is on the solution and the coordinates are displayed, even if CoordOff format isselected. To move to the same x-value for other selected functions, press } or †. To restore thefree-moving cursor, press | or ~.minimum, maximumminimum and maximum find a minimum or maximum of a function within a specified interval to atolerance of 1âL5.To find a minimum or maximum, follow these steps.1. Select 3:minimum or 4:maximum from the CALCULATE menu. The current graph is displayed.2. Select the function and set left bound, right bound, and guess as described for zero. Chapter 3: Function Graphing 87
The cursor is on the solution, and the coordinates are displayed, even if you have selectedCoordOff format; Minimum or Maximum is displayed in the bottom-left corner.To move to the same x-value for other selected functions, press } or †. To restore the free-moving cursor, press | or ~.intersectintersect finds the coordinates of a point at which two or more functions intersect using solve(. Theintersection must appear on the display to use intersect.To find an intersection, follow these steps.1. Select 5:intersect from the CALCULATE menu. The current graph is displayed with First curve? in the bottom-left corner.2. Press † or }, if necessary, to move the cursor to the first function, and then press Í. Second curve? is displayed in the bottom-left corner.3. Press † or }, if necessary, to move the cursor to the second function, and then press Í.4. Press ~ or | to move the cursor to the point that is your guess as to location of the intersection, and then press Í.The cursor is on the solution and the coordinates are displayed, even if CoordOff format isselected. Intersection is displayed in the bottom-left corner. To restore the free-moving cursor,press |, }, ~, or †.dy/dxdy/dx (numerical derivative) finds the numerical derivative (slope) of a function at a point, withH=1âL3.To find a function's slope at a point, follow these steps.1. Select 6:dy/dx from the CALCULATE menu. The current graph is displayed.2. Press } or † to select the function for which you want to find the numerical derivative.3. Press | or ~ (or enter a value) to select the X value at which to calculate the derivative, and then press Í.The cursor is on the solution and the numerical derivative is displayed.To move to the same x-value for other selected functions, press } or †. To restore the free-moving cursor, press | or ~. Chapter 3: Function Graphing 88
‰f(x)dx‰f(x)dx (numerical integral) finds the numerical integral of a function in a specified interval. It usesthe fnInt( function, with a tolerance of H=1âL3.To find the numerical integral of a function, follow these steps.1. Select 7:‰f(x)dx from the CALCULATE menu. The current graph is displayed with Lower Limit? in the bottom-left corner.2. Press } or † to move the cursor to the function for which you want to calculate the integral.3. Set lower and upper limits as you would set left and right bounds for zero. The integral value is displayed, and the integrated area is shaded. Note: The shaded area is a drawing. Use ClrDraw (Chapter 8) or any action that invokes Smart Graph to clear the shaded area. Chapter 3: Function Graphing 89
Chapter 4:Parametric GraphingGetting Started: Path of a BallGetting Started is a fast-paced introduction. Read the chapter for details.Graph the parametric equation that describes the path of a ball hit at an initial speed of 30 metersper second, at an initial angle of 25 degrees with the horizontal from ground level. How far doesthe ball travel? When does it hit the ground? How high does it go? Ignore all forces except gravity.For initial velocity v o and angle q, the position of the ball as a function of time has horizontal andvertical components. 1Horizontal: X1(t)=tv 0cos(q) Vertical: Y1(t)=tv 0sin(q)N -- gt2 - 2The vertical and horizontal vectors of the ball's motion also will be graphed.Vertical vector: X2(t)=0 Y2(t)=Y1(t)Horizontal vector: X3(t)=X1(t) Y3(t)=0Gravity constant: g=9.8 m/sec21. Press z. Press † † † ~ Í to select Par mode. Press † † ~ Í to select Simul for simultaneous graphing of all three parametric equations in this example.2. Press } } } ~ Í to go to the Format Graph screen. Press † † † ~ Í to select AxesOff, which turns off the axes. Chapter 4: Parametric Graphing 90
12. Press r to obtain numerical results and answer the questions at the beginning of this section. Tracing begins at Tmin on the first parametric equation (X1T and Y1T). As you press ~ to trace the curve, the cursor follows the path of the ball over time. The values for X (distance), Y (height), and T (time) are displayed at the bottom of the screen.Defining and Displaying Parametric GraphsTI-84 Plus Graphing Mode SimilaritiesThe steps for defining a parametric graph are similar to the steps for defining a function graph.Chapter 4 assumes that you are familiar with Chapter 3: Function Graphing. Chapter 4 detailsaspects of parametric graphing that differ from function graphing.Setting Parametric Graphing ModeTo display the mode screen, press z. To graph parametric equations, you must selectparametric graphing mode before you enter window variables and before you enter thecomponents of parametric equations.Displaying the Parametric Y= EditorAfter selecting parametric graphing mode, press o to display the parametric Y= editor.In this editor, you can display and enter both the X and Y components of up to six equations, X1Tand Y1T through X6T and Y6T. Each is defined in terms of the independent variable T. A commonapplication of parametric graphs is graphing equations over time.Selecting a Graph StyleThe icons to the left of X1T through X6T represent the graph style of each parametric equation. Thedefault in parametric mode is ç (line), which connects plotted points. Line, è (thick), ë (path),ì (animate), and í (dot) styles are available for parametric graphing. Chapter 4: Parametric Graphing 92
Defining and Editing Parametric EquationsTo define or edit a parametric equation, follow the steps in Chapter 3 for defining a function orediting a function. The independent variable in a parametric equation is T. In parametric graphingmode, you can enter the parametric variable T in either of two ways.• Press ".• Press ƒ [T].Two components, X and Y, define a single parametric equation. You must define both of them.Selecting and Deselecting Parametric EquationsThe TI-84 Plus graphs only the selected parametric equations. In the Y= editor, a parametricequation is selected when the = signs of both the X and Y components are highlighted. You mayselect any or all of the equations X1T and Y1T through X6T and Y6T.To change the selection status, move the cursor onto the = sign of either the X or Y componentand press Í. The status of both the X and Y components is changed.Setting Window VariablesTo display the window variable values, press p. These variables define the viewing window.The values below are defaults for parametric graphing in Radian angle mode.Tmin=0 Smallest T value to evaluateTmax=6.2831853... Largest T value to evaluate (2p)Tstep=.1308996... T value increment T window variables.Setting the Graph FormatTo display the current graph format settings, press y .. Chapter 3 describes the formatsettings in detail. The other graphing modes share these format settings; Seq graphing mode hasan additional axes format setting. Chapter 4: Parametric Graphing 93
Displaying a GraphWhen you press s, the TI-84 Plus plots the selected parametric equations. It evaluates the Xand Y components for each value of T (from Tmin to Tmax in intervals of Tstep), and then plotseach point defined by X and Y. The window variables define the viewing window.As the graph is plotted, X, Y, and T are updated.Smart Graph applies to parametric graphs.Window Variables and Y.VARS MenusYou can perform these actions from the home screen or a program.• Access functions by using the name of the X or Y component of the equation as a variable.• Store parametric equations.• Select or deselect parametric equations.• Store values directly to window variables.Exploring Parametric GraphsFree-Moving CursorThe free-moving cursor in parametric graphing works the same as in Func graphing.In RectGC format, moving the cursor updates the values of X and Y; if CoordOn format is selected,X and Y are displayed.In PolarGC format, X, Y, R, and q are updated; if CoordOn format is selected, R and q aredisplayed. Chapter 4: Parametric Graphing 94
TRACETo activate TRACE, press r. When TRACE is active, you can move the trace cursor along thegraph of the equation one Tstep at a time. When you begin a trace, the trace cursor is on the firstselected function at Tmin. If ExprOn is selected, then the function is displayed.In RectGC format, TRACE updates and displays the values of X, Y, and T if CoordOn format is on.In PolarGC format, X, Y, R, q and T are updated; if CoordOn format is selected, R, q, and T aredisplayed. The X and Y (or R and q) values are calculated from T.To move five plotted points at a time on a function, press y | or y ~. If you move the cursorbeyond the top or bottom of the screen, the coordinate values at the bottom of the screen continueto change appropriately.Quick Zoom is available in parametric graphing; panning is not.Moving the Trace Cursor to Any Valid T ValueTo move the trace cursor to any valid T value on the current function, enter the number. When youenter the first digit, a T= prompt and the number you entered are displayed in the bottom-left cornerof the screen. You can enter an expression at the T= prompt. The value must be valid for thecurrent viewing window. When you have completed the entry, press Í to move the cursor.ZOOMZOOM operations in parametric graphing work the same as in Func graphing. Only the X (Xmin,Xmax, and Xscl) and Y (Ymin, Ymax, and Yscl) window variables are affected.The T window variables (Tmin, Tmax, and Tstep) are only affected when you select ZStandard. TheVARS ZOOM secondary menu ZT/Zq items 1:ZTmin, 2:ZTmax, and 3:ZTstep are the zoom memoryvariables for parametric graphing.CALCCALC operations in parametric graphing work the same as in Func graphing. The CALCULATEmenu items available in parametric graphing are 1:value, 2:dy/dx, 3:dy/dt, and 4:dx/dt. Chapter 4: Parametric Graphing 95
Chapter 5:Polar GraphingGetting Started: Polar RoseGetting Started is a fast-paced introduction. Read the chapter for details.The polar equation R=Asin(Bq) graphs a rose. Graph the rose for A=8 and B=2.5, and thenexplore the appearance of the rose for other values of A and B.1. Press z to display the MODE screen. Press † † † ~ ~ Í to select Pol graphing mode. Select the defaults (the options on the left) for the other mode settings.2. Press o to display the polar Y= editor. Press 8 ˜ 2.5 " ¤ Í to define r1.3. Press q 6 to select 6:ZStandard and graph the equation in the standard viewing window. The graph shows only five petals of the rose, and the rose does not appear to be symmetrical. This is because the standard window sets qmax=2p and defines the window, rather than the pixels, as square.4. Press p to display the window variables. Press † 4 y B to increase the value of qmax to 4p.5. Press q 5 to select 5:ZSquare and plot the graph.6. Repeat steps 2 through 5 with new values for the variables A and B in the polar equation r1=Asin(Bq). Observe how the new values affect the graph. Chapter 5: Polar Graphing 96
Defining and Displaying Polar GraphsTI-84 Plus Graphing Mode SimilaritiesThe steps for defining a polar graph are similar to the steps for defining a function graph. Chapter5 assumes that you are familiar with Chapter 3: Function Graphing. Chapter 5 details aspects ofpolar graphing that differ from function graphing.Setting Polar Graphing ModeTo display the mode screen, press z. To graph polar equations, you must select Pol graphingmode before you enter values for the window variables and before you enter polar equations.Displaying the Polar Y= EditorAfter selecting Pol graphing mode, press o to display the polar Y= editor.In this editor, you can enter and display up to six polar equations, r1 through r6. Each is defined interms of the independent variable q.Selecting Graph StylesThe icons to the left of r1 through r6 represent the graph style of each polar equation. The defaultin Pol graphing mode is ç (line), which connects plotted points. Line, è (thick), ë (path), ì (animate),and í (dot) styles are available for polar graphing.Defining and Editing Polar EquationsTo define or edit a polar equation, follow the steps in Chapter 3 for defining a function or editing afunction. The independent variable in a polar equation is q. In Pol graphing mode, you can enterthe polar variable q in either of two ways.• Press ".• Press ƒ [q].Selecting and Deselecting Polar EquationsThe TI-84 Plus graphs only the selected polar equations. In the Y= editor, a polar equation isselected when the = sign is highlighted. You may select any or all of the equations. Chapter 5: Polar Graphing 97
To change the selection status, move the cursor onto the = sign, and then press Í.Setting Window VariablesTo display the window variable values, press p. These variables define the viewing window.The values below are defaults for Pol graphing in Radian angle mode.qmin=0 Smallest q value to evaluateqmax=6.2831853... Largest q value to evaluate (2p)qstep=.1308996... Increment between q values q window variables.Setting the Graph FormatTo display the current graph format settings, press y .. Chapter 3 describes the formatsettings in detail. The other graphing modes share these format settings.Displaying a GraphWhen you press s, the TI-84 Plus plots the selected polar equations. It evaluates R for eachvalue of q (from qmin to qmax in intervals of qstep) and then plots each point. The windowvariables define the viewing window.As the graph is plotted, X, Y, R, and q are updated.Smart Graph applies to polar graphs.Window Variables and Y.VARS MenusYou can perform these actions from the home screen or a program.• Access functions by using the name of the equation as a variable. These function names are available on the YVARS shortcut menu (t a). Chapter 5: Polar Graphing 98
• Store polar equations.• Select or deselect polar equations.• Store values directly to window variables.Exploring Polar GraphsFree-Moving CursorThe free-moving cursor in PolTo activate TRACE, press r. When TRACE is active, you can move the trace cursor along thegraph of the equation one qstep at a time. When you begin a trace, the trace cursor is on the firstselected function at qmin. If ExprOn format is selected, then the equation is displayed.In RectGC format, TRACE updates the values of X, Y, and q; if CoordOn format is selected, X, Y,and q are displayed. In PolarGC format, TRACE updates X, Y, R, and q; if CoordOn format isselected, R and q are displayed.To move five plotted points at a time on a function, press y | or y ~. If you move the tracecursor beyond the top or bottom of the screen, the coordinate values at the bottom of the screencontinue to change appropriately.Quick Zoom is available in Pol graphing mode; panning is not.Moving the Trace Cursor to Any Valid Theta ValueTo move the trace cursor to any valid q value on the current function, enter the number. When youenter the first digit, a q= prompt and the number you entered are displayed in the bottom-left cornerof the screen. You can enter an expression at the q= prompt. The value must be valid for thecurrent viewing window. When you complete the entry, press Í to move the cursor. Chapter 5: Polar Graphing 99
ZOOMZOOM operations in Pol graphing work the same as in Func graphing. Only the X (Xmin, Xmax, andXscl) and Y (Ymin, Ymax, and Yscl) window variables are affected.The q window variables (qmin, qmax, and qstep) are not affected, except when you selectZStandard. The VARS ZOOM secondary menu ZT/Zq items 4:Zqmin, 5:Zqmax, and 6:Zqstep arezoom memory variables for Pol graphing.CALCCALC operations in Pol graphing work the same as in Func graphing. The CALCULATE menuitems available in Pol graphing are 1:value, 2:dy/dx, and 3:dr/dq. Chapter 5: Polar Graphing 100
Chapter 6:Sequence GraphingGetting Started: Forest and TreesNote: Getting Started is a fast-paced introduction. Read the chapter for details.A small forest of 4,000 trees is under a new forestry plan. Each year 20 percent of the trees will beharvested and 1,000 new trees will be planted. Will the forest eventually disappear? Will the forestsize stabilize? If so, in how many years and with how many trees?1. Press z. Press † † † ~ ~ ~ Í to select Seq graphing mode.2. Press y . and select Time axes format and ExprOn format if necessary.3. Press o. If the graph-style icon is not ç (dot), press | |, press Í until ç is displayed, and then press ~ ~.4. Press ~ 3 to select iPart( (integer part) because only whole trees are harvested. After each annual harvest, 80 percent (.80) of the trees remain. Press Ë 8 y [u] £ " ¹ 1 ¤ to define the number of trees after each harvest. Press à 1000 ¤ to define the new trees. Press † 4000 to define the number of trees at the beginning of the program. Note: Be sure to press y [u], not t [U]. [u] is the second function of the ¬ key.5. Press p 0 to set nMin=0. Press † 50 to set nMax=50. nMin and nMax evaluate forest size over 50 years. Set the other window variables. PlotStart=1 Xmin=0 Ymin=0 PlotStep=1 Xmax=50 Ymax=6000 Xscl=10 Yscl=1000 Chapter 6: Sequence Graphing 101
6. Press r. Tracing begins at nMin (the start of the forestry plan). Press ~ to trace the sequence year by year. The sequence is displayed at the top of the screen. The values for n (number of years), X (X=n, because n is plotted on the x-axis), and Y (tree count) are displayed at the bottom. When will the forest stabilize? With how many trees?Defining and Displaying Sequence GraphsTI-84 Plus Graphing Mode SimilaritiesThe steps for defining a sequence graph are similar to the steps for defining a function graph.Chapter 6 assumes that you are familiar with Chapter 3: Function Graphing. Chapter 6 detailsaspects of sequence graphing that differ from function graphing.Setting Sequence Graphing ModeTo display the mode screen, press z. To graph sequence functions, you must select Seqgraphing mode before you enter window variables and before you enter sequence functions.Sequence graphs automatically plot in Simul mode, regardless of the current plotting-order modesetting.TI-84 Plus Sequence Functions u, v, and wThe TI-84 Plus has three sequence functions that you can enter from the keyboard: u, v, and w.They are second functions of the ¬, −, and ® keys. Press y [u] to enter u, for example.You can define sequence functions in terms of:• The independent variable n• The previous term in the sequence function, such as u(nN1)• The term that precedes the previous term in the sequence function, such as u(nN2)• The previous term or the term that precedes the previous term in another sequence function, such as u(nN1) or u(nN2) referenced in the sequence v(n).Note: Statements in this chapter about u(n) are also true for v(n) and w(n); statements about u(nN1)are also true for v(nN1) and w(nN1); statements about u(nN2) are also true for v(nN2) and w(nN2).Displaying the Sequence Y= EditorAfter selecting Seq mode, press o to display the sequence Y= editor. Chapter 6: Sequence Graphing 102
In this editor, you can display and enter sequences for u(n), v(n), and w(n). Also, you can edit thevalue for nMin, which is the sequence window variable that defines the minimum n value toevaluate.The sequence Y= editor displays the nMin value because of its relevance to u(nMin), v(nMin), andw(nMin), which are the initial values for the sequence equations u(n), v(n), and w(n), respectively.nMin in the Y= editor is the same as nMin in the window editor. If you enter a new value for nMin inone editor, the new value for nMin is updated in both editors.Note: Use u(nMin), v(nMin), or w(nMin) only with a recursive sequence, which requires an initialvalue.Selecting Graph StylesThe icons to the left of u(n), v(n), and w(n) represent the graph style of each sequence (Chapter 3).The default in Seq mode is í (dot), which shows discrete values. Dot, ç (line), and è (thick) stylesare available for sequence graphing. Graph styles are ignored in Web format.Selecting and Deselecting Sequence FunctionsThe TI-84 Plus graphs only the selected sequence functions. In the Y= editor, a sequence functionis selected when the = signs of both u(n)= and u(nMin)= are highlighted.To change the selection status of a sequence function, move the cursor onto the = sign of thefunction name, and then press Í. The status is changed for both the sequence function u(n)and its initial value u(nMin).Defining and Editing a Sequence FunctionTo define or edit a sequence function, follow the steps in Chapter 3 for defining a function. Theindependent variable in a sequence is n.In Seq graphing mode, you can enter the sequence variable in either of two ways.• Press ".• Press y N [N].You can enter the function name from the keyboard (y [u], y [v], y [w]).• To enter the function name u, press y [u] (above ¬).• To enter the function name v, press y [v] (above −). Chapter 6: Sequence Graphing 103
• To enter the function name w, press y [w] (above ®).Generally, sequences are either nonrecursive or recursive. Sequences are evaluated only atconsecutive integer values. n is always a series of consecutive integers, starting at zero or anypositive integer.Nonrecursive SequencesIn a nonrecursive sequence, the nth term is a function of the independent variable n. Each term isindependent of all other terms.For example, in the nonrecursive sequence below, you can calculate u(5) directly, without firstcalculating u(1) or any previous term.The sequence equation above returns the sequence 2, 4, 6, 8, 10, … for n = 1, 2, 3, 4, 5, … .Note: You may leave blank the initial value u(nMin) when calculating nonrecursive sequences.Recursive SequencesIn a recursive sequence, the nth term in the sequence is defined in relation to the previous term orthe term that precedes the previous term, represented by u(nN1) and u(nN2). A recursive sequencemay also be defined in relation to n, as in u(n)=u(nN1)+n.For example, in the sequence below you cannot calculate u(5) without first calculating u(1), u(2),u(3), and u(4).Using an initial value u(nMin) = 1, the sequence above returns 1, 2, 4, 8, 16, ... .Note: On the TI-84 Plus, you must type each character of the terms. For example, to enter u(nN1),press y [u] £ " ¹ À ¤.Recursive sequences require an initial value or values, since they reference undefined terms.• If each term in the sequence is defined in relation to the previous term, as in u(nN1), you must specify an initial value for the first term. Chapter 6: Sequence Graphing 104
• If each term in the sequence is defined in relation to the term that precedes the previous term, as in u(nN2), you must specify initial values for the first two terms. Enter the initial values as a list enclosed in brackets ({ }){ } with commas separating the values.The value of the first term is 0 and the value of the second term is 1 for the sequence u(n).Setting Window VariablesTo display the window variables, press p. These variables define the viewing window. Thevalues below are defaults for Seq graphing in both Radian and Degree angle modes.nMin=1 Smallest n value to evaluatenMax=10 Largest n value to evaluatePlotStart=1 First term number to be plottedPlotStep=1 Incremental n value (for graphing onlynMin must be an integer | 0. nMax, PlotStart, and PlotStep must be integers | 1.nMin is the smallest n value to evaluate. nMin also is displayed in the sequence Y= editor. nMax isthe largest n value to evaluate. Sequences are evaluated at u(nMin), u(nMin+1), u(nMin+2), ... ,u(nMax).PlotStart is the first term to be plotted. PlotStart=1 begins plotting on the first term in the sequence.If you want plotting to begin with the fifth term in a sequence, for example, set PlotStart=5. The firstfour terms are evaluated but are not plotted on the graph. Chapter 6: Sequence Graphing 105
PlotStep is the incremental n value for graphing only. PlotStep does not affect sequence evaluation;it only designates which points are plotted on the graph. If you specify PlotStep=2, the sequence isevaluated at each consecutive integer, but it is plotted on the graph only at every other integer.Selecting Axes CombinationsSetting the Graph FormatTo display the current graph format settings, press y .. Chapter 3 describes the formatsettings in detail. The other graphing modes share these format settings. The axes setting on thetop line of the screen is available only in Seq mode.Time Web uv vw uw Type of sequence plot (axes)RectGC Polar GC Rectangular or polar outputCoordOn CoordOff Cursor coordinate display on/offGridOff GridOn Grid display off or onAxesOn AxesOff Axes display on or offLableOff LabelOn Axes label display off or onExprOn ExprOff Expression display on or offSetting Axes FormatFor sequence graphing, you can select from five axes formats. The table below shows the valuesthat are plotted on the x-axis and y-axis for each axes setting. Axes Setting x-axis y-axis Time n u(n), v(n), w(n) Web u(nN1), v(nN1), w(nN1) u(n), v(n), w(n) uv u(n) v(n) vw v(n) w(n) uw u(n) w(n)Displaying a Sequence GraphTo plot the selected sequence functions, press s. As a graph is plotted, the TI-84 Plusupdates X, Y, and n.Smart Graph applies to sequence graphs (Chapter 3). Chapter 6: Sequence Graphing 106
Exploring Sequence GraphsFree-Moving CursorThe free-moving cursor in SeqThe axes format setting affects TRACE.When Time, uv, vw, or uw axes format is selected, TRACE moves the cursor along the sequenceone PlotStep increment at a time. To move five plotted points at once, press y ~ or y |.• When you begin a trace, the trace cursor is on the first selected sequence at the term number specified by PlotStart, even if it is outside the viewing window.• Quick Zoom applies to all directions. To center the viewing window on the current cursor location after you have moved the trace cursor, pressÍÍ. The trace cursor returns to nMin.In Web format, the trail of the cursor helps identify points with attracting and repelling behavior inthe sequence. When you begin a trace, the cursor is on the x-axis at the initial value of the firstselected function.Note: To move the cursor to a specified n during a trace, enter a value for n, and press Í. Forexample, to quickly return the cursor to the beginning of the sequence, paste nMin to the n= promptand press Í.Moving the Trace Cursor to Any Valid n ValueTo move the trace cursor to any valid n value on the current function, enter the number. When youenter the first digit, an n= prompt and the number you entered are displayed in the bottom-leftcorner of the screen. You can enter an expression at the n= prompt. The value must be valid for thecurrent viewing window. When you have completed the entry, press Í to move the cursor.ZOOMZOOM operations in Seq graphing work the same as in Func graphing. Only the X (Xmin, Xmax,and Xscl) and Y (Ymin, Ymax, and Yscl) window variables are affected. Chapter 6: Sequence Graphing 107
PlotStart, PlotStep, nMin, and nMax are only affected when you select ZStandard. The VARS Zoomsecondary menu ZU items 1 through 7 are the ZOOM MEMORY variables for Seq graphing.CALCThe only CALC operation available in Seq graphing is value.• When Time axes format is selected, value displays Y (the u(n) value) for a specified n value.• When Web axes format is selected, value draws the web and displays Y (the u(n) value) for a specified n value.• When uv, vw, or uw axes format is selected, value displays X and Y according to the axes format setting. For example, for uv axes format, X represents u(n) and Y represents v(n).Evaluating u, v, and wTo enter the sequence names u, v, or w, press y [u], y [v], or y [w]. You can evaluate thesenames in any of three ways.• Calculate the nth value in a sequence.• Calculate a list of values in a sequence.• Generate a sequence with u(nstart,nstop[,nstep]). nstep is optional; default is 1.Graphing Web PlotsGraphing a Web PlotTo select Web axes format, press y . ~ Í. A web plot graphs u(n) versus u(nN1),which you can use to study long-term behavior (convergence, divergence, or oscillation) of arecursive sequence. You can see how the sequence may change behavior as its initial valuechanges.Valid Functions for Web PlotsWhen Web axes format is selected, a sequence will not graph properly or will generate an error.• It must be recursive with only one recursion level (u(nN1) but not u(nN2)).• It cannot reference n directly.• It cannot reference any defined sequence except itself. Chapter 6: Sequence Graphing 108
Displaying the Graph ScreenIn Web format, press s to display the graph screen. The TI-84 Plus:• Draws a y=x reference line in AxesOn format.• Plots the selected sequences with u(nN1) as the independent variable.Note: A potential convergence point occurs whenever a sequence intersects the y=x referenceline. However, the sequence may or may not actually converge at that point, depending on thesequence's initial value.Drawing the WebTo activate the trace cursor, press r. The screen displays the sequence and the current n, X,and Y values (X represents u(nN1) and Y represents u(n)). Press ~ repeatedly to draw the webstep by step, starting at nMin. In Web format, the trace cursor follows this course.1. It starts on the x-axis at the initial value u(nMin) (when PlotStart=1).2. It moves vertically (up or down) to the sequence.3. It moves horizontally to the y=x reference line.4. It repeats this vertical and horizontal movement as you continue to press ~.Using Web Plots to Illustrate ConvergenceExample: Convergence1. Press o in Seq mode to display the sequence Y= editor. Make sure the graph style is set to í (dot), and then define nMin, u(n) and u(nMin) as shown below u(n) = -.8u(n-1) + 3.6.2. Press y . Í to set Time axes format.3. Press p and set the variables as shown below. nMin=1 Xmin=0 Ymin=L10 nMax=25 Xmax=25 Ymax=10 PlotStart=1 Xscl=1 Yscl=1 PlotStep=14. Press s to graph the sequence. Chapter 6: Sequence Graphing 109
5. Press y . and select the Web axes setting.6. Press p and change the variables below. Xmin=L10 Xmax=107. Press s to graph the sequence.8. Press r, and then press ~ to draw the web. The displayed cursor coordinates n, X (u(nN1)), and Y (u(n)) change accordingly. When you press ~, a new n value is displayed, and the trace cursor is on the sequence. When you press ~ again, the n value remains the same, and the cursor moves to the y=x reference line. This pattern repeats as you trace the web.Graphing Phase PlotsGraphing with uv, vw, and uwThe phase-plot axes settings uv, vw, and uw show relationships between two sequences. To selecta phase-plot axes setting, press y ., press ~ until the cursor is on uv, vw, or uw, and thenpress Í. Axes Setting x-axis y-axis uv u(n) v(n) vw v(n) w(n) uw u(n) w(n)Example: Predator-Prey ModelUse the predator-prey model to determine the regional populations of a predator and its prey thatwould maintain population equilibrium for the two species.This example uses the model to determine the equilibrium populations of foxes and rabbits, withinitial populations of 200 rabbits (u(nMin)) and 50 foxes (v(nMin)). Chapter 6: Sequence Graphing 110
5. Press r ~ to individually trace the number of rabbits (u(n)) and foxes (v(n)) over time (n). Note: Press a number, and then press Í to jump to a specific n value (month) while in TRACE.6. Press y . ~ ~ Í to select uv axes format.7. Press p and change these variables as shown below. Xmin=84 Ymin=25 Xmax=237 Ymax=75 Xscl=50 Yscl=108. Press r. Trace both the number of rabbits (X) and the number of foxes (Y) through 400 generations. Note: When you press r, the equation for u is displayed in the top-left corner. Press } or † to see the equation for v.Comparing TI-84 Plus and TI-82 Sequence VariablesSequences and Window VariablesRefer to the table if you are familiar with the TI-82. It shows TI-84 Plus sequences and sequencewindow variables, as well as their TI-82 counterparts.TI-84 Plus TI-82In the Y= editor: u(n) Un u(nMin) UnStart (window variable) v(n) Vn v(nMin) VnStart (window variable) w(n) not available w(nMin) not availableIn the window editor: nMin nStart Chapter 6: Sequence Graphing 112
Chapter 7:TablesGetting Started: Roots of a FunctionGetting Started is a fast-paced introduction. Read the chapter for details.Evaluate the function Y = X3 N 2X at each integer between L10 and 10. How many sign changesoccur, and at what X values?1. Press z † † † Í to set Func graphing mode.2. Press o. Press " 3 to select 3. Then press ¹ 2 " to enter the function Y1=X3N2X.3. Press y - to display the TABLE SETUP screen. Press Ì 10 Í to set TblStart=L10. Press 1 Í to set @Tbl=1. Press Í to select Indpnt: Auto (automatically generated independent values). Press † Í to select Depend: Auto (automatically generated dependent values).4. Press y 0 to display the table screen. Note: The message on the entry line, "Press + for @Tbl" is a reminder that you can change @Tbl from this table view. The entry line is cleared when you press any key.5. Press † until you see the sign changes in the value of Y1. How many sign changes occur, and at what X values? In this case, you can also see the roots of the function by finding when Y1=0. You can explore changes in X by pressing à to display the @TTbl prompt, entering a new value, and searching for your answer. Chapter 7: Tables 114
Setting Up the TableTABLE SETUP ScreenTo display the TABLE SETUP screen, press y -.TblStart, @TblTblStart (table start) defines the initial value for the independent variable. TblStart applies onlywhen the independent variable is generated automatically (when Indpnt: Auto is selected).@Tbl (table step) defines the increment for the independent variable.Indpnt: Auto, Indpnt: Ask, Depend: Auto, Depend: AskSelections Table CharacteristicsIndpnt: Auto Values are displayed automatically in both the independent-Depend: Auto variable column and in all dependent-variable columns.Indpnt: Ask The table is empty. When you enter a value for the independentDepend: Auto variable, all corresponding dependent-variable values are calculated and displayed automatically.Indpnt: Auto Values are displayed automatically for the independent variable.Depend: Ask To generate a value for a dependent variable, move the cursor to that cell and press Í.Indpnt: Ask The table is empty; enter values for the independent variable. ToDepend: Ask generate a value for a dependent variable, move the cursor to that cell and press Í.Setting Up the Table from the Home Screen or a ProgramTo store a value to TblStart, @Tbl, or Tbl[nput from the home screen or a program, select thevariable name from the VARS TABLE secondary menu. TblZnput is a list of independent-variablevalues in the current table.When you press y - in the program editor, you can select IndpntAuto, IndpntAsk,DependAuto, and DependAsk. Chapter 7: Tables 115
Defining the Dependent VariablesDefining Dependent Variables from the Y= EditorIn the Y= editor, enter the functions that define the dependent variables. Only functions that areselected in the Y= editor are displayed in the table. The current graphing mode is used. Inparametric mode, you must define both components of each parametric equation (Chapter 4).Editing Dependent Variables from the Table EditorTo edit a selected Y= function from the table editor, follow these steps.1. Press y 0 to display the table, then press ~ or | to move the cursor to a dependent- variable column.2. Press } until the cursor is on the function name at the top of the column. The function is displayed on the bottom line.3. Press Í. The cursor moves to the bottom line. Edit the function.4. Press Í or †. The new values are calculated. The table and the Y= function are updated automatically. Note: You also can use this feature to view the function that defines a dependent variable without having to leave the table. Chapter 7: Tables 116
Displaying the TableThe TableTo display the table, press y 0.Note: The table abbreviates the values, if necessary. Current cellIndependent-variable Dependent-variablevalues in the first values in the secondcolumn and third columns Current cell's full value @Tbl" is on the entry line. ThisNote: When the table first displays, the message "Press + formessage reminds you that you can press à to change @Tbl at any time. When you press any key,the message disappears.Independent and Dependent VariablesThe current graphing mode determines which independent and dependent variables are displayedin the table (Chapter 1). In the table above, for example, the independent variable X and thedependent variables Y1 and Y2 are displayed because Func graphing mode is set. Independent VariableGraphing Mode Dependent VariableFunc (function) X Y1 through Y9, and Y0Par (parametric) T X1T/Y1T through X6T/Y6TPol (polar) q r1 through r6Seq (sequence) n u(n), v(n), and w(n)Clearing the Table from the Home Screen or a ProgramFrom the home screen, select the ClrTable instruction from the CATALOG. To clear the table, pressÍ.From a program, select 9:ClrTable from the PRGM I/O menu or from the CATALOG. The table iscleared upon execution. If IndpntAsk is selected, all independent and dependent variable valueson the table are cleared. If DependAsk is selected, all dependent variable values on the table arecleared. Chapter 7: Tables 117
Scrolling Independent-Variable ValuesIf Indpnt: Auto is selected, you can press } and † in the independent-variable column to displaymore values. As you scroll the column, the corresponding dependent-variable values also aredisplayed. All dependent-variable values may not be displayed if Depend: Ask is selected.Note: You can scroll back from the value entered for TblStart. As you scroll, TblStart is updatedautomatically to the value shown on the top line of the table. In the example above, TblStart=0 and@Tbl=1 generates and displays values of X=0, …, 6; but you can press } to scroll back and displaythe table for X=M1, …, 5.Changing Table Settings from the Table ViewYou can change table settings from the table view by highlighting a value in the table, pressing Ã,and entering a new @ value.1. Press o and then press 1 t ^ 1 2 ~ " to enter the function Y1=1/2x.2. Press y 0.3. Press † † † to move the cursor to highlight 3, and then press Ã.4. Press 1 t ^ 1 2 to change the table settings to view changes in X in increments of 1/2. Chapter 7: Tables 118
5. Press Í.Displaying Other Dependent VariablesIf you have defined more than two dependent variables, the first two selected Y= functions aredisplayed initially. Press ~ or | to display dependent variables defined by other selected Y=functions. The independent variable always remains in the left column, except during a trace withparametric graphing mode and G-T split-screen mode set.Note: To simultaneously display two dependent variables on the table that are not defined asconsecutive Y= functions, go to the Y= editor and deselect the Y= functions between the two youwant to display. For example, to simultaneously display Y4 and Y7 on the table, go to the Y= editorand deselect Y5 and Y6. Chapter 7: Tables 119
6. Press Í. The tangent line is drawn; the X value and the tangent-line equation are displayed on the graph.Consider repeating this activity with the mode set tothe number of decimal places desired. The first screenshows four decimal places. The second screen showsthe decimal setting at Float.Using the DRAW MenuDRAW MenuTo display the DRAW menu, press y <. The TI-84 Plus's interpretation of these instructionsdepends on whether you accessed the menu from the home screen or the program editor ordirectly from a graph.DRAW POINTS STO1: ClrDraw Clears all drawn elements.2: Line( Draws a line segment between 2 points.3: Horizontal Draws a horizontal line.4: Vertical Draws a vertical line.5: Tangent( Draws a line segment tangent to a function.6: DrawF Draws a function.7: Shade( Shades an area between two functions.8: DrawInv Draws the inverse of a function.9: Circle( Draws a circle.0: Text( Draws text on a graph screen.A: Pen Activates the free-form drawing tool.Before Drawing on a GraphThe DRAW instructions draw on top of graphs. Therefore, before you use the DRAW instructions,consider whether you want to perform one or more of the following actions.• Change the mode settings on the mode screen.• Change the format settings on the format screen. You can press y . or use the shortcut on the mode screen to go to the format graph screen. Chapter 8: Draw Instructions 121
• Enter or edit functions in the Y= editor.• Select or deselect functions in the Y= editor.• Change the window variable values.• Turn stat plots on or off.• Clear existing drawings with ClrDraw.Note: If you draw on a graph and then perform any of the actions listed above, the graph isreplotted without the drawings when you display the graph again. Before you clear drawings, youcan store them with StorePic.Drawing on a GraphYou can use any DRAW menu instructions except DrawInv to draw on Func, Par, Pol, and Seqgraphs. DrawInv is valid only in Func graphing. The coordinates for all DRAW instructions are thedisplay's x-coordinate and y-coordinate values.You can use most DRAW menu and DRAW POINTS menu instructions to draw directly on a graph,using the cursor to identify the coordinates. You also can execute these instructions from the homescreen or from within a program. If a graph is not displayed when you select a DRAW menuinstruction, the home screen is displayed.Clearing DrawingsClearing Drawings When a Graph Is DisplayedAll points, lines, and shading drawn on a graph with DRAW instructions are temporary.To clear drawings from the currently displayed graph, select 1:ClrDraw from the DRAW menu. Thecurrent graph is replotted and displayed with no drawn elements.Clearing Drawings from the Home Screen or a ProgramTo clear drawings on a graph from the home screen or a program, begin on a blank line on thehome screen or in the program editor. Select 1:ClrDraw from the DRAW menu. The instruction iscopied to the cursor location. Press Í.When ClrDraw is executed, it clears all drawings from the current graph and displays the messageDone. When you display the graph again, all drawn points, lines, circles, and shaded areas will begone.Note: Before you clear drawings, you can store them with StorePic. Chapter 8: Draw Instructions 122
Drawing Line SegmentsDrawing a Line Segment Directly on a GraphTo draw a line segment when a graph is displayed, follow these steps.1. Select 2:Line( from the DRAW menu.2. Place the cursor on the point where you want the line segment to begin, and then press Í.3. Move the cursor to the point where you want the line segment to end. The line is displayed as you move the cursor. Press Í.To continue drawing line segments, repeat steps 2 and 3. To cancel Line(, press '.Drawing a Line Segment from the Home Screen or a ProgramLine( also draws a line segment between the coordinates (X1,Y1) and (X2,Y2). The values may beentered as expressions.Line(X1,Y1,X2,Y2)To erase a line segment, enter Line(X1,Y1,X2,Y2,0) Chapter 8: Draw Instructions 123
Drawing Horizontal and Vertical LinesDrawing a Line Directly on a GraphTo draw a horizontal or vertical line when a graph is displayed, follow these steps.1. Select 3:Horizontal or 4:Vertical from the DRAW menu. A line is displayed that moves as you move the cursor.2. Place the cursor on the y-coordinate (for horizontal lines) or x-coordinate (for vertical lines) through which you want the drawn line to pass.3. Press Í to draw the line on the graph.To continue drawing lines, repeat steps 2 and 3.To cancel Horizontal or Vertical, press '.Drawing a Line from the Home Screen or a ProgramHorizontal (horizontal line) draws a horizontal line at Y=y. y, which can be an expression but not alist.Horizontal yVertical (vertical line) draws a vertical line at X=x. x, which can be an expression but not a list.Vertical xTo instruct the TI-84 Plus to draw more than one horizontal or vertical line, separate eachinstruction with a colon ( : ).MathPrint™ Classic Chapter 8: Draw Instructions 124
Drawing Tangent LinesDrawing a Tangent Line Directly on a GraphTo draw a tangent line when a graph is displayed, follow these steps.1. Select 5:Tangent( from the DRAW menu.2. Press † and } to move the cursor to the function for which you want to draw the tangent line. The current graph's Y= function is displayed in the top-left corner, if ExprOn is selected.3. Press ~ and | or enter a number to select the point on the function at which you want to draw the tangent line.4. Press Í. In Func mode, the X value at which the tangent line was drawn is displayed on the bottom of the screen, along with the equation of the tangent line. In all other modes, the dy/dx value is displayed.5. Change the fixed decimal setting on the mode screen if you want to see fewer digits displayed for X and the equation for Y.Drawing a Tangent Line from the Home Screen or a ProgramTangent( (tangent line) draws a line tangent to expression in terms of X, such as Y1 or X2, at pointX=value. X can be an expression. expression is interpreted as being in Func mode. Chapter 8: Draw Instructions 125
Tangent(expression,value)Drawing Functions and InversesDrawing a FunctionDrawF (draw function) draws expression as a function in terms of X on the current graph. When youselect 6:DrawF from the DRAW menu, the TI-84 Plus returns to the home screen or the programeditor. DrawF is not interactive.DrawF expressionNote: You cannot use a list in expression to draw a family of curves.Drawing an Inverse of a FunctionDrawInv (draw inverse) draws the inverse of expression by plotting X values on the y-axis and Yvalues on the x-axis. When you select 8:DrawInv from the DRAW menu, the TI-84 Plus returns tothe home screen or the program editor. DrawInv is not interactive. DrawInv works in Func modeonly.DrawInv expressionNote: You cannot use a list of expressions with DrawInv. Chapter 8: Draw Instructions 126
Shading Areas on a GraphShading a GraphTo shade an area on a graph, select 7:Shade( from the DRAW menu. The instruction is pasted tothe home screen or to the program editor.Shade(lowerfunc,upperfunc[,Xleft,Xright,pattern,patres])MathPrint™ ClassicShade( draws lowerfunc and upperfunc in terms of X on the current graph and shades the area that isspecifically above lowerfunc and below upperfunc. Only the areas where lowerfunc < upperfunc areshaded.Xleft and Xright, if included, specify left and right boundaries for the shading. Xleft and Xright must benumbers between Xmin and Xmax, which are the defaults.pattern specifies one of four shading patterns.pattern=1 vertical (default)pattern=2 horizontalpattern=3 negative—slope 45¡pattern=4 positive—slope 45¡patres specifies one of eight shading resolutions.patres=1 shades every pixel (default)patres=2 shades every second pixelpatres=3 shades every third pixelpatres=4 shades every fourth pixelpatres=5 shades every fifth pixelpatres=6 shades every sixth pixelpatres=7 shades every seventh pixelpatres=8 shades every eighth pixelDrawing CirclesDrawing a Circle Directly on a GraphTo draw a circle directly on a displayed graph using the cursor, follow these steps.1. Select 9:Circle( from the DRAW menu. Chapter 8: Draw Instructions 127
2. Place the cursor at the center of the circle you want to draw. Press Í.3. Move the cursor to a point on the circumference. Press Í to draw the circle on the graph.Note: This circle is displayed as circular, regardless of the window variable values, because youdrew it directly on the display. When you use the Circle( instruction from the home screen or aprogram, the current window variables may distort the shape.To continue drawing circles, repeat steps 2 and 3. To cancel Circle(, press '.Drawing a Circle from the Home Screen or a ProgramCircle( draws a circle with center (X,Y) and radius. These values can be expressions.Circle(X,Y,radius)Note: When you use Circle( on the home screen or from a program, the current window valuesmay distort the drawn circle. Use ZSquare (Chapter 3) before drawing the circle to adjust thewindow variables and make the circle circular.Placing Text on a GraphPlacing Text Directly on a GraphTo place text on a graph when the graph is displayed, follow these steps.1. Select 0:Text( from the DRAW menu.2. Place the cursor where you want the text to begin.3. Enter the characters. Press ƒ or y 7 to enter letters and q. You may enter TI-84 Plus functions, variables, and instructions. The font is proportional, so the exact number of characters you can place on the graph varies. As you type, the characters are placed on top of the graph.To cancel Text(, press '. Chapter 8: Draw Instructions 128
Placing Text on a Graph from the Home Screen or a ProgramText( places on the current graph the characters comprising value, which can include TI-84 Plusfunctions and instructions. The top-left corner of the first character is at pixel (row,column), where rowis an integer between 0 and 57 and column is an integer between 0 and 94. Both row and column canbe expressions.Text(row,column,value,value…)value can be text enclosed in quotation marks ( " ), or it can be an expression. The TI-84 Plus willevaluate an expression and display the result with up to 10 characters.ClassicSplit ScreenOn a Horiz split screen, the maximum value for row is 25. On a G-T split screen, the maximumvalue for row is 45, and the maximum value for column is 46.Using Pen to Draw on a GraphUsing Pen to Draw on a GraphPen draws directly on a graph only. You cannot execute Pen from the home screen or a program.You can capture the image you created using TI-Connect™ software and save it to your computerfor homework or teaching material or store it as a picture file on your TI-84 Plus (see Storing GraphPictures below).To draw on a displayed graph, follow these steps.1. Select A:Pen from the DRAW menu.2. Place the cursor on the point where you want to begin drawing. Press Í to turn on the pen.3. Move the cursor. As you move the cursor, you draw on the graph, shading one pixel at a time.4. Press Í to turn off the pen. Chapter 8: Draw Instructions 129
For example, Pen was used to create the arrow pointing to the local minimum of the selectedfunction. Note: To continue drawing on the graph, move the cursor to a new position where you want to begin drawing again, and then repeat steps 2, 3, and 4. To cancel Pen, press '.Drawing Points on a GraphDRAW POINTS MenuTo display the DRAW POINTS menu, press y < ~. The TI-84 Plus's interpretation of theseinstructions depends on whether you accessed this menu from the home screen or the programeditor or directly from a graph.DRAW POINTS STO1: Pt-On( Turns on a point.2: Pt-Off( Turns off a point.3: Pt-Change( Toggles a point on or off.4: Pxl-On( Turns on a pixel.5: Pxl-Off( Turns off a pixel.6: Pxl-Change( Toggles a pixel on or off.7: pxl-Test( Returns 1 if pixel on, 0 if pixel off.Drawing Points Directly on a Graph with Pt-On(To draw a point on a graph, follow these steps.1. Select 1:Pt-On( from the DRAW POINTS menu.2. Move the cursor to the position where you want to draw the point.3. Press Í to draw the point.To continue drawing points, repeat steps 2 and 3. To cancel Pt-On(, press '. Chapter 8: Draw Instructions 130
Erasing Points with Pt-Off(To erase (turn off) a drawn point on a graph, follow these steps.1. Select 2:Pt-Off( (point off) from the DRAW POINTS menu.2. Move the cursor to the point you want to erase.3. Press Í to erase the point.To continue erasing points, repeat steps 2 and 3. To cancel Pt-Off(, press '.Changing Points with Pt-Change(To change (toggle on or off) a point on a graph, follow these steps.1. Select 3:Pt-Change( (point change) from the DRAW POINTS menu.2. Move the cursor to the point you want to change.3. Press Í to change the point's on/off status.To continue changing points, repeat steps 2 and 3. To cancel Pt-Change(, press '.Drawing Points from the Home Screen or a ProgramPt-On( (point on) turns on the point at (X=x,Y=y). Pt-Off( turns the point off. Pt-Change( toggles thepoint on or off. mark is optional; it determines the point's appearance; specify 1, 2, or 3, where: 1 = ¦ (dot; default) 2 = › (box) 3 = + (cross)Pt-On(x,y[,mark])Pt-Off(x,y[,mark])Pt-Change(x,y)Note: If you specified mark to turn on a point with Pt-On(, you must specify mark when you turn offthe point with Pt-Off(. Pt-Change( does not have the mark option.Drawing PixelsTI-84 Plus PixelsA pixel is a square dot on the TI-84 Plus display. The Pxl- (pixel) instructions let you turn on, turnoff, or reverse a pixel (dot) on the graph using the cursor. When you select a pixel instruction from Chapter 8: Draw Instructions 131
the DRAW POINTS menu, the TI-84 Plus returns to the home screen or the program editor. Thepixel instructions are not interactive.Turning On and Off Pixels with Pxl-On( and Pxl-Off(Pxl-On( (pixel on) turns on the pixel at (row,column), where row is an integer between 0 and 62 andcolumn is an integer between 0 and 94.Pxl-Off( turns the pixel off. Pxl-Change( toggles the pixel on and off.Pxl-On(row,column)Pxl-Off(row,column)Pxl-Change(row,column)Using pxl-Test(pxl-Test( (pixel test) returns 1 if the pixel at (row,column) is turned on or 0 if the pixel is turned off onthe current graph. row must be an integer between 0 and 62. column must be an integer between 0and 94.pxl-Test(row,column)Split ScreenOn a Horiz split screen, the maximum value for row is 30 for Pxl-On(, Pxl-Off(, Pxl-Change(, andpxl-Test(.On a G-T split screen, the maximum value for row is 50 and the maximum value for column is 46 forPxl-On(, Pxl-Off(, Pxl-Change(, and pxl-Test(. Chapter 8: Draw Instructions 132
Storing Graph Pictures (Pic)DRAW STO MenuTo display the DRAW STO menu, press y < |. When you select an instruction from theDRAW STO menu, the TI-84 Plus returns to the home screen or the program editor. The pictureand graph database instructions are not interactive.DRAW POINTS STO1: StorePic Stores the current picture.2: RecallPic Recalls a saved picture.3: StoreGDB Stores the current graph database.4: RecallGDB Recalls a saved graph database.Storing a Graph PictureYou can store up to 10 graph pictures, each of which is an image of the current graph display, inpicture variables Pic1 through Pic9, or Pic0. Later, you can superimpose the stored picture onto adisplayed graph from the home screen or a program.A picture includes drawn elements, plotted functions, axes, and tick marks. The picture does notinclude axes labels, lower and upper bound indicators, prompts, or cursor coordinates. Any partsof the display hidden by these items are stored with the picture.To store a graph picture, follow these steps.1. Select 1:StorePic from the DRAW STO menu. StorePic is pasted to the current cursor location.2. Enter the number (from 1 to 9, or 0) of the picture variable to which you want to store the picture. For example, if you enter 3, the TI-84 Plus will store the picture to Pic3. Note: You also can select a variable from the PICTURE secondary menu ( 4). The variable is pasted next to StorePic.3. Press Í to display the current graph and store the picture. Chapter 8: Draw Instructions 133
Recalling Graph Pictures (Pic)Recalling a Graph PictureTo recall a graph picture, follow these steps.1. Select 2:RecallPic from the DRAW STO menu. RecallPic is pasted to the current cursor location.2. Enter the number (from 1 to 9, or 0) of the picture variable from which you want to recall a picture. For example, if you enter 3, the TI-84 Plus will recall the picture stored to Pic3. Note: You also can select a variable from the PICTURE secondary menu ( 4). The variable is pasted next to RecallPic.3. Press Í to display the current graph with the picture superimposed on it. Note: Pictures are drawings. You cannot trace a curve that is part of a picture.Deleting a Graph PictureTo delete graph pictures from memory, use the MEMORY MANAGEMENT/DELETE secondary menu(Chapter 18).Storing Graph Databases (GDB)What Is a Graph Database?A graph database (GDB) contains the set of elements that defines a particular graph. You canrecreate the graph from these elements. You can store up to 10 GDBs in variables GDB1 throughGDB9, or GDB0 and recall them to recreate graphs.A GDB stores five elements of a graph.• Graphing mode• Window variables• Format settings• All functions in the Y= editor and the selection status of each• Graph style for each Y= functionGDBs do not contain drawn items or stat plot definitions.Storing a Graph DatabaseTo store a graph database, follow these steps. Chapter 8: Draw Instructions 134
1. Select 3:StoreGDB from the DRAW STO menu. StoreGDB is pasted to the current cursor location.2. Enter the number (from 1 to 9, or 0) of the GDB variable to which you want to store the graph database. For example, if you enter 7, the TI-84 Plus will store the GDB to GDB7. Note: You also can select a variable from the GDB secondary menu ( 3). The variable is pasted next to StoreGDB.3. Press Í to store the current database to the specified GDB variable.Recalling Graph Databases (GDB)Recalling a Graph DatabaseCAUTION: When you recall a GDB, it replaces all existing Y= functions. Consider storing thecurrent Y= functions to another database before recalling a stored GDB.To recall a graph database, follow these steps.1. Select 4:RecallGDB from the DRAW STO menu. RecallGDB is pasted to the current cursor location.2. Enter the number (from 1 to 9, or 0) of the GDB variable from which you want to recall a GDB. For example, if you enter 7, the TI-84 Plus will recall the GDB stored to GDB7. Note: You also can select a variable from the GDB secondary menu ( 3). The variable is pasted next to RecallGDB.3. Press Í to replace the current GDB with the recalled GDB. The new graph is not plotted. The TI-84 Plus changes the graphing mode automatically, if necessary.Deleting a Graph DatabaseTo delete a GDB from memory, use the MEMORY MANAGEMENT/DELETE secondary menu(Chapter 18). Chapter 8: Draw Instructions 135
Chapter 9:Split ScreenGetting Started: Exploring the Unit CircleGetting Started is a fast-paced introduction. Read the chapter for details.Use G-T (graph-table) split-screen mode to explore the unit circle and its relationship to thenumeric values for the commonly used trigonometric angles of 0¡ 30¡, 45¡, 60¡, 90¡, and so on.1. Press z to display the mode screen. Press † † ~ Í to select Degree mode. Press † ~ Í to select Par (parametric) graphing mode. Press † † † † ~ ~ Í to select G-T (graph- table) split-screen mode.2. Press † † † † ~ Í to display the format screen. Press † † † † † ~ Í to select ExprOff.3. Press o to display the Y= editor for Par graphing mode. Press ™ " ¤ Í to store cos(T) to X1T. Press ÷ ˜ " ¤ Í to store sin(T) to Y1T.4. Press p to display the window editor. Enter these values for the window variables. Tmin=0 Xmin=L2.3 Ymin=L2.5 Tmax=360 Xmax=2.3 Ymax=2.5 Tstep=15 Xscl=1 Yscl=15. Press r. On the left, the unit circle is graphed parametrically in Degree mode and the trace cursor is activated. When T=0 (from the graph trace coordinates), you can see from the table on the right that the value of X1T (cos(T)) is 1 and Y1T (sin(T)) is 0. Press ~ to move the cursor to the next 15¡ angle increment. As you trace around the circle in steps of 15¡, an approximation of the standard value for each angle is highlighted in the table.6. Press y - and change Indpnt to Ask. Chapter 9: Split Screen 136
7. Press y 0 to make the table portion of the split screen active.Using Split ScreenSetting a Split-Screen ModeTo set a split-screen mode, press z, and then move the cursor to Horiz or G-T and press Í.• Select Horiz (horizontal) to display the graph screen and another screen split horizontally.• Select G-T (graph-table) to display the graph screen and table screen split vertically. $ $The split screen is activated when you press any key that applies to either half of the split screen.If stat plots are turned on, the plots are shown along with the x-y plots in graphs. Press y 0to make the table portion of the split screen active and to display the list data. Press † or } tohighlight a value you want to edit, and then enter a new value directly in the table to overwrite theprevious value. Press ~ repeatedly to display each column of data (both table and list data). Chapter 9: Split Screen 137
Split-screen display with both x-y plots and stat plotsSome screens are never displayed as split screens. For example, if you press z in Horiz or G-Tmode, the mode screen is displayed as a full screen. If you then press a key that displays eitherhalf of a split screen, such as r, the split screen returns.When you press a key or key combination in either Horiz or G-T mode, the cursor is placed in thehalf of the display to which that key applies. For example, if you press r, the cursor is placedin the half where the graph is displayed. If you press y 0, the cursor is placed in the halfwhere the table is displayed.The TI-84 Plus will remain in split-screen mode until you change back to Full screen mode.Horiz (Horizontal) Split ScreenHoriz ModeIn Horiz (horizontal) split-screen mode, a horizontal line splits the screen into top and bottomhalves.The top half displays the graph.The bottom half displays any of these screens.• Home screen (four lines)• Y= editor (four lines)• Stat list editor (two rows)• Window editor (three settings)• Table editor (two rows)Moving from Half to Half in Horiz ModeTo use the top half of the split screen: Chapter 9: Split Screen 138
• Press s or r.• Select a ZOOM or CALC operation.To use the bottom half of the split screen:• Press any key or key combination that displays the home screen.• Press o (Y= editor).• Press … Í (stat list editor).• Press p (window editor).• Press y 0 (table editor).Full Screens in Horiz ModeAll other screens are displayed as full screens in Horiz split-screen mode.To return to the Horiz split screen from a full screen when in Horiz mode, press any key or keycombination that displays the graph, home screen, Y= editor, stat list editor, window editor, or tableeditor.G-T (Graph-Table) Split ScreenG-T ModeIn G-T (graph-table) split-screen mode, a vertical line splits the screen into left and right halves.The left half displays all active graphs and plots.The right half displays either table data corresponding to the graph at the left or list datacorresponding to the plot at the left.Moving from Half to Half in G-T ModeTo use the left half of the split screen:• Press s or r.• Select a ZOOM or CALC operation.To use the right half of the split screen, press y 0. If the values on the right are list data,these values can be edited similarly to using the Stat List Editor. Chapter 9: Split Screen 139
Using TRACE in G-T ModeAs you press | or ~ to move the trace cursor along a graph in the split screen's left half in G-Tmode, the table on the right half automatically scrolls to match the current cursor values. If morethan one graph or plot is active, you can press } or † to select a different graph or plot.Note: When you trace in Par graphing mode, both components of an equation (XnT and YnT) aredisplayed in the two columns of the table. As you trace, the current value of the independentvariable T is displayed on the graph.Full Screens in G-T ModeAll screens other than the graph and the table are displayed as full screens in G-T split-screenmode.To return to the G-T split screen from a full screen when in G-T mode, press any key or keycombination that displays the graph or the table.TI-84 Plus Pixels in Horiz and G-T ModesTI-84 Plus Pixels in Horiz and G-T ModesNote: Each set of numbers in parentheses above represents the row and column of a corner pixel,which is turned on.DRAW POINTS Menu Pixel InstructionsFor Pxl-On(, Pxl-Off(, Pxl-Change(, and pxl-Test(:• In Horiz mode, row must be {30; column must be {94.• In G-T mode, row must be {50; column must be {46.Pxl-On(row,column) Chapter 9: Split Screen 140
DRAW Menu Text( InstructionFor the Text( instruction:• In Horiz mode, row must be {25; column must be {94.• In G-T mode, row must be {45; column must be {46.Text(row,column,"text")PRGM I/O Menu Output( InstructionFor the Output( instruction:• In Horiz mode, row must be {4; column must be {16.• In G-T mode, row must be {8; column must be {16.Output(row,column,"text")Note: The Output( instruction can only be used within a program.Setting a Split-Screen Mode from the Home Screen or a ProgramTo set Horiz or G-T from a program, follow these steps.1. Press z while the cursor is on a blank line in the program editor.2. Select Horiz or G-T.The instruction is pasted to the cursor location. The mode is set when the instruction isencountered during program execution. It remains in effect after execution.Note: You also can paste Horiz or G-T to the home screen or program editor from the CATALOG(Chapter 15). Chapter 9: Split Screen 141
Chapter 10:MatricesGetting Started: Using the MTRX Shortcut MenuGetting Started is a fast-paced introduction. Read the chapter for details.You can use the MTRX shortcut menu (t `) to enter a quick matrix calculation on the homescreen or in the Y= editor.Note: To input a fraction in a matrix, delete the pre-populated zero first.Example: Add the following matrices: and store the result to matrix C.1. Press t ` to display the quick matrix editor. The default size of the matrix is two rows by two columns.2. Press † † to highlight OK and then press Í.3. Press 2 ~ k 3 ~ 5 ~ 8 ~ to create the first matrix.4. Press à t ` † † Í 4 ~ 3 ~ 2 ~ 1 ~ Í to create the second matrix and perform the calculation.5. Press v y Q and select 3:[C]. Chapter 10: Matrices 142
6. Press Í to store the matrix to [C].In the matrix editor (y Q), you can see thatmatrix [C] has dimension 2x2.You can press ~ ~ to display the EDIT screen andthen select [C] to edit it.Getting Started: Systems of Linear EquationsGetting Started is a fast-paced introduction. Read the chapter for details.Find the solution of X + 2Y + 3Z = 3 and 2X + 3Y + 4Z = 3. On the TI-84 Plus, you can solve asystem of linear equations by entering the coefficients as elements in a matrix, and then using rref(to obtain the reduced row-echelon form.1. Press y . Press ~ ~ to display the MATRX EDIT menu. Press 1 to select 1: [A].2. Press 2 Í 4 Í to define a 2×4 matrix. The rectangular cursor indicates the current element. Ellipses (...) indicate additional columns beyond the screen.3. Press 1 Í to enter the first element. The rectangular cursor moves to the second column of the first row. Chapter 10: Matrices 143
4. Press 2 Í 3 Í 3 Í to complete the first row for X + 2Y + 3Z = 3.5. Press 2 Í 3 Í 4 Í 3 Í to enter the second row for 2X + 3Y + 4Z = 3.6. Press y 5 to return to the home screen. If necessary, press ' to clear the home screen. Press y ~ to display the MATRX MATH menu. Press } to wrap to the end of the menu. Select B:rref( to copy rref( to the home screen.7. Press y 1 to select 1: [A] from the MATRX NAMES menu. Press ¤ Í. The reduced row-echelon form of the matrix is displayed and stored in Ans. 1X N 1Z = L3 therefore X = L3 + Z 1Y + 2Z = 3 therefore Y = 3 N 2ZDefining a MatrixWhat Is a Matrix?A matrix is a two-dimensional array. You can display, define, or edit a matrix in the matrix editor.You can also define a matrix using the MTRX shortcut menu (t `).The TI-84 Plus has 10matrix variables, [A] through [J]. You can define a matrix directly in an expression. A matrix,depending on available memory, may have up to 99 rows or columns. You can store only realnumbers in TI-84 Plus matrices. Fractions are stored as real numbers and can be used inmatrices.Selecting a MatrixBefore you can define or display a matrix in the editor, you first must select the matrix name. To doso, follow these steps.1. Press y | to display the MATRX EDIT menu. The dimensions of any previously defined matrices are displayed.2. Select the matrix you want to define. The MATRX EDIT screen is displayed. Chapter 10: Matrices 144
Accepting or Changing Matrix DimensionsThe dimensions of the matrix (row × column) are displayed on the top line. The dimensions of a newmatrix are 1 × 1. You must accept or change the dimensions each time you edit a matrix. When youselect a matrix to define, the cursor highlights the row dimension.• To accept the row dimension, press Í.• To change the row dimension, enter the number of rows (up to 99), and then press Í.The cursor moves to the column dimension, which you must accept or change the same way youaccepted or changed the row dimension. When you press Í, the rectangular cursor moves tothe first matrix element.Viewing and Editing Matrix ElementsDisplaying Matrix ElementsAfter you have set the dimensions of the matrix, you can view the matrix and enter values for thematrix elements. In a new matrix, all values are zero.Select the matrix from the MATRX EDIT menu and enter or accept the dimensions. The centerportion of the matrix editor displays up to seven rows and three columns of a matrix, showing thevalues of the elements in abbreviated form if necessary. The full value of the current element,which is indicated by the rectangular cursor, is displayed on the bottom line.This is an 8 × 4 matrix. Ellipses in the left or right column indicate additional columns. # or $ in theright column indicate additional rows.Deleting a MatrixTo delete matrices from memory, use the MEMORY MANAGEMENT/DELETE secondary menu(Chapter 18).Viewing a MatrixThe matrix editor has two contexts, viewing and editing. In viewing context, you can use the cursorkeys to move quickly from one matrix element to the next. The full value of the highlighted elementis displayed on the edit line. Chapter 10: Matrices 145
