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This course involves studying properties of natural numbers and integers. Topics include divisibility, prime numbers, the Euclidean Algorithm and the RSA Encryption system for putting messages into code.
Crossing the River With Dogs Problem Solving for College Students 9781931914147 ISBN: 1931914141 Pub Date: 2003 Publisher: Springer Verlag Summary: Students who often complain when faced with challenging word problems will be engaged as they acquire essential problem solving skills that are applicable beyond the math classroom. The authors of Crossing the River with Dogs: Problem Solving for College Students:- Use the popular approach of explaining strategies through dialogs from fictitious students- Present all the classic and numerous non-traditional problem s...olving strategies (from drawing diagrams to matrix logic, and finite differences) - Provide a text suitable for students in quantitative reasoning, developmental mathematics, mathematics education, and all courses in between - Challenge students with interesting, yet concise problem sets that include classic problems at the end of each chapter With Crossing the River with Dogs, students will enjoy reading their text and will take with them skills they will use for a lifetime. Johnson, Ken is the author of Crossing the River With Dogs Problem Solving for College Students, published 2003 under ISBN 9781931914147 and 1931914141. One hundred twenty seven Crossing the River With Dogs Problem Solving for College Students textbooks are available for sale on ValoreBooks.com, twenty three used from the cheapest price of $3.99, or buy new starting at $701931914141
Calculus for Business, Economics, Life Sciences, and Social Sciences13.98 FREE Used Very Good (1 Copy): Instructor's Edition84 FREE About the Book This accessible text is designed to help readers help themselves to excel. The content is organized into two parts: (1) A Library of Elementary Functions (Chapters 1-2) and (2) Calculus (Chapters 3-9). The book's overall approach, refined by the authors' experience with large sections of college freshmen, addresses the challenges of teaching and learning when readers' prerequisite knowledge varies greatly. Reader-friendly features such as Matched Problems, Explore & Discuss questions, and Conceptual Insights, together with the motivating and ample applications, make this text a popular choice for today's students and instructors.
More About This Textbook Overview Written in problem-solving format, this book emphasizes the purpose of an advanced calculus course by offering a more thorough presentation of some topics to which engineering and physical science students have already been exposed. By supplementing and extending these subjects, the book demonstrates how the tools and ideas developed are vital to an understanding of advanced physical
Texas Instruments TI-86 calculator. Runs on 4 AAA batteries (not included). This calculator is required for pretty much any college level math, geometry, calculus or stats class. This is a steal! Usually runs between $44-53 dollars. Get it now! $15
What's New in Version 1.4.5 iPhone Screenshots Customer Reviews Good, but graphs are sometimes buggy by Xyzzy123 I generally like this app quite a bit. Most of the time, it's excellent for graphing an equation. Unfortunately, sometimes it demonstrates significant bugs when graphing the equation. For example, if it doesn't understand the equation, it will just draw a straight line -- rather than telling you that it can't interpret the equation. As another example, sometimes it draws graphs that are simply wrong. For example, with the equation "y = (1 + x)^(1/9)", it fails to display half of the equation -- the portion to the left of negative one. Problems like this mean that you can't entirely trust the results of the graphed equations. I've attempted to report some of these bugs to the developer using the provided email address. Unfortunately, he hasn't responded. Student by Kristie62688 This app has been a godsend in helping me get through my college algebra class. In fact, it was mentioned in my textbook and that's why I bought it. It sure beats buying an expensive graphing calculator like I did back in high school, and it functions just as well. I highly recommend this app. Great App but NEEDS CALCULUS by lj_schamber This app is absolutely wonderful and I use it almost every day on all of my iOS devices, however it NEEDS Calculus functions. I'm a Mechanical Engineering major in college starting this Fall and the ability to graph derivatives, integrals, and possibly even vector/slope fields would be much appreciated.
This is a very brief introduction to differential and integral calculus, the revolutionary mathematical development of the 17th century and a basis for essentially all modern science and engineering. Fundamental concepts and several application examples will be presented. Requirements: An understanding of high-school algebra and geometry. The course includes a brief review of some needed concepts from algebra and geometry.
Singapore Math Standards 1A Text Singapore Math Standards 1A Text The Primary Mathematics Standards Editionis a series of elementary math textbooks and workbooks from the publishers of Singapore's successful Primary Mathematics series. The program aims to equip students with sound concept development, critical thinking and efficient problem-solving skills. The Primary Mathematics Standards Edition 1A-5B is ideal for both classroom and home use for 1st - 5th grades. This series is an adaptation of Primary Mathematics U.S. edition, the most popular (Singapore Math) primary math series used by schools and homeschoolers in the U.S. and Canada. Each textbook is used in conjunction with a workbook. Also available are optional Extra Practice books and Tests books. This series is recommended for those who want a solid, basic math program with a proven track record and an emphasis on concept development, mental techniques, and problem solving, along with a program that covers all the California state standards for each grade level. The books include added units on probability and data analysis primarily, as well as negative numbers and coordinate graphs, concepts not covered normally until secondary levels in Singapore and not covered in the Primary Mathematic U.S. Edition. This series also provides more re-teaching of material from earlier levels and more frequent review and is therefore recommended for students new to Primary Math but not starting at the beginning. This is primarily a direct instruction program. Students are given several approaches for solving problems and are encouraged to discuss ideas and explore additional methods.
Medley, FL Precalculus looking forward to working with you soon.Factoring is the main emphasis after Prealgebra concerns. Prerequisites include solving 1 and 2 step linear equations; adding, subtracting, multiplying, and dividing integers; combining like terms; the Order of Operations. Prerequisites include fact
The Saxon Math middle grades textbooks move students from primary grades to algebra. Each course contains a series of daily lessons covering all areas of general math. Each lesson presents a small portion of math content (called an increment) that builds on prior knowledge and understanding. Students are not required or expected to grasp a concept fully the first time it is presented. After an increment is introduced, it becomes a part of the studentís daily work for the rest of the year. Students will have many opportunities to gain understanding and to achieve mastery. This cumulative, continual practice ensures that students will retain what they have learned. The homeschool kits for Math 5/4, Math 6/5, Math 7/6, and Math 8/7 consist of a student textbook with 120 lessons and 12 investigations, a Tests and Worksheets Booklet, which includes tests and fact practice worksheets, and a Solutions Manual, which offers step-by-step solutions to all lessons, investigations, and test questions. Algebra Ĺ includes a textbook and Homeschool Packet (31 test forms in addition to answers for all textbook problems and test questions), and a Solutions Manual is also available. Saxon Teacher, a CD-ROM supplement that offers lesson instruction videos and video problem solutions in a convenient player, is also available for all middle grades math books. << Use the Product Offers on the left to navigate through this category.
MATH20602 - Numerical Analysis 1 Year: 2 - Semester: 2 - Credit Rating: 10 Requisites Pre Requisites MATH20401 or MATH20411 Aims The course unit unit aims to introduce students to theoretical and practical aspects of the numerical solution of linear and nonlinear equations, the approximation of functions by polynomials and the approximation of integrals via quadrature schemes. Brief Description Numerical analysis is concerned with finding numerical solutions to problems for which analytical solutions either do not exist or are not readily or cheaply obtainable. This course provides an introduction to the subject, focusing on the three core topics of iteration, interpolation and quadrature. The module starts with 'interpolation schemes', methods for approximating functions by polynomials, and 'quadrature schemes', numerical methods for approximating integrals, will then be explored in turn. The second half of the module looks at solving systems of linear and nonlinear equations via iterative techniques. In the case of linear systems, examples will be drawn from the numerical solution of differential equations. Students will learn about practical and theoretical aspects of all the algorithms. Insight into the algorithms will be given through MATLAB illustrations, but the course does not require any programming. Learning Outcomes On completion of this unit successful students will have: practical knowledge of a range of iterative techniques for solving linear and nonlinear systems of equations, theoretical knowledge of their convergence properties, an appreciation of how small changes in the data affect the solutions and experience with key examples arising in the solution of differential equations;
fun and easy way to learn pre-calculus Getting ready for calculus but still feel a bit confused? Have no fear. Pre-Calculus For Dummies is an un-intimidating, hands-on guide that walks you through all the essential topics, from absolute value and quadratic equations to logarithms and exponential functions to trig identities and matrix operations. With this guide's help you'll quickly and painlessly get a handle on all of the concepts — not just the number crunching — and understand how to perform all pre-calc tasks, from graphing to tackling proofs. You'll also get a new appreciation for how these concepts are used in the real world, and find out that getting a decent grade in pre-calc isn't as impossible as you thought. Updated with fresh example equations and detailed explanations Tracks to a typical pre-calculus class Serves as an excellent supplement to classroom learning If "the fun and easy way to learn pre-calc" seems like a contradiction, get ready for a wealth of surprises in Pre-Calculus For Dummies! From the Back Cover Grasp the principles and concepts you need to score high in pre–calculus Getting ready for calculus but feel confused? Have no fear! This un–intimidating guide walks you through all the essential topics, from absolute value and quadratic equations to logarithms and exponential functions to trig identities and matrix operations. You′ll understand the concepts — not just the number crunching — and see how to perform all the tasks you need to score high at exam time. Pre–calculus 101 — get a review of the Algebra II you need to know, real numbers, and how to graph, solve, and perform operations with functions What′s your angle? — take a tour of the essentials of trigonometry, from angles, right triangles, and trig ratios to graphing the parent graph of the six basic trig functions Keep it simple — discover how to simplify trig expressions and solve for an unknown variable using formulas and identities (and solve triangles that aren′t right triangles using Law of Sines and Law of Cosines) Not just plane thinking — delve into analytic geometry and system solving with the understanding of complex numbers, polar–coordinate graphing, conics, systems of equations, sequences, and more I bought this book for my son, who needs to review Pre-Calculus. He had just finished The Complete Idiot's Guide to Precalculus and really enjoyed the humor and real-life analogies in that book. This one is much drier and less intuitive, and requires my explanation from time to time. If you are buying just one book on the subject, I highly recommend the Complete Idiot's Guide instead. 10 of 10 people found the following review helpful 4.0 out of 5 starsGood book for newcomers to calculus or as a refreshers guide8 Aug 2012 I've been out of school for a number of years now and thought I would refresh myself on that most hated of subjects, calculus. So why not go for the basics and get a dummies book? To be honest I've not put a lot of stock in dummies guides before as they seemed superfluous and full of talk rather than learning. After reading this guide I may have to change my mind. This book is really aimed at students who have a good grasp of algebra and are about to take calculus so wasn't really targetted for my purposes of using it as a refresher after a number of years. After a brief introduction of what you will learn and concepts you should already know it delves straight in with functions and graphing, two concepts used heavily throughout calculus. After this the book takes you through all the usual concepts such as Trig and Geometry, each chapter building on the knowledge you've learned in the previous one. Each chapter leads you by the hand at the beginning and not so gradually introduces new concepts and techniques throughout, each with plenty of examples. The book is written well and doesn't play down to the "dummy" level, building your confidence and knowledge along the way. If you are the intended target audience of someone about to study calculus then this book gives you a good introduction to the concepts and techniques that you will use. It is not an be-all-and-end-all book on calculus and you will need other books to finish your course. This is really an introductory book. As a refresher on concepts long since forgotten (as was my purpose for buying it), it met and even exceeded my expectations. As an introductory book it does well, is easy to follow and the examples are well explained. 4 of 4 people found the following review helpful 5.0 out of 5 starsWell done. Good for review, refreshing or as a supplement. You do need to DO some math to master precalc though.7 Aug 2012 There is something just less intimidating about the Dummies series of texts than actual textbooks. I think it is the combination of the fact that they simply weigh less, the text is less dense and the tone is conversational rather than ivory tower. The particular book does a very good job of presenting the material in a non threatening and fairly humorous manner. This book is excellent.... For those who need to review the material, but at one time had previous exposure and understanding. As a supplement to a more weighty precalc text. As review before taking calc. For parents who have had the math, but need some "reminders" to be able to help kids effectively. If you have never taken precalc this book is not going to be enough to bring you up to speed. One caveat: You really do need to put pen to paper and do some problems yourself. It is a trap to believe that simply reading an explanations is enough to cement the idea in your head.
Find an IB World School Mathematics Overview Because individual students have different needs, interests and abilities, four courses in mathematics are available: mathematical studies standard level mathematics SL mathematics higher level further mathematics standard level which will become a higher level course in 2012 with first examinations in 2014. These courses are designed for different types of students: those who wish to study mathematics in depth, either as a subject in its own right or to pursue their interests in areas related to mathematics; those who wish to gain a degree of understanding and competence better to understand their approach to other subjects; and those who may not as yet be aware how mathematics may be relevant to their studies and in their daily lives. Each course is designed to meet the needs of a particular group of students. Therefore, great care should be taken to select the course that is most appropriate for an individual student. In making this selection, individual students should be advised to take account of the following types of factor. Their own abilities in mathematics and the type of mathematics in which they can be successful Their own interest in mathematics, and those particular areas of the subject that may hold the most interest for them Their other choices of subjects within the framework of the Diploma Programme Their academic plans, in particular the subjects they wish to study in future Their choice of career Teachers are expected to assist with the selection process and to offer advice to students about how to choose the most appropriate course from the four mathematics courses available. Mathematical studies SL—course details This course is available at standard level (SL) only. It caters for students with varied backgrounds and abilities. More specifically, it is designed to build confidence and encourage an appreciation of mathematics in students who do not anticipate a need for mathematics in their future studies. Students taking this course need to be already equipped with fundamental skills and a rudimentary knowledge of basic processes. The course concentrates on mathematics that can be applied to contexts related as far as possible to other subjects being studied, to common real-world occurrences and to topics that relate to home, work and leisure situations. The course includes project work, a feature unique within this group of courses: students must produce a project, a piece of written work based on personal research, guided and supervised by the teacher. The project provides an opportunity for students to carry out a mathematical investigation in the context of another course being studied, a hobby or interest of their choice using skills learned before and during the course. This process allows students to ask their own questions about mathematics and to take responsibility for a part of their own course of studies in mathematics. The students most likely to select this course are those whose main interests lie outside the field of mathematics, and for many students this course will be their final experience of being taught formal mathematics. All parts of the syllabus have therefore been carefully selected to ensure that an approach starting with first principles can be used. As a consequence, students can use their own inherent, logical thinking skills and do not need to rely on standard algorithms and remembered formulae. Students likely to need mathematics for the achievement of further qualifications should be advised to consider an alternative mathematics course. Because of the nature of mathematical studies, teachers may find that traditional methods of teaching are inappropriate and that less formal, shared learning techniques can be more stimulating and rewarding for students. Lessons that use an inquiry-based approach, starting with practical investigations where possible, followed by analysis of results, leading to the understanding of a mathematical principle and its formulation into mathematical language, are often most successful in engaging the interest of students. Furthermore, this type of approach is likely to assist students in their understanding of mathematics by providing a meaningful context and by leading them to understand more fully how to structure their work for the project. Mathematics SL—course details This course caters for students who already possess knowledge of basic mathematical concepts, and who are equipped with the skills needed to apply simple mathematical techniques correctly. The majority of these students will expect to need a sound mathematical background as they prepare for future studies in subjects such as chemistry, economics, psychology and business administration. The course focuses on introducing important mathematical concepts through the development of mathematical techniques. The intention is to introduce students to these concepts in a comprehensible and coherent way, rather than insisting on mathematical rigour. Students should wherever possible apply the mathematical knowledge they have acquired to solve realistic problems set in an appropriate context. The internally assessed component, the portfolio, offers students a framework for developing independence in their mathematical learning by engaging in mathematical investigation and mathematical modelling. Students are provided with opportunities to take a considered approach to these activities and to explore different ways of approaching a problem. The portfolio also allows students to work without the time constraints of a written examination and to develop the skills they need for communicating mathematical ideas. This course does not have the depth found in the mathematics HL course. Students wishing to study subjects with a high degree of mathematical content should therefore opt for the mathematics HL course rather than a mathematics SL course. Mathematics HL—course details This course caters for students with a good background in mathematics who are competent in a range of analytical and technical skills. The majority of these students will be expecting to include mathematics as a major component of their university studies, either as a subject in its own right or within courses such as physics, engineering and technology. Others may take this subject because they have a strong interest in mathematics and enjoy meeting its challenges and engaging with its problems. The nature of the subject is such that it focuses on developing important mathematical concepts in a comprehensible, coherent and rigorous way. This is achieved by means of a carefully balanced approach. Students are encouraged to apply their mathematical knowledge to solving problems set in a variety of meaningful contexts. Development of each topic should feature justification and proof of results. Students embarking on this course should expect to develop insight into mathematical form and structure, and should be intellectually equipped to appreciate the links between concepts in different topic areas. They should also be encouraged to develop the skills needed to continue their mathematical growth in other learning environments. The internally assessed component, the portfolio, offers students a framework for developing independence in their mathematical learning through engaging in mathematical investigation and mathematical modelling. Students will be provided with opportunities to take a considered approach to these activities, and to explore different ways of approaching a problem. The portfolio also allows students to work without the time constraints of a written examination and to develop skills in communicating mathematical ideas. Further Mathematics SL – course details Further mathematics, available as a standard level (SL) subject only, caters for students with a good background in mathematics who have attained a high degree of competence in a range of analytical and technical skills, and who display considerable interest in the subject. Most of these students will intend to study mathematics at university, either as a subject in its own right or as a major component of a related subject. In particular, the course is designed to allow students to learn about a variety of branches of mathematics in depth and also to appreciate practical applications. The nature of the subject is such that it focuses on different branches of mathematics to encourage the student to appreciate the diversity of the subject. Candidates should be equipped at this stage in their mathematical progress to begin to form an overview of the characteristics that are common to all mathematical thinking, independent of topic or branch. All categories of candidate may register for mathematics HL only or for further mathematics SL only or for both. However, candidates registering for further mathematics SL will be presumed to know the topics in the core syllabus of mathematics HL and to have studied one of the options, irrespective of whether they have also registered for mathematics HL. Examination questions are intended to be comparable in difficulty with those set on the four options in the mathematics HL course. The challenge for candidates will be to reach an equivalent level of understanding across these topics. This course is a demanding one, requiring students to study a broad range of mathematical topics through a number of different approaches and to varying degrees of depth. Students wishing to study mathematics in a less rigorous environment should therefore opt for one of the standard level courses, mathematics SL or mathematical studies SL. Find out more
Physics, StudentImproving the Game When it comes to teaching and learning physics, most pedagogical innovations were pioneered in Cutnell and Johnson's Physics--the number one algebra-based physics text for over a decade. With each new edition of Physics, Cutnell and Johnson have strived to improve the heart of the game--problem solving. Now in their new Seventh Edition, you can expect the same spirit of innovation that has made this text so successful. Here's how the Seventh Edition continues to improve the game AMP Examples (Analyzing Multi-Concept Problems) These unique new example problems show students how to combine different physics concepts algebraically to solve more difficult problems. AMP examples visually map-out why the different algebraic steps are needed and how to do the steps. GO (Guided Online) Problems in WileyPLUS These new multipart, online tutorial-style problems lead students through the key steps of solving the problems. Student responses to each problem step are recorded in the grade book, so the instructor can evaluate whether the student really has mastered the material. WileyPLUS WileyPLUS provides the technology needed to create an environment where students can reach their full potential and experience the exhilaration of academic success. WileyPLUS gives students access to a complete online version of the text, study resources and problem-solving tutorials, and immediate feedback and context-sensitive help on assignments and quizzes.
Course Completeness Legend Mathematics A degree in General Mathematics is designed to equip you with the skills necessary to be a professional problem-solver, as a mathematician is not defined by his or her knowledge of laws and theorems, but by his or her critical thinking and reasoning skills. As a Mathematics Major, your classes will range from courses on mathematical logic and induction to courses on differential equations and analysis. The core Mathematics program is designed to allow you to gain basic training in a variety of mathematical tracks, whereas the advanced electives will provide you with the choice of which higher-level mathematical subfields you wish to progress into further. The complete list of applicable fields for which Mathematics is an integral part is extensive; here we show a few of the more common specializations. Every course listed in our "Core Program" is required of math majors with the exception of MA111: Introduction to Mathematical Reasoning. This course is recommended before MA231 and MA241 or any of the Advanced Mathematics Electives other than Partial Differential Equations. In addition to the Core Program, each student is required to choose seven electives, at least five of which must come from the Advanced Mathematics Electives. At least one must be chosen from Real Analysis II, Complex Analysis, and Probability Theory. At least one must be chosen from Abstract Algebra II and Linear Algebra II. Finally, each student is required to take two supplementary electives; MA121 cannot be used to satisfy this requirement. Those choosing to take Elementary Number Theory should treat Abstract Algebra I as a pre- or co-requisite. For those interested in graduate study in math, we recommend taking as many of the advanced electives as possible, with a focus on real and complex analysis and linear and abstract algebra. For those interested in working in applied mathematics, we recommend Linear Algebra II, Probability Theory, Partial Differential Equations, Numerical Analysis, and Topics in Applied Mathematics. For those interested in statistics, we recommend Real Analysis II, Linear Algebra II, Probability Theory, and Statistics II. To fulfill the requirements for this major, you must complete all of the core program except for MA111, which is optional (8 courses) as well as 7 electives of your choosing (7 courses — 5 of which must be "Advanced Mathematics") for a total of 15 courses. Foundational Material The courses in this section are not a formal part of the Saylor.org Mathematics Major, which begins with MA101: Single-Variable Calculus I, listed beneath the "Core Program" section below. These courses are designed to bridge the gap between high school and college-level mathematics. Core Program MA101 is a college level single-variable calculus course that covers theory and applications of limits, continuity, derivatives, and integration of real valued functions of a single variable. Students can earn a Saylor.org certificate of completion with a score of 70 or higher on the Final Exam after completing the entire course. MA101-EXC is aligned with a challenge exam proctored by Excelsior College, a private, nonprofit institution with a credit-by-exam program that enables students to earn college level credit in select subject areas by passing proficiency examinations. Should you choose to take MA101-EXC, you will complete your coursework at Saylor.org, but will then have the opportunity to pay a fee in order to take the proficiency exam through Excelsior College and potentially receive transferable college credit. (Please be sure to determine whether your home institution will accept this transfer credit prior to taking the exam!) This course differs from MA101 above in that it also covers applications of integration, material that is addressed in Unit 2 of Saylor.org's MA102: Single Variable Calculus II. However, you may choose to take either MA101-EXC or MA101 above in order to fulfill your Saylor area of study requirements in Mathematics.
Exploring Calculus with a Graphing Calculator ISBN 0201555743 / 9780201555745 / 0-201-55574-3 Book summary This easy-to-use maual enhances the fundamental concepts of calculus using a graphing calculator. Intended for use in a laboratory setting, a wide range of calculus concepts are developed and investigated through a series of exploratory activities. [via]
Microsoft Math Product Guide "Microsoft Math provides a space for nurturing student learning in mathematics with dynamic visualizations. The program provides essential ingredients for classroom environments designed to challenge all students to engage in visual thinking." — Margaret L. Neiss Professor of Mathematics Education, Oregon State University Microsoft® Math is a set of mathematical tools that can help students get their work done quickly and easily while promoting a better understanding of mathematical concepts. The primary tool in Microsoft Math is a full- featured scientific calculator with extensive graphing and equation-solving capabilities. It is designed to work just like a handheld calculator, but offers a wide range of additional tools that deepen students' knowledge of complex mathematics. Overview Microsoft Math provides students and teachers with a wide range of tools, including the following: • A 2-D and 3-D graphing calculator to graph complex equations • A Step-by-step Equation Solver to walk students through math problems • An Equation Library, which puts more than 100 common equations and formulas in a single location • A Triangle Solver that helps students explore the relationships between the parts of triangles • A Unit Conversion Tool for fast, easy conversion of different measurements • An Ink Handwriting Support feature for added flexibility in how students work Microsoft Math features a host of math tools designed to help students learn mathematical concepts. This collection of tools, tutorials and instructions helps students tackle math and science problems in one central location. Teachers say that students need more help with their homework. In addition, the majority of those teachers believe the quality of work and overall performance are enhanced when students use technology both at home and at school. And, as any child will admit, math is often a stumbling block to student success. The easy-to-use interface of Microsoft Math looks like a calculator — a familiar tool — but with a difference: It moves the emphasis from solving mathematical equations to increasing understanding of math and promoting lifelong skills. Microsoft Math gives students a worksheet and a graphing space. Combined, these features enable students to explore mathematical functions to better understand them. In addition, these abilities provide an invaluable tool for teachers, who can use Microsoft Math to create visual images of many mathematical ideas. As just a few examples, students can do any of the following activities: • Walk through the steps necessary to solve common algebra problems • Dynamically manipulate graphs of functions and equations • Visualize systems of inequalities to identify regions for solutions to problems • Conduct explorations in three dimensions • Enter multiple data sets and search for patterns from the visual graphics Graphing Calculator A major component of Microsoft Math, the graphing calculator is a groundbreaking complement to the calculators already required by most schools. It is designed primarily to help students visualize and solve difficult math and science problems, whether trigonometry, statistics, algebra, or calculus. Support for calculus includes functions on limits, series, derivatives and integrals. By graphing complex equations, students get not only a better way to solve their homework problems, but also help in gaining a deeper understanding of the reasoning behind those problems. This ability is enhanced through sophisticated graphing capabilities that allow students to view, rotate and animate large 2-D and enhanced 3-D color graphs. An animated slider enables them to make changes to the equation and see the effect on the graph. In addition, reusing common numbers and expressions can save time when there are multiple problems to solve. With the graphing calculator, it's simple to add and store variables. Step-by-Step Equation Solver The Step-by-step Equation Solver generator gives students the support they need by providing them with complete walk-through solutions to many math problems in middle-school and high- school pre-algebra, algebra I, algebra II, and trigonometry classes. In addition to supporting students when they are learning on their own, this feature is also ideal within a classroom setting because it allows a teacher to quickly display a particular solution method applied to a specific math problem. Equation Library With more than 100 interactive common math equations and formulas in a single location, students can find and interact with the right equations necessary to solve problems. If they want, they can quickly graph these equations using the graphing calculator. Triangle Solver The Triangle Solver helps students explore triangles and the relationships between their parts. As a student enters the values for sides and angles, the displayed triangle changes shape to reflect those values. Once enough values are entered, the remaining sides and angles are completed and the trigonometric rules used are called out. Unit Conversion Tool The Unit Conversion Tool makes it easy for students to quickly convert units of measure such as length, area, volume, weight, temperature, pressure, energy, power, velocity and time. Ink Handwriting Support New support for Tablet and Ultra-Mobile PCs means that students can do their math homework more naturally, by writing it out. Microsoft Math contains handwriting recognition software tailored to math, giving it high recognition accuracy. As a result, students can enter mathematical expressions as they would on paper. System Requirements Microsoft Math requires the following: • A personal computer with a Pentium 600MHz or faster processor (1GHz or faster processor recommended) ® • Windows XP Service Pack 2 or later • 256 MB of RAM (512 MB or more recommended) • 450 MB of available hard disk space • Microsoft .NET Framework 2.0 (requires between 200 MB and 450 MB of hard disk space) • VGA-capable or better video card with a minimum 640x480 resolution (1024x768 recommended) "When solving a quadratic equation, Microsoft Math doesn't just churn and spit out one of the solutions. Instead, it shows you both solutions and how to obtain them using square completion or the quadratic equation." — Jonathan Briggs Math Teacher, Eastside Preparatory School Kirkland, Wash
More About This Textbook Overview Galois theory is one of the jewels of mathematics. Its intrinsic beauty, dramatic history, and deep connections to other areas of mathematics give Galois theory an unequaled richness. David Cox's Galois Theory helps readers understand not only the elegance of the ideas but also where they came from and how they relate to the overall sweep of mathematics. Galois Theory covers classic applications of the theory, such as solvability by radicals, geometric constructions, and finite fields. The book also delves into more novel topics, including Abel's theory of Abelian equations, the problem of expressing real roots by real radicals (the casus irreducibilis), and the Galois theory of origami. With intriguing Mathematical and Historical Notes that clarify the ideas and their history in detail, Galois Theory brings one of the most colorful and influential theories in algebra to life for professional algebraists and students alike. Related Subjects Meet the Author DAVID A. COX, PhD, is Professor in the Department of Mathematics at Amherst College. He has published extensively in his areas of research interest, which include algebraic geometry, number theory, and the history of mathematics. Dr. Cox is consulting editor for Wiley's Pure and Applied Mathematics book series and the author of Primes of the Form x2 + ny2
Buy Used Textbook Buy New Textbook eTextbook 180 day subscription $101.99i will be using ecampus for future purchases....March 29, 2011 by Debra i'm so glad i bought my textbook on this!! it pretty much like new!!! my friend paid like almost 200 for this at our on amazon bookstore and i paid 80something and got everything she got with her book even a free bookmark xD haha.. and i also didn't have to waste gas to go pick it up!!! i will be back on ecampus for my books needed for the next semester.. Elementary and Intermediate Algebra: 5 out of 5 stars based on 1 user reviews. Summary The Sullivan/Struve/MazzarellaAlgebra Series was written to motivate students to do the math outside of the classroom through a design and organization that models what you do inside the classroom. The left-to-right annotations in the examples provide a teachers voice through every step of the problem-solving process. The Sullivan exercise sets, which begin with Quick Checks to reinforce each example, present problem types of every possible derivation with a gradual increase in difficulty level. The "Do the Math" Workbook acts as a companion to the text and to MyMathLab by providing short warm-up exercises, guided practice examples, and additional Do the Math practice exercises for every section of the text. Sullivan Examples and Showcase Examples provide students with superior guidance and instruction when they need it most–when they are away from the instructor and the classroom. Students learn algebra by doing algebra. Throughout the textbook, the exercise sets are grouped into eight categories–some of which appear only as needed. Study Skills features are a regular theme throughout the book, anticipating students' needs and providing the voice of an instructor. Author Biography Mike Sullivan, III is a professor of mathematics at Joliet Junior College. He holds graduate degrees from DePaul University in both mathematics and economics. Mike is an author or co-author on more than 20 books, including a statistics book and a developmental mathematics series. Mike is the father of three children and an avid golfer who tries to spend as much of his limited free time as possible on the golf course.
products in this textbook are intended to provide students with a sense of the ... with special fabric for extra binding strength. The cover is a ... shape the vision for . Addison Wesley Mathematics Makes Sense through discussions and reviews of. Book Bureau (stock number 80637). Order online at ... Conceptual Framework for Kindergarten to Grade 9 Mathematics. 5. Assessment. 9. Instructional .....makesense of information, relate it to prior knowledge, and use it for new learning. This
Numerical Analysis course, presented by MIT and taught by Professor Alar Toomre, provides an introduction to numerical analysis. The material looks at the basic techniques for the efficient numerical solution of problems in science and engineering. Topics include root finding, interpolation, approximation of functions, integration, differential equations and direct and iterative methods in liner algebra. Lecture notes are included on the site. MIT presents OpenCourseWare as free educational material online. No registration or enrollment is required to use the materials.Wed, 12 Jan 2011 03:00:02 -0600Numeric Computation of Integrals Tue, 22 Jun 2010 03:00:01 -0500Double Integrals II by Lang Moore and David Smith for the Connected Curriculum Project, the purpose of this module is to use iterated double integrals in polar coordinates to carry out complicated volume computations. This is one within a much larger set of learning modules hosted by Duke University.Mon, 24 May 2010 03:00:01 -0500Double Integrals I by Lang Moore and David Smith for the Connected Curriculum Project, the purpose of this module is to investigate the definition of the double integral and to develop numerical methods for calculating double integrals. This is one within a much larger set of learning modules hosted by Duke University.Fri, 21 May 2010 03:00:02 -0500The Mathematics of Change presented at a 1993 conference focused on change and reform, with notes on the type of technology represented and the level for which each is intended.Thu, 31 Jan 2008 03:00:01 -0600Calculus Course Materials online course: learning units presented in worksheet format review the most important results, techniques and formulas in college and pre-college calculus. Logarithms and Exponential; Sequences; Series; Techniques of Integration; Local Behavior of Functions; Power Series; Fourier Series; and an Appendix of Mathematical Tables.Wed, 12Homepage of e-Calculus mathematics professor from the University of Akron has made available this online tutorial covering many of the topics of a typical first semester of calculus. Beginning with a general overview of continuous functions and fundamental operations, the tutorial progresses to limits, differentiation, and integration. Since the material is so extensive, it is divided into several documents that can be easily navigated using the hyperlinks scattered throughout the text. One shortcoming of the tutorial is its lack of illustrative figures and diagrams; however, equations are clearly shown with adequate explanation. Also provided via a link on the site is the Algebra Review in Ten Lessons.Wed, 16 May 2007 03:00:01 -0500
books.google.com - This introduction to computational geometry focuses on algorithms. Motivation is provided from the application areas as all techniques are related to particular applications in robotics, graphics, CAD/CAM, and geographic information systems. Modern insights in computational geometry are used to provide... Geometry Computational Geometry: Algorithms and Applications This introduction to computational geometry focuses on algorithms. Motivation is provided from the application areas as all techniques are related to particular applications in robotics, graphics, CAD/CAM, and geographic information systems. Modern insights in computational geometry are used to provide solutions that are both efficient and easy to understand and implement. User ratings Review: Computational Geometry: Algorithms and Applications User Review - Goodreads Read enough for an exam, great book. Review: Computational Geometry: Algorithms and Applications User Review - Goodreads Beauty is the first test. This is a very beautiful book (form) with a beautiful contents. The book explains in a very throrough way some of the fundamental algoritms in "computational geometry". You ...
The Lial series has helped thousands of readers succeed in developmental mathematics through its approachable writing style, relevant real-world examples, extensive exercise sets, and complete supplements package. The Real Number System; Linear Equations and Inequalities in One Variable; Linear Equations and Inequalities in Two Variables: Functions; Systems of Linear Equations and Inequalities; Exponents and Polynomials; Factoring and Applications; Rational Expressions and Applications; Roots and Radicals; Quadratic Equations For all readers interested in Beginning Algebra.
This is a free online course offered by the Saylor Foundation.'This introductory mathematics course is for you if you have a... see more This is a free online course offered by the Saylor Foundation.'This introductory mathematics course is for you if you have a solid foundation in arithmetic (that is, you know how to perform operations with real numbers, including negative numbers, fractions, and decimals). Numbers and basic arithmetic are used often in everyday life in both simple situations, like estimating how much change you will get when making a purchase in a store, as well as in more complicated ones, like figuring out how much time it would take to pay off a loan under interest.The subject of algebra focuses on generalizing these procedures. For example, algebra will enable you to describe how to calculate change without specifying how much money is to be spent on a purchase–it will teach you the basic formulas and steps you need to take no matter what the specific details of the situation are. Likewise, accountants use algebraic formulas to calculate the monthly loan payments for a loan of any size under any interest rate. In this course, you will learn how to work with formulas that are already known from science or business to calculate a given quantity, and you will also learn how to set up your own formulas to describe various situations by translating verbal descriptions to mathematical language. In the later units of this course, you will discover another tool used in mathematics to describe numbers and analyze relationships: graphing. You will learn that any pair of numbers can be represented by a point on a coordinate plane and that a relationship between two quantities can be represented by a line or a curve.Units 6, 7, and 8 may seem more abstract than the earlier ones, as you will deal with expressions that contain mostly variables and not too many numbers. While the procedures you will master in these units might seem to have little practical application, you have to keep in mind that they result in formulas that describe very real situations in business, accounting, and science. Knowing how to perform various operations with algebraic expressions will eventually enable you to solve quadratic and even more complex equations. You will explore a variety of real-world scenarios that can be described by these kinds of equations. For example, if a ball is thrown up in the air, solving a quadratic equation will help you find out when it will hit the ground. As another example, if you know the area of a rectangular garden, then you can use a quadratic equation to find the length of each side.' Ths is a free version of the Boundless Algebra book that can be downloaded from Amazon for a Kindle.'The Boundless Algebra... see more Ths is a free version of the Boundless Algebra book that can be downloaded from Amazon for a Kindle for״Have you forgotten most of your algebra? Algebra Touch will refresh your skills using touch-based techniques built from the... see more ״Have you forgotten most of your algebra? Algebra Touch will refresh your skills using touch-based techniques built from the ground up for your iPhone/iPad/iPod Touch. Say you have x + 3 = 5. You can drag the 3 to the other side of the equation. Enjoy the wonderful conceptual leaps of algebra, without getting bogged down by the tedium of traditional methods. Drag to rearrange, tap to simplify, and draw lines to eliminate identical terms. Easily switch between lessons and randomly-generated practice problems. Create your own sets of problems to work through in the equation editor, and have them appear on all of your devices with iCloud. (there is a version of this app for OSX as well!) Current material covers: Simplification, Like Terms, Commutativity, Order of Operations, Factorization, Prime Numbers, Elimination, Isolation, Variables, Basic Equations, Distribution, Factoring Out, Substitution.״This app costs $1.99 This is a free
in Our World "Mathematics in Our World" is designed for mathematics survey courses for non-science majors. The text covers a variety of topics designed to foster ...Show synopsis"Mathematics in Our World" is designed for mathematics survey courses for non-science majors. The text covers a variety of topics designed to foster interest in and show the applicability of mathematics. The book is written by our successful statistics author, Allan Bluman. His easy-going writing style and step-by-step approach make this text very readable and accessible to lower-level students. The text contains many pedagogical features designed to both aid the student and instill a sense that mathematics is not just adding and subtract73311820 GREAT shape! May have some minor physical...Very Good. 0073311820 Mathematics in Our World with MathZone. This book is in...Good. Mathematics in Our World with MathZone73311820 AtAGlance Books--Orders ship next business day,...New. 0073311820 AtAGlance Books--Orders ship next business day, with tracking numbers, from our warehouse in upstate NY. This book is in brand new condition. Reviews of Mathematics in Our World This book was in terrific condition and came sooner than expected for my husband's college course. Even better, the book is the teacher's edition and had some great helps for those of you who are rusty on math procedures for different real-world applications. If you need tyo brush up your math
Mathematics (T) Mathematics involves observing, representing and investigating patterns and relationships in social and physical phenomena and between mathematical objects themselves. Mathematics is the science of patterns. The mathematician seeks patterns in number, in space, in science, in computers, and in imagination. Mathematical theories explain the relation between patterns… Applications of mathematics use these patterns to explain and predict natural phenomena.' (National Statement on Mathematics for Australian Schools 1991 p4) Many students study mathematics not because they want to be a mathematician, but because they realise the benefit that mathematics can have on a broader education. Tertiary mathematical courses often assume some mathematical knowledge or explicitly require mathematical prerequisites for study in most tertiary degrees. These can include but are not limited to Commerce (MM at ANU, MA at UC), Economics (MM at ANU, MA at UC), Computer Science (SM and ANU), Engineering (SM at ANU), Finance (MM at ANU), Information Technology (MM at ANU and UC), Actuarial Studies (SM double major at ANU) and Business Informatics (MA at UC). (MM=Mathematical Methods, SM=Specialist Mathematics, MA=Mathematical Applications) Importantly, employers are recognising the benefit that students with mathematical studies can bring to their business. Skills such as analytical thinking, structured and unstructured problem solving and ability to apply rules, formulae and ordered processes to contextual situations are becoming wildly sought after and respected experiences. Our tertiary courses can prepare students for life in the workforce and tertiary studies.
Description: Calculus isn't just a required math class to weed out would-be science majors-it's a useful way to understand the patterns in physics, economics, and the natural world. With its distinctive mix of serious educational content and Japanese-style comics, The Manga Guide to Calculus will entertain you while it helps you understand the key concepts of calculus (and ace those exams). Our story begins as Noriko, a recent liberal arts grad, arrives at a branch office of the Asagake Newspaper to start her career as a journalist. With the help of her overbearing and math-minded boss Kakeru, she's finally able to do some real reporting. But Noriko soon discovers the mathematical functions behind all the stories she struggles to cover. How to integrate and differentiate trigonometric and other complicated functions Multivariate calculus and partial differentiation Taylor expansions Reluctant calculus students of all abilities will enjoy following along with Noriko as she learns calculus from Kakeru's quirky stories and examples. This charming and easy-to-read guide also includes an appendix with answers to the book's many useful exercises. This EduManga book is a translation from a bestselling series in Japan, co-published with Ohmsha, Ltd. of Tokyo,
rigorous yet elementary introduction to the theory of analytic functions of a single complex variable. While presupposing in its readership a degree of mathematical maturity, it insists on no formal prerequisites beyond a sound knowledge of calculus. Starting from basic definitions, the text slowly and carefully develops the ideas of complex analysis to the point where such landmarks of the subject as Cauchy's theorem, the Riemann mapping theorem, and the theorem of Mittag-Leffler can be treated without sidestepping any issues of rigor. The emphasis throughout is a geometric one, most pronounced in the extensive chapter dealing with conformal mapping, which amounts essentially to a "short course" in that important area of complex function theory. Each chapter concludes with a wide selection of exercises, ranging from straightforward computations to problems of a more conceptual and thought-provoking nature.
Harmonic Analysis: A Gentle Introduction Book Description: Many branches of mathematics come together in harmonic analysis, each adding richness to the subject and each giving insights into this fascinating field. Devito's Harmonic Analysis presents a comprehensive introduction to Fourier analysis and Harmonic analysis and provides numerous examples and models so that students leave with a clear understanding of the theory
Advanced Mathematical Concepts - 06 edition Summary: Advanced Mathematical Concepts, 2006 provides comprehensive coverage of all the topics covered in a full-year Pre-calculus 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 in the Teacher Wraparound Edition. Advanced Mathematical Concepts lessons develop mathematics us...show moreing 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. New Features: " A full-color design, a wide range of exercise sets, relevant special features, and an emphasis on graphing and technology invite your students to experience the excitement of understanding and applying higher-level mathematics skills. " Graphing calculator instructions is provided in the Graphing Calculator Appendix. Each Graphing Calculator Exploration provides a unique problem-solving situation. " SAT/ACT Preparation is a feature of the chapter end matter. The Glencoe Web site offers additional practice: amc.glencoe.com " Applications immediately engage your students; interest. Concepts are reinforced through a variety of examples and exercise sets that encourage students to write, read, practice, think logically, and review. " Calculus concepts and skills are integrated throughout the course50 +$3.99 s/h VeryGood AlphaBookWorks Alpharetta, GA 0078682274
A Short Course in Problem Solving The lessons help students in early grades progress from arithmetic to algebra. Simple arithmetic problems can be cast in a problem solving form. Build problem solving skills early. Middle school Middle school is the ideal place to bring about the transition from arithmetic to algebra. Using logical problem solving not only develops problem solving skills but motivates understanding and use of basic algebra. High school Word problems in high school algebra are typically difficult and discouraging for many students. This is a result of lack of familiarity with the basic ideas of problem solving. This can be corrected through use of the lessons in this sequence. College Students entering disciplines such as engineering, physics and chemistry frequently lack the most rudimentary problem solving skills. They often rely on arithmetic alone. This is quite inadequate. The needed remedial training in problem solving is available in the lessons in this sequence.
Mathematics for Physical Chemistry Reviews "The text is extremely clear and concise delivering exactly what the student needs to know in a pinch - nothing more, nothing less. It is an indispensable resource for any student of physical chemistry."--Gregory S. Engel, Harvard University "Mathematics for Physical Chemistry is a comprehensive review of many useful mathematical topics...The book would be useful for anyone studying physical chemistry."--Daniel B. Lawson, University of Michigan-Dearborn "The student will derive benefit from the clarity, and the professional from a concise compilation of techniques stressing application rather than theory.… Recommended."--John A. Wass for SCIENTIFIC COMPUTING AND INSTRUMENTATION
Find a Brookhaven, PA PrealgebraThe crucial skills gained in this course include analysis of numerical relations and spatial-visual understanding of graphing principles. The skills they learn in this course will stay with them throughout higher math education, and will prove useful in standardized testing, such as the PSAT and SAT. Pre-algebra centers on building the foundations of the student's algebra. ...I'm pleased with the new features of each release. I am confident that my students can learn to generate awesome power point presentation. Basically any math problem is solved by either adding, subtracting, multiplying or dividing.
Classroom Expectations: 1.Attendance: It is very important to attend class every day. Although you will not lose points for absences, your grade will definitely be affected. 2.Make Up Work: a.If you miss a quiz or test you will be expected to make it up. b.If you miss notes you should see a classmate to get them or talk to me. 3.Readiness: You must be in the classroom on time with all of your supplies ready. You may not be permitted to retrieve your materials and the like from your locker. 4.Do Now: a.Something new this year is what I call the 'Do Now' work. As soon as you enter the room you'll find the 'Do Now' section and the work to be done at the top left section of the blackboard, unless otherwise specified. b.You'll be given 5 minutes to finish the work in a loose leaf. c.Credit for this will be applied to the Homework 5.Homework: a.Homework is usually checked at the beginning of class and should always be done in the notebook unless otherwise specified. b.Homework should be submitted on time. There is no such thing as a late homework. Late homework is tantamount to 'no homework'. 6.Plagiarism: You are responsible for your own work. Copying of any assignment, quiz, or test will result in a zero for all parties involved. 7.Quizzes: Quizzes are either announced or unannounced. 8.Additional Information: a.Notebooks and binders will be graded at the end of each marking period. Credit for this will be applied to the Class Participation category of your Algebra grade. b.Progress reports are available bi-weekly on the First Term and by Marking Periods on the Second Term. c.We use pencil in this class most of the time. d.Mathematics is not a spectator sport! Class Participation is 20% of the final grade.
Scheme of work – Cambridge IGCSE® Mathematics (US) 0444 v1 2Y01 Cambridge IGCSEMathematics (US) 0444 1 ... Only those parts of the learning objectives or notes and exemplars not included in the core units are itemised, so this document v1 2Y01 Cambridge IGCSEMathematics (US) 0444 3 Syllabus ref Learning objectives Suggested teaching activities Learning resources particular case is about an area or a volume as the problems can be about 3D 2 Cambridge IGCSE International Mathematics 0607. Examination in June and November 2013. 1. Introduction 1.1 Why choose Cambridge? University of Cambridge International Examinations (CIE) is the world's largest provider of international IGCSEMathematics is a fully examined course which encourages the development of mathematical knowledge as a key life skill, and as a basis for more advanced study. The syllabus aims to build students' confidence by helping them develop a feel Instructions on IGCSE Examination Timetable JUNE 2013. ... books and notes, and they can talk to each other. However, they must not have any access to telephones, mobile phones, ... • Mathematics without Coursework Paper 12 - Core: 0580/12: 1h: IGCSE/ GCSE /GCE EXAMINATIONS MAY/JUNE 2011 PLEASE OBSERVE STARTING TIME FOR EVERY EXAM EXAMS IN THE OLD HALL – SEE NOTES FOR EXCEPTIONS • Morning exams start 10:00 for CIE and AQA if there are no other exams in the morning session ; Full teaching notes, plus a starter and ... This companion volume to IGCSE Mathematicsis written by the same highly-experienced author team specifically for candidates preparing for the IGCSEMathematics Core Curriculum examination.
Mathematicians seek deep truths about a purely formal world, one that may or may not have much to do with the physical world we inhabit. Through our readings, seminar discussions, and writing assignments, we'll explore that connection, the existential status of mathematical objects (What is mathematics? Do mathematical objects actually exist, and if so, where? Are mathematical systems discovered or created?), and surrounding issues as we learn more about modern mathematical practice. This course coincides with the seminar portion of the Mathematical Systems program, so students in this course will share seminars with students in that program.
{"currencyCode":"USD","itemData":[{"priceBreaksMAP":null,"buyingPrice":38.72,"ASIN":"0387201726","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":22.21,"ASIN":"0817636773","isPreorder":0}],"shippingId":"0387201726::a3exxWpPI9w3WpB0M%2BPU%2FkgEzhwNC4%2BJbF8oIKZNWUu2sccxEF%2B8%2FoM7Z8fbHf%2BE6G6%2BP5XMYeFHkKYmBPtvGMCHeXPzQOym2itfD%2BMk6NM%3D,0817636773::rfHGkjUokAr504qHkDCEXFLeewmZrUXeTWkyxwdVM6JAMzv3rZU070Iyfmd%2BWJyQd1wgtX%2FmHx3OMFG7X2WITqAjL%2BouN2cY3uw7WMY3bQ focuses mainly on the 'doing' of algebra. … The chief aim of the author is for students 'to master such skills as learning what a mathematical statement is, what a mathematical argument or proof is, how to present an argument orally … and how to converse effectively about mathematics.' … the author strives to motivate students, gradually developing their insights and abilities. … It is an excellent primer for beginners in the field of abstract algebra, especially for future school teachers." (P. Shiu, The Mathematical Gazette, Vol. 89 (516), 2005) "This is an instructional exposition which treats some elementary number theory … . It is apparent that the author has made every effort to motivate students resp. to put them in the right way. 'I love algebra. I want my students to love algebra' – I believe that the author succeeded even in this regard." (G. Kowol, Monatshefte für Mathematik, Vol. 144 (2), 2005) "This is a very elementary introduction to elementary number theory and some related topics in algebra … . The topics chosen are well suited for a student's first exposure to 'serious' mathematics (much more so, in the reviewer's opinion, than the calculus course that is the norm in almost all curricula almost everywhere)." (S. Frisch, Internationale Mathematische Nachrichten, Issue 196, 2004) "The book … represents a very special introduction to modern algebra … . focuses less on contents and more on the 'doing' of algebra. … Many proofs are left as exercises, together with detailed hints or outlines, and these exercises actually form the heart of the entire text. … Summing up, this book is a great primer for beginners in the field … . could serve well in an undergraduate course for non-mathematicians, and as a guide to self-education beyond academic training, too." (Werner Kleinert, Zentralblatt MATH, Vol. 1046 (2), 2004) "Originally conceived as a text for a course for future secondary-school mathematics teachers, this book has developed into one that could serve well in an undergraduate course in abstract algebra … . The topics studies should be of interest to all mathematics students and are especially appropriate for future teachers. … Many proofs are left as exercises, and for almost every such exercise, a detailed hint or outline of the proof is provided. These exercises form the heart of the text." (Zentralblatt für Didaktik der Mathematik, November, 2004) "Mathematics is often regarded as the study of calculation … . It combines creativity and logic in order to arrive at abstract truths. This book is intended to illustrate how calculation, creativity, and logic can be combined to solve a range of problems in algebra. … Many proofs are left as exercises, and for almost every such exercise, a detailed hint or outline of the proof is provided. These exercises form the heart of the text." (L'Enseignement Mathematique, Vol. 50 (1-2), 2004) "The book is meant to be a structurally different abstract algebra textbook. … the book is very unitary and it has a good flow. … Integers, Polynominals and Rings is a unique book, and should be extremely useful for an audience of future high school teachers. It would also be a valuable supplement for students taking a traditional abstract algebra course, especially since it is very readable." (Ioana Mihaila, MathDL, January, 2004) From the Back Cover Mathematics is often regarded as the study of calculation, but in fact, mathematics is much more. It combines creativity and logic in order to arrive at abstract truths. This book is intended to illustrate how calculation, creativity, and logic can be combined to solve a range of problems in algebra. Originally conceived as a text for a course for future secondary-school mathematics teachers, this book has developed into one that could serve well in an undergraduate course in abstract algebra or a course designed as an introduction to higher mathematics. Not all topics in a traditional algebra course are covered. Rather, the author focuses on integers, polynomials, their ring structure, and fields, with the aim that students master a small number of serious mathematical ideas. The topics studied should be of interest to all mathematics students and are especially appropriate for future teachers. One nonstandard feature of the book is the small number of theorems for which full proofs are given. Many proofs are left as exercises, and for almost every such exercise a detailed hint or outline of the proof is provided. These exercises form the heart of the text. Unwinding the meaning of the hint or outline can be a significant challenge, and the unwinding process serves as the catalyst for learning. Ron Irving is the Divisional Dean of Natural Sciences at the University of Washington. Prior to assuming this position, he served as Chair of the Department of Mathematics. He has published research articles in several areas of algebra, including ring theory and the representation theory of Lie groups and Lie algebras. In 2001, he received the University of Washington's Distinguished Teaching Award for the course on which this book is based. it offers well-thought explanations of problems. Sometimes terminology gets in the way of understanding, but for the most part it's a decent book. I'm taking a course right now that uses this book, and Irving is the professor, so it helps that he's able to expound on what he wrote in the book.
Product Details: GIK1092: Motivate Your Students! This easy-to-use workbook is chock full of stimulating activities that will jumpstart your students' interest in algebra while reinforcing the major algebra concepts. A variety of puzzles, mazes, and games will challenge students to think creatively as they sharpen their algebra skills. A special assessment section is also included to help prepare students for standardized tests Features "This workbook is chocked full of stimulating activities A special assessment section is also included" Challenge students to think creatively Jumpstart your students interest in algebra while reinforcing concepts
John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, and is the only one to have received all three of the Leroy P. Steele prizes. In honor of his eightieth birthday, this book gathers together surveys andIn this book, Claire Voisin provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The volume is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the... more...... more... new technique for generating generic models with categories... more... Real quaternion analysis is a multi-faceted subject. Created to describe phenomena in special relativity, electrodynamics, spin etc., it has developed into a body of material that interacts with many branches of mathematics, such as complex analysis, harmonic analysis, differential geometry, and differential equations. It is also a ubiquitous factor... more... The third book in Peterson's NEW series of guides for visual learners, this volume covers basic algebra topics that are essential for success on standardized tests. egghead's Guide to Algebra can also be used in tandem with Peterson's egghead's Guide to Geometry, as it teaches critical algebra skills necessary for solving geometry problems. Topics... more...
More About This Textbook Overview Choose the algebra textbook that's written so you can understand it. ALGEBRA AND TRIGONOMETRY reads simply and clearly so you can grasp the math you need to ace the test. And with Video Skillbuilder CD-ROM, you'll follow video presentations that show you step-by-step how it all works. Plus, this edition comes with iLrn, the online tool that lets you sign on, save time, and get the grade you want. With iLrn, you'll get customized explanations of the material you need to know through explanations you can understand, as well as tons of practice and step-by-step problem-solving help. Make ALGEBRA AND TRIGONOMETRY your choice today. This Enhanced Edition includes instant access to Enhanced WebAssign, the most widely-used and reliable homework system. Enhanced WebAssign presents thousands of problems, links to relevant textbook sections, video examples, problem-specific tutorials, and more, that help students grasp the concepts needed to succeed in this course. As an added bonus, the Start Smart Guide has been bound into this text. This guide contains instructions to help students learn the basics of WebAssign quickly. Editorial Reviews Booknews The authors of this college-level text work from the premise that conceptual understanding and technical skills are inextricable. They emphasize a view of mathematics as a problem-solving art (rather than a collection of facts) that will help students succeed not only in subsequent math and science courses, but in the increasingly technology-oriented career world as well. Based on their own experiences in teaching algebra, trigonometry, and pre-calculus, they include elements such as progressive exercise sets, projects that students can work on alone or in groups, vignettes about interesting mathematicians and uses of math, and examples of real-world applications
Latest Pre-algebra Stories Learn Basic Math at Educator.com's newest course. This course is in addition to 80+ subjects already available for high school, college, test prep, and professional subjects. Los Angeles, CA (PRWEB) January 02, 2014 Learning mathematics is like building a wall. Brick by brick of basic concepts add up to form a complex structure, and the strength of this wall depends on its foundation--Basic Math. Learn or review your basic math skills today with Educator.com's online course.... Two Wrong Angles Do Not Make a Right Mountain View, CA (PRWEB) January 05, 2012 When many students hear the word "algebra," their heads begin swimming with thoughts of long, complex equations chock-full of variables, coefficients, exponents, fractions and square roots. It´s true - there is indeed plenty of that. However, many of the fundamental ideas that need to be mastered before moving on to more advanced algebraic concepts involve a branch of mathematics generally... Shmoop takes the confusion out of the Pre-Algebra equation Mountain View, CA (PRWEB) December 19, 2011 Shmoop, a publisher of digital curriculum and test prep, announces the launch of its expanded free Pre-Algebra resource at Many studentws might regard math as their sworn enemy. The mere mention of a Cartesian Coordinate System may cause toes to curl. The concept of a box and whisker plot may be nothing more than a plan for keeping litter off the... By Katy Murphy EDITOR'S NOTE: This is a sampling of the Education Report, Katy Murphy's blog on Oakland schools. The below comments were chosen from more than 50 posted on the subject. Read more and post comments at July 9: The California Board of Education voted 8-1 to scrap the eighth-grade general math test and require all students to be tested in Algebra I -- likely, within the next three years. It appears the decision may have been influenced by a... NEW YORK, July 16 /PRNewswire/ -- New York City students continue to make substantial progress in math achievement since the district set out to make vast improvements six years ago. The city's curriculum includes Wright Group/McGraw-Hill's Everyday Mathematics ( in elementary schools and Glencoe/McGraw-Hill's Impact Mathematics ( for middle schools. The number of... The U.S. Department of Education's What Works Clearinghouse (WWC), the central and trusted source of scientific evidence of what works in education, just awarded its highest rating to the I CAN Learn® Algebra and Pre-Algebra programs for their "Positive Effects" in raising student test scores on No Child Left Behind (NCLB) - required high-stakes testing. According to the department's latest report, the technology-based I CAN Learn® Education...
Search multiple online resources Mathematics This illustrated work explains basic concepts of math and geometry, and provides information on historical milestones, notable mathematicians, and more. Explores the functions of math in daily life, as well as its role as a tool for measurement, data analysis, and technological development.