Select the matrix from the MATRX EDIT menu, and then enter or accept the dimensions.Using Viewing-Context KeysKey Function| or ~ Moves the cursor within the current row† or } Moves the cursor within the current column; on the top row, } moves the cursor to the column dimension; on the column dimension, } moves the cursor to the row dimensionÍ Switches to editing context; activates the edit cursor on the bottom line' Switches to editing context; clears the value on the bottom lineAny entry character Switches to editing context; clears the value on the bottom line; copies the character to the bottom liney6 Nothing{ NothingEditing a Matrix ElementIn editing context, an edit cursor is active on the bottom line. To edit a matrix element value, followthese steps.1. Select the matrix from the MATRX EDIT menu, and then enter or accept the dimensions.2. Press |, }, ~, and † to move the cursor to the matrix element you want to change.3. Switch to editing context by pressing Í, ', or an entry key.4. Change the value of the matrix element using the editing-context keys described below. You may enter an expression, which is evaluated when you leave editing context. Note: You can press ' Í to restore the value at the cursor if you make a mistake.5. Press Í, }, or † to move to another element. Chapter 10: Matrices 146
Using Editing-Context KeysKey Function| or ~ Moves the edit cursor within the value† or } Stores the value displayed on the edit line to the matrix element; switches to viewing context and moves the cursor within the columnÍ Stores the value displayed on the edit line to the matrix element; switches to viewing context and moves the cursor to the next row element' Clears the value on the bottom lineAny entry character Copies the character to the location of the edit cursor on the bottom liney6 Activates the insert cursor{ Deletes the character under the edit cursor on the bottom lineUsing Matrices with ExpressionsTo use a matrix in an expression, you can do any of the following.• Copy the name from the MATRX NAMES menu.• Recall the contents of the matrix into the expression with y K (Chapter 1).• Enter the matrix directly (see below).Entering a Matrix in an ExpressionYou can enter, edit, and store a matrix in the matrix editor. You also can enter a matrix directly inan expression.To enter a matrix in an expression, follow these steps.1. Press y [ [ ] to indicate the beginning of the matrix.2. Press y [ [ ] to indicate the beginning of a row.3. Enter a value, which can be an expression, for each element in the row. Separate the values with commas.4. Press y [ ] ] to indicate the end of a row.5. Repeat steps 2 through 4 to enter all of the rows.6. Press y [ ] ] to indicate the end of the matrix. The resulting matrix is displayed in the form: [[element1,1,...,element1,n],...,[elementm,1,...,elementm,n]] Any expressions are evaluated when the entry is executed. Chapter 10: Matrices 147
Note: • The commas that you must enter to separate elements are not displayed on output. • Closing brackets are required when you enter a matrix directly on the home screen or in an expression. • When you define a matrix using the matrix editor, it is automatically stored. However, when you enter a matrix directly on the home screen or in an expression, it is not automatically stored, but you can store it.In MathPrint™ mode, you could also use the MTRX shortcut menu to enter this kind of matrix:1. Press t ` † ~ ~ Í † Í to define the matrix dimension.2. Press 1 ~ 2 ~ 2 ~ 4 ~ 5 ~ 6 ~ to define the matrix.3. Press Í to perform the calculation.Displaying and Copying MatricesDisplaying a MatrixTo display the contents of a matrix on the home screen, select the matrix from the MATRX NAMESmenu, and then press Í.In MathPrint™ mode:• An arrow at the left or right indicates additional columns.• An arrow at the top or bottom indicates additional rows.In Classic mode:• Ellipses in the left or right column indicate additional columns. Chapter 10: Matrices 148
• # or $ in the right column indicate additional rows.In either mode, press ~, |, †, and } to scroll the matrix. You can scroll the matrix after youpress Í to calculate the matrix. If you cannot scroll the matrix, press } Í Í to repeatthe calculation.MathPrint™ ClassicNote:• You cannot copy a matrix output from the history.• Matrix calculations are not saved when you change from MathPrint™ mode to Classic mode or vice-versa.Copying One Matrix to AnotherTo copy a matrix, follow these steps.1. Press y > to display the MATRX NAMES menu.2. Select the name of the matrix you want to copy.3. Press ¿.4. Press y > again and select the name of the new matrix to which you want to copy the existing matrix.5. Press Í to copy the matrix to the new matrix name.Accessing a Matrix ElementOn the home screen or from within a program, you can store a value to, or recall a value from, amatrix element. The element must be within the currently defined matrix dimensions. Select matrixfrom the MATRX NAMES menu.[matrix](row,column) Chapter 10: Matrices 149
Using Math Functions with MatricesUsing Math Functions with MatricesYou can use many of the math functions on the TI-84 Plus keypad, the MATH menu, the MATH NUMmenu, and the MATH TEST menu with matrices. However, the dimensions must be appropriate.Each of the functions below creates a new matrix; the original matrix remains the same.Addition, Subtraction, MultiplicationTo add or subtract matrices, the dimensions must be the same. The answer is a matrix in whichthe elements are the sum or difference of the individual corresponding elements.matrixA+matrixBmatrixANmatrixBTo multiply two matrices together, the column dimension of matrixA must match the row dimensionof matrixB.matrixA…matrixBMultiplying a matrix by a value or a value by a matrix returns a matrix in which each element of matrixis multiplied by value.matrix…valuevalue…matrix Chapter 10: Matrices 150
NegationNegating a matrix returns a matrix in which the sign of every element is changed.Lmatrixabs(abs( (absolute value, MATH NUM menu) returns a matrix containing the absolute value of eachelement of matrix.abs(matrix)round(round( (MATH NUM menu) returns a matrix. It rounds every element in matrix to #decimals ( 9). If#decimals is omitted, the elements are rounded to 10 digits.round(matrix[,#decimals])InverseUse the L1 function (œ) or › L1 to invert a matrix. matrix must be square. The determinant cannotequal zero. Chapter 10: Matrices 151
1matrixLPowersTo raise a matrix to a power, matrix must be square. You can use 2 (¡), 3 (MATH menu), or ^power(›) for integer power between 0 and 255.matrix2matrix3matrix^power MathPrint™ ClassicRelational OperationsTo compare two matrices using the relational operations = and ƒ (TEST menu), they must have thesame dimensions. = and ƒ compare matrixA and matrixB on an element-by-element basis. The otherrelational operations are not valid with matrices.matrixA=matrixB returns 1 if every comparison is true; it returns 0 if any comparison is false.matrixAƒmatrixB returns 1 if at least one comparison is false; it returns 0 if no comparison is false. Chapter 10: Matrices 152
iPart(, fPart(, int(iPart( (integer part), fPart( (fractional part), and int( (greatest integer) are on the MATH NUM menu.iPart( returns a matrix containing the integer part of each element of matrix.fPart( returns a matrix containing the fractional part of each element of matrix.int( returns a matrix containing the greatest integer of each element of matrix.iPart(matrix)fPart(matrix)int(matrix)Using the MATRX MATH OperationsMATRX MATH MenuTo display the MATRX MATH menu, press y ~.NAMES MATH EDIT1: det( Calculates the determinant.2: T Transposes the matrix.3: dim( Returns the matrix dimensions.4: Fill( Fills all elements with a constant.5: identity( Returns the identity matrix.6: randM( Returns a random matrix.7: augment( Appends two matrices.8: Matr4list( Stores a matrix to a list. Chapter 10: Matrices 153
NAMES MATH EDIT9: List4matr( Stores a list to a matrix.0: cumSum( Returns the cumulative sums of a matrix.A: ref( Returns the row-echelon form of a matrix.B: rref( Returns the reduced row-echelon form.C: rowSwap( Swaps two rows of a matrix.D: row+( Adds two rows; stores in the second row.E: …row( Multiplies the row by a number.F: …row+( Multiplies the row, adds to the second row.det(det( (determinant) returns the determinant (a real number) of a square matrix.det(matrix)TransposeT (transpose) returns a matrix in which each element (row, column) is swapped with thecorresponding element (column, row) of matrix.matrixTAccessing Matrix Dimensions with dim(dim( (dimension) returns a list containing the dimensions ({rows columns}) of matrix.dim(matrix) Chapter 10: Matrices 154
Note: dim(matrix)"Ln:Ln(1) returns the number of rows. dim(matrix)"Ln:Ln(2) returns the number ofcolumns.Creating a Matrix with dim(Use dim( with ¿ to create a new matrixname of dimensions rows × columns with 0 as each element.{rows,columns}"dim(matrixname)Redimensioning a Matrix with dim(Use dim( with ¿ to redimension an existing matrixname to dimensions rows × columns. Theelements in the old matrixname that are within the new dimensions are not changed. Additionalcreated elements are zeros. Matrix elements that are outside the new dimensions are deleted.{rows,columns}"dim(matrixname)Fill(Fill( stores value to every element in matrixname.Fill(value,matrixname)identity(identity( returns the identity matrix of dimension rows × dimension columns.identity(dimension) Chapter 10: Matrices 155
Matr4list( also fills a listname with elements from a specified column# in matrix. To fill a list with a specificcolumn from matrix, you must enter column# after matrix.Matr4list(matrix,column#,listname)List4matr(List4matr( (lists stored to matrix) fills matrixname column by column with the elements from each list. Ifdimensions of all lists are not equal, List4matr( fills each extra matrixname row with 0. Complex lists arenot valid.List4matr(listA,...,list n,matrixname)cumSum(cumSum( returns cumulative sums of the elements in matrix, starting with the first element. Eachelement is the cumulative sum of the column from top to bottom.cumSum(matrix)Row OperationsMATRX MATH menu items A through F are row operations. You can use a row operation in anexpression. Row operations do not change matrix in memory. You can enter all row numbers andvalues as expressions. You can select the matrix from the MATRX NAMES menu. Chapter 10: Matrices 157
ref(, rref(ref( (row-echelon form) returns the row-echelon form of a real matrix. The number of columns mustbe greater than or equal to the number of rows.ref(matrix)rref( (reduced row-echelon form) returns the reduced row-echelon form of a real matrix. The numberof columns must be greater than or equal to the number of rows.rref(matrix)rowSwap(rowSwap( returns a matrix. It swaps rowA and rowB of matrix.rowSwap(matrix,rowA,rowB)row+(row+( (row addition) returns a matrix. It adds rowA and rowB of matrix and stores the results in rowB.row+(matrix,rowA,rowB) Chapter 10: Matrices 158
…row(…row( (row multiplication) returns a matrix. It multiplies row of matrix by value and stores the results inrow.…row(value,matrix,row)…row+(…row+( (row multiplication and addition) returns a matrix. It multiplies rowA of matrix by value, adds itto rowB, and stores the results in rowB.…row+(value,matrix,rowA,rowB) Chapter 10: Matrices 159
Chapter 11:ListsGetting Started: Generating a SequenceGetting Started is a fast-paced introduction. Read the chapter for details.Calculate the first eight terms of the sequence 1/A2. Store the results to a user-created list. Thendisplay the results in fraction form. Begin this example on a blank line on the home screen.1. Press y 9 ~ to display the LIST OPS menu.2. Press 5 to select 5:seq(, which pastes seq( to the current cursor location.3. Press t ^ Í 1 ~ ƒ [A] ¡ ~ ¢ ƒ [A] ¢ 1 ¢ 8 ¢ 1 ¤ to enter the sequence.4. Press ¿, and then press y 7 to turn on alpha-lock. Press [S] [E] [Q], and then press ƒ to turn off alpha-lock. Press 1 to complete the list name. Note: Since the seq( command creates a list, you can name give the list a name up to five characters long.5. Press Í to generate the list and store it in SEQ1. The list is displayed on the home screen. An ellipsis (...) indicates that the list continues beyond the viewing window. Press ~ repeatedly (or press and hold ~) to scroll the list and view all the list elements.6. Press y 9 to display the LIST NAMES menu. Press 7 to select 7:SEQ1 to paste ÙSEQ1 to the current cursor location. (If SEQ1 is not item 7 on your LIST NAMES menu, move the cursor to SEQ1 before you press Í.) Chapter 11: Lists 160
7. Press to display the MATH menu. Press 2 to select 2:4Dec, which pastes 4Dec to the current cursor location.8. Press Í to show the sequence in decimal form. Press ~ repeatedly (or press and hold ~) to scroll the list and view all the list elements.Naming ListsUsing TI-84 Plus List Names L1 through L6The TI-84 Plus has six list names in memory: L1, L2, L3, L4, L5, and L6. The list names L1 throughL6 are the second functions of À through ¸. To paste one of these names to a valid screen, pressy, and then press the appropriate key. L1 through L6 are stored in stat list editor columns 1through 6 when you reset memory.Creating a List Name on the Home ScreenTo create a list name on the home screen, follow these steps.1. Press y E, enter one or more list elements, and then press y F. Separate list elements with commas. List elements can be real numbers, complex numbers, or expressions.2. Press ¿.3. Press ƒ [letter from A to Z or q] to enter the first letter of the name.4. Enter zero to four letters, q, or numbers to complete the name.5. Press Í. The list is displayed on the next line. The list name and its elements are stored in memory. The list name becomes an item on the LIST NAMES menu. Note: If you want to view a user-created list in the stat list editor, you must retrieve the list in the stat list editor (Chapter 12).You also can create a list name in these four places.• At the Name= prompt in the stat list editor• At an Xlist:, Ylist:, or Data List: prompt in the stat plot editor Chapter 11: Lists 161
• At a List:, List1:, List2:, Freq:, Freq1:, Freq2:, XList:, or YList: prompt in the inferential stat editors• On the home screen using SetUpEditorYou can create as many list names as your TI-84 Plus memory has space to store.Storing and Displaying ListsStoring Elements to a ListYou can store list elements in either of two ways.• Use brackets and ¿ on the home screen.• Use the stat list editor (Chapter 12).The maximum dimension of a list is 999 elements.Note: When you store a complex number to a list, the entire list is converted to a list of complexnumbers. To convert the list to a list of real numbers, display the home screen, and then enterreal(listname)!listname.Displaying a List on the Home ScreenTo display the elements of a list on the home screen, enter the name of the list (preceded by Ù, ifnecessary), and then press Í. An ellipsis indicates that the list continues beyond the viewingwindow. Press ~ repeatedly (or press and hold ~) to scroll the list and view all the list elements.Copying One List to AnotherTo copy a list, store it to another list.Accessing a List ElementYou can store a value to or recall a value from a specific list element. You can store to any elementwithin the current list dimension or one element beyond. Chapter 11: Lists 162
listname(element)Deleting a List from MemoryTo delete lists from memory, including L1 through L6, use the MEMORY MANAGEMENT/DELETEsecondary menu (Chapter 18). Resetting memory restores L1 through L6. Removing a list from thestat list editor does not delete it from memory.Using Lists in GraphingTo graph a family of curves, you can use lists (Chapter 3) or the Transformation Graphing App.Entering List NamesUsing the LIST NAMES MenuTo display the LIST NAMES menu, press y 9. Each item is a user-created list name except forL1 through L6. LIST NAMES menu items are sorted automatically in alphanumerical order. Only thefirst 10 items are labeled, using 1 through 9, then 0. To jump to the first list name that begins with aparticular alpha character or q, press ƒ [letter from A to Z or q].Note: From the top of a menu, press } to move to the bottom. From the bottom, press † to moveto the top.When you select a list name from the LIST NAMES menu, the list name is pasted to the currentcursor location.• The list name symbol Ù precedes a list name when the name is pasted where non-list name data also is valid, such as the home screen.• The Ù symbol does not precede a list name when the name is pasted where a list name is the only valid input, such as the stat list editor's Name= prompt or the stat plot editor's XList: and YList: prompts. Chapter 11: Lists 163
Entering a User-Created List Name DirectlyTo enter an existing list name directly, follow these steps.1. Press y 9 ~ to display the LIST OPS menu.2. Select B:Ù, which pastes Ù to the current cursor location. Ù is not always necessary. Note: You also can paste Ù to the current cursor location from the CATALOG.3. Enter the characters that comprise the list name.Attaching Formulas to List NamesAttaching a Formula to a List NameYou can attach a formula to a list name so that each list element is a result of the formula. Whenexecuted, the attached formula must resolve to a list.When anything in the attached formula changes, the list to which the formula is attached isupdated automatically.• When you edit an element of a list that is referenced in the formula, the corresponding element in the list to which the formula is attached is updated.• When you edit the formula itself, all elements in the list to which the formula is attached are updated.For example, the first screen below shows that elements are stored to L3, and the formula L3+10 isattached to the list name ÙADD10. The quotation marks designate the formula to be attached toÙADD10. Each element of ÙADD10 is the sum of an element in L3 and 10.The next screen shows another list, L4. The elements of L4 are the sum of the same formula that isattached to L3. However, quotation marks are not entered, so the formula is not attached to L4.On the next line, L6!L3(1):L3 changes the first element in L3 to L6, and then redisplays L3. Chapter 11: Lists 164
The last screen shows that editing L3 updated ÙADD10, but did not change L4. This is because theformula L3+10 is attached to ÙADD10, but it is not attached to L4.Note: To view a formula that is attached to a list name, use the stat list editor (Chapter 12).Attaching a Formula to a List on the Home Screen or in a ProgramTo attach a formula to a list name from a blank line on the home screen or from a program, followthese steps.1. Press ƒ [ã], enter the formula (which must resolve to a list), and press ƒ [ã] again. Note: When you include more than one list name in a formula, each list must have the same dimension.2. Press ¿.3. Enter the name of the list to which you want to attach the formula. • Press y, and then enter a TI-84 Plus list name L1 through L6. • Press y 9 and select a user.created list name from the LIST NAMES menu. • Enter a user.created list name directly using Ù.4. Press Í.Note: The stat list editor displays a formula-lock symbol next to each list name that has an attachedformula. Chapter 12 describes how to use the stat list editor to attach formulas to lists, editattached formulas, and detach formulas from lists.Detaching a Formula from a ListYou can detach (clear) an attached formula from a list in several ways.For example:• Enter ã ã !listname on the home screen.• Edit any element of a list to which a formula is attached.• Use the stat list editor (Chapter 12). Chapter 11: Lists 165
• Use ClrList or ClrAllList to detach a formula from a list (Chapter 18).Using Lists in ExpressionsUsing Lists in an ExpressionYou can use lists in an expression in any of three ways. When you press Í, any expression isevaluated for each list element, and a list is displayed.• Use L1–L6 or any user-created list name in an expression.• Enter the list elements directly.• Use y K to recall the contents of the list into an expression at the cursor location (Chapter 1).Note: You must paste user-created list names to the Rcl prompt by selecting them from theLIST NAMES menu. You cannot enter them directly using Ù.Using Lists with Math FunctionsYou can use a list to input several values for some math functions. See Appendix A specify forinformation about where a list is valid. The function is evaluated for each list element, and a list isdisplayed.• When you use a list with a function, the function must be valid for every element in the list. In graphing, an invalid element, such as L1 in ‡({1,0,L1}), is ignored. This returns an error. This graphs X…‡(1) and X…‡(0), but skips X…‡(L1).• When you use two lists with a two-argument function, the dimension of each list must be the same. The function is evaluated for corresponding elements. Chapter 11: Lists 166
• When you use a list and a value with a two-argument function, the value is used with each element in the list.LIST OPS MenuLIST OPS MenuTo display the LIST OPS menu, press y 9 ~.NAMES OPS MATH1: SortA( Sorts lists in ascending order.2: SortD( Sorts lists in descending order.3: dim( Sets the list dimension.4: Fill( Fills all elements with a constant.5: seq( Creates a sequence.6: cumSum( Returns a list of cumulative sums.7: @List( Returns difference of successive elements.8: Select( Selects specific data points.9: augment( Concatenates two lists.0: List4matr( Stores a list to a matrix.A: Matr4list( Stores a matrix to a list.B: Ù Designates the list-name data type.With one list, SortA( and SortD( sort the elements of listname and update the list in memory.SortA(listname) SortD(listname) Chapter 11: Lists 167
With two or more lists, SortA( and SortD( sort keylistname, and then sort each dependlist by placing itselements in the same order as the corresponding elements in keylistname. All lists must have thesame dimension.SortA(keylistname,dependlist1[,dependlist2,...,dependlist n])SortD(keylistname,dependlist1[,dependlist2,...,dependlist n])Note:• In the example, 5 is the first element in L4, and 1 is the first element in L5. After SortA(L4,L5), 5 becomes the second element of L4, and likewise, 1 becomes the second element of L5.• SortA( and SortD( are the same as SortA( and SortD( on the STAT EDIT menu (Chapter 12).• You cannot sort a locked list.Using dim( to Find List Dimensionsdim( (dimension) returns the length (number of elements) of list.dim(list)Using dim( to Create a ListYou can use dim( with ¿ to create a new listname with dimension length from 1 to 999. Theelements are zeros.length!dim(listname)Using dim( to Redimension a ListYou can use dim with ¿ to redimension an existing listname to dimension length from 1 to 999.• The elements in the old listname that are within the new dimension are not changed.• Extra list elements are filled by 0.• Elements in the old list that are outside the new dimension are deleted. Chapter 11: Lists 168
length!dim(listname)Fill(Fill( replaces each element in listname with value.Fill(value,listname)Note: dim( and Fill( are the same as dim( and Fill( on the MATRX MATH menu (Chapter 10).seq(seq( (sequence) returns a list in which each element is the result of the evaluation of expression withregard to variable for the values ranging from begin to end at steps of increment. variable need not bedefined in memory. increment can be negative; the default value for increment is 1. seq( is not validwithin expression. Complex lists are not valid.seq(expression,variable,begin,end[,increment])cumSum(cumSum( (cumulative sum) returns the cumulative sums of the elements in list, starting with thefirst element. list elements can be real or complex numbers.cumSum(list)@List(@List( returns a list containing the differences between consecutive elements in list. @List subtractsthe first element in list from the second element, subtracts the second element from the third, and Chapter 11: Lists 169
so on. The list of differences is always one element shorter than the original list. list elements canbe a real or complex numbers.@List(list)Select(Select( selects one or more specific data points from a scatter plot or xyLine plot (only), and thenstores the selected data points to two new lists, xlistname and ylistname. For example, you can useSelect( to select and then analyze a portion of plotted CBL 2™/CBL™ or CBR™ data.Select(xlistname,ylistname)Note: Before you use Select(, you must have selected (turned on) a scatter plot or xyLine plot.Also, the plot must be displayed in the current viewing window.Before Using Select(Before using Select(, follow these steps.1. Create two list names and enter the data.2. Turn on a stat plot, select " (scatter plot) or Ó (xyLine), and enter the two list names for Xlist: and Ylist: (Chapter 12).3. Use ZoomStat to plot the data (Chapter 3). MathPrint™ ClassicUsing Select( to Select Data Points from a PlotTo select data points from a scatter plot or xyLine plot, follow these steps.1. Press y 9 ~ 8 to select 8:Select( from the LIST OPS menu. Select( is pasted to the home screen. Chapter 11: Lists 170
2. Enter xlistname, press ¢, enter ylistname, and then press ¤ to designate list names into which you want the selected data to be stored.3. Press Í. The graph screen is displayed with Left Bound? in the bottom-left corner.4. Press } or † (if more than one stat plot is selected) to move the cursor onto the stat plot from which you want to select data points.5. Press | and ~ to move the cursor to the stat plot data point that you want as the left bound.6. Press Í. A 4 indicator on the graph screen shows the left bound. Right Bound? is displayed in the bottom-left corner. Chapter 11: Lists 171
7. Press | or ~ to move the cursor to the stat plot point that you want for the right bound, and then press Í. The x-values and y-values of the selected points are stored in xlistname and ylistname. A new stat plot of xlistname and ylistname replaces the stat plot from which you selected data points. The list names are updated in the stat plot editor.Note: The two new lists (xlistname and ylistname) will include the points you select as left bound andright bound. Also, left-bound x-value { right-bound x-value must be true.augment(augment( concatenates the elements of listA and listB. The list elements can be real or complexnumbers.augment(listA,listB)List4matr(List4matr( (lists stored to matrix) fills matrixname column by column with the elements from each list.If the dimensions of all lists are not equal, then List4matr( fills each extra matrixname row with 0.Complex lists are not valid. Chapter 11: Lists 172
List4matr(list1,list2, ... ,list n,matrixname)Matr4list(Matr4list( (matrix stored to lists) fills each listname with elements from each column in matrix. If thenumber of listname arguments exceeds the number of columns in matrix, then Matr4list( ignoresextra listname arguments. Likewise, if the number of columns in matrix exceeds the number oflistname arguments, then Matr4list( ignores extra matrix columns.Matr4list(matrix,listname1,listname2, . . . ,listname n)Matr4list( also fills a listname with elements from a specified column# in matrix. To fill a list with aspecific column from matrix, you must enter a column# after matrix.Matr4list(matrix,column#,listname)Ù preceding one to five characters identifies those characters as a user-created listname. listnamemay comprise letters, q, and numbers, but it must begin with a letter from A to Z or q.ÙlistnameGenerally, Ù must precede a user-created list name when you enter a user-created list namewhere other input is valid, for example, on the home screen. Without the Ù, the TI-84 Plus maymisinterpret a user-created list name as implied multiplication of two or more characters.Ù need not precede a user-created list name where a list name is the only valid input, for example,at the Name= prompt in the stat list editor or the Xlist: and Ylist: prompts in the stat plot editor. Ifyou enter Ù where it is not necessary, the TI-84 Plus will ignore the entry. Chapter 11: Lists 173
LIST MATH MenuLIST MATH MenuTo display the LIST MATH menu, press y 9 |.NAMES OPS MATH1: min( Returns minimum element of a list.2: max( Returns maximum element of a list.3: mean( Returns mean of a list.4: median( Returns median of a list.5: sum( Returns sum of elements in a list.6: prod( Returns product of elements in list.7: stdDev( Returns standard deviation of a list.8: variance( Returns the variance of a list.min(, max(min( (minimum) and max( (maximum) return the smallest or largest element of listA. If two lists arecompared, it returns a list of the smaller or larger of each pair of elements in listA and listB. For acomplex list, the element with smallest or largest magnitude (modulus) is returned.min(listA[,listB])max(listA[,listB])MathPrint™ ClassicNote: min( and max( are the same as min( and max( on the MATH NUM menu.mean(, median(mean( returns the mean value of list. median( returns the median value of list. The default value forfreqlist is 1. Each freqlist element counts the number of consecutive occurrences of thecorresponding element in list. Complex lists are not valid. Chapter 11: Lists 174
mean(list[,freqlist])median(list[,freqlist])MathPrint™ Classicsum(, prod(sum( (summation) returns the sum of the elements in list. start and end are optional; they specify arange of elements. list elements can be real or complex numbers.prod( returns the product of all elements of list. start and end elements are optional; they specify arange of list elements. list elements can be real or complex numbers.sum(list[,start,end]) prod(list[,start,end])Sums and Products of Numeric SequencesYou can combine sum( or prod( with seq( to obtain:upper upperG expression(x) expression(x)x=lower x=lowerTo evaluate G 2 (N–1) from N=1 to 4:stdDev(, variance(stdDev( returns the standard deviation of the elements in list. The default value for freqlist is 1. Eachfreqlist element counts the number of consecutive occurrences of the corresponding element in list.Complex lists are not valid. Chapter 11: Lists 175
stdDev(list[,freqlist])MathPrint™ Classicvariance( returns the variance of the elements in list. The default value for freqlist is 1. Each freqlistelement counts the number of consecutive occurrences of the corresponding element in list.Complex lists are not valid.variance(list[,freqlist])MathPrint™ Classic Chapter 11: Lists 176
4. Press 6 Ë 5 Í to store the first pendulum string length (6.5 cm) in L1. The rectangular cursor moves to the next row. Repeat this step to enter each of the 12 string length values in the table.5. Press ~ to move the rectangular cursor to the first row in L2. Press Ë 51 Í to store the first time measurement (.51 sec) in L2. The rectangular cursor moves to the next row. Repeat this step to enter each of the 12 time values in the table.6. Press o to display the Y= editor. If necessary, press ' to clear the function Y1. As necessary, press }, Í, and ~ to turn off Plot1, Plot2, and Plot3 from the top line of the Y= editor (Chapter 3). As necessary, press †, |, and Í to deselect functions.7. Press y , 1 to select 1:Plot1 from the STAT PLOTS menu. The stat plot editor is displayed for plot 1.8. Press Í to select On, which turns on plot 1. Press † Í to select " (scatter plot). Press † y d to specify Xlist:L1 for plot 1. Press † y e to specify Ylist:L2 for plot 1. Press † ~ Í to select + as the Mark for each data point on the scatter plot.9. Press q 9 to select 9:ZoomStat from the ZOOM menu. The window variables are adjusted automatically, and plot 1 is displayed. This is a scatter plot of the time-versus-length data.Since the scatter plot of time-versus-length data appears to be approximately linear, fit a line to thedata.10. Press … ~ 4 to select 4:LinReg(ax+b) (linear regression model) from the STAT CALC menu. LinReg(ax+b) is pasted to the home screen. Chapter 12: Statistics 178
11 LinReg(ax+b). Note: You can also use the YVARS (t a)shortcut menu to select Y1.12. Press Í to execute LinReg(ax+b). The linear regression for the data in L1 and L2 is calculated. Values for a and b are displayed on the home screen. The linear regression equation is stored in Y1. Residuals are calculated and stored automatically in the list name RESID, which becomes an item on the LIST NAMES menu. Note: - You can control the number of decimal places displayed by changing the decimal mode setting. - The statistics reported are not stored in the history on the home screen.13. Press s. The regression line and the scatter plot are displayed.The regression line appears to fit the central portion of the scatter plot well. However, a residualplot may provide more information about this fit.14. Press … 1 to select 1:Edit. The stat list editor is displayed. Press ~ and } to move the cursor onto L3. Press y 6. An unnamed column is displayed in column 3; L3, L4, L5, and L6 shift right one column. The Name= prompt is displayed in the entry line, and alpha-lock is on.15. Press y 9 to display the LIST NAMES menu. If necessary, press † to move the cursor onto the list name RESID.16. Press Í to select RESID and paste it to the stat list editor's Name= prompt. Chapter 12: Statistics 179
17. Press Í. RESID is stored in column 3 of the stat list editor. Press † repeatedly to examine the residuals.Notice that the first three residuals are negative. They correspond to the shortest pendulum stringlengths in L1. The next five residuals are positive, and three of the last four are negative. The lattercorrespond to the longer string lengths in L1. Plotting the residuals will show this pattern moreclearly.18. Press y , 2 to select 2:Plot2 from the STAT PLOTS menu. The stat plot editor is displayed for plot 2.19. Press Í to select On, which turns on plot 2. Press † Í to select " (scatter plot). Press † y d to specify Xlist:L1 for plot 2. Press † ãRä ãEä ãSä ãIä ãDä (alpha-lock is on) to specify Ylist:RESID for plot 2. Press † Í to select › as the mark for each data point on the scatter plot.20. Press o to display the Y= editor. Press | to move the cursor onto the = sign, and then press Í to deselect Y1. Press } Í to turn off plot 1.21. Press q 9 to select 9:ZoomStat from the ZOOM menu. The window variables are adjusted automatically, and plot 2 is displayed. This is a scatter plot of the residuals.Notice the pattern of the residuals: a group of negative residuals, then a group of positiveresiduals, and then another group of negative residuals.The residual pattern indicates a curvature associated with this data set for which the linear modeldid not account. The residual plot emphasizes a downward curvature, so a model that curves Chapter 12: Statistics 180
down with the data would be more accurate. Perhaps a function such as square root would fit. Trya power regression to fit a function of the form y = a … xb.22. Press o to display the Y= editor. Press ' to clear the linear regression equation from Y1. Press } Í to turn on plot 1. Press ~ Í to turn off plot 2.23. Press q 9 to select 9:ZoomStat from the ZOOM menu. The window variables are adjusted automatically, and the original scatter plot of time- versus-length data (plot 1) is displayed.24. Press … ~ ƒ ãAä to select A:PwrReg from the STAT CALC menu. PwrReg is pasted to the home screen PwrReg. Note: You can also use the YVARS (t a)shortcut menu to select Y1.25. Press Í to calculate the power regression. Values for a and b are displayed on the home screen. The power regression equation is stored in Y1. Residuals are calculated and stored automatically in the list name RESID.26. Press s. The regression line and the scatter plot are displayed.The new function y=.192x.522 appears to fit the data well. To get more information, examine aresidual plot.27. Press o to display the Y= editor. Press | Í to deselect Y1. Press } Í to turn off plot 1. Press ~ Í to turn on plot 2. Note: Step 19 defined plot 2 to plot residuals (RESID) versus string length (L1). Chapter 12: Statistics 181
28. Press q 9 to select 9:ZoomStat from the ZOOM menu. The window variables are adjusted automatically, and plot 2 is displayed. This is a scatter plot of the residuals.The new residual plot shows that the residuals are random in sign, with the residuals increasing inmagnitude as the string length increases.To see the magnitudes of the residuals, continue with these steps.29. Press r. Press ~ and | to trace the data. Observe the values for Y at each point. With this model, the largest positive residual is about 0.041 and the smallest negative residual is about L0.027. All other residuals are less than 0.02 in magnitude.Now that you have a good model for the relationship between length and period, you can use themodel to predict the period for a given string length. To predict the periods for a pendulum withstring lengths of 20 cm and 50 cm, continue with these steps.30. Press ~ 1 to display the VARS Y-VARS FUNCTION secondary menu, and then press 1 to select 1:Y1. Y1 is pasted to the home screen. Note: You can also use the YVARS (t a)shortcut menu to select Y1.31. Press £ 20 ¤ to enter a string length of 20 cm. Press Í to calculate the predicted time of about 0.92 seconds. Based on the residual analysis, we would expect the prediction of about 0.92 seconds to be within about 0.02 seconds of the actual value. Chapter 12: Statistics 182
32. Press y [ to recall the Last Entry. Press | | | 5 to change the string length to 50 cm.33. Press Í to calculate the predicted time of about 1.48 seconds. Since a string length of 50 cm exceeds the lengths in the data set, and since residuals appear to be increasing as string length increases, we would expect more error with this estimate. Note: You also can make predictions using the table with the TABLE SETUP settings Indpnt:Ask and Depend:Auto (Chapter 7).Setting Up Statistical AnalysesUsing Lists to Store DataData for statistical analyses is stored in lists, which you can create and edit using the stat listeditor. The TI-84 Plus has six list variables in memory, L1 through L6, to which you can store datafor statistical calculations. Also, you can store data to list names that you create (Chapter 11).Setting Up a Statistical AnalysisTo set up a statistical analysis, follow these steps. Read the chapter for details.1. Enter the statistical data into one or more lists.2. Plot the data.3. Calculate the statistical variables or fit a model to the data.4. Graph the regression equation for the plotted data.5. Graph the residuals list for the given regression model.Displaying the Stat List EditorThe stat list editor is a table where you can store, edit, and view up to 20 lists that are in memory.Also, you can create list names from the stat list editor.To display the stat list editor, press …, and then select 1:Edit from the STAT EDIT menu. Chapter 12: Statistics 183
The top line displays list names. L1 through L6 are stored in columns 1 through 6 after a memoryreset. The number of the current column is displayed in the top-right corner.The bottom line is the entry line. All data entry occurs on this line. The characteristics of this linechange according to the current context.The center area displays up to seven elements of up to three lists; it abbreviates values whennecessary. The entry line displays the full value of the current element.Using the Stat List EditorEntering a List Name in the Stat List EditorTo enter a list name in the stat list editor, follow these steps.1. Display the Name= prompt in the entry line in either of two ways. • Move the cursor onto the list name in the column where you want to insert a list, and then press y 6. An unnamed column is displayed and the remaining lists shift right one column. • Press } until the cursor is on the top line, and then press ~ until you reach the unnamed column. Note: If list names are stored to all 20 columns, you must remove a list name to make room for an unnamed column. The Name= prompt is displayed and alpha-lock is on.2. Enter a valid list name in any of four ways. • Select a name from the LIST NAMES menu (Chapter 11). • Enter L1, L2, L3, L4, L5, or L6 from the keyboard. • Enter an existing user-created list name directly from the keyboard. • Enter a new user-created list name.3. Press Í or † to store the list name and its elements, if any, in the current column of the stat list editor. Chapter 12: Statistics 184
To begin entering, scrolling, or editing list elements, press †. The rectangular cursor is displayed. Note: If the list name you entered in step 2 already was stored in another stat list editor column, then the list and its elements, if any, move to the current column from the previous column. Remaining list names shift accordingly.Creating a Name in the Stat List EditorTo create a name in the stat list editor, follow these steps.1. Display the Name= prompt.2. Press [letter from A to Z or q] to enter the first letter of the name. The first character cannot be a number.3. Enter zero to four letters, q, or numbers to complete the new user-created list name. List names can be one to five characters long.4. Press Í or † to store the list name in the current column of the stat list editor. The list name becomes an item on the LIST NAMES menu (Chapter 11).Removing a List from the Stat List EditorTo remove a list from the stat list editor, move the cursor onto the list name and then press {. Thelist is not deleted from memory; it is only removed from the stat list editor.Notes:• To delete a list name from memory, use the MEMORY MANAGEMENT/DELETE secondary menu (Chapter 18).• If you archive a list, it will be removed from the stat list editor.Removing All Lists and Restoring L1 through L6You can remove all user-created lists from the stat list editor and restore list names L1 through L6to columns 1 through 6 in either of two ways.• Use SetUpEditor with no arguments.• Reset all memory (Chapter 18). Chapter 12: Statistics 185
Clearing All Elements from a ListYou can clear all elements from a list in any of five ways.• Use ClrList to clear specified lists.• In the stat list editor, press } to move the cursor onto a list name, and then press ' Í.• In the stat list editor, move the cursor onto each element, and then press { one by one.• On the home screen or in the program editor, enter 0!dim(listname) to set the dimension of listname to 0 (Chapter 11).• Use ClrAllLists to clear all lists in memory (Chapter 18).Editing a List ElementTo edit a list element, follow these steps.1. Move the cursor onto the element you want to edit.2. Press Í to move the cursor to the entry line. Note: If you want to replace the current value, you can enter a new value without first pressing Í. When you enter the first character, the current value is cleared automatically.3. Edit the element in the entry line. • Press one or more keys to enter the new value. When you enter the first character, the current value is cleared automatically. You can use the shortcut menus to enter values. When you use n/d to enter a fraction, it is not displayed as a stacked fraction in the list. Instead, the fraction has a thick bar separating the numerator and denominator. Thick-bar fraction on the list editor entry line: Thin-bar fraction on the home screen (regular division): Note: Order of operations applies to fractions. For example, evaluates to because the order of operations dictates that division is performed before addition. To evaluate , enter with parentheses around the numerator. • Press ~ to move the cursor to the character before which you want to insert, press y 6, and then enter one or more characters. • Press ~ to move the cursor to a character you want to delete, and then press { to delete the character. To cancel any editing and restore the original element at the rectangular cursor, press ' Í. Chapter 12: Statistics 186
Note: You can enter expressions and variables for elements.4. Press Í, }, or † to update the list. If you entered an expression, it is evaluated. If you entered only a variable, the stored value is displayed as a list element. When you edit a list element in the stat list editor, the list is updated in memory immediately.Attaching Formulas to List NamesAttaching a Formula to a List Name in Stat List EditorYou can attach a formula to a list name in the stat list editor, and then display and edit thecalculated list elements. When executed, the attached formula must resolve to a list. Chapter 11describes in detail the concept of attaching formulas to list names.To attach a formula to a list name that is stored in the stat list editor, follow these steps.1. Press … Í to display the stat list editor.2. Press } to move the cursor to the top line.3. Press | or ~, if necessary, to move the cursor onto the list name to which you want to attach the formula. Note: If a formula in quotation marks is displayed on the entry line, then a formula is already attached to the list name. To edit the formula, press Í, and then edit the formula.4. Press ƒ ããä, enter the formula, and press ƒ ããä. Note: If you do not use quotation marks, the TI-84 Plus calculates and displays the same initial list of answers, but does not attach the formula for future calculations. Note: Any user-created list name referenced in a formula must be preceded by an Ù symbol (Chapter 11). Chapter 12: Statistics 187
5. Press Í. The TI-84 Plus calculates each list element and stores it to the list name to which the formula is attached. A lock symbol is displayed in the stat list editor, next to the list name to which the formula is attached. lock symbolUsing the Stat List Editor When Formula-Generated Lists Are DisplayedWhen you edit an element of a list referenced in an attached formula, the TI-84 Plus updates thecorresponding element in the list to which the formula is attached (Chapter 11).When a list with a formula attached is displayed in the stat list editor and you edit or enter elementsof another displayed list, then the TI-84 Plus takes slightly longer to accept each edit or entry thanwhen no lists with formulas attached are in view.Note: To speed editing time, scroll horizontally until no lists with formulas are displayed, orrearrange the stat list editor so that no lists with formulas are displayed.Handling Errors Resulting from Attached FormulasOn the home screen, you can attach to a list a formula that references another list with dimension0 (Chapter 11). However, you cannot display the formula-generated list in the stat list editor or onthe home screen until you enter at least one element to the list that the formula references.All elements of a list referenced by an attached formula must be valid for the attached formula. Forexample, if Real number mode is set and the attached formula is log(L1), then each element of L1must be greater than 0, since the logarithm of a negative number returns a complex result.When you use the shortcut menus, all values must be valid for use in the templates. For example,if you use the n/d template, both the numerator and demoninator must be integers.Notes:• If an error menu is returned when you attempt to display a formula-generated list in the stat list editor, you can select 2:Goto, write down the formula that is attached to the list, and then press ' Í to detach (clear) the formula. You then can use the stat list editor to find the Chapter 12: Statistics 188
source of the error. After making the appropriate changes, you can reattach the formula to a list.• If you do not want to clear the formula, you can select 1:Quit, display the referenced list on the home screen, and find and edit the source of the error. To edit an element of a list on the home screen, store the new value to listname(element#) (Chapter 11).Detaching Formulas from List NamesDetaching a Formula from a List NameYou can detach (clear) a formula from a list name in several ways.For example:• In the stat list editor, move the cursor onto the name of the list to which a formula is attached. Press Í ' Í. All list elements remain, but the formula is detached and the lock symbol disappears.• In the stat list editor, move the cursor onto an element of the list to which a formula is attached. Press Í, edit the element, and then press Í. The element changes, the formula is detached, and the lock symbol disappears. All other list elements remain.• Use ClrList. All elements of one or more specified lists are cleared, each formula is detached, and each lock symbol disappears. All list names remain.• Use ClrAllLists (Chapter 18). All elements of all lists in memory are cleared, all formulas are detached from all list names, and all lock symbols disappear. All list names remain.Editing an Element of a Formula-Generated ListAs described above, one way to detach a formula from a list name is to edit an element of the listto which the formula is attached. The TI-84 Plus protects against inadvertently detaching theformula from the list name by editing an element of the formula-generated list.Because of the protection feature, you must press Í before you can edit an element of aformula-generated list.The protection feature does not allow you to delete an element of a list to which a formula isattached. To delete an element of a list to which a formula is attached, you must first detach theformula in any of the ways described above.Switching Stat List Editor ContextsStat List Editor ContextsThe stat list editor has four contexts.• View-elements context• View-names context Chapter 12: Statistics 189
• Edit-elements context• Enter-name contextThe stat list editor is first displayed in view-elements context. To switch through the four contexts,select 1:Edit from the STAT EDIT menu and follow these steps.1. Press } to move the cursor onto a list name and switch to view-names context. Press ~ and | to view list names stored in other stat list editor columns.2. Press Í to switch to edit-elements context. You may edit any element in a list. All elements of the current list are displayed in braces ( { } ) in the entry line. Press ~ and | to view more list elements.3. Press Í again to switch to view-elements context. Press ~, |, †, and } to view other list elements. The current element's full value is displayed in the entry line.4. Press Í again to switch back to edit-elements context. You may edit the current element in the entry line.5. Press } until the cursor is on a list name, then press y 6 to switch to enter-name context.6. Press ' to switch to view-names context.7. Press † to switch back to view-elements context. Chapter 12: Statistics 190
Stat List Editor ContextsView-Elements ContextIn view-elements context, the entry line displays the list name, the current element's place in thatlist, and the full value of the current element, up to 12 characters at a time. An ellipsis (...) indicatesthat the element continues beyond 12 characters.To page down the list six elements, press ƒ †. To page up six elements, press ƒ }. Todelete a list element, press {. Remaining elements shift up one row. To insert a new element,press y 6. 0 is the default value for a new element.Edit-Elements ContextIn edit-elements context, the data displayed in the entry line depends on the previous context.• When you switch to edit-elements context from view-elements context, the full value of the current element is displayed. You can edit the value of this element, and then press † and } to edit other list elements.• When you switch to edit-elements context from view-names context, the full values of all elements in the list are displayed. An ellipsis indicates that list elements continue beyond the screen. You can press ~ and | to edit any element in the list.Note: In edit-elements context, you can attach a formula to a list name only if you switched to itfrom view-names context. Chapter 12: Statistics 191
View-Names ContextIn view-names context, the entry line displays the list name and the list elements.To remove a list from the stat list editor, press {. Remaining lists shift to the left one column. Thelist is not deleted from memory.To insert a name in the current column, press y 6. Remaining columns shift to the right onecolumn.Enter-Name ContextIn enter-name context, the Name= prompt is displayed in the entry line, and alpha-lock is on.At the Name= prompt, you can create a new list name, paste a list name from L1 to L6 from thekeyboard, or paste an existing list name from the LIST NAMES menu (Chapter 11). The Ù symbol isnot required at the Name= prompt.To leave enter-name context without entering a list name, press '. The stat list editorswitches to view-names context.STAT EDIT MenuSTAT EDIT MenuTo display the STAT EDIT menu, press ….EDIT CALC TESTS1: Edit... Displays the stat list editor.2: SortA( Sorts a list in ascending order.3: SortD( Sorts a list in descending order.4: ClrList Deletes all elements of a list.5: SetUpEditor Stores specified lists in the stat list editor. Chapter 12: Statistics 192
SortA( and SortD( each can sort in either of two ways.• With one listname, SortA( and SortD( sort the elements in listname and update the list in memory.• With two or more lists, SortA( and SortD( sort keylistname, and then sort each dependlist by placing its elements in the same order as the corresponding elements in keylistname. This lets you sort two-variable data on X and keep the data pairs together. All lists must have the same dimension.The sorted lists are updated in memory.SortA(listname)SortD(listname)SortA(keylistname,dependlist1[,dependlist2,...,dependlist n])SortD(keylistname,dependlist1[,dependlist2,...,dependlist n])Note: SortA( and SortD( are the same as SortA( and SortD( on the LIST OPS menu.ClrListClrList clears (deletes) from memory the elements of one or more listnames. ClrList also detachesany formula attached to a listname.ClrList listname1,listname2,...,listname nNote: To clear from memory all elements of all list names, use ClrAllLists (Chapter 18).SetUpEditorWith SetUpEditor you can set up the stat list editor to display one or more listnames in the order thatyou specify. You can specify zero to 20 listnames.Additionally, if you want to use listnames which happen to be archived, the SetUp Editor willautomatically unarchive the listnames and place them in the stat list editor at the same time.SetUpEditor [listname1,listname2,...,listname n] Chapter 12: Statistics 193
SetUpEditor with one to 20 listnames removes all list names from the stat list editor and then storeslistnames in the stat list editor columns in the specified order, beginning in column 1.MathPrint™ClassicIf you enter a listname that is not stored in memory already, then listname is created and stored inmemory; it becomes an item on the LIST NAMES menu.Restoring L1 through L6 to the Stat List EditorSetUpEditor with no listnames removes all list names from the stat list editor and restores list namesL1 through L6 in the stat list editor columns 1 through 6.Regression Model FeaturesRegression Model FeaturesSTAT CALC menu items 3 through C are regression models. The automatic residual list andautomatic regression equation features apply to all regression models. Diagnostics display modeapplies to some regression models.Automatic Residual ListWhen you execute a regression model, the automatic residual list feature computes and stores theresiduals to the list name RESID. RESID becomes an item on the LIST NAMES menu (Chapter 11). Chapter 12: Statistics 194
The TI-84 Plus uses the formula below to compute RESID list elements. The next sectiondescribes the variable RegEQ.RESID = Ylistname N RegEQ(Xlistname)Automatic Regression EquationEach regression model has an optional argument, regequ, for which you can specify a Y= variablesuch as Y1. Upon execution, the regression equation is stored automatically to the specified Y=variable and the Y= function is selected.MathPrint™ MathPrint™Classic ClassicRegardless of whether you specify a Y= variable for regequ, the regression equation always isstored to the TI-84 Plus variable RegEQ, which is item 1 on the VARS Statistics EQ secondarymenu.Note: For the regression equation, you can use the fixed-decimal mode setting to control thenumber of digits stored after the decimal point (Chapter 1). However, limiting the number of digitsto a small number could affect the accuracy of the fit.Diagnostics Display ModeWhen you execute some regression models, the TI-84 Plus computes and stores diagnosticsvalues for r (correlation coefficient) and r2 (coefficient of determination) or for R2 (coefficient ofdetermination). You can control whether these values are displayed by turning StatDiagnostics onor off on the mode screen.r and r2 are computed and stored for these regression models.LinReg(ax+b) LnReg PwrRegLinReg(a+bx) ExpReg Chapter 12: Statistics 195
R2 is computed and stored for these regression models.QuadReg CubicReg QuartRegThe r and r2 that are computed for LnReg, ExpReg, and PwrReg are based on the linearlytransformed data. For example, for ExpReg (y=ab^x), r and r2 are computed on ln y=ln a+x(ln b).By default, these values are not displayed with the results of a regression model when youexecute it. However, you can set the diagnostics display mode by executing the DiagnosticOn orDiagnosticOff instruction. Each instruction is in the CATALOG (Chapter 15).• To turn diagnostics on or off from the mode screen, select On or Off for StatDiagnostics. The default is Off.• To set DiagnosticOn or DiagnosticOff from the home screen, press y N, and then select the instruction for the mode you want. The instruction is pasted to the home screen. Press Í to set the mode.When DiagnosticOn is set, diagnostics are displayed with the results when you execute aregression model.MathPrint™ClassicWhen DiagnosticOff is set, diagnostics are not displayed with the results when you execute aregression model.MathPrint™Classic Chapter 12: Statistics 196
STAT CALC MenuSTAT CALC MenuTo display the STAT CALC menu, press … ~.EDIT CALC TESTS1: 1-Var Stats Calculates 1-variable statistics.2: 2-Var Stats Calculates 2-variable statistics.3: Med-Med Calculates a median-median line.4: LinReg(ax+b) Fits a linear model to data.5: QuadReg Fits a quadratic model to data.6: CubicReg Fits a cubic model to data.7: QuartReg Fits a quartic model to data.8: LinReg(a+bx) Fits a linear model to data.9: LnReg Fits a logarithmic model to data.0: ExpReg Fits an exponential model to data.A: PwrReg Fits a power model to data.B: Logistic Fits a logistic model to data.C: SinReg Fits a sinusoidal model to data.D: Manual Linear Fit Fits a linear equation interactively to a scatter plot.For each STAT CALC menu item, if neither Xlistname nor Ylistname is specified, then the default listnames are L1 and L2. If you do not specify freqlist, then the default is 1 occurrence of each listelement.Frequency of Occurrence for Data PointsFor most STAT CALC menu items, you can specify a list of data occurrences, or frequencies(freqlist).Each element in freqlist indicates how many times the corresponding data point or data pair occursin the data set you are analyzing.For example, if L1={15,12,9,14} and ÙFREQ={1,4,1,3}, then the TI-84 Plus interprets the instruction1-Var Stats L1, ÙFREQ to mean that 15 occurs once, 12 occurs four times, 9 occurs once, and 14occurs three times.Each element in freqlist must be ' 0, and at least one element must be > 0.Noninteger freqlist elements are valid. This is useful when entering frequencies expressed aspercentages or parts that add up to 1. However, if freqlist contains noninteger frequencies, Sx andSy are undefined; values are not displayed for Sx and Sy in the statistical results. Chapter 12: Statistics 197
1-Var Stats1-Var Stats (one-variable statistics) analyzes data with one measured variable. Each element infreqlist is the frequency of occurrence for each corresponding data point in Xlistname. freqlistelements must be real numbers > 0.1-Var Stats [Xlistname,freqlist]2-Var Stats2-Var Stats (two-variable statistics) analyzes paired data. Xlistname is the independent variable.Ylistname is the dependent variable. Each element in freqlist is the frequency of occurrence for eachdata pair (Xlistname,Ylistname).2-Var Stats [Xlistname,Ylistname,freqlist]Med-Med (ax+b)Med-Med (median-median) fits the model equation y=ax+b to the data using the median-medianline (resistant line) technique, calculating the summary points x1, y1, x2, y2, x3, and y3. Med-Meddisplays values for a (slope) and b (y-intercept).Med-Med [Xlistname,Ylistname,freqlist,regequ]LinReg (ax+b)LinReg(ax+b) (linear regression) fits the model equation y=ax+b to the data using a least-squares fit.It displays values for a (slope) and b (y-intercept); when DiagnosticOn is set, it also displays valuesfor r2 and r.LinReg(ax+b) [Xlistname,Ylistname,freqlist,regequ]QuadReg (ax2+bx+c)QuadReg (quadratic regression) fits the second-degree polynomial y=ax2+bx+c to the data. Itdisplays values for a, b, and c; when DiagnosticOn is set, it also displays a value for R2. For threedata points, the equation is a polynomial fit; for four or more, it is a polynomial regression. At leastthree data points are required.QuadReg [Xlistname,Ylistname,freqlist,regequ] Chapter 12: Statistics 198
CubicReg—(ax 3+bx 2+cx+d)CubicReg (cubic regression) fits the third-degree polynomial y=ax 3+bx 2+cx+d to the data. Itdisplays values for a, b, c, and d; when DiagnosticOn is set, it also displays a value for R2. For fourpoints, the equation is a polynomial fit; for five or more, it is a polynomial regression. At least fourpoints are required.CubicReg [Xlistname,Ylistname,freqlist,regequ]QuartReg—(ax 4+bx 3+cx 2+ dx+e)QuartReg (quartic regression) fits the fourth-degree polynomial y=ax 4+bx 3+cx 2+dx+e to the data. Itdisplays values for a, b, c, d, and e; when DiagnosticOn is set, it also displays a value for R2. Forfive points, the equation is a polynomial fit; for six or more, it is a polynomial regression. At leastfive points are required.QuartReg [Xlistname,Ylistname,freqlist,regequ]LinReg—(a+bx)LinReg(a+bx) (linear regression) fits the model equation y=a+bx to the data using a least-squares fit.It displays values for a (y-intercept) and b (slope); when DiagnosticOn is set, it also displays valuesfor r2 and r.LinReg(a+bx) [Xlistname,Ylistname,freqlist,regequ]LnReg—(a+b ln(x))LnReg (logarithmic regression) fits the model equation y=a+b ln(x) to the data using a least-squares fit and transformed values ln(x) and y. It displays values for a and b; when DiagnosticOn isset, it also displays values for r2 and r.LnReg [Xlistname,Ylistname,freqlist,regequ]ExpReg—(ab x)ExpReg (exponential regression) fits the model equation y=abx to the data using a least-squares fitand transformed values x and ln(y). It displays values for a and b; when DiagnosticOn is set, it alsodisplays values for r2 and r.ExpReg [Xlistname,Ylistname,freqlist,regequ] Chapter 12: Statistics 199
PwrReg—(axb)PwrReg (power regression) fits the model equation y=axb to the data using a least-squares fit andtransformed values ln(x) and ln(y). It displays values for a and b; when DiagnosticOn is set, it alsodisplays values for r2 and r.PwrReg [Xlistname,Ylistname,freqlist,regequ]Logistic—c/(1+a…e-bx)Logistic fits the model equation y=c/(1+a…eLbx) to the data using an iterative least-squares fit. Itdisplays values for a, b, and c.Logistic [Xlistname,Ylistname,freqlist,regequ]SinReg—a sin(bx+c)+dSinReg (sinusoidal regression) fits the model equation y=a sin(bx+c)+d to the data using aniterative least-squares fit. It displays values for a, b, c, and d. At least four data points are required.At least two data points per cycle are required in order to avoid aliased frequency estimates.SinReg [iterations,Xlistname,Ylistname,period,regequ]iterations is the maximum number of times the algorithm will iterate to find a solution. The value foriterations can be an integer ' 1 and 16; if not specified, the default is 3. The algorithm may find asolution before iterations is reached. Typically, larger values for iterations result in longer executiontimes and better accuracy for SinReg, and vice versa.A period guess is optional. If you do not specify period, the difference between time values inXlistname must be equal and the time values must be ordered in ascending sequential order. If youspecify period, the algorithm may find a solution more quickly, or it may find a solution when it wouldnot have found one if you had omitted a value for period. If you specify period, the differencesbetween time values in Xlistname can be unequal.Note: The output of SinReg is always in radians, regardless of the Radian/Degree mode setting. Chapter 12: Statistics 200
SinReg Example: Daylight Hours in Alaska for One YearCompute the regression model for the number of hours of daylight in Alaska during one year.MathPrint™Classic 1 periodWith noisy data, you will achieve better convergence results when you specify an accurateestimate for period. You can obtain a period guess in either of two ways.• Plot the data and trace to determine the x-distance between the beginning and end of one complete period, or cycle. The illustration above and to the right graphically depicts a complete period, or cycle.• Plot the data and trace to determine the x-distance between the beginning and end of N complete periods, or cycles. Then divide the total distance by N.After your first attempt to use SinReg and the default value for iterations to fit the data, you may findthe fit to be approximately correct, but not optimal. For an optimal fit, executeSinReg 16,Xlistname,Ylistname,2p/b where b is the value obtained from the previous SinReg execution.Manual Linear FitManual Linear Fit allows you to visually fit a linear function to a scatter plot. Manual Linear Fit is anoption in the … / menu. Chapter 12: Statistics 201
After entering List data and viewing the StatPlot, select the Manual-Fit function.1. Press … to display the Stat menu. Press ~ to select CALC. Press † several times to scroll down to select D:Manual-Fit. Press Í. This displays a free-floating cursor at the center of the display screen2. Press the cursor navigation keys (} † | ~ ) to move the cursor to the desired location. Press Í to select the first point.3. Press the cursor navigation keys (} † | ~ ) to move the cursor to the second location. Press Í. This displays a line containing the two points selected.The linear function is displayed. The Manual-Fit Line equation displays in the form of Y=mX+b.The current value of the first parameter (m) is highlighted in the symbolic expression.Modify parameter valuesPress the cursor navigation keys ( | ~ ) to move from the first parameter (m) or (b) the secondparameter. You can press Í and type a new parameter value. Press Í to display the newparameter value. When you edit the value of the selected parameter, the edit can include insert,delete, type over, or mathematical expression.The screen dynamically displays the revised parameter value. Press Í to complete themodification of the selected parameter, save the value, and refresh the displayed graph. Thesystem displays the revised parameter value in the symbolic expression Y=mX+B, and refreshesthe graph with the updated Manual-Fit Line.Select y 5 to finish the Manual Fit function. The calculator stores the current mX+bexpression into Y1 and makes that function active for graphing. You can also select Manual-Fitwhile on the Home screen. You can then enter a different Y-Var such as Y4 and then press Í.This takes you to the Graph screen and then pastes the Manual-Fit equation in the specified Y-Var.In this example, Y4.Statistical VariablesThe statistical variables are calculated and stored as indicated below. To access these variablesfor use in expressions, press , and select 5:Statistics. Then select the VARS menu shown in Chapter 12: Statistics 202
Statistical Analysis in a ProgramEntering Stat DataYou can enter statistical data, calculate statistical results, and fit models to data from a program.You can enter statistical data into lists directly within the program (Chapter 11).Statistical CalculationsTo perform a statistical calculation from a program, follow these steps.1. On a blank line in the program editor, select the type of calculation from the STAT CALC menu.2. Enter the names of the lists to use in the calculation. Separate the list names with a comma.3. Enter a comma and then the name of a Y= variable, if you want to store the regression equation to a Y= variable.Statistical PlottingSteps for Plotting Statistical Data in ListsYou can plot statistical data that is stored in lists. The six types of plots available are scatter plot,xyLine, histogram, modified box plot, regular box plot, and normal probability plot. You can defineup to three plots.To plot statistical data in lists, follow these steps.1. Store the stat data in one or more lists.2. Select or deselect Y= functions as appropriate.3. Define the stat plot.4. Turn on the plots you want to display.5. Define the viewing window.6. Display and explore the graph. Chapter 12: Statistics 204
ScatterScatter (")plots plot the data points from Xlist and Ylist as coordinate pairs, showing each point asa box ( › ), cross ( + ), or dot ( ¦ ). Xlist and Ylist must be the same length. You can use the samelist for Xlist and Ylist.xyLinexyLine (Ó)is a scatter plot in which the data points are plotted and connected in order ofappearance in Xlist and Ylist. You may want to use SortA( or SortD( to sort the lists before you plotthem.HistogramHistogram (Ò) plots one-variable data. The Xscl window variable value determines the width ofeach bar, beginning at Xmin. ZoomStat adjusts Xmin, Xmax, Ymin, and Ymax to include all values,and also adjusts Xscl. The inequality (Xmax N Xmin) à Xscl 47 must be true. A value that occurs onthe edge of a bar is counted in the bar to the right.ModBoxplotModBoxplot (Õ) (modified box plot) plots one-variable data, like the regular box plot, exceptpoints that are 1.5 … Interquartile Range beyond the quartiles. (The Interquartile Range is definedas the difference between the third quartile Q3 and the first quartile Q1.) These points are plottedindividually beyond the whisker, using the Mark (› or + or ¦) you select. You can trace these points,which are called outliers. Chapter 12: Statistics 205
The prompt for outlier points is x=, except when the outlier is the maximum point (maxX) or theminimum point (minX). When outliers exist, the end of each whisker will display x=. When nooutliers exist, minX and maxX are the prompts for the end of each whisker. Q1, Med (median), andQ3 define the boxBoxplotBoxplot (Ö)(regular box plot) plots one-variable data. The whiskers on the plot extend from theminimum data point in the set (minX) to the first quartile (Q1) and from the third quartile (Q3) to themaximum point (maxX). The box is defined by Q1, Med (median), and Q3NormProbPlotNormProbPlot (Ô) (normal probability plot) plots each observation X in Data List versus thecorresponding quantile z of the standard normal distribution. If the plotted points lie close to astraight line, then the plot indicates that the data are normal.Enter a valid list name in the Data List field. Select X or Y for the Data Axis setting.• If you select X, the TI-84 Plus plots the data on the x-axis and the z-values on the y-axis. Chapter 12: Statistics 206
Data Data Plot Type XList YList Mark Freq List Axis Ô NormProbPlot œ œ _ œ _ _5. Enter list names or select options for the plot type. • Xlist (list name containing independent data) • Ylist (list name containing dependent data) • Mark (› or + or ¦) • Freq (frequency list for Xlist elements; default is 1) • Data List (list name for NormProbPlot) • Data Axis (axis on which to plot Data List)Displaying Other Stat Plot EditorsEach stat plot has a unique stat plot editor. The name of the current stat plot (Plot1, Plot2, or Plot3)is highlighted in the top line of the stat plot editor. To display the stat plot editor for a different plot,press } and ~ to move the cursor onto the name in the top line, and then press Í. The statplot editor for the selected plot is displayed, and the selected name remains highlighted.Turning On and Turning Off Stat PlotsPlotsOn and PlotsOff allow you to turn on or turn off stat plots from the home screen or a program.With no plot number, PlotsOn turns on all plots and PlotsOff turns off all plots. With one or moreplot numbers (1, 2, and 3), PlotsOn turns on specified plots, and PlotsOff turns off specified plots.PlotsOff [1,2,3]PlotsOn [1,2,3]Note: You also can turn on and turn off stat plots in the top line of the Y= editor (Chapter 3). Chapter 12: Statistics 208
Defining the Viewing WindowStat plots are displayed on the current graph. To define the viewing window, press p andenter values for the window variables. ZoomStat redefines the viewing window to display allstatistical data points.Tracing a Stat PlotWhen you trace a scatter plot or xyLine, tracing begins at the first element in the lists.When you trace a histogram, the cursor moves from the top center of one column to the top centerof the next, starting at the first column.When you trace a box plot, tracing begins at Med (the median). Press | to trace to Q1 and minX.Press ~ to trace to Q3 and maxX.When you press } or † to move to another plot or to another Y= function, tracing moves to thecurrent or beginning point on that plot (not the nearest pixel).The ExprOn/ExprOff format setting applies to stat plots (Chapter 3). When ExprOn is selected, theplot number and plotted data lists are displayed in the top-left corner.Statistical Plotting in a ProgramDefining a Stat Plot in a ProgramTo display a stat plot from a program, define the plot, and then display the graph.To define a stat plot from a program, begin on a blank line in the program editor and enter data intoone or more lists; then, follow these steps.1. Press y , to display the STAT PLOTS menu.2. Select the plot to define, which pastes Plot1(, Plot2(, or Plot3( to the cursor location.3. Press y , ~ to display the STAT TYPE menu. Chapter 12: Statistics 209
4. Select the type of plot, which pastes the name of the plot type to the cursor location.5. Press ¢. Enter the list names, separated by commas.6. Press ¢ y , | to display the STAT PLOT MARK menu. (This step is not necessary if you selected 3:Histogram or 5:Boxplot in step 4.) Select the type of mark (› or + or ¦) for each data point. The selected mark symbol is pasted to the cursor location.7. Press ¤ Í to complete the command line.Displaying a Stat Plot from a ProgramTo display a plot from a program, use the DispGraph instruction (Chapter 16) or any of the ZOOMinstructions (Chapter 3). Chapter 12: Statistics 210
Chapter 13:Inferential Statistics and DistributionsGetting Started: Mean Height of a PopulationGetting Started is a fast-paced introduction. Read the chapter for details.Suppose you want to estimate the mean height of a population of women given the randomsample below. Because heights among a biological population tend to be normally distributed, a tdistribution confidence interval can be used when estimating the mean. The 10 height valuesbelow are the first 10 of 90 values, randomly generated from a normally distributed population withan assumed mean of 165.1 centimeters and a standard deviation of 6.35 centimeters(randNorm(165.1,6.35,90) with a seed of 789).Height (in centimeters) of Each of 10 Women169.43 168.33 159.55 169.97 159.79 181.42 171.17 162.04 167.15 159.531. Press … Í to display the stat list editor. Press } to move the cursor onto L1, and then press y 6 to insert a new list. The Name= prompt is displayed on the bottom line. The Ø cursor indicates that alpha-lock is on. The existing list name columns shift to the right. Note: Your stat editor may not look like the one pictured here, depending on the lists you have already stored.2. Enter [H] [G] [H] [T] at the Name= prompt, and then press Í to create the list to store the women's height data. Press † to move the cursor into the first row of the list. HGHT(1)= is displayed on the bottom line. Press Í.3. Press 169 Ë 43 to enter the first height value. As you enter it, it is displayed on the bottom line. Press Í. The value is displayed in the first row, and the rectangular cursor moves to the next row. Enter the other nine height values the same way. Chapter 13: Inferential Statistics and Distributions 211
4. Press … | to display the STAT TESTS menu, and then press † until 8:TInterval is highlighted.5. Press Í to select 8:TInterval. The inferential stat editor for TInterval is displayed. If Data is not selected for Inpt:, press | Í to select Data. Press † y 9 and press † until HGHT is highlighted and then press Í. Press † † Ë 99 to enter a 99 percent confidence level at the C-Level: prompt.6. Press † to move the cursor onto Calculate, and then press Í. The confidence interval is calculated, and the TInterval results are displayed on the home screen.Interpreting the resultsThe first line, (159.74,173.94), shows that the 99 percent confidence interval for the populationmean is between about 159.74 centimeters and 173.94 centimeters. This is about a 14.2centimeters spread.The .99 confidence level indicates that in a very large number of samples, we expect 99 percent ofthe intervals calculated to contain the population mean. The actual mean of the populationsampled is 165.1 centimeters, which is in the calculated interval.The second line gives the mean height of the sample v used to compute this interval. The third linegives the sample standard deviation Sx. The bottom line gives the sample size n.To obtain a more precise bound on the population mean m of women's heights, increase thesample size to 90. Use a sample mean v of 163.8 and sample standard deviation Sx of 7.1calculated from the larger random sample. This time, use the Stats (summary statistics) inputoption.1. Press … | 8 to display the inferential stat editor for TInterval. Press ~ Í to select Inpt:Stats. The editor changes so that you can enter summary statistics as input. Chapter 13: Inferential Statistics and Distributions 212
2. Press † 163 Ë 8 Í to store 163.8 to v. Press 7 Ë 1 Í to store 7.1 to Sx. Press 90 Í to store 90 to n.3. Press † to move the cursor onto Calculate, and then press Í to calculate the new 99 percent confidence interval. The results are displayed on the home screen.If the height distribution among a population of women is normally distributed with a mean m of165.1 centimeters and a standard deviation s of 6.35 centimeters, what height is exceeded by only5 percent of the women (the 95th percentile)?4. Press ' to clear the home screen. Press y = to display the DISTR (distributions) menu.5. Press 3 to paste invNorm( to the home screen. Press Ë 95 ¢ 165 Ë 1 ¢ 6 Ë 35 ¤ Í. .95 is the area, 165.1 is m, and 6.35 is s.The result is displayed on the home screen; it shows that five percent of the women are taller than175.5 centimeters.Now graph and shade the top 5 percent of the population.6. Press p and set the window variables to these values. Xmin=145 Ymin=L.02 Xres=1 Xmax=185 Ymax=.08 Xscl=5 Yscl=07. Press y = ~ to display the DISTR DRAW menu. Chapter 13: Inferential Statistics and Distributions 213
8. Press Í to paste ShadeNorm( to the home screen. Press y Z ¢ 1 y D 99 ¢ 165 Ë 1 ¢ 6 Ë 35 ¤. Ans (175.5448205 from step 11) is the lower bound. 1â99 is the upper bound. The normal curve is defined by a mean m of 165.1 and a standard deviation s of 6.35.9. Press Í to plot and shade the normal curve. Area is the area above the 95th percentile. low is the lower bound. up is the upper bound.Inferential Stat EditorsDisplaying the Inferential Stat EditorsWhen you select a hypothesis test or confidence interval instruction from the home screen, theappropriate inferential statistics editor is displayed. The editors vary according to each test orinterval's input requirements. Below is the inferential stat editor for T-Test.Note: When you select the ANOVA( instruction, it is pasted to the home screen. ANOVA( does nothave an editor screen.Using an Inferential Stat EditorTo use an inferential stat editor, follow these steps.1. Select a hypothesis test or confidence interval from the STAT TESTS menu. The appropriate editor is displayed.2. Select Data or Stats input, if the selection is available. The appropriate editor is displayed.3. Enter real numbers, list names, or expressions for each argument in the editor.4. Select the alternative hypothesis (ƒÄ, <, or >) against which to test, if the selection is available.5. Select No or Yes for the Pooled option, if the selection is available.6. Select Calculate or Draw (when Draw is available) to execute the instruction. • When you select Calculate, the results are displayed on the home screen. Chapter 13: Inferential Statistics and Distributions 214
• When you select Draw, the results are displayed in a graph.This chapter describes the selections in the above steps for each hypothesis test and confidenceinterval instruction.Select Data or Select anStats input alternative hypothesisEnter valuesfor arguments Select Calculate or Draw outputSelecting Data or StatsMost inferential stat editors prompt you to select one of two types of input. (1-PropZInt and2-PropZTest, 1-PropZInt and 2-PropZInt, c2-Test, c2GOF-Test, LinRegTInt, and LinRegTTest do not.)• Select Data to enter the data lists as input.• Select Stats to enter summary statistics, such as v, Sx, and n, as input.To select Data or Stats, move the cursor to either Data or Stats, and then press Í.Entering the Values for ArgumentsInferential stat editors require a value for every argument. If you do not know what a particularargument symbol represents, see the Inferential Statistics Input Descriptions tables.When you enter values in any inferential stat editor, the TI-84 Plus stores them in memory so thatyou can run many tests or intervals without having to reenter every value.Selecting an Alternative Hypothesis (ă < >)Most of the inferential stat editors for the hypothesis tests prompt you to select one of threealternative hypotheses.• The first is a ƒ alternative hypothesis, such as mƒm0 for the Z-Test.• The second is a < alternative hypothesis, such as m1<m2 for the 2-SampTTest.• The third is a > alternative hypothesis, such as p1>p2 for the 2-PropZTest.To select an alternative hypothesis, move the cursor to the appropriate alternative, and then pressÍ.Selecting the Pooled OptionPooled (2-SampTTest and 2-SampTInt only) specifies whether the variances are to be pooled for thecalculation. Chapter 13: Inferential Statistics and Distributions 215
• Select No if you do not want the variances pooled. Population variances can be unequal.• Select Yes if you want the variances pooled. Population variances are assumed to be equal.To select the Pooled option, move the cursor to Yes, and then press Í.Selecting Calculate or Draw for a Hypothesis TestAfter you have entered all arguments in an inferential stat editor for a hypothesis test, you mustselect whether you want to see the calculated results on the home screen (Calculate) or on thegraph screen (Draw).• Calculate calculates the test results and displays the outputs on the home screen.• Draw draws a graph of the test results and displays the test statistic and p-value with the graph. The window variables are adjusted automatically to fit the graph.To select Calculate or Draw, move the cursor to either Calculate or Draw, and then press Í. Theinstruction is immediately executed.Selecting Calculate for a Confidence IntervalAfter you have entered all arguments in an inferential stat editor for a confidence interval, selectCalculate to display the results. The Draw option is not available.When you press Í, Calculate calculates the confidence interval results and displays theoutputs on the home screen.Bypassing the Inferential Stat EditorsTo paste a hypothesis test or confidence interval instruction to the home screen without displayingthe corresponding inferential stat editor, select the instruction you want from the CATALOG menu.Appendix A describes the input syntax for each hypothesis test and confidence interval instruction.Note: You can paste a hypothesis test or confidence interval instruction to a command line in aprogram. From within the program editor, select the instruction from either the CATALOG(Chapter 15) or the STAT TESTS menu.STAT TESTS MenuSTAT TESTS MenuTo display the STAT TESTS menu, press … |. When you select an inferential statisticsinstruction, the appropriate inferential stat editor is displayed. Chapter 13: Inferential Statistics and Distributions 216
Most STAT TESTS instructions store some output variables to memory. For a list of these variables,see the Test and Interval Output Variables table.EDIT CALC TESTS1: Z-Test... Test for 1 m, known s2: T-Test... Test for 1 m, unknown s3: 2-SampZTest... Test comparing 2 m's, known s's4: 2-SampTTest... Test comparing 2 m's, unknown s's5: 1-PropZTest... Test for 1 proportion6: 2-PropZTest... Test comparing 2 proportions7: ZInterval... Confidence interval for 1 m, known s8: TInterval... Confidence interval for 1 m, unknown s9: 2-SampZInt... Confidence interval for difference of 2 m's, known s's0: 2-SampTInt... Confidence interval for difference of 2 m's, unknown s'sA: 1-PropZInt... Confidence interval for 1 proportionB: 2-PropZInt... Confidence interval for difference of 2 proportionsC: c2-Test... Chi-square test for 2-way tablesD: c2-GOF Test... Chi-square Goodness of Fit testE: 2-SampÛTest... Test comparing 2 s'sF: LinRegTTest... t test for regression slope and rG: LinRegTInt... Confidence interval for linear regression slope coefficient bH: ANOVA( One-way analysis of varianceNote: When a new test or interval is computed, all previous output variables are invalidated.Inferential Stat Editors for the STAT TESTS InstructionsIn this chapter, the description of each STAT TESTS instruction shows the unique inferential stateditor for that instruction with example arguments.• Descriptions of instructions that offer the Data/Stats input choice show both types of input screens.• Descriptions of instructions that do not offer the Data/Stats input choice show only one input screen.The description then shows the unique output screen for that instruction with the example results.• Descriptions of instructions that offer the Calculate/Draw output choice show both types of screens: calculated and graphic results.• Descriptions of instructions that offer only the Calculate output choice show the calculated results on the home screen. Chapter 13: Inferential Statistics and Distributions 217
1-PropZInt1-PropZInt (one-proportion z confidence interval; item A) computes a confidence interval for anunknown proportion of successes. It takes as input the count of successes in the sample x and thecount of observations in the sample n. The computed confidence interval depends on the user-specified confidence level. Input: Calculated results:2-PropZInt2-PropZInt (two-proportion z confidence interval; item B) computes a confidence interval for thedifference between the proportion of successes in two populations (p1Np2). It takes as input thecount of successes in each sample (x1 and x2) and the count of observations in each sample(n1 and n2). The computed confidence interval depends on the user-specified confidence level. Input: Calculated results: Chapter 13: Inferential Statistics and Distributions 226
c2-Testc2-Test (chi-square test; item C) computes a chi-square test for association on the two-way table ofcounts in the specified Observed matrix. The null hypothesis H 0 for a two-way table is: noassociation exists between row variables and column variables. The alternative hypothesis is: thevariables are related.Before computing a c2-Test, enter the observed counts in a matrix. Enter that matrix variable nameat the Observed: prompt in the c2.Test editor; default=[A]. At the Expected: prompt, enter the matrixvariable name to which you want the computed expected counts to be stored; default=[B]. Matrix Note: Press y ú ~ ~ 1 to editor: select 1:[A] from the MATRX EDIT menu. Input: Note: Press y ú †] Í to display matrix [B]. Calculated results: Drawn results:c2GOF-Testc2GOF-Test (Chi Square Goodness of Fit; item D) performs a test to confirm that sample data isfrom a population that conforms to a specified distribution. For example, c2 GOF can confirm thatthe sample data came from a normal distribution. Chapter 13: Inferential Statistics and Distributions 227
automatically stored to the specified Y= equation. In the example below, the regression equation isstored to Y1, which is then selected (turned on).In the example:L3={38, 56, 59, 64, 74}L4={41, 63, 70, 72, 84} Input: Calculated results:When LinRegTTest is executed, the list of residuals is created and stored to the list name RESIDautomatically. RESID is placed on the LIST NAMES menu.Note: For the regression equation, you can use the fix-decimal mode setting to control the numberof digits stored after the decimal point (Chapter 1). However, limiting the number of digits to a smallnumber could affect the accuracy of the fit.LinRegTIntLinRegTInt computes a linear regression T confidence interval for the slope coefficient b. If theconfidence interval contains 0, this is insufficient evidence to indicate that the data exhibits a linearrelationship. Chapter 13: Inferential Statistics and Distributions 230
In the example:list 1={4, 5, 6, 7, 8}list 2={1, 2, 3, 3.5, 4.5} LinRegTInt input Note: Press … ~ ~ to screen: select TESTS. Press † several times to select G:LinRegTint... Press Í. Press † several times to select Calculate. Press Í. Calculated results:Xlist, Ylist is the list of independent and dependent variables. The list containing the Freq(frequency) values for the data is stored in List. The default is 1. All elements must be realnumbers. Each element in the Freq list is the frequency of occurence for each corresponding datapoint in the input list specified in the List fields. RegEQ (optional) is the designated Yn variable forstoring the regression equation. StoreRegEqn (optional) is the designated variable for storing theregression equation. The C level is the Confidence level probability with default = .95.ANOVA(ANOVA( (one-way analysis of variance; item H) computes a one-way analysis of variance forcomparing the means of two to 20 populations. The ANOVA procedure for comparing these meansinvolves analysis of the variation in the sample data. The null hypothesis H0: m1=m2=...=mk istested against the alternative Ha: not all m1...mk are equal.ANOVA(list1,list2[,...,list20]) Chapter 13: Inferential Statistics and Distributions 231
In the example:L1={7 4 6 6 5}L2={6 5 5 8 7}L3={4 7 6 7 6} Input: Calculated results:Note: SS is sum of squares and MS is mean square.Inferential Statistics Input DescriptionsThe tables in this section describe the inferential statistics inputs discussed in this chapter. Youenter values for these inputs in the inferential stat editors. The tables present the inputs in thesame order that they appear in this chapter.Input Descriptionm0 Hypothesized value of the population mean that you are testing.s The known population standard deviation; must be a real number > 0.List The name of the list containing the data you are testing.Freq The name of the list containing the frequency values for the data in List. Default=1. All elements must be integers | 0.Calculate/Draw Determines the type of output to generate for tests and intervals. Calculate displays the output on the home screen. In tests, Draw draws a graph of the results.v, Sx, n Summary statistics (mean, standard deviation, and sample size) for the one-sample tests and intervals. Chapter 13: Inferential Statistics and Distributions 232
Input Descriptions1 The known population standard deviation from the first population for the two-sample tests and intervals. Must be a real number > 0.s2 The known population standard deviation from the second population for the two-sample tests and intervals. Must be a real number > 0.List1, List2 The names of the lists containing the data you are testing for the two-sample tests and intervals. Defaults are L1 and L2, respectively.Freq1, Freq2 The names of the lists containing the frequencies for the data in List1 and List2 for the two-sample tests and intervals. Defaults=1. All elements must be integers | 0.v1, Sx1, n1, v2, Sx2, n2 Summary statistics (mean, standard deviation, and sample size) for sample one and sample two in the two-sample tests and intervals.Pooled Specifies whether variances are to be pooled for 2-SampTTest and 2-SampTInt. No instructs the TI-84 Plus not to pool the variances. Yes instructs the TI-84 Plus to pool the variances.p0 The expected sample proportion for 1-PropZTest. Must be a real number, such that 0 < p0 < 1.x The count of successes in the sample for the 1-PropZTest and 1-PropZInt. Must be an integer | 0.n The count of observations in the sample for the 1-PropZTest and 1-PropZInt. Must be an integer > 0.x1 The count of successes from sample one for the 2-PropZTest and 2-PropZInt. Must be an integer | 0.x2 The count of successes from sample two for the 2-PropZTest and 2-PropZInt. Must be an integer | 0.n1 The count of observations in sample one for the 2-PropZTest and 2-PropZInt. Must be an integer > 0.n2 The count of observations in sample two for the 2-PropZTest and 2-PropZInt. Must be an integer > 0.C-Level The confidence level for the interval instructions. Must be ' 0 and < 100. If it is ' 1, it is assumed to be given as a percent and is divided by 100. Default=0.95.Observed (Matrix) The matrix name that represents the columns and rows for the observed values of a two-way table of counts for the c2-Test and c2GOF-Test. Observed must contain all integers | 0. Matrix dimensions must be at least 2×2.Expected (Matrix) The matrix name that specifies where the expected values should be stored. Expected is created upon successful completion of the c2-Test and c2GOF-Test.df df (degree of freedom) represents (number of sample categories) - (number of estimated parameters for the selected distribution + 1). Chapter 13: Inferential Statistics and Distributions 233
normalpdf(x[,m,s]) Note: For this example, Xmin = 28 Xmax = 42 Xscl = 1 Ymin = 0 Ymax = .2 Yscl = .1Note: For plotting the normal distribution, you can set window variables Xmin and Xmax so that themean m falls between them, and then select 0:ZoomFit from the ZOOM menu.normalcdf(normalcdf( computes the normal distribution probability between lowerbound and upperbound for thespecified mean m and standard deviation s. The defaults are m=0 and s=1.normalcdf(lowerbound,upperbound[,m,s])invNorm(invNorm( computes the inverse cumulative normal distribution function for a given area under thenormal distribution curve specified by mean m and standard deviation s. It calculates the x valueassociated with an area to the left of the x value. 0 area 1 must be true. The defaults are m=0 ands=1.invNorm(area[,m,s])invT(invT( computes the inverse cumulative Student-t probability function specified by Degree ofFreedom, df for a given Area under the curve. Chapter 13: Inferential Statistics and Distributions 236
Üpdf(x,numerator df,denominator df) Note: For this example, Xmin = 0 Xmax = 5 Ymin = 0 Ymax = 1Fcdf(Ücdf( computes the Ü distribution probability between lowerbound and upperbound for the specifiednumerator df (degrees of freedom) and denominator df. numerator df and denominator df must be integers> 0.Ücdf(lowerbound,upperbound,numerator df,denominator df)binompdfbinompdf( computes a probability at x for the discrete binomial distribution with the specifiednumtrials and probability of success (p) on each trial. x can be an integer or a list of integers. 0p1must be true. numtrials must be an integer > 0. If you do not specify x, a list of probabilities from 0 tonumtrials is returned. The probability density function (pdf) is: fx = p 1 – p n x n–x ,x = 0,1,...,n x where n = numtrialsbinompdf(numtrials,p[,x])binomcdf(binomcdf( computes a cumulative probability at x for the discrete binomial distribution with thespecified numtrials and probability of success (p) on each trial. x can be a real number or a list ofreal numbers. 0p1 must be true. numtrials must be an integer > 0. If you do not specify x, a list ofcumulative probabilities is returned. Chapter 13: Inferential Statistics and Distributions 239
binomcdf(numtrials,p[,x])poissonpdf(poissonpdf( computes a probability at x for the discrete Poisson distribution with the specifiedmean m, which must be a real number > 0. x can be an integer or a list of integers. The probabilitydensity function (pdf) is: – x f x = e x! ,x = 0,1,2,...poissonpdf(m,x)poissoncdf(poissoncdf( computes a cumulative probability at x for the discrete Poisson distribution with thespecified mean m, which must be a real number > 0. x can be a real number or a list of realnumbers.poissoncdf(m,x)geometpdf(geometpdf( computes a probability at x, the number of the trial on which the first success occurs,for the discrete geometric distribution with the specified probability of success p. 0p1 must betrue. x can be an integer or a list of integers. The probability density function (pdf) is: x–1 fx = p1 – p ,x = 1,2,...geometpdf(p,x) Chapter 13: Inferential Statistics and Distributions 240
geometcdf(geometcdf( computes a cumulative probability at x, the number of the trial on which the firstsuccess occurs, for the discrete geometric distribution with the specified probability of success p.0p1 must be true. x can be a real number or a list of real numbers.geometcdf(p,x)MathPrint™ ClassicDistribution ShadingDISTR DRAW MenuTo display the DISTR DRAW menu, press y = ~. DISTR DRAW instructions draw varioustypes of density functions, shade the area specified by lowerbound and upperbound, and display thecomputed area value.To clear the drawings, select 1:ClrDraw from the DRAW menu (Chapter 8).Note: Before you execute a DISTR DRAW instruction, you must set the window variables so that thedesired distribution fits the screen.DISTR DRAW1: ShadeNorm( Shades normal distribution.2: Shade_t( Shades Student-t distribution.3: Shadec2( Shades c2 distribution.4: ShadeÜ( Shades Üdistribution.Note: L1â99 and 1â99 specify infinity. If you want to view the area left of upperbound, for example,specify lowerbound=L1â99.ShadeNorm(ShadeNorm( draws the normal density function specified by mean m and standard deviation s andshades the area between lowerbound and upperbound. The defaults are m=0 and s=1. Chapter 13: Inferential Statistics and Distributions 241
Chapter 14:ApplicationsThe Applications MenuThe TI-84 Plus comes with several applications already installed and listed on the APPLICATIONSmenu. These applications include the following:FinanceTopics in Algebra 1Science ToolsCatalog Help 1.1CellSheet™Conic GraphingInequality GraphingTransformation GraphingVernier EasyData™DataMatePolynomial Root Finder and Simultaneous Equation SolverStudyCards™LearningCheck™Except for the Finance application, you can add and remove applications as space permits. TheFinance application is built into the TI-84 Plus code and cannot be deleted.Many other applications in addition to the ones mentioned above, including language localizationapplications, are included on your TI-84 Plus. Press ŒÎ to see the complete list of applicationsthat came with your calculator.You can download additional TI-84 Plus software applications from education.ti.com that allow youto customize your calculator's functionality even further. The calculator reserves 1.54 M of spacewithin ROM memory specifically for applications.Guidebooks for applications are on the Texas Instruments Web site at: education.ti.com/guides.Steps for Running the Finance ApplicationFollow these basic steps when using the Finance application.1. Press Œ Í to select the Finance application. Chapter 14: Applications 244
2. Select from list of functions.Getting Started: Financing a CarGetting Started is a fast-paced introduction. Read the chapter for details.You have found a car you would like to buy. You can afford payments of 250 per month for fouryears. The car costs 9,000. Your bank offers an interest rate of 5%. What will your payments be?Can you afford it?1. Press z † ~ ~ ~ Í to set the fixed-decimal mode setting to 2. (The TI-84 Plus will display all numbers with two decimal places.)2. Press Œ Í to select 1:Finance from the APPLICATIONS menu.3. Press Í to select 1:TVM Solver from the CALC VARS menu. The TVM Solver is displayed.4. Enter the data: N (number of payments)= 48 I% (interest rate)=5 PV (present value)=9000 FV (future value)=0 P/Y (payments per year)=12 C/Y (compounding periods per year)=125. Select PMT:END, which indicates that payments are due at the end of each period.6. Move the cursor to PMT and press ƒ . Can you afford the payment? Chapter 14: Applications 245
Getting Started: Computing Compound InterestAt what annual interest rate, compounded monthly, will 1,250 accumulate to 2,000 in 7 years?Note: Because there are no payments when you solve compound interest problems, PMT must beset to 0 and P/Y must be set to 1.1. Press Œ Í to select 1:Finance from the APPLICATIONS menu.2. Press Í to select 1:TVM Solver from the CALC VARS menu. The TVM Solver is displayed.3. Enter the data: N=7 PV=M1250 PMT=0 FV=2000 P/Y=1 C/Y=124. Move the curstor to æ and press ƒ . YYou need to look for an interest rate of 6.73% to grow 1250 to 2000 in 7 years.Using the TVM SolverUsing the TVM SolverThe TVM Solver displays the time-value-of-money (TVM) variables. Given four variable values,the TVM Solver solves for the fifth variable.The FINANCE VARS menu section describes the five TVM variables (Ú, æ, PV, PMT, and FV) andP/Y and C/Y.PMT: END BEGIN in the TVM Solver corresponds to the FINANCE CALC menu items Pmt_End(payment at the end of each period) and Pmt_Bgn (payment at the beginning of each period).To solve for an unknown TVM variable, follow these steps.1. Press Œ Í Í to display the TVM Solver. The screen below shows the default values with the fixed-decimal mode set to two decimal places. Chapter 14: Applications 246
2. Enter the known values for four TVM variables. Note: Enter cash inflows as positive numbers and cash outflows as negative numbers.3. Enter a value for P/Y, which automatically enters the same value for C/Y; if P/Y ƒ C/Y, enter a unique value for C/Y.4. Select END or BEGIN to specify the payment method.5. Place the cursor on the TVM variable for which you want to solve.6. Press ƒ . The answer is computed, displayed in the TVM Solver, and stored to the appropriate TVM variable. An indicator square in the left column designates the solution variable.Using the Financial FunctionsEntering Cash Inflows and Cash OutflowsWhen using the TI-84 Plus financial functions, you must enter cash inflows (cash received) aspositive numbers and cash outflows (cash paid) as negative numbers. The TI-84 Plus follows thisconvention when computing and displaying answers.FINANCE CALC MenuTo display the FINANCE CALC menu, press ÎŒ Í.CALC VARS1: TVM Solver... Displays the TVM Solver.2: tvm_Pmt Computes the amount of each payment.3: tvm_¾æ Computes the interest rate per year.4: tvm_PV Computes the present value.5: tvm_òÚ Computes the number of payment periods.6: tvm_FV Computes the future value.7: npv( Computes the net present value. Chapter 14: Applications 247
CALC VARS8: irr( Computes the internal rate of return.9: bal( Computes the amortization sched. balance.0: GPrn( Computes the amort. sched. princ. sum.A: GInt( Computes the amort. sched. interest sum.B: 4Nom( Computes the nominal interest rate.C: 4Eff( Computes the effective interest rate.D: dbd( Calculates the days between two dates.E: Pmt_End Selects ordinary annuity (end of period).F: Pmt_Bgn Selects annuity due (beginning of period).Use these functions to set up and perform financial calculations on the home screen.TVM SolverTVM Solver displays the TVM Solver.Calculating Time Value of Money (TVM)Calculating Time Value of MoneyUse time-value-of-money (TVM) functions (menu items 2 through 6) to analyze financialinstruments such as annuities, loans, mortgages, leases, and savings.Each TVM function takes zero to six arguments, which must be real numbers. The values that youspecify as arguments for TVM functions are not stored to the TVM variables.Note: To store a value to a TVM variable, use the TVM Solver or use ¿ and any TVM variable onthe FINANCE VARS menu.If you enter less than six arguments, the TI-84 Plus substitutes a previously stored TVM variablevalue for each unspecified argument.If you enter any arguments with a TVM function, you must place the argument or arguments inparentheses. Chapter 14: Applications 248
tvm_Pmttvm_Pmt computes the amount of each payment.tvm_Pmt[(òÚ,¾æ,PV,FV,P/Y,C/Y)]Note: In the example above, the values are stored to the TVM variables in the TVM Solver. Thepayment (tvm_Pmt) is computed on the home screen using the values in the TVM Solver. Next, theinterest rate is changed to 9.5 to illustrate the effect on the payment amount.tvm_I%tvm_æ computes the annual interest rate.tvm_¾æ [(Ú,PV,PMT,FV,P/Y,C/Y)] ClassicMathPrint™tvm_PVtvm_PV computes the present value.tvm_PV[(Ú,¾æ,PMT,FV,P/Y,C/Y)]MathPrint™ Classictvm_Ntvm_Ú computes the number of payment periods. Chapter 14: Applications 249
tvm_Ú[(æ¾,PV,PMT,FV,P/Y,C/Y)]MathPrint™ Classictvm_FVtvm_FV computes the future value.tvm_FV[(Ú,¾æ,PV,PMT,P/Y,C/Y)]MathPrint™ ClassicCalculating Cash FlowsCalculating a Cash FlowUse the cash flow functions (menu items 7 and 8) to analyze the value of money over equal timeperiods. You can enter unequal cash flows, which can be cash inflows or outflows. The syntaxdescriptions for npv( and irr( use these arguments.• interest rate is the rate by which to discount the cash flows (the cost of money) over one period.• CF0 is the initial cash flow at time 0; it must be a real number.• CFList is a list of cash flow amounts after the initial cash flow CF0.• CFFreq is a list in which each element specifies the frequency of occurrence for a grouped (consecutive) cash flow amount, which is the corresponding element of CFList. The default is 1; if you enter values, they must be positive integers < 10,000.For example, express this uneven cash flow in lists.2000 2000 2000 4000 4000 -3000 Chapter 14: Applications 250
CF0 = 2000CFList = {2000,L3000,4000}CFFreq = {2,1,2}npv(, irr(npv( (net present value) is the sum of the present values for the cash inflows and outflows. Apositive result for npv indicates a profitable investment.npv(interest rate,CF0,CFList[,CFFreq])irr( (internal rate of return) is the interest rate at which the net present value of the cash flows isequal to zero.irr(CF0,CFList[,CFFreq]) 1000 0 5000 3000-2000 -2500Calculating AmortizationCalculating an Amortization ScheduleUse the amortization functions (menu items 9, 0, and A) to calculate balance, sum of principal, andsum of interest for an amortization schedule.bal(bal( computes the balance for an amortization schedule using stored values for æ, PV, and PMT.npmt is the number of the payment at which you want to calculate a balance. It must be a positiveinteger < 10,000. roundvalue specifies the internal precision the calculator uses to calculate thebalance; if you do not specify roundvalue, then the TI-84 Plus uses the current Float/Fix decimal-mode setting. Chapter 14: Applications 251
bal(npmt[,roundvalue])GPrn(, GInt(GPrn( computes the sum of the principal principal; if you do not specify roundvalue, the TI-84 Plus uses thecurrent Float/Fix decimal-mode setting.Note: You must enter values for æ, PV, PMT, and before computing the principal.GPrn(pmt1,pmt2[,roundvalue])GInt( computes the sum of the interest interest; if you do not specify roundvalue, the TI-84 Plus uses thecurrent Float/Fix decimal-mode setting.GInt(pmt1,pmt2[,roundvalue])Amortization Example: Calculating an Outstanding Loan BalanceYou want to buy a home with a 30-year mortgage at 8 percent APR. Monthly payments are 800.Calculate the outstanding loan balance after each payment and display the results in a graph andin the table.1. Press z. Press † ~ ~ ~ Í to set the fixed-decimal mode setting to 2. Press † † ~ Í to select Par graphing mode.2. Press Î Œ Í Í to display the TVM Solver. Chapter 14: Applications 252
3. Press 360 to enter number of payments. Press † 8 to enter the interest rate. Press † † Ì 800 to enter the payment amount. Press † 0 to enter the future value of the mortgage. Press † 12 to enter the payments per year, which also sets the compounding periods per year to 12. Press † † Í to select PMT:END.4. Move the cursor to the PV prompt and then press ƒ to solve for the present value.5. Press o to display the parametric Y= editor. Turn off all stat plots. Press " to define X1T as T. Press † Œ Í 9 " ¤ to define Y1T as bal(T).6. Press p to display the window variables. Enter the values below. Tmin=0 Xmin=0 Ymin=0 Tmax=360 Xmax=360 Ymax=125000 Tstep=12 Xscl=50 Yscl=100007. Press r to draw the graph and activate the trace cursor. Press ~ and | to explore the graph of the outstanding balance over time. Press a number and then press Í to view the balance at a specific time T.8. Press y - and enter the values below. TblStart=0 @Tbl=129. Press y 0 to display the table of outstanding balances (Y1T).10. Press z and select G-T split-screen mode, so that the graph and table are displayed simultaneously. Press r to display X1T (time) and Y1T (balance) in the table. Chapter 14: Applications 253
Calculating Interest ConversionCalculating an Interest ConversionUse the interest conversion functions (menu items B and C) to convert interest rates from anannual effective rate to a nominal rate (4Nom( ) or from a nominal rate to an annual effective rate(4Eff( ).4Nom(4Nom( computes the nominal interest rate. effective rate and compounding periods must be realnumbers. compounding periods must be >0.4Nom(effective rate,compounding periods)4Eff(4Eff( computes the effective interest rate. nominal rate and compounding periods must be real numbers.compounding periods must be >0.4Eff(nominal rate,compounding periods)Finding Days between Dates/Defining Payment Methoddbd(Use the date function dbd( (menu item D) to calculate the number of days between two dates usingthe actual-day-count method. date1 and date2 can be numbers or lists of numbers within the rangeof the dates on the standard calendar.Note: Dates must be between the years 1950 through 2049.dbd(date1,date2)You can enter date1 and date2 in either of two formats.• MM.DDYY (United States)• DDMM.YY (Europe) Chapter 14: Applications 254
The decimal placement differentiates the date formats.MathPrint™ ClassicDefining the Payment MethodPmt_End and Pmt_Bgn (menu items E and F) specify a transaction as an ordinary annuity or anannuity due. When you execute either command, the TVM Solver is updated.Pmt_EndPmt_End (payment end) specifies an ordinary annuity, where payments occur at the end of eachpayment period. Most loans are in this category. Pmt_End is the default.Pmt_EndOn the TVM Solver's PMT:END BEGIN line, select END to set PMT to ordinary annuity.Pmt_BgnPmt_Bgn (payment beginning) specifies an annuity due, where payments occur at the beginning ofeach payment period. Most leases are in this category.Pmt_BgnOn the TVM Solver's PMT:END BEGIN line, select BEGIN to set PMT to annuity due.Using the TVM VariablesFINANCE VARS MenuTo display the FINANCE VARS menu, press Œ Í ~. You can use TVM variables in TVMfunctions and store values to them on the home screen.CALC VARS1: Ú Total number of payment periods2: æ Annual interest rate3: PV Present value4: PMT Payment amount5: FV Future value6: P/Y Number of payment periods per year Chapter 14: Applications 255
CALC VARS7: C/Y Number of compounding periods/yearN, I%, PV, PMT, FVÚ, æ, PV, PMT, and FV are the five TVM variables. They represent the elements of commonfinancial transactions, as described in the table above. æ is an annual interest rate that isconverted to a per-period rate based on the values of P/Y and C/Y.P/Y and C/YP/Y is the number of payment periods per year in a financial transaction.C/Y is the number of compounding periods per year in the same transaction.When you store a value to P/Y, the value for C/Y automatically changes to the same value. To storea unique value to C/Y, you must store the value to C/Y after you have stored a value to P/Y.The EasyData™ ApplicationThe Vernier EasyData™ application by Vernier Software & Technology allows you to view andanalyze real-world data when the TI-84 Plus is connected to data collection devices such as TexasInstruments CBR 2é, CBL 2é, Vernier LabProê, Vernier USB sensors, Vernier Go!éMotion, orVernier Motion Detector Unit. The TI-84 Plus comes with the EasyData™ App already installed.Note: The application will only work with Vernier auto-ID sensors when using CBL 2é andVernier LabProê.The EasyData™ App will autolaunch on your TI-84 Plus if you plug in a USB sensor such as theCBR 2é or Vernier USB Temperature sensor.Steps for Running the EasyData™ AppFollow these basic steps when using the EasyData™ App. Chapter 14: Applications 256
Starting the EasyData™ App1. Attach your data collection device to your TI-84 Plus. Make sure the cables are firmly connected.2. If the EasyData™ App has not auto-launched, press Œ and the } or † to select the EasyData™ App.3. Press Í. The EasyData™ information screen is displayed for about three seconds followed by the main screen.Quitting the EasyData™ App1. To quit the EasyData™ App, select Quit (press s). The Ready to quit? screen is displayed, which indicates that the collected data has been transferred to lists L1 through L4 on the TI-84 Plus.2. Press OK (press s) to quit.EasyData™ SettingsChanging EasyData™ settingsThe EasyData™ App displays the most commonly used settings before data collection begins.To change a predefined setting:1. From the main screen in the EasyData™ App, choose Setup and select 2: Time Graph. The current settings are displayed on the calculator. Note: If using a motion detector, settings for 3: Distance Match and 4: Ball Bounce in the Setup menu are preset and cannot be changed.2. Select Next (press q) to move to the setting you want to change. Press ' to clear a setting.3. Repeat to cycle through the available options. When the option is correct, select Next to move to the next option.4. To change a setting, enter 1 or 2 digits, and then select Next (press q).5. When all the settings are correct, select OK (press s) to return to the main menu.6. Select Start (press q) to begin collecting data.Restoring the EasyData™ App to the default settingsThe default settings are appropriate for a wide variety of sampling situations. If you are unsure ofthe best settings, begin with the default settings, and then adjust the settings for your specificactivity.To restore the default settings in the EasyData™ App while a data collection device is connectedto the TI-84 Plus, choose File and select 1:New. Chapter 14: Applications 257
Starting and Stopping Data CollectionStarting Data CollectionTo start sampling, select Start (press q). Sampling will automatically stop when the number ofsamples set in the Time Graph Settings menu is reached. The TI-84 Plus will then display a graphof the sampled data.Stopping Data CollectionTo stop sampling before it automatically stops, select Stop (press and hold q) at any timeduring the sampling process. When sampling stops, a graph of the sampled data is displayed.Saving Collected DataCollected data is automatically transferred to the TI-84 Plus and stored in lists L1 through L4 whendata collection is complete. When you exit the EasyData™ App, a prompt reminds you of the listsin which time, distance, velocity, and acceleration are stored.This manual describes basic operation for the EasyData2™ application. For more informationabout the EasyData2™ App, visit Chapter 14: Applications 258
Chapter 15:CATALOG, Strings, Hyperbolic FunctionsBrowsing the TI-84 Plus CATALOGWhat Is the CATALOG?The CATALOG is an alphabetical list of all functions and instructions on the TI-84 Plus. You alsocan access each CATALOG item from a menu or the keyboard, except:• The six string functions• The six hyperbolic functions• The solve( instruction without the equation solver editor (Chapter 2)• The inferential stat functions without the inferential stat editors (Chapter 13)Note: The only CATALOG programming commands you can execute from the home screen areGetCalc(, Get(, and Send(.Selecting an Item from the CATALOGTo select a CATALOG item, follow these steps.1. Press y N to display the CATALOG. The 4 in the first column is the selection cursor.2. Press † or } to scroll the CATALOG until the selection cursor points to the item you want. • To jump to the first item beginning with a particular letter, press that letter; alpha-lock is on. • Items that begin with a number are in alphabetical order according to the first letter after the number. For example, 2-PropZTest( is among the items that begin with the letter P. • Functions that appear as symbols, such as +, L1, <, and ‡(, follow the last item that begins with Z. To jump to the first symbol, !, press [q].3. Press Í to paste the item to the current screen. Chapter 15: CATALOG, Strings, Hyperbolic Functions 259
Note:• From the top of the CATALOG menu, press } to move to the bottom. From the bottom, press † to move to the top.• When your TI-84 Plus is in MathPrint™ mode, many functions will paste the MathPrint™ template on the home screen. For example, abs( pastes the absolute value template on the home screen instead of abs(. MathPrint™ ClassicEntering and Using StringsWhat Is a String?A string is a sequence of characters that you enclose within quotation marks. On the TI-84 Plus, astring has two primary applications.• It defines text to be displayed in a program.• It accepts input from the keyboard in a program.Characters are the units that you combine to form a string.• Each number, letter, and space counts as one character.• Each instruction or function name, such as sin( or cos(, counts as one character; the TI-84 Plus interprets each instruction or function name as one character.Entering a StringTo enter a string on a blank line on the home screen or in a program, follow these steps.1. Press ƒ [ã] to indicate the beginning of the string.2. Enter the characters that comprise the string. • Use any combination of numbers, letters, function names, or instruction names to create the string. • To enter a blank space, press ƒ O. • To enter several alpha characters in a row, press y 7 to activate alpha-lock.3. Press ƒ [ã] to indicate the end of the string. ãstringã4. Press Í. On the home screen, the string is displayed on the next line without quotations. An ellipsis (...) indicates that the string continues beyond the screen. To scroll to see the entire string, press ~ and |. Chapter 15: CATALOG, Strings, Hyperbolic Functions 260
Note: A string must be enclosed in quotation marks. The quotation marks do not count as stringcharacters.Storing Strings to String VariablesString VariablesThe TI-84 Plus has 10 variables to which you can store strings. You can use string variables withstring functions and instructions.To display the VARS STRING menu, follow these steps.1. Press to display the VARS menu. Move the cursor to 7:String.2. Press Í to display the STRING secondary menu.Storing a String to a String VariableTo store a string to a string variable, follow these steps.1. Press ƒ [ã], enter the string, and press ƒ [ã].2. Press ¿.3. Press 7 to display the VARS STRING menu.4. Select the string variable (from Str1 to Str9, or Str0) to which you want to store the string. Chapter 15: CATALOG, Strings, Hyperbolic Functions 261
The string variable is pasted to the current cursor location, next to the store symbol (!).5. Press Í to store the string to the string variable. On the home screen, the stored string is displayed on the next line without quotation marks.Displaying the Contents of a String VariableTo display the contents of a string variable on the home screen, select the string variable from theVARS STRING menu, and then press Í. The string is displayed.String Functions and Instructions in the CATALOGDisplaying String Functions and Instructions in the CATALOGString functions and instructions are available only from the CATALOG. The table below lists thestring functions and instructions in the order in which they appear among the other CATALOGmenu items. The ellipses in the table indicate the presence of additional CATALOG items.CATALOG ... Equ4String( Converts an equation to a string. ... expr( Converts a string to an expression. ... inString( Returns a character's place number. ... length( Returns a string's character length. ... String4Equ( Converts a string to an equation. sub( Returns a string subset as a string. ... Chapter 15: CATALOG, Strings, Hyperbolic Functions 262
ConcatenationTo concatenate two or more strings, follow these steps.1. Enter string1, which can be a string or string name.2. Press Ã.3. Enter string2, which can be a string or string name. If necessary, press à and enter string3, and so on. string1+string2+string3...4. Press Í to display the strings as a single string.Selecting a String Function from the CATALOGTo select a string function or instruction and paste it to the current screen, follow the steps forselecting an item from the CATALOG.Equ4String(Equ4String( converts an equation to a string. The equation must be store in a VARS Y-VARSvariable. Yn contains the equation. Strn (from Str1 to Str9, or Str0) is the string variable to which youwant the equation to be stored.Equ4String(Yn,Strn)expr(expr( converts the character string contained in string to an expression and executes it. string can bea string or a string variable. Chapter 15: CATALOG, Strings, Hyperbolic Functions 263
expr(string)inString(inString( returns the character position in string of the first character of substring. string can be a stringor a string variable. start is an optional character position at which to start the search; the defaultis 1.inString(string,substring[,start])Note: If string does not contain substring, or start is greater than the length of string, inString( returns 0.length(length( returns the number of characters in string. string can be a string or string variable.Note: An instruction or function name, such as sin( or cos(, counts as one character.length(string)String4Equ(String4Equ( converts string into an equation and stores the equation to Yn. string can be a string orstring variable. String4Equ( is the inverse of Equ4String(.String4Equ(string,Yn) Chapter 15: CATALOG, Strings, Hyperbolic Functions 264
sub(sub( returns a string that is a subset of an existing string. string can be a string or a string variable.begin is the position number of the first character of the subset. length is the number of characters inthe subset.sub(string,begin,length)Entering a Function to Graph during Program ExecutionIn a program, you can enter a function to graph during program execution using these commands.Note: When you execute this program, enter a function to store to Y3 at the ENTRY= prompt. Chapter 15: CATALOG, Strings, Hyperbolic Functions 265
Hyperbolic Functions in the CATALOGHyperbolic FunctionsThe hyperbolic functions are available only from the CATALOG. The table below lists thehyperbolic functions in the order in which they appear among the other CATALOG menu items. Theellipses in the table indicate the presence of additional CATALOG items.CATALOG ... cosh( Hyperbolic cosine cosh-1( Hyperbolic arccosine ... sinh( Hyperbolic sine sinh-1( Hyperbolic arcsine ... tanh( Hyperbolic tangent tanh-1( Hyperbolic arctangent ...sinh(, cosh(, tanh(sinh(, cosh(, and tanh( are the hyperbolic functions. Each is valid for real numbers, expressions,and lists.sinh(value)cosh(value)tanh(value)sinh-1(, cosh-1(, tanh-1(sinh-1( is the hyperbolic arcsine function. cosh-1( is the hyperbolic arccosine function. tanh-1( is thehyperbolic arctangent function. Each is valid for real numbers, expressions, and lists. Chapter 15: CATALOG, Strings, Hyperbolic Functions 266
Chapter 16:ProgrammingGetting Started: Volume of a CylinderGetting Started is a fast-paced introduction. Read the chapter for details.A program is a set of commands that the TI-84 Plus executes sequentially, as if you had enteredthem from the keyboard. Create a program that prompts for the radius R and the height H of acylinder and then computes its volume.1. Press ~ ~ to display the PRGM NEW menu.2. Press Í to select 1:Create New. The Name= prompt is displayed, and alpha-lock is on. Press C Y L I N D E R, and then press Í to name the program CYLINDER. You are now in the program editor. The colon ( : ) in the first column of the second line indicates the beginning of a command line.3. Press ~ 2 to select 2:Prompt from the PRGM I/O menu. Prompt is copied to the command line. Press ƒ R ¢ ƒ H to enter the variable names for radius and height. Press Í.4. Press y B ƒ R ¡ ƒ H ¿ ƒ V Í to enter the expression pR 2H and store it to the variable V.5. Press ~ 3 to select 3:Disp from the PRGM I/O menu. Disp is pasted to the command line. Press y 7 [ã] V O L U M E O I S [ã] ƒ ¢ ƒ V Í to set up the program to display the text VOLUME IS on one line and the calculated value of V on the next.6. Press y 5 to display the home screen. Chapter 16: Programming 268
7. Press to display the PRGM EXEC menu. The items on this menu are the names of stored programs.8. Press Í to paste prgmCYLINDER to the current cursor location. (If CYLINDER is not item 1 on your PRGM EXEC menu, move the cursor to CYLINDER before you press Í.)9. Press Í to execute the program. Enter 1.5 for the radius, and then press Í. Enter 3 for the height, and then press Í. The text VOLUME IS, the value of V, and Done are displayed. Repeat steps 7 through 9 and enter different values for R and H.Creating and Deleting ProgramsWhat Is a Program?A program is a set of one or more command lines. Each line contains one or more instructions.When you execute a program, the TI-84 Plus performs each instruction on each command line inthe same order in which you entered them. The number and size of programs that the TI-84 Pluscan store is limited only by available memory.What Is New with Operating System 2.53MP?• Programs created with OS 2.43 and earlier should run correctly but may give unexpected results when you run them using OS 2.53MP. You should test programs created with earlier OS versions to make sure you get the desired results.• Programs can run in Classic or MathPrint™ mode.• Shortcut menus are available wherever the MATH menu can be accessed.• MathPrint™ templates are not available for programs. All input and output is in Classic format.• You can use fractions in programs, but you should test the program to make sure that you get the desired results.• The spacing of the display may be slightly different in MathPrint™ mode than in Classic mode. If you prefer the spacing in Classic mode, set the mode using a command in your program. Screen shots for the examples in this chapter were taken in Classic mode. Chapter 16: Programming 269
Creating a New ProgramTo create a new program, follow these steps.1. Press | to display the PRGM NEW menu.2. Press Í to select 1:Create New. The Name= prompt is displayed, and alpha-lock is on.3. Press a letter from A to Z or q to enter the first character of the new program name. Note: A program name can be one to eight characters long. The first character must be a letter from A to Z or q. The second through eighth characters can be letters, numbers, or q.4. Enter zero to seven letters, numbers, or q to complete the new program name.5. Press Í. The program editor is displayed.6. Enter one or more program commands.7. Press y 5 to leave the program editor and return to the home screen.Managing Memory and Deleting a ProgramTo check whether adequate memory is available for a program you want to enter:1. Press y L to display the MEMORY menu.2. Select 2:Mem Mgmt/Del to display the MEMORY MANAGEMENT/DELETE menu (Chapter 18).3. Select 7:Prgm to display the PRGM editor.The TI-84 Plus expresses memory quantities in bytes.You can increase available memory in one of two ways. You can delete one or more programs oryou can archive some programs.To increase available memory by deleting a specific program:1. Press y L and then select 2:Mem Mgmt/Del from the MEMORY menu.2. Select 7:Prgm to display the PRGM editor (Chapter 18). Chapter 16: Programming 270
3. Press } and † to move the selection cursor (4) next to the program you want to delete, and then press {. The program is deleted from memory. Note: You will receive a message asking you to confirm this delete action. Select 2:yes to continue. To leave the PRGM editor screen without deleting anything, press y 5, which displays the home screen.To increase available memory by archiving a program:4. Press y L and then select 2:Mem Mgmt/Del from the MEMORY menu.5. Select 2:Mem Mgmt/Del to display the MEM MGMT/DEL menu.6. Select 7:Prgm... to display the PRGM menu.7. Press Í to archive the program. An asterisk will appear to the left of the program to indicate it is an archived program. To unarchive a program in this screen, put the cursor next to the archived program and press Í. The asterisk will disappear. Note: Archive programs cannot be edited or executed. In order to edit or execute an archived program, you must first unarchive it.Entering Command Lines and Executing ProgramsEntering a Program Command LineYou can enter on a command line any instruction or expression that you could execute from thehome screen. In the program editor, each new command line begins with a colon. To enter morethan one instruction or expression on a single command line, separate each with a colon.Note: A command line can be longer than the screen is wide.While in the program editor, you can display and select from menus. You can return to the programeditor from a menu in either of two ways.• Select a menu item, which pastes the item to the current command line. — or —• Press '.When you complete a command line, press Í. The cursor moves to the next command line. Chapter 16: Programming 271
Programs can access variables, lists, matrices, and strings saved in memory. If a program stores anew value to a variable, list, matrix, or string, the program changes the value in memory duringexecution.You can call another program as a subroutine.Executing a ProgramTo execute a program, begin on a blank line on the home screen and follow these steps.1. Press to display the PRGM EXEC menu.2. Select a program name from the PRGM EXEC menu. prgmname is pasted to the home screen (for example, prgmCYLINDER).3. Press Í to execute the program. While the program is executing, the busy indicator is on.Last Answer (Ans) is updated during program execution. Last Entry is not updated as eachcommand is executed (Chapter 1).The TI-84 Plus checks for errors during program execution. It does not check for errors as youenter a program.Breaking a ProgramTo stop program execution, press É. The ERR:BREAK menu is displayed.• To return to the home screen, select 1:Quit.• To go where the interruption occurred, select 2:Goto.Editing ProgramsEditing a ProgramTo edit a stored program, follow these steps.1. Press ~ to display the PRGM EDIT menu.2. Select a program name from the PRGM EDIT menu. Up to the first seven lines of the program are displayed. Note: The program editor does not display a $ to indicate that a program continues beyond the screen.3. Edit the program command lines. • Move the cursor to the appropriate location, and then delete, overwrite, or insert. • Press ' to clear all program commands on the command line (the leading colon remains), and then enter a new program command. Chapter 16: Programming 272
Note: To move the cursor to the beginning of a command line, press y |; to move to the end,press y ~. To scroll the cursor down seven command lines, press ƒ †. To scroll the cursorup seven command lines, press ƒ }.Inserting and Deleting Command LinesTo insert a new command line anywhere in the program, place the cursor where you want the newline, press y 6, and then press Í. A colon indicates a new line.To delete a command line, place the cursor on the line, press ' to clear all instructions andexpressions on the line, and then press { to delete the command line, including the colon.Copying and Renaming ProgramsCopying and Renaming a ProgramTo copy all command lines from one program into a new program, follow steps 1 through 5 forCreating a New Program, and then follow these steps.1. Press y K. Rcl is displayed on the bottom line of the program editor in the new program (Chapter 1).2. Press | to display the PRGM EXEC menu.3. Select a name from the menu. prgmname is pasted to the bottom line of the program editor.4. Press Í. All command lines from the selected program are copied into the new program.Copying programs has at least two convenient applications.• You can create a template for groups of instructions that you use frequently.• You can rename a program by copying its contents into a new program.Note: You also can copy all the command lines from one existing program to another existingprogram using RCL.Scrolling the PRGM EXEC and PRGM EDIT MenusThe TI-84 Plus sorts PRGM EXEC and PRGM EDIT menu items automatically into alphanumericalorder. Each menu only labels the first 10 items using 1 through 9, then 0.To jump to the first program name that begins with a particular alpha character or q, press ƒ[letter from A to Z or q].Note: From the top of either the PRGM EXEC or PRGM EDIT menu, press } to move to the bottom.From the bottom, press † to move to the top. To scroll the cursor down the menu seven items,press ƒ †. To scroll the cursor up the menu seven items, press ƒ }. Chapter 16: Programming 273
PRGM CTL (Control) InstructionsPRGM CTL MenuTo display the PRGM CTL (program control) menu, press from the program editor only.CTL I/O EXEC1: If Creates a conditional test.2: Then Executes commands when If is true.3: Else Executes commands when If is false.4: For( Creates an incrementing loop.5: While Creates a conditional loop.6: Repeat Creates a conditional loop.7: End Signifies the end of a block.8: Pause Pauses program execution.9: Lbl Defines a label.0: Goto Goes to a label.A: IS>( Increments and skips if greater than.B: DS<( Decrements and skips if less than.C: Menu( Defines menu items and branches.D: prgm Executes a program as a subroutine.E: Return Returns from a subroutine.F: Stop Stops execution.G: DelVar Deletes a variable from within program.H: GraphStyle( Designates the graph style to be drawn.I: OpenLib( No longer used.J: ExecLib( No longer used.These menu items direct the flow of an executing program. They make it easy to repeat or skip agroup of commands during program execution. When you select an item from the menu, the nameis pasted to the cursor location on a command line in the program.To return to the program editor without selecting an item, press '.Controlling Program FlowProgram control instructions tell the TI-84 Plus which command to execute next in a program. If,While, and Repeat check a defined condition to determine which command to execute next.Conditions frequently use relational or Boolean tests (Chapter 2), as in: Chapter 16: Programming 274
If A<7:A+1!AorIf N=1 and M=1:Goto ZIfUse If for testing and branching. If condition is false (zero), then the command immediately following Ifis skipped. If condition is true (nonzero), then the next command is executed. If instructions can benested.:If condition:command (if true):commandProgram OutputIf-ThenThen following an If executes a group of commands if condition is true (nonzero). End identifies theend of the group of commands.:If condition:Then:command (if true):command (if true):End:commandProgram OutputIf-Then-ElseElse following If-Then executes a group of commands if condition is false (zero). End identifies the endof the group of commands.:If condition:Then:command (if true) Chapter 16: Programming 275
:command (if true):Else:command (if false):command (if false):End:commandProgram OutputNote: In OS 2.53MP, the program name displays again when you press Í to repeat theprogram.For(For( loops and increments. It increments variable from begin to end by increment. increment is optional(default is 1) and can be negative (end<begin). end is a maximum or minimum value not to beexceeded. End identifies the end of the loop. For( loops can be nested.:For(variable,begin,end[,increment]):command (while end not exceeded):command (while end not exceeded):End:commandProgram OutputWhileWhile performs a group of commands while condition is true. condition is frequently a relational test(Chapter 2). condition is tested when While is encountered. If condition is true (nonzero), the programexecutes a group of commands. End signifies the end of the group. When condition is false (zero), theprogram executes each command following End. While instructions can be nested.:While condition:command (while condition is true):command (while condition is true) Chapter 16: Programming 276
:End:commandProgram OutputRepeatRepeat repeats a group of commands until condition is true (nonzero). It is similar to While, but conditionis tested when End is encountered; therefore, the group of commands is always executed at leastonce. Repeat instructions can be nested.:Repeat condition:command (until condition is true):command (until condition is true):End:commandProgram OutputEndEnd identifies the end of a group of commands. You must include an End instruction at the end ofeach For(, While, or Repeat loop. Also, you must paste an End instruction at the end of each If-Thengroup and each If-Then-Else group.PausePause suspends execution of the program so that you can see answers or graphs. During thepause, the pause indicator is on in the top-right corner. Press Í to resume execution.• Pause without a value temporarily pauses the program. If the DispGraph or Disp instruction has been executed, the appropriate screen is displayed.• Pause with value displays value on the current home screen. value can be scrolled. Chapter 16: Programming 277
Pause [value]Program OutputLbl, GotoLbl (label) and Goto (go to) are used together for branching.Lbl specifies the label for a command. label can be one or two characters (A through Z, 0 through99, or q).Lbl labelGoto causes the program to branch to label when Goto is encountered.Goto labelProgram Output Chapter 16: Programming 278