OR1-2008 Course: MS 221, Fall 2009 School: East Los Angeles College Word Count: 19387 Rating: Document Preview MAT2009 OPERATIONSOPERATIONS MAT2009 that are available. Operations Research (OR), otherwise known as Management Science, can be traced back to the early twentieth century. The most notable developments in this field took place during World War II when scientists were recruited by the military to help allocate scarce lot of interest in applying OR methods outside the military and the subject expanded greatly during this period. Nowadays, computers can solve very large scale OR problems of enormous complexity. Many of the techniques that we shall study in the first part of the course are of relatively recent origin. The Simplex algorithm for Linear Programming, which is the principal technique for solving many OR problems, was formulated by George Dantzig in 1947. This method has been applied to problems in a wide variety of disciplines including finance, production planning, timetabling and aircraft scheduling. Later in the course we consider optimization of non-linear functions. The methods here are based on calculus. Non-linear problems with equality constraints will be solved using the method of Lagrange Multipliers, named after Joseph Louis Lagrange (1736 - 1812). A similar method for inequality constraints, developed by Karush in 1939 and by Kuhn and Tucker in 1951, is beyond the scope of this course but will be covered in the Level 3 Mathematical Economics module. Books which may be useful for this course include: Introduction to Operations Research, by F Hillier and G Lieberman Operations Research: An Introduction, by H A Taha (Prentice Hall) Operations Research: Applications and Algorithms, by Wayne L Winston (Thomson) Operations Research: Principles and Practice, by Ravindran, Phillips and Solberg (Wiley) Schaum's Outline of Operations Research, by R Bronson (McGraw-Hill) (McGraw Hill) (McGraw-Hill) Linear and Non-linear Programming by S Nash and A Sofer 1.2 The graphical method In a Linear Programming (LP) problem, we aim to optimize an objective function which is a linear combination of some decision variables x1 , . . . , xn . These variables are restricted by a set of constraints, expressed as linear equations or inequalities. When there are just two decision variables, the constraints can be illustrated graphically in the (x1 , x2 ) plane. We represent x1 on the horizontal axis and x2 on the vertical axis. To find the region defined by ax1 + bx2 c, first draw the straight line with equation c c ax1 + bx2 = c. This line crosses the axes at x1 = and x2 = . a b If b > 0, then ax1 + bx2 c defines the region on and below the line and ax1 + bx2 c defines the region on and above the line. If b < 0 then this situation is reversed. It is usual to shade along the side of the line away from the region that is being defined. A strict inequality (< or > rather than or ) defines the same region, but the line itself is not included. x1 < c to the left of the vertical line x1 = c, and x1 > c to the right of this line. x2 < c below the horizontal line x2 = c, and x2 > c above this line. A simple way of deciding which side of a line satisfies a given inequality is to consider the origin (0, 0). For example, 2x1 - 3x2 5 defines that side of the line 2x1 - 3x2 = 5 which contains the origin, since 2(0) - 3(0) 5. If the origin lies on the line, consider a convenient point which is not on the line, e.g. (1, 1). Example 1: Containers problem To produce two types of container, A and B, a company uses two machines, M1 and M2 . Producing one of container A uses M1 for 2 minutes and M2 for 4 minutes. Producing one of container B uses M1 for 8 minutes and M2 for 4 minutes. The profit made on each container is 30 for type A and 45 for type B. Determine the production plan that maximizes the total profit per hour. Identify the decision variables. Let x1 type A and x2 type B containers be produced per hour. Formulate the objective function and state whether it is to be maximized or minimized. Let z be the profit generated per hour, so we must maximize z = 30x1 + 45x2 . List the constraints. The allowable values of the decision variables are restricted by 2x1 + 8x2 4x1 + 4x2 x1 x2 60, 60, 0, 0. (at most 60 minutes on M1 in an hour) (at most 60 minutes on M2 in an hour) (non-negativity) (non-negativity) Different values of the objective function correspond to straight lines with equations of the form 30x1 + 45x2 = c for varying values of c. 2 These lines are all parallel, with gradient - . 3 2 Terminology Ordered pairs (x1 , x2 ) which satisfy all the constraints are called feasible points. The set of all feasible points is called the feasible region or solution space. Points where at least one constraint fails to hold are called infeasible points. These points lie outside the feasible region. The optimum or optimal value of the objective function is the maximum or minimum value, whichever the problem requires. A feasible point at which the optimal value occurs is called an optimal point, and gives the optimal solution. The following diagram shows the feasible region for the Containers problem. 15 10 x2 5 0 0 5 10 15 x1 20 25 30 There are infinitely many feasible solutions. An optimal one can be found by considering the slope, and direction of increase, of the objective function z = 30x1 + 45x2 . All lines corresponding to fixed values of z are parallel to 2x1 +3x2 = 6 (for example), which crosses the axes at (3, 0) and (0, 2). To find the maximum profit we translate this `profit line' in the direction of increasing z, keeping it parallel, until moving it further would take it completely outside the feasible region. We see that the optimum occurs at a corner point (or extreme point) of the feasible region, at the intersection of the lines that correspond to the two machine constraints. Hence the optimal values of x1 and x2 are the solutions of the simultaneous equations 2x1 + 8x2 = 60 and 4x1 + 4x2 = 60, i.e. (x1 , x2 ) = (10, 5). The optimal value of the objective function is then z = 525. Thus the optimal production plan is to produce 10 type A containers and 5 type B containers per hour. This plan yields the maximum profit, which is 525 per hour. It can be shown (see Proposition 1.3) that if a LP problem has an optimal solution then there is an extreme point of the feasible region where this solution occurs. There can sometimes be optimal solutions which do not occur at extreme points. If two vertices P and Q of the feasible region give equal optimal values for the objective function then the same optimal value occurs at all points on the straight line segment P Q. 45 < p < 45 then the optimal Suppose the profit on container A is changed to p. If 4 45 solution still occurs at (10, 5). If p = or p = 45 there are multiple optimal points, all 4 45 yielding the same profit. If p > 45 then (15, 0) is optimal. If p < then (0, 7.5) is optimal, 4 but 7.5 containers cannot be made. The best integer solution can be found by inspection. 3 Example 2 : Rose-growing problem A market gardener grows red and white rose bushes. The red and white bushes require an area of 5 dm2 and 4 dm2 per bush respectively. Each red bush costs 8 per year to grow and each white bush costs 2 per year. The labour needed per year for a red bush is 1 person-hour, whereas for a white bush it is 5 person-hours. The reds each yield a profit of 2, and the whites 3, per bush per year. The total land available is at most 6100 dm2 , and the total available finance is 8000. The labour available is at most 5000 person-hours per year. How many bushes of each type should be planted to maximize the profit? Step 1 Summarise the information (dm2 ) Red 5 8 1 2 White 4 2 5 3 Max. resource available 6100 8000 5000 Area Finance () Labour (person-hours) Profit Step 2 Define the decision variables. Let x1 = number of red rose bushes and x2 = number of white rose bushes. Step 3 Specify the objective function. We aim to maximize the profit function. Thus the problem is to Maximize z = 2x1 + 3x2 Step 4 Identify the constraints and the non-negativity conditions. We must have 5x1 + 4x2 6100, 8x1 + 2x2 8000, x1 + 5x2 5000, x1 0, x2 0. Step 5 Sketch the feasible region. 2000 1500 x2 1000 500 0 500 1000 x1 1500 2000 Step 6 Method 1 Sketch a line of the form z = constant and translate this line, in the direction in which z increases, until it no longer intersects the feasible region. The dotted lines in the diagram are z = 2000 and z = 5000. The points on z = 2000 are feasible but not optimal, whereas those on z = 5000 are all infeasible. The optimal point lies between these lines at (500, 900), where z = 3700. 4 Method 2 Evaluate z at each of the extreme points and see where it is greatest. The extreme points of the feasible region (going clockwise) are (x1 , x2 ) = (0, 0), (0, 1000), (500, 900), (900, 400), and (1000, 0). These can be calculated by solving pairs of simultaneous equations. If you just read them off from the graph, you should check that they satisfy the equations of the lines on which they lie. The optimum occurs at (x1 , x2 ) = (500, 900), and the objective function is then z = 2 500 + 3 900 = 3700. Final answer: Growing 500 red and 900 white rose bushes is optimal, with profit 3700. Note that the original problem was stated in words: it did not mention the variables x1 and x2 or the objective function z. Your final answer should not use these symbols, as they have been introduced to formulate the mathematical model. Example 3 : Cattle feed problem During the winter, farmers feed cattle a combination of oats and hay. The following table shows the nutritional value of oats and hay, and gives the daily nutritional requirements of one cow: Units of protein Units of fat Units of calcium Units of phosphorus Cost per unit () per unit of hay 13.2 4.3 0.02 0.04 0.66 per unit of oats 34.0 5.9 0.09 0.09 2.08 daily requirement per cow 65.0 14.0 0.12 0.15 Find the optimal feeding plan and identify any redundant constraint(s). Decision variables Suppose the farmer mixes x1 units of hay with x2 units of oats. Objective function The aim is to make the feeding cost per cow as low as possible, so we must Minimize z = 0.66x1 + 2.08x2 . Constraints and non-negativity conditions The following constraints must be satisfied in order that the cows are fed an adequate diet: 13.2x1 4.3x1 0.02x1 0.04x1 x1 + 34.0x2 + 5.9x2 + 0.09x2 + 0.09x2 x2 65.0 14.0 0.12 0.15 0 0 (protein) (fat) (calcium) (phosphorus) (non-negativity) (non-negativity) Note that the inequalities here all have `' signs. 5 2.5 2 1.5 x2 1 0.5 0 1 2 3 x1 4 5 6 The feasible region is shown above note that it is unbounded. However an optimum does exist, as we are aiming to minimize the objective function. Extreme points (calculated to two decimal places where appropriate) A (x1 , x2 ) = (6, 0) C (x1 , x2 ) = (1.35, 1.39) z = 3.96 z = 3.78 B (x1 , x2 ) = (3.48, 0.56) D (x1 , x2 ) = (0, 2.37) z = 3.46 z = 4.96 Thus the minimum cost of feeding one cow is approximately 3.46. This is obtained with 3.48 units of hay and 0.56 units of oats. The phosphorus constraint is redundant, as we can see from the diagram. Any mixture containing enough protein will automatically contain enough phosphorus. 1.3 Convexity and extreme points Definitions The feasible region for a LP problem is a subset of the Euclidean vector space Rn , whose elements we call points. The norm of x = (x1 , . . . , xn ) is \|x\| = x1 2 + + xn 2 . An equation of the form a1 x1 + + an xn = b, or at x = b, defines a hyperplane in Rn . A hyperplane in R2 is a straight line; in R3 it is a plane. Any hyperplane determines the half-spaces given by at x b and at x b. Consider the expression (1 - r)x + ry where 0 r 1. Geometrically, this weghted average of x and y represents a point on the straight line segment between x and y, and is called a convex linear combination of x and y. More generally, a convex linear combination of the points x1 , . . . , xm is an expression of the form r1 x1 + + rm xm , where r1 + + rm = 1 and ri 0 for i = 1, . . . , m. A convex set is a set S such that if x, y S then (1 - r)x + ry S for all r [0, 1]. It then follows that a convex linear combination of any number of elements of S is also in S. A convex set can be interpreted geometrically as a region S such that if A and B are any two points in S, every point on the straight line segment AB lies in S. Examples of convex sets include: Rn itself; any vector subspace of Rn ; any interval of the real line; any hyperplane; any half-space; the interior of a circle or sphere. 6 We can often use a graph to decide whether a subset of R2 is convex, e.g. {(x, y) : y x2 } is convex whereas {(x, y) : y x3 } is not convex. If S1 , . . . , Sn are convex sets, their intersection S1 Sn is also convex. An extreme point, corner point or vertex of a convex set S is a point z S such that there are no two distinct points x, y S with z = (1 - r)x + ry where r (0, 1). A neighbourhood of a point x Rn is a set of the form {y Rn : \|y - x\| < } for some > 0. This can also be called an -neighbourhood or an open ball of radius . Let S be a subset of Rn . A point x in S is an interior point of S if some neighbourhood of x is contained in S. If every point of S is an interior point then S is an open set. A point y, not necessarily in S, is a boundary point of S if every neighbourhood of y contains a point in S and a point not in S. The set of all boundary points of S is called the boundary of S. If every boundary point of S is in S then S is a closed set. Every point in S is either an interior point or a boundary point of S. Some sets are neither open nor closed. Others, such as Rn itself, are both open and closed. A set S Rn is bounded if there exists a real number M such that \|x\| < M for all x S. If S is bounded and closed then S is said to be a compact set. If S is not bounded it has a direction of unboundedness, i.e. a vector u such that for all x S and k 0, x + ku S. (In general there are infinitely many such directions.) The feasible region for a LP problem with n decision variables is the intersection of a finite number of half-spaces in Rn . Hence it is convex. The region is closed, since it includes the hyperplanes which form its boundary, and it has a finite number of extreme points. Proposition 1.1 (Weierstrass's Theorem, or the Extreme Value Theorem) Let f be a continuous real-valued function defined on a non-empty compact subset S of Rn . Then f(x) attains minimum and maximum values on S, i.e. there exist xm , xM S such that - < f(xm ) f(x) f(xM ) < for all x S. Example Consider the problem: x1 + x2 6, - x2 + 2x3 4, Maximize z = 4x1 + x2 + 3x3 , subject to 0 x2 4, x3 0. 0 x1 4, 7 The feasible set is a convex polyhedron or simplex with seven plane faces, as shown. As it is closed and bounded, and any linear function is continuous, Weierstrass's theorem tells us that z attains a greatest and least value over this region. Clearly the minimum occurs at (0, 0, 0). It turns out that z is maximum at (4, 2, 3). Considering other objective functions with the same feasible region, x1 +x2 +x3 is maximum at (2, 4, 4), while x1 - 3x2 - x3 is maximum at (4, 0, 0). Proposition 1.2 Let S be the intersection of a finite number of half-planes in Rn . Let v1 , . . . , vm be the extreme points of S. Then x S if and only if x = r1 v1 + + rm vm + u where r1 + + rm = 1, each ri 0, and u is 0 if S is bounded and a direction of unboundedness otherwise. The proof of the `if' part of the above result is in the Exercises. The `only if' part is more difficult to prove, but if we assume it we can show the following: Proposition 1.3 (The Extreme Point Theorem) Let S be the feasible region for a linear programming problem. 1. If S is non-empty and bounded then an optimal solution exists and there is an extreme point of S at which it occurs. 2. If S is non-empty and not bounded, and if an optimal solution to the LP problem exists, then an optimal solution occurs at an extreme point of S. Proof Let the objective function be z = c1 x1 + + cn xn = ct x. S is closed so in case 1 it is compact, hence by Weierstrass's theorem z attains its maximum and minimum values at some points in S. In case 2 we are assuming that the required optimum is attained. Suppose z is to be maximized, and takes its maximum value over S at x . Let the extreme points of S be v1 , . . . , vm . Suppose there is no extreme point where z is maximum, so ct vj < ct x for j = 1, . . . , m. By Proposition 1.2, x = r1 v1 + + rm vm + u where r1 + + rm = 1, each ri 0, and u is 0 in case 1 or a direction of unboundedness in case 2. Thus ct x = ct (r1 v1 + + rm vm + u) = r1 ct v1 + + rm ct vm + ct u < r1 ct x + + rm ct x + ct u = (r1 + + rm )ct x + ct u = ct x + ct u, since r1 + + rm = 1. If u = 0 we have ct x < ct x , giving a contradiction, so z is maximum at an extreme point. If u = 0 then we have ct x < ct x + ct u, so ct u > 0. Now x + ku S for any k > 0. ct (x + ku) = ct x + kct u, which tends to as k , so z is unbounded on S, contradicting the assumption that z takes its maximum value in S. Hence again z is maximum at an extreme point. If z is to be minimized, the same reasoning can be applied to -z. 8 Exercises 1 1. Solve the following Linear Programming problems graphically. (a) Maximize z = 2x1 + 3x2 subject to 2x1 + 5x2 10, x1 - 4x2 -1, x1 0, x2 0. (b) Minimize z = 4x2 - 5x1 subject to x1 + x2 10, -2x1 + 3x2 -6, 6x1 - 4x2 13. 2. The objective function z = px + qy, where p > 0, q > 0, is to be maximized subject to the constraints 3x + 2y 6, x 0, y 0. Find the maximum value of z in terms of p and q. (There are different cases, depending on the relative sizes of p and q.) 3. A company makes two products, A and B, using two components X and Y . To produce 1 unit of A requires 5 units of X and 2 units of Y . To produce 1 unit of B requires 6 units of X and 3 units of Y . At most 85 units of X and 40 units of Y are available per day. The company makes a profit of 12 on each unit of A and 15 on each unit of B. Assuming that all the units produced can be sold, find the number of units of each product that should be made per day to optimze the profit. If the profit on B is fixed, how low or high would the profit on A have to become before the optimal production schedule changed? 4. A brick company manufactures three types of brick in each of its two kilns. Kiln A can produce 2000 standard, 1000 oversize and 500 glazed bricks per day, whereas kiln B can produce 1000 standard, 1500 oversize and 2500 glazed bricks per day. The daily operating cost for kiln A is 400 and for kiln B is 320. The brickyard receives an order for 10000 standard, 9000 oversize and 7500 glazed bricks. Determine the production schedule (i.e. the number of days for which each kiln should be operated) which will meet the demand at minimum cost (assuming both kilns can be operated immediately) in each of the following separate cases. (Note: the kilns may be used for fractions of a day.) (a) there is no time limit, (b) kiln A must be used for at least 5 days, (c) there are at most 2 days available on kiln A and 9 days on kiln B. 5. A factory can assemble mobile phones and laptop computers. The maximum amount of the workforce's time that can be spent on this work is 10 hours per day. Before the phones and laptops can be assembled, the component parts must be purchased. The maximum value of the stock that can be held for a day's assembly work is 2200. In the factory, a mobile phone takes 10 minutes to assemble using 10 worth of components whereas a laptop takes 1 hour 40 minutes to assemble using 500 worth of components. The profit made on a mobile phone is 5 and the profit on a laptop is 100. (a) Summarise the above information in a table. 9 (b) Assuming that the factory can sell all the phones and laptops that it assembles, formulate the above information into a Linear Programming problem. (c) Determine the number of mobile phones and the number of laptops that should be made in a day to maximise profit. (d) The market for mobile phones is saturating. In response, retailers are dropping prices which means reduced profits. How low can the profit on a mobile phone go before the factory should switch to assembling only laptops? 6. In the Containers problem (Example 1), suppose we write the machine constraints as 2x1 + 8x2 + x3 = 60 and 4x1 + 4x2 + x4 = 60, where x3 and x4 are the number of minutes in an hour for which M1 and M2 are not used, so x3 0 and x4 0. Show that the objective function can be written as 1 1 z = 30 10 + x3 - x4 6 3 1 1 + 45 5 - x3 + x4 . 6 12 By simplifying this, deduce that that the maximum value of z occurs when x3 = x4 = 0 and state this maximum value. 7. By sketching graphs and using the fact that the straight line joining any two points of a convex set lies in the set, decide which of the following subsets of R2 are convex. (a) {(x, y) : xy 1, x 0, y 0}, (c) {(x, y) : y - x2 1}, (e) {(x, y) : x2 - y 2 = 1}, (b) {(x, y) : xy -1, x 0}, (d) {(x, y) : 2x2 + 3y 2 < 6}, (f) {(x, y) : y ln x, x > 0}. 8. Let a = (a1 , . . . , an ) be a fixed element of Rn and let b be a real constant. Let x1 and x2 lie in the half-space at x b, so that at x1 b and at x2 b. Prove that for all r [0, 1], at ((1 - r)x1 + rx2 ) b. Deduce that the half-space is a convex set. 9. Let S and T be convex sets. Show that their intersection S T is a convex set. (Hint : let x, y S T , so x, y S and x, y T . Why must (1 - r)x + ry be in S T for 0 r 1?) Generalise this to show that if S1 , . . . , Sn are convex sets then S = S1 . . . Sn is convex. Deduce that the feasible region for a linear programming problem is convex. 10. Let S be the feasible region for a Linear Programming problem. If an optimal solution to the problem does not exist, what can be deduced about S from Proposition 1.3? 11. Prove that every extreme point of a convex set S is a boundary point of S. (Method: suppose x is in S and is not a boundary point. Then some neighbourhood of x must be contained in S (why?) Deduce that x is a convex linear combination of two points in this neighbourhood and so x is not an extreme point.) 12. Prove the `if' part of Proposition 1.3 as follows. Let the half-planes defining S be a1 t x b1 , . . . , ak t x bk , so x = vi satisfies all these inequalities for i = 1, . . . , m. Show that x = r1 v1 + +rm vm also satisfies all the inequalities, where r1 + +rm = 1 and each ri 0. Deduce that if S is unbounded and u is a direction of unboundedness then r1 v1 + + rm vm + u S. 10 Chapter 2 The Simplex Method 2.1 Matrix formulation of Linear Programming problems If a LP problem has more than two decision variables then a graphical solution is impracticable. We therefore develop an algebraic, rather than geometric, approach. We write x 0 to mean that every entry of the vector x is positive or zero. x is then called a non-negative vector. The set of non-negative vectors in Rn is denoted by Rn . + x y means that every entry of x is less than or equal to the corresponding entry of y. Thus x y is equivalent to y - x 0. If x and y are non-negative vectors then xt y 0, and if x + y = 0 then x = 0 and y = 0. If v 0 and x y then vt x vt y. Recall that for matrices A and B, (AB)t = Bt At . For vectors x and y, we have that xt y = yt x and xt Ay = yt At x (these are all scalars). We shall sometimes need to work with partitioned matrices. If two matrices can be split into blocks which are conformable for matrix multiplication, then A11 A12 A21 A22 B11 B12 B21 B22 = A11 B11 + A12 B21 A11 B12 + A12 B22 A21 B11 + A22 B21 A21 B12 + A22 B22 . A typical LP problem seeks values of the decision variables x1 , x2 ,. . . , xn to optimize the objective function z = c1 x1 + c2 x2 + + cn xn subject to the constraints a11 x1 + a12 x2 + + a1n xn a21 x1 + a22 x2 + + a2n xn . . . b1 b2 am1 x1 + am2 x2 + + amn xn bm and the non-negativity conditions x1 0, x2 0, . . . , xn 0. A maximization LP problem can be written in matrix-vector form as 11 Maximize z = ct x subject to Ax b and x0 where A is the m n matrix with (i, j) entry x1 . x = . , b = . xn (LP1) aij and b1 . , c = . . bm c1 . . . . cn The problem (LP1) is said to be feasible if the constraints are consistent, i.e. there exists some x Rn such that Ax b. The vector x is then a feasible solution of (LP1). If it + maximizes z, it is the optimal solution. The problem is unbounded if there is no finite maximum over the feasible region, i.e. there exists a sequence of vectors {xk } satisfying the constraints such that ct xk as k . The standard form of a LP problem is defined as follows: 1. The objective function is to be maximized. 2. All constraints are equations with non-negative right-hand sides. 3. All the variables are non-negative. Any LP problem can be converted into standard form by the following methods: Minimizing f(x) is equivalent to maximizing -f(x). Thus, the problem Minimize c1 x1 + c2 x2 + + cn xn is equivalent to Maximize - c1 x1 - c2 x2 - - cn xn subject to the same constraints, and the optimum occurs at the same values of x1 , . . . , xn . Any equation with a negative right hand side can be multiplied through by -1 so that the right-hand side becomes positive. Remember that if an inequality is multiplied through by a negative number, the inequality sign must be reversed. A constraint of the form can be converted to an equation by adding a non-negative slack variable to the left-hand side of the constraint. A constraint of the form can be converted to an equation by subtracting a non-negative surplus variable from the left-hand side of the constraint. 2.1.1 Example Suppose we start with the problem Minimize subject to and z = -3x1 + 4x2 - 5x3 3x1 + 2x2 - 3x3 4 2x1 - 3x2 + x3 -5 x1 0, x2 0, x3 0. Rewrite the second constraint with a non-negative right-hand side: -2x1 + 3x2 - x3 5. 12 Then add a slack variable x4 in the first constraint and subtract a surplus variable x5 in the second one. In standard form the problem is Maximize subject to and z = 3x1 - 4x2 + 5x3 3x1 + 2x2 - 3x3 + x4 = 4 -2x1 + 3x2 - x3 - x5 = 5 xj 0 for j = 1, . . . , 5. The slack and surplus variables represent the differences between the left and right hand sides of the inequalities. If a slack or surplus variable is zero at a point then the corresponding constraint is said to be active or binding or tight at that point. A variable is unrestricted if it is allowed to take both positive and negative values. An unrestricted variable xj can be expressed in terms of two non-negative variables by substituting xj = xj - xj where both xj and xj are non-negative. The substitution must be used throughout, i.e. in all the constraints and in the objective function. 2.1.2 Example Suppose we have to maximize z = x1 - 2x2 subject to the constraints x1 + x2 4, 2x1 + 3x2 5, x1 0 and x2 unrestricted in sign. To express the problem in standard form, let x2 = x2 - x2 where x2 0, x2 0. Introducing a slack variable x3 and a surplus variable x4 , the problem in standard form is: Maximize z = x1 - 2x2 + 2x2 subject to x1 + x2 - x2 + x3 = 4, 2x + 3x2 - 3x2 - x4 = 5 and x1 , x2 , x2 , x3 , x4 0. If a LP problem contains n non-negative decision variables and m inequality constraints then we need m additional (slack or surplus) variables and so the total number of variables becomes n + m. In this case the problem (LP1) can be written in the form ~~ Maximize z = ct x ~x subject to A~ = b ~ and x0 (LP2) x1 . . . c1 . . . , c = cn . ~ 0 . . . 0 xn ~ ~ where A is the m (n + m) matrix (A \| Im ), x = xn+1 . . . xn+m The constraints are now expressed as m linearly independent equations in n + m unknowns, which represent m hyperplanes in Rn+m . Their general solution depends on n arbitrary parameters, which we can set equal to zero to get a particular solution. ~ A feasible solution of (LP2) is a vector v Rn+m such that Av = b. + ~ If b 0 then (LP2) is in standard form and has an obvious feasible solution x = 0 b . 13 2.1.3 Example Suppose we have to maximize z = x1 + 2x2 + 3x3 subject to 2x1 + 4x2 + 3x3 10, 3x1 + 6x2 + 5x3 15 and x1 0, x2 0, x3 0. Introducing slack variables x4 , x5 0, this can be written as x1 x1 x2 x2 x3 subject to 2 4 3 1 0 x3 = Maximize z = 1 2 3 0 0 3 6 5 0 1 x4 x4 x5 x5 where x1 , . . . , x5 0. 10 15 , Setting any 3 variables to zero, if the resulting equations are consistent we can solve for the other two, e.g. (1, 2, 0, 0, 0) and (0, 0, 0, 10, 15) are feasible solutions for (x1 , x2 , x3 , x4 , x5 ). Note that setting x3 = x4 = x5 = 0 gives 2x1 + 4x2 = 10, 3x1 + 6x2 = 15 which are the same equation and do not have unique solutions for x1 and x2 . A unique solution of (LP2) obtained by setting n variables to zero is called a basic solution. If it is also feasible, i.e. non-negative, it is a basic feasible solution (bfs). The n variables set to zero are called non-basic variables; the remaining m (some of which may be zero) are basic variables. The set of basic variables is called a basis. In the above example, (0, 0, 10/3, 0, -5/3) is a basic infeasible solution. (1, 2, 0, 0, 0) is a feasible solution but not a bfs. (0, 0, 3, 1, 0) is a basic feasible solution with basis {x3 , x4 }. ~ It can be proved that x is a basic feasible solution of (LP2) if and only if it is an extreme point of the feasible region in Rn+m . The vector x consisting of the first n components of ~ x is then an extreme point of the original feasible region in Rn . In any LP problem that has an optimal solution, we know by Proposition 1.3 that an optimal solution exists at an extreme point. Hence we are looking for the basic feasible solution which optimizes the objective function. Suppose we have a problem with n = 25 original variables and m = 15 constraints, giving rise to 15 slack (or surplus) variables. In each basic feasible solution, 25 variables are set 40 equal to zero. There are possible sets of 25 variables which could be equated to 25 zero, which is more than 4.0 1010 combinations. Clearly an efficient method for choosing the sets of variables to set to zero is required! Suppose that we are able to find an initial bfs, say x1 , we have a way of checking whether a bfs xi is optimal, and we have a way of moving from a non-optimal bfs xi to another, xi+1 , that yields a better value of the objective function. Combining these three separate steps will yield an algorithm for solving the LP problem. If x1 is optimal, then we are done. If it is not, then we move from x1 to x2 , which is better than x1 by definition. If x2 is not optimal then we move to x3 and so on. Since the number of extreme points is finite and we always move towards a better one, we must ultimately find the optimal one. The Simplex method is based on this principle. 14 2.2 The Simplex algorithm Consider the Linear Programming problem: Maximize z = 12x1 + 15x2 subject to 5x1 + 6x2 85, 2x1 + 3x2 40 and x1 0, x2 0. To use the Simplex method we must first write the problem in standard form: Maximize subject to and z = 12x1 + 15x2 5x1 + 6x2 + x3 = 85 2x1 + 3x2 + x4 = 40 x1 0, x2 0. (1) (2) The two equations in four unknowns have infinitely many solutions. Setting any two of x1 , x2 , x3 , x4 to zero gives a unique solution for the other two, hence a basic solution of the problem. If we take x1 = 0, x2 = 0, x3 = 85, x4 = 40 we certainly have a basic feasible solution; since it gives z = 0 it is clearly not optimal. z can be made larger by increasing x1 or x2 . 1 Equation (2) gives x2 = (40 - 2x1 - x4 ). 