IS>(IS>( (increment and skip) adds 1 to variable. If the answer is > value (which can be an expression),the next command is skipped; if the answer is { value, the next command is executed. variable cannotbe a system variable.:IS>(variable,value):command (if answer value):command (if answer > value)Program OutputNote: IS>( is not a looping instruction.DS<(DS<( (decrement and skip) subtracts 1 from variable. If the answer is < value (which can be anexpression), the next command is skipped; if the answer is | value, the next command is executed.variable cannot be a system variable.:DS<(variable,value):command (if answer ' value):command (if answer < value)Program OutputNote: DS<( is not a looping instruction.Menu(Menu( sets up branching within a program. If Menu( is encountered during program execution, themenu screen is displayed with the specified menu items, the pause indicator is on, and executionpauses until you select a menu item.The menu title is enclosed in quotation marks ( " ). Up to seven pairs of menu items follow. Eachpair comprises a text item (also enclosed in quotation marks) to be displayed as a menu selection,and a label item to which to branch if you select the corresponding menu selection.Menu("title","text1",label1,"text2",label2, . . .)Program Output Chapter 16: Programming 279
The program above pauses until you select 1 or 2. If you select 2, for example, the menudisappears and the program continues execution at Lbl B.prgmUse prgm to execute other programs as subroutines. When you select prgm, it is pasted to thecursor location. Enter characters to spell a program name. Using prgm is equivalent to selectingexisting programs from the PRGM EXEC menu; however, it allows you to enter the name of aprogram that you have not yet created.prgmnameNote: You cannot directly enter the subroutine name when using RCL. You must paste the namefrom the PRGM EXEC menu.ReturnReturn quits the subroutine and returns execution to the calling program, even if encounteredwithin nested loops. Any loops are ended. An implied Return exists at the end of any program thatis called as a subroutine. Within the main program, Return stops execution and returns to thehome screen.StopStop stops execution of a program and returns to the home screen. Stop is optional at the end of aprogram.DelVarDelVar deletes from memory the contents of variable.DelVar variable Chapter 16: Programming 280
GraphStyle(GraphStyle( designates the style of the graph to be drawn. function# is the number of the Y= functionname in the current graphing mode. graphstyle is a number from 1 to 7 that corresponds to thegraph style, as shown below.1 = ç (line) 5 = ë (path)2 = è (thick) 6 = ì (animate)3 = é (shade above) 7 = í (dot)4 = ê (shade below)GraphStyle(function#,graphstyle)For example, GraphStyle(1,5) in Func mode sets the graph style for Y1 to ë (path; 5).Not all graph styles are available in all graphing modes. For a detailed description of each graphstyle, see the Graph Styles table in Chapter 3.PRGM I/O (Input/Output) InstructionsPRGM I/O MenuTo display the PRGM I/O (program input/output) menu, press ~ from within the programeditor only.CTL I/O EXEC1: Input Enters a value or uses the cursor.2: Prompt Prompts for entry of variable values.3: Disp Displays text, value, or the home screen.4: DispGraph Displays the current graph.5: DispTable Displays the current table.6: Output( Displays text at a specified position.7: getKey Checks the keyboard for a keystroke.8: ClrHome Clears the display.9: ClrTable Clears the current table.0: GetCalc( Gets a variable from another TI-84 Plus.A: Get( Gets a variable from CBL 2™ or CBR™.B: Send( Sends a variable to CBL 2 or CBR.These instructions control input to and output from a program during execution. They allow you toenter values and display answers during program execution.To return to the program editor without selecting an item, press '. Chapter 16: Programming 281
Displaying a Graph with InputInput without a variable displays the current graph. You can move the free-moving cursor, whichupdates X and Y (and R and q for PolarGC format). The pause indicator is on. Press Í toresume program execution.InputProgram OutputStoring a Variable Value with InputInput with variable displays a ? (question mark) prompt during execution. variable may be a realnumber, complex number, list, matrix, string, or Y= function. During program execution, enter avalue, which can be an expression, and then press Í. The value is evaluated and stored tovariable, and the program resumes execution.Input [variable]You can display text or the contents of Strn (a string variable) of up to 16 characters as a prompt.During program execution, enter a value after the prompt and then press Í. The value isstored to variable, and the program resumes execution.Input ["text",variable]Input [Strn,variable]Program Output Chapter 16: Programming 282
Note: When a program prompts for input of lists and Yn functions during execution, you mustinclude the braces ( { } ) around the list elements and quotation marks ( " ) around theexpressions.PromptDuring program execution, Prompt displays each variable, one at a time, followed by =?. At eachprompt, enter a value or expression for each variable, and then press Í. The values are stored,and the program resumes execution.Prompt variableA[,variableB,...,variable n]Program OutputNote: Y= functions are not valid with Prompt.Displaying the Home ScreenDisp (display) without a value displays the home screen. To view the home screen during programexecution, follow the Disp instruction with a Pause instruction.DispDisplaying Values and MessagesDisp with one or more values displays the value of each.Disp [valueA,valueB,valueC,...,value n]• If value is a variable, the current value is displayed.• If value is an expression, it is evaluated and the result is displayed on the right side of the next line.• If value is text within quotation marks, it is displayed on the left side of the current display line. ! is not valid as text.Program OutputIf Pause is encountered after Disp, the program halts temporarily so you can examine the screen.To resume execution, press Í. Chapter 16: Programming 283
Note: If a matrix or list is too large to display in its entirety, ellipses (...) are displayed in the lastcolumn, but the matrix or list cannot be scrolled. To scroll, use Pause value.DispGraphDispGraph (display graph) displays the current graph. If Pause is encountered after DispGraph, theprogram halts temporarily so you can examine the screen. Press Í to resume execution.DispTableDispTable (display table) displays the current table. The program halts temporarily so you canexamine the screen. Press Í to resume execution.Output(Output( displays text or value on the current home screen beginning at row (1 through 8) and column(1 through 16), overwriting any existing characters.Note: You may want to precede Output( with ClrHome.Expressions are evaluated and values are displayed according to the current mode settings.Matrices are displayed in entry format and wrap to the next line. ! is not valid as text.Output(row,column,"text")Output(row,column,value)Program OutputFor Output( on a Horiz split screen, the maximum value for row is 4. Chapter 16: Programming 284
getKeygetKey returns a number corresponding to the last key pressed, according to the key code diagrambelow. If no key has been pressed, getKey returns 0. Use getKey inside loops to transfer control,for example, when creating video games.Program Output Note: , Œ, , and Í were pressed during program execution.Note: You can press É at any time during execution to break the program.TI-84 Plus Key Code DiagramClrHome, ClrTableClrHome (clear home screen) clears the home screen during program execution.ClrTable (clear table) clears the values in the table during program execution.GetCalc(GetCalc( gets the contents of variable on another TI-84 Plus and stores it to variable on the receivingTI-84 Plus. variable can be a real or complex number, list element, list name, matrix element, matrixname, string, Y= variable, graph database, or picture. Chapter 16: Programming 285
GetCalc(variable[,portflag])By default, the TI-84 Plus uses the USB port if it is connected. If the USB cable is not connected, ituses the I/O port. If you want to specify either the USB or I/O port, use the following portflagnumbers:portflag=0 use USB port if connected;portflag=1 use USB port;portflag=2 use I/O portNote: GetCalc( does not work between TI.82 and TI-83 Plus or a TI.82 and TI-84 Plus calculators.Get(, Send(Get( gets data from the CBL 2™ or CBR™ and stores it to variable on the receiving TI-84 Plus.variable can be a real number, list element, list name, matrix element, matrix name, string,Y= variable, graph database, or picture.Get(variable)Note: If you transfer a program that references the Get( command to the TI-84 Plus from a TI.82,the TI-84 Plus will interpret it as the Get( described above. Use GetCalc( to get data from anotherTI-84 Plus.Send( sends the contents of variable to the CBL 2™ or CBR™. You cannot use it to send to anotherTI-84 Plus. variable can be a real number, list element, list name, matrix element, matrix name,string, Y= variable, graph database, or picture. variable can be a list of elements.Send(variable) Note: This program gets sound data and time in seconds from CBL 2™.Note: You can access Get(, Send(, and GetCalc( from the CATALOG to execute them from thehome screen (Chapter 15).Calling Other Programs as SubroutinesCalling a Program from Another ProgramOn the TI-84 Plus, any stored program can be called from another program as a subroutine. Enterthe name of the program to use as a subroutine on a line by itself.You can enter a program name on a command line in either of two ways.• Press | to display the PRGM EXEC menu and select the name of the program prgmname is pasted to the current cursor location on a command line. Chapter 16: Programming 286
• Select prgm from the PRGM CTL menu, and then enter the program name.prgmnameWhen prgmname is encountered during execution, the next command that the program executes isthe first command in the second program. It returns to the subsequent command in the firstprogram when it encounters either Return or the implied Return at the end of the second program.Program OutputSubroutine ( Notes about Calling ProgramsVariables are global.label used with Goto and Lbl is local to the program where it is located. label in one program is notrecognized by another program. You cannot use Goto to branch to a label in another program.Return exits a subroutine and returns to the calling program, even if it is encountered within nestedloops.Running an Assembly Language ProgramYou can run programs written for the TI-84 Plus in assembly language. Typically, assemblylanguage programs run much faster and provide greater control than than the keystroke programsthat you write with the built-in program editor.Note: Because an assembly langauge program has greater control over the calculator, if yourassembly language program has error(s), it may cause your calculator to reset and lose all data,programs, and applications stored in memory.When you download an assembly language program, it is stored among the other programs as aPRGM menu item. You can:• Transmit it using the TI-84 Plus communication link (Chapter 19).• Delete it using the MEM MGMT DEL screen (Chapter 18).To run an assembly Program, the syntax is: Asm(assemblyprgmname) Chapter 16: Programming 287
If you write an assembly language program, use the two instructions below from the CATALOG toidentify and compile the program.Instructions CommentsAsmComp(prgmASM1, Compiles an assembly language program written inprgmASM2) ASCII and stores the hex versionAsmPrgm Identifies an assembly language program; must be entered as the first line of an assembly language programTo compile an assembly program that you have written:1. Follow the steps for writing a program (16-4) but be sure to include AsmPrgm as the first line of your program.2. From the home screen, press y N and then select AsmComp( to paste it to the screen.3. Press to display the PRGM EXEC menu.4. Select the program you want to compile. It will be pasted to the home screen.5. Press ¢ and then select prgm from the CATALOG.6. Key in the name you have chosen for the output program. Note: This name must be unique — not a copy of an existing program name.7. Press ¤ to complete the sequence. The sequence of the arguments should be as follows: AsmComp(prgmASM1, prgmASM2)8. Press Í to compile your program and generate the output program. Chapter 16: Programming 288
Chapter 17:ActivitiesThe Quadratic FormulaNote: This example uses MathPrint™ mode for real answers and Classic mode for non-real(complex) results. You can also use the Polynomial Root Finder/Simultaneous Equation Solverapplication to solve these types of problems with a quick set-up. This application comes preloadedon your TI-84 Plus and can be downloaded from education.ti.com.Use the quadratic formula to solve the quadratic equations 2x2 N 11x + 14 = 0 and2x2 N 6x + 5 = 0.Graphing the FunctionsBefore you begin, look at the graphs of the functions to see the approximate locations of thesolutions.1. Press o to display the Y= editor.2. Press 2 " ¡ ¹ 11 " Ã 14 for Y1, and then press Í.3. Press 2 " ¡ ¹ 6 " Ã 5 for Y2.4. Press q and select 4:ZDecimal. The graph of the functions displays.You can see that the graph of the firstfunction, 2x2 N 11x + 14 = 0, crosses thex-axis, so it has a real solution. The graph ofthe second function does not cross thex-axis, so it has a complex solution. Chapter 17: Activities 289
Entering a CalculationBegin with the equation 2x2 N 11x + 14 = 0.1. Press 2 ¿ ƒ A to store the coefficient of the x2 term.2. Press ƒ [:]. The colon allows you to enter more than one instruction on a line.3. Press Ì 11 ¿ ƒ B to store the coefficient of the X term. Press ƒ [:] to enter a new instruction on the same line. Press 14 ¿ ƒ C to store the constant.4. Press Í to store the values to the variables A, B, and C.5. The last value you stored is shown on the right side of the display. The cursor moves to the next line, ready for your next entry.6. Press ƒ ^ 1 Ì ƒ B Ã y C ƒ B ¡¹ 4 ƒ A ƒ C ~~2 ƒ A to enter the expression for one of the solutions for the quadratic formula, 2 – b b – 4ac -------------------------------------- 2a7. Press Í to find one solution for the equation 2x2 N 11x + 14 = 0. The answer is shown on the right side of the display. The cursor moves to the next line, ready for you to enter the next expression.Converting to a DecimalYou can show the solution as a fraction.1. Press ƒ ^ 4 to select 4F3 4D from the FRAC shortcut menu. Chapter 17: Activities 290
2. Press Í to convert the result to a decimal.To save keystrokes, you can scroll up to find an expression you entered, copy it, and then edit it fora new calculation.3. Press } to highlight and then press Í to paste it to the entry line.4. Press | until the cursor is on the + sign in the formula. Press ¹ to edit the quadratic-formula expression to become .5. Press Í to find the other solution for the quadratic equation 2x2 N 11x + 14 = 0.Displaying Complex ResultsNow solve the equation 2x2 N 6x + 5 = 0. When you set a+bi complex number mode, the TI-84Plus displays complex results.1. Press z † † † † † † (6 times), and then press ~ to highlight a+bi. Press Í to select a+bi complex-number mode.2. Press y 5 to return to the home screen, and then press ' to clear it. Chapter 17: Activities 291
3. Press 2 ¿ ƒ A ƒ [:] Ì 6 ¿ ƒ B ƒ [:] 5 ¿ ƒ C Í. The coefficient of the x2 term, the coefficient of the X term, and the constant for the new equation are stored to A, B, and C, respectively.4. Enter the quadratic formula using Classic entry: £ Ì ƒ B à y C ƒ B ¡¹4 ƒA ƒC ~¤¥£2 ƒ A ¤. Because the solution is a complex number, you have to enter the formula using the division operation instead of using the n/d shortcut template. Complex numbers are not valid in the n/d template in input or output and will cause Error: Data Type to display.5. Press Í to find one solution for the equation 2x2 N 6x + 5 = 0.6. Press } to highlight the quadratic- formula expression, and then press Í to paste it to the entry line.7. Press | until the cursor is on the + sign in the formula. Press ¹ to edit the quadratic-formula expression to become .8. Press Í to find the other solution for the quadratic equation: 2x2 N 6x + 5 = 0. Chapter 17: Activities 292
Box with LidDefining a FunctionTake a 20 cm × 25 cm. sheet of paper and cut X × X squares from two corners. Cut X × 12½ cmrectangles from the other two corners as shown in the diagram below. Fold the paper into a boxwith a lid. What value of X would give your box the maximum volume V? Use the table and graphsto determine the solution.Begin by defining a function that describesthe volume of the box. XFrom the diagram: 20 A2X + A = 202X + 2B = 25 X B X BV = A…B…X 25Substituting:V = (20 N 2X) (25à2 N X) X1. Press o to display the Y= editor, which is where you define functions for tables and graphing.2. Press £ 20 ¹ 2 " ¤ £ 25 t ^ 1 2 ~ ¹ " ¤ " Í to define the volume function as Y1 in terms of X. " lets you enter X quickly, without having to press ƒ. The highlighted = sign indicates that Y1 is selected.Defining a Table of ValuesThe table feature of the TI-84 Plus displays numeric information about a function. You can use atable of values from the function you just defined to estimate an answer to the problem.1. Press y - to display the TABLE SETUP menu.2. Press Í to accept TblStart=0.3. Press 1 Í to define the table increment @Tbl=1. Leave Indpnt: Auto and Depend: Auto so that the table will be generated automatically. Chapter 17: Activities 293
4. Press y 0 to display the table. Notice that the maximum value for Y1 (box's volume) occurs when X is about 4, between 3 and 5.5. Press and hold † to scroll the table until a negative result for Y1 is displayed. Notice that the maximum length of X for this problem occurs where the sign of Y1 (box's volume) changes from positive to negative, between 10 and 11.6. Press y -. Notice that TblStart has changed to 5 to reflect the first line of the table as it was last displayed. (In step 5, the first value of X displayed in the table is 5.)Zooming In on the TableYou can adjust the way a table is displayed to get more information about a defined function. Withsmaller values for @Tbl, you can zoom in on the table. You can change the values on the TBLSETscreen by pressing y - or by pressing à on the TABLE screen1. Press y 0.2. Press } to move the cursor to highlight 3.3. Press Ã. The @Tbl displays on the entry line.4. Enter Ë 1 Í. The table updates, showing the changes in X in increments of 0.1. Notice that the maximum value for Y1 in this table view is 410.26, which occurs at X=3.7. Therefore, the maximum occurs where 3.6<X<3.8.5. With X=3.6 highlighted, press Ã Ë 01 Í to set @Tbl=0.01. Chapter 17: Activities 294
6. Press † and } to scroll the table. Four equivalent maximum values are shown, 410.26 at X=3.67, 3.68, 3.69, and 3.70.7. Press † or } to move the cursor to 3.67. Press ~ to move the cursor into the Y1 column. The value of Y1 at X=3.67 is displayed on the bottom line in full precision as 410.261226.8. Press † to display the other maximum. The value of Y1 at X=3.68 in full precision is 410.264064, at X=3.69 is 410.262318 and at X=3.7 is 410.256. The maximum volume of the box would occur at 3.68 if you could measure and cut the paper at .01-centimeter increments.Setting the Viewing WindowYou also can use the graphing features of the TI-84 Plus to find the maximum value of a previouslydefined function. When the graph is activated, the viewing window defines the displayed portion ofthe coordinate plane. The values of the window variables determine the size of the viewingwindow.1. Press p to display the window editor, where you can view and edit the values of the window variables. The standard window variables define the viewing window as shown. Xmin, Xmax, Ymin, and Ymax define the boundaries of the display. Xscl and Yscl define the distance between tick marks on the X and Y axes. Xres controls resolution. Chapter 17: Activities 295
2. Press 0 Í to define Xmin.3. Press 20 ¥ 2 to define Xmax using an expression. Note: For this example, the division sign is used for the calculation. However, you can use n/d entry format where fraction output can be experienced, depending on mode settings.4. Press Í. The expression is evaluated, and 10 is stored in Xmax. Press Í to accept Xscl as 1.5. Press 0 Í 500 Í 100 Í 1 Í to define the remaining window variables.Displaying and Tracing the GraphNow that you have defined the function to be graphed and the window in which to graph it, you candisplay and explore the graph. You can trace along a function using the TRACE feature.1. Press s to graph the selected function in the viewing window. The graph of Y1=(20N2X)(25à2NX)X is displayed.2. Press ~ to activate the free-moving graph cursor. The X and Y coordinate values for the position of the graph cursor are displayed on the bottom line.3. Press |, ~, }, and † to move the free- moving cursor to the apparent maximum of the function. As you move the cursor, the X and Y coordinate values are updated continually.4. Press r. The trace cursor is displayed on the Y1 function. The function that you are tracing is displayed in the top-left corner.5. Press | and ~ to trace along Y1, one X dot at a time, evaluating Y1 at each X. Chapter 17: Activities 296
You also can enter your estimate for the maximum value of X.6. Press 3 Ë 8. When you press a number key while in TRACE, the X= prompt is displayed in the bottom-left corner.7. Press Í. The trace cursor jumps to the point on the Y1 function evaluated at X=3.8.8. Press | and ~ until you are on the maximum Y value. This is the maximum of Y1(X) for the X pixel values. The actual, precise maximum may lie between pixel values.Zooming In on the GraphTo help identify maximums, minimums, roots, and intersections of functions, you can magnify theviewing window at a specific location using the ZOOM instructions.1. Press q to display the ZOOM menu. This menu is a typical TI-84 Plus menu. To select an item, you can either press the number or letter next to the item, or you can press † until the item number or letter is highlighted, and then press Í.2. Press 2 to select 2:Zoom In. The graph is displayed again. The cursor has changed to indicate that you are using a ZOOM instruction.3. With the cursor near the maximum value of the function, press Í. The new viewing window is displayed. Both XmaxNXmin and YmaxNYmin have been adjusted by factors of 4, the default values for the zoom factors.4. Press | and ~ to search for the maximum value. Chapter 17: Activities 297
5. Press p to display the new window settings. Note: To return to the previous graph, press q ~ 1:ZPrevious.Finding the Calculated MaximumYou can use a CALCULATE menu operation to calculate a local maximum of a function. To do this,pick a point to the left of where you think the maximum is on the graph. This is called the leftbound. Next, pick a point to the right of the maximum. This is called the right bound. Finally, guessthe maximum by moving the cursor to a point between the left and right bounds. With thisinformation, the maximum can be calculated by the methods programmed in the TI-84 Plus.1. Press y / to display the CALCULATE menu. Press 4 to select 4:maximum. The graph is displayed again with a Left Bound? prompt.2. Press | to trace along the curve to a point to the left of the maximum, and then press Í. A 4 at the top of the screen indicates the selected bound. A Right Bound? prompt is displayed.3. Press ~ to trace along the curve to a point to the right of the maximum, and then press Í. A 3 at the top of the screen indicates the selected bound. A Guess? prompt is displayed.4. Press | to trace to a point near the maximum, and then press Í. Chapter 17: Activities 298
Or, press 3 Ë 8, and then press Í toenter a guess for the maximum.When you press a number key in TRACE,the X= prompt is displayed in the bottom-left corner.Notice how the values for the calculatedmaximum compare with the maximumsfound with the free-moving cursor, thetrace cursor, and the table.Note: In steps 2 and 3 above, you canenter values directly for Left Bound andRight Bound, in the same way asdescribed in step 4. Chapter 17: Activities 299
Comparing Test Results Using Box PlotsProblemAn experiment found a significant difference between boys and girls pertaining to their ability toidentify objects held in their left hands, which are controlled by the right side of their brains, versustheir right hands, which are controlled by the left side of their brains. The TI Graphics teamconducted a similar test for adult men and women.The test involved 30 small objects, which participants were not allowed to see. First, they held 15of the objects one by one in their left hands and guessed what they were. Then they held the other15 objects one by one in their right hands and guessed what they were. Use box plots to comparevisually the correct-guess data from this table.Each row in the table represents the results observed for one subject. Note that 10 women and 12men were tested. Correct Guesses Women Women Men Men Left Right Left Right 8 4 7 12 9 1 8 6 12 8 7 12 11 12 5 12 10 11 7 7 8 11 8 11 12 13 11 12 7 12 4 8 9 11 10 12 11 12 14 11 13 9 5 9Procedure1. Press … 5 to select 5:SetUpEditor. Enter list names WLEFT, WRGHT, MLEFT, and MRGHT, separated by commas. Press Í. The stat list editor now contains only these four lists. (See Chapter 11: Lists for detailed instructions for using the SetUpEditor.)2. Press … 1 to select 1:Edit.3. Enter into WLEFT the number of correct guesses each woman made using her left hand (Women Left). Press ~ to move to WRGHT and enter the number of correct guesses each woman made using her right hand (Women Right).4. Likewise, enter each man's correct guesses in MLEFT (Men Left) and MRGHT (Men Right). Chapter 17: Activities 300
5. Press y ,. Select 1:Plot1. Turn on plot 1; define it as a modified box plot Õ that uses Xlist as WLEFT. Move the cursor to the top line and select Plot2. Turn on plot 2; define it as a modified box plot that uses Xlist as WRGHT. (See Chapter 12: Statistics for detailed information on using Stat Plots.)6. Press o. Turn off all functions.7. Press p. Set Xscl=1 and Yscl=0. Press q 9 to select 9:ZoomStat. This adjusts the viewing window and displays the box plots for the women's results.8. Press r. Women's left-hand data Women's right-hand data Use | and ~ to examine minX, Q1, Med, Q3, and maxX for each plot. Notice the outlier to the women's right-hand data. What is the median for the left hand? For the right hand? With which hand were the women more accurate guessers, according to the box plots?9. Examine the men's results. Redefine plot 1 to use MLEFT, redefine plot 2 to use MRGHT. Press r. Men's left-hand data Men's right-hand data Press | and ~ to examine minX, Q1, Med, Q3, and maxX for each plot. What difference do you see between the plots?10. Compare the left-hand results. Redefine plot 1 to use WLEFT, redefine plot 2 to use MLEFT, and then press r to examine minX, Q1, Med, Q3, and maxX for each plot. Who were the better left-hand guessers, men or women?11. Compare the right-hand results. Define plot 1 to use WRGHT, define plot 2 to use MRGHT, and then press r to examine minX, Q1, Med, Q3, and maxX for each plot. Who were the better right-hand guessers? In the original experiment boys did not guess as well with right hands, while girls guessed equally well with either hand. This is not what our box plots show for adults. Do you think that this is because adults have learned to adapt or because our sample was not large enough? Chapter 17: Activities 301
Graphing Piecewise FunctionsProblemThe fine for speeding on a road with a speed limit of 45 kilometers per hour (kph) is 50; plus 5 foreach kph from 46 to 55 kph; plus 10 for each kph from 56 to 65 kph; plus 20 for each kph from 66kph and above. Graph the piecewise function that describes the cost of the ticket.The fine (Y) as a function of kilometers per hour (X) is: ,which simplifies to:Procedure1. Press z. Select Func and Classic.2. Press o. Turn off all functions and stat plots. Enter the Y= function to describe the fine. Use the TEST menu operations to define the piecewise function. Set the graph style for Y1 to í (dot).3. Press p and set Xmin=L2, Xscl=10, Ymin=L5, Yscl=10 and @X=1. Ignore Xmax and Ymax; they are set in step 4. Chapter 17: Activities 302
4. Press y 5 to return to the home screen. Store 5 to @Y. @X and @Y are on the VARS Window X/Y secondary menu. @X and @Y specify the horizontal and vertical distance between the centers of adjacent pixels. Integer values for @X and @Y produce nice values for tracing.5. Press r to plot the function. At what speed does the ticket exceed 250? Chapter 17: Activities 303
Graphing InequalitiesProblemGraph the inequality 0.4x3 N 3x + 5 < 0.2x + 4. Use the TEST menu operations to explore the valuesof X where the inequality is true and where it is false.Note: You can also investigate graphing inequalities using the Inequality Graphing application. Theapplication is pre-loaded on your TI-84 Plus and can be downloaded from education.ti.com.Procedure1. Press z. Select Dot, Simul, and the default settings. Setting Dot mode changes all graph style icons to í (dot) in the Y= editor.2. Press o. Turn off all functions and stat plots. Enter the left side of the inequality as Y4 and the right side as Y5.3. Enter the statement of the inequality as Y6. This function evaluates to 1 if true or 0 if false. Note: You can use the YVARS shortcut menu to paste Y4 and Y5 in the Y= editor.4. Press q 6 to graph the inequality in the standard window.5. Press r † † to move to Y6. Then press | and ~ to trace the inequality, observing the value of Y. When you trace, you can see that Y=1 indicates that Y4<Y5 is true and that Y=0 indicates that Y4<Y5 is false.6. Press o. Turn off Y4, Y5, and Y6. Enter equations to graph only the inequality. Chapter 17: Activities 304
7. Press r. Notice that the values of Y7 and Y8 are zero where the inequality is false. You only see the intervals of the graph where Y4<Y5 because intervals that are false are multiplied by 0 (Y6†Y4 and Y6†Y5) Chapter 17: Activities 305
Solving a System of Nonlinear EquationsProblemUsing a graph, solve the equation x3N2x=2cos(x). Stated another way, solve the system of twoequations and two unknowns: y = x 3N2x and y = 2cos(x). Use ZOOM factors to control the decimalplaces displayed on the graph and use y / 5:intersect to find an approximate solution.Procedure1. Press z. Select the default mode settings. Press o. Turn off all functions and stat plots. Enter the functions.2. Press q 4 to select 4:ZDecimal. The display shows that two solutions may exist (points where the two functions appear to intersect).3. Press q ~ 4 to select 4:SetFactors from the ZOOM MEMORY menu. Set XFact=10 and YFact=10.4. Press q 2 to select 2:Zoom In. Use |, ~, }, and † to move the free-moving cursor onto the apparent intersection of the functions on the right side of the display. As you move the cursor, notice that the X and Y values have one decimal place.5. Press Í to zoom in. Move the cursor over the intersection. As you move the cursor, notice that now the X and Y values have two decimal places.6. Press Í to zoom in again. Move the free-moving cursor onto a point exactly on the intersection. Notice the number of decimal places.7. Press y / 5 to select 5:intersect. Press Í to select the first curve and Í to select the second curve. To guess, move the trace cursor near the intersection. Press Í. What are the coordinates of the intersection point?8. Press q 4 to select 4:ZDecimal to redisplay the original graph.9. Press q. Select 2:Zoom In and repeat steps 4 through 8 to explore the apparent function intersection on the left side of the display. Chapter 17: Activities 306
Using a Program to Create the Sierpinski TriangleSetting up the ProgramThis program creates a drawing of a famous fractal, the Sierpinski Triangle, and stores the drawingto a picture. To begin, press ~ ~ 1. Name the program SIERPINS, and then press Í.The program editor is displayed.Note: After you run this program, press y . † † † Í to turn on the axes in the graphscreen.ProgramPROGRAM:SIERPINS:FnOff :ClrDraw:PlotsOff:AxesOff:0!Xmin:1!Xmax Set viewing window.:0!Ymin:1!Ymax:rand!X:rand!Y:For(K,1,3000) Beginning of For group.:rand!N:If N1 à3:Then:.5X!X If/Then group:.5Y!Y:End:If 1 à3 <N and N2 à3:Then:.5(.5+X)!X If/Then group.:.5(1+Y)!Y:End:If 2 à3 <N:Then:.5(1+X)!X If/Then group.:.5Y!Y:End:Pt-On(X,Y) Draw point.:End End of For group.:StorePic 6 Store picture.After you execute the program above, you can recall and display the picture with the instructionRecallPic 6. Chapter 17: Activities 307
Chapter 17: Activities 308
Graphing Cobweb AttractorsProblemUsing Web format, you can identify points with attracting and repelling behavior in sequencegraphing.Procedure1. Press z. Select Seq and the default mode settings. Press y .. Select Web format and the default format settings.2. Press o. Clear all functions and turn off all stat plots. Enter the sequence that corresponds to the expression Y = K X(1NX). u(n)=Ku(nN1)(1Nu(nN1)) u(nMin)=.013. Press y 5 to return to the home screen, and then store 2.9 to K.4. Press p. Set the window variables. nMin=0 Xmin=0 Ymin=M.26 nMax=10 Xmax=1 Ymax=1.1 PlotStart=1 Xscl=1 Yscl=1 PlotStep=15. Press r to display the graph, and then press ~ to trace the cobweb. This is a cobweb with one attractor.6. Change K to 3.44 and trace the graph to show a cobweb with two attractors.7. Change K to 3.54 and trace the graph to show a cobweb with four attractors. Chapter 17: Activities 309
Using a Program to Guess the CoefficientsSetting Up the ProgramThis program graphs the function A sin(BX) with random integer coefficients between 1 and 10.Try to guess the coefficients and graph your guess as C sin(DX). The program continues until yourguess is correct.Note: This program changes the graph window and graph styles. After you run the program, youcan change individual settings as needed or you can press y L 7 2 2 to return to defaultsettings.Programs typically do not restore your settings in MODE, Y=, WINDOW and other locations thatwere used by the program. This is dependent on who created the program.ProgramPROGRAM:GUESS:PlotsOff :Func:FnOff :Radian:ClrHome:"Asin(BX)"!Y1 Define equations.:"Csin(DX)"!Y2:GraphStyle(1,1) Set line and path graph styles.:GraphStyle(2,5):FnOff 2:randInt(1,10)!A:randInt(1,10)!B Initialize coefficients.:0!C:0!D:L2p!Xmin:2p!Xmax:pà2!Xscl:L10!Ymin Set viewing window.:10!Ymax:1!Yscl:DispGraph:Pause Display graph.:FnOn 2:Lbl Z:Prompt C,D Prompt for guess.:DispGraph:Pause Display graph. Chapter 17: Activities 310
:If C=A:Text(1,1,"C IS OK"):If CƒA:Text(1,1,"C IS Display results.WRONG"):If D=B:Text(1,50,"D IS OK"):If DƒB:Text(1,50,"D ISWRONG"):DispGraph:Pause Display graph.:If C=A and D=B:Stop Quit if guesses are correct.:Goto ZNote: The Guess My Coefficients App is an educational game that challenges you to enter thecorrect coeffiecients for graphs of linear, quadratic and absolute value functions. This app isavailable at education.ti.com. Chapter 17: Activities 311
Graphing the Unit Circle and Trigonometric CurvesProblemUsing parametric graphing mode, graph the unit circle and the sine curve to show the relationshipbetween them.Any function that can be plotted in Func mode can be plotted in Par mode by defining the Xcomponent as T and the Y component as F(T).Procedure1. Press z. Select Par, Simul, and the default settings.2. Press p. Set the viewing window. Tmin=0 Xmin=L2 Ymin=L3 Tmax=2p Xmax=7.4 Ymax=3 Tstep=.1 Xscl=pà2 Yscl=13. Press o. Turn off all functions and stat plots. Enter the expressions to define the unit circle centered on (0,0).4. Enter the expressions to define the sine curve.5. Press r. As the graph is plotting, you may press Í to pause and Í again to resume graphing as you watch the sine function "unwrap" from the unit circle.Note:• You can generalize the unwrapping. Replace sin(T) in Y2T with any other trig function to unwrap that function. Chapter 17: Activities 312
• You can graph the functions again by turning the functions off and then turning them back on on the Y= editor or by using the FuncOFF and FuncON commands on the home screen. Chapter 17: Activities 313
Finding the Area between CurvesProblemFind the area of the region bounded by:f(x) = 300x / (x2 + 625)g(x) = 3cos(.1x)x = 75Procedure1. Press z. Select the default mode settings.2. Press p. Set the viewing window. Xmin=0 Ymin=L5 Xres=1 Xmax=100 Ymax=10 Xscl=10 Yscl=13. Press o. Turn off all functions and stat plots. Enter the upper and lower functions. Y1=300Xà(X2+625) Y2=3cos(.1X)4. Press y / 5 to select 5:Intersect. The graph is displayed. Select a first curve, second curve, and guess for the intersection toward the left side of the display. The solution is displayed, and the value of X at the intersection, which is the lower limit of the integral, is stored in Ans and X.5. Press y 5 to go to the home screen. Press y < 7 and use Shade( to see the area graphically. Shade(Y2,Y1,Ans,75)6. Press y 5 to return to the home screen. Enter the expression to evaluate the integral for the shaded region. fnInt(Y1NY2,X,Ans,75) The area is 325.839962. Chapter 17: Activities 314
Using Parametric Equations: Ferris Wheel ProblemProblemUsing two pairs of parametric equations, determine when two objects in motion are closest to eachother in the same plane.A ferris wheel has a diameter (d) of 20 meters and is rotating counterclockwise at a rate (s) of onerevolution every 12 seconds. The parametric equations below describe the location of a ferriswheel passenger at time T, where a is the angle of rotation, (0,0) is the bottom center of the ferriswheel, and (10,10) is the passenger's location at the rightmost point, when T=0.X(T) = r cos a where a = 2pTs and r = dà2Y(T) = r + r sin aA person standing on the ground throws a ball to the ferris wheel passenger. The thrower's arm is atthe same height as the bottom of the ferris wheel, but 25 meters (b) to the right of the ferris wheel'slowest point (25,0). The person throws the ball with velocity (v0) of 22 meters per second at an angle(q) of 66¡ from the horizontal. The parametric equations below describe the location of the ball attime T.X(T) = b N Tv 0 cosq 2Y(T) = Tv 0 sinq N (gà2) T 2 where g = 9.8 m/secProcedure1. Press z. Select Par, Simul, and the default settings. Simul (simultaneous) mode simulates the two objects in motion over time.2. Press p. Set the viewing window. Tmin=0 Xmin=L13 Ymin=0 Tmax=12 Xmax=34 Ymax=31 Tstep=.1 Xscl=10 Yscl=103. Press o. Turn off all functions and stat plots. Enter the expressions to define the path of the ferris wheel and the path of the ball. Set the graph style for X2T to ë (path). Note: Try setting the graph styles to ë X1T and ì X2T, which simulates a chair on the ferris wheel and the ball flying through the air when you press s. Chapter 17: Activities 315
4. Press s to graph the equations. Watch closely as they are plotted. Notice that the ball and the ferris wheel passenger appear to be closest where the paths cross in the top-right quadrant of the ferris wheel.5. Press p. Change the viewing window to concentrate on this portion of the graph. Tmin=1 Xmin=0 Ymin=10 Tmax=3 Xmax=23.5 Ymax=25.5 Tstep=.03 Xscl=10 Yscl=106. Press r. After the graph is plotted, press ~ to move near the point on the ferris wheel where the paths cross. Notice the values of X, Y, and T.7. Press † to move to the path of the ball. Notice the values of X and Y (T is unchanged). Notice where the cursor is located. This is the position of the ball when the ferris wheel passenger passes the intersection. Did the ball or the passenger reach the intersection first? You can use r to, in effect, take snapshots in time and explore the relative behavior of two objects in motion. Chapter 17: Activities 316
Demonstrating the Fundamental Theorem of CalculusProblem 1Using the functions fnInt( and nDeriv( from the FUNC shortcut menu or the MATH menu to graphfunctions defined by integrals and derivatives demonstrates graphically that: and thatProcedure 11. Press z. Select the default settings.2. Press p. Set the viewing window. Xmin=.01 Ymin=L1.5 Xres=3 Xmax=10 Ymax=2.5 Xscl=1 Yscl=13. Press o. Turn off all functions and stat plots. Enter the numerical integral of 1àT from 1 to X and the function ln(X). Set the graph style for Y1 to ç (line) and Y2 to ë (path).4. Press r. Press |, }, ~, and † to compare the values of Y1 and Y2.5. Press o. Turn off Y1 and Y2, and then enter the numerical derivative of the integral of 1àX and the function 1àX. Set the graph style for Y3 to ç (line) and Y4 to è (thick). Chapter 17: Activities 317
6. Press r. Again, use the cursor keys to compare the values of the two graphed functions, Y3 and Y4.Problem 2Explore the functions defined by x x x – 2 t 0 t 2 t 2 2 2 y = dt, dt , and dtProcedure 21. Press o. Turn off all functions and stat plots. Use a list to define these three functions simultaneously. Store the function in Y5.2. Press q 6 to select 6:ZStandard. The graphs are displayed as each calculation of the integral and derivative occurs at the pixel point, which may take some time.3. Press r. Notice that the functions appear identical, only shifted vertically by a constant.4. Press o. Enter the numerical derivative of Y5 in Y6. Chapter 17: Activities 318
5. Press r. Notice that although the three graphs defined by Y5 are different, they share the same derivative. Chapter 17: Activities 319
Computing Areas of Regular N-Sided PolygonsProblemUse the equation solver to store a formula for the area of a regular N-sided polygon, and thensolve for each variable, given the other variables. Explore the fact that the limiting case is the areaof a circle, pr2.Consider the formula A = NB 2 sin(pàN) cos(pàN) for the area of a regular polygon with N sides ofequal length and B distance from the center to a vertex. N = 4 sides N = 8 sides N = 12 sidesProcedure1. Press t B to select B:Solver from the MATH menu. Either the equation editor or the interactive solver editor is displayed. If the interactive solver editor is displayed, press } to display the equation editor.2. Enter the formula as 0=ANNB2sin(p / N)cos(p / N), and then press Í. The interactive solver editor is displayed.3. Enter N=4 and B=6 to find the area (A) of a square with a distance (B) from center to vertex of 6 centimeters.4. Press } } to move the cursor onto A, and then press ă . The solution for A is displayed on the interactive solver editor.5. Now solve for B for a given area with various number of sides. Enter A=200 and N=6. To find the distance B, move the cursor onto B, and then press ƒ . Chapter 17: Activities 320
6. Enter N=8. To find the distance B, move the cursor onto B, and then press ƒ . Find B for N=9, and then for N=10.Find the area given B=6, and N=10, 100, 150, 1000, and 10000. Compare your results with p62 (thearea of a circle with radius 6), which is approximately 113.097.7. Enter B=6. To find the area A, move the cursor onto A, and then press ƒ . Find A for N=10, then N=100, then N=150, then N=1000, and finally N=10000. Notice that as N gets large, the area A approaches pB2.Now graph the equation to see visually how the area changes as the number of sides gets large.8. Press z. Select the default mode settings.9. Press p. Set the viewing window. Xmin=0 Ymin=0 Xres=1 Xmax=200 Ymax=150 Xscl=10 Yscl=1010. Press o. Turn off all functions and stat plots. Enter the equation for the area. Use X in place of N. Set the graph styles as shown. Chapter 17: Activities 321
11. Press r. After the graph is plotted, press 100 Í to trace to X=100. Press 150 Í. Press 188 Í. Notice that as X increases, the value of Y converges to p62, which is approximately 113.097. Y2=pB2 (the area of the circle) is a horizontal asymptote to Y1. The area of an N-sided regular polygon, with r as the distance from the center to a vertex, approaches the area of a circle with radius r (pr 2) as N gets large. Chapter 17: Activities 322
Computing and Graphing Mortgage PaymentsProblemYou are a loan officer at a mortgage company, and you recently closed on a 30-year homemortgage at 8 percent interest with monthly payments of 800. The new home owners want to knowhow much will be applied to the interest and how much will be applied to the principal when theymake the 240th payment 20 years from now.Procedure1. Press z and set the fixed-decimal mode to 2 decimal places. Set the other mode settings to the defaults.2. Press Œ Í Í to display the TVM Solver. Enter these values. Note: Enter a positive number (800) to show PMT as a cash inflow. Payment values will be displayed as positive numbers on the graph. Enter 0 for FV, since the future value of a loan is 0 once it is paid in full. Enter PMT: END, since payment is due at the end of a period.3. Move the cursor onto the PV= prompt, and then press ƒ . The present value, or mortgage amount, of the house is displayed at the PV= prompt.Now compare the graph of the amount of interest with the graph of the amount of principal for eachpayment.4. Press z. Set Par and Simul.5. Press o. Turn off all functions and stat plots. Enter these equations and set the graph styles as shown. Chapter 17: Activities 323
Note: GPrn( and GInt( are located on the FINANCE menu (APPS 1:FINANCE).6. Press p. Set these window variables. Tmin=1 Xmin=0 Ymin=0 Tmax=360 Xmax=360 Ymax=1000 Tstep=12 Xscl=10 Yscl=100 Note: To increase the graph speed, change Tstep to 24.7. Press r. After the graph is drawn, press 240 Í to move the trace cursor to T=240, which is equivalent to 20 years of payments. The graph shows that for the 240th payment (X=240), 358.03 of the 800 payment is applied to principal (Y=358.03). Note: The sum of the payments (Y3T=Y1T+Y2T) is always 800.8. Press † to move the cursor onto the function for interest defined by X2T and Y2T. Enter 240. The graph shows that for the 240th payment (X=240), 441.97 of the 800 payment is interest (Y=441.97).9. Press y 5 Œ Í 9 to paste 9:bal( to the home screen. Check the figures from the graph.At which monthly payment will the principal allocation surpass the interest allocation? Chapter 17: Activities 324
Note: Some Apps take up several App slots.Displaying the About ScreenAbout displays information about the TI-84 Plus Operating System (OS) Version, Product Number,Product Identification (ID), and Flash Application (App) Certificate Revision Number. To display theAbout screen, press y L and then select 1:About.Displays the type of Displays the Productgraphing calculator. ID. Each Flash-based graphing calculator has a unique product ID,Displays the OS which you may need ifversion. As new you contact technicalsoftware upgrades support. You can alsobecome available, use this 14 digit ID toyou can register your calculatorelectronically at education.ti.com, orupgrade your unit. identify your calculator in the event that it is lost or stolen.Displaying the MEMORY MANAGEMENT/DELETE MenuMem Mgmt/Del displays the MEMORY MANAGEMENT/DELETE menu. The two lines at the top reportthe total amount of available RAM (RAM FREE) and Archive (ARC FREE) memory. By selectingmenu items on this screen, you can see the amount of memory each variable type is using. Thisinformation can help you determine if you need to delete variables from memory to make room fornew data, such as programs or Apps.To check memory usage, follow these steps.1. Press y L to display the MEMORY menu. Note: The # and $ in the top or bottom of the left column indicate that you can scroll up or down to view more variable types.2. Select 2:Mem Mgmt/Del to display the MEMORY MANAGEMENT/DELETE menu. The TI-84 Plus expresses memory quantities in bytes. Chapter 18: Memory and Variable Management 326
3. Select variable types from the list to display memory usage. Notes: Real, List, Y-Vars, and Prgm variable types never reset to zero, even after memory is cleared. Apps are independent applications which are stored in Flash ROM. AppVars is a variable holder used to store variables created by Apps. You cannot edit or change variables in AppVars unless you do so through the application which created them.To leave the MEMORY MANAGEMENT/DELETE menu, press either y 5 or '. Both optionsdisplay the home screen. Chapter 18: Memory and Variable Management 327
Deleting Items from MemoryDeleting an ItemTo increase available memory by deleting the contents of any variable (real or complex number,list, matrix, Y= variable, program, Apps, AppVars, picture, graph database, or string), follow thesesteps.1. Press y L to display the MEMORY menu.2. Select 2:Mem Mgmt/Del to display the MEMORY MANAGEMENT/DELETE menu.3. Select the type of data you want to delete, or select 1:All for a list of all variables of all types. A screen is displayed listing each variable of the type you selected and the number of bytes each variable is using. For example, if you select 4:List, the LIST editor screen is displayed.4. Press } and † to move the selection cursor (4) next to the item you want to delete, and then press {. The variable is deleted from memory. You can delete individual variables one by one from this screen. No warning will be given to verify the deletion. Note: If you are deleting programs or Apps, you will receive a message asking you to confirm this delete action. Select 2:Yes to continue. To leave any variable screen without deleting anything, press y 5, which displays the home screen. You cannot delete some system variables, such as the last-answer variable Ans and the statistical variable RegEQ. Chapter 18: Memory and Variable Management 328
Clearing Entries and List ElementsClear EntriesClear Entries clears the contents of the ENTRY (last entry on home screen) storage area. To clearthe ENTRY storage area, follow these steps.1. Press y L to display the MEMORY menu.2. Select 3:Clear Entries to paste the instruction to the home screen.3. Press Í to clear the ENTRY storage area.To cancel Clear Entries, press '.Note: If you select 3:Clear Entries from within a program, the Clear Entries instruction is pasted tothe program editor, and the Entry (last entry) is cleared when the program is executed.ClrAllListsClrAllLists sets the dimension of each list in RAM to 0.To clear all elements from all lists, follow these steps.1. Press y L to display the MEMORY menu.2. Select 4:ClrAllLists to paste the instruction to the home screen.3. Press Í to set the dimension of each list in memory to 0.To cancel ClrAllLists, press '.ClrAllLists does not delete list names from memory, from the LIST NAMES menu, or from the statlist editor.Note: If you select 4:ClrAllLists from within a program, the ClrAllLists instruction is pasted to theprogram editor. The lists are cleared when the program is executed. Chapter 18: Memory and Variable Management 329
Archiving and UnArchiving VariablesArchiving and UnArchiving VariablesArchiving lets you store data, programs, or other variables to the user data archive (ARC) wherethey cannot be edited or deleted inadvertently. Archiving also allows you to free up RAM forvariables that may require additional memory.Archived variables cannot be edited or executed. They can only be seen and unarchived. Forexample, if you archive list L1, you will see that L1 exists in memory but if you select it and pastethe name L1 to the home screen, you won't be able to see its contents or edit it.Note: Not all variables may be archived. Not all archived variables may be unarchived. Forexample, system variables including r, t, x, y, and q cannot be archived. Apps and Groups alwaysexist in Flash ROM so there is no need to archive them. Groups cannot be unarchived. However,you can ungroup or delete them. Archive? UnArchive?Variable Type Names (yes/no) (yes/no)Real numbers A, B, ... , Z yes yesComplex A, B, ... , Z yes yesnumbersMatrices [A], [B], [C], ... , [J] yes yesLists L1, L2, L3, L4, L5, L6, yes yes and user-defined namesPrograms yes yesFunctions Y1, Y2, . . . , Y9, Y0 no not applicableParametric X1T and Y1T, ... , X6T no notequations and Y6T applicablePolar functions r1, r2, r3, r4, r5, r6 no not applicableSequence u, v, w no notfunctions applicableStat plots Plot1, Plot2, Plot3 no not applicableGraph databases GDB1, GDB2,... yes yesGraph pictures Pic1, Pic2, ... , Pic9, yes yes Pic0Strings Str1, Str2, . . . Str9, Str0 yes yesTables TblStart, @Tbl, no not TblInput applicableApps Applications see Note no aboveAppVars Application variables yes yes Chapter 18: Memory and Variable Management 330
Archive? UnArchive? Variable Type Names (yes/no) (yes/no) Groups see Note no above Variables with minX, maxX, RegEQ, no not reserved names and others applicable System variables Xmin, Xmax, and others no not applicableArchiving and unarchiving can be done in two ways:• Use the 5:Archive or 6:UnArchive commands from the MEMORY menu or CATALOG.• Use a Memory Management editor screen.Before archiving or unarchiving variables, particularly those with a large byte size (such as largeprograms) use the MEMORY menu to:• Find the size of the variable.• See if there is enough free space. For: Sizes must be such that: Archive Archive free size > variable size UnArchive RAM free size > variable sizeNote: If there is not enough space, unarchive or delete variables as necessary. Be aware thatwhen you unarchive a variable, not all the memory associated with that variable in user dataarchive will be released since the system keeps track of where the variable has been and where itis now in RAM.Even if there appears to be enough free space, you may see a Garbage Collection message whenyou attempt to archive a variable. Depending on the usability of empty blocks in the user dataarchive, you may need to unarchive existing variables to create more free space.To archive or unarchive a list variable (L1) using the Archive/UnArchive options from the MEMORYmenu:1. Press y L to display the MEMORY menu.2. Select 5:Archive or 6:UnArchive to place the command in the Home screen.3. Press y d to place the L1 variable in the Home screen. Chapter 18: Memory and Variable Management 331
4. Press Í to complete the archive process.Note: An asterisk will be displayed to the left of the Archived variable name to indicate it isarchived.To archive or unarchive a list variable (L1) using a Memory Management editor:1. Press y L to display the MEMORY menu.2. Select 2:Mem Mgmt/Del to display the MEMORY MANAGEMENT/DELETE menu.3. Select 4:List to display the LIST menu.4. Press Í to archive L1. An asterisk will appear to the left of L1 to indicate it is an archived variable. To unarchive a variable in this screen, put the cursor next to the archived variable and press Í. The asterisk will disappear. Chapter 18: Memory and Variable Management 332
Resetting the TI-84 PlusRAM ARCHIVE ALL MenuReset displays the RAM ARCHIVE ALL menu. This menu gives you the option of resetting allmemory (including default settings) or resetting selected portions of memory while preservingother data stored in memory, such as programs and Y= functions. For instance, you can choose toreset all of RAM or just restore the default settings. Be aware that if you choose to reset RAM, alldata and programs in RAM will be erased. For archive memory, you can reset variables (Vars),applications (Apps), or both of these. Be aware that if you choose to reset Vars, all data andprograms in archive memory will be erased. If you choose to reset Apps, all applications in archivememory will be erased.When you reset defaults on the TI-84 Plus, all defaults in RAM are restored to the factory settings.Stored data and programs are not changed.These are some examples of TI-84 Plus defaults that are restored by resetting the defaults.• Mode settings such as Normal (notation); Func (graphing); Real (numbers); and Full (screen)• Y= functions off• Window variable values such as Xmin=L10, Xmax=10, Xscl=1, Yscl=1, and Xres=1• STAT PLOTS off• Format settings such as CoordOn (graphing coordinates on); AxesOn; and ExprOn (expression on)• rand seed value to 0Displaying the RAM ARCHIVE ALL MenuTo display the RAM ARCHIVE ALL menu on the TI-84 Plus, follow these steps.1. Press y L to display the MEMORY menu.2. Select 7:Reset to display the RAM ARCHIVE ALL menu.Resetting RAM MemoryResetting all RAM restores RAM system variables to factory settings and deletes all nonsystemvariables and all programs. Resetting RAM defaults restores all system variables to defaultsettings without deleting variables and programs in RAM. Resetting all RAM or resetting defaultsdoes not affect variables and applications in user data archive.Note: Before you reset all RAM memory, consider restoring sufficient available memory by deletingonly selected data. Chapter 18: Memory and Variable Management 334
To reset all RAM memory or RAM defaults on the TI-84 Plus, follow these steps.1. From the RAM ARCHIVE ALL menu, select 1:All RAM to display the RESET RAM menu or 2:Defaults to display the RESET DEFAULTS menu.2. If you are resetting RAM, read the message below the RESET RAM menu. • To cancel the reset and return to the HOME screen, press Í. • To erase RAM memory or reset defaults, select 2:Reset. Depending on your choice, the message RAM cleared or Defaults set is displayed on the home screen.Resetting Archive MemoryWhen resetting archive memory on the TI-84 Plus, you can choose to delete from user dataarchive all variables, all applications, or both variables and applications.To reset all or part of user data archive memory, follow these steps.1. From the RAM ARCHIVE ALL menu, press ~ to display the ARCHIVE menu.2. Select one of the following: 1:Vars to display the RESET ARC VARS menu. 2:Apps to display the RESET ARC APPS menu. Chapter 18: Memory and Variable Management 335
3:Both to display the RESET ARC BOTH menu.3. Read the message below the menu. • To cancel the reset and return to the HOME screen, press Í. • To continue with the reset, select 2:Reset. A message indicating the type of archive memory cleared will be displayed on the HOME screen.Resetting All MemoryWhen resetting all memory on the TI-84 Plus, RAM and user data archive memory is restored tofactory settings. All nonsystem variables, applications, and programs are deleted. All systemvariables are reset to default settings.Before you reset all memory, consider restoring sufficient available memory by deleting onlyselected data.To reset all memory on the TI-84 Plus, follow these steps.1. From the RAM ARCHIVE ALL menu, press ~ ~ to display the ALL menu.2. Select 1:All Memory to display the RESET MEMORY menu.3. Read the message below the RESET MEMORY menu. • To cancel the reset and return to the HOME screen, press Í. • To continue with the reset, select 2:Reset. The message MEM cleared is displayed on the HOME screen.When you clear memory, the contrast sometimes changes. If the screen is faded or blank, adjustthe contrast by pressing y } or †. Chapter 18: Memory and Variable Management 336
Grouping and Ungrouping VariablesGrouping VariablesGrouping allows you to make a copy of two or more variables residing in RAM and then store themas a group in user data archive. The variables in RAM are not erased. The variables must exist inRAM before they can be grouped. In other words, archived data cannot be included in a group.Once grouped, the variables can be deleted from RAM to open memory. When the variables areneeded later, they can be ungrouped for use.To create a group of variables:1. Press y L to display the MEMORY menu.2. Select 8:Group to display GROUP UNGROUP menu.3. Press Í to display the GROUP menu.4. Enter a name for the new group and press Í. Note: A group name can be one to eight characters long. The first character must be a letter from A to Z or q. The second through eighth characters can be letters, numbers, or q.5. Select the type of data you want to group. You can select 1:All+ which shows all variables of all types available and selected. You can also select 2:All- which shows all variables of all types available but not selected. A screen is displayed listing each variable of the type you selected. Chapter 18: Memory and Variable Management 337
For example, suppose some variables have been created in RAM, and selecting 2:All- displays the following screen.6. Press } and † to move the selection cursor (4) next to the first item you want to copy into a group, and then press Í. A small square will remain to the left of all variables selected for grouping. Repeat the selection process until all variables for the new group are selected and then press ~ to display the DONE menu.7. Press Í to complete the grouping process.Note: You can only group variables in RAM. You cannot group some system variables, such as thelast-answer variable Ans and the statistical variable RegEQ.Ungrouping VariablesUngrouping allows you to make a copy of variables in a group stored in user data archive andplace them ungrouped in RAM. Chapter 18: Memory and Variable Management 338
DuplicateName MenuDuring the ungrouping action, if a duplicate variable name is detected in RAM, the DUPLICATENAME menu is displayed.DuplicateName1: Rename Prompts to rename receiving variable.2: Overwrite Overwrites data in receiving duplicate variable.3: Overwrite All Overwrites data in all receiving duplicate variables.4: Omit Skips ungrouping of sending variable.5: Quit Stops ungrouping at duplicate variable.Notes about Menu Items:• When you select 1:Rename, the Name= prompt is displayed, and alpha-lock is on. Enter a new variable name, and then press Í. Ungrouping resumes.• When you select 2:Overwrite, the unit overwrites the data of the duplicate variable name found in RAM. Ungrouping resumes.• When you select 3: Overwrite All, the unit overwrites the data of all duplicate variable names found in RAM. Ungrouping resumes.• When you select 4:Omit, the unit does not ungroup the variable in conflict with the duplicated variable name found in RAM. Ungrouping resumes with the next item.• When you select 5:Quit, ungrouping stops, and no further changes are made.To ungroup a group of variables:1. Press y L to display the MEMORY menu.2. Select 8:Group to display the GROUP UNGROUP menu.3. Press ~ to display the UNGROUP menu. Chapter 18: Memory and Variable Management 339
4. Press } and † to move the selection cursor (4) next to the group variable you want to ungroup, and then press Í. The ungroup action is completed.Note: Ungrouping does not remove the group from user data archive. You must delete the group inuser data archive to remove it. Chapter 18: Memory and Variable Management 340
Garbage CollectionGarbage Collection MessageIf you use the user data archive extensively, you may see a Garbage Collect? message. Thisoccurs if you try to archive a variable when there is not enough free contiguous archive memory.The Garbage Collect? message lets you know an archive will take longer than usual. It also alertsyou that the archive will fail if there is not enough memory.The message can also alert you when a program is caught in a loop that repetitively fills the userdata archive. Select No to cancel the garbage collection process, and then find and correct theerrors in your program.When YES is selected, the TI-84 Plus will attempt to rearrange the archived variables to makeadditional room.Responding to the Garbage Collection Message• To cancel, select 1:No.• If you select 1:No, the message ERR:ARCHIVE FULL will be displayed.• To continue archiving, select 2:Yes.• If you select 2:Yes, the process message Garbage Collecting... or Defragmenting... will be displayed.Note: The process message Defragmenting... is displayed whenever an application marked fordeletion is encountered. Garbage collection may take up to 20 minutes, depending on how muchof archive memory has been used to store variables.After garbage collection, depending on how much additional space is freed, the variable may ormay not be archived. If not, you can unarchive some variables and try again.Why Is Garbage Collection Necessary?The user data archive is divided into sectors. When you first begin archiving, variables are storedconsecutively in sector 1. This continues to the end of the sector.An archived variable is stored in a continuous block within a single sector. Unlike an applicationstored in user data archive, an archived variable cannot cross a sector boundary. If there is notenough space left in the sector, the next variable is stored at the beginning of the next sector.Typically, this leaves an empty block at the end of the previous sector. Chapter 18: Memory and Variable Management 341
variable A Sector 1 variable B Empty block variable D variable C Sector 2Depending on its size,variable D is stored in Sector 3one of these locations.Each variable that you archive is stored in the first empty block large enough to hold it.This process continues to the end of the last sector. Depending on the size of individual variables,the empty blocks may account for a significant amount of space. Garbage collection occurs whenthe variable you are archiving is larger than any empty block.How Unarchiving a Variable Affects the ProcessWhen you unarchive a variable, it is copied to RAM but it is not actually deleted from user dataarchive memory. Unarchived variables are "marked for deletion," meaning they will be deletedduring the next garbage collection. Sector 1 variable A After you unarchive variables B and C, they continue to take Sector 2 up space. variable D Sector 3If the MEMORY Screen Shows Enough Free SpaceEven if the MEMORY screen shows enough free space to archive a variable or store an application,you may still get a Garbage Collect? message or an ERR: ARCHIVE FULL message.When you unarchive a variable, the Archive free amount increases immediately, but the space isnot actually available until after the next garbage collection.If the Archive free amount shows enough available space for your variable, there probably will beenough space to archive it after garbage collection (depending on the usability of any emptyblocks). Chapter 18: Memory and Variable Management 342
The Garbage Collection ProcessThe garbage collection process:• Deletes unarchived variables Sector 1 from the user data archive. variable A• Rearranges the remaining variable D variables into consecutive blocks. Sector 2Note: Power loss during garbage collection may cause all memory (RAM and Archive) to bedeleted.Using the GarbageCollect CommandYou can reduce the number of automatic garbage collections by periodically optimizing memory.This is done by using the GarbageCollect command.To use the GarbageCollect command, follow these steps.1. From the HOME screen, press y N to display the CATALOG.2. Press † or } to scroll the CATALOG until the selection cursor points to the GarbageCollect command or press G to skip to the commands starting with the letter G.3. Press Í to paste the command to the HOME screen.4. Press Í to display the Garbage Collect? message.5. Select 2:Yes to begin garbage collection. Chapter 18: Memory and Variable Management 343
ERR:ARCHIVE FULL MessageEven if the MEMORY screen shows enoughfree space to archive a variable or store anapplication, you may still get an ERR:ARCHIVE FULL message.An ERR:ARCHIVE FULL message may be displayed:• When there is insufficient space to archive a variable within a continuous block and within a single sector.• When there is insufficient space to store an application within a continuous block of memory.When the message is displayed, it will indicate the largest single space of memory available forstoring a variable and an application.To resolve the problem, use the GarbageCollect command to optimize memory. If memory is stillinsufficient, you must delete variables or applications to increase space. Chapter 18: Memory and Variable Management 344
Chapter 19:Communication LinkGetting Started: Sending VariablesGetting Started is a fast-paced introduction. Read the chapter for details.Create and store a variable and a matrix, and then transfer them to another TI-84 Plus.1. On the home screen of the sending unit, press 5 Ë 5 ¿ ƒ Q. Press Í to store 5.5 to Q.2. Press t ` † † Í to display the 2x2 matrix template. Press 1 ~ 2 ~ 3 ~ 4 ~ to enter the values. Press ¿ y > 1 Í to store the matrix to [A].3. On the sending unit, press y L to display the MEMORY menu.4. On the sending unit, press 2 to select 2:Mem Mgmt/Del. The MEMORY MANAGEMENT menu is displayed.5. On the sending unit, press 5 to select 5:Matrix. The MATRIX editor screen is displayed.6. On the sending unit, press Í to archive [A]. An asterisk (ä) will appear, signifying that [A] is now archived.7. Connect the graphing calculators with the USB unit-to-unit cable. Push both ends in firmly.8. On the receiving unit, press y 8 ~ to display the RECEIVE menu. Press 1 to select 1:Receive. The message Waiting... is displayed and the busy indicator is on. Chapter 19: Communication Link 345