3 Now express z in terms of x1 and x4 only: z = 12x1 + 15x2 = 12x1 + 5(40 - 2x1 - x4 ) = 200 + 2x1 - 5x4 . If we let x1 = x4 = 0 we find z = 200. This is a great improvement on 0, and corresponds 40 to increasing x2 to so that the second constraint holds as an equation. We have moved 3 from the origin to another vertex of the feasible region. Now x3 = 5, so there are still 5 units of slack in the first constraint. All the xj are 0 so we still have a bfs. z can be improved further by increasing x1 . Eliminating x2 between the constraint equations: (1) - 2 (2) gives x1 = 5 - x3 + 2x4 . We then have z = 200 + 2x1 - 5x4 = 200 + 2(5 - x3 + 2x4 ) - 5x4 = 210 - 2x3 - x4 . As all the variables are non-negative, increasing x3 or x4 above zero would make z smaller than 210. Hence the maximum value of z is 210 and this occurs when x3 = x4 = 0, i.e. there is no slack in either constraint. Then x1 = 5, x2 = 10. We have moved round the feasible region to the vertex where the two constraint lines intersect. The above working can be set out in an abbreviated way. The objective function is written as z - 12x1 - 15x2 = 0. This and the constraints are three equations in five unknowns z, x1 , x2 , x3 , x4 . This system of linear equations holds at every feasible point. We can obtain an equivalent system of equations by combining these equations together, so long as the resulting equations remain linearly independent. This can be carried out most easily by writing the equations in the form of an augmented matrix and carrying out row operations, as in Gaussian elimination. This matrix is written in a way which helps us to identify the basic variables at each stage, called a simplex tableau. Eventually we should get an expression for the objective function of the form z = k- j xj where each j 0, such as z = 210 - 2x3 - x4 in the example above. Then increasing any of the xj will decrease z, so we have arrived at a maximum value of z. 15 A simplex tableau consists of a grid with headings for each of the variables and a row for each equation, including the objective function. Under the heading `Basic' are the variables which yield a basic feasible solution when they take the values in the `Solution' column and the others are all zero. The `=' sign comes immediately before the `Solution' column. You will find various forms of the tableau in different books. Some omit the z column and/or the `Basic' column. The objective function is often placed at the top. Some writers define the standard form to be a minimizing rather than a maximizing problem. For the problem on the previous page, the initial tableau is: Basic x3 x4 z z 0 0 1 x1 5 2 -12 x2 6 3 -15 x3 1 0 0 x4 0 1 0 Solution 85 40 0 This tableau represents the initial basic feasible solution x1 = x2 = 0, x3 = 85, x4 = 40 giving z = 0. Hence x3 and x4 are the basic variables at this stage. The negative values in the z row tell us that this solution is not optimal: z -12x1 -15x2 = 0 so z can be made larger by increasing x1 or x2 from 0. Increasing x2 is likely to give the best increase in z, as the largest coefficient in z is that of x2 . We carry out row operations on the tableau so as to make x2 basic. One of the entries in the x2 column must become 1 and the others must become 0. The right-hand sides must all remain non-negative. Thus we choose the entry `3' as the `pivot'. Divide Row 2 by 3 to make the pivot 1. Then combine multiples of this row (only) with each of the other rows so that all other entries in the pivot column become zero: Basic x3 x2 z z 0 0 1 x1 1 2/3 -2 x2 0 1 0 x3 1 0 0 x4 -2 1/3 5 Solution 5 40/3 200 R1 := R1 - 2R2 R2 := R2 /3 R3 := R3 + 5R2 40 , x3 = 5, x4 = 0. x4 has left the basis 3 (it is the `departing variable') and x2 has entered the basis (it is the `entering variable'). This represents the bfs z = 200 when x1 = 0, x2 = Now the bottom row says z - 2x1 + 5x4 = 200 so we can still increase z by increasing x1 . Thus x1 must enter the basis. The entry `1' in the top left of the tableau is the new pivot, and all other entries in its column must become 0. x3 leaves the basis. Basic x1 x2 z z 0 0 1 x1 1 0 0 x2 0 1 0 x3 1 -2/3 2 x4 -2 5/3 1 Solution 5 10 210 2 R2 := R2 - 3 R1 R3 := R3 + 2R1 Now there are no negative numbers in the z row, which says z + 2x3 + x4 = 210. Increasing a non-basic variable from 0 cannot now increase z, so we have the optimal value z = 210 when x3 = x4 = 0. The tableau shows that x1 = 5, x2 = 10 are the values at which the maximum occurs. 16 The rose-growing problem revisited Consider the rose-growing problem from Chapter 1. Step 1 Write the problem in standard form. Maximise z = 2x1 + 3x2 = 6100 5x1 + 4x2 + x3 8x1 + 2x2 + x4 = 8000 subject to x1 + 5x2 + x5 = 5000 and xj 0 for j = 1, . . . , 5. The slack variables x3 , x4 , x5 represent the amount of spare area, finance and labour that are available. The extreme points of the feasible region are: Extreme point O A B C D x1 0 0 500 900 1000 x2 0 1000 900 400 0 x3 6100 2100 0 0 1100 x4 8000 6000 2200 0 0 x5 5000 0 0 2100 4000 Objective function z 0 3000 3700 3000 2000 The boundary lines each have one of x1 , . . . , x5 equal to zero. Hence the vertices of the feasible region, which occur where two boundary lines intersect, correspond to solutions in which two of the five variables in (LP2) are zero. 1200 1000 800 x2 600 400 200 0 200 400 600 x1 800 1000 1200 The simplex method systematically moves around the boundary of this feasible region, starting at the origin and improving the objective function at every stage. Step 2 Form the initial tableau. Basic x3 x4 x5 z z 0 0 0 1 x1 5 8 1 -2 x2 4 2 5 -3 x3 1 0 0 0 x4 0 1 0 0 x5 0 0 1 0 Solution 6100 8000 5000 0 Notice that the bottiom row comes from writing the objective function as z - 2x1 - 3x2 = 0. 17 The tableau represents a set of equations which hold simultaneously at every feasible solution. The `=' sign in the equations occurs immediately before the solution column. There are 5 - 3 = 2 basic variables. Each column headed by a basic variable has entry 1 in just one row and 0 in all the other rows. In the above tableau, x1 and x2 are non-basic; if they are set equal to zero, an initial basic feasible solution can be read directly from the tableau: x3 = 6100, x4 = 8000 and x5 = 5000, giving z = 0. Step 3 Test for optimality. Are all the coefficients in the z-row non-negative? 1. If yes, then stop the optimal solution has been reached. 2. If no, then go to Step 4. The bottom row of the above tableau says z - 2x1 - 3x2 = 0, so clearly z can be increased by increasing x1 or x2 . Step 4 Choose the variable to enter the basis: the entering variable. This will be increased from its current value of 0. z = 2x1 + 3x2 , so we should be able to increase z the most if we increase x2 from 0. (The rate of change of z is greater with respect to x2 ; this is called a steepest ascent method.) Thus we choose the column with the most negative entry in the z-row. Suppose this is in the column headed xj . Then xj is the entering variable. Here the entering variable is x2 , as this has the most negative entry (-3) in the z-row. Step 5 Choose the variable to leave the basis: the departing variable. This will be decreased to 0 from its current value. Column j now needs to become `basic', so that it has 1 in some row i and 0 in every other row. To avoid making the right-hand side negative, we need aij > 0. If there are no positive entries in column j then stop the problem is unbounded and has no solution. Otherwise a multiple of row i must be added to every other row k so as to make the entries in column j zero. The operations on rows i and k must be: bi Ri := a1 Ri 1 aij bi aij ij - a a Rk := Rk - akj Ri for all k = i akj bk 0 bk - akj bi ij ij To keep all the right-hand sides non-negative requires bk - this certainly holds if akj 0. If akj > 0 then the condition gives akj bi 0 for all k. Since aij > 0, aij bi bk for all k, so row i has to be the row in which aij akj bi aij > 0 and the row quotient i = is minimum. This row is the pivot row. aij The element aij in the selected row and column is called the pivot element. The basic variable corresponding to this row is the departing variable. It becomes 0 at this iteration. Step 6 Form a new tableau. Divide the entire pivot row by the pivot element, to obtain a 1 in the pivot position. Make every other entry in the pivot column zero, including the entry in the z-row, by carrying out row operations in which multiples of the pivot row (only) are added to or subtracted from the other rows. Then go to Step 3. 18 Applying this procedure to the rose-growing problem, we have: Initial tableau Basic x3 x4 x5 z z 0 0 0 1 x1 5 8 1 -2 x2 4 2 5 -3 x3 1 0 0 0 x4 0 1 0 0 x5 0 0 1 0 Solution 6100 8000 5000 0 i 6100 4 = 1525 8000 2 = 4000 5000 5 = 1000 (smallest) The entering variable is x2 , as this has the most negative coefficient in the z-row. Calculating the associated row quotients i shows that x5 is the departing variable, i.e. the x5 row is the pivot row for the row operations. The pivot element is 5. Fill in the next two tableaux: Second tableau Basic z 0 0 0 1 z 0 0 0 1 x1 x2 x3 x4 x5 Solution i z Final tableau Basic x1 x2 x3 x4 x5 Solution z The coefficients of the non-basic variables in the z-row are both positive. This shows that we have reached the optimum make sure you can explain why! The algorithm now stops and the optimal values can be read off from the final tableau. zmax = 3700 when (x1 , x2 , x3 , x4 , x5 ) = (500, 900, 0, 2200, 0). Summary We move from one bfs to the next by taking one variable out of the basis and bringing one variable into the basis. The entering variable is chosen by looking at the numbers in the z-row of the current tableau. If they are all non-negative then we already have the optimum value. Otherwise we choose the non-basic column with the most negative number in the z-row of the current tableau. The departing variable is determined (usually uniquely) by finding the basic variable which will be the first to reach zero as the entering variable increases. This is identified by finding the row with a positive entry in the pivot column such that the row quotient i is minimum. The version of the algorithm described here can be used only on problems which are in standard form. It relies on having an initial basic feasible solution. This is easy to find when all the constraints are of the `' type. However, in some cases such as the cattle feed problem this is not the case and a modification of the method is needed. 19 2.2.1 1. Further examples Minimize -2x1 - 4x2 + 5x3 , subject to x1 + 2x2 + x3 5, 2x1 + x2 - 4x3 6, 3x1 - 2x2 -3, xj 0 for j = 1, 2, 3. In standard form, the problem is: Maximize z = 2x1 + 4x2 - 5x3 , subject to x1 + 2x2 + x3 + x4 = 5, 2x1 + x2 - 4x3 + x5 = 6, Basic z 0 0 0 1 z 0 0 0 1 z 0 0 0 1 x1 x2 x3 x4 - 3x1 + 2x2 + x6 = 3, all xj 0. x5 x6 Solution z Basic x1 x2 x3 x4 x5 x6 Solution z Basic x1 x2 x3 x4 x5 x6 Solution z From the last tableau z = 10 - 7x3 - 2x4 - 0x6 , so any increase in a non-basic variable (x3 , x4 or x6 ) would decrease z. Hence this tableau is optimal. z has a maximum value of 10, so the minimum of the given function is -10 when x1 = 0.5, x2 = 2.25, x3 = 0. x5 = 2.75 shows that strict inequality holds in the second constraint: 2x1 + x2 - 4x3 falls short of 6 by 2.75. The other constraints are active at the optimum. 2. Maximize z = 4x1 + x2 + 3x3 , subject to - x2 + 2x3 4, x1 4, x2 4, xj 0 for j = 1, 2, 3. - x2 + 2x3 + x5 = 4, x1 + x6 = 4, x2 + x7 = 4. Basic z 0 0 0 0 1 z 0 0 0 0 1 x1 x2 x3 x4 x5 x6 x7 Solution x1 + x2 6, In standard form, the constraints become: x1 + x2 + x4 = 6, z Basic x1 x2 x3 x4 x5 x6 x7 Solution z 20 Basic z Basic z 0 0 0 0 1 z 0 0 0 0 1 x1 x2 x3 x4 x5 x6 x7 Solution x1 x2 x3 x4 x5 x6 x7 Solution z 5 3 3 The entries in the objective row are all positive; this row reads z + x4 + x5 + x6 = 27. 2 2 2 (Note that z is always expressed in terms of the non-basic variables in each tableau.) Thus z has a maximum value of 27 when x4 = x5 = x6 = 0 and x1 = 4, x2 = 2, x3 = 3, x7 = 2. Only one slack variable is basic in the optimal solution, so the at this optimal point the first three constraints are active they hold as equalities while the fourth inequality x2 4 is inactive. Indeed, x2 is strictly less than 4, by precisely 2 which is the amount of slack in this constraint. From the final tableau, each basic variable can be expressed in terms of the non-basic 1 1 1 1 1 1 variables, e.g. the x3 row says x3 + x4 + x5 - x6 = 3, so x3 = 3 - x4 - x5 + x6 . 2 2 2 2 2 2 2.3 Degeneracy When one (or more) of the basic variables in a basic feasible solution is zero, both the problem and that bfs are said to be degenerate. In a degenerate problem, the same bfs may correspond to more than one basis. This will occur when two rows have the same minimum quotient i . Selecting either of the associated variables to become non-basic results in the one not chosen, which therefore remains basic, becoming equal to zero in the next basic feasible solution. Degeneracy reveals that the LP problem has at least one redundant constraint. The problem of degeneracy is easily dealt with in practice; just keep going! If two or more of the row quotients i are equal, pick any one of them and proceed to the next tableau. There will then be a 0 in the `Solution' column. Follow the usual rules: the minimum i may be 0, so long as aij > 0. Even if the value of z does not increase, a new basis has been found and we should eventually get to the optimum. Example: Consider the following problem: Maximize z = 3x1 - x2 2x1 - x2 4 x1 - 2x2 2 subject to x1 + x2 5 and xj 0, j = 1, 2. 21 Degeneracy Graphically, we see that the optimum occurs when x1 = 3, x2 = 2. The vertex (2, 0) has three lines passing through it. Since, in 2-dimensions, only two lines are needed to define an extreme point, this point is overdetermined and one of the constraints is redundant. In standard form, the problem is: x2 0 0 1 2 3 4 5 6 1 2 3 x1 4 5 6 Maximize z = 3x1 - x2 2x1 - x2 + x3 = 4 x1 - 2x2 + x4 = 2 subject to x1 + x2 + x5 = 5 and xj 0 for j = 1, . . . , 5. Basic x3 x4 x5 z z 0 0 0 1 x1 2 1 1 -3 x2 -1 -2 1 1 x3 1 0 0 0 x4 0 1 0 0 x5 0 0 1 0 Solution 4 2 5 0 i 2 2 5 The entering variable must be x1 . The departing variable could be either of x3 or x4 . We will arbitrarily choose x4 to depart. The next tableau is: Basic x3 x1 x5 z z 0 0 0 1 x1 0 1 0 0 x2 3 -2 3 -5 x3 1 0 0 0 x4 -2 1 -1 3 x5 0 0 1 0 Solution 0 2 3 6 i 0 1 Now x3 is basic but takes the value x3 = 0, so we are at a degenerate bfs. The algorithm has not finished as there is a negative entry in the z-row. The next iteration gives: Basic x2 x1 x5 z z 0 0 0 1 x1 0 1 0 0 x2 1 0 0 0 x3 1/3 2/3 -1 5/3 x4 -2/3 -1/3 1 -1/3 x5 0 0 1 0 Solution 0 2 3 6 i 3 This solution is degenerate again, and the objective function has not increased. There is still a negative entry in the z-row. Pivoting on the x5 row gives: Basic x2 x1 x4 z z 0 0 0 1 x1 0 1 0 0 x2 1 0 0 0 x3 -1/3 1/3 -1 4/3 x4 0 0 1 0 x5 2/3 1/3 1 1/3 Solution 2 3 3 7 The algorithm has now terminated. zmax = 7 when x1 = 3, x2 = 2. 22 Both the second and third tableaux represent the point (2, 0, 0, 0, 3). The only difference is that the decision variables are classified differently as basic and nonbasic at the two stages. In a degenerate problem, it is possible (though unlikely) that the Simplex algorithm could return at some iteration to a previous tableau. Once caught in this cycle, it will go round and round without improving z. Several methods have been proposed for preventing cycling. One of the simplest is the `smallest subscript rule' or `Bland's rule', which states that the simplex method will not cycle provided that whenever there is more than one candidate for the entering or leaving variable, the variable with the smallest subscript is chosen (e.g. x3 in preference to x4 ). 2.4 Theory of the Simplex method We now make some observations which provide more insight into how and why the Simplex algorithm works. In the rose growing problem we had the initial and final tableaux: Initial tableau Basic x3 x4 x5 z Final tableau Solution 500 2200 900 3700 z 0 0 0 1 x1 5 8 1 -2 x2 4 2 5 -3 x3 1 0 0 0 x4 0 1 0 0 x5 0 0 1 0 Solution 6100 8000 5000 0 In the initial tableau, the 3 3 identity matrix appears below the slack variables. In 5/21 0 -4/21 the final tableau another 3 3 matrix M = -38/21 1 22/21 appears in this -1/21 0 5/21 position. The middle three rows of the initial tableau have been pre-multiplied by M to give these rows in the final tableau. M tells us the row operations that would convert the middle rows of the initial tableau 38 22 directly to those in the final tableau, e.g. R2 := - R1 + R2 + R3 . This applies 21 21 500 6100 to all columns including the `solution' column, where M 8000 = 2200 . 900 5000 In the final tableau, the columns of the identity matrix occur under x1 , x4 , x2 respectively. If we the corresponding columns of the initial tableau together to form a put 5 0 4 matrix B = 8 1 2 then MB = I so M = B-1 . 1 0 5 23 In the bottom row of the tableau, the initial row of zeros under the slack variables has 1 1 become 0 in the final tableau. This row vector multiplies each column in 3 3 the middle section of the initial tableau to give the values that have been added to 4 1 1 2 the bottom row of the final tableau, e.g. 0 + (-3) = 0. 3 3 5 5/21 0 -4/21 0 -38/21 1 22/21 0 The entire initial tableau is pre-multiplied by the matrix P = -1/21 0 5/21 0 1/3 0 1/3 1 to give the final tableau. To generalize this, suppose we are maximizing z = ct x subject to Ax b (where b 0) and x 0. The initial tableau has the following form, where Im is the m m identity matrix. Main variables x1 xn A -c1 -cn Slack variables xn+1 xn+m Im 0 0 Basic xn+1 . . . xn+m z z 0 . . . 0 1 Solution b1 . . . bm 0 Any subsequent tableau may be represented as follows, for a suitable matrix B: Main variables x1 xn B-1 A d1 dn Slack variables xn+1 xn+m B-1 y1 ym Basic z 0 . . . 0 1 Solution B-1 b z z When this tableau is optimal, z = zmax and d1 , . . . , dn , y1 , . . . , ym are all 0. If we put the `z' column immediately before the solution column then, where the (m 1) + 0 . . B-1 . . (m + 1) identity matrix was initially, there would now be P = 0 y1 ym 1 Let yt = (y1 ym ). The entire initial tableau is pre-multiplied by P to give the new tableau: B-1 yt 0 1 A -ct I 0t b = 0 B-1 A yt A - ct B-1 yt B-1 b . tb y We see that z = yt b, which can also be written as bt y. 24 For optimality we must have y 0 and yt A - ct 0, i.e. At y c. We shall meet these conditions again when we study the dual of a LP problem. Now suppose that at a particular stage k of the Simplex algorithm the basic variables, reading down the list in the `Basic' column, are xB1 , . . . , xBm and the non-basic variables are xN 1 , . . . , xN n . The initial tableau can be rearranged as follows: Stage k basic variables xB1 xBm B -cB1 -cBm Stage k non-basic variables xN 1 xN n N -cN 1 -cN n Basic xn+1 . . . xn+m z z 0 . . . 0 1 Solution b 0 Thus the objective function can be written as z = cB1 xB1 + + cBm xBm + cN 1 xN 1 + + cN n xN n . Let cB = (cB1 cBm )t , cN = (cN 1 cN n )t . With the columns arranged in this order, the process of transforming the initial tableau to the stage k tableau is: B-1 yt 0 1 B -cB t N -cN t b = 0 I y t B - cB t B-1 N yt N - cN t B-1 b tb y We must have zeros at the bottom of the basic columns, so yt B - cB t = 0. As B is non-singular, we can rewrite this equation as yt = cB t B-1 . Every entry in the z-row is positive at the optimum, so yt N - cN t 0, i.e. cB t B-1 N - cN t 0. Then zmax = yt b = cB t B-1 b. 2.4.1 Example Consider the LP problem: Maximize z = 2x1 + 4x2 - 5x3 , subject to x1 + 2x2 + x3 5, 2x1 + x2 - 4x3 6, The initial tableau is Basic x4 x5 x6 z The optimal tableau is Basic x1 x5 x2 z z 0 0 0 1 x1 x2 x3 x4 x5 x6 1 0 1/4 1/4 0 -1/4 0 0 -39/8 -7/8 1 3/8 0 1 3/8 3/8 0 1/8 0 0 7 2 0 0 25 Solution 1/2 11/4 9/4 10 z 0 0 0 1 x1 x2 x3 x4 x5 x6 1 2 1 1 0 0 2 1 -4 0 1 0 -3 2 0 0 0 1 -2 -4 5 0 0 0 Solution 5 6 3 0 - 3x1 + 2x2 3. The basic variables reading down the `Basic' column are x1 , x5x2 , so B is formed from , 1 0 2 columns 1, 5, 2 respectively of (A \| I3 ). Thus B = 2 1 1 . -3 0 2 2 0 -2 x1 1/2 1 3 . At the optimal point x5 = B-1 b = 11/4 . B-1 = -7 8 8 x2 3 0 1 9/4 2 1/2 cB = 0 , so zmax = cB t B-1 b = 2 0 4 11/4 = 10. 4 9/4 1 1 0 -4 0 0 , from columns 3, 4, 6 of (A \| I3 ), and cN = (-5 0 0). We can take N = 0 0 1 t B-1 N - c t = (7 2 0), the z-row entries in the non-basic columns. Then cB N 1 9 11 The maximum value of z is 10, when x1 = , x2 = , x3 = 0, x4 = 0, x5 = . 2 4 4 We have been assuming that only extreme points of the feasible region need to be considered as possible solutions. The proof of this is now given; you are not expected to learn it. Proposition 2.1 v is a basic feasible solution of (LP2) if and only if v is an extreme point ~x ~ of the set of feasible solutions S = {~ : A~ = b, x 0} in Rn+m . x Proof () Suppose v is a bfs of (LP2) and is distinct feasible solutions u, w S such that, for ~ We can permute the columns of A, as described vB entries of v, u, w correspondingly so that 0 not an extreme point of S, so there are some r (0, 1), v = (1 - r)u + rw. earlier, to get (B \| N) and permute the uB wB = (1 - r) +r uN wN As 0 < r < 1 and u, w are non-negative, it follows that uN = wN = 0. ~ ~ ~ As v, u, w are feasible, Av = Au = Aw = b, so (B \| N) vB 0 = (B \| N) uB uN = (B \| N) wB wN = b. Thus BvB = BuB = BwB = b. As B is non-singular, vB = uB = wB = B-1 b. Hence v = u = w, contradicting the assumption that these points are distinct, so v is an extreme point. () Suppose v is in S, so is feasible, but is not a bfs of (LP2). ~ Av = b, so we can write (B \| N ) v = b, where v contains all the non-zero entries of 0 v. If v = v then B is not unique, but the following reasoning applies to any choice of B . If B were non-singular then v = B-1 b, in which case v would be a bfs. Hence B is singular, so there is a vector p such that B p = 0. As the entries of v are strictly positive, we can find > 0 so that v - p 0 and v p v + p 0. Then B (v p) = B v A p = B v 0 = b, so (B \| N ) = b. 0 26 v v p are permuted feasible solution vectors. But is their 0 0 mean, so v is a convex linear combination of two points in S, hence v is not an extreme point of S. It follows that any extreme point of S is a bfs of (LP2). Thus both of 2.5 Complexity of algorithms Although we are doing calculations manually in order to understand the method, in practice the simplex algorithm is designed to be implmented on a computer. The development of more efficient algorithms is a key feature of research in OR. Suppose an algorithm is used on a problem in which there are n of some feature (e.g. variables, constraints). We say the algorithm is O(f(n)), and call f(n) the complexity or order of the algorithm, if the number of arithmetic operations required in finding an optimal solution is bounded above by some multiple of f(n). For example, if an algorithm requires up to 3n3 + 2n2 + 5 operations then we say it is O(n3 ), as for large enough n we have 3n3 + 2n2 + 5 4n3 . Such an algorithm has cubic complexity. If an algorithm is O(na ) for some a, we say it has polynomial complexity, or the problem can be solved in polynomial time. If it is O(an ) for some a then it has exponential complexity. The complexity can be thought of as a measure of the cost involved in implementing an algorithm (e.g. the cost of computer time). Clearly polynomial is preferable to exponential complexity, because the latter increases far more steeply as n increases. In practice, the simplex algorithm typically requires about 2m iterations to solve a problem with m constraints. Taking the number of variables, n, into account, the number of operations per iteration is polynomial in m and n; in fact, it can be shown that it is O(m3 + mn). The algorithm is more sensitive to the number of constraints than the number of variables, so it is best to keep the number of constraints as small as possible by eliminating any redundant ones. However, in the worst case a problem with n variables and n constraints can require 2n - 1 iterations. This is proved by constructing a problem with 2n possible basic feasible solutions and showing that the simplex algorithm, implemented according to the standard rules, works through every bfs before reaching the optimum. Thus the algorithm actually has exponential order. This result is due to Klee and Minty (1972). A version of their problem is given in the exercises. Problems that require this maximum number of operations are rare, but the fact that they exist has prompted research into other ways of solving LP problems. One such is Karmarkar's interior point algorithm. Instead of starting at an extreme point, this starts at a point inside the feasible region and works out towards the optimal vertex. This algorithm has polynomial complexity and can be more efficient for problems with a very large number of variables. 27 Exercises 2 1. Use the Simplex method to solve Maximize subject to and z = 4x1 + 8x2 5x1 + x2 8 3x1 + 2x2 4 x1 0, x2 0. Express objective the function in terms of the non-basic variables in the final tableau, and hence explain how you know that the solution is optimal. 2. Use the Simplex method to solve Maximize z = 5x1 + 4x2 + 3x3 2x1 + 3x2 + x3 5 4x1 + x2 + 2x3 11 subject to 3x1 + 4x2 + 2x3 8 and xj 0 for j = 1, 2, 3. Write down a matrix P that pre-multiplies the whole initial tableau to give the final tableau. After you have studied Section 2.4 in the notes: write down, from the optimal tableau of this problem, the matrices B, B-1 and N and the vectors y, cB , cN as defined on pages 24 - 25. Verify that the optimal solution is B-1 b and that zmax = cB t B-1 b. 3. Express the following LP problem in standard form. (Do not solve it.) Minimize z = 2x1 + 3x2 x1 + x2 -1 2x1 + 3x2 5 3x1 - 2x2 3 subject to and x1 unrestricted in sign, x2 0. 4. The following tableau arose in the solution of a LP problem: Basic z x1 x2 x3 x4 x5 x6 x7 0 0 4/3 2/3 0 1 0 -1/3 0 0 1/3 2/3 1 0 1 -1/3 0 1 -1/3 1/6 1/2 0 0 1/6 z 1 0 -5/3 -4/3 -1 0 0 5/3 Solution 4 10 4 12 (a) Which variables are basic? What basic feasible solution does the tableau represent? Is it optimal? Explain your answer by writing the z-row as an equation. (b) Starting from the given tableau, proceed to find an optimal solution. (c) From the optimal tableau, express the objective function and each of the basic variables in terms of the non-basic variables. 28 5. A furniture manufacturer makes chairs and settees, producing up to 80 chairs and 48 settees per week. The items are sold in suites: Mini - two chairs, Family - three chairs and one settee, Grand - three chairs and two settees. The profits are 20, 30 and 70 per suite, respectively. The total profit is to be maximized. Use the Simplex algorithm to find the maximum profit that can be made in one week. State the profit and the number of each type of suite that should be made. 6. Suppose one of the constraints in a LP problem is an equation rather than an inequality, as in Minimize 3x1 + 2x2 + 4x3 3x1 + 4x2 + 2x3 6 3x1 + 5x2 + 7x3 10 subject to x1 + x2 + x3 = 1 and x1 0, x2 0. Use the third constraint to express x3 in terms of x1 and x2 . Hence eliminate x3 from the objective function and the other two constraints. Introduce two slack variables x4 and x5 , and solve the problem by the Simplex algorithm. Check that your solution satisfies all the constraints. How must this approach be modified if also x3 0? 7. Solve the following problem using the simplex tableau method, making your decision variable for I1 basic at the first iteration. Identify any degenerate basic feasible solutions your calculations produce. Sketch the solution space and explain why degeneracy occurs. A foundry produces two kinds of iron, I1 and I2 , by using three raw materials R1 , R2 and R3 . Maximise the daily profit. Raw material R1 R2 R3 Profit per tonne Amount required per tonne of I1 of I2 2 1 1 1 0 1 150 300 Daily raw material availability (tonnes) 16 8 3.5 8. Prove that v is an extreme point of the feasible region R for (LP1) if and only if v is an extreme point of the feasible region S for (LP2). b - Av 9. (Klee-Minty problem) Solve the following by the simplex algorithm, showing that every possible basic feasible solution occurs in the iterations. Maximize subject to z = 100x1 + 10x2 + x3 x1 1 20x1 + x2 100 200x1 + 20x2 + x3 10000 x1 0, x2 0, x3 0. and 29 Chapter 3 Further Simplex Methodology 3.1 Sensitivity analysis Suppose some feature of a LP problem changes after we have found the optimal solution. Do we have to solve the problem again from scratch or can the optimum of the original problem be used as an aid to solving the new problem? These questions are addressed by sensitivity analysis, also called post-optimal analysis. If a resource is used up completely then the slack variable in the constraint representing this resource is zero, i.e. the constraint is active at the optimal solution. This type of resource is called scarce. In contrast, if the slack variable is non-zero (i.e. the constraint is not active) then this resource is not used up totally in the optimal solution. Resources of this kind are called abundant. Increasing the availability of an abundant resource will not in itself yield an improvement in the optimal solution. However, increasing a scarce resource will improve the optimal solution. With the notation of Chapter 2, zmax = yt b. Suppose bi is increased by a small amount bi . We say that the ith constraint has been relaxed by bi . Then as long as the same variables remain basic at the optimal solution, zmax = y1 b1 + + yi (bi + bi ) + + ym bm , i.e. zmax has increased by yi bi . yi is thus the approximate increase in the optimum value of z that results from allowing an extra 1 unit of the ith resource. yi is called the shadow price of the ith resource. It does not tell us by how much the resource can be increased while maintaining the same rate of improvement. If the set of constraints which are active at the optimal solution changes, then the shadow price is no longer applicable. The shadow prices yi can be read directly from the optimal tableau: they are the numbers in the z-row at the bottom of the slack variable columns. 3.1.1 Example: the rose-growing problem modified To illustrate these concepts we again consider the rose-growing problem, with the initial and final tableaux as obtained in Chapter 2. The optimal solution is (500, 900, 0, 2200, 0). The slack variables x3 , x4 and x5 were introduced in the land, finance and labour resource constraints respectively. Thus land and labour are scarce resources, and finance is an abundant resource. We now consider various modifications to the problem and its solution. 30 1. Shadow prices x3 x4 and x5 have coefficients 1/3, 0 and 1/3 in the z-row of the final tableau, so these are the shadow prices of land, finance and labour respectively. The shadow prices give the rate of improvement of z with each resource, within certain bounds. 1 The maximum profit z increases at the rate of 3 per extra unit of land or labour. The shadow price of finance is zero it is an abundant resource at the optimum. Labour is a scarce resource so making extra labour available will increase profit. The shadow price of labour is 1 , which means that 1 extra person-hour of labour would 3 result in the profit increasing by 0.33. Hence if more than 0.33 per hour is paid in wages, the grower will be worse off. They would certainly have to pay more than this so it is not economical to hire extra labour. 2. Changing the resources The rose grower has the option to buy some more land. What is the maximum area that should be purchased if the other constraints remain the same? Suppose extra land of area k dm2 was purchased. Then the initial tableau would be the same as before except that b1 = 6100 + k in the `solution' column. Thus the `solution' column in the 5/21 0 -4/21 -38/21 1 22/21 -1/21 0 5/21 1/3 0 1/3 optimal tableau becomes 5 0 6100 + k 500 + 21 k 38 0 8000 2200 - 21 k = 1 0 5000 900 - 21 k 1 1 0 3700 + 3 k . Keeping keeping the `solution' column non-negative requires x1 0, x2 0, x4 0, so k 2200 21/38 1215.8. Thus the maximum area the grower should consider buying is 1215.8 dm2 . At this point all the finance is used up as well as all the land and labour, and any extra increase in land will not improve the optimal profit. We have seen that the shadow price of land is 1 . Thus the maximum profit when this 3 extra land is added will be 3700 + 1 1215.8 4105.27. 3 3. Changing a coefficient in the objective function The profit margin alters on one of the roses. Is the same solution still optimal? Changing the objective function does not alter the solution space, so cannot affect feasibility. However, it may affect whether the solution is still optimal. Suppose the profit on a red rose changes by t. The new objective function is z = (2 + t)x1 + 3x2 = z + tx1 . Thus the bottom row of the final tableau has -t in the x1 column. To get zero there we must add t times the top row to the z row, which becomes 1 \| 0 0 5t 1 + 3 21 0 1 4t - 3 21 \|\| 3700 + 500t . The solution remains optimal at x3 = x5 = 0 if none of these z-row entries is negative. Therefore the original solution is still optimal, with zmax = 3700 + 500t, provided 1 5t 1 4t 7 7 + 0 and - 0, i.e. - t . 3 21 3 21 5 4 3 15 Thus the profit 2 + t can vary between and . 5 4 31 4. Adding an extra variable. Suppose the grower has the option of growing an extra variety, pink roses, with the following requirements per bush: 7 dm2 area, 6 finance, and 3.5 hours labour. Let the profit per pink bush be t, and suppose that x6 pink rose bushes are grown. Original problem Maximize z = 2x1 + 3x2 5x1 + 4x2 + x3 8x1 + 2x2 + x4 x1 + 5x2 + x5 = = = 6100 8000 5000 Maximize New problem z = 2x1 + 3x2 + tx6 = 6100 = 8000 = 5000 subject to 5x1 + 4x2 + 7x6 + x3 8x1 + 2x2 + 6x6 + x4 subject to 7 x1 + 5x2 + 2 x6 + x5 and and xj 0, j = 1, . . . , 5. xj , xk 0, j = 1, 2, 6; k = 3, 4, 5. The initial tableau for the new problem is: Basic x3 x4 x5 z z 0 0 0 1 x1 5 8 1 -2 x2 4 2 5 -3 x3 1 0 0 0 x4 0 1 0 0 x5 0 0 1 0 x6 7 6 7/2 -t Solution 6100 8000 5000 0 The x6 column of the final tableau becomes: 5/21 0 -4/21 0 -38/21 1 22/21 0 -1/21 0 5/21 0 1/3 0 1/3 1 7 1 6 -3 = 7/2 1/2 -t 7/2 - t . Hence the current solution, which involves growing no pink roses at all, is optimal provided 7 - t 0, i.e. t 7 . 