9. On the sending unit, press y 8 to display the SEND menu.10. Press 2 to select 2:AllN. The AllN SELECT screen is displayed.11. Press † until the selection cursor ( 4 ) is next to [A] MATRX. Press Í.12. Press † until the selection cursor is next to Q REAL. Press Í. A square dot next to [A] and Q indicates that each is selected to send.13. On the sending unit, press ~ to display the TRANSMIT menu.14. On the sending unit, press 1 to select 1:Transmit and begin transmission. The receiving unit displays the message Receiving....When the items are transmitted, both units display the name and type of each transmitted variable. Chapter 19: Communication Link 346
TI-84 Plus LINKThis chapter describes how to communicate with compatible TI units. The TI-84 Plus has a USBport to connect and communicate with another TI-84 Plus or TI-84 Plus Silver Edition. A USBunit-to-unit cable is included with the TI-84 Plus.The TI-84 Plus also has an I/O port using a I/O unit-to-unit cable to communicate with:• TI-83 Plus Silver Edition • TI-82• TI-83 Plus • TI-73• TI-83 • CBL 2™ or a CBR™You can send items from a calculator with an older OS to a calculator with OS 2.53MP. However,you may receive a version error if you send items from a calculator with OS 2.53MP to a calculatorwith an older OS. Transferring files between calculators works best if both calculators have thelatest operating system software installed. For example, if you send a list that contains fractions(OS 2.53MP) to a calculator with OS 2.43, a version error displays because OS 2.43 does notsupport fractions.Connecting Two Graphing Calculators with a USB Unit-to-Unit Cable or an I/O Unit-to-UnitCableUSB Unit-to-Unit CableThe TI-84 Plus USB link port is located at thetop right edge of the graphing calculator.1. Firmly insert either end of the USB unit-to-unit cable into the USB port.2. Insert the other end of the cable into the other graphing calculator's USB port.I/O Unit-to-Unit CableThe TI-84 Plus I/O link port is located at thetop left edge of the graphing calculator.1. Firmly insert either end of the I/O unit-to-unit cable into the port.2. Insert the other end of the cable into the other graphing calculator's I/O port. Chapter 19: Communication Link 347
TI-84 Plus to a TI-83 Plus using I/O Unit-to-Unit CableThe TI-84 Plus I/O link port is located at thetop left edge of the graphing calculator. TheTI-83 Plus I/O link port is located at thebottom edge of the graphing calculator.3. Firmly insert either end of the I/O unit-to-unit cable into the port.4. Insert the other end of the cable into the other graphing calculator's I/O port.Linking to the CBL/CBR SystemThe CBL 2™ system and the CBR™ system are optional accessories that also connect to a TI-84Plus with the I/O unit-to-unit cable. With a CBL 2™ system or CBR™ system and a TI-84 Plus, youcan collect and analyze real-world data.Linking to a ComputerWith TI Connect™ software and the USB computer cable that is included with your TI-84 Plus, youcan link the graphing calculator to a personal computer. Chapter 19: Communication Link 348
2. Select the menu item that describes the data type to send. The corresponding SELECT screen is displayed.3. Press } and † to move the selection cursor ( 4 ) to an item you want to select or deselect.4. Press Í to select or deselect the item. Selected names are marked with a 0. Note: An asterisk (ä) to the left of an item indicates the item is archived.5. Repeat steps 3 and 4 to select or deselect additional items.Sending the Selected ItemsAfter you have selected items to send on the sending unit and set the receiving unit to receive,follow these steps to transmit the items. To set the receiving unit, see Receiving Items.1. Press ~ on the sending unit to display the TRANSMIT menu.2. Confirm that Waiting... is displayed on the receiving unit, which indicates it is set to receive.3. Press Í to select 1:Transmit. The name and type of each item are displayed line-by-line on the sending unit as the item is queued for transmission, and then on the receiving unit as each item is accepted. Note: Items sent from the RAM of the sending unit are transmitted to the RAM of the receiving unit. Items sent from user data archive (flash) of the sending unit are transmitted to user data archive (flash) of the receiving unit.After all selected items have been transmitted, the message Done is displayed on both calculators.Press } and † to scroll through the names.Sending to a TI-84 Plus Silver Edition or TI-84 PlusYou can transfer variables (all types), programs, and Flash applications to another TI-84 PlusSilver Edition or TI-84 Plus. You can also backup the RAM memory of one unit to another.Note: Keep in mind that the TI-84 Plus has less Flash memory than the TI-84 Plus Silver Edition. Chapter 19: Communication Link 350
• Variables stored in RAM on the sending TI-84 Plus Silver Edition will be sent to the RAM of the receiving TI-84 Plus Silver Edition or TI-84 Plus.• Variables and applications stored in the user data archive of the sending TI-84 Plus Silver Edition will be sent to the user data archive of the receiving TI-84 Plus Silver Edition or TI-84 Plus.After sending or receiving data, you can repeat the same transmission to additional TI-84 PlusSilver Edition or TI-84 Plus units—from either the sending unit or the receiving unit—withouthaving to reselect data to send. The current items remain selected. However, you cannot repeattransmission if you selected All+ or All..To send data to an additional TI-84 Plus Silver Edition or a TI-84 Plus:1. Use a USB unit-to-unit cable to link two units together.2. On the sending unit press y 8 and select a data type and items to SEND.3. Press ~ on the sending unit to display the TRANSMIT menu.4. On the other unit, press y 8 ~ to display the RECEIVE menu.5. Press Í on the receiving unit.6. Press Í on the sending unit. A copy of the selected item(s) is sent to the receiving unit.7. Disconnect the link cable only from the receiving unit and connect it to another unit.8. Press y 8 on the sending unit.9. Select only the data type. For example, if the unit just sent a list, select 4:LIST. Note: The item(s) you want to send are pre-selected from the last transmission. Do not select or deselect any items. If you select or deselect an item, all selections or deselections from the last transmission are cleared.10. Press ~ on the sending unit to display the TRANSMIT menu.11. On the new receiving unit, press y 8 ~ to display the RECEIVE menu.12. Press Í on the receiving unit.13. Press Í on the sending unit. A copy of the selected item(s) is sent to the receiving unit.14. Repeat steps 7 through 13 until the items are sent to all additional units.Sending to a TI-83 Plus or TI-83 Plus Silver EditionYou can send all variables from a TI-84 Plus to a TI-83 Plus or TI-83 Plus Silver Edition exceptFlash applications with new features, or programs with new features in them.If archived variables on the TI-84 Plus are variable types recognized and used on the TI-83 Plus orTI-83 Plus Silver Edition, you can send these variables to the TI-83 Plus or TI-83 Plus SilverEdition. They will be automatically sent to the RAM of the TI-83 Plus or TI-83 Plus Silver Editionduring the transfer process. It will send to archive if the item is from archive.To send data to a TI-83 Plus or TI-83 Plus Silver Edition:1. Use an I/O unit-to-unit cable to link the two units together.2. Set the TI-83 Plus or TI-83 Plus Silver Edition to receive. Chapter 19: Communication Link 351
3. Press y 8 on the sending TI-84 Plus to display the LINK SEND menu.4. Select the menu of the items you want to transmit.5. Press ~ on the sending TI-84 Plus to display the LINK TRANSMIT menu.6. Confirm that the receiving unit is set to receive.7. Press Í on the sending TI-84 Plus to select 1:Transmit and begin transmitting. Chapter 19: Communication Link 352
Receiving ItemsLINK RECEIVE MenuTo display the LINK RECEIVE menu, press y 8 ~.SEND RECEIVE1: Receive Sets unit to receive data transmission.Receiving UnitWhen you select 1:Receive from the LINK RECEIVE menu on the receiving unit, the messageWaiting... and the busy indicator are displayed. The receiving unit is ready to receive transmitteditems. To exit the receive mode without receiving items, press É, and then select 1:Quit from theError in Xmit menu.When transmission is complete, the unit exits the receive mode. You can select 1:Receive again toreceive more items. The receiving unit then displays a list of items received. Press y 5 to exitthe receive mode.DuplicateName MenuDuring transmission, if a variable name is duplicated, the DuplicateName menu is displayed on thereceiving unit.DuplicateName1: Rename Prompts to rename receiving variable.2: Overwrite Overwrites data in receiving variable.3: Omit Skips transmission of sending variable.4: Quit Stops transmission at duplicate variable.When you select 1:Rename, the Name= prompt is displayed, and alpha-lock is on. Enter a newvariable name, and then press Í. Transmission resumes.When you select 2:Overwrite, the sending unit's data overwrites the existing data stored on thereceiving unit. Transmission resumes.When you select 3:Omit, the sending unit does not send the data in the duplicated variable name.Transmission resumes with the next item.When you select 4:Quit, transmission stops, and the receiving unit exits receive mode. Chapter 19: Communication Link 353
Receiving from a TI-84 Plus Silver Edition or TI-84 PlusThe TI-84 Plus Silver Edition and the TI-84 Plus are totally compatible. Keep in mind, however thatthe TI-84 Plus has less Flash memory than a TI-84 Plus Silver Edition.You cannot send memory backups between the TI-84 Plus product family and the TI-83 Plusproduct family.Receiving from a TI-83 Plus Silver Edition or TI-83 PlusThe TI-84 Plus product family and the TI-83 Plus product family are compatible with a fewexceptions.Receiving from a TI-83You can transfer all variables and programs from a TI-83 to a TI-84 Plus if they fit in the RAM of theTI-84 Plus. The RAM of the TI-84 Plus is slightly less than the RAM of the TI-83. Chapter 19: Communication Link 354
Backing Up RAM MemoryWarning: H:Back Up overwrites the RAM memory and mode settings in the receiving unit. Allinformation in the RAM memory of the receiving unit is lost.Note: Archived items on the receiving unit are not overwritten.You can backup the contents of RAM memory and mode settings (no Flash applications orarchived items) to another TI-84 Plus Silver Edition. You can also backup RAM memory and modesettings to a TI-84 Plus. The backup calculator must also have OS 2.53MP installed.To perform a RAM memory backup:1. Use a USB unit-to-unit cable to link two TI-84 Plus units, or a TI-84 Plus and a TI-84 Plus Silver Edition together.2. On the sending unit press y 8 and select H:Back Up. The MEMORYBACKUP screen displays.3. On the receiving unit, press y 8 ~ to display the RECEIVE menu.4. Press Í on the receiving unit.5. Press Í on the sending unit. A WARNING — Backup message displays on the receiving unit.6. Press Í on the receiving unit to continue the backup. — or — Press 2:Quit on the receiving unit to cancel the backup and return to the LINK SEND menu Note: If a transmission error is returned during a backup, the receiving unit is reset.Memory Backup CompleteWhen the backup is complete, both the sending graphing calculator and receiving graphingcalculator display a confirmation screen. Chapter 19: Communication Link 355
Error ConditionsA transmission error occurs after one or two seconds if:• A cable is not attached to the sending unit.• A cable is not attached to the receiving unit. Note: If the cable is attached, push it in firmly and try again.• The receiving unit is not set to receive transmission.• You attempt a backup between a TI-73, TI-82, TI-83, TI-83 Plus, or TI-83 Plus Silver Edition.• You attempt a data transfer from a TI-84 Plus to a TI-83 Plus, TI-83 Plus Silver Edition, TI-83, TI-82, or TI-73 with variables or features not recognized by the TI-83 Plus, TI-83 Plus Silver Edition, TI-83, TI-82, or TI-73. New variable types and features not recognized by the TI-83, TI-83 Plus, TI-82, or TI-73 include applications, application variables, grouped variables, new variable types, or programs with new features in them such as Archive, UnArchive, SendID, SendOS, Asm(, AsmComp(, AsmPrgm, checkTmr(, ClockOff, ClockOn, dayOfWk(, getDate, getDtFmt, getDtStr(, getTime, getTmFmt, getTmStr, isClockOn, randIntNoRep(, setDate(, setDtFmt(, setTime(, setTmFmt(, startTmr, summation(, timeCnv and fractions.• You attempt a data transfer from a TI-84 Plus to a TI-82 with data other than real lists L1 through L6 or without using menu item 5:Lists to TI82.• You attempt a data transfer from a TI-84 Plus to a TI-73 with data other than real numbers, pics, real lists L1 through L6 or named lists with q as part of the name.Although a transmission error does not occur, these two conditions may prevent successfultransmission.• You try to use Get( with a graphing calculator instead of a CBL 2™ system or CBR™ system.• You try to use GetCalc( with a TI-83 instead of a TI-84 Plus or TI-84 Plus Silver Edition.Insufficient Memory in Receiving Unit• During transmission, if the receiving unit does not have sufficient memory to receive an item, the Memory Full menu is displayed on the receiving unit.• To skip this item for the current transmission, select 1:Omit. Transmission resumes with the next item.• To cancel the transmission and exit receive mode, select 2:Quit. Chapter 19: Communication Link 356
Appendix A:Functions and InstructionsFunctions return a value, list, or matrix. You can use functions in an expression. Instructions initiatean action. Some functions and instructions have arguments. Optional arguments and accompanyingcommas are enclosed in brackets ( [ ] ). For details about an item, including argument descriptionsand restrictions, turn to the page listed on the right side of the table.From the CATALOG, you can paste any function or instruction to the home screen or to a commandline in the program editor. However, some functions and instructions are not valid on the homescreen. The items in this table appear in the same order as they appear in the CATALOG.† indicates either keystrokes that are valid in the program editor only or ones that paste certaininstructions when you are in the program editor. Some keystrokes display menus that are availableonly in the program editor. Others paste mode, format, or table-set instructions only when you arein the program editor.Function or Key orInstruction/Arguments Result Keys/Menu or Screen/Itemabs(value) Returns the absolute value of a real number, expression, list, or matrix. NUM 1:abs(abs(complex value) Returns the magnitude of a complex number or list. CPX 5:abs(valueA and valueB Returns 1 if both valueA and valueB are ƒ 0. valueA and y: valueB can be real numbers, expressions, or lists. LOGIC 1:andangle(value) Returns the polar angle of a complex number or list of complex numbers. CPX 4:angle(ANOVA(list1,list2 Performs a one-way analysis of variance for comparing the …[,list3,...,list20]) means of two to 20 populations. TESTS H:ANOVA(Ans Returns the last answer. yZArchive Moves the specified variables from RAM to the user data yL archive memory. 5:ArchiveAsm(assemblyprgmname) Executes an assembly language program. yN Asm(AsmComp(prgmASM1, Compiles an assembly language program written in ASCII y NprgmASM2) and stores the hex version. AsmComp(AsmPrgm Must be used as the first line of an assembly language yN program. AsmPrgmaugment(matrixA, Returns a matrix, which is matrixB appended to matrixA as y>matrixB) new columns. MATH 7:augment( Appendix A: Functions and Instructions 357
Function or Key orInstruction/Arguments Result Keys/Menu or Screen/Itemaugment(listA,listB) Returns a list, which is listB concatenated to the end of y9 listA. OPS 9:augment(AUTO Answer Displays answers in a similar format as the input. z Answers: AUTOAxesOff Turns off the graph axes. †y. AxesOffAxesOn Turns on the graph axes. †y. AxesOna+bi Sets the mode to rectangular complex number mode †z (a+bi). a+bibal(npmt[,roundvalue]) Computes the balance at npmt for an amortization Œ 1:Finance schedule using stored values for PV, æ, and PMT and CALC rounds the computation to roundvalue. 9:bal(binomcdf(numtrials,p Computes a cumulative probability at x for the discrete y=[,x]) binomial distribution with the specified numtrials and DISTR probability p of success on each trial. B:binomcdf(binompdf(numtrials,p Computes a probability at x for the discrete binomial y=[,x]) distribution with the specified numtrials and probability p of DISTR success on each trial. A:binompdf(checkTmr(starttime) Returns the number of seconds since you used startTmr yN to start the timer. The starttime is the value displayed by checkTmr( startTmr.c2cdf(lowerbound, Computes the c2 distribution probability between y=upperbound,df) lowerbound and upperbound for the specified degrees of DISTR freedom df. 8:c2cdf(c2pdf(x,df) Computes the probability density function (pdf) for the c2 y = distribution at a specified x value for the specified degrees DISTR of freedom df. 7:c2pdf(c2LTest(observedmatrix, Performs a chi-square test. drawflag=1 draws results; †…expectedmatrix drawflag=0 calculates results. TESTS[,drawflag]) C:c2LTest(c2GOF-Test(observedlist, Performs a test to confirm that sample data is from a †…expectedlist,df) population that conforms to a specified distribution. TESTS D:c2GOFLTest(Circle(X,Y,radius) Draws a circle with center (X,Y) and radius. y< DRAW 9:Circle(CLASSIC Displays inputs and outputs on a single line, such as z 1/2+3/4. CLASSIC Appendix A: Functions and Instructions 358
Function or Key orInstruction/Arguments Result Keys/Menu or Screen/ItemcumSum(list) Returns a list of the cumulative sums of the elements in y9 list, starting with the first element. OPS 6:cumSum(cumSum(matrix) Returns a matrix of the cumulative sums of matrix y> elements. Each element in the returned matrix is a MATH cumulative sum of a matrix column from top to bottom. 0:cumSum(dayOfWk(year,month, Returns an integer from 1 to 7, with each integer yNday) representing a day of the week. Use dayOfWk( to dayOfWk( determine on which day of the week a particular date 1:Sunday would occur. The year must be 4 digits; month and day can 2:Monday be 1 or 2 digit. 3:Tuesday...dbd(date1,date2) Calculates the number of days between date1 and date2 Œ 1:Finance using the actual-day-count method. CALC D:dbd(DEC Answers Displays answers as integers or decimal numbers. z Answers: DECvalue4Dec Displays a real or complex number, expression, list, or matrix in decimal format. MATH 2:4DecDegree Sets degree angle mode. †z DegreeDelVar variable Deletes from memory the contents of variable. † CTL G:DelVarDependAsk Sets table to ask for dependent-variable values. †y- Depend: AskDependAuto Sets table to generate dependent-variable values †y- automatically. Depend: Autodet(matrix) Returns determinant of matrix. y> MATH 1:det(DiagnosticOff Sets diagnostics-off mode; r, r2, and R2 are not displayed yN as regression model results. DiagnosticOffDiagnosticOn Sets diagnostics-on mode; r, r2, and R2 are displayed as yN regression model results. DiagnosticOndim(listname) Returns the dimension of listname. y9 OPS 3:dim(dim(matrixname) Returns the dimension of matrixname as a list. y> MATH 3:dim( Appendix A: Functions and Instructions 360
Function or Key orInstruction/Arguments Result Keys/Menu or Screen/Itemgeometcdf(p,x) Computes a cumulative probability at x, the number of the y= trial on which the first success occurs, for the discrete DISTR geometric distribution with the specified probability of F:geometcdf( success p.geometpdf(p,x) Computes a probability at x, the number of the trial on which y = the first success occurs, for the discrete geometric DISTR distribution with the specified probability of success p. E:geometpdf(Get(variable) Gets data from the CBL 2™ or CBR™ System and stores it † in variable. I/O A:Get(GetCalc(variable Gets contents of variable on another TI-84 Plus and stores it † [,portflag]) to variable on the receiving TI-84 Plus. By default, the TI-84 I/O Plus uses the USB port if it is connected. If the USB cable 0:GetCalc( is not connected, it uses the I/O port. portflag=0 use USB port if connected; portflag=1 use USB port; portflag=2 use I/O port.getDate Returns a list giving the date according to the current value y N of the clock. The list is in {year,month,day} format. getDategetDtFmt Returns an integer representing the date format that is yN currently set on the device. getDtFmt 1 = M/D/Y 2 = D/M/Y 3 = Y/M/DgetDtStr(integer) Returns a string of the current date in the format specified yN by integer, where: getDtStr( 1 = M/D/Y 2 = D/M/Y 3 = Y/M/DgetTime Returns a list giving the time according to the current value y N of the clock. The list is in {hour,minute,second} format. The getTime time is returned in the 24 hour format.getTmFmt Returns an integer representing the clock time format that yN is currently set on the device. getTmFmt 12 = 12 hour format 24 = 24 hour formatgetTmStr(integer) Returns a string of the current clock time in the format yN specified by integer, where: getTmStr( 12 = 12 hour format 24 = 24 hour formatgetKey Returns the key code for the current keystroke, or 0, if no † key is pressed. I/O 7:getKeyGoto label Transfers control to label. † CTL 0:Goto Appendix A: Functions and Instructions 364
Function or Key orInstruction/Arguments Result Keys/Menu or Screen/ItemInput [variable] Prompts for value to store to variable. †Input ["text",variable] I/O 1:InputInput [Strn,variable] Displays Strn and stores entered value to variable. † I/O 1:InputinString(string,substring Returns the character position in string of the first yN[,start]) character of substring beginning at start. inString(int(value) Returns the largest integer a real or complex number, expression, list, or matrix. NUM 5:int(GInt(pmt1,pmt2 Computes the sum, rounded to roundvalue, of the interest Œ 1:Finance[,roundvalue]) amount between pmt1 and pmt2 for an amortization CALC schedule. A:GInt(invNorm(area[,m,s]) Computes the inverse cumulative normal distribution y= function for a given area under the normal distribution DISTR curve specified by m and s. 3:invNorm(invT(area,df) Computes the inverse cumulative student-t probability y= function specified by degree of freedom, df for a given area DISTR under the curve. 4:invT(iPart(value) Returns the integer part of a real or complex number, expression, list, or matrix. NUM 3:iPart(irr(CF0,CFList[,CFFreq]) Returns the interest rate at which the net present value of Œ 1:Finance the cash flow is equal to zero. CALC 8:irr(isClockOn Identifies if clock is ON or OFF. Returns 1 if the clock is yN ON. Returns 0 if the clock is OFF. isClockOn:IS>(variable,value) Increments variable by 1; skips commandA if variable>value. † :commandA CTL:commands A:IS>(Ùlistname Identifies the next one to five characters as a user-created y 9 list name. OPS B:ÙLabelOff Turns off axes labels. †y. LabelOffLabelOn Turns on axes labels. †y. LabelOnLbl label Creates a label of one or two characters. † CTL 9:Lbl Appendix A: Functions and Instructions 366
Function or Key orInstruction/Arguments Result Keys/Menu or Screen/Itemlcm(valueA,valueB) Returns the least common multiple of valueA and valueB, which can be real numbers or lists. NUM 8:lcm(length(string) Returns the number of characters in string. yN length(Line(X1,Y1,X2,Y2) Draws a line from (X1,Y1) to (X2,Y2). y< DRAW 2:Line(Line(X1,Y1,X2,Y2,0) Erases a line from (X1,Y1) to (X2,Y2). y< DRAW 2:Line(LinReg(a+bx 8:LinReg(a+bx)LinReg(ax+b 4:LinReg(ax+b)LinRegTInt [Xlistname, Performs a linear regression and computes the t †…Ylistname,freqlist, confidence interval for the slope coefficient b. TESTSconfidence level, regequ] G:LinRegTIntLinRegTTest [Xlistname, Performs a linear regression and a t-test. alternative=L1 is †…Ylistname,freqlist, <; alternative=0 is ƒ; alternative=1 is >. TESTSalternative,regequ] F:LinRegTTest@List(list) Returns a list containing the differences between y9 consecutive elements in list. OPS 7:@List(List 4 matr(listname1,..., Fills matrixname column by column with the elements from y9listname n,matrixname) each specified listname. OPS 0:List 4 matr(ln(value) Returns the natural logarithm of a real or complex number, μ expression, or list.LnReg [Xlistname, Fits a logarithmic regression model to Xlistname and …Ylistname,freqlist, Ylistname with frequency freqlist, and stores the regression CALCregequ] equation to regequ. 9:LnReglog(value) Returns logarithm of a real or complex number, « expression, or list.logBASE(value, base) Returns the logarithm of a specifed value determined from a specified base: logBASE(value, base). A: logBASELogistic [Xlistname, Fits a logistic regression model to Xlistname and Ylistname …Ylistname,freqlist, with frequency freqlist, and stores the regression equation CALCregequ] to regequ. B:Logistic Appendix A: Functions and Instructions 367
Function or Key orInstruction/Arguments Result Keys/Menu or Screen/ItemManual-Fit equname Fits a linear equation to a scatter plot. … CALC D:Manual-FitMATHPRINT Displays most entries and answers the way they are z displayed in textbooks, such as . MATHPRINTMatr4list(matrix, Fills each listname with elements from each column in y9listnameA,...,listname n) matrix. OPS A:Matr4list(Matr4list(matrix, Fills a listname with elements from a specified column# in y9column#,listname) matrix. OPS A:Matr4list(max(valueA,valueB) Returns the larger of valueA and valueB. NUM 7:max(max(list) Returns largest real or complex element in list. y9 MATH 2:max(max(listA,listB) Returns a real or complex list of the larger of each pair of y9 elements in listA and listB. MATH 2:max(max(value,list) Returns a real or complex list of the larger of value or each y9 list element. MATH 2:max(mean(list[,freqlist]) Returns the mean of list with frequency freqlist. y9 MATH 3:mean(median(list[,freqlist]) Returns the median of list with frequency freqlist. y9 MATH 4:median(Med-Med [Xlistname, Fits a median-median model to Xlistname and Ylistname …Ylistname,freqlist, with frequency freqlist, and stores the regression equation CALCregequ] to regequ. 3:Med-MedMenu("title","text1", Generates a menu of up to seven items during program †label1[,...,"text7",label7]) execution. CTL C:Menu(min(valueA,valueB) Returns smaller of valueA and valueB. NUM 6:min(min(list) Returns smallest real or complex element in list. y9 MATH 1:min( Appendix A: Functions and Instructions 368
Function or Key orInstruction/Arguments Result Keys/Menu or Screen/Itemmin(listA,listB) Returns real or complex list of the smaller of each pair of y9 elements in listA and listB. MATH 1:min(min(value,list) Returns a real or complex list of the smaller of value or y9 each list element. MATH 1:min(valueA nCr valueB Returns the number of combinations of valueA taken valueB at a time. PRB 3:nCrvalue nCr list Returns a list of the combinations of value taken each element in list at a time. PRB 3:nCrlist nCr value Returns a list of the combinations of each element in list taken value at a time. PRB 3:nCrlistA nCr listB Returns a list of the combinations of each element in listA taken each element in listB at a time. PRB 3:nCrn/d Displays results as a simple fraction. t^ 1: n/d or NUM D: n/dnDeriv(expression, Returns approximate numerical derivative of expression variable,value[,H]) with respect to variable at value, with specified H. MATH 8:nDeriv(4 n/d 3 4 Un/d Converts the results from a fraction to mixed number or t^ from a mixed number to a fraction, if applicable. 3: 4 n/d 3 4 Un/d or NUM A: 4 n/d 3 4 Un/d4Nom(effective rate, Computes the nominal interest rate. Œ 1:Financecompounding periods) CALC B:4Nom(Normal Sets normal display mode. †z Normalnormalcdf(lowerbound, Computes the normal distribution probability between y=upperbound[,m,s]) lowerbound and upperbound for the specified m and s. DISTR 2:normalcdf( Appendix A: Functions and Instructions 369
Function or Key orInstruction/Arguments Result Keys/Menu or Screen/Itemnormalpdf(x[,m,s]) Computes the probability density function for the normal y= distribution at a specified x value for the specified m and s. DISTR 1:normalpdf(not(value) Returns 0 if value is ƒ 0. value can be a real number, y: expression, or list. LOGIC 4:not(valueA nPr valueB Returns the number of permutations of valueA taken valueB at a time. PRB 2:nPrvalue nPr list Returns a list of the permutations of value taken each element in list at a time. PRB 2:nPrlist nPr value Returns a list of the permutations of each element in list taken value at a time. PRB 2:nPrlistA nPr listB Returns a list of the permutations of each element in listA taken each element in listB at a time. PRB 2:nPrnpv(interest rate,CF0, Computes the sum of the present values for cash inflows Œ 1:FinanceCFList[,CFFreq]) and outflows. CALC 7:npv(valueA or valueB Returns 1 if valueA or valueB is ƒ 0. valueA and valueB can y: be real numbers, expressions, or lists. LOGIC 2:orOutput(row,column, Displays text beginning at specified row and column. †"text") I/O 6:Output(Output(row,column, Displays value beginning at specified row and column. †value) I/O 6:Output(Param Sets parametric graphing mode. †z ParPause Suspends program execution until you press Í. † CTL 8:PausePause [value] Displays value; suspends program execution until you press † Í. CTL 8:PausePlot#(type,Xlistname, Defines Plot# (1, 2, or 3) of type Scatter or xyLine for †y,Ylistname,mark) Xlistname and Ylistname using mark. STAT PLOTS 1:Plot1- 2:Plot2- 3:Plot3- Appendix A: Functions and Instructions 370
Function or Key orInstruction/Arguments Result Keys/Menu or Screen/Itempxl-Test(row,column) Returns 1 if pixel (row, column) is on, 0 if it is off; y< 0 row 62 and 0 column 94. POINTS 7:pxl-Test(P4Rx(r,q) Returns X, given polar coordinates r and q or a list of polar y ; coordinates. ANGLE 7:P4Rx(P4Ry(r,q) Returns Y, given polar coordinates r and q or a list of polar y ; coordinates. ANGLE 8:P4Ry(QuadReg [Xlistname, Fits a quadratic regression model to Xlistname and …Ylistname,freqlist, Ylistname with frequency freqlist, and stores the regression CALCregequ] equation to regequ. 5:QuadRegQuartReg [Xlistname, Fits a quartic regression model to Xlistname and Ylistname …Ylistname,freqlist, with frequency freqlist, and stores the regression equation CALCregequ] to regequ. 7:QuartRegRadian Sets radian angle mode. †z Radianrand[(numtrials)] Returns a random number between 0 and 1 for a specified number of trials numtrials. PRB 1:randrandBin(numtrials,prob Generates and displays a random real number from a [,numsimulations]) specified Binomial distribution. PRB 7:randBin(randInt( lower,upper Generates and displays a random integer within a range [,numtrials]) specified by lower and upper integer bounds for a specified PRB number of trials numtrials. 5:randInt(randIntNoRep(lowerint, Returns a random ordered list of integers from a lower upperint) integer to an upper integer which may include the lower PRB integer and upper integer. 8:randIntNoRep(randM(rows,columns) Returns a random matrix of rows (1-99) × columns (1-99). y> MATH 6:randM(randNorm(m,s Generates and displays a random real number from a [,numtrials]) specified Normal distribution specified by m and s for a PRB specified number of trials numtrials. 6:randNorm(re^qi Sets the mode to polar complex number mode (re^qi). †z re^qiReal Sets mode to display complex results only when you enter † z complex numbers. Realreal(value) Returns the real part of a complex number or list of complex numbers. CPX 2:real( Appendix A: Functions and Instructions 373
Function or Key orInstruction/Arguments Result Keys/Menu or Screen/ItemRecallGDB n Restores all settings stored in the graph database variable y < GDBn. STO 4:RecallGDBRecallPic n Displays the graph and adds the picture stored in Picn. y< STO 2:RecallPiccomplex value 4Rect Displays complex value or list in rectangular format. CPX 6:4RectRectGC Sets rectangular graphing coordinates format. †y. RectGCref(matrix) Returns the row-echelon form of a matrix. y> MATH A:ref(remainder(dividend, Reports the remainder as a whole number from a division divisor) of two whole numbers where the divisor is not zero. NUM 0:remainder(remainder(list, divisor) Reports the remainder as a whole number from a division of two lists where the divisor is not zero. NUM 0:remainder(remainder(dividend, list) Reports the remainder as a whole number from a division of two whole numbers where the divisor is a list. NUM 0:remainder(remainder(list, list) Reports the remainder as a whole number from a division of two lists. NUM 0:remainder(:Repeat condition Executes commands until condition is true. †:commands CTL:End 6:Repeat:commandsReturn Returns to the calling program. † CTL E:Returnround(value[,#decimals]) Returns a number, expression, list, or matrix rounded to #decimals ( 9). NUM 2:round(ärow(value,matrix,row) Returns a matrix with row of matrix multiplied by value and y> stored in row. MATH E:ärow(row+(matrix,rowA,rowB) Returns a matrix with rowA of matrix added to rowB and y> stored in rowB. MATH D:row+( Appendix A: Functions and Instructions 374
Function or Key orInstruction/Arguments Result Keys/Menu or Screen/ItemSetUpEditor Removes all list names from the stat list editor, and then … restores list names L1 through L6 to columns 1 through 6. EDIT 5:SetUpEditorSetUpEditor listname1 Removes all list names from the stat list editor, then sets it …[,listname2,..., up to display one or more listnames in the specified order, EDITlistname20] starting with column 1. 5:SetUpEditorShade(lowerfunc, Draws lowerfunc and upperfunc in terms of X on the current y <upperfunc[,Xleft,Xright, graph and uses pattern and patres to shade the area DRAWpattern,patres]) bounded by lowerfunc, upperfunc, Xleft, and Xright. 7:Shade(Shadec2(lowerbound, Draws the density function for the c2 distribution specified y =upperbound,df) by degrees of freedom df and shades the area between DRAW lowerbound and upperbound. 3:Shadec2(ShadeÜ(lowerbound, Draws the density function for the Û distribution specified y=upperbound, by numerator df and denominator df and shades the area DRAWnumerator df, between lowerbound and upperbound. 4:ShadeÜ(denominator df)ShadeNorm(lowerbound, Draws the normal density function specified by m and s y=upperbound[,m,s]) and shades the area between lowerbound and upperbound. DRAW 1:ShadeNorm(Shade_t(lowerbound, Draws the density function for the Student-t distribution y=upperbound,df) specified by degrees of freedom df, and shades the area DRAW between lowerbound and upperbound. 2:Shade_t(Simul Sets mode to graph functions simultaneously. †z Simulsin(value) Returns the sine of a real number, expression, or list. ˜sinL1(value) Returns the arcsine of a real number, expression, or list. y?sinh(value) Returns the hyperbolic sine of a real number, expression, yN or list. sinh(sinhL1 (value) Returns the hyperbolic arcsine of a real number, yN expression, or list. sinhL1(SinReg [iterations, Attempts iterations times to fit a sinusoidal regression model …Xlistname,Ylistname, to Xlistname and Ylistname using a period guess, and stores CALCperiod,regequ] the regression equation to regequ. C:SinRegsolve(expression, Solves expression for variable, given an initial guess and †variable,guess, lower and upper bounds within which the solution is sought. MATH{lower,upper}) 0:solve(SortA(listname) Sorts elements of listname in ascending order. y9 OPS 1:SortA(SortA(keylistname, Sorts elements of keylistname in ascending order, then y9dependlist1[,dependlist2, sorts each dependlist as a dependent list. OPS...,dependlist n]) 1:SortA( Appendix A: Functions and Instructions 377
Function or Key orInstruction/Arguments Result Keys/Menu or Screen/ItemSortD(listname) Sorts elements of listname in descending order. y9 OPS 2:SortD(SortD(keylistname,dependl Sorts elements of keylistname in descending order, then y9ist1[,dependlist2, sorts each dependlist as a dependent list. OPS..., dependlist n]) 2:SortD(startTmr Starts the clock timer. Store or note the displayed value, yN and use it as the argument for checkTmr( ) to check the startTmr elapsed time.stdDev(list[,freqlist]) Returns the standard deviation of the elements in list with y9 frequency freqlist. MATH 7:stdDev(Stop Ends program execution; returns to home screen. † CTL F:StopStore: value!variable Stores value in variable. ¿StoreGDB n Stores current graph in database GDBn. y< STO 3:StoreGDBStorePic n Stores current picture in picture Picn. y< STO 1:StorePicString4Equ(string,Y= var) Converts string into an equation and stores it in Y= var. yN String4Equ(sub(string,begin,length) Returns a string that is a subset of another string, from yN begin to length. sub(sum(list[,start,end]) Returns the sum of elements of list from start to end. y9 MATH 5:sum(summation G(expression Displays the MathPrint™ summation entry template and [,start,end]) returns the sum of elements of list from start to end, where NUM start <= end. 0: summation G(tan(value) Returns the tangent of a real number, expression, or list. štanL1(value) Returns the arctangent of a real number, expression, or yA list.Tangent(expression, Draws a line tangent to expression at X=value. y<value) DRAW 5:Tangent(tanh(value) Returns hyperbolic tangent of a real number, expression, or y N list. tanh(tanhL1(value) Returns the hyperbolic arctangent of a real number, yN expression, or list. tanhL1( Appendix A: Functions and Instructions 378
Function or Key orInstruction/Arguments Result Keys/Menu or Screen/Itemtvm_PV[(Ú,æ,PMT,FV, Computes the present value. Œ 1:FinanceP/Y,C/Y)] CALC 4:tvm_PVUnArchive Moves the specified variables from the user data archive yL memory to RAM. 6:UnArchive To archive variables, use Archive.Un/d Displays results as a mixed number, if applicable. NUM C: Un/duvAxes Sets sequence graphs to plot u(n) on the x-axis and v(n) †y. on the y-axis. uvuwAxes Sets sequence graphs to plot u(n) on the x-axis and w(n) †y. on the y-axis. uw1-Var Stats [Xlistname, Performs one-variable analysis on the data in Xlistname …freqlist] with frequency freqlist. CALC 1:1-Var Stats2-Var Stats [Xlistname, Performs two-variable analysis on the data in Xlistname …Ylistname,freqlist] and Ylistname with frequency freqlist. CALC 2:2-Var Statsvariance(list[,freqlist]) Returns the variance of the elements in list with frequency y 9 freqlist. MATH 8:variance(Vertical x Draws a vertical line at x. y< DRAW 4:VerticalvwAxes Sets sequence graphs to plot v(n) on the x-axis and w(n) †y. on the y-axis. vwWeb Sets sequence graphs to trace as webs. †y. Web:While condition Executes commands while condition is true. †:commands CTL:End 5:While:commandvalueA xor valueB Returns 1 if only valueA or valueB = 0. valueA and valueB y: can be real numbers, expressions, or lists. LOGIC 3:xorZBox Displays a graph, lets you draw a box that defines a new †q viewing window, and updates the window. ZOOM 1:ZBoxZDecimal Adjusts the viewing window so that @X=0.1 and @Y=0.1, †q and displays the graph screen with the origin centered on ZOOM the screen. 4:ZDecimal Appendix A: Functions and Instructions 380
Function or Key orInstruction/Arguments Result Keys/Menu or Screen/ItemZFrac 1/2 Sets the window variables so that you can trace in q increments of , if possible. Sets @X and @Y to . ZOOM B:ZFrac1/2ZFrac 1/3 Sets the window variables so that you can trace in q increments of , if possible. Sets @X and @Y to . ZOOM C:ZFrac1/3ZFrac 1/4 Sets the window variables so that you can trace in q increments of , if possible. Sets @X and @Y to . ZOOM D:ZFrac1/4ZFrac 1/5 Sets the window variables so that you can trace in q increments of , if possible. Sets @X and @Y to . ZOOM E:ZFrac1/5ZFrac 1/8 Sets the window variables so that you can trace in q increments of , if possible. Sets @X and @Y to . ZOOM F:ZFrac1/8ZFrac 1/10 Sets the window variables so that you can trace in q increments of , if possible. Sets @X and @Y to . ZOOM G:ZFrac1/10ZInteger Redefines the viewing window using these dimensions: †q @X=1 Xscl=10 ZOOM @Y=1 Yscl=10 8:ZIntegerZInterval s[,listname, Computes a z confidence interval. †…freqlist,confidence level] TESTS(Data list input) 7:ZIntervalZInterval s,v,n Computes a z confidence interval. †…[,confidence level] TESTS(Summary stats input) 7:ZIntervalZoom In Magnifies the part of the graph that surrounds the cursor †q location. ZOOM 2:Zoom InZoom Out Displays a greater portion of the graph, centered on the †q cursor location. ZOOM 3:Zoom OutZoomFit Recalculates Ymin and Ymax to include the minimum and † q maximum Y values, between Xmin and Xmax, of the ZOOM selected functions and replots the functions. 0:ZoomFitZoomRcl Graphs the selected functions in a user-defined viewing †q window. MEMORY 3:ZoomRclZoomStat Redefines the viewing window so that all statistical data †q points are displayed. ZOOM 9:ZoomStat Appendix A: Functions and Instructions 381
Appendix B:Reference InformationVariablesUser VariablesThe TI-84 Plus uses the variables listed below in various ways. Some variables are restricted tospecific data types.The variables A through Z and q are defined as real or complex numbers. You may store to them.The TI-84 Plus can update X, Y, R, q, and T during graphing, so you may want to avoid using thesevariables to store nongraphing data.The variables (list names) L1 through L6 are restricted to lists; you cannot store another type ofdata to them.The variables (matrix names) [A] through [J] are restricted to matrices; you cannot store anothertype of data to them.The variables Pic1 through Pic9 and Pic0 are restricted to pictures; you cannot store another typeof data to them.The variables GDB1 through GDB9 and GDB0 are restricted to graph databases; you cannot storeanother type of data to them.The variables Str1 through Str9 and Str0 are restricted to strings; you cannot store another type ofdata to them.Except for system variables, you can store any string of characters, functions, instructions, orvariables to the functions Yn, (1 through 9, and 0), XnT/YnT (1 through 6), rn (1 through 6), u(n), v(n),and w(n) directly or through the Y= editor. The validity of the string is determined when the function isevaluated.Archive VariablesYou can store data, programs or any variable from RAM to user data archive memory where theycannot be edited or deleted inadvertantly. Archiving also allows you to free up RAM for variables thatmay require additional memory. The names of archived variables are preceded by an asterisk (*)indicating they are in user data archive.System VariablesThe variables below must be real numbers. You may store to them. Since the TI-84 Plus canupdate some of them, as the result of a ZOOM, for example, you may want to avoid using thesevariables to store nongraphing data.• Xmin, Xmax, Xscl, @X, XFact, Tstep, PlotStart, nMin, and other window variables. Appendix B: Reference Information 386
CP = compounding periods Nom = nominal rateDays between DatesWith the dbd( function, you can enter or compute a date within the range Jan. 1, 1950, throughDec. 31, 2049.Actual/actual day-count method (assumes actual number of days per month and actual numberof days per year):dbd( (days between dates) = Number of Days II - Number of Days I Number of Days I = (Y1-YB) 365 + (number of days MB to M1) + DT1 + Y1 – YB ------------------------ 4 Number of Days II = (Y2-YB) 365 + (number of days MB to M2) + DT2 + Y2 – YB ------------------------ 4where: M1 = month of first date DT1 = day of first date Y1 = year of first date M2 = month of second date DT2 = day of second date Y2 = year of second date MB = base month (January) DB = base day (1) YB = base year (first year after leap year) Appendix B: Reference Information 395
Important Things You Need to Know About Your TI-84 PlusTI-84 Plus ResultsThere may be a number of reasons that your TI-84 Plus is not displaying the expected results;however, the most common solutions involve order of operations or mode settings. Your calculatoruses an Equation Operating System™ (EOS™) which evaluates the functions in an expression inthe following order:1. Functions that precede the argument, such as square root, sin(, or log(2. Functions that are entered after the argument, such as exponents, factorial, r, ¡, and conversions3. Powers and roots, such as 2^5, or 5*square root(32)4. Permutations (nPr) and combinations (nCr)5. Multiplication, implied multiplication, and division6. Addition and subtraction7. Relational functions, such as > or <8. Logic operator and9. Logic operators or and xorRemember that EOS™ evaluates from left to right and calculations within parentheses areevaluated first. You should use parentheses where the rules of algebra may not be clear. In OS2.53 MP, parentheses may be pasted in an expression to indicate how the input is interpreted.If you are using trigonometric functions or performing polar and rectangular conversions, theunexpected results may be caused by an angle mode setting. The Radian and Degree angle modesettings control how the TI-84 Plus interprets angle values.To change the angle mode settings, follow these steps:1. Press z to display the Mode settings.2. Select Degree or Radian.3. Press Í to save the angle mode setting.ERR:DIM MISMATCH ErrorYour TI-84 Plus displays the ERR:DIM MISMATCH error if you are trying to perform an operationthat references one or more lists or matrices whose dimensions do not match. For example,multiplying L1*L2, where L1={1,2,3,4,5} and L2={1,2} produces an ERR:DIM MISMATCH errorbecause the number of elements in L1 and L2 do not match. Appendix B: Reference Information 396
ERR:INVALID DIM ErrorThe ERR:INVALID DIM error message may occur if you are trying to graph a function that does notinvolve the stat plot features. The error can be corrected by turning off the stat plots. To turn thestat plots off, press y , and then select 4:PlotsOff.Link-Receive L1 (or any file) to Restore MessageYour TI-84 Plus displays the Link-Receive L1 (or any file) to Restore message if it has been disabledfor testing, and not re-enabled. To restore your calculator to full functionality after testing, link toanother TI-84 Plus and transfer any file to the disabled calculator, or use TI Connect™ software todownload a file from your computer to your TI-84 Plus.To transfer a file from another TI-84 Plus:1. On the receiving unit, press y 8 and then select RECEIVE.2. On the sending calculator, Press y 8.3. Select a file to send by selecting a category, and then selecting a file to send.4. Select TRANSMIT to send the file.Contrast FeatureIf the contrast setting is too dark (set to 9) or too dim (set to 0) the unit may appear as if it ismalfunctioning or turned off. To adjust the contrast, press and release y, and then press and hold} or †.TI-84 Plus Identification CodeYour graphing calculator has a unique identification (ID) code that you should record and keep.You can use this 14 digit ID to register your calculator at education.ti.com or identify your calculatorin the event that it is lost or stolen. A valid ID includes numbers 0 through 9 and the letters Athrough F. Appendix B: Reference Information 397
You can view the calculator's Operating System, Product Number, ID, and Certificate RevisionNumber from the About screen. To display the About screen, press y L and then select1:About.Your unique product ID code: _____________________________BackupsYour TI-84 Plus is similar to a computer, in that it stores files and Apps that are important to you. Itis always a good idea to back up your graphing calculator device files and Apps using theTI Connect™ software and a USB computer cable. You can find the specific procedures forbacking up your calculator's device files and Apps in the TI Connect™ Help file.AppsTI-84 Plus Software Applications (Apps) is software that you can add to your calculator in thesame way you would add software to your computer. Apps let you customize your calculator forpeak performance in specific areas of study. You can find apps for the TI-84 Plus ateducation.ti.com.TI-Cares KnowledgeBaseThe TI-Cares KnowledgeBase provides 24-hour access through the Web to find answers tofrequently asked questions. The TI-Cares KnowledgeBase searches its repository of knownsolutions and presents you with the solutions that are most likely to solve your problem. You cansearch the TI-Cares KnowledgeBase at education.ti.com/support. Appendix B: Reference Information 398
Error ConditionsWhen the TI-84 Plus detects an error, it returns an error message as a menu title, such asERR:SYNTAX or ERR:DOMAIN. This table contains each error type, possible causes, andsuggestions for correction. The error types listed in this table are each preceded by ERR: on yourgraphing calculator display. For example, you will see ERR:ARCHIVED as a menu title when yourgraphing calculator detects an ARCHIVED error type.Error Type Possible Causes and Suggested RemediesARCHIVED You have attempted to use, edit, or delete an archived variable. For example, the expression dim(L1) produces an error if L1 is archived.ARCHIVE FULL You have attempted to archive a variable and there is not enough space in archive to receive it.ARGUMENT A function or instruction does not have the correct number of arguments. See Appendix A for function and instruction syntax, stdDev(list[,freqlist]) might be entered as stdDev(L1) or stdDev(L1,L2) since the frequency list or freqlist is optional.BAD ADDRESS You have attempted to send or receive an application and an error (e.g. electrical interference) has occurred in the transmission.BAD GUESS • In a CALC operation, you specified a Guess that is not between Left Bound and Right Bound. • For the solve( function or the equation solver, you specified a guess that is not between lower and upper. • Your guess and several points around it are undefined. Examine a graph of the function. If the equation has a solution, change the bounds and/or the initial guess.BOUND • In a CALC operation or with Select(, you defined Left Bound > Right Bound. • In fMin(, fMax(, solve(, or the equation solver, you entered lower ' upper.BREAK You pressed the É key to break execution of a program, to halt a DRAW instruction, or to stop evaluation of an expression. Appendix B: Reference Information 399
Error Type Possible Causes and Suggested RemediesDATA TYPE You entered a value or variable that is the wrong data type. • For a function (including implied multiplication) or an instruction, you entered an argument that is an invalid data type, such as a complex number where a real number is required. See Appendix A and the appropriate chapter. • In an editor, you entered a type that is not allowed, such as a matrix entered as an element in the stat list editor. See the appropriate chapter. • You attempted to store an incorrect data type, such as a matrix, to a list.DIM MISMATCH Your calculator displays the ERR:DIM MISMATCH error if you are trying to perform an operation that references one or more lists or matrices whose dimensions do not match. For example, multiplying L1*L2, where L1={1,2,3,4,5} and L2={1,2} produces an ERR:DIM MISMATCH error because the number of elements in L1 and L2 do not match.DIVIDE BY 0 • You attempted to divide by zero. This error is not returned during graphing. The TI-84 Plus allows for undefined values on a graph. • You attempted a linear regression with a vertical line.DOMAIN • You specified an argument to a function or instruction outside the valid range. This error is not returned during graphing. The TI-84 Plus allows for undefined values on a graph. See Appendix A. • You attempted a logarithmic or power regression with a LX or an exponential or power regression with a LY. • You attempted to compute GPrn( or GInt( with pmt2 < pmt1.DUPLICATE You attempted to create a duplicate group name.Duplicate Name A variable you attempted to transmit cannot be transmitted because a variable with that name already exists in the receiving unit.EXPIRED You have attempted to run an application with a limited trial period which has expired.Error in Xmit • The TI-84 Plus was unable to transmit an item. Check to see that the cable is firmly connected to both units and that the receiving unit is in receive mode. • You pressed É to break during transmission. • You attempted to perform a backup from a TI.82 to a TI-84 Plus. • You attempted to transfer data (other than L1 through L6) from a TI-84 Plus to a TI.82. • You attempted to transfer L1 through L6 from a TI-84 Plus to a TI.82 without using 5:Lists to TI82 on the LINK SEND menu. Appendix B: Reference Information 400
Error Type Possible Causes and Suggested RemediesID NOT FOUND This error occurs when the SendID command is executed but the proper graphing calculator ID cannot be found.ILLEGAL NEST • You attempted to use an invalid function in an argument to a function, such as seq( within expression for seq(.INCREMENT • The increment in seq( is 0 or has the wrong sign. This error is not returned during graphing. The TI-84 Plus allows for undefined values on a graph. • The increment in a For( loop is 0.INVALID • You attempted to reference a variable or use a function where it is not valid. For example, Yn cannot reference Y, Xmin, @X, or TblStart. • You attempted to reference a variable or function that was transferred from the TI.82 and is not valid for the TI-84 Plus For example, you may have transferred UnN1 to the TI-84 Plus from the TI.82 and then tried to reference it. • In Seq mode, you attempted to graph a phase plot without defining both equations of the phase plot. • In Seq mode, you attempted to graph a recursive sequence without having input the correct number of initial conditions. • In Seq mode, you attempted to reference terms other than (nN1) or (nN2). • You attempted to designate a graph style that is invalid within the current graph mode. • You attempted to use Select( without having selected (turned on) at least one xyLine or scatter plot.INVALID DIM • The ERR:INVALID DIM error message may occur if you are trying to graph a function that does not involve the stat plot features. The error can be corrected by turning off the stat plots. To turn the stat plots off, press y , and then select 4:PlotsOff. • You specified a list dimension as something other than an integer between 1 and 999. • You specified a matrix dimension as something other than an integer between 1 and 99. • You attempted to invert a matrix that is not square.ITERATIONS • The solve( function or the equation solver has exceeded the maximum number of permitted iterations. Examine a graph of the function. If the equation has a solution, change the bounds, or the initial guess, or both. • irr( has exceeded the maximum number of permitted iterations. • When computing æ, the maximum number of iterations was exceeded. Appendix B: Reference Information 401
Error Type Possible Causes and Suggested RemediesLABEL The label in the Goto instruction is not defined with a Lbl instruction in the program.LINK L1 (or any The calculator has been disabled for testing. To restoreother file) to full functionality, use TI Connect™ software toRestore download a file to your calculator from your computer, or transfer any file to your calculator from another TI-84 Plus. (See the instructions under Important Things to Know about your TI-84 Plus, earlier in this chapter.)MEMORY Memory is insufficient to perform the instruction or function. You must delete items from memory before executing the instruction or function. Recursive problems return this error; for example, graphing the equation Y1=Y1. Branching out of an If/Then, For(, While, or Repeat loop with a Goto also can return this error because the End statement that terminates the loop is never reached.MemoryFull • You are unable to transmit an item because the receiving unit's available memory is insufficient. You may skip the item or exit receive mode. • During a memory backup, the receiving unit's available memory is insufficient to receive all items in the sending unit's memory. A message indicates the number of bytes the sending unit must delete to do the memory backup. Delete items and try again.MODE You attempted to store to a window variable in another graphing mode or to perform an instruction while in the wrong mode; for example, DrawInv in a graphing mode other than Func.NO SIGN CHNG • The solve( function or the equation solver did not detect a sign change. • You attempted to compute æ when FV, (Ú…PMT), and PV are all ' 0, or when FV, (Ú…PMT), and PV are all _ 0. • You attempted to compute irr( when neither CFList nor CFO is > 0, or when neither CFList nor CFO is < 0.NONREAL ANS In Real mode, the result of a calculation yielded a complex result. This error is not returned during graphing. The TI-84 Plus allows for undefined values on a graph.OVERFLOW You attempted to enter, or you have calculated, a number that is beyond the range of the graphing calculator. This error is not returned during graphing. The TI-84 Plus allows for undefined values on a graph.RESERVED You attempted to use a system variable inappropriately. See Appendix A. Appendix B: Reference Information 402
Error Type Possible Causes and Suggested RemediesSINGULAR MAT • A singular matrix (determinant = 0) is not valid as the argument for L1. • The SinReg instruction or a polynomial regression generated a singular matrix (determinant = 0) because it could not find a solution, or a solution does not exist. This error is not returned during graphing. The TI-84 Plus allows for undefined values on a graph.SINGULARITY expression in the solve( function or the equation solver contains a singularity (a point at which the function is not defined). Examine a graph of the function. If the equation has a solution, change the bounds or the initial guess or both.STAT You attempted a stat calculation with lists that are not appropriate. • Statistical analyses must have at least two data points. • Med-Med must have at least three points in each partition. • When you use a frequency list, its elements must be ' 0. • (Xmax N Xmin) à Xscl must be' 47 for a histogram.STAT PLOT You attempted to display a graph when a stat plot that uses an undefined list is turned on.SYNTAX The command contains a syntax error. Look for misplaced functions, arguments, parentheses, or commas stdDev(list[,freqlist]) might be entered as stdDev(L1) or stdDev(L1,L2) since the frequency list or freqlist is optional.TOL NOT MET You requested a tolerance to which the algorithm cannot return an accurate result.UNDEFINED You referenced a variable that is not currently defined. For example, you referenced a stat variable when there is no current calculation because a list has been edited, or you referenced a variable when the variable is not valid for the current calculation, such as a after Med-Med.VALIDATION Electrical interference caused a link to fail or this graphing calculator is not authorized to run the application. Appendix B: Reference Information 403
Error Type Possible Causes and Suggested RemediesVARIABLE You have tried to archive a variable that cannot be archived or you have tried to unarchive an application or group. Examples of variables that cannot be archived include: • Real numbers LRESID, R, T, X, Y, Theta, Statistic variables under Vars, STATISTICS menu, Yvars, and the AppIdList.VERSION You have attempted to receive an incompatible variable version from another graphing calculator.WINDOW A problem exists with the window variables.RANGE • You defined Xmax Xmin or Ymax Ymin. • You defined qmax qmin and qstep > 0 (or vice versa). • You attempted to define Tstep=0. • You defined Tmax Tmin and Tstep > 0 (or vice versa). • Window variables are too small or too large to graph correctly. You may have attempted to zoom in or zoom out to a point that exceeds the TI-84 Plus's numerical range.ZOOM • A point or a line, instead of a box, is defined in ZBox. • A ZOOM operation returned a math error. Appendix B: Reference Information 404
Accuracy InformationComputational AccuracyTo maximize accuracy, the TI-84 Plus carries more digits internally than it displays. Values arestored in memory using up to 14 digits with a two-digit exponent.• You can store a value in the window variables using up to 10 digits (12 for Xscl, Yscl, Tstep, and qstep).• Displayed values are rounded as specified by the mode setting with a maximum of 10 digits and a two-digit exponent.• RegEQ displays up to 14 digits in Float mode. Using a fixed-decimal setting other than Float causes RegEQ results to be rounded and stored with the specified number of decimal places.Xmin is the center of the leftmost pixel, Xmax is the center of the next-to-the-rightmost pixel. (Therightmost pixel is reserved for the busy indicator.) @X is the distance between the centers of twoadjacent pixels.• In Full screen mode, @X is calculated as (Xmax N Xmin) à 94. In G-T split-screen mode, @X is calculated as (Xmax N Xmin) à 46.• If you enter a value for @X from the home screen or a program in Full screen mode, Xmax is calculated as Xmin + @X É… 94. In G-T split-screen mode, Xmax is calculated as Xmin + @X É… 46.Ymin is the center of the next-to-the-bottom pixel; Ymax is the center of the top pixel. @Y is thedistance between the centers of two adjacent pixels.• In Full screen mode, @Y is calculated as (Ymax N Ymin) à 62. In Horiz split-screen mode, @Y is calculated as (Ymax N Ymin) à 30. In G-T split-screen mode, @Y is calculated as (Ymax N Ymin) à 50.• If you enter a value for @Y from the home screen or a program in Full screen mode, Ymax is calculated as Ymin + @Y É… 62. In Horiz split-screen mode, Ymax is calculated as Ymin + @Y … 30. In G-T split-screen mode, Ymax is calculated as Ymin + @Y É … 50.Cursor coordinates are displayed as eight-character numbers (which may include a negative sign,decimal point, and exponent) when Float mode is selected. X and Y are updated with a maximumaccuracy of eight digits.minimum and maximum on the CALCULATE menu are calculated with a tolerance of 1âL5; ‰f(x)dx iscalculated at 1âL3. Therefore, the result displayed may not be accurate to all eight displayed digits.For most functions, at least five accurate digits exist. For fMin(, fMax(, and fnInt( on the MATH menuand solve( in the CATALOG, the tolerance can be specified.Function LimitsFunction Range of Input Valuessin x, cos x, tan x 0 |x| < 10 12 (radian or degree) Appendix B: Reference Information 405
Appendix C:Service and Warranty InformationTexas Instruments Support and ServiceFor general informationHome Page: education.ti.comKnowledgeBase and education.ti.com/supporte-mail inquiries:Phone: (800) TI-CARES / (800) 842-2737 For U.S., Canada, Mexico, Puerto Rico, and Virgin Islands onlyInternational education.ti.com/internationalinformation:For product (hardware) serviceCustomers in the U.S., Canada, Mexico, Puerto Rico and Virgin Islands: Always contact TexasInstruments Customer Support before returning a product for service.All other customers: Refer to the leaflet enclosed with this product (hardware) or contact your localTexas Instruments retailer/distributor.Battery InformationWhen to Replace the BatteriesThe TI-84 Plus uses five batteries: four AAA alkaline batteries and one button cell backup battery.The backup battery provides auxiliary power to retain memory while you replace the AAAbatteries.When the battery voltage level drops below a usable level, the TI-84 Plus: Displays this message when Displays this message when you attempt you turn on the unit. to download an application. Message A Message B Appendix C: Service and Warranty Information 407
After Message A is first displayed, you can expect the batteries to function for about one or twoweeks, depending on usage. (This one-week to two-week period is based on tests with alkalinebatteries; the performance of other types of batteries may vary.)If Message B is displayed, you must replace the batteries immediately to successfully download anapplication.Effects of Replacing the BatteriesDo not remove both types of batteries (AAA and backup ) at the same time. Do not allow thebatteries to lose power completely. If you follow these guidelines and the steps for replacingbatteries, you can replace either type of battery without losing any information in memory.Battery PrecautionsTake these precautions when replacing batteries.• Do not leave batteries within reach of children• Do not mix new and used batteries. Do not mix brands (or types within brands) of batteries.• Do not mix rechargeable and nonrechargeable batteries.• Install batteries according to polarity (+ and N) diagrams.• Do not place nonrechargeable batteries in a battery recharger.• Properly dispose of used batteries immediately. Do not leave them within the reach of children.• Do not incinerate or dismantle batteries.Disposing of used batteries safely and properlyDo not mutilate, puncture, or dispose of batteries in fire. The batteries can burst or explode,releasing hazardous chemicals. Discard used batteries according to local regulations.Replacing the BatteriesTo replace the batteries, follow these steps.1. Turn off the graphing calculator. Replace the slide cover over the keyboard to avoid inadvertently turning on the graphing calculator. Turn the back of the unit toward you.2. Hold the graphing calculator upright, push downward on the latch on the top of the battery cover, and then pull the cover toward you. Note: To avoid loss of information stored in memory, you must turn off the graphing calculator. Do not remove the AAA batteries and the backup battery simultaneously.3. Replace all four AAA alkaline batteries simultaneously. Or, replace the backup battery. • To replace the AAA alkaline batteries, remove all four discharged AAA batteries and install new ones according to the polarity (+ and N) diagram in the battery compartment. Appendix C: Service and Warranty Information 408
• To replace the backup battery, remove the screw from the backup battery cover, and then remove the cover. Install the new battery, + side up. Replace the cover and secure it with the screw.4. Replace the battery compartment cover. Turn the graphing calculator on and adjust the display contrast, if necessary, by pressing y } or †. Appendix C: Service and Warranty Information 409
In Case of DifficultyHandling a DifficultyTo handle a difficulty, follow these steps.1. If you cannot see anything on the screen, you may need to adjust the graphing calculator contrast. To darken the screen, press and release y, and then press and hold } until the display is sufficiently dark. To lighten the screen, press and release y, and then press and hold † until the display is sufficiently light.2. If an error menu is displayed, follow these steps: • Note the error type (ERR:error type). • Select 2:GOTO, if it is available. The previous screen is displayed with the cursor at or near the error location. • Deteremine the error. • Correct the expression. Refer to the Error Conditions table for details about specific errors, if necessary.3. If the busy indicator (dotted line) is displayed, a graph or program has been paused; the TI-84 Plus is waiting for input. Press Í to continue or press É to break.4. If a checkerboard cursor ( # ) is displayed, then either you have entered the maximum number of characters in a prompt, or memory is full. If memory is full: • Press y L 2 to display the MEMORY MANAGEMENT / DELETE menu. • Select the type of data you want to delete, or select 1:All for a list of all variables of all types. A screen is displayed listing each variable of the type you selected and the number of bytes each variable is using. • Press } and † to move the selection cursor (4) next to the item you want to delete, and then press {.5. If the graphing calculator does not seem to work at all, be sure the alkaline batteries are fresh and that they are installed properly.6. If the TI-84 Plus does not function even though you are sure that the batteries are fresh, you can try manually resetting it. • Remove all of the AAA batteries from the graphing calculator. • Press and hold the É key for ten seconds. • Replace the batteries. • Turn on the unit. When you reset your graphing calculator, the contrast sometimes changes. If the screen is faded or blank, adjust the contrast by pressing y and releasing } or †.7. If the above solutions do not work you can reset all of the memory. The RAM, user data archive memory, and system variables are restored to factory settings when you reset all memory. All nonsystem variables, applications (Apps), and programs are deleted. Appendix C: Service and Warranty Information 410
• Press y L to display the MEMORY menu.• Select 7:Reset to display the RAM ARCHIVE ALL menu.• Press ~ ~ to display the ALL menu.• Select 1:All Memory to display the RESET MEMORY menu.• To continue with the reset, select 2:Reset. The message Mem cleared is displayed on the home screen. Appendix C: Service and Warranty Information 411 |
OneStone® Pebbles Surfaces of Revolution
The Surfaces of Revolution Pebble
provides a high-quality visualization for what is often percieved to be one of the
more difficult topics in the calculus syllabus. The 2D display shows a user-defined
2D curve, while the 3D display shows the surface created by rotating the 2D curve
about an axis. The curve and surface can be animated by varying any constant in
the expression, as described on the Using Pebbles
page.