2 2 However, if t > 7 then this solution is no longer optimal. For example if t = 2 the original final tableau, with the x6 column added, is x6 1 -3 1/2 -1 Solution 500 2200 900 3700 9 2 then This is not optimal, but the optimum is obtained after one more Simplex iteration. 5. Adding an extra constraint. Suppose the maximum number of roses that can be sold is 1340, so the constraint x1 + x2 1340 is added to the system. The current optimum is not feasible as x1 + x2 = 500 + 900 > 1340. Adding a slack variable x6 gives x1 + x2 + x6 = 1340. Re-write the new constraint in terms of the current non-basic variables x3 and x5 , 500 - 5 4 x3 + x5 21 21 + 900 + 32 1 5 x3 - x5 21 21 + x6 = 1340 4 1 x3 - x5 + x6 = -60. In fact, this calculation can be done within the tableau: 21 21 if we add a row for x1 + x2 + x6 = 1340 into the previously optimal tableau we get: so - Basic x1 x4 x2 x6 z z 0 0 0 0 1 x1 1 0 0 1 0 x2 0 0 1 1 0 x3 5/21 -38/21 -1/21 0 1/3 x4 0 1 0 0 0 x5 -4/21 22/21 5/21 0 1/3 x6 0 0 0 1 0 Solution 500 2200 900 1340 3700 This is not a good simplex tableau as it does not have four basic variable columns. However, subtracting rows 1 and 3 from row 4 gives: Basic x1 x4 x2 x6 z z 0 0 0 0 1 x1 1 0 0 0 0 x2 0 0 1 0 0 x3 5/21 -38/21 -1/21 -4/21 1/3 x4 0 1 0 0 0 x5 -4/21 22/21 5/21 -1/21 1/3 x6 0 0 0 1 0 Solution 500 2200 900 -60 3700 Notice that the x6 row now contains the equation that we derived previously. This tableau would be optimal if the negative entry in the solution column were not there. We perform row operations keeping the z-row entries non-negative until all the numbers in the `Solution' column are non-negative. In the above, the basic variable x6 = -60, so we need x6 to leave the basis. How do we choose the entering variable? Our choice is restricted to the nonbasic variables that have negative coefficients in the pivot row, since we want to add a positive multiple of the pivot row to the z-row without creating any negative entries. To ensure this, choose the variable that yields the minimum absolute value (modulus) of the z-row coefficient divided by the pivot row entry aij . 4 1 In this case 3 - 21 < give the tableau 1 3 1 - 21 , so x3 is the entering variable. Row operations Basic x1 x4 x2 x3 z z 0 0 0 0 1 x1 1 0 0 0 0 x2 0 0 1 0 0 x3 0 0 0 1 0 x4 0 1 0 0 0 x5 -1/4 3/2 1/4 1/4 1/4 x6 5/4 -19/2 -1/4 -21/4 7/4 Solution 425 2770 915 315 3595 This solution is now optimal, as the z-row entries are non-negative and each bj 0. Introducing the new constraint reduces the optimum value of the objective function to 3595. This occurs when x1 = 425 and x2 = 915. The above procedure, keeping the z-row coefficients positive and carrying out row operations until the basic variables are non-negative, is called the dual simplex method . It can be used, on its own or in combination with the normal simplex method, on problems which are not quite in standard form because some of the right-hand sides are negative. The dual simplex method provides one way of solving problems for which no initial basic feasible solution is apparent. There are other variations on the simplex method which can be used for such purposes, such as the Two-Phase method (see a textbook). 33 3.1.2 Example: the garments problem A company makes three types of garment. The constraints are given by: Type A B C No. of units x1 x2 x3 Amount available Labour (hours) per unit 1 2 3 55 hours 2 ) per unit Material (m 3 1 4 80 m2 Profit () per unit 7 6 9 The company wants to make the largest possible profit subject to the constraints on labour and materials, so they must ................................. z = subject to and x1 0, x2 0, x3 0. Adding slack variables, the constraints become: Solve by the Simplex algorithm: Basic z x1 x2 x3 x4 x5 Solution z Basic z x1 x2 x3 x4 x5 Solution z Basic z x1 x2 x3 x4 x5 Solution z Basic z x1 x2 x3 x4 x5 Solution z so the maximum profit is when of type A, of type B and of type C are made. From the initial to the final tableau, the middle two rows are multiplied by B-1 = (using the columns for . This is the inverse of B = , respectively in the initial tableau.) 34 Taking cB = cB t B-1 b = , b= = , = zmax . We now consider the effect of changing some features of the problem. 1. Shadow prices The shadow prices of labour and material are Both resources are .................... The maximum profit increases at the rate of per extra hour of labour and per extra m2 of material, within certain limits. 2. Changing the resources Suppose p hours of labour and q m2 of material are available. The `solution' column in the initial tableau is . so in the final tableau it is and respectively. = For the solution to remain optimal when x1 > 0, x2 > 0, x3 = 0 we need 0 and q and p . Thus q , x2 = then zmax = , x3 = If p is kept at 55, this gives when x1 = When q = 0, so q Provided the amount of material is within this range, zmax = 3. Changing a coefficient in the objective function Suppose the profit on type A changes by t per unit. This does not alter the feasible region, but it may affect the optimal solution. The new objective function is z = ( )x1 + 6x2 + 9x3 . Thus the bottom row of the initial tableau is (1 \| ................ - 6 - 9 0 0 \|\| 0), so in the final tableau it is (1 \| .............. 0 4 11/5 8/5 \|\| 249). For x1 to remain basic we must have 0 at the bottom of the x1 column, so add ........ times Row ....... to Row ....... to get (1 \| .......... .......... .......... .............. .............. \|\| ......................). , provided t without affecting the optimal The original solution (21, 17, 0, 0, 0) is still optimal if no z-row entry is negative. Therefore the original solution is optimal, with zmax = 0, 0, 0, i.e. to The profit on type A can vary from quantities to produce. 35 4. Adding an extra variable. Suppose the company can produce a fourth garment, type D, requiring 4 hours of labour and 2 m2 of material and yielding k profit per unit. Let x0 units of type D be produced. The new problem is to maximize z = subject to = 55, where xj 0 for j = 0, . . . , 5. x0 The initial tableau is as before with the addition of a column 3/5 -1/5 0 -1/5 2/5 0 11/5 8/5 1 , so we have = 80, x0 in the final tableau. = Thus if k , the z-row remains non-negative and the current solution is optimal; D should not be made. Suppose we take k = 13. Then the previously optimal tableau becomes Basic x2 x1 z z 0 0 1 x0 x1 0 1 0 x2 1 0 0 x3 1 1 4 x4 3/5 -1/5 11/5 x5 -1/5 2/5 8/5 Solution 17 21 249 After one more iteration this becomes optimal: Basic z 0 0 1 units of D should be made. x0 x1 x2 x3 x4 x5 Solution z so units of A and (If half-units are not possible, Integer Programming is needed.) 5. Adding an extra constraint. Suppose that in the original 3-product problem there is a further restriction: 3x1 + x2 + 2x3 50, i.e. 3x1 + x2 + 2x3 + x6 = 50 where x6 0 is a slack variable. 3(21) + 17 + 2(0) > 50, so the previous optimal point (21, 17, 0) is not feasible. Adding the new constraint as a row in the original optimal tableau gives: Basic z 0 0 0 1 x1 0 1 0 x2 1 0 0 x3 1 1 4 36 x4 3/5 -1/5 11/5 x5 -1/5 2/5 8/5 x6 0 0 0 Solution 17 21 249 x6 z This does not represent a valid Simplex iteration because there is only one basic column. Subtracting Row 1 and 3 times Row 2 from Row 3 gives: Basic x2 x1 x6 z z 0 0 0 1 x1 0 1 0 x2 1 0 0 x3 1 1 4 x4 3/5 -1/5 11/5 x5 -1/5 2/5 8/5 x6 0 0 0 Solution 17 21 249 This represents a basic but infeasible solution. It would be optimal if the negative entry in the solution column were not there. In this situation we use the dual simplex method, as described on page 33. x6 = , so x6 must leave leave the basis. To keep the z-row entries non-negative, choose the entering variable as follows: Where there is a negative number in the pivot row, divide the z-row entry by this and choose the variable for which the modulus of this quotient is smallest. Here \| \| < \| \| so the entering variable is Basic x2 x1 z Making z 0 0 0 1 x1 x2 x3 x4 x5 x6 Solution of type A and of type B is now optimal, giving profit 3.1.3 The dual simplex method The differences between the original simplex method and the dual simplex method can be summarised as follows: Original (primal) simplex Starts from a basic feasible solution. All bj 0. At least one z-row entry is negative. Seek to make all z-row entries 0 Consider the problem: Minimize 5x1 + 4x2 subject to 3x1 + 2x2 6, x1 + 2x2 4, x1 0, x2 0. In standard form the problem would contain surplus variables, whose columns in the simplex tableau would contain -1. To use the dual simpled method we write the problem so that there are correct basic columns, even though the right-hand sides are negative: Maximize subject to and z = -5x1 - 4x2 -3x1 - 2x2 + x3 = -6 -x1 - 2x2 + x4 = -4 xj 0, j = 1, . . . , 4. 37 Dual simplex Starts from a basic infeasible solution. At least one bj < 0. All z-row entries 0. Seeks to make all bj 0 keeping the z-row 0. We start with a basic infeasible solution: Basic x3 x4 z z 0 0 1 x1 -3 -1 5 x2 -2 -2 4 x3 1 0 0 x4 0 1 0 Dual simplex algorithm: Solution Find the most negative number in the `solution' column. -6 cj For this row i find the smallest absolute value . -4 aij 0 Use aij as the pivot in the usual way. x4 0 1 0 Solution 2 -2 -10 Second Iteration Basic z x1 x2 x3 x4 Solution x1 0 1 0 -1/2 1/2 1 x2 0 0 1 1/4 -3/4 3/2 z 1 0 0 3/2 1/2 -11 This is an optimal, feasible tableau. First Iteration Basic z x1 x2 x1 0 1 2/3 x4 0 0 -4/3 z 1 0 2/3 This is still infeasible. x3 -1/3 -1/3 5/3 z = -5x1 - 4x2 has maximum value -11, so 5x1 + 4x2 has minimum value 11, when x1 = 1, x2 = 3/2. 3.2 The dual of a Linear Programming problem Recall the Containers problem from Chapter 1. Suppose now that the company delegates its production to a contractor who pays them y1 and y2 per minute for the use of machines M1 and M2 respectively. Let w be the total hourly charge for using the two machines. The contractor wants to make this hourly charge as small as possible, but must ensure that the company is paid at least as much as it originally made in profit for each container produced: 30 per Type A and 45 per Type B. Thus the contractor's problem is to minimize w = 60y1 + 60y2 subject to the constraints 2y1 + 4y2 30, 8y1 + 4y2 45, where y1 0, y2 0. The feasible region lies in the first quadrant above the boundary lines, as illustrated: 14 12 10 y2 8 6 4 2 0 2 4 6 8 10 y1 12 14 16 18 w is minimum where the lines cross, at (2.5, 6.25). Here w = 60 2.5 + 60 6.25 = 525. Thus the contractor should pay 2.50 per minute for M1 and 6.25 per minute for M2 , so that the company gets 525 per hour the same as the profit when it made the containers itself! We have solved the dual of the original problem. Every linear programming problem has an associated problem called its dual. For now we will restrict attention to pairs of problems of the following form: Primal: maximize z = ct x subject to Ax b and x 0 Dual: minimize w = bt y subject to At y c and y 0 (P) (D) To obtain the dual problem from the primal problem we swap c and b, replace A by its transpose At , replace `' with `' in the constraints, and replace `maximize' with `minimize'. The non-negativity restrictions remain. 38 3.2.1 Example The following is a primal-dual pair of LP problems: Primal Maximize subject to and z = 6x1 + 4x2 3x1 + x2 2x1 + 2x2 5 4 Minimize subject to and Dual w = 5y1 + 4y2 3y1 + 2y2 y1 + 2y2 6 4 x1 , x2 0. y1 , y2 0. The primal problem can be solved easily using the standard simplex algorithm. The optimal solution is zmax = 11 when (x1 , x2 , x3 , x4 ) = (3/2, 1/2, 0, 0). We solved the dual problem by the dual simplex algorithm in Section 3.1.3. The optimal solution is wmin = 11 when (y1 , y2 , y3 , y4 ) = (1, 3/2, 0, 0). Notice that the objective functions of the primal and dual problems have the same optimum value. Furthermore, in the optimal dual tableau, the objective row coefficients of the slack variables are equal to the optimal primal decision variables. Consider the cattle-feed problem in Chapter 1. Suppose a chemical company offers the farmer synthetic nutrients at a cost of y1 per unit of protein, y2 per unit of fat, y3 per unit of calcium and y4 per unit of phosphorus. The cost per unit of hay substitute is thus (13.2y1 +4.3y2 +0.02y3 +0.04y4 ). To be economic to the farmer, this must not be more than 0.66. Similarly, considering the oats substitute, 34.0y1 + 5.9y2 + 0.09y3 + 0.09y4 2.08. For feeding one cow, the company will receive (65.0y1 + 14.0y2 + 0.12y3 + 0.15y4 ), which it will wish to maximize. Thus the company's linear programming problem is : Maximize subject to and w = 65.0y1 + 14.0y2 + 0.12y3 + 0.15y4 13.2y1 + 4.3y2 + 0.02y3 + 0.04y4 0.66 34.0y1 + 5.9y2 + 0.09y3 + 0.09y4 2.08 yj 0, j = 1, ..., 4. We see that the dual of this is the farmer's original problem. As we shall show next, in fact the two problems are the duals of each other. Proposition 3.1 The dual of the dual problem (D) is the primal problem (P). Proof The dual problem (D) can be written as follows: maximize (-b)t y subject to (-A)t y -c and y 0. The dual of this is: minimize (-c)t x subject to (-At )t x -b and x 0. 39 This is equivalent to: maximize ct x subject to Ax b and x 0, which is the same as the primal problem (P). Thus the dual of either problem may be constructed according to the following rules: Primal Maximize ct x Minimize ct x Constraints Ax b Constraints Ax b x0 Dual Minimize bt y Maximize bt y Constraints At y c Constraints At y c y 0. For every primal constraint there is a dual variable. For every primal variable there is a dual constraint. the constraint coefficients of a primal variable form the left-side coefficients of the corresponding dual constraint; the objective coefficient of the same variable becomes the right-hand side of the dual constraint. We now investigate how the solutions of the primal and dual problems are related, so that by solving one we automatically solve the other. Proposition 3.2 (The weak duality theorem) Let x be any feasible solution to the primal problem (P) and let y be any feasible solution to the dual problem (D). (i) ct x bt y. (ii) If ct x = bt y then x and y are optimal solutions to the primal and dual problems. Proof (i) We have x 0, y 0, Ax b, At y c. Thus ct x (At y)t x = yt Ax yt b = bt y. (ii) From (i) no primal feasible x can give ct x > bt y, so if ct x = bt y then x maximizes z. A similar argument shows that y minimizes w. Proposition 3.3 (The strong duality theorem) If either the primal or dual problem has a finite optimal solution, then so does the other, and the optimum values of the primal and dual objective functions are equal, i.e. zmax = wmin . Proof Let x be a finite optimal solution to the primal, so that zmax = ct x = z say. We have seen that the initial tableau is pre-multiplied as follows to give the final tableau: B-1 yt 0 1 A -ct I 0t B-1 A yt A - ct B-1 yt B-1 b . tb y b = 0 40 As this is optimal, yt A - ct 0, so At y c, and y 0. Hence y is feasible for the dual problem. Now z = yt b = bt y. But also z = ct x so x, y are feasible solutions which give equal values of the primal and dual objective functions respectively. Thus by Proposition 3.2 (ii), x, y are optimal solutions and z is the optimal value of both z and w. As each problem is the dual of the other, the same reasoning applies if we start with a finite optimal solution y to the dual. The above shows that the entries of y are in fact the optimal values of the main variables in the dual problem. (They are also the shadow prices for the primal constraints). Furthermore the entries of yt A - ct are the values of the dual surplus variables at the optimum, so we shall denote them by ym+1 , . . . , ym+n . Thus the optimal primal tableau contains the following information: Primal main x1 xn Primal slack xn+1 xn+m Basic Primal basic variables z z 0 0 1 ym+1 ym+n Values of dual surplus variables y1 ym Values of dual main variables Solution Values of primal basic variables Optimum of objective functions (primal and dual) The optimal dual solutions may therefore be read off the optimal primal tableau without further calculations. Furthermore, since the dual of the dual is the primal, it does not matter which problem we solve the optimal solution of one will give us the optimal solution of the other. This is important, as if we are presented with a `difficult' primal problem, it may be easier to solve it by tackling its dual: if the primal constraints are all of the `' form then (P) cannot be solved by the normal simplex algorithm, but (D) can; if the primal problem has many more constraints than variables then the dual has many fewer constraints than variables, and will in general be quicker to solve. Proposition 3.4 If either the primal (P) or the dual (D) has an unbounded optimal solution then the other has no feasible solution. Proof: Suppose the dual has a feasible solution y. Then for any primal feasible solution x, ct x bt y, so bt y is an upper bound on solutions of the primal. Similarly, if the primal has a feasible solution this places a lower bound on solutions of the dual. It follows that if either problem is unbounded then the other does not have a feasible solution. Proposition 3.4 identifies some cases where the duality results do not hold, i.e. we cannot say that the primal and dual LP problems have the same optimal values of their objective functions: 41 1. Primal problem unbounded and dual problem infeasible. 2. Primal problem infeasible and dual problem unbounded. 3. Primal and dual problems both infeasible. 3.3 Complementary slackness Complementary slackness is a very important and useful consequence of the relationship between the primal and dual optima. We continue to work with the primal-dual pair (P) and (D). In the final tableau for the primal problem, if xi is non-basic then xi = 0 at the optimum. Now suppose xi is basic in the optimal tableau. Then there is a zero in the z row at the bottom of the xi column, so a certain dual variable is zero at the optimum. If xi is a main variable in the primal, this dual variable is the surplus variable ym+i . For i = 1, . . . , n, either xi = 0 or ym+1 = 0. If xi is a slack variable in the primal then i = n + j for some j and the corresponding dual variable is the main variable yj . For j = 1, . . . , m, either yj = 0 or xn+j = 0. Thus in every case xi ym+i = 0 and xn+j yj = 0. So at the optimal solution, ith primal main variable ith dual surplus variable = 0, jth primal slack variable jth dual main variable = 0. These relationships are called the complementary slackness equations. Thus (x1 xn \| xn+1 xn+m )(ym+1 ym+n \| y1 ym )t = 0, since the scalar product of two non-negative vectors is 0 iff the product of each corresponding pair of entries is 0. Now the entries of b - Ax are the primal slack variables xn+1 , . . . , xn+m and the entries of At y - c are the dual surplus variables ym+1 , . . . , ym+n , so complementary slackness asserts that at the optimal solution, yt (b - Ax) = 0 and xt (At y - c) = 0. An interpretation of complementary slackness is that if the shadow price of a resource is non-zero then the associated constraint is active at the optimum, i.e. the resource is scarce, but if the constraint is not active (the resource is abundant) then its shadow price is zero. Proposition 3.5 (The complementary slackness theorem) A necessary and sufficient condition for x and y to be optimal for the primal and dual problems (P) and (D) is that x is primal feasible, y is dual feasible, and x and y satisfy the complementary slackness conditions yt (b - Ax) = 0 and xt (At y - c) = 0. Proof: By the duality theorems, x and y are optimal iff they are feasible and ct x = bt y. Now ct x = bt y yt b - xt c = 0 yt b - yt Ax + xt At y - xt c = 0 0 0 0 0 (since yt Ax = xt At y) yt (b - Ax) + xt (At y - c) = 0 yt (b - Ax) = 0 and xt (At y 42 - c) = 0. 3.3.1 Examples 1. Consider the rose-growing problem in Chapter 1. The solution of this was (x1 , x2 , x3 , x4 , x5 ) = (500, 900, 0, 2200, 0). The dual problem is : Minimize w = 6100y1 + 8000y2 + 5000y3 subject to 5y1 + 8y2 + y3 2, 4y1 + 2y2 + 5y3 3, yi 0 for i = 1, 2, 3. By the strong duality theorem we know that the minimum value of w is the same as the maximum of z in the primal problem, namely 3700. We can read off from the final tableau that y1 = y3 = 1/3, y2 = y4 = y5 = 0, where y4 , y5 are the surplus variables in the two constraints of the dual. If we solved the primal problem graphically, we would only know x1 = 500, x2 = 900. Complementary slackness then tells us that x1 y4 = x2 y5 = x3 y1 = x4 y2 = x5 y3 = 0 so 500y4 = 900y5 = 0y1 = 2200y2 = 0y3 = 0, and since all the yj are 0 it follows that y2 = y4 = y5 = 0. Thus both dual constraints are active at the optimum, so 5y1 + y3 = 2, 4y1 + 5y3 = 3. Solving these gives y1 = y3 = 1/3. 2. Suppose we wish to verify that (x1 , x2 ) = (10, 5) maximizes z = 30x1 + 45x2 subject to the constraints in the Containers Problem of Chapter 1. (10, 5) is certainly feasible for the primal, i.e. it satisfies the constraints. Then z = 525. The dual is: Minimize w = 60y1 + 60y2 subject to 2y1 + 4y2 30, 8y1 + 4y2 45, y1 0, y2 0. When equality holds in both constraints, y1 = 5/2 and y2 = 25/4. Thus (x1 , x2 , x3 , x4 ) = (10, 5, 0, 0) is primal feasible, (y1 , y2 , y3 , y4 ) = (5/2, 25/4, 0, 0) is dual feasible, and (10, 5, 0, 0).(0, 0, 5/2, 25/4) = 0, i.e. the complementary slackness conditions hold. By Theorem 3.5, these solutions are optimal for the primal and dual problems. We can further check that when y1 = 5/2 and y2 = 25/4, w = 525. 3. Consider again the Cattle Feed problem and its dual from Chapter 1. From the original solution, zmin = 3.46 when x1 = 3.48, x2 = 0.56. If x3 , x4 , x5 , x6 are the surplus variables in the four constraints, then x3 = x5 = 0 at the optimum as the first and third constraint are active, but x4 , x6 are non-zero. Let y5 , y6 be the slack variables in the two dual constraints. By complementary slackness, (x1 , x2 , \| x3 , x4 , x5 , x6 ).(y5 , y6 , \| y1 , y2 , y3 , y4 ) = 0. Thus y2 = y4 = y5 = y6 = 0 at the dual optimum. Hence 13.2y1 + 0.02y3 = 0.66, 34.0y1 + 0.09y3 = 2.08, and solving these gives y1 = 0.035, y3 = 9.87. We conclude that the company should charge 0.035 per unit of synthetic protein, 9.87 per unit of synthetic calcium, give away synthetic fat and phosphorus free, and thus charge 3.46 for feeding one cow. No price structure can bring them a higher income without costing the farmer more than before. In accordance with the Strong Duality Theorem, if the farmer and the company both behave rationally (i.e. optimally) then the costs of the normal and synthetic feeding plans are the same. Of course, in practice other considerations might influence the farmer's decision. If we solved the dual problem by the simplex algorithm, the solutions for x1 , . . . , x6 in the primal problem could be read off from the bottom row of the optimal tableau. The complementary slackness conditions have the following interpretation here: 43 (a) If (Ax)i < bi then yi = 0. This means the farmer should buy zero of any nutrient that is overpriced compared to its synthetic equivalent. (b) If (At y)j > cj then xj = 0. Thus the company should charge zero for any nutrient that is over-supplied in the normal feeding plan. 3.4 Asymmetric duality The dual problems in Equations (P) and (D) are said to represent symmetric duality. We now examine the situation where the variables in the primal problem are unrestricted in sign. Thus the primal problem is: Maximize z = ct x subject to Ax b, x unrestricted. (AP) We can convert this into a LP problem in standard form by letting x = x - x , where x 0 and x 0. The constraints then become A(x - x ) b. This can be written as (A \| - A)x b, where x = (x1 xn x1 xn )t . The objective function is z = ct x - ct x = (ct \| - ct )x, and clearly x 0. Thus the problem becomes Maximize z = (ct \| - ct )x subject to (A \| - A)x b, x 0. The dual of this problem can be written as: Minimize w = bt y subject to At -At y c -c , y 0. Now if -At y -c then At y c. If this is true simultaneously with At y c then we must have At y = c. Thus the dual of the unrestricted problem (3.4.1) is Minimize w = bt y subject to At y = c, y 0. (AD) Conversely, the dual of (AD), a problem with equality constraints, is (AP), a problem in which the variables are unrestricted. (Of course, every LP problem can be expressed as one with equality constraints by including slack and surplus variables.) 3.4.1 Example Consider the following asymmetric pair of primal-dual problems : Primal Maximize subject to and z = x1 - 2x2 x1 + x2 -2x1 - 3x2 4 -5 Minimize subject to and Dual w = 4y1 - 5y2 y1 - 2y2 y1 - 3y2 = = 1 -2 x1 , x2 unrestricted. y1 , y2 0. Clearly, in the dual the solution can only occur where the two equations hold, i.e. where y1 = 7, y2 = 3. Thus w has a minimum value of 13, and this must also be the maximum value of z in the primal problem. By complementary slackness we find that x1 = 7, x2 = -3. 44 Exercises 3 1. A company which manufactures three products A, B and C, needs to solve the following LP problem in order to maximize their profit. Maximize subject to and z = 3x1 + x2 + 5x3 6x1 + 3x2 + 5x3 45 3x1 + 4x2 + 5x3 30 xj 0 for j = 1, 2, 3. x1 , x2 and x3 are the amounts of A, B and C to be produced. The first constraint is a labour constraint, and the second is a material constraint. The company solves the problem and obtains an optimal solution in which x1 and x3 are basic. (a) Find the company's optimal solution. (b) How much can c2 , the unit profit for B, be increased above 1 without affecting the original optimal solution? (c) Find the range of values of c1 , the unit profit for A, for which x1 and x3 are still basic at the optimal solution. When this is the case, express the maximum value of z in terms of c1 . (d) Find an optimal solution when b2 , the amount of material available, is 60 units. (e) The constraint 3x1 + 2x2 + 3x3 25 is added to the original problem. How does this affect the original optimal solution? (f) A new product D has a unit profit of 5, and its labour and material requirements are 3 units and 4 units respectively. Is it profitable to produce D? (g) An additional 15 units of material are available for 10. What should be done? 2. A firm can manufacture four products at its factory. Production is limited by the machine hours available and the number of special components available. The data are given in the table below. Note that production of fractions of a unit is possible. Product 2 3 4 3 8 4 2 1 3 25 40 55 45 80 85 Availability Up to 90 machine hours per day Up to 80 components per day Machine hours per unit Components per unit Production costs ( per unit) Sales income ( per unit) 1 1 2 20 30 (a) Formulate this as a linear programming problem, where xj is the daily production of product j and the objective is to maximize the daily profit (income minus production costs). Find the optimal solution using the simplex tableau method, and state the optimal profit. (b) Write down the shadow prices of machine hours and components, briefly explaining their significance. (c) The firm can increase the available machine hours by up to 10 hours per day by hiring extra machinery. The cost of this would be 40 per day. Use sensitivity analysis to decide whether they should hire it, and if so, find the new production schedule. 45 (d) The production costs of products 1 and 4 are changed by t per unit. Within what range of values can t lie if the original production schedule is to remain optimal? Find the corresponding range of values of the maximum profit. (e) Due to a problem at the distributors, the total daily amount produced has to be limited to 25 units. Implement the dual simplex algorithm to find a new production schedule which meets this restriction. (f) After production has returned to normal (i.e. the original solution is optimal again) the firm considers manufacturing a new product that would require 3 machine hours and 4 components per unit. The production costs would be 45 and sales income 75 per unit. Use sensitivity analysis to decide whether they should go ahead, and if so what the optimum production schedule would be. 3. In each case formulate the dual problem and verify that the given solution is optimal by showing that primal feasibility, dual feasibility and complementary slackness all hold. (a) Maximize 19x1 + 16x2 subject to the constraints x1 + 4x2 20, 3x1 + 2x2 15, x1 0, x2 0. x1 0, x2 0. Solution (x1 , x2 ) = (2, 9/2) Solution (x1 , x2 ) = (1, 0) (b) Minimize 8x1 + 11x2 subject to the constraints 2x1 - 2x2 2, x1 + 4x2 -5, 4. The optimal solution of the problem Maximize 6x1 + 4x2 + 10x3 x1 + 2x2 + x3 20 3x1 + 2x3 24 subject to 2x1 + 2x2 22 and x1 , x2 , x3 0. occurs at (x1 , x2 , x3 ) = (0, 4, 12). Deduce the solution of the dual problem. 5. Formulate the dual of each of the two problems in Section 2.2.1 and solve them from the optimal primal tableaux using the theory of duality and complementary slackness. 6. By finding and solving the dual problem (without using the simplex algorithm), find the maximum value of z = 5x1 + 7x2 + 8x3 + 4x4 subject to x1 + x3 6, x1 + 2x4 5, x2 + x3 9, x2 + x4 3, where x1 , x2 , x3 , x4 are unrestricted in sign. 7. Use asymmetric duality to find the solutions (if any) of the following: (a) Maximize subject to and 46 z = x1 + 2x2 2x1 - 3x2 1 x1 + 4x2 5 x1 , x2 unrestricted. (b) Maximize subject to and 8. When the problem Maximize 12x1 + 6x2 + 4x3 4x1 + 2x2 + x3 60 2x1 + 3x2 + 3x3 50 subject to x1 + 3x2 + x3 45 and x1 , x2 , x3 0. z = 3x1 + 7x2 + 5x3 2x1 + 5x2 + 4x3 9 2x1 + 4x2 + 2x3 7 x1 , x2 , x3 unrestricted. is solved by the simplex method, using slack variables x4 , x5 , x6 respectively in the three constraints, the final tableau is Basic x1 x3 x6 z z 0 0 0 1 x1 1 0 0 0 x2 3/10 4/5 19/10 4/5 x3 0 1 0 0 x4 3/10 -1/5 -1/10 14/5 x5 -1/10 2/5 -3/10 2/5 x6 0 0 1 0 Solution 13 8 24 188 (a) State the optimal solution and the values of x1 , . . . , x6 at the optimum. (b) Write down the dual problem, usimg y1 , y2 , y3 for the dual main variables and y4 , y5 , y6 for the dual surplus variables. (c) Using the above tableau, write down the optimal solution of the dual problem and give the values of y1 , . . . , y6 at the optimum. (d) Show how complementary slackness occurs in these solutions. (e) Convert the dual problem to a maximization problem and solve it by the Dual Simplex method. (f) Comment on the relationships between the two optimal tableaux. 9. By solving the dual problem graphically, solve the LP problem: Minimize subject to and 4x1 + 3x2 + x3 3x1 + 8x2 + 2x3 3 2x1 + 5x2 + 3x3 5 x1 , x2 , x3 0. Give the values of all the dual and primal variables (main, slack and surplus) at the optimum. 10. Find the dual of the problem Maximize ct x subject to Ax = b, x 0. 47 Chapter 4 The Transportation Problem 4.1 Formulation of the model The transportation model is concerned with finding the minimum cost of transporting a single commodity from a given number of sources (e.g. factories) to a given number of destinations (e.g. warehouses). Any destination can receive its demand from more than one source. The objective is to find how much should be shipped from each source to each destination so as to minimize the total transportation cost. Sources Supply a1 a2 . . . am S1 S2 . . . Sm c11 Destinations D1 D2 . . . Dn Demand b1 b2 . . . bn cmn The figure represents a transportation model with m sources and n destinations. Each source or destination is represented by a point. The route between a source and destination is represented by a line joining the two points. The supply available at source i is ai , and the demand required at destination j is bj . The cost of transporting one unit between source i and destination j is cij . Let xij denote the amount transported from source i to destination j. Then the problem is m n Minimize n z= i=1 j=1 cij xij , subject to j=1 m xij ai for i = 1, . . . , m xij bj for j = 1, . . . , n, i=1 and where xij 0 for all i and j. The first constraint says that the sum of all shipments from a source cannot exceed the available supply. The second constraint says that the sum of all shipments to a destination must be at least as large as the demand. 