This Pebble accepts all the standard math
operators plus the variable X or Y; expressions may use one or
the other, but not both. Functions are graphed according to the following rules:
Functions in which X is the only independent variable are drawn in the 2D Graph
as the curve Y = F(X). This curve is then rotated about the X axis to form the surface
shown in the 3D graph.
Functions in which Y is the only independent variable are drawn in the 2D Graph
as X = F(Y). This curve is then rotated about the Y axis to form the surface shown
in the 3D graph.
Click the 'Launch...' link below to start the Surfaces of Revolution Pebble. The
process of installing and running a Pebble for the first time is explained on the
Installing Pebbles page. Running
one after the first time is quicker: the JOGL package doesn't have to be downloaded,
plus something other than the generic 'Starting Java' image is displayed during
start-up.
A special thanks to Dr. Bryan Shader of the University of Wyoming for suggesting
this topic |
Dr John Kelliher
This book is available for download with iBooks on your Mac or iOS device, and with iTunes on your computer. Books can be read with iBooks on your Mac or iOS device.
Description
This eBook introduces the rational and irrational numbers, whole numbers, fractions, terminating or recurring decimals as well as never-ending non-repeating decimals as well as their manipulation.This eBook is part of our range of Grades 9 & 10 math eBooks that are aligned with the North American math curriculum.Our Grades 9 & 10 math eBooks comprise three principle sections. These are, notably:•Number and Algebra•Geometry and Measures•StatisticsIn addition, there exists a Publications Guide.Our math eBooks are produced such as that as well as a Publications Guide, and three principle publications corresponding to the principle sections (Number and Algebra, Geometry and Measures and Statistics) there are individual modules produced within each principle section which are published as eBooks.Rational and Irrational Numbers is a module within the Number and Algebra principle section of our Grades 9 & 10 publications. |
A study was performed using a convenience sample of 90 students at a northeastern community college to determine gender differences of math anxiety and its effect on math avoidance. Four sections of an introductory English class were given aWe present a collection of algorithms which are used to construct Hamiltonian paths
and cycles in hypercube networks with faulty nodes. In graph theoretical terms, the
hypercube is a connected, undirected graph composed of edges and nodes, or...
Numerous studies can be found regarding the development of mathematics anxiety with secondary and college level students. However, little can be found providing evidence of mathematics anxiety in elementary school students. In an attempt to offer...
The purpose of this study was to examine how mathematical discourse affects the achievement level of girls across third, fourth, and fifth grades. The Experimental Group sample of 26 students (13 boys, 13 girls) all participated in the Project M³With the rules and regulations the federal government has put in place, financial institutions must consistently improve their anti-money laundering programs. Financial institutions use alert monitoring systems to combat a number of moneyEach year thousands of students are tracked into mathematics classes. In these particular classes, students may struggle or find their mathematics skills less academically able than their classmates and give up on the tasks that are introduced to...
The age a child should be when beginning formal education has become a focus of interest. How old should a child be? Is there a correlation between age and performance on mathematical strategies? Kindergarten programs have become more knowledge and...
The purpose of this study is to determine if a personality test combined with a mathematics placement test can be used to identify students that may need additional interventions as early in the school year as possible. The school in which thisThe debate over the effectiveness of a "reform" mathematics curriculum verses a "traditional" mathematics curriculum in student achievement has been widely discussed and studied. Standards-based, or reform, programs emphasize applications to solve...
The purpose of this study was to examine if there is a correlation between cognitive development levels and CMT's scores across fourth, fifth, and sixth grade. The sample size in this study consisted of 30 students (10 from grade four, 10 from... |
intended to be a thorough introduction to the subject of order and lattices and can be used for a course at the graduate or advanced undergraduate level or for independent study. Prerequisites consist mostly of a bit of mathematical maturity, such as that provided by a basic undergraduate course in abstract algebra. |
Wednesday, September 19, 2007
The Education of T.C. Mits by Lillian R. Lieber
T.C Mits stands for The Celebrated Man In The Street, so this book applies to people with minimal knowlage of math. This book is writen in freeverse poetry and is written in two parts. The first part talks about the old way of math: algebra,geometry, calculus, and trig, so if you haven't studied these branches of math yet, you might not get the book. The second part talks about integrating math into the real world. The second portion of the book explains how math can fit into politics and other subjects of that nature.
I was confused most of the entire way. I do not believe that the boook was writen for young adults because, unless you have taken Calculus before, you are lost on page one. One positive thing about the book was the fact that there was a math problem at almost every chapter, so it kept you interested and wanting to find out the answers. Unfortunatly for the readers, the book was written a long time ago, so there might be some issues currently with the computation of the math. This book does go into detail about math and how math applies to politics, so I would not recomend this book for anyone who dislikes math OR has never taken any kind of math besides Algebra 2.
Rating (0 - 10 scale): 3 |
A Classical Introduction to Galois Theory
Overview
This book provides an introduction to Galois theory and focuses on one central theme - the solvability of polynomials by radicals. Both classical and modern approaches to the subject are described in turn in order to have the former (which is relatively concrete and computational) provide motivation for the latter (which can be quite abstract). The theme of the book is historically the reason that Galois theory was created, and it continues to provide a platform for exploring both classical and modern concepts. This book examines a number of problems arising in the area of classical mathematics, and a fundamental question to be considered is: For a given polynomial equation (over a given field), does a solution in terms of radicals exist? That the need to investigate the very existence of a solution is perhaps surprising and invites an overview of the history of mathematics. The classical material within the book includes theorems on polynomials, fields, and groups due to such luminaries as Gauss, Kronecker, Lagrange, Ruffini and, of course, Galois. These results figured prominently in earlier expositions of Galois theory, but seem to have gone out of fashion. This is unfortunate since, aside from being of intrinsic mathematical interest, such material provides powerful motivation for the more modern treatment of Galois theory presented later in the book. Over the course of the book, three versions of the Impossibility Theorem are presented: the first relies entirely on polynomials and fields, the second incorporates a limited amount of group theory, and the third takes full advantage of modern Galois theory. This progression through methods that involve more and more group theory characterizes the first part of the book. The latter part of the book is devoted to topics that illustrate the power of Galois theory as a computational tool, but once again in the context of solvability of polynomial equations by radicals.
Author Information
STEPHEN C. NEWMAN, MD, MSc, is Professor Emeritus of Psychiatry at the University of Alberta, Canada. He has published widely in psychiatric epidemiology and epidemiologic methods. Dr. Newman is the author of Biostatistical Methods in Epidemiology (Wiley).
Customer Reviews
9781118091395
There are no customer reviews available at this time. Would you like to write a review? |
Downloads
More Info
Project Goal(s):
The goal of the project is to increase chemistry students' knowledge
of mathematics and its role in chemistry. In particular, we want to
enable the students to use the tools and language of mathematics to
solve scientific problems.
The project has the following specific components:
We developed a new placement exam to identify students who cannot solve
multi-step non-algorithmic word problems at the precalculus level.
We developed a math course which teaches problem solving in the
sciences.
We developed lab materials for Organic Chemistry which reinforce
mathematical skills and skills they have learned in General
Chemistry.
We developed modules on Linear Algebra and Differential Equations in
Physical Chemistry
We developed an interdisciplinary course on Chirality for mathematics
and chemistry students.
Investigators:
Principal Investigator:
Erica L Flapan, Pomona College
Enhancing the Mathematical Understanding of Students in Chemistry
Contact Information:
Placement Exam:
We believe the main difficulty facing beginning chemistry students is their mathematics skills.
We identified the following areas of mathematics which are used in General
Chemistry:
Logarithms and anti-logarithms
Percentages, proportions, and averages
Scientific Notation
Factoring, solving equations and systems of equations
Functional relations, proportionality, linear relations
We have developed a new Chemistry placement exam which tests
students' ability to
do multi-step non-algorithmic problems involving these areas of
mathematics.
This exam contains no questions on chemistry.
Our assessment consultant, Barbara Gonzalez, together with her students evaluated the effectiveness of the materials we developed. Their findings may be downloaded below.
For a copy of the exam, interested professionals may contact
Kathy Sheldon with your name, institution, and department.
Some conditions will accompany use of the exam, such as
keeping the exam confidential.
Problem Solving Course:
We developed a new Math course entitled 'Problem Solving in the
Sciences.' The course was geared for students who performed poorly
on the Chemistry
placement exam. The course taught the students to use precalculus
level mathematics to
solve problems that occur in the sciences. The course was organized
around scientific
rather than mathematical themes. Students in the course learned to
use their own
mathematical reasoning rather than relying on algorithms that they
were taught.
We developed a collection of 11 worksheets for this class, which teach
the students to use
mathematical techniques to solve hard real world problems related to a
variety of
scientific disciplines as well as economics and psychology.
Organic Chemistry Module:
In order to reinforce and strengthen the Organic Chemistry students' ability
to use mathematical and scientific reasoning together, we developed a
sophomore organic laboratory in which students first derive the
calculus of a kinetic rate process and then use the expression to study
a series of alkyl halides being subject to a dehydrohalogenation
reaction.
Physical Chemistry Module:
We have developed three mathematical modules for the physical chemistry
course: one preliminary module on background material (Preliminary Module), one on
differentiability of functions of several variables (Module I), and
another on partial
differential equations and boundary value problems (Module II).
The Preliminary Mathematics Module for Physical Chemistry is focused on Calculus and
Approximations. Calculus is a powerful tool for working with functions
when only partial information about the function is known. In this
module, the students review basic ideas of differential calculus by
focusing on local approximations of functions. Linearization and Taylor
polynomials are introduced as methods for using calculus to approximate
unknown quantities, functions, and solutions to differential equations.
Module I is intended to be
a refresher of multivariable calculus concepts on differentiability.
Its focus is on the chain
rule and its consequences (differentials and change of variables).
The main goal of the
first module is to enhance students' understanding and competence on
these concepts so
that when they encounter them in thermodynamics and quantum mechanics,
they have
enough confidence when applying them.
Module II is more advanced and
more difficult. It
introduces the students through the development of the vibrating
string problem to the
method of separation of variables and eigenfunctions expansion which
are crucial in
dealing with problems in quantum mechanics.
Chirality:
Dan O'Leary (Chemistry) and Erica Flapan (Mathematics) developed and co-taught a new
interdiscilinary course on
chirality. The syllabus for this new course may be downloaded below. For more information contact Erica Flapan. |
MathematicsCalc is arbitrary precision C-like arithmetic system that is a calculator, an algorithm prototyper and mathematical research tool. Calc comes with a rich set of builtin mathematical and programmatic functions |
CliffsNotes Algebra II Practice Pack (Cliffnotes)
Book Description: Your guide to a higher score in Algebra IIWhy CliffsNotes?Go with the name you know and trustGet the information you need-fast!About the Contents:PretestHelps you pinpoint where you need the most help and directs you to the corresponding sections of the book Topic Area ReviewsMath basicsFactoring and solving equationsFunction operations and transformationsPolynomialsExponential and logarithmic functionsGraphingOther equationsConic sectionsSystems of equations and inequalitiesSystems of linear equations with three or more variablesCustomized Full-Length ExamCovers all subject areas
Featured Bookstores
Buyback (Sell directly to one of these merchants and get cash immediately) |
Grapher
Center for Cultural Design presents this site on graffiti artwork as it relates to mathematics. Graffiti artists often use geometric concepts when creating their work. The site includes interactive software that allows students to create their own graffiti while illustrating geometric patterns.Wed, 22 Dec 2010 03:00:01 -06Functions Defined by Data
by Lawrence Moore and David Smith for the Connected Curriculum Project, the purpose of this module is to carry out an exploration of functions defined by data; to learn about data entry and plotting operations. This is one lesson in a much larger set of learning modules hosted by Duke University.Tue, 4 May 2010 03:00:04 -0500Excel: Converting Radians to Degrees
this animated and interactive object, students read how to use the DEGREES() function to convert radians to degrees. Target Audience: 2-4 Year College StudentsWed, 14 Oct 2009 03:00:02 -0500Computational Geometry Pages
comprehensive directory of computational geometry resources both on and off the Internet. General Resources, Literature, Research and Teaching, Events, Software, other links. Also found at 27 Dec 2007 03:00:03Geometry (MathPages)
notes" by Kevin Brown on geometry: constructing the heptadecagon, what mirrors do, the golden pentagon, the grazing goat and the lune, Napoleon's theorem, chess boards, Diophantine geodesic boxes, Zeno's mice and the logarithmic spiral, and many more.Tue, 11 Dec 2007 03:00:02Projective Geometry
site departs from the common themes taught in general geometry classes and introduces projective geometry, which has to do with special properties resulting from the intersection of lines, planes, and points. The coincidence of such elements is what is referred to as an incidence, and this is the basis of the topic. The site makes extensive use of animated figures to demonstrate principles involved in projective geometry, such as path curves, pivot transforms, and the curious concept of counter space. The author does a good job of explaining what is depicted in the figures as well as the underlying theory.Fri, 20 Jul 2007 03:00:02 -0500 |
Mathematics for Business - 8th edition
Summary: The Eighth Edition of Mathematics for Business continues to provide solid, practical, and current coverage of the mathematical topics students must master to succeed in business today. The text begins with a review of basic mathematics and goes on to introduce key business topics in an algebra-based context.
Chapter 1, Problem Solving and Operations with Fractions, starts off with a section devoted to helping students become better problem solvers and critical...show more thinker while reviewing basic math skills. Optional scientific calculator boxes are integrated throughout and financial calculator boxes are presented in later chapters to help students become more comfortable with technology as they enter the business world. The text incorporates applications pertaining to a wide variety of careers so students from all disciplines can relate to the material. Each chapter opener features a real-world application.
Features
Current financial data used throughout the text.
Real-world applications within exercise sets are now called out by topical headings for each problem so that students immediately see the relevance of the problems to their lives.
Introduction to problem solving in Section 1.1 helps students learn how to think through solving common problems. The emphasis on problem-solving skills is carried through the text so that students can enter the business world with critical thinking skills and apply what they have learned.
Chapter openers now incorporate an application with a real-world graph or figure so students can understand how the chapter content pertains to actual business situations.
Financial calculator boxes that explain how to solve examples using a financial calculator are now integrated into later chapters to familiarize students with the technology they will be using in the business world.
A Metric System Appendix, complete with examples and exercises, explains the metric system and teaches students to convert between US and metric units of measurement.
'Net Assets emphasize the World Wide Web and keep students current on how businesses adapt to technology.
Cumulative Reviews help students review groups of related chapter topics and reinforce their understanding of the material.Quality School Texts OH Coshocton, OH
2006-06-22 |
Course Description
Math Analysis
MATH ANALYSIS
Course Rationale:
Virtually everyone uses or consumes statistical material every day and most people do so without training in the proper use or
potential abuse of statistical information. This course is designed as a general-purpose introduction to the field of statistics and
probability. Students will need a working knowledge of algebra in order to successfully solve the problems in this course. The
primary objective of this course is to enable students to be wiser users and more critical consumers of statistical material.
Course Description:
The primary goal of this course is to enhance statistical literacy. This includes mathematical formulas and arithmetic computation,
logical reasoning and problem solving abilities and terminology and vocabulary acquisition. This course uses a non-theoretical
approach in which concepts are explained intuitively and supported by examples.
337
Resources:
Adopted Text: __________________________________________________________________________
Websites:
Classroom Assessment Item Bank
Additional Assessment Problems:
Abbreviations:
Subjects GLE Grade Level Expectations BT Bloom's Taxonomy
CA Communication Arts Math Strands K Knowledge
MA Mathematics NO Number and Operations C Comprehension
SC Science AR Algebraic Relationships Ap Application
SS Social Studies GSR Geometric Spatial Relationships An Analysis
M Measurements S Synthesis
DP Data and Probability E Evaluation
338
Math Analysis properties of MA 4 NO Ap >The students will apply properties of Students will be assessed on their
operations 1.6 2.C.11 logarithms to simplify expressions or ability to apply properties of 80%
1.10 solve equations when completing the logarithms to simplify expressions
• Apply properties of teacher led lesson "Exponents and or solve equations when
logarithms to simplify Logarithms Discussion" completing a performance based
expressions or solve assessment
equations ussions/exp.html
Attachment A
>The students will be introduced to the
properties of logarithms. The three main
properties (logarithm of a variable or Additional Assessment:
number raised to a power, logarithm of Word Problems
a product, and logarithm of a quotient)
will be demonstrated using variables es/expoprob.htm
and later with numbers. They will do a
number of exercises in which they will
have to simplify expressions that
include logarithms and exponents.
>The students will apply properties of
logarithms when completing a variety of
on-line interactive programs.
ra2/index.html
Algebra 2; Exponentials & Logarithms
Integrated Skills: Workplace Readiness
339 operations on real MA 1 NO Ap >The students will apply operations to Students will be assessed on their
and complex numbers MA 4 2.D.11 complex numbers when completing the ability to apply operations to 80%
MA 5 lessons and quizzes located at: matrices and complex numbers,
• Apply operations to 1.4 using mental computation or
matrices and complex 3.4 paper-and pencil calculations for
numbers, using mental MAP B simple cases and technology for
computation or paper- Operations more complicated cases with an
and-pencil calculations Complex Numbers on-line assessment.
for simple cases and
technology for more
complicated cases ons/vmch16/vmch16_2.html
Integrated Skills: Workplace Readiness, Technology
340Estimate and justify MA 1 NO E >The students will judge the Students will be assessed on their
solutions 3.8 3.D. reasonableness of numerical ability to judge the 80%
9-12 computations and their results when reasonableness of numerical
• Judge the completing the following lesson computations and their results
reasonableness of "Spending—Past, Present and when completing a constructed
numerical computations Future" response item.
and their results
Classroom Assessment Item Bank
Systems of Linear Equations
Integrated Skills: Workplace Readiness
341Use proportional MA 1 NO Ap >The students will solve problems Students will be assessed on their
reasoning MA 4 3.E. involving proportions and percents ability to solve problems involving 80%
3.3 9-12 when completing the 2 lesson unit proportions when completing a
• Solve problems "Making Sense of Percent textbook assignment and/or
involving proportions Concentrations" classroom activities.
Lessons, Grades 9-12
Integrated Skills: Workplace Readiness
342Create and analyze MA 4 AR C >The students will generalize patterns Students will be assessed on their
patterns 1.6 1.B using explicitly or recursively defined ability to generalize patterns using 80%
3.5 9-12 functions when investigating parabolas, explicitly or recursively defined
• Generalize patterns ellipses and hyperbolas. functions with a textbook
using explicitly or assignment and/or classroom
recursively defined activities.
functions Exploring Conics Through Paper-
Folding and Geometer's Sketchpad
>The students will generalize patterns
using explicitly or recursively defined
functions when completing the unit
"Trout Pond"
Lessons, Grades 9-12
Integrated Skills: Workplace Readiness
343Classify objects and MA 4 AR A >The students will compare and Students will be assessed on their
representations 1.6 1.C contrast forms of patterns. Introduce ability to compare and contrast 80%
9-12 the unit by asking students where they various forms of representations
Compare and contrast notice numerical patterns and what of patterns when completing a
various forms of predictions they make based on their constructed response question.
representations of observations.
patterns Below are some possible responses: Attachment B
* house numbers
* streets and avenues
* savings account with regular deposits
* family tree growth if everyone had two
children
* gossip
- Find an example that demonstrates a
pattern and draw a picture that
illustrates five 'reoccurrences' of the
pattern
>The students will compare and
contrast various forms of numeric,
algebraic, and graphical representations
when completing the following lesson
"Successive Discounts"
Lessons Grades 9-12
Integrated Skills: Workplace Readiness
344Identify and compare MA 4 AR A >The students will understand and Students will be assessed on their
functions 1.6 1.D compare the properties of exponential ability to understand and compare 80%
3.6 11 functions in the lesson "Exponential the properties of linear, quadratic,
Understand and Functions and the Number e" exponential, logarithmic and
compare the properties rational functions with a textbook
of linear, quadratic, hlabs/exp&log/index.htm# assignment and or worksheet.
exponential,
logarithmic and >The students will understand and
rational functions compare the properties of rational High School
(include asymptotes) functions and asymptotes when Algebra I/II
completing the on-line tutorial on Logarithms and Exponential
asymptotes. Functions
ymtote.htm
>The students will understand and
compare the properties of linear and
rational functions and asymptotes when
completing the on-line interactive lesson
with assessments.
1s02/lecture/intlec1.html
>The students will understand and
compare the properties of logarithmic
equations when completing the on-line
tutorial.
velog.htm
Integrated Skills: Technology
345 the effects of MA 4 AR C >The students will describe the effects Students will be assessed on their
parameter changes 1.6 1.E.11 of parameter changes in exponential ability to describe the effects of 80%
4.1 functions when completing the following parameter changes in logarithmic
Describe the effects of lesson: "National Debt and Wars" and exponential functions when
parameter changes in completing a textbook assignment
logarithmic and Lessons Grades 9-12 and/or worksheet.
exponential functions
> The students will describe the effects Attachment C
of parameter changes in functions when
completing the following lesson And/or
"Shrinking Candles, Running Water,
Folding Boxes".
High School
Lessons Grades 9-12 Algebra I/II
Logarithms
>The students will describe the effects Solving Exponential Equations
of parameter changes in exponential
functions when completing the lesson
"Shedding Light on the Subject"
Lessons Grades 9-12
Integrated Skills: Workplace Readiness
346Represent mathematical MA 4 AR Ap >The students will use symbolic algebra Students will be assessed on their
situations MA 6 2.A.11 to represent and solve problems when ability to use symbolic algebra to 80%
1.6 completing the lesson: "Which Trig represent and solve problems that
Use symbolic algebra 3.1 Ratio?" involve exponential and
to represent and solve logarithmic relationships, including
problems that involve recursive and parametric
exponential and relationships with a worksheet.
logarithmic >Students will use symbolic algebra to
relationships, including represent and solve problems that Attachment D
recursive and involve exponential relationships with
parametric fractional exponents.
relationships
MAP B
#3 Operations
Evaluate Expressions With
Fractional Exponents
Integrated Skills: Workplace Readiness
347 and use MA 4 AR C >The students will describe and use Students will be assessed on their
mathematical 3.1 2.B. Ap algebraic manipulations when working ability to describe and use 80%
manipulation 4.1 11-12 with algebra tiles to demonstrate the algebraic manipulations, including
inverse of functions and integer inverse of functions, composition
Describe and use exponents. of functions and rules of integer
algebraic exponents on a worksheet.
manipulations, Attachment E
including inverse of Attachment F
functions, composition >The students will describe and use
of functions and rules of algebraic manipulations when working
integer exponents with inverse of functions at the following
website:
ction/inversefunction.html
>The students will describe and use
algebraic manipulations to complete a
worksheet on exponents
MAP B
#4 Modeling/Multiple Representations
Use Positive, Negative and Zero
Exponents
>The students will describe and use
inverse of functions with the on-line
interactive activity Inverse Functions
located at the following website:
s.html
Integrated Skills:
348 equivalent forms MA 4 AR Ap > The students will use and solve Students will be assessed on their
1.6 2.C.11 equivalent forms of equations and ability to use and solve equivalent 80%
Use and solve 3.4 inequalities when completing a series of forms of equations and
equivalent forms of stations described in the lesson: inequalities when completing the
equations and on-line assessments located at:
inequalities
(exponential, MAP B
logarithmic and #7 Patterns and Functions ebra.html
rational) D.6 Absolute Value Equations Solving Equations
D.8 Absolute Value Inequalities
>The students will use and solve
equivalent forms of equations and
inequalities with exponential functions
when completing the following lesson:
/activities/exploremath.html
Exponential Functions
Integrated Skills: Workplace Readiness
349 systems MA 4 AR Ap >The students will use and solve Students will be assessed on their
1.6 2.D.11 systems of linear and quadratic ability to use and solve systems of 80%
Use and solve systems equations when completing the linear and quadratic equations or
of linear and quadratic following lesson: "Building inequalities with 2 variables on a
equations or Connections" worksheet.
inequalities with 2
variables http;//illuminations.nctm.org Attachment G
Lessons Grades 9-12
>The students will use and solve
systems of linear and quadratic
equations when completing the
following lesson: "Quadratic
Functions (standard form)"
/quadratics.htm
Integrated Skills:
350
Algebraic Relationships
3. Use mathematical models to represent and understand quantitative relationships mathematical models MA 4 AR K >The students will identify quantitative Students will be assessed on
1.6 3.A. C relationships and determine the type(s) of their ability to identify 80%
Identify quantitative 3.6 11-12 functions when completing the following quantitative relationships and
relationships and lesson: "Understanding Inverse Linear determine the type(s) of
determine the type(s) of Functions" functions that might model the
functions that might situation to solve the problem
model the situation to with a performance based
solve the problem assessment "Animators R Us"
(including recursive >The students will identify quantitative
forms) relationships and determine the type(s) of
functions that might model the situation to Classroom Assessment Item
solve the problem when completing the Bank
lesson "Modeling Orbital Debris
Problems"
Lessons Grades 9-12
Integrated Skills: Technology, Workplace Readiness
351
Algebraic Relationships
4. Analyze change in various contextsAnalyze change MA 4 AR A >The students will analyze exponential Students will be assessed on their
1.6 4.A.11 and logarithmic functions by ability to analyze exponential and 80%
Analyze exponential 4.1 investigating rates of change and logarithmic functions by
and logarithmic intercepts when completing the lesson investigating rates of change,
functions by "Pedal Power" located at the following intercepts and asymptotes when
investigating rates of website: completing a worksheet.
change, intercepts and
asymptotes Attachment H and I
Lessons Grades 9-12
>The students will analyze exponential
and logarithmic functions by
investigating rates of change and
intercepts when completing the lesson
"Interest Rates and Growth of
Money"
/activities/gc-interestrates/home.html
Integrated Skills: Technology
352
Geometric and Spatial Relationships
2. Specify locations and describe spatial relationships using coordinate geometry and other representational systems coordinate systems MA 2 GSR Ap >The students will use vectors to represent Students will be assessed on their
3.6 2.A.11 and analyze problems when completing the ability to use vectors to represent 80%
Use vectors to 4.1 on-line lesson "Graphing Vector and analyze problems involving
represent and analyze Calculator" velocity and direction with a
problems involving constructed response question.
velocity and direction
Attachment J
Integrated Skills:
353 2 GSR Ap >The students will use and apply matrices Students will use and apply
objects 1.10 3.A.11 to represent transformations when matrices to represent translations, 80%
completing the lesson Matrix/ reflections, rotations, and dilations
Use and apply matrices Spreadsheet/Transformation located at when completing a group
to represent the following webpage. performance task.
translations, reflections,
rotations, and dilations t.htm Attachment K
and/or
math.mit.edu/~djk/18_022/chapter16/conten
ts.html
Integrated Skills:
354 4 GSR Ap >The students will perform simple Students will be assessed on their 80%
functions 3.1 3.B.11 transformations and their compositions on ability to perform simple
linear and quadratic functions when transformations and their
Perform simple completing the lesson located at the compositions on linear, quadratic,
transformations and following website: logarithmic and exponential
their compositions on functions when completing a
linear, quadratic, worksheet.
logarithmic and MAP B
exponential functions #7 Patterns and Functions Attachment L
B. Ways to Represent and Work With
Functions
Use Transformations to Investigate the
Relationships Between Functions
Integrated Skills:
355
Measurement
2. Apply appropriate techniques, tools and formulas to determine measurements precision MA 2 M Ap >The students will apply concepts of Students will be assessed on their
1.6 2.D.11 successive approximation when completing ability to apply concepts of 80%
Apply concepts of 3.4 a lesson chosen by the teacher. successive approximation on a
successive textbook assignment.
approximation
=category&code=12&off=0&tag=920043892
0658
Integrated Skills: Technology
356Formulate questions MA 3 DP Ap >The students will formulate question, Students will be assessed on their
1.2 1.A. design a study and collect data about ability to formulate questions, 80%
Formulate questions, 9-12 foreign currency and budgets design studies and collect data
design studies and about a characteristic with a
collect data about a classroom activity.
characteristic ssons/archive.html
Foreign Exchange
Integrated Skills:
357 and interpret MA 3 DP C >The students will describe the Students will describe the
data 1.2 1.C.11 characteristics of well designed studies, characteristics of well designed 80%
3.1 including the role of randomization in survey studies, including the role of
Describe the when completing the New York Time lesson randomization in survey and
characteristics of well "Opinions, Please!" experimental research when
designed studies, completing the survey from the
including the role of lesson Opinions, Please!
randomization in survey ssons/archive.html
and experimental And/or
research >The students will describe the
characteristics of well designed studies, when completing the student
including the role of randomization in survey designed survey from the
when completing a survey of their own classroom activity
about why people should take Algebra.
Information for the survey can be found at
the following website:
ath.htm
>The students will design a study, gather
data, display the data and make inferences
from the resulting data. Students could
draw inferences from other student's data.
Students will determine if the inferences are
legitimate.
Integrated Skills: Technology
358Describe and analyze MA 3 DP Ap >The students will apply statistical concepts Students will be assessed on their
data 1.10 2.A. when looking at the relationship between ability to apply statistical concepts 80%
3.4 10-12 diet, exercise and weight loss. to solve problems and distinguish
Apply statistical between a statistic and a
concepts to solve parameter when completing the
problems and independent activity in the given
distinguish between a ssons/archive.html classroom exercise.
statistic and a Weight Training
parameter
Integrated Skills:
359Compare data MA 3 DP Ap >The students will display the distribution of Students will be assessed on their
representations 1.8 2.B.11 one-variable data when completing the four ability to display the distribution of 80%
1.10 lesson unit "The Regression Line and one variable quantitative data,
Given one-variable Correlation" describe its shape and calculate
quantitative data, summary statistics when working
display the distribution, with the unit "The Regression
describe its shape and Line and Correlation" and/or a
calculate summary >The students given one-variable textbook assignment.
statistics quantitative data will display the distribution
when completing the lesson "Exploring
Period…Period" located at the following
website:
vities/exploremath.html
Integrated Skills: Technology
360 data MA 3 DP A >The students will determine a type of Students will be assessed on their
algebraically 1.6 2.C.11 function which models the data on a ability to determine a type of 80%
scatterplot when completing the lesson function which models the data on
Given a scatterplot, "Analyzing Birth Rates" a given scatterplot from the
determine a type of website:
function which models
the data vities/excel.html math/activities/excel.html
>The students will determine a type of
function which models the data on a
scatterplot of a environmental issue.
Integrated Skills:
361 Ap >The students will use simulations to Students will be assessed on their
inferences 1.2 3.A.11 describe the variability of sample statistics ability to use simulations to 80%
from a known population and to construct describe the variability of sample
Use simulations to sampling distribution in the lesson statistics from a known population
describe the variability "Buffon's Needle" and to construct sampling
of sample statistics distribution with a constructed
from a known response.
population and to n.html
construct sampling
distribution >The students will use simulations to Probability
describe the variability of sample statistics
from a known population and to construct
sampling distribution when working with a
variety of applets involving Experimenting
Paradoxes.
stat.stanford.edu/~susan/surprise/
Integrated Skills:
362 E >Students will evaluate published reports Students will be assessed on their
inferences 1.5 3.A.12 that are based on data when completing a ability to evaluate published 80%
New York Times Lesson "A Valid reports that are based on data by
Evaluate published Conclusion?" and/or the lesson "It's Gas, examining the design of the study,
reports that are based Gas, Gas" the appropriateness of the data
on data by examining analysis, and the validity of
the design of the study, conclusions when completing
the appropriateness of s/archive.html classroom activities.
the data analysis, and
the validity of
conclusions
Integrated Skills:
363 basic statistical MA 3 DP >The students will describe how basic Students will be assessed on their
techniques 1.4 3.B.12 statistical are used in the workplace when ability to describe how basic 80%
completing a New York Times lesson statistical techniques are used in
Describe how basic activity. the workplace when completing
statistical techniques classroom activities.
are used in the s/archive.html
workplace Wheeling and Dealing
No New Workers Need Apply
Working It Out
Integrated Skills: Workplace Readiness
364 compute and interpret Students will be assessed on their
probability 3.1 4.A.11 the expected value of random variables ability to compute and interpret the 80%
when completing the lesson "Hypothesis expected value of random
Compute and interpret Testing Project" variables with the completion of
the expected value of the project from the activity.
random variables
>The students will compute and interpret
the expected value of random variables
when completing an on-line interactive
lesson.
vities/ex-randomevents/
Integrated Skills:
365 use simulations to Students will be assessed on their
probability 1.2 4.A.12 construct probability distributions with a TI- ability to use simulations to 80%
82 calculator in the SuccessLink lesson construct empirical probability
Use simulations to "Exploring Limits of Rational Functions distributions when completing
construct empirical Using the TI-82 Graphics Calculator" classroom activities.
probability
distributions
>The students will use simulations to
represent medicine in a body when
completing the SuccessLink lesson
"Exploring Limits and Recursive
Sequences Through a Classroom
Simulation"
Integrated Skills:
366 and describe MA 2 DP Ap >The students will use and describe how to Students will be assessed on their
compound events 3.1 4.B.11 compute the probability of a compound ability to use and describe how to 80%
event when completing the lesson "Stick or compute the probability of a
Use and describe how Switch" compound event
to compute the
probability of a
compound event Lessons Grades 9-12
Integrated Skills:
367 |
An introduction to cartography emphasizing map projections, their properties, applications and basic mathematics. Concepts...
see more
An introduction to cartography emphasizing map projections, their properties, applications and basic mathematics. Concepts addressed can be presented at a basic level or expanded to explore the use of more advanced mathematics.
A free online interactive geometry textbook. The site has two primary audiences: teachers using the animations in...
see more
A free online interactive geometry textbook. The site has two primary audiences: teachers using the animations in class with a projector to demonstrate geometry concepts, and students who need a reference source when the teacher is not around. The scope of the site is all geometry concepts up to and including high school and college general ed. Each topic has a java applet that both animates the concepts and allows the user to interact with it to gain a deeper understanding. The idea is to go beyong what a static paper textbook can offer, and make the content more available without carrying heavy books everywhere offers terms and formulas in a mathematical dictionary appropriate for students taking algebra and calculus...
see more
This site offers terms and formulas in a mathematical dictionary appropriate for students taking algebra and calculus courses. The main page contains an A to Z clickable menu displaying all of the terms in the site's dictionary that begin with that particular letter. This site is an interactive math dictionary with enough math words, math terms, math formulas, pictures, diagrams, tables, and examples from beginning algebra to calculus.
This website explains the method the Mayans used in calculating their numbers. At first the subject can be...
see more
This website explains the method the Mayans used in calculating their numbers. At first the subject can be confusing given that the math is depicted with lines and dots, but the website excels at breaking the lines and dots down, showing you exactly how to calculate what each group of symbols means, and how to add them together. It is possibly one of the better sites for understanding Mayan Math.
Since 1994, the Michigan Journal of Community Service Learning (MJCSL) has been the premiere national, peer-reviewed journal...
see more
Since 1994, the Michigan Journal of Community Service Learning (MJCSL) has been the premiere national, peer-reviewed journal publishing articles written by faculty and service-learning educators on research, theory, pedagogy, and other issues related to academic (curriculum-based) service-learning in higher education. The Special Journal Issue: Service-Learning Course Design Workbook is a valuable tool for faculty constructing new service-learning courses.
This is an online textbook in Adobe PDF format. It begins with an introduction to Euclidian Three-Space and vector algebra...
see more
This is an online textbook in Adobe PDF format. It begins with an introduction to Euclidian Three-Space and vector algebra and covers traditional calculus topics from derivatives up through Stoke's Theorem. Many nice exercises are included.
The readings on this web site were designed as part of the IT Multivariable Calculus and Vector Analysis course at the...
see more
The readings on this web site were designed as part of the IT Multivariable Calculus and Vector Analysis course at the University of Minnesota. Students in this course are expected to read some of these documents (those marked with an asterisk * in the lecture list) before attending the lecture on the topic. The intent was to allow lecturers in the course spend more lecture time helping students understand and apply the material and less time on simply presenting the theory.The remaining pages are a loosely organized collection of lecture notes, example problems, and other resources for students in the course. As no effort has been made to turn this into a comprehensive source of information on multivariable calculus and vector analysis, the coverage of different topics is uneven, with some important topics (such as Lagrange multipliers) missing altogether. Moreover, some of the readings not marked by asterisks assume content that is presented in lecture and not in the online readings. Nonetheless, I hope that what is available will be helpful for those trying to learn multivariable calculus and vector analysis.One can view these readings more like a lecture than a textbook. They are not a replacement of a mathematics textbook because they don't cover all the theoretical details behind the main ideas. For the same reason, they should be easier to understand than a textbook. Many of the readings contain interactive graphics that I term concept visualization tools (or CVTs). |
ARTICLE TOOLS
Abstract
Algebraic geometry is an area of mathematics that concerns the interaction between algebra and goemetry. This article discusses some main algebraic and geometric objects of interest in algebraic geometry. Computational methods and abstract theories are discussed, and some main applications to computer science are indicated. |
Customer Reviews
Mathematics for the TradesAugust 2, 2011 by Myles Dalquist
This is not a complicated, in depth, advanced algebra textbook. This textbook contains the simple, common sense skills that you will need in any job. It has easy-to-follow-lessons and plenty of practice problems. This textbook is very well priced compared to similar textbooks. It is an easy read that is straight to the point and easy to understand.
The authors interviewed trades workers, apprentices, teachers, and training program directors to ensure realistic problems and applications and added over 100 new exercises to this edition. Geometry, triangle trigonometry, and advanced algebra. For individuals who will need technical math skills to succeed in a wide variety of trades. |
This video explains the mean value theorem. The mean value
theorem is essentially just the average slope of the graph in a closed interval. This is just the mean (average) value theorem which states that the average slope is equal to a prime of a p...oint c. The video is 16:47 in length and is done with a smart board and narrator.[more]
[BEST VIEWED IN FULL-SCREEN MODE] The instructor introduces the use of basic calculus to determine the high or low point of the vertical component of a roadway design. This application provides a practical use of derivatives that are critical in hi...ghway design when locating a roadway over or under features (such
as pipelines or bridges). This application can be replicated by a teacher with local information (using commonly available online mapping tools, such as Google Earth) to aid in the understanding of the topic. The instructor,
Daniel Findley, is a licensed Professional Engineer, focused on Transportation
Engineering. He has a PhD in Civil Engineering from NC State University
and is a Senior Research Associate at the Institute for Transportation Research and Education and is an Adjunct Assistant Professor in the Department of Civil Construction and Environmental Engineering at NC State. (4:46)[more]
This video is about the correction of questions that were put in college Calculus 1 Final exam. The solution of each question was explained in detail and accompanied with the concept. Duration: 1:07:59
This video is about the correction of questions that were put in college Calculus 1 Final exam. The solution of each question was explained in detail and accompanied with the concept. Duration: 1:08:46
Students will be given examples of parabolas and a method for determining the look of a graph describing a parabola. The student will have an opportunity to use and manipulate the quadratic formula in determining a graph. The values of the formula wi...ll also be determined after reading a graph |
Topic outline
Mathematics
Faculty
Why Study Mathematics?
It's all about looking for patterns and relationships in numbers and shapes.
It's about solving puzzles and problems.
It uses and develops our natural skills of logic and reasoning.
It gives us the tools to present concise and elegant analysis of the world around us, for instance through the use of algebra or statistics.
What's more… it's useful!
Whether you are out shopping, cooking a meal, organising your finances, or just doing a Sudoku, you will use mathematical skills.
In school you need Mathematics in lots of subjects: Physics, Chemistry, Biology, ICT, Geography, Business Studies, Graphics, Electronics, Food; even Art and Music have links to Mathematics.
Lots of people use Mathematics in their jobs too: Scientists, Doctors, Nurses, Teachers, Statisticians, Bankers, Insurers, Engineers, Architects, Surveyors, Environmentalists, Computer Programmers, Website Designers, Designers and Business people to name just a few.
Mathematics at Soham Village College
The Mathematics scheme of work for each year is based on a modular (topic) approach. Lessons are designed to encourage you to investigate and discuss mathematics and to develop your problem solving and thinking skills. SMART boards are used extensively to introduce new concepts and skills. Lessons will often involve whole class discussion, group work, computer or practical activities, as well as opportunities for you to consolidate concepts and skills through exercises and problem solving.
In Key Stage 3 (Years 7, 8 & 9) the scheme of work is differentiated into 4 tiers which correspond to the National Curriculum tests at the end of Year 9.
The Mathematics course is designed to help you to develop confidence in, and a positive attitude towards, Mathematics and to recognise the importance of mathematics in your own life and to society.
Through your study of Data Handling, Number, Algebra, Geometry and Measure, you will develop skills in the following areas:
Applying Mathematics: You will develop knowledge, skills and understanding of mathematical methods and concepts and apply these techniques in mathematical situations as well as making links to other subjects
Problem Solving: You will develop and refine problem solving strategies and build the confidence and skills required to tackle unfamiliar challenges.
Thinking & Reasoning Skills: You will develop independent thinking skills and learn to reason mathematically, making deductions and inferences and drawing conclusions.
The Functional Elements of Mathematics: You will develop confidence in using mathematics in real life, interpreting, processing and communicating the mathematics of everyday situations.
Assessment
GCSE Mathematics is a linear course with 2 examinations at the end of the course. Both papers assess the full range of assessment objectives in Data Handling, Number & Algebra, Geometry & Measures. |
Mathematics for Carpentry and the Construction Trades, Third Edition, offers a unique approach based on the authors' experience in building construction and applied education. Loaded with photographs and detailed drawings, the text illustrates the underlying mathematics in each step of the building process. The text's problems, infused with the authors' real industry experience, provide students with relevant examples of problems they will face in the construction and carpentry trades. Problems include step-by-step summary explanations of their solutions with the necessary steps highlighted for easy identification. After giving students a solid foundation in math, the text then leads them through the steps of a construction project and applying the mathematical skills involved in completing the project.
Discrete Mathematics and its Applications, Seventh Edition…from computer science to data networking, to psychology, to chemistry, to engineering, to linguistics, to biology, to business, and to many other important fields practical introduction to the techniques needed to produce high-quality mathematical illustrations is suitable for anyone with basic knowledge of coordinate geometry. Bill Casselman combines a completely self-contained step-by-step introduction to the graphics programming language PostScript with an analysis of the requirements of good mathematical illustrations. The many small simple graphics projects can also be used in courses in geometry, graphics, or general mathematics |
Many people wonder whether students need to know algebra in order to be successful in today's world. What do you think? Look at the suggested readings and research some of the arguments on both sides of the issue. Is "Algebra for Everyone" a worthwhile goal?
Problem H2
The following quick sketch graphs describe two light aircraft, A and B (note: the graphs have not been drawn accurately):
a.
The first graph shows that aircraft B is more expensive than aircraft A. What else does it say?
b.
Are the following statements true or false?
1. The older aircraft is cheaper.
2. The faster aircraft is smaller.
3. The larger aircraft is older.
4. The cheaper aircraft carries fewer passengers.
c.
Copy the graphs below. On each graph, mark and label two points to represent A and B.
Using your sketches, explain why a bottle with straight sloping sides does not give a straight line graph (that is., explain why the ink bottle does not correspond to graph G in Problem C9).
Is it possible to draw two different bottles that give the same height-volume graph? Try to draw some examples. Close Tip
Problems H2 and H3 taken from The Language of Functions and Graphs, by Malcolm Swan and the Shell Centre Team (Nottingham, U.K.: Shell Centre Publications, 1999), p. 986. |
Elementary Geometry for College Students
9780618645251
ISBN:
061864525X
Edition: 4 Pub Date: 2006 Publisher: Houghton Mifflin College Div
Summary: Building on the success of its first three editions, the Fourth Edition of this market-leading text covers the important principles and real-world applications of plane geometry, with additional chapters on solid geometry, analytic geometry, and an introduction to trigonometry. Strongly influenced by both NCTM and AMATYC standards, the text takes an inductive approach that includes integrated activities and tools to ...promote hands-on application and discovery."New!" Tables provide visual connections between figures and concepts and help students better assess their level of mastery and test readiness."New!" Chapter Tests have been added to the end of every chapter."New!" Proofs have been varied to include written and visual proofs, as well as comparisons, to support students with different learning styles."New!" Exercise sets in the Student Study Guide, with cross-references to the text, offer additional practice and review."New!" Technology-related margin features encourage the use of the Geometer's Sketchpad, graphing calculators, and further explorations."New!" Coverage now includes Section 2.6, "Symmetry and Transformations.""New!" Technology Package includes the HM ClassPrep CD with computerized test bank (powered by Diploma)."Updated!" The number of Exercises and Explorations has been increased.Highly visual approach begins with the presentation of an idea, followed by the examination and development of a theory, verification of the theory through deduction, and finally, application of the principles to the real world.Discovery features reinforce the text's inductive approach: activities integrated throughout enable students to discover geometry concepts on their own, andsection tools provide with hands-on application of geometric conceptsApplications reinforce the connection of geometry to the real world: high-interest "Chapter Openers" introduce the principal notion of the chapter and relate to the real world and "A Perspective On..." sections conclude each chapter, providing sketches that are interesting, sometimes historical, and always informative.Summaries of constructions, postulates, and theorems are provided, and an easy-to-navigate numbering system for postulates and theorems provides a user-friendly structure. In response to user feedback, paragraph proofs feature more prominently in this edition.Comprehensive appendices include "Algebra Review and An Introduction to Logic." A glossary of terms, a summary of applications in the text, and selected answers are also provided in the back of the text.
Koeberlein, Geralyn M. is the author of Elementary Geometry for College Students, published 2006 under ISBN 9780618645251 and 061864525X. One hundred seventy four Elementary Geometry for College Students textbooks are available for sale on ValoreBooks.com, sixty three used from the cheapest price of $38.95, or buy new starting at $69.63 |
Give students a solid high school introduction to math with Horizons Algebra 1. The colorful student workbook reviews all foundational pre-algebra concepts before covering topics such as: common factors, exponents, radical expressions, writing linear equations from graphs, polynomials, binomials, FOIL method, quadratic inequalities, and more. Grade 8. The student workbook includes a set of lesson review boxes accompanied by questions that provide practice for previously taught concepts and the concepts taught in the lesson. Exploring Math Through... sections help students understand how ordinary people use algebraic math, providing concrete examples of how math is useful in life. Students will need to supply paper to work the problems. 333 pages, softcover. The teacher's guide includes...
Less sidBuy Basic Math & Pre-Algebra Super Review by The Editors of REA and Read this Book on Kobo's Free Apps. Discover Kobo's Vast Collection of Ebooks Today - Over 3 Million Titles, Including 2 Million Free Ones!
ISBN13: 9781600320163 ISBN: 1600320163 UPC: 000000889920 Released: Nov 07, 2006 Group: Homeschool Format: Other Reading/Grade Level: 11th Description: Algebra 2 covers all topics that are traditionally covered in second-year algebra, as well as a considerable amount of geometry. In fact, students completing Algebra 2 will have studied the equivalent of one semester of informal geometry. Time is spent developing geometric concepts and writing proof outlines. Real-world problems are included alon sidebar hints and notes; and Lesson Practice questions with lesson reference numbers underneath the question number. Online connections are given throughout for additional help. Real-world applications and continual practice & review provide the time needed to master each concept, helping students to...
Less 2: An Essential Part of Every Math Curriculum eBook
Free Shipping
/algebra-readiness-made/Cpd8jBnEfJLBQbMZEnF7aQ==/info
Kobo eBooks
(
In stock
)
Store not rated
Free Shipping
Buy Algebra Readiness Made Easy: Grade 2Buy 50 Fill-in Math Word Problems: Algebra: Engaging Story Problems for Students to Read, Fill-in, Solve, and Sharpen Their Math Skills by Krech, Bob and Read this Book on Kobo's Free Apps. Discover Kobo's Vast Collection of Ebooks Today - Over 3 Million Titles, Including 2 Million Free Ones!
Provide students with a college-prep math course that will give them the foundation they need to successfully move into higher levels of math. Saxon Algebra 1, 4th Edition covers all of the traditional first-year algebra topics while helping students build higher-order thinking skills, real-world application skills, reasoning, and an understanding of interconnecting math strands. Saxon Algebra 1 focuses on algebraic thinking through multiple representations, including verbal, numeric, symbolic, and graphical, while graphing calculator labs model mathematical situations. Incremental lessons include a Warm Up activity; New Concepts section that introduces new concepts through examples with sidebar hints and notes; and Lesson Practice questions with lesson reference numbers underneath the...
Less
Buy The Math Dude's Quick and Dirty Guide to Algebra by Jason Marshall and Read this Book on Kobo's Free Apps. Discover Kobo's Vast Collection of Ebooks Today - Over 3 Million Titles, Including 2 Million Free Ones!
Buy Basic Math and Pre-Algebra Workbook For Dummies by Mark Zegarelli and Read this Book on Kobo's Free Apps. Discover Kobo's Vast Collection of Ebooks Today - Over 3 Million Titles, Including 2 Million Free Ones!
Buy Bob Miller's Basic Math and Pre-Algebra for the Clueless, 2nd Ed. by Miller, Bob 3: An Essential Part of Every Math Curriculum eBook
Free Shipping
/algebra-readiness-made/Xz7bRE8Wi4txWR1_E6XKiA==/info
Kobo eBooks
(
In stock
)
Store not rated
Free Shipping
Buy Algebra Readiness Made Easy: Grade 3 1: An Essential Part of Every Math Curriculum eBook
Free Shipping
/algebra-readiness-made/delafEs_vk57LHVejOuOWA==/info
Kobo eBooks
(
In stock
)
Store not rated
Free Shipping
Buy Algebra Readiness Made Easy: Grade 1The Prentice-Hall mathematics series is designed to help students develop a deeper understanding of math through an emphasis on thinking, reasoning, and problem-solving. A mix of print and digital materials helps engage students with visual and dynamic activities alongside textbook instruction. Algebra 2 reviews key Algebra 1 concepts before moving on to traditional advanced Algebra topics; the comprehensive table of contents also allows one to easily include trigonometry, statistics, or precalculus readiness. Grade 11. In the Getting Ready to Learn portion of the textbook lesson, Check your readiness exercises help students see where they might need to review before the lesson. Check skills you'll need list out the skills used in the lesson, and new vocabulary is listed before it's...
Less
$11.79
Kobo eBooksAlgebra Readiness Made Easy: Grade 6: An Essential Part of Every Math Curriculum eBook
Free Shipping
/algebra-readiness-made/nJKQQL2vszKkbGDiZrKPTg==/info
Kobo eBooks
(
In stock
)
Store not rated
Free Shipping
Buy Algebra Readiness Made Easy: Grade 6 4: An Essential Part of Every Math Curriculum eBook
Free Shipping
/algebra-readiness-made/osGWOQdMAdg_aesobzGnAw==/info
Kobo eBooks
(
In stock
)
Store not rated
Free Shipping
Buy Algebra Readiness Made Easy: Grade 4 5: An Essential Part of Every Math Curriculum eBook
Free Shipping
/algebra-readiness-made/s97ZlDCWEXhB9p-2jlZawg==/info
Kobo eBooks
(
In stock
)
Store not rated
Free Shipping
Buy Algebra Readiness Made Easy: Grade 5 |
Designed specifically for middle school students, the Spectrum Geometry 6-8 Workbook helps students apply vital math skills to everyday life. The Spectrum Series of books was developed by experts in education and have been a favorite among parents and teachers alike. This geometry workbook strengthens basic skills by focusing on points, lines, rays, angles, triangles, polygons, circles, perimeter, area, and more! Includes simple instructions and an answer key. 128 pages. Ages 11-13+ |
calculate 1.Combination nCr. 2.Permutation nPr. 3.Factorial n! Its great oppurtunity to get the answers for your solutions. No need to go through the tiring solutions on the paper. Just give the values for the fields and here is your answer. Its an app for students and others concerns as well.
ArrangementCALC is a simple calculator for calculating ordering options, using factorial, permutations and combinations formulae.
FUNCTIONALITY Calculate: - Possible ways to arrange items - Possible ways to arrange items if up to two items are identical - Possible ways to arrange items when chosen in subsets - Ability to specify if order matters and/or repeats are allowed
Aptitude Chapters Problems on Trains Time and Distance Height and Distance Time and Work Simple Interest Compound Interest Profit and Loss Partnership Percentage Problems on Ages Calendar Clock Average Area Volume and Surface Area Permutation and Combination Numbers Problems on Numbers Problems on H.C.F and L.C.M Decimal Fraction Simplification Square Root and Cube Root Surds and Indices Ratio and Proportion Chain Rule Pipes and Cistern Boats and Streams Alligation or Mixture Logarithm Races and Games Stocks and Shares Probability True Discount Banker's Discount Odd Man Out and SeriesCalculator LS serves the needs for most students from high school throughout college. All scientific and trigonometry functions are included.
Calculator LS has functions for combination and permutations, as well as simple probability distribution functions needed for elementary statistics classes, including the normal distribution, the inverse normal, the t-distribution, and even the inverse t-distribution!
Improve your quantitative ability skills with the all new Maths Courses App by GradeStack. This app brings to you multiple courses to help you improve your Maths skills, these courses also help you prepare for exams such as GMAT, GRE, SAT, CAT, CMAT.
Maths Tricks & Tips Learn tricks to amaze others as you use mental math to solve tough math problems. The "Maths Tips and Tricks" app makes you do magic with Maths and flaunt your new earned knowledge in front of your friends.
A fun way to improve your math skills in addition, subtraction, multiplication, division and squares and cube roots.
a. Study the tricks of trade when it comes to fast calculations b. Take our easy to use flashcards and practice math tricks everyday, solve multiple questions and improve your learning. c. Or just take the timed challenge and see how you score in your math skills d. Are you a Math Wiz - Take a complete math workout and get scores you can flaunt to your friends
Have fun while you learn the tricks of trade and excel in your ability of cracking any calculation
General Aptitude Helps you master the complex skills of Quantitative Ability, suitable for those preparing for competitive exams. such as - GMAT, GRE, SAT, CAT etc.
With over 40 chapters covering multiple topics in Maths, this is the ultimate course for improving
This is a very interesting book to read.. It covers very useful topics #Cell Biology & Genetics #Cell Biology & Genetics #Nucleus #Nucleic Acid #Nucleic Acid which provides you the deep knowledge about cell biology & genetics.
Dev-C++is a full-featured integrated development environment (IDE), which is able to create Windows or DOS-based C/C++ programs using the Mingw compiler system (included with the package), or the Cygwin compiler.
#Dev-C++ allows you to write, compile and run a C or C++ program.
#C++ programming language is an enhanced version of C language that provides object-oriented programming (OOP)capabilities.
Our approach has been reveal the logic of organic chemistry by being selective in the topic we cover,as well as thorough and patient in developing them. Different interesting topics readers encounter describes "Organometallic Compounds" "Organosulphur compounds", "Amino Acids, Peptides, Proteins and Nucleic Acids", "Synthetic Polymers " much more than this.This is the book which gives you the deep knowledge about molecular biology. This book covers #Watson and crick model of DNA #Plant tissue culture #Concept of cellular totipotency #Biotechnology #Recombinant DNA technology
Public Adminstration as independent Subject of a social science has recent origin. Traditionly Public Adminstration was considered as a part of political science. But in Modern age the nature of state-under went change and it became from police stale to social service state.
Macroeconomics is an analysis of a country's economic structure and performance and the government's policies in affecting its economic conditions. Economists are interested to know the factors that contribute towards a country's economic growth because if the economy progresses, ...... interesting?????
Read more thoughtful and interesting topics in this book and get knowledge about macro economy. |
The pythagorean Theorem Curriculum:
Math Grade Level:
9-12
Description:
This webquest is an activity that will enhance student's knowledge about using math skills in real-life application focusing on the Pythagorean Theorem. Keywords:right triangle, hypotenuse and legs Author(s): Judith Sabol
Prom on a budget! Curriculum:
Math Grade Level:
9-12
Description:
Students will create a poster presentation of their schools prom using a budget of 10,000 dollars. Keywords:math, budget, prom, 10 thousand dollars and party Author(s): Haley Tennant
Stopping the Quad Squad Curriculum:
Math Grade Level:
9-12
Description:
Use a digital portfolio to illustrate how to produce an equivalent form of an expression to explain properties of the quantity represented by the expression, factor a quadratic expression to reveal the zeros of the function it defines, and to complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Keywords:Quadratic, Expression, Factor, Function and Parabola Author(s): Johnathan Blue
Mathz is Fun Curriculum:
Math Grade Level:
9-12
Description:
This WebQuest seeks to guide students through various ways in which they can use solids found in their environment in a meaning full manner. Keywords:Solids, Regular, Irregular, Polyhedron and 3-Dimensional Author(s): Melissa Shields
Modeling Quadratic Behavior Curriculum:
Math Grade Level:
9-12
Description:
This WebQuest is designed to facilitate the understanding of quadratic functions by associating them with real-life applications. The students will explore various applications on quadratics. They will then be required to model free-fall, an application of quadratics, by creating a stop motion animation. Keywords:quadratics, applications, models, real-life, free-fall, stop motion, animation, mathematics, high school and functions Author(s): Elias Youhanna
Unit Circle Discovery Curriculum:
Math Grade Level:
9-12
Description:
The unit circle is generally a circle used in trigonometry with a radius of one. Here students will learn how to work with the unit circle. They will complete tasks that are designed to cause an understanding of the relationship between right triangle trigonometry and unit circle trigonometry. Students will also use angle measures in degrees and radians interchangeably as well as will be able to recognize and use the reciprocal relationships of the six trigonometric functions. Keywords:Unit Circle, degrees, radians, trig, trigonometry, pre-calculus and algebra Author(s): Rebecca Snider
Using Area and Perimeter Curriculum:
Math Grade Level:
9-12
Description:
This webquest will add the students' knowledge in manipulating the Area and Perimeter to design a house. Keywords:area and perimeter in designing a house Author(s): Marnelie Quines
Number Theory Curriculum:
Math Grade Level:
9-12
Description:
Number theory (or arithmetic) is a branch of pure mathematics devoted primarily to the study of the integers, sometimes called "The Queen of Mathematics". Keywords:Math Number Theory and its Application Webquest Author(s): Remelie Acob
Solving two-step equations Curriculum:
Math Grade Level:
9-12
Description:
Students will be able to solve a two step equation. Students will start with simple one step equations, using correct vocabulary, and find solutions using the correct procedures. They will end by being able to apply prior knowledge for solving a two-step equation. Keywords:Solving two-step equations Author(s): Neil Furnish
Exponential Growth and Decay Curriculum:
Math Grade Level:
9-12
Description:
This wequest will help you determine if a problem is exponential growth or exponential decay by using equations. Keywords:Helpful, Resourceful, Math, Growth and Decay Author(s): De'shawna Mosley
Studying Tessellations Curriculum:
Math Grade Level:
9-12
Description:
This WebQuest is designed to allow students to not only learn about the four types of symmetry, but also create their own tessellation and identify different examples of tessellations in nature. Keywords:pattern, tessellations, math and geometry Author(s): Ashley G
Learning Concepts in Math Geometry Curriculum:
Math Grade Level:
9-12
Description:
Through this WebQuest, all concepts in Math would be accesible to everyone. Worksheets are available here to practice one's Math skills. Keywords:All in Math Geometry Author(s): Donna Marie Boter
Exploring the Math with Automobile Ownership and Driving Data Curriculum:
Math Grade Level:
9-12
Description:
Owning and Operating an automobile is a tremendous responsibility. In this unit, students will examine and work with some of the math behind automobile ownership. Student will explore strategies to pricing a used vehicle, better understand the terminology used in obtaining adequate insurance coverage, and compute data using formulas and unit conversion to gain a better understanding of driver safety. Keywords:automobiles, pricing, safety, insurance and driving Author(s): Angela Rice
SMART HOME APPLIANCES:-for a better tomorrow Curriculum:
Math Grade Level:
9-12
Description:
•Children will collect data information from the Electricity board for the past 20 years for the yearly electricity consumption in their locality. They will also calculate the rate of increase in the consumption of electricity in the city for every 5 years based on the collected data. Observe and assess how the demand of electricity has changed over the years.• Students will then take two home appliances (Refrigerator and Air Conditioner) which are available in many star ratings in the market.... Keywords:DATA ANALYSIS, STATISTICS, FREQUENCY POLYGON, HISTOGRAM and LINE GRAPH AND BAR GRAPH Author(s): Sushirta Sachdeva, Gurpreet Bhatnagar, Anamika Verma
Logical Nonsense Curriculum:
Math Grade Level:
9-12
Description:
Students will be looking at a scene at the tea party, from the famous children's novel, "Alice's Adventures in Wonderland." Students will be looking at logic sentences. Students will have to decide if the sentences are true or false, using converse, inverse and contrapositive definitions. Keywords:Mathematics, geometry, logic, converse, inverse and contrapositive Author(s): Colleen Coon
Mathematics Quest Solutions Curriculum:
Math Grade Level:
9-12
Description:
The WebQuest is designed to assist students in enhancing their techniques in handling topics in Mathematics. The topics that I will be focusing on is Statistics. Keywords:Data, information, gathering, Pie Chart and Bar Chart and Pictograph Author(s): Edward Carter
The Concept Of Pythagoras' Theorem Curriculum:
Math Grade Level:
9-12
Description:
This WebQuest outlines the concept of Pythagoras'Theorem. It shows how to find the length of a side of right angled triangle by finding the square root of the sum of the squares of the other two sides. Keywords:Pythagoras' Theorem, Rectangles, Right- angle Triangles, Area, Formula, Hypotenuse, Adjacent and Opposite Author(s): Monique Bookal
Volume of a Solid of Revolution Curriculum:
Math Grade Level:
9-12
Description:
This WebQuest is designed for students taking AP Calculus. The topic is how to determine the volume of a solid of revolution by using the disk method and the washer method Keywords:calculus, volume, disk and washer Author(s): Will Kellogg |
Pages to are hidden for
"MITRES_18_001_strang_16"
Please download to view full document
13099588328676782
Contents
CHAPTER 14 Multiple Integrals
14.1 Double Integrals
14.2 Changing to Better Coordinates
14.3 Triple Integrals
14.4 Cylindrical and Spherical Coordinates
CHAPTER 15 Vector Calculus
15.1 Vector Fields
15.2 Line Integrals
15.3 Green's Theorem
15.4 Surface Integrals
15.5 The Divergence Theorem
15.6 Stokes' Theorem and the Curl of F
CHAPTER 16 Mathematics after Calculus
16.1 Linear Algebra
16.2 Differential Equations
16.3 Discrete Mathematics
Study Guide For Chapter 1
Answers to Odd-Numbered Problems
Index
Table of Integrals
C H A P T E R 16
Mathematics after Calculus
I would like this book to do more than help you pass calculus. (I hope it does that
too.) After calculus you will have choices- Which mathematics course to take next?-
and these pages aim to serve as a guide. Part of the answer depends on where you
are going-toward engineering or management or teaching or science or another
career where mathematics plays a part. The rest of the answer depends on where the
courses are going. This chapter can be a useful reference, to give a clearer idea than
course titles can do:
Linear Algebra Differential Equations Discrete Mathematics
Advanced Calculus (with Fourier Series) Numerical Methods Statistics
Pure mathematics is often divided into analysis and algebra and geometry. Those
parts come together in the "mathematical way of thinking9'-a mixture of logic
and ideas. It is a deep and creative subject-here we make a start.
Two main courses after calculus are linear algebra and differential equations.
I hope you can take both. To help you later, Sections 16.1 and 16.2 organize them
by examples. First a few words to compare and contrast those two subjects.
Linear algebra is about systems of equations. There are n variables to solve for. A
change in one affects the others. They can be prices or velocities or currents or
concentrations-outputs from any model with interconnected parts.
Linear algebra makes only one assumption-the model must be linear. A change
in one variable produces proportional changes in all variables. Practically every
subject begins that way. (When it becomes nonlinear, we solve by a sequence of linear
equations. Linear programming is nonlinear because we require x >, 0 )Elsewhere J
.
wrote that "Linear algebra has become as basic and as applicable as calculus, and
fortunately it is easier." I recommend taking it.
A differential equation is continuous (from calculus), where a matrix equation is
discrete (from algebra). The rate dyldt is determined by the present state y-which
changes by following that rule. Section 16.2 solves y' = cy + s(t) for economics and
life sciences, and y" + by' + cy =f(t) for physics and engineering. Please keep it and
refer to it.
16 Mathematics after Calculus
A third key direction is discrete mathematics. Matrices are a part, networks and
algorithms are a bigger part. Derivatives are not a part-this is closer to algebra. It
is needed in computer science. Some people have a knack for counting the ways a
computer can send ten messages in parallel-and for finding the fastest way.
Typical question: Can 25 states be matched with 25 neighbors, so one state in each
f
pair has an even number o letters? New York can pair with New Jersey, Texas with
Oklahoma, California with Arizona. We need rules for Hawaii and Alaska. This
matching question doesn't sound mathematical, but it is.
Section 16.3 selects four topics from discrete mathematics, so you can decide if you
want more.
Go back for a moment to calculus and differential equations. A completely realistic
problem is seldom easy, but we can solve models. (Developing a good model is a skill
in itself.) One method of solution involves complex numbers:
any function u(x + iy) solves uxx u,,+ =0 (Laplace equation)
any function eik("")
+ solves u,, - c2uxx 0 (wave equation).
=
From those building blocks we assemble solutions. For the wave equation, a signal
starts at t = 0.It is a combination of pure oscillations eikx.The coefficients in that
combination make up the Fourier transform-to tell how much of each frequency is
in the signal. A lot of engineers and scientists would rather know those Fourier
coefficients than f(x).
A Fourier series breaks the signal into Z a, cos kx or Z b, sin kx or T. ekeikx.
These sums can be infinite (like power series). Instead of values of f(x), or derivatives
at the basepoint, the function is described by a,, b,, c,. Everything is computed by
the "Fast Fourier Transform." This is the greatest algorithm since Newton's method.
A radio signal is near one frequency. A step function has many frequencies. A delta
function has every frequency in the same amount: 6(x) = Z cos kx. Channel 4 can't
broadcast a perfect step function. You wouldn't want to hear a delta function.
We mentioned computing. For nonlinear equations this means Newton's method.
f
For Ax = b it means elimination-algorithms take the place o formulas. Exact solu-
tions are gone-speed and accuracy and stability become essential. It seems right to
make scientific computing a part of applied mathematics, and teach the algorithms
with the theory. My text Introduction to Applied Mathematics is one step in this
direction, trying to present advanced calculus as it is actually used.
We cannot discuss applications and forget statistics. Our society produces oceans
of data-somebody has to draw conclusions. To decide if a new drug works, and if
oil spills are common or rare, and how often to have a checkup, we can't just guess.
I am astounded that the connection between smoking and health was hidden for
centuries. It was in the data! Eventually the statisticians uncovered it. Professionals
can find patterns, and the rest of us can understand (with a little mathematics) what
has been found.
One purpose in studying mathematics is to know more about your own life.
Calculus lights up a key idea: Functions. Shapes and populations and heart signals
and profits and growth rates, all are given by functions. They change in time. They
have integrals and derivatives. To understand and use them is a challenge-
mathematics takes effort. A lot of people have contributed, in whatever way they
could-as you and I are doing. We may not be Newton or Leibniz or Gauss or
Einstein, but we can share some part of what they created.
16.1 Vector Spaces and Llnear Algebra 599
16.1 Vector Spaces and Linear Algebra
You have met the idea of a matrix. An m by n matrix A has m rows and n columns
(it is square if m = n). It multiplies a vector x that has n components. The result is a
vector Ax with m components. The central problem of linear algebra is to go back-
ward: From Ax = bfind x. That is possible when A is square and invertible. Otherwise
there is no solution x-or there are infinitely many.
The crucial property of matrix multiplication is linearity. If Ax = b and AX = B
+ +
then A times x X is b B. Also A times 2x is 2b. In general A times cx is cb.
In particular A times 0 is 0 (one vector has n zeros, the other vector has m zeros). The
. whole subject develops from linearity. Derivatives and integrals obey linearity too.
Question 1 What are the solutions to Ax = O One solution is x = 0. There
?
y=o
01
may be other solutions and they fill up the "nullspace":
requires
x=o
11[JL
z
A
=
x=
also allows y = - 1
z=
2
3
When there are more unknowns than equations-when A has more columns than
rows-the system Ax = 0 has many solutions. They are not scattered randomly
around! Another solution is X = 4, Y= - 2, Z = 6. This lies on the same line as
(2, - 1,3) and (0,0,O). Always the solutions to Ax = 0 form a "space" of vectors-
which brings us to a central idea of linear algebra.
Note These pages are not concentrating on the mechanics of multiplying or invert-
ing matrices. Those are explained in all courses. My own teaching emphasizes that
f f
Ax is a combination o the columns o A. The solution x = A-'b is computed by
elimination. Here we explain the deeper idea of a vector space-and especially the
particular spaces that control Ax = 6. I cannot go into the same detail as in my book
on Linear Algebra and Its Applications, where examples and exercises develop the
new ideas. Still these pages can be a useful support.
All vectors with n components lie in n-dimensional space. You can add them and
subtract them and multiply them by any c. (Don't multiply two vectors and never
write llx or 1/A). The results x + X and x - X and cx are still vectors in the space.
Here is the important point:
The line o solutions to Ax = 0 is a "subspace"-a vector space in its own right.
f
+
The sum x X has components 6, - 3,9-which is another solution. The difference
x - X is a solution, and so is 4x. These operations leave us in the subspace.
The nullspace consists of all solutions to Ax = 0. It may contain only the zero
vector (as in the first example). It may contain a line of vectors (as in the second
example). It may contain a whole plane of vectors (Problem 5). In every case x + X
and x - X and cx are also in the nullspace. We are assigning a new word to an old
idea-the equation x - 2y = 0 has always been represented by a line (its nullspace).
Now we have 6-dimensional subspaces of an %dimensional vector space.
Notice that x2 - y = 0 does not produce a subspace (a parabola instead). Even the
x and y axes together, from xy = 0, do not form a subspace. We go off the axes when
we add (1,O) to (0, 1). You might expect the straight line x - 2y = 1 to be a subspace,
but again it is not so. When x and y are doubled, we have X - 2Y = 2. Then (X, Y)
is on a different line. Only Ax = 0 is guaranteed to produce a subspace.
16 Mathematics after Calculus
Figure 16.1 shows the nullspace and "row space." Check dot products (both zero).
Fig. 16.1 The nullspace is perpendicular to the rows of A (the columns of AT).
Question 2 When A multiplies a vector x, what subspace does Ax lie in? The
product Ax is a combination of the columns of A-hence the name "column space":
No choice of x can produce Ax = (0,0, 1). For this A, all combinations of the columns
end in a . The column space is like the xy plane within xyz space. It is a
subspace of m-dimensional space, containing every vector b that is a combination of
the columns:
The system Ax = b has a solution exactly when b is in the column space.
When A has an inverse, the column space is the whole n-dimensional space. The
nullspace contains only x = 0. There is exactly one solution x = A 1 b . This is the
good case-and we outline four more key topics in linear algebra.
1. Basis and dimension of a subspace. A one-dimensional subspace is a line. A plane
has dimension two. The nullspace above contained all multiples of (2, - 1, 3)-by
knowing that "basis vector" we know the whole line. The column space was a plane
containing column 1 and column 2. Again those vectors are a "basis"-by knowing
the columns we know the whole column space.
Our 2 by 3 matrix has three columns: (1,O) and (2, 3) and (0, 1). Those are not a
basis for the column space! This space is only a plane, and three vectors are too
many. The dimension is two. By combining (1,O) and (0, 1) we can produce the other
vector (2, 3). There are only two independent columns, and they form a basis for this
column space.
In general: When a subspace contains r independent vectors, and no more, those
vectors are a basis and the dimension is r. "Independent" means that no vector is a
combination of the others. In the example, (1,O) and (2, 3) are also a basis. A subspace
has many bases, just as a plane has many axes.
2. Least squares. If Ax = b has no solution, we look for the x that comes closest.
Section 1 1.4 found the straight line nearest to a set of points. We make the length
of Ax - b as small as possible, when zero length is not possible. No vector solves
16.1 Vector Spaces and Llnear Algebra
Ax = b, when b is not in the column space. So b is projected onto that space. This
leads to the "normal equations" that produce the best x:
When a rectangular matrix appears in applications, its transpose generally comes
too. The columns of A are the rows of AT. The rows of A are the columns of AT.
Then AT^ is square and symmetric-equal to its transpose and vital for applied
mathematics.
3. Eigenvalues (for square matrices only). Normally Ax points in a direction
different from x. For certain special eigenvectors, Ax is parallel to x. Here is a 2 by
2 matrix with two eigenvectors-in one case Ax = 5x and in the other Ax = 2x;
3 2 1
Ax=Ax: [ 1 4][]=[:]=5[:]
1 and [: :][-:]=[-:]=2[-:].
The multipliers 5 and 2 are the eigenvalues of A. An 8 by 8 matrix has eight eigen-
values, which tell what the matrix is doing (to the eigenvectors). The eigenvectors are
uncoupled, and they go their own way. A system of equations dyldt = Ay acts like
one equation-when y is an eigenvector:
d~ildt= ~ 1 + ~ 2
3 2
dyddt = yi + 4 ~ 2
has the solution
yl = eSt
y2 = eSt
which is est [:I.
The eigenvector is (1, 1).The eigenvalue A = 5 is in the exponent. When you substitute
y1 and y2 the differential equations become 5est = 5est. The fundamental principle
for dyldt = cy still works for the system dyldt = Ay: Look for pure exponential solu-
tions. The eigenvalue "lambda" is the growth rate in the exponent.
I have to add: Find the eigenvectors also. The second eigenvector (2, - 1) has
eigenvalue i 2. A second solution is y1 = 2e2', y2 = - e". Substitute those into the
=
equation-they are even better at displaying the general rule:
If x
Ax = A then d/dt(ehx) = ~ ( e ~ ~ x ) . pure exponentials are y = eAtx.
The
The four entries of A pull together for the eigenvector. So do the 64 entries of an
8 by 8 matrix-again e"x solves the equation. Growth or decay is decided by A > 0
or K < 0. When A = k + iw is a complex number, growth and oscillation combine in
e ~ = e k t e i ~ t = ekt(cos
t wt + i sin at).
Subspaces govern static problems Ax = b. Eigenvalues and eigenvectors
govern dynamic problems dyldt = Ay. Look for exponentials y = eUx.
4. Determinants and inverse matrices. A 2 by 2 matrix has determinant D = ad - bc.
This matrix has no inverse if D = 0.Reason: A-' divides by D:
This pattern extends to n by n matrices, but D and A - l become more 'complicated.
For 3 by 3 matrices D has six terms. Section 11.5 identified D as a triple product
a (b x c) of the columns. Three events come together in the singular case: D is zero
and A has no inverse and the columns lie in a plane. The opposite events produce the
"nonsingular" case: D is nonzero and A- ' exists. Then Ax = b is solved by x = A- b.
16 Mathematics after Calculus
D is also the product of the pivots and the product of the eigenvalues. The pivots
arise in elimination-the practical way to solve Ax = b without A - ' . To find eigen-
values we turn Ax = Ax into ( A - i1)x = 0 . By a nice twist of fate, this matrix A - A1
has D = 0.Go back to the example:
[f :]-A[: :]=[j;' 4iA] has D = ( i - q ( 4 - A ) - 2 = A 2 - 7 A l O .
The equation A2 - 71. + 10 = 0 gives 1 = 5 and A = 2. The eigenvalues come first, to
.
make D = 0.Then ( A - 51)x = 0 and (A - 2I)x = 0 yield the eigenvectors. These x's
go into y = e"x to solve differential equations-which come next.
16.1 EXERCISES
Read-through questions
If Ax = b and A X = B, then A times 2x + 3X equals a .
If Ax = 0 and A X = 0 then A times 2x + 3 X equals b .
In this case x and X are in the c of A, and so is the
combination d . The nullspace contains all solutions to
e . It is a subspace, which means
(independent)(dependent).
f . If x = (1, 1, 1) is
in the nullspace then the columns add to g , so they are
7 Change Problem 1 to A x =
r:1
L A
(a) Find any particular
solution x,. (b) Add any x , from the nullspace and show that
Another subspace is the h space of A, containing all x , + xo is also a solution.
combinations of the columns. The system A x = b can be
solved when b is i . Otherwise the best solution comes
from A T ~ = i . Here AT is the k
x matrix, whose
rows are I . The nullspace of AT contains all solutions to
8 Change Problem 1 to Ax =
Graph the lines x ,
Ll +
L
+ 2x2 = 1 and 2 x ,
and find all solutions.
4x2 = 0 in a plane.
m . The n space of AT (row space of A) is the fourth
9 Suppose A X , = b and A x , = 0 . Then by linearity
fundamental subspace. Each subspace has a basis containing
as many o vectors as possible. The number of vectors in &P + xo) = -Conclusion: The sum of a particular
.
the basis is the P of the subspace. solution x , and any nullvector xo is .
10 Suppose Ax = b and A X , = b. Then by linearity
When Ax =Ax, the number ;. is an q and x is an
I
r . The equation dyldt = A y has the exponential solution A(x - x,) = . The difference between solutions is a
y = s . A 7 by 7 matrix has t eigenvalues, whose
vector in . Conclusion: Every solution has the form
x = x , + x o , one particular solution plus a vector in the
product is the u D. If D is nonzero the matrix A has an
nullspace.
v . Then Ax = b is solved by x = w . The formula
for D contains 7! = 5040 terms, so x is better computed by 11 Find three vectors b in the column space of E. Find all
x . On the other hand Ax = i.x means that A - >.I has vectors b for which Ex = b can be solved.
determinant v . The eigenvalue is computed before the
12 I Ax = 0 then the rows of A are perpendicular to x. Draw
f
Z .
the row space and nullspace (lines in a plane) for A above.
13 Compute CCT and C T C . Why not C 2 ?
Find the nullspace in 1-6. Along with x go all cx. 14 Show that C x = b has no solution, if b = (-1, 1,l). Find
the best solution from C TC X = cTb.
12 - 6
1 A= 2 ] (solve A x = 0) 15 CT has three columns. How many are independent?
2 4 '=[-6 31
Which ones?
1 0 16 Find two independent vectors that are in the column space
1 0 1 of C but are not columns of C.
3 C = 0 1 (solve C x = 0)
1 2 cT=[o 1 21 17 For which of the matrices A B C E F are the columns a
basis for the column space?
16.2 Differential Equations 603
18 Explain the reasoning: If the columns of a matrix A are I
27 Compute the determinant of E - A. Find all A's that make
independent, the only solution to Ax = 0 is x = 0. this determinant zero. Which eigenvalue is repeated?
19 Which of the matrices ABCEF have nonzero deter- 28 Which previous problem found eigenvectors for Ex = Ox?
minants? Find an eigenvector for Ex = 3x.
20 Find a basis for the full three-dimensional space using 29 Find the eigenvalues and eigenvectors of A.
only vectors with positive components. 30 Explain the reasoning: A matrix has a zero eigenvalue if
[:;I-
and only if its determinant is zero.
' '
21 Find the matrix F - for which F F - = I =
31 Find the matrix H whose eigenvalues are 0 and 4 with
eigenvectors (1, 1)and (1, - 1).
~
22 Verify that (determinant of F ) = (determinant of F ~ ) . 32 If Fx = I x then multiplying both sides by gives
23 (Important) Write down F - A and compute its determi-
I F-'x = A-'x. If F has eigenvalues 1 and 3 then F-' has
nant. Find the two numbers A that make this determinant eigenvalues . The determinants of F and F are
zero. For those two numbers find eigenvectors x such that
Fx = Ax. 33 True or false, with a reason or an example.
24 Compute G = F 2. Find the determinant of G - A and the
such that Gx = Ax. Conclusion: if Fx = A then F ~ X A2x.
x
I
two A's that make it zero. For those I's find eigenvectors x
=
25 From Problem 23 find two exponential solutions to the
(b) [ O 2] has
0 0
[:I
(a) The solutions to Ax = b form a subspace.
in its nullspace and column space.
(c) ATA has the same entry in its upper right and lower
equation dyldt = Fy. Then find a combination of those left corners.
solutions that starts from yo = (1,O) at t = 0. x
(d) If Ax = A then y = e" solves dyldt = Ay.
26 From Problem 24 find two solutions to dyldt = Gy. Then (e) If the columns of A are not independent, their combi-
find the solution that starts from yo = (2, 1). nations still form a subspace.
16.2 Differential Equations
We just solved differential equations by linear algebra. Those were special systems
dyJdt = Ay, linear with constant coefficients. The solutions were exponentials,
involving eU. The eigenvalues of A were the "growth factors" A. This section solves
other equations-by no means all. We concentrate on a few that have important
applications.
Return for a moment to the beginning-when direct integration was king:
In 1, y(t) is the integral of s(t). In 2, y(t) is the integral of cy(t). That sounds circular-
it only made sense after the discovery of y = ec'. This exponential has the correct
derivative cy. To find it by integration instead of inventing it, separate y from t:
Separation and integration also solve 3: j dy/u(y)= 5 c(t)dt. The model logistic equation
has u = y - y2 = quadratic. Equation 2 has u = y = linear. Equation 1is also a special
case with u = 1 = constant. But 2 and 1 are very different, for the following reason.
The compound interest equation y' = cy is growing from inside. The equation
y' = s(t) is growing from outside. Where c is a "growth rate," s is a "source." They
don't have the same meaning, and they don't have the same units. The combination
y' = cy + s was solved in Chapter 6, provided c and s are constant-but applications
force us to go further.
16 Mathematics after Calculus
In three examples we introduce non-constant source terms.
E A P E 1 Solve dyldt = cy
XML + s with the new source term s = ekt.
Method Substitute y = B@, with an "undetermined coefficient" B to make it right:
kBek' = cBek' + ekt yields B = l/(k - c).
The source ek' is the driving term. The solution Bekt is the response. The exponent is
the same! The key idea is to expect ek' in the response.
Initial condition To match yo at t = 0, the solution needs another exponential.
It is the free response Aec', which satisfies dyldt = cy with no source. To make
y = Aec' + Bek' agree with yo, choose A = yo - B:
Final solution y = (yo - B)ec' + Bekt = yoect+ (ek' - ec')/(k- c). (1)
Exceptional case B = l / ( k - c) grows larger as k approaches c. When k = c the
method breaks down-the response Bek' is no longer correct. The solution (1)
approaches 010, and in the limit we get a derivative. It has an extra factor t:
ekt - ect
-- d
- change in ect + -(ect) tect.
=
k -c change in c dc
The correct response is tectwhen k = c. This is the form to substitute, when the driving
rate k equals the natural rate c (called resonance).
Add the free response yoec' to match the initial condition.
E A P E 2 Solve dyldt = cy + s with the new source term s = cos kt.
XML
Substitute y = B sin kt + D cos kt. This has two undetermined coefficients B and D:
kB cos kt - kD sin kt = c(B sin kt + D cos kt) + cos kt. (3)
Matching cosines gives kB = cD + 1. The sines give - kD = cB. Algebra gives B, D, y:
B=- C
D=- k c sin kt + k cos kt
k2 + c2 k2 + c2 y= k2+c2 (4)
Question Why do we need both B sin kt and D cos kt in the response to cos kt?
First Answer Equation (3) is impossible if we leave out B or D.
Second Answer cos kt is f eikt+ i e -"'. So eiktand e - "' are both in the response.
E A P E 3 Solve dyldt = cy + s with the new source term s = tekt.
XML
Method Look for y = ~ e +~ ' Problem 13 determines B and D. Add Aec' as
Dtek'.
needed, to match the initial value yo.
SECOND-ORDER EQUATIONS
The equation dyldt = cy is jrst-order. The equation d2y/dt2= - cy is second-order.
The first is typical of problems in life sciences and economics-the rate dyldt depends
on the present situation y. The second is typical of engineering and physical sciences-
the acceleration d2y/dt2enters the equation.
If you put money in a bank, it starts growing immediately. If you turn the wheels
of a car, it changes direction gradually. The path is a curve, not a sharp corner.
Newton's law is F = ma, not F = mu.
16.2 Differential Equations
A mathematician compares a straight line to a parabola. The straight line crosses
the x axis no more than once. The parabola can cross twice. The equation
+
ax2 + bx c = 0 has two solutions, provided we allow them to be complex or equal.
These are exactly the possibilities we face below: two real solutions, two complex
solutions, or one solution that counts twice. The quadratic could be x2 - 1 or x2 + 1
or x2. The roots are 1 and - 1, i and - i, 0 and 0.
In solving diflerential equations the roots appear in the exponent, and are called A.
E A P E 4 y" = + y: solutions y = et
XML and y = e-' A = 1, - 1
XML
E A P E 5 y" = 0 y: solutions y = 1 and y = t iZ = 0 , 0
E A P E 6 y" = -y: solutions y = cos t and y = sin t A = i, -i
XML
Where are the complex solutions? They are hidden in Example 6, which could be
written y = eitand y = e-". These satisfy y" = - y since i2 = - 1. The use of sines and
cosines avoids the imaginary number i, but it breaks the pattern of e".
Example 5 also seems to break the pattern-again eUis hidden. The solution y = 1
is eO'. The other solution y = t is teot. The zero exponent is repeated-another excep-
tional case that needs an extra factor t.
Exponentials solve every equation with constant coeficients and zero right hand side:
To solve ay" + by' + cy = 0 substitute y = e" and find A.
This method has three steps, leading to the right exponents A = r and A = s:
+
1. With y = eU the equation is aA2ee" bAe" + ceAt 0. Cancel e".
=
2. Solve aA2 + b + c = 0. Factor or use the formula A = (- b f Jbi-rlac)/2a.
A
3. Call those roots A = r and A= s. The complete solution is y = Aert + Best.
The pure exponentials are y = er' and y = e". Depending on r and s, they grow or
decay or oscillate. They are combined with constants A and B to match the two
+
conditions at t = 0. The initial state yo equals A B. The initial velocity yb equals
rA + sB (the derivative at t = 0).
E A P E 7 Solve y" - 3y'
XML + 2y = 0 with yo = 5 and yb = 4.
Step 1 substitutes y = e". The equation becomes i2e" - 3Ae" + 2e" = 0. Cancel e".
Step 2 solves 1 - 31. + 2 = 0. Factor into (A - 1)(A- 2) = 0. The exponents r, s are 1,2.
'.
+
Step 3 produces y = Aet + ~ e ~ ' . initial conditions give A + B = 5 and 1A 2B = 4.
The
The constants are A = 6 and B = - 1. The solution is y = 6et - elt.
This solution grows because there is a positive A. The equation is "unstable." It
becomes stable when the middle term - 3y' is changed to + 3y'. When the damping
is positive the solution decays. The 1's are negative:
+
E A P E 8 (A2 31 + 2) factors into (1 + 1)(1+ 2). The exponents are - 1 and -2.
XML
The solution is y = Ae-' + Be-2t. It decays to zero for any initial condition.
E A P E 9-10
XMLS Solve y" + 2y' + 2y = 0 and y" + 2y' + y = 0. How do they differ?
Key difference A2 + 2A + 2 has complex roots, L2 + 23, + 1 has a repeated root:
A2+21+2=0 gives A=-1+i ( 1 + 1 ) ~ = 0 gives A=-1,-1.
The - 1 in all these R's means decay. The i means oscillation. The first exponential
is e(- I+ i)t , which splits into e-' (decay) times eit (oscillation). Even better, change eit
16 Mathematics after Calculus
and e-" into cosines and sines:
= Ae(-1 +i)t + ge(-1-i)t = e-'(a cos t + b sin t).
(5)
At t = 0 this produces yo = a. Then matching yb leads to b.
Example 10 has r = s = - 1 (repeated root). One solution is e-' as usual. The
second solution cannot be another e-'. Problem 21 shows that it is te-'-again the
exceptional case multiplies by t! The general solution is y = Ae-' Bte-'. +
Without the damping term 2yf,these examples are y" + 2y = 0 or y" + y = 0-pure
oscillation. A small amount of damping mixes oscillation and decay. Large damping
gives pure decay. The borderline is when A is repeated (r = s). That occurs when
b2 - 4ac in the square root is zero. The borderline between two real roots and two
complex roots is two repeated roots.
The method of solution comes down to one idea: Substitute y = eU.The equations
apply to mechanical vibrations and electrical circuits (also other things, but those
two are of prime importance). While describing these applications I will collect the
information that comes from A.
SPRINGS AND CIRCUITS: MECHANICAL AND ELECTRICAL ENGINEERING
A mass is hanging from a spring. We pull it down an extra distance yo and give it a
starting velocity yb. The mass moves up or down, obeying Newton's law: mass times
acceleration equals spring force plus damping force:
my" = - ky - dy' or my" + dy' + ky = 0. (6)
This is free oscillation. The spring force - ky is proportional to the stretching y
(Hooke's law). The damping acts like a shock absorber or air resistance-it takes
out energy. Whether the system goes directly toward zero or swings back and forth
is decided by the three numbers m, d, k. They were previously called a, b, c.
16A The solutions e" to my" + dy' + ky = 0 are controlled by the roots of
+
mA2 d l + k = 0. With d > 0 there is damping and decay. From J62-4mk
there may be oscillation:
overdamping: d > 4mk gives real roots and pure decay (Example 8)
underdamping: d < 4mk gives complex roots and oscillation (Example 9)
I critical damping: d2 = 4mk gives a real repeated root -d/2m (Example 10)
We are using letters when the examples had numbers, but the results are the same:
d l
m i 2 + d 1 + k = 0 hasroots r , s = - - f --,/;iT-4mk.
2m 2m
Overdamping has no imaginary parts or oscillations:y = Ae" + Bes'. Critical damping
has r = s and an exceptional solution with an extra t: y = Ae" + Bte". (This is only a
solution when r = s.) Underdamping has decay from - d/2m and oscillation from the
imaginary part. An undamped spring (d = 0) has pure oscillation at the natural fre-
quency wo = Jklm.
AN these possibilities are in Figure 16.2, created by Alar Toomre. At the top is pure
oscillation (d = 0 and y = cos 2t). The equation is y" + dy' + 4y = 0 and d starts to
grow. When d reaches 4, the quadratic is A2 + 41 + 4 or (1 + 2)2. The repeated root
16.2 Differential Equations 607
1
0
)lex
yo = 1
yo,= 0
Fig. 16.2
yields e-2t and te-2t. After that the oscillation is gone. There is a smooth transition
from one case to the next-as complex roots join in the repeated root and split into
real roots.
At the bottom right, the final value y(27n) increases with large damping. This was
a surprise. At d = 5 the roots are -1 and - 4. At d = 8.5 the roots are - ' and - 8.
The small root gives slow decay (like molasses). As d -+ oo the solution approaches
y= 1.
If we are serious about using mathematics, we should take advantage of anything
that helps. For second-order equations, the formulas look clumsy but the examples
are quite neat. The idea of e"' is absolutely basic. The good thing is that electrical
circuits satisfy the same eqution. There is a beautiful analogy between springs and
circuits:
mass m , inductance L
damping constant d *- resistance R
elastic constant k - 1/(capacitance C)
The resistor takes out energy as the shock absorber did-converting into heat by
friction. Without resistance we have pure oscillation. Electric charge is stored in
the capacitor (like potential energy). It is released as current (like kinetic energy).
It is stored up again (like a stretched spring). This continues at a frequency
c0 = 1/-LC (like the spring's natural frequency k/r). These analogies turn
mechanical engineers into electrical engineers and vice versa.
The equation for the current y(t) now includes a driving term on the right:
(7)
L
dt + R y + -C y dt = applied voltage = V sin cot.
To match networks with springs, differentiate both sides of (7):
Ly" + Ry' + y/C = Vwt cos cot. (8)
The oscillations are free when V = 0 and forced when V A0. The free oscillations
eAt are controlled by LA2 + RA + 1/C = 0.Notice the undamped case R = 0 when
16 Mathematics after Calculus
A = i/-. This shows the natural frequency w, = l / p . Damped free oscilla-
tions are in the exercises-what is new and important is the forcing from the right
hand side. Our last step is to solve equation (8).
PARTICULAR SOLUTIONS-THE METHOD OF UNDETERMINED COEFFICIENTS
The forcing term is a multiple of cos wt. The "particular solution" is a multiple
of cos o t plus a multble of sin wt. To discover the undetermined coefficients in
y = a cos cot + b sin cot, substitute into the differential equation (8):
- Lw2(a cos wt + b sin ot) + Rw(- a sin wt + b cos wt)
+ (a cos wt + b sin wt)/C = Vw cos wt.
The terms in cos wt and the terms in sin o t give two equations for a and b:
-a h 2 + bRw + a/C = Vw and -b ~ - aRw + blC = 0.
w ~ (9)
EXAMPLE 11 Solve y" + y = cos o t . The oscillations are forced at frequency w. The
oscillations are free (y" + y = 0) at frequency 1. The solution contains both.
Particular solution Set y = a cos o t + b sin wt at the driving frequency w, and (9)
becomes
-aw2+O+a= 1 and -bo2-O+b=O.
The second equation gives b = 0. No sines are needed because the problem has no
dyldt. The first equation gives a = 1/(1 - w2), which multiplies the cosine:
y = (cos wt)/(1 - w2) solves y" + y = cos wt. (10)
General solution Add to this particular solution any solution to y" + y = 0:
Problem of resonance When the driving frequency is o = 1, the solution (1 1)
becomes meaningless-its denominator is zero. Reason: The natural frequency in
cos t and sin t is also 1. A new particular solution comes from t cos t and t sin t.
The key to success is to know the form for y. The table displays four right hand
sides and the correct y's for any constant-coefficient equation:
Right hand side Particular solution
ekt y = Bek' (same exponent)
cos wt or sin o t y = a cos a t + b sin wt (include both)
polynomial in t y = polynomial of the same degree
ektcos wt or ektsin wt y = aektcos wt + bektsin wt
Exception If one of the roots A for free oscillation equals k or ico or 0 or k + iw, the
corresponding y in the table is wrong. The proposed solution would give zero on the
right hand side. The correct form for y includes an extra t. All particular solutions
are computed by substituting into the differential equation.
Apology Constant-coefficient equations hardly use calculus (only e"). They reduce
l
directly to algebra (substitute y, solve for i and a and b). I find the S-curve from the
logistic equation much more remarkable. The nonlinearity of epidemics or heartbeats
or earthquakes demands all the calculus we know. The solution is not so predictable.
The extreme of unpredictability came when Lorenz studied weather prediction and
discovered chaos.
NUMERICAL METHODS
Those four pages explained how to solve linear equations with constant coefficients:
Substitute y = eat. The list of special solutions becomes longer in a course on
differential equations. But for most nonlinear problems we enter another world-
where solutions are numerical and approximate, not exact.
In actual practice, numerical methods for dyldt =f (t, y) divide in two groups:
1. Single-step methods like Euler and Runge-Kutta
2. Multistep methods like Adams-Bashforth
The unknown y and the right side f can be vectors with n components. The notation
stays the same: the step is At = h, the time t, is nh, and y, is the approximation to
the true y at that time. We test the first step, to find y, from yo = 1. The equation is
dyldt = y, so the right side is f = y and the true solution is y = et.
Notice how the first value off (in this case 1) is used inside the second f:
TEST y, =1 + +h[l+ (1 + h)] = 1 + h + $h2
At time h the true solution equals eh. Its infinite series is correct through h2
for Improved Euler (a second-order method). The ordinary Euler method
+
yn+ = yn h (t,, y,) is first-order. TEST: y, = 1 + h. Now try Runge-Kutta
f
(a fourth-order method):
Now the first value off is used in the second (for k,), the second is used in the third,
and then k3 is used in k,. The programming is easy. Check the accuracy with another
test on dyldt = y:
h2 h3 h4
=1 + h + - + - + -.
2 6 24
This answer agrees with eh through h4.
These formulas are included in the book so that you can apply them directly-
for example to see the S-shape from the logistic equation with f = cy - by2.
Multistep formulas are simpler and quicker, but they need a single-step method to
get started. Here is y, in a fourth-order formula that needs yo, y,, y,, y,. Just shift
,
all indices for y,, y,, and y, + :
h
Multistep y4 = y3 + - [55yi - 59y; + 37y; - 9ybl.
24
The advantage is that each step needs only one new evaluation of y; =f(t,, y,).
Runge-Kutta needs four evaluations for the same accuracy.
Stability is the key requirement for any method. Now the good test is y' = - y. The
solution should decay and not blow up. Section 6.6 showed how a large time step
makes Euler's method unstable-the same will happen for more accurate formulas.
The price of total stability is an "implicit method" like y, = yo + + h ( ~ b y;), +
where the unknown y, appears also in y; . There is an equation to be solved at every
step. Calculus is ending as it started-with the methods of Isaac Newton.
610 16 Mathematics after Calculus
16.2 EXERCISES
Read-through questions 14 Solve y' = y + t following Example 3 (c = 1 and k = 0).
5t
The solution to y' -5y = 10 is y = Ae + B. The homo-
geneous part Ae 5' satisfies y'-5y = a . The particu-
lar solution B equals b . The initial condition Yo is Problems 15-28 are about second-order linear equations.
matched by A = c . For y'-5y = ekt the right form 15 Substitute y = ea' into y" + 6y' + 5y = 0. (a) Find all it's.
is y =Ae + d . For y'-5y = cos t the form is (b) The solution decays because . (c) The general
y = Ae5'+ e + f solution with constants A and B is
The equation y"+4y'+ 5y=0 is second-order because 16 Substitute y = eat into y" + 9y = 0. (a) Find all it's. (b) The
g . The pure exponential solutions come from the roots solution oscillates because . (c) The general solution
of h , which are r= i and s= I . The general with constants a and b is
solution is y = k . Changing 4y' to I yields pure
oscillation. Changing to 2y' yields = - 1 + 2i, when the F
17 Substitute y = eAt into y" + 2y' + 3y = O0. ind both it's.
solutions become y= m . This oscillation is The solution oscillates as it decays because . The
(over)(under)(critically) damped. A spring with m = 1, d = 2, general solution with A and B and et is . The
k = 5 goes (back and forth)(directly to zero). An electrical general solution with e-' times sine and cosine is
network with L = 1, R = 2, C = also n 18 Substitute y = eat into y" + 6y' + 9y = 0. (a) Find all 's.
One particular solution of y" + 4y = e' is e' times (b) The general solution with e and teA is
o . If the right side is cos t, the form of y, is p . If the 19 For y"+dy'+y=0 find the type of damping at
right side is 1 then y, = q . If the right side is r we d=0, 1, 2, 3.
have resonance and y, contains an extra factor s
20 For y"+2y'+ky=0 find the type of damping at
k=0, 1,2.
Problems 1-14 are about first-order linear equations.
21 If A2+ b + c = 0 has a repeated root prove it is =
I Substitute y = Be 3 ' into y' - y = 8e3' to find a particular - b/2. In this case compute y" + by' + cy when y = teA'.
solution.
22 A2+ 3 + 2 = 0 has roots -1 and -2 (not repeated). Show
2 Substitute y = a cos 2t + b sin 2t into y' + y = 4 sin 2t to that te-' does not solve y" + 3y' + 2y = 0.
find a particular solution.
23 Find y = a cos t + b sin t to solve y" + y'+ y = cos t.
3 Substitute y = a + bt + ct2 into y' + y = 1 + t2 to find a
24 Find y = a cos ot + b sin cot to solve y" + y' + y = sin ot.
particular solution.
25 Solve y" + 9y = cos 5t with Yo= 0 and yO= 0. The solution
4 Substitute y = aetcos t + be'sin t into y' = 2e'cos t to find
contains cos 3t and cos 5t.
a particular solution.
26 The difference cos 5t - cos 3t equals 2 sin 4t sin t. Graph
5 In Problem 1 we can add Ae' because this solves the equa-
it to see fast oscillations inside slow oscillations (beats).
tion . Choose A so that y(0) = 7.
27 The solution to y"+o2y=coscot with yo=0 and
6 In Problem 2 we can add Ae - t, which solves
y = 0 is what multiple of cos ot-cos cot? The formula
Choose A to match y(0)= 0.
breaks down when o =
7 In Problem 3 we add to match y(O)= 2.
28 Substitute y = Ae i "' into the circuit equation
i'
Ly' + Ry + y dt/C = Ve . Cancel ei"' to find A. Its denomi-
8 In Problem 4 we can add y = A. Why?
nator is the impedance.
9 Starting from Yo= 0 solve y' = ek ' and also solve y' = 1.
Show that the first solution approaches the second as k -0.
10 Solve y' - y = ek' starting from Yo= 0. What happens to Problems 29-32 have the four right sides in the table (end of
your formula as k - 1? By l'H6pital's rule show that y section). Find Ypa,,icularby using the correct form.
approaches te' as k -, 1.
29 y"+ 3y = e5' 30 y" + 3y = sin t
11 Solve y' - y = e' + cos t. What form do you assume for y
with two terms on the right side? 31 y"+2y= l+t 32 y" + 2y = e' cos t.
33 Find the coefficients of y in Problems 29-31 for which the
12 Solve y' + y = e' + t. What form to assume for y?
forms in the table are wrong. Why are they wrong? What new
13 Solve y' = cy + te' following Example 3 (c - 1). forms are correct?
16.3 Dlscrete Mathematics: Algorithms 611
34 The magic factor t entered equation (2). The series for 40 In one sentence tell why y" = 6 y has exponential solutions
+ + + +
ek' - eCt starts with 1 kt 4k2t2minus 1 ct ic2t2.Divide but y" = 6y2 does not. What power y = xn solves this
by k - c and set k = c to start the series for te". equation?
35 Find four exponentials y = e for d 4y/dt4- y = 0.
" 41 The solution to dy/dt =f (t), with no y on the right side,
is y = j f (t) dt. Show that the Runge-Kutta method becomes
36 Find a particular solution to d4y/dt + y = et. Simpson's Rule.
37 The solution is y = ~ e - + Bte-2t when d = 4 in
~ '
42 Test all methods on the logistic equation y' = y - y2 to
Figure 16.2. Choose A and B to match yo = 1 and yb = 0.
see which gives y, = 1 most accurately. Start at the inflection
How large is y(271)?
point yo = 4 with h = &. Begin the multistep method with
38 When d reaches 5 the quadratic for Figure 16.2 is exact values of y = (1 + e-')- l.
+
A2 + 5A 4 = ( + l)(A + 4). Match y = Ae-I + Bed4' to
A
43 Extend the tests of Improved Euler and Runge-Kutta to
yo = 1 and yb = 0. How large is y(2n)?
y' = - y with yo = 1. They are stable if 1y, 1 < 1 How large
.
'
39 When the quadratic for Figure 16.2 has roots -r and can h be?
-4/r, the solution is y = Ae-" + 44 Apply Runge-Kutta to y' = - 100y + 100 sin t with
(a) Match the initial conditions yo = 1 and yb = 0. yo = 0 and h = .02. Increase h to .03 to see that instability
(b) Show that y approaches 1 as r + 0. is no joke.
Discrete Mathematics: Algorithms
Discrete mathematics is not like calculus. Everything isfinite. I can start with the 50
states of the U.S. I ask if Maine is connected to California, by a path through
neighboring states. You say yes. I ask for the shortest path (fewest states on the way).
You get a map and try all possibilities (not really all-but your answer is right). Then
I close all boundaries between states like Illinois and Indiana, because one has an
even number of letters and the other has an odd number. Is New York still connected
to Washington? You ask what kind of game this is-but I hope you will read on.
Far from being dumb, or easy, or useless, discrete mathematics asks good questions.
It is important to know the fastest way across the country. It is more important to
know the fastest way through a phone network. When you call long distance, a quick
connection has to be found. Some lines are tied up, like Illinois to Indiana, and there
is no way to try every route.
The example connects New York to New Jersey (7 letters and 9). Washington is
' connected to Oregon (10 letters and 6). As you read those words, your mind jumps
to this fact-there is no path from New York with 7 letters to Washington with 10.
Somewhere you must get stuck. There might be a path between all states with an
odd number of letters-I doubt it. Graph theory gives a way to find out.
GRAPHS
A model for a large part of finite mathematics is a graph. It is not the graph of
y =f(x). The word "graph" is used in a totally different way, for a collection of nodes
and edges. The nodes are like the 50 states. The edges go between two nodes-the
neighboring states. A network of computers fits this model. So do the airline connec-
tions between cities. A pair of cities may or may not have an edge between them-
depending on flight schedules. The model is determined by V and E.
16 Mathematics after Calculus
DEFINITION A graph is a set V of nodes (or vertices) and a set E of edges.
EXAMPLE 1 How many edges are possible with n nodes, in a complete graph?
The first node has edges to the n - 1 other nodes. (An edge to itself is not allowed.)
The second node has n - 2 new edges. The third node has another n - 3. The total
count of edges, when none are missing, is the sum from Section 5.3:
1 2 + + --- + (n - 1) = in(n - 1) edges in a complete graph.
Fifty states have 25 -49 = 1225 possible edges. The "neighboring states graph" has
less than 200. A line of 6 nodes has 5 edges, out of 4 6 5 = 15 possible.
EXAMPLE 2 Which states with an odd number of letters are reachable from New
York? Boundaries to states like Pennsylvania (12 letters) are closed.
Method of solution Start from New York (7). There is an edge to Connecticut (11).
That touches Massachusetts (13), which is a neighbor of Vermont (7). But we missed
Rhode Island, and how do we get back? The order depends on our search method-
and two methods are specially important.
Depth f i s t search (DFS) "From the current state, go to one new state if possible."
But what do we do from Vermont, when New Hampshire (12) is not allowed? The
answer is: backtrack to Massachusetts. That becomes the next current state.
We label every state as we reach it, to show which state we came from. Then VT
has the label MA, and we easily cross back. From MA we go to RI. Then backtrack
to MA and CT and NY. At every step I searched for a new state with no success.
From NY we see NJ (9). Finally we are in a corner.
The depth first search is ended, by a barrier of even states. Unless we allow Ontario
and keep going to Minnesota.
Breadthfist search (BFS) "From the current state, add all possible new states to the
bottom of the list. But take the next current state from the top of the list." There is
no need to backtrack.
From NY we reach VT and MA and CT and NJ. What comes next?
Where DFS moves from the last possible state, breadth first search moves from
thefirst possible state. No move from VT is possible-so we "scan" from Massachu-
setts. We see Rhode Island (barely). That ends BFS.
The same six states are reached both ways. Only the order is different. DFS is last
in-&st out. BFS is f i s t in-fist out. You have the same choice in drawing a family
tree-follow a path as far as it goes and backtrack, or list all brothers and sisters
before their children. The BFS graph in Figure 16.3 is a tree. So is the DFS graph,
using forward edges only.
MA
DFS from NY
i
CT BFS from NY - CT 3
NJ NJ
Fig. 16.3 Search trees from New York. The minimum spanning tree.
16.3 Discrete Mathematics: Algorithms
DEFINITION A tree is a connected graph with no loops. Its N nodes are connected
by N - 1 edges. If N = n, so every node is in the tree, it is a spanning tree.
A A
The path from V to KY to TN to NC to V is a loop (or cycle). If one of those four
edges is removed, we have a tree. If two edges are removed, we have two trees (a
small forest).
XML
E A P E 3 .411ow an edge between neighboring states only when one state is even
and the other is odd. Are the lower 48 states connected?
Start anywhere-say California. Apply either type of search-maybe DFS. Go to
Arizona (7) then Utah (4) then WY (9) then CO (8) then NM then OK then TX.
(I am writing this on an airplane, looking at the map.) We will never get to Florida!
It is blocked by Alabama and Georgia.
The search creates a tree, but not a spanning tree. This graph is not connected.
An odd-to-even graph is special and important. It is called "bipartite," meaning
two parts. The odd states are in one part, the even states are in the other. All edges
go between parts. No edges are within a part.?
XML
E A P E 4 Is there a "complete matching" between 25 even and 25 odd states?
This requires neighboring states to be paired off (with no repetition).
Method 1 Start pairing them OR CA-AZ, UT-WY, NV-ID, NE-SD, WA-MT.
What about Oregon? Maybe it should have been paired with Idaho. Then Nevada
could pair with Arizona. Trial and error goes nowhere fast.
Method 2 Think first. The four states CA-OR-WA-NV are even. This whole group
is only connected to three odd states (AZ, ID, MT). The matching is impossible.
This is Hall's Theorem. In a course on graphs, it would be proved. Our purpose here
is to see the ideas and questions in discrete mathematics, more than the proofs.
H REY
T E G E D ALGORITHM
Put back all edges between neighboring states. The nodes could be provinces of
Canada or states of Australia. If they are countries of Europe-Asia-Africa (or the
Americas), we need a new map. The essential thing is the new problem.
In a network each edge has a "length." A positive number cij is assigned to the edge
from node i to node j. In an economics problem, cij is the cost. In a flow problem it
is the capacity, in an electrical circuit it is the conductance. We look for paths that
minimize these "lengths."
R BE
P O L M Find the minimum spanning tree. Connect all nodes by a tree with the
smallest possible total length.
The six cheapest highways connecting seven cities form a minimum spanning tree. It
is cheapest to build, not cheapest to drive-you have to follow the tree. Where there
tExactly half the states have an even number of letters (a real trivia question). This is the little-
known reason for admitting Alaska and Hawaii.
16 Mathematics after Calculus
is no edge we set cij = GO (or an extremely large value, in an actual code). Then the
algorithm works with a complete network-all n(n - 1)/2 edges are allowed. How
does it find the minimum spanning tree in Figure 16.3c?
Method 1 Always add the shortest edge that goes out from the current tree.
Starting from node s, this rule chooses edges of length 1,2, 7,4, 3. Now it skips 5,
which would close a loop. It chooses 6, for total length 23.
Method 2 Add edges in order, from shortest to longest. Reject an edge that closes a
loop. Several trees grow together (a forest). At the end we have a minimum spanning
tree.
This variation chooses edge lengths in the order 1 , 2 , 3 , 4 , 6 (rejecting 5), 7. In our
network both methods produce the same tree. When many edges have equal length,
there can be many shortest trees.
These methods are examples of the Greedy Algorithm: Do the best thing at every
step. Don't look ahead. Stick to a decision once it is made. In most network problems
the Greedy Algorithm is not optimal-in this spanning tree problem it is.
Method 2 looks faster than Method 1. Sort the edges by length, and go down the
list. Just avoid loops. But sorting takes time! It is a fascinating problem in itself-
bubble sort or insertion sort or heapsort. We go on to a final example of discrete
mathematics and its algorithms.
PROBLEM Find the shortest path from the source node s to each other node.
The shortest path may not go along the minimum spanning tree. In the figure, the
best path going east has length 1 + 8. There is a new shortest path tree, in which the
source plays a special role as the "root."
How do we find shortest paths? Listing all possibilities is more or less insane. A
good algorithm builds out from the source, selecting one new edge at every step.
After k steps we know the distances dl, ..., d, to the k nearest nodes.
Algorithm: Minimize di + cij over all settled nodes i and all remaining nodes j.
The best new node j is a distance cij from a settled node, which is a distance di from
the source. In the example network, the first edges are 1,2,7. Next is 8. The northeast
node is closest to the source at this step. The final tree does not use edges 3, 5,6-
even though they are short.
These pages were written to show you the algorithmic part of discrete mathematics.
The other part is algebra-permutations, partitions, groups, counting problems,
generating functions. There is no calculus, but that's fair. The rest of the book was
written to show what calculus can do-I hope very much that you enjoyed it.
Thank you for reading, and thinking, and working.
Read-through questions To find a path from node i to node j, two search methods
A graph is a set V of a and a set E of b . With 6 are h . As nodes are reached, DFS looks out from the
nodes, a complete graph has c edges. A spanning tree i node for a new one. BFS looks out from j . DFS
has only d . A tree is defined as e , and it is spanning must be prepared to k to earlier nodes. In case of fire,
if r . It has a path between each pair of nodes. BFS locates all doors from the room you are in before I .
16.3 Discrete Mcrthemcrtics: Algorithms 615
In a bipartite graph, all edges go from one part to m .
A matching is impossible if k nodes in one part are connected
to n nodes in the other part. The edges in a network have
o cij. A minimum spanning tree is P . It can be found
by the q algorithm, which accepts the shortest edge to a
new node without worrying about r .
1 Start from one node of a hexagon (six nodes, six edges).
Number the other nodes by (a) breadth first search (b) depth
first search.
2 Draw two squares with one node in common (7-node
graph). From that node number all others by DFS and BFS.
Indicate backtracks. 16 Find the loop in network B. Then find a minimum span-
3 How many spanning trees in the hexagon graph? ning tree by Method 1 and Method 2.
4 Draw a spanning tree in the two-square graph. How many 17 How many spanning trees in graph B? It has one loop.
spanning trees does it have? 18 Show that a graph cannot have O,1,2,3, and 4 edges
5 Define a connected graph. If a graph has 7 edges and 9 going into its five nodes.
nodes, prove that it is not connected. 19 If the only edges into a node have lengths 6 and 8, can
6 Define a loop. If a connected graph has 8 edges and 9 they both be in a minimum spanning tree?
nodes, prove that it has no loops. 20 In Problem 19, prove that a minimum spanning tree con-
7 Find the shortest path (minimum number of edges) from tains edge (6) if it contains edge (8).
Maine to California. 21 True or false, with reason or example.
8 Which state is farthest (how many edges are needed) from (a) In a complete network, the minimum spanning tree
the state you are in? Why would it come last in BFS? contains the n - 1 shortest edges.
(b) If a graph has 9 nodes and 9 edges, it has a loop.
9 List the steps of BFS from your state to Georgia or
Colorado or New Jersey. (There are edges Hawaii-California (c) A graph with a complete matching must be connected.
and Alaska- Washington.) 22 Draw a tree that is perfect for (a) DFS; (b) BFS.
10 With edges between odd neighboring states and between 23 The adjacency matrix has aij = 1 if there is an edge from
even neighbors, what is the largest connected set of states? node i to node j. Write down this matrix for graphs A and B.
Map required.
24 In a complete network start with dij = cij. Show that the
11 With edges only from odd to even neighbors, how many dij at the end of this program are shortest distances:
states can be matched? (Answer unknown to author-please for i = l to n d o
advise.)
for j = 1 to n do
12 A matching is a forest of two-node trees. Give another for k = 1 to n do
description. +
dij = max(dij, dik dkj)
13 Find the minimum spanning tree for network A. 25 How many spanning trees in graph A?
14 Find the shortest path tree from the center of network A. 26 A maximum spanning tree has greatest possible length.
Give an algorithm to find it.
15 Is there a complete matching between left and right nodes
in graph B? If not, which group of nodes has too few 27 Write a code that will find a spanning tree (or stop), given
connections? a list of edges like (1, 2), (1, 3), (4, 7), ....
MIT OpenCourseWare
Resource: Calculus Online Textbook
Gilbert Strang
The following may not correspond to a particular course on MIT OpenCourseWare, but has been
provided by the author as an individual learning resource.
For information about citing these materials or our Terms of Use, |
IGCSEMathematics (4400) London Examinations November 2004 delivered locally, recognised globally ... Mathematics 4400, November 2004 PAPER 4H General Comments There were few errors which occurred regularly and hardly any at all on the first half of the
Curriculum Description for London IGCSEMathematics (4400) IGCSEMathematics will • meet the needs of students of all abilities • provide a solid basis for AS and Advanced GCE or equivalent qualifications
MathematicsIGCSE Introduction 1 Maths IGCSE Introduction Welcome to your MathematicsIGCSE course! This introduction contains all the information you need to be able ... Paper 2F or 4H 2 hours – 50% of the total marks In all examination papers:
GCSE Mathematics (1380) Paper 4H . Edexcel is one of the leading examining and awarding bodies in the UK and throughout the world. We provide a wide range of qualifications including academic, vocational, occupational and specific programmes for employers.
IGCSEMathematics Overview IGCSE in Mathematics aims to give students a foundation in mathematical skills and develop their knowledge and understanding of how to use and apply mathematical techniques and concepts to solve |
Schaum's Outline of Mathematical Methods for Business and Economics reviews the mathematical tools, topics, and techniques essential for success in business and economics today. The theory and solved problem format of each chapter provides concise explanations illustrated by examples, plus numerous problems with fully worked-out solutions. And you don't have to know advanced math beyond what you learned high school. The pedagogy enables you to progress at your own pace and adapt the book to your own needs.
Description:
This text offers the ideal approach for economics and business
students seeking to understand the mathematics relevant to them. Each chapter demonstrates basic mathematical techniques, while also explaining the economic analysis and business context where each is used.Now in ... |
Integrating
more in-class projects in which students do not have a "blueprint"
solution, but must figure out how to apply knowledge:
Using similar triangles to measure tall outdoor objects
Using trigonometry to measure flagpole height
Combining algebraic and geometric techniques to figure out the weight of the pillars in the hallway
Using
calculus techniques to cut out a box that will hold the maximum
possible number of packing peanuts (using actual packing peanuts to
test)
Notes, HW, and
assessments invariably contain problems designed to have more than one
possible approach; we often review multiple paths, sometimes students
will informally demonstrate their different ideas
P6. Connects and relates knowledge in and across disciplines.
English
- always put heavy emphasis on vocabulary, word origins, and Greek and
Latin roots that apply in non-math situations (this often ties to World
Languages as well); require students to explain things in their own
words, usually using (eek!) complete sentences
History - occasional historical notes about Ancient Greeks and Geometry
Physics
- coordinating with Kat so that we construct a Geometric proof of a
major optics equation while she's presenting the scientific side of it
Philosophy - our concepts of proof and
validity are a decent introduction to Logic; in Honors Geometry we also
looked at some of Zeno's paradoxes that question the discrete vs
continuous nature of space and time
Higher Mathematics
(yes, it's different enough that I'm counting it as cross-disciplinary) -
Problems of the Week introduce Honors Geometry students to Abstract
Algebra (Finite and Cyclic Group Theory) and its applications in
Cryptography and Combinatorics. |
Algebra and Functions Worksheets Section for SAT Exam Prep
Find in this section exercises on algebra I and algebra II. All these exercise will surely help you to get ready and ACE your SAT exam. They are exercises and free printable worksheets, video solutions available upon request by email totally for free. The algerba exercises cover the following sections:
Algebra, functions, equations, measurement, data analysis, statistics, probability, sequences and series, transformations, numbers and operations, geometry and various other mathematics topics.
To cover all your SAT exam math program, we propose to the user to follow the planning below and get started. It's a 4 weeks planning, each week enough to cover one module.