48 When the total supply is equal to the total demand (i.e. n m i=1 ai = m n j=1 bj ) then the trans- portation model is said to be balanced. In a balanced transportation model, each of the constraints is an equation, i.e. j=1 xij = ai for i = 1, . . . , m and i=1 xij = bj for j = 1, . . . , n. The following is an example of a balanced transportation problem: Factory 1 Factory 2 Demand Warehouse 1 c11 c21 7 Warehouse 2 c12 c22 10 Warehouse 3 c13 c23 13 Supply 20 10 Total supply = 20 + 10 = 30 = 7 + 10 + 13 = Total demand. A transportation model in which the total supply and total demand are not equal is called unbalanced. It is always possible to balance an unbalanced transportation problem. Suppose the demand at warehouse 1 above is 9 units. Then the total supply and total demand are unequal, and the problem is unbalanced. In this case it is not possible to satisfy all the demand at each destination simultaneously. We reformulate the model as follows: since demand exceeds supply by 2 units, we introduce a dummy source which has a capacity of 2. The amount sent from this dummy source to a destination represents the shortfall at that destination. If supply exceeds demand then a dummy destination is added to absorb the surplus units. Any units shipped from a source to a dummy destination represent a surplus at that source. If a penalty cost is incurred for each unit of unsatisfied demand or unused supply, then the transportation cost is set equal to the penalty cost. If there is no penalty cost, the transportation cost is set equal to zero. If no units may be assigned to a dummy or a particular route, allocate a cost M . This represents a number larger than any other in the problem think of it as a million! From now on, we will discuss balanced transportation problems only, as any unbalanced problem can always be balanced by introducing a dummy. 4.2 Solution of the transportation problem m i=1 A balanced transportation problem has ai = n j=1 bj . Hence one constraint is a linear combination of the others, so there are n + m - 1 independent constraint equations. It is not practicable to use the standard simplex method to solve the transportation problem. However, there is an efficient tableau-based method which makes use of the dual problem. Starting the algorithm: finding an initial basic feasible solution Here we examine ways of constructing initial basic feasible solutions, i.e. allocations with m + n - 1 basic variables. 49 Method 1: The North-West Corner Method Consider the problem represented by the following transportation tableau. The number in the bottom right of cell (i, j) is cij , the cost of transporting 1 unit from source i to destination j. Supply 10 12 0 Demand 5 0 7 14 15 20 9 16 15 11 20 18 10 15 25 5 The north-west corner method proceeds as follows: Assign as much as possible to the cell in the top-left of the tableau: 1. x11 = 5. Cross out column 1. Amount left in row 1 is 10. 2. x12 = 10. Cross out row 1. Have 5 units left in column 2. 3. x22 = 5. Cross out column 2. Leaves 20 units in row 2. 4. x23 = 15. Cross out column 3. Have 5 units in row 2. 5. x24 = 5. Cross out row 2. Leaves 5 units in column 4. 6. x34 = 5. Cross out row 3 or column 4. Only one row or column remains, so stop. This provides the basic feasible solution x11 = 5, x12 = 10, x22 = 5, x23 = 15, x24 = 5, x34 = 5. The remaining variables are non-basic and therefore equal to zero. The solution has m + n - 1 basic variables as required. The values of the basic variables xij are entered in the top left of each cell. There should always be m + n - 1 of these; in certain (degenerate) cases some of them may be zero. They must always add up to the total supply and demand in each row and column. Note that some books position the data differently in the cells of the tableau. Method 2: The Least-Cost Method This method usually provides a better initial basic feasible solution than the North-West Corner method. 50 Assign as much as possible to the cell with the smallest unit cost in the entire tableau. If there is a tie then choose arbitrarily. It may be necessary to assign 0, Supply 10 12 0 Demand 5 0 7 14 15 20 9 16 15 11 20 18 10 15 25 5 1. x12 = 15. Cross out column 2. Amount left in row 1 is 0. 2. x31 = 5. Cross out column 1. Have 0 units left in row 3. 3. x23 = 15. Cross out column 3. Leaves 10 units in row 2. 4. x14 = 0. Cross out row 1. Still 10 units in column 4. 5. x34 = 0. Cross out row 3. Still 10 units in column 4. 6. x24 = 10. Cross out column 4 or row 2. Only one row or column remains, so stop. Checking for optimality and iterating the algorithm So far, we have only looked at ways of obtaining an initial basic feasible solution to the balanced transportation problem. We now develop a method for checking whether the current basic feasible solution is optimal, and a way of moving to a better basic feasible solution if the current solution is not optimal. For illustrative purposes, we will start the algorithm using the initial bfs that was provided by the North-West Corner method. Usually, initial basic feasible solutions obtained by the Least-Cost method (or other methods given in many text-books, such as Vogel's method) will give better starting configurations. Using asymmetric duality, the dual of the transportation problem can be written as m n Maximize subject to and w= i=1 ai i + j=1 bj j , i + j cij for each i and j i , j unresricted in sign. Introducing slack variables sij , the constraints can be written as i + j + sij = cij , i = 1, . . . , m, j = 1, . . . , n. 51 By the complementary slackness conditions we must have xij sij = 0, i.e. xij (cij - i - j ) = 0 for all i and j. As before, primal feasibility, dual feasibility and complementary slackness are necessary and sufficient for optimality. This is the underlying strategy for solving the problem. We assign 's and 's which satisfy i + j = cij to the rows and columns containing the basic variables. (This comes from the complementary slackness condition xij sij = 0.) The degeneracy in the original problem means that we are free to choose one of the 's or 's arbitrarily. For simplicity, it is usual to set 1 = 0. The values of sij = cij - i - j are entered in the top right of the cells. If all the sij values are non-negative, we have an optimal solution. Carrying out this procedure, the transportation tableau becomes: 10 0 7 5 5 10 5 12 0 7 14 0 10 0 15 9 5 16 18 2 20 5 20 13 11 We test for optimality by checking whether sij = cij - i - j 0 for all i and j, i.e. in all cells. (This is the dual feasibility condition). If this holds for every cell of the tableau then the optimum has been reached. Otherwise, choose the cell with the most negative value of sij . This identifies the variable to enter the basis. In this case the entering variable is x31 . Determining the leaving variable We construct a closed loop that starts and ends at the entering variable, and links it to basic variables by a succession of horizontal and vertical segments. It does not matter whether the loop is clockwise or anticlockwise. Initial tableau 10 0 7 5 5 10 5 12 0 7 14 0 10 0 15 9 5 16 18 2 20 5 20 13 11 We now see how large the entering variable can be made without violating the feasibility conditions. Suppose x31 increases from zero to some level > 0. Then x11 must change to 5 - to preserve the demand constraint in column 1. This has a knock on effect for x12 which therefore changes to 10 + . This process continues for all the corners of the loop. 52 The departing variable is chosen from among the corners of the loop which decrease when the entering variable increases above zero level. It is the one with the smallest current value, as this will be the first to reach zero as the entering variable increases. Any further increase in the entering variable past this value leads to infeasibility. We may choose any one of x11 , x22 or x34 as the departing variable here. Arbitrarily, we choose x34 . The entering variable x31 can increase to 5 and feasibility will be preserved. Second tableau 0 7 -10 5 0 14 16 18 0 10 0 12 7 10 0 15 0 15 9 2 20 10 20 13 11 Notice that some of the basic variables are zero valued this solution is degenerate. However, this causes no problem to the general method of solving the problem. As before, we construct 's and 's which satisfy i + j = cij for the basic variables. Then we check for optimality as before. This tableau is not optimal because sij 0 does not hold for all the cells. The most negative value of sij occurs for x21 , so this is the entering variable. Next we construct a loop. Thus can only be as large as zero. (This is bound to happen because of the degeneracy of the current solution). We let x11 be the departing variable. Third tableau 5 0 7 -5 0 12 5 0 14 16 18 10 0 7 0 15 0 15 9 2 20 10 20 13 11 Again, this is a degenerate solution, as some of the basic variables are equal to zero. We construct 's and 's as before, and then check for optimality. The tableau is not optimal, and x14 is the entering variable. The loop construction shows that can be as large as 10, and that x24 is the departing variable. Fourth tableau 5 0 7 -5 0 12 5 0 14 16 18 5 10 10 7 0 15 9 20 20 0 2 11 10 11 This is now optimal because i + j cij , i.e. sij 0, in every cell. The minimum cost is therefore given by 5 0 + 10 11 + 0 12 + 10 7 + 15 9 + 5 0 = 315, which occurs when x12 = 5, x14 = 10, x22 = 10, x23 = 15, x31 = 5, and all the other decision variables are equal to zero. 53 The dual objective function is a1 1 + a2 2 + a3 3 + b1 1 + b2 2 + b3 3 + b4 4 = 0 + 175 - 25 + 25 + 0 + 30 + 110 = 315. The primal and dual functions have the same value at feasible points, which confirms that the value is optimal. 4.2.1 Example This example emphasizes the connection between the transportation algorithm and the primal-dual linear programming problems which underlie the method. Three factories F1 , F2 , F3 produce 15000, 25000 and 15000 units respectively of a commodity. Three warehouses W1 , W2 , W3 require 20000, 19000 and 16000 units respectively. The cost of transporting from Fi to Wj is cij per unit, where c11 = 12, c12 = 7, c13 = 10, c21 = 10, c22 = 8, c23 = 6, c31 = 9, c32 = 15, c33 = 8. If xij thousand units are transported from Fi to Wj , the total cost 1000z is given by z = 12x11 + 7x12 + 10x13 + 10x21 + 8x22 + 6x23 + 9x31 + 15x32 + 8x33 which must be minimized subject to the constraints x11 + x12 + x13 x21 + x22 + x23 x11 x12 x13 where xij 0 for i, j = 1, 2, 3. By Asymmetric Duality, the dual of this problem is : Maximize w = 151 + 252 + 153 + 201 + 192 + 163 subject to 1 + 1 12 1 + 2 7 1 + 3 10 2 + 1 10 2 + 2 8 2 + 3 6 3 + 1 9 3 + 2 15 3 + 3 8 i.e. i.e. i.e. i.e. i.e. i.e. i.e. i.e. i.e. 1 1 1 2 2 2 3 3 3 + + + + + + + + + 1 + s11 2 + s12 3 + s13 1 + s21 2 + s22 3 + s23 1 + s31 2 + s32 3 + s33 = = = = = = = = = 12 7 10 10 8 6 9 15 8 + x21 + x22 + x23 x31 + x32 + x33 + x31 + x32 + x33 = = = = = = 15 25 15 20 19 16 where sij 0 for i, j = 1, 2, 3 but i , j are unrestricted in sign. Complementary slackness tells us that xij sij = 0 for all i, j. 12 10 9 7 8 15 10 6 8 54 12 10 9 7 8 15 10 6 8 12 10 9 7 8 15 10 6 8 12 10 9 7 8 15 10 6 8 We choose to find an initial bfs by the north-west corner method. After three iterations, all the sij are non-negative so we have primal feasibility, dual feasibility and complementary slackness. Hence the optimal solution has been found. The minimum cost is 418,000. The primal solution is (x11 , . . . , x13 , . . .) = (0, 15, 0, 5, 4, 16, 15, 0, 0, \| 0, 0, 0, 0, 0, 0). The last six 0's represent unnecessary slack variables in the primal problem; they are included only to show that complementary slackness does indeed hold when we look at the dual solution (1 , 2 , 3 , 1 , 2 , 3 , s11 , . . . , s33 ) = (0, 1, 0, 9, 7, 5, \| 3, 0, 5, 0, 0, 0, 0, 8, 3). The problem is now modified as follows: The demand at W2 is increased to 28. There is no link between F2 and W2 . All the demand at W3 must be satisfied. We add a dummy source F4 with capacity 9. The costs c22 and c43 are set equal to a large number M . This is a standard method for ensuring that the allocation to a particular cell is always zero. M is to be thought of as larger than any other number in the problem. The tableau becomes: 12 10 9 0 7 M 15 0 10 6 8 M 12 10 9 0 7 M 15 0 10 6 8 M 12 10 9 0 7 M 15 0 10 6 8 M 12 10 9 0 7 M 15 0 10 6 8 M We can find an initial allocation by the least-cost method, or by adapting the existing optimal tableau. The minimum cost is 450,000; this is uniquely determined, though the allocation which produces it may not be. 55 Exercises 4 1. For the transportation problem given by the following tableau, find an initial basic feasible solution by the least-cost method and proceed to find an optimal solution. Supply 2 1 3 7 4 5 6 8 Demand 5 6 4 2. Formulate the transportation problem in Question 1 in linear programming form. Also state the dual problem. From your final tableau, write down the values of all the primal and dual variables at the optimal solution. Show how this provides a check on your answer. 3. For the transportation problem given by the following tableau, find an initial basic feasible solution by the North-West corner method and then find an optimal solution. 10 5 15 Demand 5 15 10 10 9 10 8 12 2 12 15 12 4 20 10 10 5 Supply 8 7 10 The supply at Source 3 is now reduced from 10 to 6. There is a penalty of 5 for each unit required but not supplied. Find the new optimal solution. 4. Three refineries with maximum daily capacities of 6, 5, and 8 million gallons of oil supply three distribution areas with daily demands of 4, 8 and 7 million gallons. Oil is transported to the three distribution areas through a network of pipes. The transportation cost is 1 p per 100 gallons per mile. The mileage table below shows that refinery 1 is not connected to distribution area 3. Formulate the problem as a transportation model and solve it. [Hint: Let the transportation cost for the nonconnected route be equal to some large value M say and then proceed as normal.] Distribution 1 2 120 180 300 100 200 250 Area 3 -- 80 120 Refinery 1 2 3 5. In Question 4, suppose additionally that the capacity of refinery 3 is reduced to 6 million gallons. Also, distribution area 1 must receive all its demand, and any shortage at areas 2 and 3 will result in a penalty of 5 pence per gallon. Formulate the problem as a transportation model and solve it. 6. In Question 4, suppose the daily demand at area 3 drops to 4 million gallons. Any surplus production at refineries 1 and 2 must be diverted to other distribution areas by tanker. The resulting average transportation costs per 100 gallons are 1.50 from refinery 1 and 2.20 from refinery 2. Refinery 3 can divert its surplus oil to other chemical processes within the plant. Formulate the problem as a transportation model and solve it. 56 Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education. The New LegislatureBy Patrick Johnston The power blackouts of 2001 turned lights off all over California, but at the State Capitol the lights went on in the heads of new legislators. They realized that there was no point feeling insecure about lawma FEDERALIST NO.39 James MadisonThe last paper having concluded the observations which were meant to introduce a candid survey of the plan of government reported by the convention, we now proceed to the execution of that part of the undertaking. The Each student will be required to participate in one group analysis. Because of the unanticipated large class, we are adding 3 groups to the pool. They will consider policy responses to the recent terrorist attacks: foreign policy, domestic policy, an Nomination of CandidatessssU.S. Constitution Article I, Sect. 4 The times, places and manner of holding elections for Senators and Representatives, shall be prescribed in each state by the legislature thereof. Article I, Section 5 Each House Lecture P1: Introduction to C#include <stdio.h> int main(void) { printf("This is a C program.\n"); return 0; }Learning to ProgramProgramming is learned with practice and patience.sDon't expect to learn solely from these lectures. Do exercises Lecture P10: WAR Card GamePrinceton UniversityCOS 126General Computer ScienceFall 2002http:/ a program to play the card game "War." Goals.sPractice with linked lists and pointers. Appreciat Lecture P9: Pointers and Linked Lists"The name of the song is called 'Haddocks' Eyes.' " "Oh, that's the name of the song, is it?" Alice said, trying to feel interested. "No, you don't understand," the Knight said, looking a little vexed. "That's wh Lecture S2: Artificial IntelligenceOverviewA whirlwind tour of Artificial Intelligence. We spend just one lecture, but there are:Many courses on AI, but only one at Princeton :- ( Professorships of AI. Entire university departments of AI. Lecture T235: NP-CompletenessCS Building West Wall, Circa 2001OverviewLecture T3:What is an algorithm? Turing machine Which problems can be solved on a computer? not the halting problemLecture T4:Which algorithms will be useful in prac
Bob Miller's Geometry for the Clueless - 2 edition Summary: An easy-to-use guide that takes the fear out ofgeometryBob Miller's Geometry for the Clueless tackles a subjectmore than three million students face every year.Miller acts as a private tutor, painstakingly coveringthe high school curriculum as well as post secondarycourses in geometry. Great condition for a used book! Minimal wear. 100% Money Back Guarantee. Shipped to over one million happy customers. Your purchase benefits world literacy! $19.03 +$3.99 s/h VeryGood THE-BOOK-FACTORY Sleaford, 200520 +$3.99 s/h VeryGood Booksnsuch Allamuchy, NJ 2006 First Printing full # line.clean white pages no writing no highlighting good tight binding Free delivery confirmation ..no remainder mark Very good covers.slight curling to cover. I ship daily I...show more use padded envelopes 7282010B70A-52
University Place ExcelCarpentry requires measuring, multiplication, estimating, and geometry. Shopping requires estimating to ensure that you can afford what you are spending and that the bill is correct. Household budgets, selecting a loan, comparing savings and investments require understanding percentages.
Web Resources Lesson Plans Title: Supply and Demand Description:Standard(s): [MA2013] (8) 10: Analyze and solve pairs of simultaneous linear equations. [8-EE8] [MA2013] AL1 (9-12) 20: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. [A-REI6] Supply and Demand Title: Graphing Lines Description: Students explore the world of lines by investigating the relationships between linear equations, slope, and graphs of lines. This interactive tool requires Javascript. Standard(s): [MA2013] (7) 2: Recognize and represent proportional relationships between quantities. [7-RP2] [MA2013] (8) 10: Analyze and solve pairs of simultaneous linear equations. [8-EE8] [MA2013] AL1 (9-12) 17: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. [A-REI3Graphing Lines Students explore the world of lines by investigating the relationships between linear equations, slope, and graphs of lines. This interactive tool requires Javascript.
Polynomial Jeopardy is a review of polynomials including: key terms, addition, subtraction, multiplication & factoring of polynomials, F.O.I.L., and solving of equations. The activity is presented in a game-like format that may increase student engagement. This site provides about 35 graphical applets on topics relative to Algebra, Precalculus, Calculus, and Statistics. These... see more This site provides about 35 graphical applets on topics relative to Algebra, Precalculus, Calculus, and Statistics. These are designed for classroom demonstrations of various mathematical/statistical concepts. From the popular Annenberg/CPB Channel workshop "Private Universe Project in Mathematics," this Shockwave simulation provides... see more From the popular Annenberg/CPB Channel workshop "Private Universe Project in Mathematics," this Shockwave simulation provides a hands-on way to introduce combinatorics through inquiry: You have two colors of cubes available with which to build towers. Your homework task is to make as many different looking towers as is possible, each exactly four cubes high. A tower always points up, with the little knob on top. Find a way to convince yourself and others that you have found all possible towers four cubes high and that you have no duplicates. This applet is a web based lab that explores the properties of rational functions. The purpose of this lab is to help the... see more This applet is a web based lab that explores the properties of rational functions. The purpose of this lab is to help the student to learn to predict the shape of the graph of a rational function, and in particular to locate its various vertical asymptotes (spikes) and its horizontal or slant asymptotes (spears). It is one in a series of other precalculus labs by the same author. The directions for using Graph Explorer are contained in the Cartesian Coordinates applet.
AMATYC (American Mathematical Association of Two-Year Colleges) is the only organization exclusively devoted to providing a national forum for the improvement of mathematics instruction in the first two years of... This is an introductory course in discrete mathematics oriented toward students interested in computer science and engineering. The course divides roughly into thirds: fundamental concepts of mathematics: definitions,... Numerous programs and initiatives to create gender equity in the areas of science, technology, engineering, and mathematics (STEM) have been implemented only to lose effectiveness or fade away. Had these programs had... This course, presented by Massachusetts Institute of Technology, introduces students to the modeling, quantification, and analysis of uncertainty. Topics covered include: formulation and solution in sample space-random... Most technical students find math and English to be their biggest obstacle in technical education. Learning Communities are a way to combine math and English in the core classes. This website provides many key steps to...
Over the next few years, advanced level mathematics will undergo dramatic changes, as the Department of Education and Science introduces new syllabuses and methods of assessment. Because A-level mathematics has scarcely altered in a generation, many see the changes as long overdue. They will go some way towards making A level more compatible with GCSE mathematics, and will include an increase in the use of coursework for assessment, a greater emphasis on reading maths texts for comprehension, more use of problem solving and practical work, and a higher profile being given to the history of mathematics. It is into this changing climate that Mathematics Review has been launched. When I received the advance publicity for it, my first reaction was one of amazement that no publisher had
Faculty Schedule Description This course is a survey of the contemporary mathematical process to be mastered by students at K-8 levels and an awareness of the problem-solving methodologies of teaching concepts, including modification of the curriculum for the exceptional student. Students are assigned to classrooms for fieldwork. This course includes meeting the needs of the adolescent learner. This course is part of the spring block experience. Prerequisite: MTH 150.
Short Description for Engineering Mathematics Engineering Mathematics is the bestselling book of its kind with over half a million copies worldwide. Its unique programmed approach takes you through the mathematics with a wealth of worked examples and exercises. The online Personal Tutor guides you through hundreds of practice questions with instant feedback. Full description Full description for Engineering Mathematics Engineering Mathematics is the best-selling introductory mathematics text for students on science and engineering degree and pre-degree courses. Sales of previous editions stand at more than half a million copies. It is suitable for classroom use and self-study. Its unique programmed approach takes students through the mathematics they need in a step-by-step fashion with a wealth of examples and exercises. The book is divided into two sections with the Foundation section starting at Level 0 of the IEng syllabus and the main section extending over all elements of a first year undergraduate course and into many second year courses. The book therefore suits a full range of abilities and levels of access. The Online Personal Tutor guides students through exercises in the same step-by-step fashion as the book, with hundreds of full workings to questions.
Textbook Text Topics Chapter 1 – Sections 1.1-1.4 Chapter 2 – Sections 2.1-2.5 Chapter 3 – Sections 3.1-3.4 Chapter 4 – Sections 4.1-4.5 (The coverage of Section 4.2 should deal primarily with the recursive definitions of number sequences such as the Fibonacci and Lucas Numbers.) Elementary proofs dealing with number theory seemed to go quite well last year for everyone, I believe. Hopefully, we'll see the same level of success this time around.
Business Math Essentials - 00 edition Summary: Empowering users with the basic mathematical skills necessary to effectively compete in today's workforce, this easy-to-follow book offers a simple and systematic learning approach with short units, an abundance of step-by-step examples, and many visual aids. Reviews fundamental operations of arithmetic, and develop readers' ability to apply basic math skills to common business situations; solve common business problems involving discounts, payroll, interest, markup,...show more depreciation, inventory, and banking; and work mentally with speed and accuracy. Provides step-by-step examples that progress from very simple to more challenging concepts; integrates a substantial amount of boxed information and reminder notes with arrow icons pointing to concepts and illustrations; and provides immediate reinforcement with review quizzes at the end of each major unit topic, plus end-of-unit exercises and word problems. For anyone entering or in a profession that requires skills in basic business mathematics
Initial chapters cover ... More About This Book Initial chapters cover functions and graphs, straight lines and conic sections, new coordinate systems, the derivative, using the derivative, integration and using the integral. The last four chapters focus on derivatives of transcendental functions, patterns for integrations, series expansion of functions, and differential equations. Throughout, the writing style is clear, readable and informal. Examples are abundant and have complete worked solutions. Practice problems appear in the body of the text in each section; these are relatively easy and are intended to be worked by the student as soon as they are encountered. Each new type of example in the text is followed by a practice problem that allows the student to gain immediate reinforcement in applying the problem-solving technique illustrated by the example. Answers to all practice problems are given at the back of the book, many with worked-out solutions. Other learning aids include the division of complex problem-solving processes into a series of step-by-step tasks, numerous exercises at the end of each section and a Status Check at the end of each chapter that helps students review what they have learned. Additional review exercises and a glossary, with definitions and page references, 2002 Great Book! I found this book to extremely clear cut, in almost all aspects of the first two semesters of Calculus. It even gives a foreshadowing of the third semester of Calculus, with their explanations of Double-Integrals. Well done. -Andrew McDaniel 1 out of 1 people found this review helpful. Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged.
The user reviews definitions of important algebra terms. After viewing further explanations and some examples, users can interactively test their understanding of the definitions of important algebra... More: lessons, discussions, ratings, reviews,... The 3-D animated video helps students understand the wordings in the following distance, rate, and time word problem: Two space jets named Dragon and Eagle start from Mars and fly in opposite dire... More: lessons, discussions, ratings, reviews,... The user reads about the definition of an equation, the use of variables, and how to write an equation from a sentence. Examples are given as well as an online quiz to practice the skill of matching
97800713901 of the courses in the junior high, high school, and college mathematics curriculum require a thorough grounding in the fundamentals, principles, and techniques of basic math and pre-algebra, yet many students have difficulty grasping the necessary concepts. Utilizing the author's acclaimed and patented fail-safe methodology for making mathematics easy to understand, Bob Miller's Basic Math and Pre-Algebra for the Clueless enhances students' facility in these techniques and in understanding the basics. This valuable new addition to Bob Miller's Clueless series provides students with the reassuring help they need to master these fundamental techniques, gives them a solid understanding of how basic mathematics works, and prepares them to perform well in any further mathematics courses they take
Seattle school leaders "discover" the math book they wanted Seattle Public Schools has decided to stick with reform math. With the School Board's 4-3 vote May 6 for the "Discovering Algebra" and "Discovering Geometry" textbooks, the battle is finished. Reform won. It was an odd battle. The side that lost wanted to define terms, make distinctions and fight. The side that won — the establishment's side — did not want to do any of that. "The vote is not about which textbook I think is best," said board member Cheryl Chow, who voted to adopt the reform texts. Here is how the rejected text, Prentice-Hall's "Geometry," introduces the concept of a chord: "A segment whose end points are on a circle is a chord." The book immediately says what a chord is, what one looks like, and what laws it follows. The reform book doesn't tell you that. You're supposed to discover it. "Discovering Geometry" shows you segments that are chords and some that aren't. It asks you to write your own definition. Then it says, "Discuss your definitions with others in your group." The teacher is to stand back: "Ask questions to guide students," says the teacher's book, "but try to refrain from correcting the definitions yourself." The publisher, Key Curriculum Press, calls this "learning through cooperative group activities." You will not find axioms, theorems and proofs in "Discovering Geometry" until the last chapter. The publisher is proud of this. It says it is better for students to be shielded from abstract logic while they get a sense of geometry by looking at pictures, folding paper, measuring angles and discussing it with the group. To opponents, this is all backward. Ted Nutting, who teaches AP calculus the traditional way at Ballard High School, recalled the year he tried a reform text. "It was a disaster," he told the School Board May 6. Nutting told them he sent his daughter to Holy Names Academy so that she could learn real math. Opponents also argued that straightforward books were better even if math were taught in a roundabout way, because books are used at home. "Parents appreciate a clear textbook," said Cliff Mass, professor of meteorology at the University of Washington. That argument resonates. Board member Harium Martin-Morris told me he had been baffled by his daughter's reform-math text. He voted against the reform texts, along with board members Michael DeBell and Mary Bass. The other side, however, did not make a case for the reform text. They argued instead that the "Discovering" books had been recommended by a committee, and that the board should respect the committee. Board member Steve Sundquist said, "I should probably not be telling educators how to teach." They argued that textbooks aren't that important anyway. Board member Peter Maier said the books "allow a variety of teaching methods." They argued that textbook adoption was too important to waste any more time. "How many classes are we willing to graduate while we disagree over textbooks?" said board member Sherry Carr. So the dominant paradigm — and reform math is that — continues. Dominance has its privileges. The supporters of reform math did not have to define their terms or label themselves. They did not have to make a logical argument or show any data. They engaged in what you might call a cooperative group activity, and led themselves to discover the books that were wanted.