Math Section Name
Preparation Timing
Day of the weeks
Numbers and Operations
16h (2h/days)
days 1-7
Algebra and Functions
20h (2h/days)
days 8-18
Geometry and Measurement
14h (2h/days)
days 19-25
Data Analysis, statistics and Probability
SAT FREE TIPS FOR MATH
The best strategy for the SAT test Math exam is to get the maximum right answer and the minimum wrong answer. This will probably seem to you as obvious but learn that the SAT scoring deducts ¼ point for each incorrect multiple choice answer. This simply means it is better not to answer a question when you are not sure of the answer. You may take some few risks but really don't try to answer randomly.
Let list some tips for the SAT exam math section. This can be considered as a SAT free online preparation. Read more...
They are many tools available on this site and our partners sites to help you prepare your SAT math exam. math games, math puzzles, SAT free practice questions, SAT free math study guide, SAT free math problem solving. All these tools are offered to you for free.
Begin your SAT exam Math section preparation now on our website. All along we are going to help the
user to prepared for the Math section of his SAT exam. Most of the people has great difficulties to pass
the SAT math exam.With our free online math preparation solution, we offer free math quick revision
notes, math quizzes, ath exercises, math worksheets, SAT exam tips and even an online math function
plottingsoftware.
Here you will find most of the detailed solutions to the worksheets proposed in the math worksheet section. Download and print all the free online printable math worksheets on algebra, geometirc, statistics, probabilities, Data analysis and find the solutions here. suggestion. Read more...
SAT Exam Online Registration
SAT exam math section special e-books and tutorial CDs for a 4 weeks full preparation program. These CDs covers all the SAT math exam: Algebra, Numbers and Operations, Geometric, Data analysis, Statistics and Probabilities. Email or Order now Read more... |
Experiencing Geometry : On Plane and Sphere - 96 edition
Summary: In Experiencing Geometry on Plane and Sphere, Henderson invites readers to explore the basic ideas of geometry beyond the formulation of proofs. The text conveys a distinctive approach, stimulating readers to develop a broader, deeper understanding of mathematics through active participation including discovery, discussion and writing about fundamental ideas. It provides a series of interesting, challenging problems, then encourages readers to gather their r...show moreeasonings and understandings of each problem and discuss their findings in an open forum. Features |
Brings together the best of both new and traditional curricula in an effort to meet the needs of even more instructors teaching calculus. This book ...Show synopsisBrings together the best of both new and traditional curricula in an effort to meet the needs of even more instructors teaching calculus. This book includes the Rule of Four, an emphasis on modeling, exposition that students can read and understand and a flexible approach to technology. It also features conceptual and modeling problems (US). Glued binding. 753 p. Contains:...New. Trade paperback (US). Glued binding. 753 p. Contains: Tables, black & white, Figures. Audience: General/trade. International edition SOFTCOVER-Brand New-6th edition-Different ISBN and cover but Exact content as US Edition, only difference may be end of chapter review questions order may be changed-Ships on the next business day-Customer Satisfaction Guaranteed. Title may be slightly different in this international editionYipes! This book may be a great reference for people who know calculus but for me, just learning? It's a struggle and a challenge. It's sparse on the text -- example problems frequently skip steps or lack descriptions of what happened between step "n" and "n + 1". It suffers from a lack of |
Each subtopic below contains a
lesson page, an interactive student practice page, and a
teacher resource page. Sections denoted with * have Graphing Calculator references.
Note: Certain topics in the NYS Learning Standards cross
strands, such as "parabolas" occurring under both the Algebra and Geometry
strands. This site presents such topics in only one location.
The materials (text,
graphics, video clips, etc) from this Integrated Algebra website are
protected by copyright law.
The materials are for classroom or personal use only.
The materials are not to be publically distributed in part or whole.
The materials are not to be reposted to the internet in part or
whole. Thank you. |
This lesson, created by Amar Patel of the University of Illinois - Urbana-Champaign, introduces simple linear regression with several Excel spreadsheet examples such as temperature versus cricket chirps, height versus...
The author of Curious Math, Clay Ford, enjoys mathematics. While Ford claims not to be an expert by any means he maintains a fun website full of math tricks and trivia. Examples of postings include how to \"quickly... |
books.google.com - Prior to the nineteenth century, algebra meant the study of the solution of polynomial equations. By the twentieth century it came to encompass the study of abstract, axiomatic systems such as groups, rings, and fields. This presentation provides an account of the history of the basic concepts, results,... History of Abstract Algebra |
Wickenburg Physics math is all about logical analysis. You need to get into the mind set of using variables instead of raw numbers. This is unlike any questions asked in regular math classes |
More About
This Textbook
Overview
This book is a short, focused introduction to Mathematica, the comprehensive software system for doing mathematics. Written for the novice, this engaging book contains an explanation of essential Mathematica commands, as well as the rich Mathematica interface for preparing polished technical documents. Mathematica can be used to graph functions, solve equations, perform statistics tests, and much more. In addition, it incorporates word processing and desktop publishing features for combining mathematical computations with text and graphics, and producing polished, integrated, interactive documents. You can even use it to create documents and graphics for the Web. This book explains everything you need to know to begin using Mathematica to do all these things and more. Written for Mathematica version 3, this book can also be used with earlier versions of the software. Intermediate and advanced users may even find useful information here, especially if they are making the switch to version 3 from an earlier |
Welcome to Singapore Math––the leading math program in the world! This workbook features math practice and activities for sixth grade students based on the Singapore Math method. Level A is designed for the first semester and Level B is for the secondDirectly correlated to Singapore Math textbooks, this comprehensive practice series allows learners to practice various types of math problems while developing their thinking and analytical skills. Learning objectives and unit assessments are included to ensure that students obtain a thorough understanding of each concept. Perfect as a supplement to classroom work or as a homeschool resource, these workbooks will boost confidence in problem-solving and critical-thinking skills.
{"currencyCode":"USD","itemData":[{"priceBreaksMAP":null,"buyingPrice":8.99,"ASIN":"0768239958","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":8.99,"ASIN":"0768240050","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":9.79,"ASIN":"0768240158","isPreorder":0}],"shippingId":"0768239958::twPtIDX4x9w1vVErSO%2BgJhPqUgt4E318gE8ARAsjx71lToMu2%2B2tTo6UG%2BBS4eZKAWEfCYIfrAAMsP9WrpEQ3WQG8XWhmHRvCHKJ8hDjuB0aWzMuiMssUw%3D%3D,0768240050::qJpN6t85XGQOEpS7neLGZUK2OpNkpHI03vdIaT9ZGTIBmgBoSrUUmJsYEgqCvsrVStVSRmMfUzDpU00oT%2BhmhShXhZgoM9SJbB4q%2B%2BDkj7Y9k3WzdjjzgQ%3D%3D,0768240158::A%2B7lsKEgt97gVzvezNoiNlPUReb%2FGyEQng8lFUInfAbWG8LwF9kvDhLriLApeOaXKzBFLBNFHvog9g2ovdeIEjBV7RuQWOEfe29Q1HkQP76ubIb6qDefWhether this book is from the "real" Singapore math publisher or not, I have no idea.
What I do know is this book is brilliant. Their approach to story problems is ground-breaking. This approach would have even helped me in my college math courses! They present a very clear, intuitive way of solving difficult story problems for kids using a very simple model. In fact, the use of their model is even faster than Algebra in most cases.
What's amazing is my daughter actually enjoys using this book at home with me to supplement the sub-standard math education she's getting in her NYC public school. The approach to math is so much more interesting and intuitive. I wish I was taught math this way.
The book has a very useful front three pages for the tutor/parent/instructor showing the elegant way to solve the problems. Then it has a variety of sections that cover the basics that 6th grade kids (In the US) should know how to do. The answers in the back are both accurate and complete, showing exactly how they solved each problem.
This is a fantastic supplemental resource for parents frustrated with the slow pace of math learning in U.S. schools.
(I review this workbook the same as the other, 5b) This is the workbook that you use with the Singapore math textbook. It is a good workbook, has plenty of opportunities for practicing. However, if you don't have, or are not, using the Singapore textbook, this workbook is only somewhat useful. I purchased it and the other level workbook (5a and 5b) to see if they could supplement the workbooks my district uses for math already (we use enVision). They're nice, laid out well, clean, but there's most of the problems/work in the workbooks is the same as what we have. Two differences: 1-the word problems in these two workbooks are better than what we have, 2-there's a little more practice than our workbooks. So, should you buy? If you already have workbooks and are looking to supplement for your class, probably not. If you have no other workbooks, then these could be useful. If you're looking for good Singapore math material to supplement, try looking at the word problems or mental math books. Also, if you're looking for more quick practice to aid memorization and automaticity, try the Singapore sprints (if you can't find them here, you can get them on Singaporemath.com, but you'll pay a lot more (around $30).
I love these Singapore books! All the home school mom's have raved about these books, so I decided to get them to keep my children's math skills fresh during the summer. They are amazing! Worth every penny! |
Sense: Teaching and Learning Mathematics with Understanding
This book presents several key principles for teaching mathematics for understanding that you can use to reflect on your own teaching, make more ...Show synopsisThis book presents several key principles for teaching mathematics for understanding that you can use to reflect on your own teaching, make more informed decisions, and develop more effective systems of instruction Making Sense: Teaching and Learning Mathematics with...Good. Making Sense: Teaching and Learning Mathematics with Understanding |
Having the right answer doesn't guarantee understanding. This book helps physics students learn to take an informed and intuitive approach to solving problems. It assists undergraduates in developing their skills and provides them with grounding in important mathematical methods. Starting with a r... read more
Fundamentals of Mathematical Physics by Edgar A. Kraut Indispensable for students of modern physics, this text provides the necessary background in mathematics to study the concepts of electromagnetic theory and quantum mechanics. 1967 Survey of Physical Theory by Max Planck In this classic of scientific literature, the Nobel Laureate and creator of the quantum revolution explores the basics of physics, concluding with an engrossing narrative of how he developed quantum theory. 1925 editionNumerical Methods by Germund Dahlquist, Åke Björck Practical text strikes balance between students' requirements for theoretical treatment and the needs of practitioners, with best methods for both large- and small-scale computing. Many worked examples and problems. 1974 edition.
Methods of Applied Mathematics by Francis B. Hildebrand Offering a number of mathematical facts and techniques not commonly treated in courses in advanced calculus, this book explores linear algebraic equations, quadratic and Hermitian forms, the calculus of variations, more.
Methods of Quantum Field Theory in Statistical Physics by A. A. Abrikosov, L. P. Gorkov, I. E. Dzyaloshinski, Richard A. Silverman This comprehensive introduction to the many-body theory was written by three renowned physicists and acclaimed by American Scientist as "a classic text on field theoretic methods in statistical physics."
Product Description:
Having the right answer doesn't guarantee understanding. This book helps physics students learn to take an informed and intuitive approach to solving problems. It assists undergraduates in developing their skills and provides them with grounding in important mathematical methods. Starting with a review of basic mathematics, the author presents a thorough analysis of infinite series, complex algebra, differential equations, and Fourier series. Succeeding chapters explore vector spaces, operators and matrices, multivariable and vector calculus, partial differential equations, numerical and complex analysis, and tensors. Additional topics include complex variables, Fourier analysis, the calculus of variations, and densities and distributions. An excellent math reference guide, this volume is also a helpful companion for physics students as they work through their assignments |
ers, Polynomials, and Rings: A Course in Algebra
This introduction to modern algebra differs from texts in this area in fundamental ways. The author's primary goal is to have the reader learn to ...Show synopsisThis introduction to modern algebra differs from texts in this area in fundamental ways. The author's primary goal is to have the reader learn to work with mathematics through reading, writing, speaking, and listening. The choice of content is important, but he regards it as a vehicle, not as an end in itself. It is the raw material through which the readers develop the ability to understand and communicate mathematics. One non-standard feature of the book is that the author proves only a few of the theorems. Most proofs are left as exercises, and these exercises can form the core of a course based on this book Integers, Polynomials, and Rings: A Course in Algebra ...Good. Integers, Polynomials, and Rings: A Course in Algebra (Undergraduate Texts in Mathematics) |
to master basic arithmetic subjects, principles, and formulas! Master Math: Basic Math and Pre-Algebra is a comprehensive reference guide that explains and clarifies mathematic principles in a simple, easy-to-follow style and format. Beginning with the most basic fundamental topics and progressing through to the more advanced, Master Math: Basic Math and Pre-Algebra explains the principles and operations of arithmetic, provides step-by-step procedures and solutions, and presents examples and applications. A complete table of contents and a comprehensive index enable you to quickly find specific topics, and the approachable style and format facilitate an understanding of what can be intimidating and tricky skills. Perfect for both students who need some extra help or rusty professionals who want to brush up on their basic math skills, Master Math: Basic Math and Pre-Algebra will help you master everything from fractions and decimals to roots and radicals. |
"From speaking and dealing with countless new people, from pupils in committees at school, to guests and visitors, I feel more confident in meeting new people. Also, having to take five assemblies at school developed my public speaking skills. "
Mathematics 2 (H) 40 hours (Mandatory) - Topics include quadratic theory and the remainder theorem, basic integration, use of the addition formulae in trigonometry and the equation of a circle and tangency.
Mathematics 3 (H) 40 hours - Topics include vectors in three dimensions and the scalar product, further differentiation and integration, properties of exponential and logarithmic functions and further trigonometric relationships |
The elementary functions (sine, cosine, tan,
exponentials,
and logarithms) are the most commonly used mathematical functions in
science
and engineering. Computing these functions quickly and accurately is a
major goal in computer arithmetic.
This new book gives the concepts and
background
necessary to understand and build algorithms for computing these
functions,
presenting and structuring the algorithms (hardware-oriented as well as
software-oriented), and discusses issues related to the accurate
floating-point
implementation.
The purpose is not to give "cookbook recipes"
that allow one to implement some given function, but to provide the
reader
with the knowledge that is necessary to build, or adapt, algorithms to
their specific computing environment.
Topics and Features
Background material reviewed in Chapter 2,
Computer
Arithmetic
Polynomial and rational approximations
Table based methods
Shift-and-add algorithms thoroughly covered
in
part two
The CORDIC algorithm
Range reduction and accuracy covered in part
three
The book provides an up-to-date presentation of
the
information needed to understand and accurately use mathematical
functions
and algorithms in computational work and design. Graduates,
professionals
and researchers in scientific computing, software engineering and
computer
engineering will find the book a useful reference and resource.
This fascinating book describes the techniques used by high level
compilers and by pocket book calculators to generate values of the
common
elementary mathematical functions. Both the theory and the
implementation
details of the algorithms are explained in sufficient detail to satisfy
the curious or to inform the professional. ASLIB Book Guide |
This book presents a broad overview of computer graphics (CG), its history, and the hardware tools it employs. Covering a substantial number of concepts and algorithms, the text describes the techniques, approaches, and algorithms at the core of this field.
This book focuses on five hot research directions in 3D model analysis and processing in computer science: compression, feature extraction, content-based retrieval, irreversible watermarking and reversible watermarking.
Transformations and Projections in Computer Graphics provides a thorough background, discussing the mathematics of perspective in a detailed, yet accessible style. It also reviews nonlinear projections in depth, including fisheye, panorama, and map projections frequently used to enhance digital images.
An ideal course book for mathematics undergraduates and graduates alike, this is a complete introduction to vector analysis/ Each topic covered is given a practical application within computer graphics. |
Basic College Mathematics - 4th edition
Summary: Elayn Martin-Gay firmly believes that every student can succeed, and her developmental math textbooks and video resources are motivated by this belief. Basic College Mathematics, Fourth Edition was written to help readers effectively make the transition from arithmetic to algebra. The new edition offers new resources like the Student Organizer and now includes Student Resources in the back of the book to help students on their quest for success7448.5648.95 |
This book will provide a unique and invaluable source for researchers and graduate students. From the Reviews: "This book is definitely a milestone in the literature of integer programming and combinatorial optimization. more...
This book presents a systematic, rigorous and comprehensive account of the theory and applications of incomplete block designs. All major aspects of incomplete block designs are considered by consolidating vast amounts of material from the literature - the classical incomplete block designs, like the balanced incomplete block (BIB) and partially balanced... more...
Features recent advances and new applications in graph edge coloring Reviewing recent advances in the Edge Coloring Problem, Graph Edge Coloring: Vizing's Theorem and Goldberg's Conjecture provides an overview of the current state of the science, explaining the interconnections among the results obtained from important graph theory studies. The... more...
Who knew how fun and useful exploring graphs could be? Graphs in Action explores different types of graphs and their practical applications and explains how graphs are used to record business growth and productivity. more...
As the school year comes to an end, the students in this book recall their favorite things and activities from the year. They use graphs and charts to record and show their favorites. Join them as they relive their experiences through the year. 32pp. more... |
SEE BULLETIN FOR PREREQUISITES.
THIS SECTION WILL USE COMPUTERS. NO CALCULATOR REQUIRED |
In my experience, those who do, have many gaps from previous years. My job is to fill in those gaps, while building their understanding of the current material. The goal is to get students to make connections between numerical operations and application of that to abstract concepts. |
Discrete Mathematics Mathematical Reasoning and Proof with Puzzles, Patterns, and Games
9780471476023
ISBN:
0471476021
Pub Date: 2005 Publisher: John Wiley & Sons Inc
Summary: Did you know that games and puzzles have given birth to many of today's deepest mathematical subjects? Now, with Douglas Ensley and Winston Crawley's Introduction to Discrete Mathematics, you can explore mathematical writing, abstract structures, counting, discrete probability, and graph theory, through games, puzzles, patterns, magic tricks, and real-world problems. You will discover how new mathematical topics can ...be applied to everyday situations, learn how to work with proofs, and develop your problem-solving skills along the way. Online applications help improve your mathematical reasoning. Highly intriguing, interactive Flash-based applications illustrate key mathematical concepts and help you develop your ability to reason mathematically, solve problems, and work with proofs. Explore More icons in the text direct you to online activities at Improve your grade with the Student Solutions Manual. A supplementary Student Solutions Manual contains more detailed solutions to selected exercises in the text.
Ensley, Douglas E. is the author of Discrete Mathematics Mathematical Reasoning and Proof with Puzzles, Patterns, and Games, published 2005 under ISBN 9780471476023 and 0471476021. Five hundred forty three Discrete Mathematics Mathematical Reasoning and Proof with Puzzles, Patterns, and Games textbooks are available for sale on ValoreBooks.com, one hundred twenty used from the cheapest price of $80.50, or buy new starting at $173.41ISBN-13:9780471476023
ISBN:0471476021
Pub Date:2005 Publisher:John Wiley & Sons Inc
Valore Books is the best place for cheap Discrete Mathematics Mathematical Reasoning and Proof with Puzzles, Patterns, and Games rentals, or used and new condition books that can be mailed to you in no time. |
Created by Lang Moore for the Connected Curriculum Project, the purpose of this module is to provide an introduction to the elementary complex transcendental functions -- the exponential, sine, and cosine functions.... study convergence of Fourier approximations of periodic functions. This is one lesson within a much larger set of the Connected Curriculum Project, the purposes of this module are to experiment with matrix operations, especially multiplication, inversion, and determinants, and to explore... |
Everyday Math Demystified, 2nd Edition
Overview
Solve your math troubles with DeMYSTiFieD
If you cannot tell the difference between your Roman and Arabic numerals, or if when someone asks 'what is pi' you say "delicious," you need Everyday Math DeMYSTiFieD, Second Edition, to unravel these fundamental concepts and theories at your own pace.
This practical guide eases you into basic math, starting key ideas.
Author Information
Stan Gibilisco has authored or co-authored more than 50 nonfiction books in the fields of electronics, general science, mathematics, and computing. He has worked as a technical writer in industry, as a shortwave radio broadcast station technician, as a radio-frequency design engineer, and as a magazine editor. One of Stan's books, the Encyclopedia of Electronics (TAB Books, 1985), was named by the American Library Association (ALA) in its list of "Best References of the 1980s." Another of his books, the McGraw-Hill Encyclopedia of Personal Computing (McGraw-Hill, 1995), was named as a "Best Reference of 1996" by the ALA. Stan is considered the "father" of the DeMYSTiFieD series.
Customer Reviews
0071790144
There are no customer reviews available at this time. Would you like to write a review? |
Preface Linearalgebra has in recent years become an essential part of the mathematical background required by mathematicians and mathematics teachers, engineers, computer scientists, physicists, economists, and
Schaums Outline of LinearAlgebra (4th edition) by Seymour Lipschutz and Marc Lipson. (This is an exercise textbook in case you liked to ex your LinearAlgebra muscles). Prerequisites: Calculus I (Math-UA:0121) with a C or higher (or the equivalent).
SCHAUM'S OUTLINE OF Theory and Problems of COLLEGE MATHEMATICS THIRDEDITION Algebra DiscreteMathematics Precalculus IntroductiontoCalculus FRANKAYRES,Jr.,Ph.D. Formerly Professor andHead ... This is a linear function and its graph is a straight line.For this graph only two points are necessary.
The Schaum's Outline of LinearAlgebra contains 1. useful summaries of the most important material and a large number of solved problems covering a wide range of topics. Consequently, it is an excellent tool for reviewing and practicing course material.
LinearAlgebra. Notation R,R+,Rn realnumbers,realsgreaterthan0,n-tuplesofreals N,C naturalnumbersf0;1;2;:::g,complexnumbers (a::b),[a::b] openinterval,closedinterval ... Algebra course is an ideal spot to work on this transition to more rigor. It
Schaum'sOutlineofLinearAlgebra, fourth edition, by Lipschutz and Lipson, McGraw-Hill. Prerequisites: Calculus 1 and 2 with a grade of C or higher. ... to linearalgebra, an introduction to ordinary differential equations, and the application of
Use the vocabulary of linearalgebra to discuss these statements. Abstract algebra skills, including ... Schaum's Outline of Linare Algebra The material of the course is standard; any textbook titled linearalgebra will cover the computational
† From LinearAlgebra and Vector Calculus at Texas A&M: { Sections 1.1{1.2 † From Schaum's Outline of Beginning LinearAlgebra: { Sections 2.1{2.9 Required problems. Turn in a solution for each of the following problems. 1. Find all solutions to the following system of linear equations:
SCHAUM'S SOLVED PROBLEMS SERIES l 3000 SOLVED PROBLEMS IN PRECALCULUS Philip Schmidt, Ph.D. State University of New York at New Paltz ... of a Function / 3.4 Step Functions and Continuity / 3.5 Linear Functions / 3.6 The Algebra of Functions
Also recommended are Schaum's Outline Linear Algebraby Lipschutz, and LinearAlgebra by Gilbert Strang. You may not use the electronic form in class, as such readers are usually wifi capable, and that is not allowed during class time.
Schaum's Outlines, McGraw-Hill, 2009 ... linearalgebra well. As well as basic complex variables and (some) probability. The goal of this course is not to teach fundamental concepts of linearalgebra (which you are assumed to
Schaum's Outlines, McGraw-Hill, 2009 ... linearalgebra, the course reading and homework will not be assigned in the order of presentation given in the textbook. 6 Assumed Programming Skills ! It is assumed that students know Matlab or an equivalent
• Schaum's Outline of LinearAlgebra, Seymour Lipschutz and Marc Lipson, 3rd Ed., McGraw-Hill, 2000, ISBN 0-071-36200-2 All books are available at and at Fast and free shipping, and discounts are available.
Additional resources: Schaum's Outlines: LinearAlgebra by S. Lipschutz and M. Lipson is a cheap, helpful book. In addition, the following textbooks are on reserve in the Mathematics Library: LinearAlgebra with Applications by Steven Leon.
To introduce the basic topics of linearalgebra such as matrices, vector spaces, linear transformations, bases and dimension. To develop the ... Theory and problems of LinearAlgebra. Schaum's Outline Series, 2000. Author: 00012123 Created Date:
Schaum's Outline of College Algebra is a complete, concise guide to college algebra, for studen ts ... LinearAlgebra, Arithmetic and Topics in Algebra, or Functions and Graphs. Features • Outline format supplies a concise guide to the standard college course in college algebra |
For the past seven years, Professor John Woodward and his colleagues at the University of Puget Sound have been creating materials for this unique site. Designed as a way to bring together resources to assist...
A course designed to familiarize high school and beginning college mathematics teachers with advanced algebra content and teaching strategies. The approach stresses modeling and solving real world problems and develops...
"In the Classroom" highlights how some schools and organizations use Mathematica extensively in their curricula. The section on "Collaborative Initiatives" illustrates how businesses have teamed up with Wolfram Research...
With a development team that includes several well-regarded mathematicians and other such folk, the S.O.S. Mathematics website is a high-quality resource for persons who might find themselves in need of a bit of...
With the kind support of the National Science Foundation, the Shodor Education Foundation continues to provide a wide set of resources designed to assist educators with the formidable task of teaching young people about... |
A+ National Pre-apprenticeship Maths and Literacy for Hospitality by Andrew Spencer
Book Description
Pre-apprenticeship Maths and Literacy for Hospitality is a write-in workbook that helps to prepare students seeking to gain a Hospitality Apprenticeship. It combines practical, real-world scenarios and terminology specifically relevant to the Hospitality industry, and provides students with the mathematical skills they need to confidently pursue a career in the Hospitality trade. Mirroring the format of current apprenticeship entry assessments, Pre-apprenticeship Maths and Literacy for Hospitality includes hundreds of questions to improve students' potential of gaining a successful assessment outcome of 75-80% and above. This workbook will therefore help to increase students' eligibility to obtain a Hospitality Apprenticeship. Pre-apprenticeship Maths and Literacy for Hospitality also supports and consolidates concepts that students studying VET (Vocational Educational Training) may use, as a number of VCE VET programs are also approved pre-apprenticeships. This workbook is also a valuable resource for older students aiming to revisit basic literacy and maths in their preparation to re-enter the workforce at the apprenticeship level.
Buy A+ National Pre-apprenticeship Maths and Literacy for Hospitality book by Andrew Spencer from Australia's Online Bookstore, Boomerang Books.
You might also like...
This major work looks at the evolution of the hospitality industry and hospitality management, with source material drawn from dedicated hospitality books and journals as well as a range of complementary sources from allied discipline areas.
With articles and photographs from the Sheffield Star and Sheffield Telegraph, Peter Tuffrey has recorded subjects and incidents ranging from pub closures to murders, from retirements to renovations and from pub bombings to pub ghosts. Many traditional pubs are pictured and documented in decline or just before demolition.
Written for the professional bartender, this book contains nearly 1,500 different cocktails and shooters. It provides tips and tricks, bar terminology, measurements, how to set up a bar, glassware, responsible serving issues, garnishes, bar games and tricks, famous toasts, and much more. It includes a section on non-alcoholic drinks.
Books By Author Andrew Spencer
Current approaches to morphology, Andrew Spencer argues, are flawed. He uses intermediate types of lexical relatedness in different languages to develop a morphologically-informed model of the lexical entry. He uses this to build a model of lexical relatedness consistent with paradigm-based models. A book for all morphologists and lexicographers.
Helps learners' improve their Maths and English skills and help prepare for Level 1 and Level 2 Functional Skills exams. This title enables learners to improve their maths and English skills and real-life questions and scenarios are written with an automotive context to help learners find essential Maths and English theory understandable Hairdressing context beauty therapy context |
The Math Trailblazers program provides a careful balance of concepts and skills with intent to prepare children for the 21st century. It contains unit lessons on Populations and Samples, Big Numbers, Fractions and Ratios, Division and Data, Investigating Fractions, Geometry, Decimals and Probability, Applications: An Assessment Unit, Connections to Division, Maps and Coordinates, Using Fractions, Ration and Proportion, Using Circles, Developing Formulas with Geometry & Bringing it All Together.
Math Triumphs is an intensive intervention resource for students who are two or more years below grade level. The series accompanies Glencoe Algebra 1,Geometry, and Algebra 2 and provides step-by-step intervention, vocabulary support, and data-driven decision making to help students succeed in high school mathematicsMathematica Cookbook helps you master the application's core principles by walking you through real-world problems. Ideal for browsing, this book includes recipes for working with numerics, data structures, algebraic equations, calculus, and statistics. You'll also venture into exotic territory with recipes for data visualization using 2D and 3D graphic tools, image processing, and music.
Although Mathematica 7 is a highly advanced computational platform, the recipes in this book make it accessible to everyone -- whether you're working on high school algebra, simple graphs, PhD-level computation, financial analysis, or advanced engineering models.
Learn how to use Mathematica at a higher level with functional programming and pattern matching
Delve into the rich library of functions for string and structured text manipulation
Learn how to apply the tools to physics and engineering problems
Draw on Mathematica's access to physics, chemistry, and biology data
Get techniques for solving equations in computational finance
Learn how to use Mathematica for sophisticated image processing
Process music and audio as musical notes, analog waveforms, or digital sound samples
Electromagnetic complex media are artificial materials that affect the propagation of electromagnetic waves in surprising ways not usually seen in nature. Because of their wide range of important applications, these materials have been intensely studied over the past twenty-five years, mainly from the perspectives of physics and engineering. But a body of rigorous mathematical theory has also gradually developed, and this is the first book to present that theory. Designed for researchers and advanced graduate students in applied mathematics, electrical engineering, and physics, this book introduces the electromagnetics of complex media through a systematic, state-of-the-art account of their mathematical theory. The book combines the study of well-posedness, homogenization, and controllability of Maxwell equations complemented with constitutive relations describing complex media. The book treats deterministic and stochastic problems both in the frequency and time domains. It also covers computational aspects and scattering problems, among other important topics. Detailed appendices make the book self-contained in terms of mathematical prerequisites, and accessible to engineers and physicists as well as mathematicians.
The book is a selection of invited chapters, all of which deal with various aspects of mathematical and statistical models and methods in reliability. Written by renowned experts in the field of reliability, the contributions cover a wide range of applications, reflecting recent developments in areas such as survival analysis, aging, lifetime data analysis, artificial intelligence, medicine, carcinogenesis studies, nuclear power, financial modeling, aircraft engineering, quality control, and transportation.
Mathematical and Statistical Models and Methods in Reliability is an excellent reference text for researchers and practitioners in applied probability and statistics, industrial statistics, engineering, medicine, finance, transportation, the oil and gas industry, and artificial intelligence.
Computational methods are rapidly becoming major tools of theoretical, pharmaceutical, materials, and biological chemists. Accordingly, the mathematical models and numerical analysis that underlie these methods have an increasingly important and direct role to play in the progress of many areas of chemistry. This book explores the research interface between computational chemistry and the mathematical sciences. In language that is aimed at non-specialists, it documents some prominent examples of past successful cross-fertilizations between the fields and explores the mathematical research opportunities in a broad cross-section of chemical research frontiers. It also discusses cultural differences between the two fields and makes recommendations for overcoming those differences and generally promoting this interdisciplinary workMathematical biomedicine is a rapidly developing interdisciplinary field of research that connects the natural and exact sciences in an attempt to respond to the modeling and simulation challenges raised by biology and medicine. There exist a large number of mathematical methods and procedures that can be brought in to meet these challenges and this book presents a palette of such tools ranging from discrete cellular automata to cell population based models described by ordinary differential equations to nonlinear partial differential equations representing complex time- and space-dependent continuous processes. Both stochastic and deterministic methods are employed to analyze biological phenomena in various temporal and spatial settings. This book illustrates the breadth and depth of research opportunities that exist in the general field of mathematical biomedicine by highlighting some of the fascinating interactions that continue to develop between the mathematical and biomedical sciences. It consists of five parts that can be read independently, but are arranged to give the reader a broader picture of specific research topics and the mathematical tools that are being applied in its modeling and analysis. The main areas covered include immune system modeling, blood vessel dynamics, cancer modeling and treatment, and epidemiology. The chapters address topics that are at the forefront of current biomedical research such as cancer stem cells, immunodominance and viral epitopes, aggressive forms of brain cancer, or gene therapy. The presentations highlight how mathematical modeling can enhance biomedical understanding and will be of interest to both the mathematical and the biomedical communities including researchers already working in the field as well as those who might consider entering it. Much of the material is presented in a way that gives graduate students and young researchers a starting point for their own work.
"The first textbook on mathematical methods focusing on techniques for optical science and engineering, this text is ideal for upper division undergraduate and graduate students in optical physics. Containing detailed sections on the basic theory, the textbook places strong emphasis on connecting the abstract mathematical concepts to the optical systems to which they are applied. It covers many topics which usually only appear in more specialized books, such as Zernike polynomials, wavelet and fractional Fourier transforms, vector spherical harmonics, the z-transform, and the angular spectrum representation. Most chapters end by showing how the techniques covered can be used to solve an optical problem. Essay problems based on research publications and numerous exercises help to further strengthen the connection between the theory and its applications"--
Mathematical Methods for Signal and Image Analysis and Representation presents the mathematical methodology for generic image analysis tasks. In the context of this book an image may be any m-dimensional empirical signal living on an n-dimensional smooth manifold (typically, but not necessarily, a subset of spacetime). The existing literature on image methodology is rather scattered and often limited to either a deterministic or a statistical point of view. In contrast, this book brings together these seemingly different points of view in order to stress their conceptual relations and formal analogies.
Furthermore, it does not focus on specific applications, although some are detailed for the sake of illustration, but on the methodological frameworks on which such applications are built, making it an ideal companion for those seeking a rigorous methodological basis for specific algorithms as well as for those interested in the fundamental methodology per se.
Covering many topics at the forefront of current research, including anisotropic diffusion filtering of tensor fields, this book will be of particular interest to graduate and postgraduate students and researchers in the fields of computer vision, medical imaging and visual perception.
Mathematical Modelling in One Dimension demonstrates the universality of mathematical techniques through a wide variety of applications. Learn how the same mathematical idea governs loan repayments, drug accumulation in tissues or growth of a population, or how the same argument can be used to find the trajectory of a dog pursuing a hare, the trajectory of a self-guided missile or the shape of a satellite dish. The author places equal importance on difference and differential equations, showing how they complement and intertwine in describing natural phenomena.
Mathematical Models with Applications is intended to build on previous courses, including Algebra I, and to place emphasis on bringing about a deeper understanding of those mathematical relationships that will help students gain mathematical literacy in the real world and simultaneously help them build a strong foundation for future study in mathematics and other disciplines. The main goals for this textbook are to teach students how to problem-solve, communicate mathematically, create and interpret mathematical representations and models, and make efficient and appropriate use of technology to solve problems.
How heavy is that cloud? Why can you see farther in rain than in fog? Why are the droplets on that spider web spaced apart so evenly? If you have ever asked questions like these while outdoors, and wondered how you might figure out the answers, this is a book for you. An entertaining and informative collection of fascinating puzzles from the natural world around us, A Mathematical Nature Walk will delight anyone who loves nature or math or both. John Adam presents ninety-six questions about many common natural phenomena--and a few uncommon ones--and then shows how to answer them using mostly basic mathematics. Can you weigh a pumpkin just by carefully looking at it? Why can you see farther in rain than in fog? What causes the variations in the colors of butterfly wings, bird feathers, and oil slicks? And why are large haystacks prone to spontaneous combustion? These are just a few of the questions you'll find inside. Many of the problems are illustrated with photos and drawings, and the book also has answers, a glossary of terms, and a list of some of the patterns found in nature. About a quarter of the questions can be answered with arithmetic, and many of the rest require only precalculus. But regardless of math background, readers will learn from the informal descriptions of the problems and gain a new appreciation of the beauty of nature and the mathematics that lies behind it.
A clear need exists for substantial improvement in mathematics proficiency in U.S. schools. The RAND Mathematics Study Panel was convened to inform the U.S. Department of Education's Office of Educational Research and Improvement on ways to improve the quality and usability of education research and development (R&D). The panel identified three areas for focused R&D: development of teachers' mathematical knowledge used in teaching; teaching and learning of skills needed for mathematical thinking and problem-solving; and teaching and learning of algebra from kindergarten through the 12th grade.
Mathematical Statistics for Economics and Business, Second Edition, provides a comprehensive introduction to the principles of mathematical statistics which underpin statistical analyses in the fields of economics, business, and econometrics. The selection of topics in this textbook is designed to provide students with a conceptual foundation that will facilitate a substantial understanding of statistical applications in these subjects. This new edition has been updated throughout and now also includes a downloadable Student Answer Manual containing detailed solutions to half of the over 300 end-of-chapter problems.
After introducing the concepts of probability, random variables, and probability density functions, the author develops the key concepts of mathematical statistics, most notably: expectation, sampling, asymptotics, and the main families of distributions. The latter half of the book is then devoted to the theories of estimation and hypothesis testing with associated examples and problems that indicate their wide applicability in economics and business. Features of the new edition include: a reorganization of topic flow and presentation to facilitate reading and understanding; inclusion of additional topics of relevance to statistics and econometric applications; a more streamlined and simple-to-understand notation for multiple integration and multiple summation over general sets or vector arguments; updated examples; new end-of-chapter problems; a solution manual for students; a comprehensive answer manual for instructors; and a theorem and definition map.
This book has evolved from numerous graduate courses in mathematical statistics and econometrics taught by the author, and will be ideal for students beginning graduate study as well as for advanced undergraduates |
Elementary Algebra for College Students - 8th edition
Summary: Today's students are visual learners, and Angel/Runde offers a visual presentation to help them succeed in math. Visual examples and diagrams are used to explain concepts and procedures. New Understanding Algebra boxes and an innovative color coding system for variables and notation keep students focused. Short, clear sentences reinforce the presentation of each topic and help students overcome language barriers to learn math. Real Numbers; Solving Linear Equations and Inequalities; ...show moreApplications of Algebra; Exponents and Polynomials; Factoring; Rational Expressions and Equations; Graphing Linear Equations; Systems of Linear Equations; Roots and Radicals; Quadratic Equations For all readers interested in algebraALL ANSWERS INCLUDED.Identical to student edition.Black tape on cover. NO CD OR ACCESS CODE.SHIPS FAST!! SAME DAY OR W/N 24 HOURS.EXPEDITED SHIPPING AVAILABLE TOO!!
$183.74 +$3.99 s/h
New
ShopSpell Wharton, NJ
Brand New item. 100% Guaranteed. Reliable customer service.
$184.60 +$3.99 s/h
New
PaperbackshopUS Secaucus, NJ
New Book. Shipped from US within 4 to 14 business days. Established seller since 2000
$189.27 +$3.99 s/h
New
indoo Avenel, NJ
BRAND NEW
$226.03 +$3.99 s/h
New
Supreme Bookstore San Jose, CA
1-3-10 Hardback 8242.56 +$3.99 s/h
New
Lyric Vibes Geneva, IL
Hardcover New 0321620933 New Condition ~~~ Right off the Shelf-BUY NOW & INCREASE IN KNOWLEDGE...
$269.92 |
Part of MIT\'s innovative OpenCourseWare Project, that provides materials from MIT classes to the public on the web, the site provides materials from a course on interpreting environmental data to derive meaningful...
Mathlets are a collection of over 40 Java applets written by Dr. Tom Leathrum, a professor of mathematics at Jacksonville State University. These handy utilities perform basic calculator and graphing functions, as well...
Algorithms in the "Real World" is a computer science class at Carnegie Mellon University. While the Course Versions links have information primarily related to each semester's offerings, the rest of the material on this...
In this technical paper published in August 2002, researchers show that optical emanations from telecommunications devices can be monitored remotely. Light emitting diodes, often used as indicators in modems and other...
This unit from the Yale-New Haven Teachers Institute is "an attempt to develop a unit in mathematics that will provide topics for students interested in the aviation trades." The unit can be used to cover all areas of... |
An undergraduate textbook devoted exclusively to relationships between mathematics and art, Viewpoints is ideally suited for math-for-liberal-arts courses and mathematics courses for fine arts majors. The textbook contains a wide variety of classroom-tested activities and problems, a series of essays by contemporary artists written especially for the book, and a plethora of pedagogical and learning opportunities for instructors and students.
Viewpoints focuses on two mathematical areas: perspective related to drawing man-made forms and fractal geometry related to drawing natural forms. Investigating facets of the three-dimensional world in order to understand mathematical concepts behind the art, the textbook explores art topics including comic, anamorphic, and classical art, as well as photography, while presenting such mathematical ideas as proportion, ratio, self-similarity, exponents, and logarithms. Straightforward problems and rewarding solutions empower students to make accurate, sophisticated drawings. Personal essays and short biographies by contemporary artists are interspersed between chapters and are accompanied by images of their work. These fine artists--who include mathematicians and scientists--examine how mathematics influences their art. Accessible to students of all levels, Viewpoints encourages experimentation and collaboration, and captures the essence of artistic and mathematical creation and discovery |
*ESSENTIAL MATH SKILLS FOR ENGINEERS - 09 edition
Summary: Just the math skills you need to excel in the study or practice of engineering Good math skills are indispensable for all engineers regardless of their specialty, yet only a relatively small portion of the math that engineering students study in college mathematics courses is used on a frequent basis in the study or practice of engineering. That's why Essential Math Skills for Engineers focuses on only these few critically essential math skills that students need in order to advanc...show moree in their engineering studies and excel in engineering practice. Essential Math Skills for Engineers features concise, easy-to-follow explanations that quickly bring readers up to speed on all the essential core math skills used in the daily study and practice of engineering. These fundamental and essential skills are logically grouped into categories that make them easy to learn while also promoting their long-term retention. Among the key areas covered are:
With the thorough understanding of essential math skills gained from this text, readers will have mastered a key component of the knowledge needed to become successful students of engineering. In addition, this text is highly recommended for practicing engineers who want to refresh their math skills in order to tackle problems in engineering with confidence |
…
To learn and understand mathematics, students must engage in the process of doing mathematics. Emphasizing active learning, Abstract Algebra: An Inquiry-Based Approach not only teaches abstract algebra but also provides a deeper understanding of what mathematics is, how it is done, and howBuilding on rudimentary knowledge of real analysis, point-set topology, and basic algebra, Basic Algebraic Topology provides plenty of material for a two-semester course in algebraic topology.
The book first introduces the necessary fundamental concepts, such as relative homotopy, fibrations and …
This book introduces the study of algebra induced by combinatorial objects called directed graphs. These graphs are used as tools in the analysis of graph-theoretic problems and in the characterization and solution of analytic problems. The book presents recent research in operator algebra theory …
Through many examples and real-world applications, Practical Linear Algebra: A Geometry Toolbox, Third Edition teaches undergraduate-level linear algebra in a comprehensive, geometric, and algorithmic way. Designed for a one-semester linear algebra course at the undergraduate level, the book gives …
Quadratic Irrationals: An Introduction to Classical Number Theory gives a unified treatment of the classical theory of quadratic irrationals. Presenting the material in a modern and elementary algebraic setting, the author focuses on equivalence, continued fractions, quadratic characters, quadratic …
Near Rings, Fuzzy Ideals, and Graph Theory explores the relationship between near rings and fuzzy sets and between near rings and graph theory. It covers topics from recent literature along with several characterizations.
After introducing all of the necessary fundamentals of algebraic systems, the …
Topology is a large subject with many branches broadly categorized as algebraic topology, point-set topology, and geometric topology. Point-set topology is the main language for a broad variety of mathematical disciplines. Algebraic topology serves as a powerful tool for studying the problems in …
Group inverses for singular M-matrices are useful tools not only in matrix analysis, but also in the analysis of stochastic processes, graph theory, electrical networks, and demographic models. Group Inverses of M-Matrices and Their Applications highlights the importance and utility of the group |
MATH& 254Calculus IV•
5 Cr.
Department
Division
Description:
Extends the concepts of calculus to vector-valued functions and functions of several variables. Partial derivatives are included. Fulfills the quantitative or symbolic reasoning course requirement at BC. Recommended: MATH& 152.
Outcomes:
After completing this class, students should be able to:
to visualize, plot and interpret
points
lines
vectors
curves
surfaces in 3D
to translate among rectangular, cylindrical and spherical coordinate systems and state some advantages and disadvantages of each system
to perform basic vector operations and apply these operations
to interpret the fundamental ideas of rates of change and accumulation for curves in higher dimensions:
tangent vectors
arc length
curvature
to interpret the fundamental ideas of rates of change and accumulation for surfaces in higher dimensions: |
Textbook lessons are divided into three sections. The first section is "power-up practice," which covers basic fact and mental math exercises which improve speed, accuracy, the ability to do mental math, and the ability to solve complicated problems. The second part of the lesson is the "New Concept," which introduces a new math concept through examples, and provides a chance for students to solve similar problems. Thirdly, the "Written Practice" section reviews previously taught concepts. One "Investigation" per session is included; "Investigations" are variations of the daily lesson and often involve activities that take up an entire class.
The included Power Up Workbook provides consumable pages for students to complete the Power Up exercises from the textbook, including the Facts Practice, Jump Start, Mental Math, and Problem Solving sections. The textbook may refer students to problems within this Power Up workbook, or the text may contain necessary problems and instructions (such as the mental math problems), which students will need to complete the exercises in this workbook.
The Solutions Manual arranges answers by section and lesson, and includes complete step-by-step solutions to the Lesson Practice, Written Practice, and Early Finishers questions, as well as the questions and practice items in the Investigations. It does not contain the answers to the Power-Up Workbook, which are currently unavailable.
The Homeschool Testing Book features reproducible cumulative tests which are available after every five lessons after lesson 10. Tests are designed to let students learn and practice concepts before being tested, helping them build confidence. Tests, a testing schedule, test answer forms, test analysis form, and test solutions are included. The three optional Test Solution Answer Forms provide the appropriate workspace for students to "show their work." The answer key shows the final solution only, not the steps taken to arrive at the answer.
What level is this curriculum? Is this more difficult than Saxon 3? My child is not yet ready for 5/4 but we are finding Saxon 3 too easy.
asked 5 months, 3 weeks ago
by
namastemama
St Louis, Mo
on Saxon Math Intermediate 3 Complete Homeschool Kit
0points
0out of0found this question helpful.
2 answers
Answers
answer 1
Intermediate 3 sounds like a better fit for your child. It is more advanced but it is still a 3rd grade program.
answered 2 weeks, 5 days ago
by
lspinkrose
St. Johns, FL
0points
0out of0found this answer helpful.
answer 2
This is intended for students that are beyond needing the one-on-one attention provided by the Saxon Math 3 program, but are not ready for Math 5/4. It sounds like it would be a perfect fit for your situation.
My daughter just completed Saxon Math 2. Should we got to Math 3 or Intermediate Math 3. Are they the same thing packaged differently?
asked 2 months, 2 weeks ago
by
Margie
on Saxon Math Intermediate 3 Complete Homeschool Kit
0points
0out of0found this question helpful.
2 answers
Answers
answer 1
No, they are not the same. The Intermediate Math is more advanced than the regular Saxon Math 3 program.
answered 2 weeks, 5 days ago
by
lspinkrose
0points
0out of0found this answer helpful.
answer 2
The "Intermediate Math 3" program is designed to be used by the student more independently than item WW20111 "Math 3 Home Study Kit". If you are looking for a curriculum that is teacher intensive like the Saxon Math 2 program, you will want WW20111 "Math 3 Home Study Kit". If your daughter is ready for more independent learning, you can use this kit.
Lesson Activities?
We have been working with this kit all semester, and occasionally the lesson refers to a Lesson Activity, which sounds like it is part of a separate book. There are money manipulatives to cut out, as well as fractions, etc. I cannot seem to find this stuff anywhere in the Power Up, Test Forms, Answers, or Textbook. Help???
asked 2 years, 3 months ago
by
Clatie Lou Fischer
on Saxon Math Intermediate 3 Complete Homeschool Kit
+1point
1out of1found this question helpful.
1 answer
Answers
answer 1
Printable copies of the supplemental materials are available at: The password for the document is the last word on page 8 of the Homeschool Testing Book. |
More About
This Textbook
Overview
This book makes it easier to use and learn Mathematica by substituting menus and dialog boxes for typing commands. Access Mathematica's power instantly by pointing and clicking in the simplified environment provided on the CD enclosed in this book. The Joy of Mathematica is both a manual for the software and an introduction to using Joy and Mathematica for mathematics and its applications to other fields. The Joy of Mathematica can be used as a supplement to mathematics texts in calculus, differential equations, and linear algebra. Note: Mathematica 3.0 or higher is required to use the accompanying software. The CD:
* Runs on both Windows and Power Macintosh platforms
* Is optimized for Mathematica 4.0
* Requires that Mathematica's kernel and front end be on the same computer
* Includes a palette for easy entry of common mathematical notation The Book:
* Contains ready-to-use exercises and labs for the mathematics classroom
* Now includes more coverage of multivariable calculus and differential equations, in addition to single-variable calculus and linear algebra About the Authors: Alan Shuchat (Ph.D., University of Michigan) and Fred Shultz (Ph.D., University of Wisconsin) are both Professors of Mathematics at Wellesley College in Wellesley, Massachusetts. They have each won awards at Wellesley for excellence in teaching. This book was developed initially to make more effective use of Mathematica in their own courses and has won wide acceptance since the publication of its First Edition in 1994.
Audience: For the mathematics professional or interested layperson engaged in learning Mathematica and it's practical applications.
Related Subjects
Table of Contents
A Brief Tour of Joy.
More About Joy.
Graphing in Two Dimensions.
Manipulating Expressions.
Solving Equations.
Working with Functions of One Variable.
Differentiating Functions of One Variable.
Integrating Functions of One Variable.
Working with Sequences and Series.
Graphing in Three Dimensions.
Working with Functions of Several Variables.
Differentiating Functions of Several Variables.
Integrating Functions of Several Variables.
Working with Vector Fields.
Solving Differential Equations.
Working with Vectors and Matrices.
Functions.
Limits and Continuity.
Derivatives in One Variable.
Integrals in One Variable.
Sequences and Series.
Parameterized Curves.
Surfaces and Level Sets.
Derivatives in Several Variables.
Multiple Integrals.
Differential Equations.
Systems of Differential Equations.
Matrices and Linear Equations.
Vector Spaces and Linear Transformations |
Rules In Finding Special Products In Math
Harcourt School Publishers Zombieland is a 2009 American zombie comedy film directed by Ruben Fleischer from a screenplay written by Rhett Reese and Paul Wernick. The film stars Jesse Eisenberg. Judaism 101: Kashrut: Jewish Dietary Laws Wrightslaw is leading website about special education law and advocacy, with thousands of articles, cases, and free resources about hundreds of special education.
Practical Algebra Lessons. Purplemath Provides a complete web based educational environment for K-12 mathematics. Math.com - World of Math Online Securities Act of 1933. Often referred to as the "truth in securities" law, the Securities Act of 1933 has two basic objectives: require that investors receive. Math Homework Help - Answers to Math Problems - Hotmath Safety Advisory for Consumer Fireworks. DOT Rule for Passengers Traveling with Lithium Batteries. DOT's rule on carrying lithium batteries during air travel, which.
Xylem Flow Control - Let's Solve Water. The new site for Jabsco, Rule, Flojet, Midland-ACS, Alcon, LVM & Totton products
Basic Math Help, Algebra, and Trigonometry Video Lessons Instructional math help video lessons online and on CD.
Health Insurance Made Simple | Golden Rule Insurance Company Visit Harcourt's marketplace, parent store, learn about products, and interact with online activities and resources.
Safe Travel - DOT Math Worksheets: Create your own math worksheets with our new Worksheet Generator. Each worksheet is interactive, with a timer and instant scoring. Math-U-See - Complete Math Curriculum for Homeschool and. Free math lessons and math homework help from basic math to algebra, geometry and beyond. Students, teachers, parents, and everyone can find solutions to their math. FindCalifornia Code - Official California Legislative. Practical algebra lessons that emphasize the practicalities of understanding the questions and intelligently and simply arriving at the answers.
Macmillan/McGraw-Hill Everyday Mathematics Online Professional Development Registration Now Open The Everyday Mathematics online professional development modules are now open for.
Wrightslaw Special Education Law and Advocacy So even though your math book may totally dismiss the topic of finding square roots without a calculator, you can consider to let them practice at least the first.
Exponents: Basic Rules. Purplemath 1 As an equation; 2 The Taylor principle; 3 Alternative versions of the rule; 4 Empirical relevance; 5 Criticisms; 6 See also; 7 References; 8 External links |
Catalog Description – This is a Beginning through Intermediate Algebra course. This course is intended to develop student proficiency and confidence in basic algebraic skills such as simplifying algebraic expressions, solving equations, factoring, and simplifying radical expressions. This course can be used to prepare students for College Algebra as well as to satisfy the general education basic algebra entrance requirement.
Algebra I is a college degree requirement for students who have enrolled at Pitt-Johnstown and have scored at the basic algebra entrance level on the Math Placement Exam.
Grade Option – Math 0001 is graded as Satisfactory/Unsatisfactory (S/U) and, as such, will not be computed in your QPA. However, you will receive three university credits when you successfully complete the course. A satisfactory grade is earned when a minimum grade of 75% overall and a minimum of 60% on the final exam is attained. Text - Developing Skills in Algebra, 6th edition, by Nanney, Cable, Tully, Shustrick, and Wilson
Calculator Policy - Given that our program emphasizes analytic skills and techniques while not ignoring the usefulness of numerical and graphical methods and the benefits of combinations of techniques, the Mathematics Department has agreed upon the use of certain types of calculators for its courses. For Math 0001, a scientific calculator is required. The students in this class will not be permitted to use calculators on quizzes or exams. We believe that students should have and be able to apply basic arithmetic skills and facts as they relate to the algebraic skills and concepts that are the objectives of this course. For example, to learn simplification of rational expressions students first must be able to simplify arithmetic fractions. Therefore, for much of the course, students are not permitted to use their calculators. However, where applicable, we will show the proper use of the scientific calculator (e.g. order of operations, radical approximations, etc.). |
Saul Stahl's book begins with a condensed overview of synthetic Euclidean geometry, which includes a good selection of theorems about plane shapes. As with most of the subsequent chapters, historical observations are interspersed with the expository narrative, and this first chapter concludes with a motivated introduction to Hilbert's axioms. Therefore, to this extent, Euclidean geometry is viewed from a modern axiomatic perspective.
Next is a rather formal introduction to the topic of Euclidean Rigid Motions in which the notion of invariance is somewhat understated. By this I mean that, unlike the first chapter, none of the theorems pertaining to plane figures are derived by such methods. In other words, the treatment seems to be all process and no product.
But there is no such imbalance in the ensuing introduction to inversive geometry. And, although no historical or mathematical motivation is provided for the introduction of this topic, the treatment is concise, and notion of invariance comes across very clearly. Inversions are then used to derive many interesting properties of circles.
Omissions from the usual Kleinian hierarchy include affine and projective geometry. But the pay-off is a fast track into the heart of hyperbolic geometry, which is achieved by an elegant introduction to the Poincaré half-plane. From then on, the notions of hyperbolic length, angle and rigid motions are clearly spelled out.
In this book, you can always see the wood for the trees — both mathematically and historically. In this vein, chapter 5 compares the structure of Euclidean with that of hyperbolic geometry. The next three chapters explore further aspects of the hyperbolic universe and then, in chapter 9, heavier mathematical machinery appears in the form of Moebius transformations. The second half of the book reveals the charms of spherical and elliptic geometry, the differential geometry of surfaces, the unit disk and Beltrami-Klein models
Among the many attractive features of Saul Stahl's book is that it doesn't overwhelm the incipient geometer with a myriad of mathematical techniques. For instance, the chapter on differential geometry assumes no knowledge of the geometry of curves, and many of the ideas are conveyed by analogy. The various diagrams and illustrations also add to the book's aesthetic appeal.
Obviously, this book is highly recommended as the basis of an introductory course on non-Euclidean geometry. It portrays geometry as a subject with an interesting history and exciting possibilities. Each section of the book concludes with a range of exercises, from the routine to the more challenging. Finally, the fact no solutions are provided is one thing, but the difficulty of accessing the online instructors manual is altogether more puzzling.
Peter Ruane finds that, as he approaches the boundary of this temporal universe, the days fly by more quickly, and his height is annually diminishing by a miniscule amount. |
Book summary
The theory of elliptic curves involves a blend of algebra, geometry, analysis, and number theory. This book stresses this interplay as it develops the basic theory, providing an opportunity for readers to appreciate the unity of modern mathematics. The books accessibility, the informal writing style, and a wealth of exercises make it an ideal introduction for those interested in learning about Diophantine equations and arithmetic geometry. |
WWW
Resources: Subject Resources
Mathematics
Resources: Associations and Organizations
The Society
is the largest professional organization in mathematics. The web
site leads mathematics specialists and enthusiasts to conferences,
career and education information, job opportunities and reference
tools. Information about MathSciNet is available. The site also
contains a directory of institutions in the mathematical sciences
and a membership directory of all persons who are or were members
of AMS, MAA, SIAM, AWATYC, and AWM.
This is the
largest organization instrumental in the field of collegiate mathematics.
It seeks to motivate interest in this subject area by providing
information on contemporary mathematics, recent developments in
research, and conferences and events. The web site provides information
on electronic journals, professional development, students and student
chapters of MAA, employment opportunities, and, scholarships and
awards.
The goals of
this organization focus on the promotion of mathematical research
that lead to new processes for science and industry. It also seeks
to provide the exchange of information and ideas between mathematicians,
engineers and scientists. The organization's nine journals are available
over the web at the SIAM web site. It sponsors many specialized
conferences, annual meetings, courses, workshops and activity groups.
This non-profit
organization promotes women dedicated to the fields in the mathematical
sciences. AWM provides workshops, travel grants, and awards. The
web site lists biographies of successful women in the fields of
mathematics and sciences. It also contains a section on careers
in mathematics that provide several links to grants, job advertisements,
programs and prizes, organizations, and mailing lists for people
within the mathematical science field of study.
The ASA's mission
is to promote excellence in the application of statistics. The ASA's
web site is a excellent resource for information on publications
in statistics, membership, professional development, awards and
grants, and job opportunities in statistics. It also provides links
to resources to other ASA chapters, sections and committees, as
well as other statistical organizations in the U.S. and the world.
Provides an
online resources page with links to educational and career information,
research grants, and mentoring programs within the field of mathematics,
engineering and science. The purpose and goal of this organization
is to bring together parents, educators, and women professionals
to promote and support young women in the fields of science, engineering,
mathematics, and technology. |
Saxon Math 1 Home Study Teachers Manual First Edition
$64.40 Sale: $51.52 Save: 20% off
Saxon math programs produce confident students who are not only able to correctly compute, but also to apply concepts to new situations. These materials gently develop concepts, and the practice of those concepts is extended over a considerable period of time. This is called 'incremental development and continual review.' Material is introduced in easily understandable pieces (increments), allowing students to grasp one facet of a concept before the next one is introduced. Both facets are then practiced together until another one is introduced. This feature is combined with continual review in every lesson throughout the year. Topics are never dropped but are increased in complexity and practiced every day, providing the time required for concepts to become totally familiar. The teacher's manual This first edition Teacher's Manual is for |
Buy Used Textbook
eTextbook
We're Sorry Not Available
New Textbook
We're Sorry Sold Out
More New and Used from Private Sellers
Starting at $4 its complete, interactive, objective-based approach, Introductory Algebra: An Applied Approach, is a best-seller in this market. The Seventh Edition provides mathematically sound and comprehensive coverage of the topics considered essential in a beginning algebra course. The text includes chapter-opening Prep Tests, updated applications, and a new design. A robust Instructor's Annotated Edition features a comprehensive selection of instructor support materials. The Aufmann Interactive Method is incorporated throughout the text, ensuring that students interact with and master the concepts as they are presented. This approach is especially important in the context of rapidly growing distance-learning and self-paced laboratory situations. New! Study Tips margin notes provide point-of-use advice and refer students back to the AIM for Success preface for support where appropriate. Integrating Technology (formerly Calculator Notes) margin notes provide suggestions for using a calculator in certain situations. For added support and quick reference, a scientific calculator screen is displayed on the inside back cover of the text. Enhanced! More bulleted annotations have been added to the solution steps of examples and to the You Try It solutions in the appendix. Enhanced! Examples have been clearly labeled How To, making them more prominent to the student. Enhanced! More operation application problems integrated into the Applying the Concepts exercises encourage students to judge which operation is needed to solve a word problem. New! Nearly 100 new photos add real-world appeal and motivation. Revised! The Chapter Summary has been reformatted to include an example column, offering students increased visual support. Enhanced! In response to instructor feedback, the number of Chapter Review Exercises and Cumulative Review Exercises has increased. Enhanced! This edition features additional coverage of time (Chapter 8), bytes (Chapter 9), and temperature (Chapter 11). Aufmann Interactive Method (AIM) Every section objective contains one or more sets of matched-pair examples that encourage students to interact with the text. The first example in each set is completely worked out; the second example, called 'You Try It,' is for the student to work. By solving the You Try It, students practice concepts as they are presented in the text. Complete worked-out solutions to these examples in an appendix enable students to check their solutions and obtain immediate reinforcement of the concept. While similar texts offer only final answers to examples, the Aufmann texts' complete solutions help students identify their mistakes and prevent frustration. Integrated learning system organized by objectives. Each chapter begins with a list of learning objectives that form the framework for a complete learning system. The objectives are woven throughout the text (in Exercises, Chapter Tests, and Cumulative Reviews) and also connect the text with the print and multimedia ancillaries. This results in a seamless, easy-to-navigate learning system. AIM for Success Student Preface explains what is required of a student to be successful and demonstrates how the features in the text foster student success. AIM for Success can be used as a lesson on the first day of class or as a project for students to c
Table of Contents
Chapters 2–6 are followed by Cumulative Review Exercises
Prealgebra Review
Introduction to Integers
Addition and Subtraction of Integers
Multiplication and Division of Integers
Exponents and the Order of Operations Agreement
Factoring Numbers and Prime Factorization
Addition and Subtraction of Rational Numbers
Multiplication and Division of Rational Numbers
Concepts from Geometry Focus on Problem Solving: Inductive Reasoning Projects and Group Activities: The +/- Key on a Calculator
Variable Expressions
Evaluating Variable Expressions
Simplifying Variable Expressions
Translating Verbal Expressions into Variable Expressions
Focus on Problem Solving: From Concrete to Abstract Projects and Group Activities: Prime and Composite Numbers |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.