Linear Algebra A linear algebra textbook that fosters mathematical thinking, problem solving abilities, and exposure to real world applications. It focuses on the ...Show synopsisA linear algebra textbook that fosters mathematical thinking, problem solving abilities, and exposure to real world applications. It focuses on the aspects of linear algebra that are likely to have practical value to the student while not compromising the intrinsic mathematical form of the subject
Algebra & Geometry My oldest is in 8th grade and we are using Art of Problem Solving's Algebra 1 this year. I choose it because it is challenging. DD has always done well in math. However, this year her confidence is really lacking. While I appreciate the format of the text and the way the problems go beyond "regurgitation" of a process, I think that she is needing more of the plug and chug style than this book is offering. For example, we are currently working with lines. There are several equations that she is working with; they are all related. (standard form, point/slope form, and slope/intercept form). The exercises usually include 3 problems that seem like a typical algebra book. Then there are 3 or 4 that go deeper. Usually, she can get 2 or 3 of the deeper ones. She does usually get the 3 typical ones correct. However, it isn't enough to make her feel like she has a grasp on it. She is worried that in a week or two (when we have moved to a new unit) that she will no longer remember what it was that we were doing. Right now she says that she uses too much brain energy deciding which formula is relevant. I realize that learning something new can literally make your brain exhausted, but I think we should be getting enough practice so that by the end of the unit, she isn't always feeling that way. So, enough rambling. . . my question: is there a place (online or a workbook to buy) that would be a source of extra algebra problems. I know where to get extra work for all the earlier math, but I haven't found a place for algebra. I end up making things up myself--which is honestly time consuming and I sometimes want the problems to come out with a nice, clean answer instead of what my random numbers create. If there isn't a quick solution, what are the other algebra programs out there that seem good? I never cared for Saxon, but maybe I should revisit it. Is their algebra as tight of a spiral as their 6/5. Finally, if we move on to Geometry next year (typical in our area to do Algebra 1, Geometry, and then Algebra 2), I am not sure if I will use Art of Problem Solving again. What are some decent Geometry curriculums? I found the same thing with AoPS. My middle dd, who is strong in math, moved into it after finishing Singapore Primary. It was good, but working through it on her own without a group of fellow students who were in it together, never quite getting to the point of ease with each new level of challenge, she thought she wasn't doing well and got bogged down. I assumed it was because she was young (I think she was 11 or 12 at the time), but maybe the same issue would have arisen even a couple of years later. She ended up using our local Canadian school's textbook the next year, something called MathPower. She had decided to enter school at that point and went a step down in level because the school could only place her one grade level up the way their schedule worked. Perhaps it was good, because her confidence recovered quickly. My dd11 is currently doing about the same level of math as your dd, but again we are using a Canadian school curriculum at home ("Math Makes Sense 9"). If we didn't have our local school willing to provide us with their curriculum, I think we'd have returned to what I used with my eldest dd which was one of the Singapore secondary programs. They're not as challenging as AoPS, and have more repetition and consolidation work, but they're not overly spirally and they take a conceptual approach rather than a formulaic one. The disadvantage for you might be that they do math in an integrated way like a typical K-8 curriculum, not separating Algebra out as a separate year-long course, etc. We used New Math Counts, which isn't available anymore, but I would take a good look at what has replaced it and see whether it would work for you. We've also tried Life of Fred and Teaching Textbooks. I wouldn't recommend either. Life of Fred wasn't challenging enough and contained far too much fluff for my kids. Teaching Textbooks was very much like Saxon but packaged in a slicker presentation: it was glacial in pace and formulaic in approach, yet impossible to skim through.
Getting Started with "Math in ONE" Code use for Windows, Android versions (AG) - Applies to Android graphing version.All (A) functions are included in (AG) (AP) - Applies to Android programing Calculator version. All (AG) functions are included in (AP). If no symbol is indicated, then instructions apply to both Windows and Android products. (W) There are two main resources to help you learn about "Math in ONE" which are found in the "Reference Manual" and the convenient descriptions built into the program. Two examples are presented below. When using "Math in ONE", these descriptions appear with the right click of your mouse on an object (e.g. checkbox, button,...), then left click on the pop up button. Refer to the example on the left. For all the pull down menus (e.g."Functions"), simply place your mouse over an item to review its description. Refer to the example on the right. (AP) Help Options: For an explanation of the features underlying each button, press and hold the button of interest for one second, a new window with a description of the button function will appears. (AP) Sliding finger in arrow direction will display new functions assign to the buttons. (A) To get to 'Help Index or Setting' window, from main view press 'option menu (om or Op Men)' button then select 'help' or 'Setting'. Thank you for your interest in "Math in ONE"! Please refer to this reference manual for further guidelines. We appreciate all of your feedback regarding this site and the "Math in ONE" program. We will make all the effort to satisfy you. We hope you enjoy working with this product!
Gateway to Modern Geometry: The Poincare Half-Plane - 2nd edition Summary: Stahl's Second Edition continues to provide students with�the elementary and constructive development of modern geometry that brings them closer to current geometric research.� At the same time, repeated use is made of high school geometry, algebra, trigonometry, and calculus, thus reinforcing�the students' understanding of these disciplines as well as enhancing their perception of mathematics as a unified endeavor. This distinct approach makes these advanced geometry principle...show mores accessible to undergraduates and graduates alike207.75 Used Currently Sold Out New $124.62 Save $83.13 (40%) FREE shipping over $25 In stock 30-day returns Condition: Brand New Order this book in the next 11 hours and 47
Algebra 2 introduces independent and dependent variables and how their solution can be determined by for linear relationships for two or three variables. Algebra 2 also gives an overview of more complex mathematical functions like basic trigonometric functions, power functions, logarithms, and ...
Basic Maths for Dummies Overview Whether you are returning to school, studying for an adult numeracy test, helping your kids with homework, or seeking the confidence that a firm maths foundation provides in everyday encounters, Basic Maths For Dummies, UK Edition, provides the content you need to improve your basic maths skills. Based upon the Adult Numeracy Core Curriculum, this title covers such topics as: Getting started with the building blocks of maths and setting yourself up for success Dealing with decimals, percentages and tackling fractions without fear Sizing Up weights, measures, and shapes How to handle statistics and gauge probability Filled with real-world examples and written by a PhD-level mathematician who specialises in tutoring adults and students, Basic Maths For Dummies also provides practical advice on overcoming maths anxiety and a host of tips, tricks, and memory aids that make learning maths (almost) painless - and even fun. Author Information Colin Beveridge, PhD, holds a doctorate in mathematics from the University of St Andrews. He gave up a position as a researcher at Montana State University (working with NASA, among other projects) to become a full-time maths tutor, helping adults and GCSE, A-level, and university students overcome their fear of maths - a position he finds 'far more enjoyable than real work'. Customer Reviews 9781119975625 There are no customer reviews available at this time. Would you like to write a review?
tic Trigonometry with Applications Featuring rich applications and integrated coverage of graphing utilities, this hands-on trigonometry text guides students step by step, from the ...Show synopsisFeaturing rich applications and integrated coverage of graphing utilities, this hands-on trigonometry text guides students step by step, from the right triangle to the unit-circle definitions of the trigonometric functions. Examples with matched problems illustrate almost every concept and encourage students to be actively involved in the learning process. Key pedagogical elements, such as annotated examples, think boxes, caution warnings, and reviews help students comprehend and retain the material
eTextbook New Textbook Related Products Summary Success in your calculus course starts here! James StewartsTable of Contents Functions and Models Four Ways to Represent a Function Mathematical Models: A Catalog of Essential Functions New Functions from Old Functions Graphing Calculators and Computers Review Principles of Problem Solving Limits The Tangent and Velocity Problems The Limit of a Function Calculating Limits Using the Limit Laws The Precise Definition of a Limit Continuity Review Problems Plus Derivatives Derivatives and Rates of Change Writing Project: Early Methods for Finding Tangents The Derivative as a Function Differentiation Formulas Applied Project: Building a Better Roller Coaster Derivatives of Trigonometric Functions The Chain Rule Applied Project: Where Should a Pilot Start Descent? Implicit Differentiation
ISBN13:978-0030169779 ISBN10: 0030169771 This edition has also been released as: ISBN13: 978-0030244865 ISBN10: 0030244862 Summary: Unlike traditional calculus texts that emphasize only an algebraic (symbolic) approach to learning the concepts, Ostebee/Zorn balances the symbolic approach with numerical and graphical ones. These three viewpoints give students a richer understanding of the fundamental concepts. Because important theorems and proofs are included, this text is appropriate for math and physical science majors. Technology (graphing calculators...show more or computers) has a supporting role as an exploratory tool, rather than as an end in itself. Among the text's other distinctive features are the unusual variety of exercises, many which are graphical in nature; the informal, reader-friendly exposition; the attention to careful definitions and theorem statements; and the seamless integration of text and graphics in the exposition....show lessEdition/Copyright: 97 Cover: Hardcover Publisher: Saunders College Division Year Published: 1997 International: No View Table of Contents Volume 2 contains Chapters 6 through 14 and selections from Chapters 3 and 4. Most chapters end with a Chapter Summary. Sequences and Their Limits Infinite Series, Convergence, and Divergence Testing for Convergence; Estimating Limits Absolute Convergence; Alternating Series Power Series Power Series as Functions Maclaurin and Taylor Series
Differential equations play an important role in the physical sciences. This treatise introduces the standard techniques for finding solutions to these equations and explores orthogonal polynomials,... More > Green's functions, and perturbation theory. These techniques form the basis of mathematical analysis used throughout the physical sciences. This book is suitable for an introductory course on differential equations for undergraduate students. Topics include first and second order differential equations, systems of differential equations, orthogonal polynomials, Green's functions, perturbation theory, and analysis of nonlinear equations. These topics provide the foundation for understanding problems in the physical sciences for both graduate and undergraduate students.< Less
MathLAN has been developed to meet a broad range of curricular needs of both students and faculty: GNU/Linux software includes programming tools that integrate various steps of coding, compiling, testing and debugging in a straightforward way. MathLAN supports compilers and interpreters for Java, Scheme, C, C++, Python, Perl, PHP, Ruby, Lua, Common Lisp, FORTRAN, Ada, Icon, and other languages, as well as a variety of programming tools and environments. Our students and faculty have easy access to the major Internet information services (the World Wide Web, ftp, ssh, and e-mail). MathLAN's World Wide Web server provides access to more than fifteen thousand local documents. High-resolution color graphics, driven by software that is both powerful and easy to use, make it possible to display data, functions, and mathematical structures in an intuitive way. These capabilities are used in a wide variety of courses -- pre-calculus, calculus, linear algebra, statistics, and modeling. Our workstations provide enough processing power to run outstanding mathematical packages that perform algebraic, symbolic, or graphical operations on functions, statistical data sets, and other mathematical objects. The faculty of the Department of Mathematics and Statistics have successfully integrated these computing tools into our courses, particularly at the first- and second-year levels, with the objective of strengthening students' intuitive understanding of mathematical ideas. About 1000 students, faculty, staff members, and recent graduates of Grinnell College currently maintain accounts on MathLAN. Each classroom in the Department of Computer Science and the Department of Mathematics and Statistics contains a MathLAN workstation linked to an Eiki digital projection system, for presentations and demonstrations. In addition, five of our classrooms are equipped with student workstations, for use in class activities, laboratory sessions, and workshops. We also support two open laboratories, each containing nineteen workstations (one of which can be similarly linked to a projection system when the lab is used for a class). The open laboratories and one of the computer-equipped classrooms are open for student use from 8 a.m. to midnight on Mondays, Tuesdays, Wednesdays, and Thursdays, from 8 a.m. to 5 p.m. on Fridays, from noon to 6 p.m. on Saturdays, and from noon to midnight on Sundays. In the evenings and on weekends, a consultant is present to answer questions and provide general assistance.
More About This Textbook Overview Many students of physical and applied science and of engineering find difficulty in copying with the mathematics necessary for the quantitative manipulation of the physical concepts they are atudying in their main course. This book is designed to help first and second year under-graduates at universities and polytechnics, as well as technical college students, to find their feet in the important mathematical methods they will need. Throughout the text the physical relevance of the mathematics is constantly stressed and, where it is helpful, use has been made of pictorial mathematics and qualitative verbal descriptions instead of over-compact mathematical symbolism. Topics are presented in three stages: a qualitative introduction, a more formal presentation and an explicit check or worked example. There are many exercises included in the text which are aimed at testing a student's understanding and building his confidence progressively throughout each piece of
Microsoft Releases Math 4.0 Free Microsoft has released a new version of its math education software Mathematics 4.0, making it available as a free download for the first time. By Dian Schaffhauser 03/10/11 Microsoft said the new version of its math program has been downloaded 250,000 times since its quiet January 2011 release. Microsoft Mathematics 4.0, designed for students in middle school, high school, and early college, is intended to teach users how to solve equations while bolstering their understanding of fundamental math and science concepts. Although the company charged for its last version, this latest edition is free. The new program works on computers running Windows XP, Vista, and 7, as well as Windows Server 2003 and 2008. The software includes a graphing calculator capable of plotting in 2D and 3D, a formulas and equations library, a triangle solver, a unit conversion tool, and ink handwriting support for tablet or ultra-mobile PC use. One new feature enables a user to create a custom movie where a 3D graphed image shifts among multiple shapes as variables change. An 18-page step-by-step guide provides basic documentation to use the program's functions. Microsoft Mathematics 4.0 is available now. Further information can be found here
Algebra to Go: Student Edition Handbook (Softcover) A resource providing explanations, charts, graphs, and numerous examples to help students understand and retain algebraic concepts.A resource providing explanations, charts, graphs, and numerous examples to help students understand and retain algebraic
J.P. McCarthy: Math Page Marks Pending… Week 6. Then we started Laplace Methods. Week 7 In Monday's tutorial we will look at doing some partial fractions. I am going to rob and adapt these exercises, this and this question are looking for but not finding intuitive ways to think about the Laplace transform 6 We finished our of vectors by looking at the application to work and moments. We also did a review of MATH6015 differentiation. Week 7 understanding the dot and cross productLATE EDIT: I just realised that I included no questions like Q.1 on P.32 of the notes… they are also examinable (matrix arithmetic). Also Q. 7 should read "Verify your answer for using Cramer's Rule" The test will be from 7.05-8 pm in Week 8 (Wednesday 19 March). I won't give a sample but instead look here for a selection of test & exam questions. You can find a summary of Chapter 1 methods here. Also I explained that some questions involving finding the inverse of 3×3 matrices using the Gauss-Jordan Algorithm can end up very messy in terms of the numbers. Questions like P. 44 Q.1-5 are fine as are P. 52 Q. 2, 3 (iv)-(vi) 6 We solved a number of linear systems using the inverse matrix method. Note however that this cannot work if the solution is not unique… we used a property of matrices called the determinant to decide in advance if a certain class of linear system, homogenous systems, had a unique solution or not. If the matrix (of coefficient) is invertible then the solution has to be unique and we saw that the matrix (of coefficients) is invertible if and only if the determinant is non-zero. Finally we will looked at Cramer's Rule: a method for solving for only one of the variables of a linear system. We then began talking about statistics and got as far as discussing the difference between the mean, the median and the mode. Week 7 We will go as far into the chapter on statistics as possible. In Maple I will allow you to revise for the test with the aid of Maple. I will write a worksheet which includes the sample questions. Remember Maple runs from 6-7.10 and 8.50-10. The theory class runs from 7.15-8.45the sample questions p.52 Q. 3 (iv)-(vi) P.62 Q. 1 and 4 P. 68 Q. 1-4 the determinantHomework Will be discussed in class on Monday. Week 9 Week 10 Exercises For the Week 10 tutorial you should look at Q.44, 45, 52, 53, 56-60 solving a complex numbers problem geometrically 5… the answer is we have to do partial pivoting. We also began the section on the powerful theory of the Laplace Transform. Week 6 In Monday's tutorial we should be looking at our sample paper — you should print off a copy for yourself. Otherwise look at the sample in the notes but replace the partial pivoting question by a Jacobi/Gauss-Siedel question answer which says that partial pivoting can actually be really bad in theory (give inaccurate solutions), but almost always works well 5 We continued working with the dot product and then introduced the cross product. Week 6 generalising the vector product to more than three dimensionsDue to the lost night in the storm, I am now putting the test in Week 8. Ye will get a sample test (format only) in week 6. It will be on Chapter 1. 5 We found some matrix inverses using the Gauss-Jordan Algorithm and saw how matrix inverses are useful in solving matrix equations. In particular, we saw that we can rewrite a linear system as a matrix equation. In Maple, we saw how Maple handles linear systems with no and infinite solutions. We also used the Maple Tutor to help us do some matrix manipulations. Week 6 We will solve a number of linear systems using this inverse matrix method. Note however that this cannot work if the solution is not unique… we will use a property of matrices called the determinant to decide in advance if a certain class of linear system, homogenous systems, have a unique solution or not. If the matrix (of coefficient) is invertible then the solution has to be unique and we will see that the matrix (of coefficients) is invertible if and only if the determinant is non-zero. Finally we will look at Cramer's Rule: a method for solving for only one of the variables of a linear systemp.44 exercises p.46 exercises p. 52 Q. 1, 3(i),(ii) & 4 matrix inversesMCQ Results As discussed previously, the MCQ results will come out the day you hand in the homework: 11 April 2014. I will send the homework next week. Week 8 Week 9 Exercises I have emailed ye a copy of the exercises. For the Week 9 tutorial you should look at Q.46 and Q. 47 the iterates of the tangent function 4 In the tutorials we looked at Gaussian Elimination. Also if you download Maple (see below), there is a Maple Tutor that is easy to use and will help you. Open up Maple and go to Tools -> Tutors ->Linear Algebra -> Gaussian Elimination which asks what are the applications of linear systems Learning
To learn about groups, rings, and fields, and the basics of how they are studied. To develop skills in abstracting the general principles underlying specific mathematical operations and structures. To become proficient at writing proofs, and more generally, at discussing math in a clear and precise manner. Courses and the Community: Teaming up with Help Yourself by Ben Newton, Professor of Mathematics and Computer Science The Help Yourself Program at Beloit College has organized tutoring and college preparation sessions, as well as weekend workshops for local high school and middle school students since 1986. The program has a fantastic track record, with a 100% high school graduation rate, and 98% of participants in recent years going on to attend a 4-year college immediately upon graduation. Many Beloit College faculty and students have been involved in the program in various capacities over the years, but until recently this involvement has not typically been connected to the college curriculum. This spring I was very fortunate to be able to take part in a pilot project that sought to link several of the weekend workshops with existing Beloit College courses. The project was organized by Carol Wickersham, and besides myself, the participating faculty were Scott Espeseth (Art), Katie Johnson (Biology), and Jingjing Lou (Education). The selected courses were ones that were already scheduled to be taught in the spring semester. Students in each of these courses were put in charge of the design and implementation of one Saturday workshop for about 20-30 high school or middle school students. The workshops were to be based around the content of the courses that they were taking, and their work in this effort would essentially amount to a major course assignment, among their other readings, homework, exams, etc. The course that I included as part of this project was Math 215: Abstract Algebra, which is typically taken by sophomore and junior math majors or junior and senior math minors. This spring, 16 students were enrolled. Students in the course were divided into three teams, each responsible for a separate activity which the Help Yourself students would rotate between during the workshop. My expectation was that the activities would be designed with various mathematical concepts from our course in mind. On the back-end, I hoped that students would be able use the language of abstract algebra (i.e. technical terms such as 'associativity' and 'isomorphism') in describing their activities to me, but on the front-end, the goal was to create fun and challenging puzzles and contests that the Help Yourself students could fully engage in without having to learn numerous abstract definitions or master complicated techniques. The teams showed remarkable creativity and resourcefulness in putting their activities together. One team used the theory of permutation groups to create a contest in which two teams of Help Yourself students raced to rearrange playing cards according to various rules. Another team created Sudoku-like puzzles to illustrate various properties of binary operations, and the third designed a wonderful collection of icons and abstract patterns which they put on cards for a game based on the concepts of sets and relations. Overall, I'd have to say that our workshop on March 19 was quite successful. One of my goals for the Help Yourself students was to impart a sense of the usefulness of abstract thinking in recognizing and describing patterns, and to make them aware that mathematics is as much about these processes as it is about techniques for numerical computation. From my perspective, it seemed like there was at least some appreciation of this point, but I think that workshop also served an even more important purpose, which was to allow the Help Yourself students to interact with Beloit College students in a campus setting and in the context of one of their courses. Of course, the day was not without its small hiccups. Even with preparation and rehearsal in a classroom setting, it proved difficult to fully envision what the activities would be like in practice. More than one team had a hard time, especially with their first group, getting on the same page as the Help Yourself Students about what the students were being asked to do. All three teams reported, however, that their last group of the day had the most success in completing the challenges put before them. This might be due to improvements in the teaching and explanations of the Math 215 students, or it might be due to the Help Yourself students starting to get a feel for the kinds of abstract thinking that they were being asked to do. Hopefully, both of these factors played a part. Finally, a large majority of the Math 215 students said that they enjoyed taking part in the workshop, but there were mixed opinions about whether they would have chosen to spend the amount of time that we did in preparation for it. The in-class time that we spent was spread out over multiple days, but I'd estimate that it amounted to between two and three Tuesday/Thursday class periods in total. Several students indicated that they would have preferred to use this time in a way that was more directly focused on the content of the course. A central conception behind the entire pilot project was that the efforts involved in preparing for the workshop could be an important part of the content learning of a given course, as opposed to being a separate but still valuable exercise. In the case of Math 215, I felt that this process could really solidify students' knowledge of key ideas by having them reframe them in another context. In retrospect, it may be that some of the more advanced students in the class were already at this level of understanding before the preparations began, but I don't think that this means that there is nothing to be gained for these students in terms of course content in such a situation. For example, the team designing the card-rearranging activity encountered the problem of determining which of the puzzles that they came up with actually had a solution. This turned out to be an interesting and thought-provoking problem that was just at the right level for our course. I think that it is a very worthwhile goal for instructors taking part in this type of project in the future to find a way to structure assignments so that the project not only benefits students needing a little extra help in mastering core concepts, but also provides difficult challenges for students who are further along in their understanding.
resource hub dedicated to the learning student. I have dished out free question sets (with full detailed solutions) here, together with personally written summary notes for various topics including differentiation, integration, AP/GP, vectors, complex numbers etc. Hope it helps. Peace
Book summary An introduction to complex analysis for students with some knowledge of complex numbers from high school. It contains sixteen chapters, the first eleven of which are aimed at an upper division undergraduate audience. The remaining five chapters are designed to complete the coverage of all background necessary for passing PhD qualifying exams in complex analysis. Topics studied include Julia sets and the Mandelbrot set, Dirichlet series and the prime number theorem, and the uniformization theorem for Riemann surfaces, with emphasis placed on the three geometries: spherical, euclidean, and hyperbolic. Throughout, exercises range from the very simple to the challenging. The book is based on lectures given by the author at several universities, including UCLA, Brown University, La Plata, Buenos Aires, and the Universidad Autonomo de Valencia, Spain. [via]
This sequel to the authors' text Elementary Mathematics for Teachers (EMT)is designed for the second semester of a mathematics course for prospective elementary teachers that istaught by mathematics faculty.This text takes prospective teachers through the development of measurement and geometry in grades K-8; it also includes material on probability and data analysis. Elementary Geometry for Teachers covers both the mathematics and other aspects of the K-8 geometry curriculum.For this purpose, this text --- like EMT --- is used in conjunction with six school textbooks from Singapore (two of these are also used with EMT).The homework sets include exercises that ask students to read a section in a Primary Math book, do the problems, and then study the material from a teachers' perspective, thinking about which skills are developed, how the problems are organized, what the prerequisite knowledge is, what order topics are developed, etc. Features: The material focuses directly on the mathematics relevant to elementary teachers. The focus is always on mathematics; this is not a ``teaching methods'' course. The text is divided into short sections, each with a homework set, of a size appropriate for a single class session. Prospective teachers are asked to write "Teacher's Solutions" to problems and to write "Elementary Proofs".These are special ways of presenting specific geometry content to elementary and middle school students that are standard in some of the world's most highly-regarded curricula.The emphasis is on building and perfecting skill at writing clear, concise solutions. The Primary Textbooks serve as teacher guides.They provide examples and activities that teachers can use in their classrooms and that help teachers understand what is important in K-8 geometry. The Primary Math books were chosen because of their clarity, organization, low cost and their exceptional fidelity to mathematics.Studying the Primary Math books prepares teachers for teaching from any elementary school materials.Furthermore, as prospective teachers work though these books, they are constantly aware that the pace, the breadth and the difficulty of the problems in the Primary Math books are at a higher level than what they experienced in their own elementary education.They come away with new expectations about the mathematics capabilities of elementary students. Elementary Geometry for Teachers pays special attention to two themes: Developing skills at solving problems involving measurements.International comparisons indicate that US students are especially weak at solving problems involving measurements.These skills are important prerequisites for middle and high school science. Elementary Geometry for Teachers builds teachers' facility at solving such problems by following the Primary Math curriculum through the grades. The problem below is one of a sequence of Grade 5 "tank problems". A rectangular tank, 40 cm long and 20 cm wide, originally contains water to a depth of 9 cm.When a stone is added, the water rises to a depth of 15 cm, covering the stone.What is the volume of the stone in liters? Unknown angle problems.One reason for studying geometry is to acquire skill at logical reasoning.The Primary Math books develop geometric reasoning in depth.In grades 4--6, students are introduced to a specific collection of geometric facts (e.g. the sum of interior angles of a triangle is 180°and opposite angles in a parallelogram are equal. These are used to solve entertaining puzzles like the one below. As they work through Elementary Geometry for Teachers, teachers solve such problems and learn to write Teacher Solutions that display the reasoning. The Primary Mathematics books were developed by the Curriculum Planning and Development Division of Singapore's Ministry of Education, and published by Marshall Cavedish. While these books were initially created for Singapore elementary students, they have been adapted for use in the United States and other countries. We will refer to them as "Primary Math 3B", and so on. The Primary Mathematics series is printed as one course book per semester, each with an accompanying workbooks. The semesters are labeled 'A' and 'B' , so '5A' refers to the first semester of Grade 5. In each grade, the first semester focuses mainly on numbers and arithmetic, while the second semester focuses more on measurement and geometry. For teachers using Primary Mathematics Standards Edition textbooks and workbooks, here is a link to EGFT homework adaption for the Standards Edition.This homework adaption may be printed out and used at no cost by teachers using the EGFT textbook. They may not be sold or incorporated into any other document. List of universities and colleges using Elementary Mathematics and Elementary Geometry for Teachers California State University Los Angeles, California California State University Northridge, California East Tennessee State University, Tennessee Fitchburg State University, Massachusetts Greenville College, Illinois Indiana University, Indiana Kansas State University, Kansas Longwood University, Virginia Los Angeles Pierce College, California Louisiana State University, Louisiana Michigan State University, Michigan Middlesex Community College, Massachusetts Northeast State Technical Community College, Tennessee Oklahoma State University, Oklahoma Pennsylvania State University Altoona, Pennsylvania Salem State University, Massachusetts Texas Tech University, Texas Tusculum College, Tennessee University of Colorado Denver, Colorado University of Memphis, Tennessee University of Michigan, Michigan University of Northern Colorado, Colorado University of Wisconsin-Madison, Wisconsin Wartburg College, Iowa Wayne State College, Nebraska Weber State University, Utah Westchester Community College, New York Worcester State College, Massachusetts Our recommendation: The Elementary Geometry for Teachers is written primarily for elementary teachers. A number of universities will be using this book as course material in classes for students taking mathematics education. This book is also suitable for individuals who would like to learn more about teaching elementary mathematics.
Function Tables and Maps Introduction In Chapter 1, addition was spoken of as a "function" because it "does something" to the numbers it is applied to and produces some result. Multiplication was also referred to as a function, but the notion of function is actually much broader than these two examples alone might suggest. For example, the average or normal weight of a woman depends on her height, and is therefore a function of her height. In fact if one were told that the normal weight for a height of 57 inches is 113 pounds, the normal weight for a height of 58 inches is 115 pounds, and so on, then one could evaluate the function "normal weight" for any given height by simply consulting the list of corresponding heights and weights. It is usually more convenient to present the necessary information about a function such as "normal weight" not by a long English sentence as begun above, but by a table of the form shown in Figure 2.1. Table of Normal Weights vs. Heights H 57 113 W E 58 115 E I 59 117 I G 60 120 G H 61 123 H T 62 126 T 63 130 I 64 134 I N 65 137 N 66 141 I 67 145 P N 68 149 O C 69 153 U H 70 157 N E 71 161 D S 72 165 S The quantity (or quantities) to which a function or verb is applied is (are) called the argument (or arguments) of the function. For example, in the expression 34 the number 3 is the left (or first) argument of the verb and 4 is the right (or second) argument. Evaluation of the "normal weight" function (represented by Table 2.1) for a given argument (say 68 inches) is performed by finding the argument 68 in the first column and reading the weight (149 pounds) which occurs in the same row. The domain of a function is the collection of all arguments for which it is defined. Addition is, of course, defined for any pair of numbers, but the function "normal weight" is certainly not defined for heights such as 2 inches or 200 inches. For practical purposes, the domain of a function such as "normal weight" is simply the collection of arguments in the table we happen to possess, even though information for other arguments might be available elsewhere. For example, the domain of the function of Table 2.1 is the set of integers from 57 to 72, that is, the set of integers 57+i.16. The rangeof a function is the collection of all the results of the function. For example, the range of the function of Figure 2.1 is the set of integers 113, 115, 117, 120, etc. occurring in the second column. #1-2 A table of normal weights often shows several columns of weights, one for small framed people, one for medium, and one for large. Such a table appears below. In such a case the weight is a function of two arguments, the height and the "frame-class"; the first argument determines the column in which the result appears. Thus the normal weight of a small-boned, 66-inch woman is 133 pounds. #3-4 Normal Weight as a Function of Two Arguments Small Medium Large H 57 105 113 121 W E 58 107 115 123 E I 59 109 117 125 I G 60 112 120 128 G H 61 115 123 131 H T 62 118 126 135 T 63 122 130 139 I 64 126 134 143 I N 65 129 137 147 N 66 133 141 151 I 67 137 145 155 P N 68 141 149 158 O C 69 145 153 162 U H 70 149 157 165 N E 71 153 161 169 D S 72 157 165 173 S An arithmetic verb can also be represented by a table, as is illustrated below for the case of multiplication. Since the domain of multiplication includes all numbers, no table can represent the entire multiplication verb; this table, for example, applies only to the domain of the first few integers. The multiplication sign in the upper left corner is included simply to indicate the arithmetic verb which the table represents. In any table, the first column represents the domain of the first argument and the first row represents the domain of the second argument; the rest is called the body of the table. For example the body of this table is that part bordered on the left and top by the solid lines. Multiplication Table Verb Right Domain Name * 1 2 3 4 5 6 7 8 9 10 1 1 2 3 4 5 6 7 8 9 10 2 2 4 6 8 10 12 14 16 18 20 3 3 6 9 12 15 18 21 24 27 30 Left 4 4 8 12 16 20 24 28 32 36 40 Domain 5 5 10 15 20 25 30 35 40 45 50 6 6 12 18 24 30 36 42 48 54 60 7 7 14 21 28 35 42 49 56 63 70 8 8 16 24 32 40 48 56 64 72 80 In any table representing a function of two arguments, any one column of the body (taken together with the column of arguments not in the body) represents a function of one argument. For example, the second column of the body of the normal weight table for two arguments represents the same function of one argument as does the original normal weight table. Thus any function of two arguments can be thought of as a collection of functions of one argument. For example, the second column of the body of the times table above represents the "times two" function, the third column represents the "times three" function, etc. Similarly, one row of the body of a function table represents a function of one argument. For of example, the fifth row of the body of Figure 2.2 gives weights as a function of "frame" for 61 inch women. #5-10 Reading Function Tables The basic rule for reading a function table is very simple: to evaluate a function, find the row in which the value of the first argument occurs (in the first column, not in the body of the table) and find the column in which the second argument occurs (in the first row) and select the element at the intersection of the selected row and the selected column. However, just as there is more to reading an English sentence than pronouncing the individual words, so a table can be "read" so as to yield useful information about a function beyond that obtained by simply evaluating it for a few cases. For example, the second normal weight table can be "read" so as to answer the following questions: Can two women of different heights have the same normal weight? For a given frame type, does normal weight always increase with increasing height? For a given height, does normal weight increase with frame type? How many inches of height produce (about) the same change in weight as the change from small to large frame? Does this change remain about the same throughout the table? Arithmetic verbs are more orderly than a function such as that represented by the table above, and the patterns that can be detected in reading their function tables are more striking and interesting. Consider, for example, an attempt to read this table to answer the following questions: The second column of the body (which was previously remarked to represent the "times two" verb) contains the numbers 2 4 6, etc., which are encountered in "counting by twos". Can a similar statement be made about the other columns? Is there any relation between corresponding rows and columns of the body, e.g., between the third row and the third column? Can every result in the body be obtained in at least two different ways? Are there any results which can be obtained in only two ways? Similarly, one can construct a verb table for addition and read it to determine answers to the following questions: In how many different ways can the result 6 be obtained by addition? Does the result 6 occur in the table in some pattern and if so does a similar pattern apply to other results such as 7, 8, etc.? What is the relation between two successive rows of the table? Because of the patterns they exhibit, verb tables can be very helpful in gaining an understanding of unfamiliar mathematical verbs. For this reason they will be used extensively in succeeding chapters. Expressions for Producing Verb Tables If a=.1 2 3 4 5 6 7 8 b=.1 2 3 4 5 6 7 8 9 10 then the expression a/byields the body of the multiplication table as follows: The general rule is that the symbol for a verb followed by the symbol / produces the appropriate verb table when applied to any arguments a and b. The expression a+/b may be read as "the addition table for a and b" or "a addition table b", or even as "a plus slash b". Similarly, "a/b", may be read as "a times table b", etc. When we want to create a table using the same list for both arguments, in the form a f/a, we have the option of abbreviating the expression in the form f/~a, using the reflexive adverb. The saving is not great when the argument is a simple variable, but it becomes more significant when the argument is an expression. Use of the reflexive adverb is akin to the use of reflexive pronouns in English and other languages, in forms such as "Take care of yourself." For example: It is important to note that the expression a+/b produces only the body of the addition table, to which one may add a first column consisting of a and a first row consisting of b if this is found to make the table easier to read. Here is an adverb for creating simple verb tables. For now, we can just use it. How it works is explained in Chapter Nine, Verb, Adverb, and Conjunction Definition. It is also important to note the difference between the expression a/b, which yields the multiplication table, and the expression ab, which yields the element-by-element product of a and b. For example: a=.1 3 5 b=.2 4 6 ab 2 12 30 a/b 2 4 6 6 12 18 10 20 30 #12-13 The body of a table alone does not define a function. For example, the following tables define two distinct functions although the bodies of the tables are identical: + 2 3 4 5 f 2 3 5 7 2 4 5 6 7 6 4 5 6 7 3 5 6 7 8 5 5 6 7 8 4 6 7 8 9 4 6 7 8 9 5 7 8 9 10 3 7 8 9 10 The name of the function represented by the first table is + (as shown in the upper left-hand corner), and the table can be used to evaluate expressions as shown on the left below: 5 + 3 is 8 5 f 3 is 6 4 + 5 is 9 4 f 5 is 8 3 + 3 is 6 3 f 3 is 8 The function represented by the second table is called f (as indicated in the upper left corner) and the expressions on the right above show the evaluation of the function f for the same arguments used on the left. Since the results differ, the two tables represent different functions. The complete specification of a function table therefore requires the specification of four items: The left domain (i.e., the domain of the left argument) The right domain The body of the table The name of the function From these four items the table can be constructed and used as illustrated below: Left domain: 3 4 5 6 Right domain: 11 9 7 5 3 1 Body: 5+(3i.4)+/(2i.6) Name: g Function Right Domain Name g 11 9 7 5 3 1 3 10 12 14 16 18 20 Left Domain 4 13 15 17 19 21 23 5 16 18 20 22 24 26 6 19 21 23 25 27 29 4 g 5 is 19 6 g 9 is 21 26 g 9 is 42 #14-16 The <. and >. Verbs The advantages of the verb table can perhaps be better appreciated by applying it to some unfamiliar verbs than by applying it to verbs such as addition and multiplication which are probably already well understood by the reader. For this purpose we will now introduce several simple new verbs, denoted by <. and >., which will also be found to be very useful in later work. It is sometimes instructive to introduce a new function as a puzzle—the reader must determine the general rule for evaluating the function by examining the results obtained when it is applied to certain chosen arguments. For example, the function <. can be applied to certain arguments with the results shown below: 3<.8 3 47<.32 32 If one performs enough such experiments it should be possible to guess the general rule for the function. In attempting such a guess it is helpful to organize the experiments in some systematic way, and the body of the function table provides precisely the sort of organization needed. For example: From the foregoing the reader should be able to state the definition of the function and from that statement be able to apply it correctly to any pair of arguments. The functions <. is called the minimum function because it yields the smaller of its two arguments. The maximum function is denoted by >. and is defined analogously. The body of its function table appears below: The [ and ] Verbs Let us treat the verbs denoted by [ and ] similarly as puzzles. For example: 1[2 1 1]2 2 1[0 1 2 3 1 1]0 1 2 3 0 1 2 3 0 1 2 3[1 0 1 2 3 Clearly [ and ] are not scalar (rank 0) verbs. Given a list argument and a scalar argument, they can yield either a scalar or a list. Can you tell what the rule is for them yet? To find out more, let us make the scalar versions of these verbs, that is ["0 and ]"0, and try them out: We have seen the effect of the dyadic [ and ] verbs on individual numbers and on lists of numbers. The [ verb, called left, returns its left argument unchanged, while the ] verb returns ins right argument unchanged. #19-20 In the monadic case, each returns its argument unchanged. In this case, they are both known as same. For example: ]1 1 [1 1 ]2 1 4 2 1 4 [2 1 4 2 1 4 Normally an assignment statement does not result in a value being displayed. Sometimes this is what we want, and at other times we want to make the assignment and then immediately display the value. We can always do it this way: a=.5 6 7 a 5 6 7 However, J allows us to abbreviate this process using the ] verb, known as right or same, described above. When applied to a single value, it returns its argument, thus: ]a=.5 6 7 5 6 7 We will use this abbreviation for assign and display throughout the rest of this book. In general, ]y returns the value of y. Similarly, x]y returns y. The right function is not provided just for this simple convenience, but has other important uses that will appear later. Similarly, there is a left function [ . As you might suppose, x[y is x. However, when [ has no left argument, it returns the argument it does have, so that [y is y. The Power Function Another very useful function is called the power function and is denoted by ^ . The body of its function table is shown below: The power function is defined in terms of multiplication in much the same way as multiplication is defined in terms of addition. To appreciate how multiplication is defined as "repeated additions", consider the following expressions: In general, m to the power n (that is, m^n) is obtained by multiplying together n factors each having the value m. The special case of m^2 occurs so frequently in algebra and elsewhere that we abbreviate it as :m, read square m. #21-24 Maps Figure 2.4 shows a map which represents the "times two" function. The rule for evaluating a function represented by a map is very simple: locate the specified argument in the top row, then follow the arrow from that argument to the result at the head of the arrow in the bottom row; e.g. the result for the argument 3 is 6. Map of "Times Two" Function Figure 2.4 The rules for constructing a map are also simple. First consider all of the values in the domain of the function together with all of the results, and choose the smallest number and the largest number from among them. Write a row of numbers beginning with the smallest and continuing through each of the integers in order up to the largest. Repeat the same numbers in a row directly below the first row. For each argument in the top row now draw an arrow to the corresponding result in the bottom row. Just is it is often helpful to read tables, so is it helpful to read such maps. Consider the four maps shown below. From the first it is clear that in the map of addition of 2, the arrows are all parallel. From the map below this it is clear that the same is true for addition of 3, and that the slope of the arrow depends on the amount added. The maps on the right show multiplication; here the arrows are not parallel, and the distance between successive arrowheads is seen to be equal to the multiplier. Maps for Addition and Multiplication Figure 2.5 It is sometimes useful to show the maps of a sequence of functions such as the following. i=.1 2 3 4 5 6 2i 2 4 6 8 10 12 8+2i 10 12 14 16 18 20 The appropriate maps are shown in the figure below. The broken lines show the map of the overall result produced, that is, the map of the function 8+2*i . Maps of a Sequence of Functions Maps will be used in the next chapter to introduce the function subtraction and the new negative numbers which this function produces. #25-26 There has been error in communication with Booktype server. Not sure right now where is the problem.
This book provides a comprehensive introduction to various mathematical approaches to achieving high-quality software. An introduction to mathematics that is essential for sound software engineering is provided as well as a discussion of various mathematical methods that are used both in academia and industry. The mathematical approaches considered include: Z specification language Vienna Development Methods (VDM) Irish school of VDM (VDM) approach of Dijkstra and Hoare classical engineering approach of Parnas Cleanroom approach developed at IBM software reliability, and unified modelling language (UML). Additionally, technology transfer of the mathematical methods to industry is considered. The book explains the main features of these approaches and applies mathematical methods to solve practical problems. Written with both student and professional in mind, this book assists the reader in applying mathematical methods to solve practical problems that are relevant to software engineers. Table of Contents Table of Contents Introduction. Software Engineering Mathematics. Logic for Software Engineering. Z Specification Language. Vienna Development Method. Irish School of VDM. Dijkstra and Hoare. The Parnas Way. Cleanroom and Software Reliability. Unified Modeling Language. Technology Transfer. Glossary
Abstract Algebra - 3rd edition Summary: Highly regarded by instructors in past editions for its sequencing of topics as well as its concrete approach, slightly slower beginning pace, and extensive set of exercises, the latest edition of Abstract Algebra extends the thrust of the widely used earlier editions as it introduces modern abstract concepts only after a careful study of important examples. Beachy and Blair's clear narrative presentation responds to the needs of inexperienced students who stumble ov...show moreer proof writing, who understand definitions and theorems but cannot do the problems, and who want more examples that tie into their previous experience. The authors introduce chapters by indicating why the material is important and, at the same time, relating the new material to things from the student's background and linking the subject matter of the chapter to the broader picture. Instructors will find the latest edition pitched at a suitable level of difficulty and will appreciate its gradual increase in the level of sophistication as the student progresses through the book. Rather than inserting superficial applications at the expense of important mathematical concepts, the Beachy and Blair solid, well-organized treatment motivates the subject with concrete problems from areas that students have previously encountered, namely, the integers and polynomials over the real numbers. ...show less Sets Construction of the Number Systems Basic Properties of the Integers Induction Complex Numbers Solution of Cubic and Quartic Equations Dimension of a Vector Space
9780321654274 ISBN: 0321654277 Edition: 3 Pub Date: 2010 Publisher: Addison Wesley Summary: Beckmann, Sybilla is the author of Mathematics for Elementary Teachers with Activity Manual (3rd Edition), published 2010 under ISBN 9780321654274 and 0321654277. Two hundred ninety eight Mathematics for Elementary Teachers with Activity Manual (3rd Edition) textbooks are available for sale on ValoreBooks.com, one hundred eighty eight used from the cheapest price of $16.79, or buy new starting at $83
Engineers require a solid knowledge of the relationship between engineering applications and underlying mathematical theory. However, most books do not present sufficient theory, or they do not fully explain its importance and relevance in understanding those applications. Advanced Engineering Mathematics with Modeling Applications employs a balanced approach to address this informational void, providing a solid comprehension of mathematical theory that will enhance understanding of applications – and vice versa. With a focus on modeling, this book illustrates why mathematical methods work, when they apply, and what their limitations are. Designed specifically for use in graduate-level courses, this book: Emphasizes mathematical modeling, dimensional analysis, scaling, and their application to macroscale and nanoscale problems Explores eigenvalue problems for discrete and continuous systems and many applications Develops and applies approximate methods, such as Rayleigh-Ritz and finite element methods Presents applications that use contemporary research in areas such as nanotechnology Apply the Same Theory to Vastly Different Physical ProblemsPresenting mathematical theory at an understandable level, this text explores topics from real and functional analysis, such as vector spaces, inner products, norms, and linear operators, to formulate mathematical models of engineering problems for both discrete and continuous systems. The author presents theorems and proofs, but without the full detail found in mathematical books, so that development of the theory does not obscure its application to engineering problems. He applies principles and theorems of linear algebra to derive solutions, including proofs of theorems when they are instructive. Tying mathematical theory to applications, this book provides engineering students with a strong foundation in mathematical terminology and methods.
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Revolutionary Teaching: Professor Encourages Students to Use Mathematica on Tests Alain Carmasol, Associate Professor, Ecole Nationale d'ingénieurs This content requires JavaScript and Adobe Flash Player 10 or higher. If you are using a browser with JavaScript or Flash disabled, please enable them now. Otherwise, please install the latest version of the free Flash Player. The Mathematica Edge Use built-in functions for Taylor series and Fourier series analysis for applications in vibrations, system dynamics, and more Model real-world systems that include parts from multiple physical domains, such as mechanics, electronics, and control systems, with Wolfram SystemModeler "The majority of traditional programming environments only work in digital, machine-like precision. However, Mathematica's primary function gives you the ability to do analytical calculations—a big plus you won't find in any other traditional environment." Challenge As associate professor at the National Engineering Institute in Metz, Alain Carmasol needs to present mathematical examples without distracting students from basic concepts and methods. Hand calculations are time consuming, often preventing students from seeing the larger picture. Solution Mathematica allows Carmasol and his students to get back to the basics of analytical approximations and easily implement the concepts in examples. For instance, he uses it in lectures to show the solution of the equation of vibrating cords using the classic technique of the Fourier series, and to show the approximate solution of partial derivative equations. Benefits According to Carmasol, students are able to replace things in very few lines of code with one or two functions—things that, in normal environments, would take dozens or hundreds of lines of code. Mathematica works so well in the engineering classroom that he encourages students to use it while taking exams. Carmasol believes that making students do lines and lines of calculations by hand no longer has any value.
Introduction to Problem Solving Grades 3-5 9780325009704 ISBN: 0325009708 Edition: 2 Pub Date: 2007 Publisher: Heinemann Summary: Susan O'Connell is the editor of Heinemann's Math Process Standards series, as well as the author its volumes Introduction to Problem Solving (grades PreK - 2, 3 - 5, and 6 - 8) and Introduction to Communication (grades PreK - 2, 3 - 5, and 6 - 8). She also wrote the popular Now I Get It (Heinemann, 2005). Sue has a varied background, including years as a classroom teacher, a school-based instructional specialist, a ...testing coordinator, a talented-and-gifted teacher, a district school-improvement specialist, and a university professional-development schools coordinator. Currently she is a project consultant for a federal teacher-quality grant in the College of Education at the University of Maryland. Additionally, she is an educational consultant, conducts mathematics seminars for teachers throughout the country, and a Heinemann Professional Development Provider. O'Connell, Susan is the author of Introduction to Problem Solving Grades 3-5, published 2007 under ISBN 9780325009704 and 0325009708. One hundred ten Introduction to Problem Solving Grades 3-5 textbooks are available for sale on ValoreBooks.com, six used from the cheapest price of $12.31, or buy new starting at $382007. An excellent copy with clean pages that are free of writing. INCLUDES CD-ROM. Booksavers receives donated books and recycles them in a variety of ways. Proceeds benefit [more] 2007. An excellent copy with clean pages that are free of writing. INCLUDES CD-ROM. Booksavers receives donated books and recycles them in a variety of ways. Proceeds benefit the work of Mennonite Central Committee (MCC) in the U.S. and around
About the Math Center The Math Center, located in LIB 101, is the place to go for help with math homework and to meet with a tutor. The Math Center is staffed by students with math expertise in a number of subjects, ranging from basic math thru calculus. The Math Center Operates Homework Help Sessions every week as well as scheduled workshops on select math topics. Homework Help Sessions These sessions are scheduled to provide walk in assistance for students while they are completing thier homework assignments. Each week several hours are dedicated to Homework Help. If you need ongoing or more intensive help, you should schedule an appointment with a tutor. Homework Help is for students enrolled in MATH 030, MATH 060, MATH 101S, MATH 102, MATH 109, MATH 110, and MATH 123.Homework Helpis conducted in the Math Center (LIB 101). Mondays, Tuesdays, Thursdays 3 – 7 pm Wednesdays 1 – 7 pm Monday Math Help Every Monday, Math Specialists and tutors hold problem solving sessions to help students in MATH 101S. This time is dedicated to helping students learn to (algebra) problem solve. Participants are assigned to work out challening problems so that they may discover the proper steps as well as how to work through problems successfully. These sessions, open to students enrolled in MATH 101S, meet in LIB SEM (library 2nd floor). Monday (afternoons) 3 - 4:30 pm Monday (evenings) 6:30 – 8 pm Friday Math Labs Friday Math Labs are provided for students enrolled in Math 101s, 102, 108, 109, and 110. Friday Math Labs are conducted by Math Specialists to provide extra instruction and support to students to ensure students have a solid foundation from algebra through statistics. Friday Math Lab is required for students enrolled in 101S and strongly recommended for any student having difficulty in the other math courses. Through the lab, students will have an opportunity to address questions and issues in greater detail than possible during regular class sessions. Because of limited spacing and seating in Friday labs, students are served on a first-come, first-served basis.
Lectures on Gas Theory by Ludwig Boltzmann A masterpiece of theoretical physics, this classic contains a comprehensive exposition of the kinetic theory of gases. It combines rigorous mathematic analysis with a pragmatic treatment of physical and chemical applications. Generalized Functions and Partial Differential Equations by Avner Friedman This self-contained text details developments in the theory of generalized functions and the theory of distributions, and it systematically applies them to a variety of problems in partial differential equations. 1963A Survey of Industrial Mathematics by Charles R. MacCluer Students learn how to solve problems they'll encounter in their professional lives with this concise single-volume treatment. It employs MATLAB and other strategies to explore typical industrial problems. 2000 editionHow to Solve Applied Mathematics Problems by B. L. Moiseiwitsch This workbook bridges the gap between lectures and practical applications, offering students of mathematics, engineering, and physics the chance to practice solving problems from a wide variety of fields. 2011 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. 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
New Equitable Syllabus Guides (All Subjects) for X Standard available in Sura Books Salient Features : Complete solutions to Text Book Exercises Classification of Additional and Textual Problems Newly introduced SMS - Tips for Students Chapter-wise Unit Tests and Model Question Papers Note from publishers: It gives me great pride and pleasure in bringing to you Sura's Mathematics Guide for X Standard (Based on New Uniform Syllabus System of Education). As this common syllabus frame work has been introduced only this year, the students and teachers have to carefully understand the concepts, theorems, examples and exercises. A deep understanding of the examples and exercises is rudimentary to have an insight into the use of formulae and theorems. Sura's Maths X Standard Guide encompasses all the requirements of the students to understand the concepts, examples and exercises. It will be a teaching companion to teachers and a learning companion to students. As the guide has been framed based on the 'Blue Print' and 'Question Paper Pattern', it provides a precise and clear understanding of examples and exercises from the examination perspective. Exhaustive additional exercises will help students practise and learn effectively all the chapters of the textbook. The newly introduced chapter-wise SMS (Success Mantra Site) provides smart learning tips to help students remember the important concepts, formulae and theorems in that particular chapter. The table 'The Classification of Text Book Exercises' has been introduced for the first time in our guide to help students and teachers classify the textual exercises based on the Question Paper Pattern. Though these salient features are available in our Sura's Mathematics Guide for X-Standard, I cannot negate the indispensable role of the teachers in assisting the student to understand the Maths concepts and examples in each chapter. I sincerely believe this guide satisfies the needs of the students and bolsters the teaching methodologies of the teachers. I pray the almighty to bless the students for consummate success in their examinations. Some special features of this book are : Each chapter begins with a small introduction. These introductions are designed to create interest in the aesthetic components of Mathematics. SMS - Success Mantra Site. (Smart tips for students) CAT - Classification of Additional and Textual problems. 3 'R's - Relax, Recreate, Rejuvenate. (Just for Relaxation) Unit Tests Model Papers I wish all students best of luck to score 100% in Mathematics. - Publishers Samacheer Kalvi Thittam New Equitable Syllabus Guides (All Subjects) for X Standard available in Sura Books
Numbers, Groups and Codes 9780521540506 ISBN: 052154050X Edition: 2 Pub Date: 2004 Publisher: Cambridge Univ Pr Summary: A thoroughly revised and updated version of the popular textbook on abstract algebra. The material is introduced with clarity and reference to problems and concepts that students will easily understand. With many examples and exercises, it will serve as the ideal introduction to this important and ubiquitous subject. Humphreys, J. F. is the author of Numbers, Groups and Codes, published 2004 under ISBN 97805...21540506 and 052154050X. Three hundred thirty one Numbers, Groups and Codes textbooks are available for sale on ValoreBooks.com, one hundred four used from the cheapest price of $39.99, or buy new starting at $61.81
Rent Textbook Buy Used Textbook Buy New Textbook Currently Available, Usually Ships in 24-48 Hours $165.10 eTextbook We're Sorry Not Available More New and Used from Private Sellers Starting at $30Clear and Concise. Varberg focuses on the most critical concepts. This popular calculus text remains the shortest mainstream calculus book available yet coversallrelevant material needed by, and appropriate to, the study of calculus at this level. It's conciseness and clarity helps you focus on, and understand, critical concepts in calculus without them getting bogged down and lost in excessive and unnecessary detail. It is accurate, without being excessively rigorous, up-to-date without being faddish. Table of Contents Preface ix Preliminaries 1 (54) Real Numbers, Estimation, and Logic 1 (7) Inequalities and Absolute Values 8 (8) The Rectangular Coordinate System 16 (8) Graphs of Equations 24 (5) Functions and Their Graphs 29 (6) Operations on Functions 35 (6) Trigonometric Functions 41 (10) Chapter Review 51 (4) Review and Preview Problems 54 (1) Limits 55 (38) Introduction to Limits 55 (6) Rigorous Study of Limits 61 (7) Limit Theorems 68 (5) Limits Involving Trigonometric Functions 73 (4) Limits at Infinity; Infinite Limits 77 (5) Continuity of Functions 82 (8) Chapter Review 90 (3) Review and Preview Problems 92 (1) The Derivative 93 (58) Two Problems with One Theme 93 (7) The Derivative 100 (7) Rules for Finding Derivatives 107 (7) Derivatives of Trigonometric Functions 114 (4) The Chain Rule 118 (7) Higher-Order Derivatives 125 (5) Implicit Differentiation 130 (5) Related Rates 135 (7) Differentials and Approximations 142 (5) Chapter Review 147 (4) Review and Preview Problems 150 (1) Applications of the Derivative 151 (64) Maxima and Minima 151 (4) Monotonicity and Concavity 155 (7) Local Extrema and Extrema on Open Intervals 162 (5) Practical Problems 167 (11) Graphing Functions Using Calculus 178 (7) The Mean Value Theorem for Derivatives 185 (5) Solving Equations Numerically 190 (7) Antiderivatives 197 (6) Introduction to Differential Equations 203 (6) Chapter Review 209 (6) Review and Preview Problems 214 (1) The Definite Integral 215 (60) Introduction to Area 215 (9) The Definite Integral 224 (8) The First Fundamental Theorem of Calculus 232 (11) The Second Fundamental Theorem of Calculus and the Method of Substitution
MATHEMATICS OF FINANCE >CANADI Description: Zima and Brown continue to identify a generic approach to problem solving with a wide range of interest rates within the problems presented in the text. They also provided the following set of pedagogical and financial tools. This text emphasizesMore... Buy it from: $3.00 Customers Also Bought Zima and Brown continue to identify a generic approach to problem solving with a wide range of interest rates within the problems presented in the text. They also provided the following set of pedagogical and financial tools. This text emphasizes the point that the most important aspect for the student is to be able to visualize the problem. Timeline diagrams help the student to determine how to solve the problem from first principles.They emphasize the use of calculators and Excel spreadsheets (solutions provided where appropriate) in problem-solving techniques, and include Internet-based resources and tools.Exercises for each topic in the text are stratified into fundamental learning exercises in Part A, and more challenging and theoretical problems in Part B. Each chapter closes with the Summary and Review Exercises, and, in many chapters, the Review Exercises include one or more Case Studies presenting more complex real-world
0131400231 9780131400238 Elementary Algebra for College Students:For freshman-level, one- or two- semester courses in Developmental Algebra.The Angel Series continues to offer proven pedagogy sound exercise sets and superior student support. An emphasis on the practical applications of algebra motivates students and encourages them to see algebra as an important part of their daily lives. The student-friendly writing style uses short, clear sentences and easy-to-understand language, and the outstanding pedagogical program makes the material easy to follow and comprehend. The new editions continue to place a strong emphasis on problem solving, incorporating it as a theme throughout the texts. Angel's solid exercise sets are recognized by reviewers as of the highest standard providing a large number of problems, paired exercises, and a broad and increasing range of difficulty. Back to top Rent Elementary Algebra for College Students 6th edition today, or search our site for Allen R. textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Prentice Hall.
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0495382671Product Details ISBN-13: 9780495382676 Publisher: Cengage Learning Publication date: 3/14/2007 Edition description: Solution M Edition number: 8 Pages: 320 Product dimensions: 8.40 (w) x 10.70 (h) x 0.70 (d) Meet the Author Charles P. "Pat" McKeague earned his B.A. in Mathematics from California State University, Northridge, and his M.S. in Mathematics from Brigham Young University. A well-known author and respected educator, he is a full-time writer and a part-time instructor at Cuesta College. He has published twelve textbooks in mathematics covering a range of topics from basic mathematics to trigonometry. An active member of the mathematics community, Professor McKeague is a popular speaker at regional conferences, including the California Mathematics Council for Community Colleges, the American Mathematical Association of Two-Year Colleges, the National Council of Teachers of Mathematics, the Texas Mathematics Association of Two-Year Colleges, the New Mexico Mathematics Association of Two-Year Colleges, and the National Association for Developmental Education. He is a member of the American Mathematics Association for Two-Year Colleges, the Mathematics Association of America, the National Council of Teachers of Mathematics, and the California Mathematics Council for Community Colleges
Wavelets and their Applications in Computer Graphics Wavelets and their Applications in Computer Graphics by Alain Fournier 1995 Number of pages: 239 Description: This course is intended to give the necessary mathematical background on wavelets, and explore the main applications, both current and potential, to computer graphics. The emphasis is put on the connection between wavelets and the tools and concepts which should be familiar to any skilled computer graphics person: Fourier techniques, pyramidal schemes, spline representations.
Elementary Linear Algebra - 9th edition Summary: Presents the fundamentals of linear algebra in the clearest possible way, examining basic ideas by means of computational examples and geometrical interpretation. This substantial revision includes greater focus on relationships between concepts, smoother transition to abstraction, early exposure to linear transformations and eigenvalues, more emphasize on visualization, new material on least squares and QR-decomposition and a greater number of proofs. Exercise sets ...show morebegin with routine drill problems, progress to problems with more substance and conclude with theoretical problems. ...show less 047166960111.9378 +$3.99 s/h Good SellBackYourBook Aurora, IL 0471669601 Item in good condition. Textbooks may not include supplemental items i.e. CDs, access codes etc... All day low prices, buy from us sell to us we do it all!!
Maths Olympiad Interested in Category?Leave your details and we will help you reach the best Colleges Snapshot About the Exam The Mathematics Olympiad is organized by NBHM (National Board for Higher Mathematics) and is currently run in association with the Homi Bhabha Centre for Science Education, Mumbai. NBHM then selects and trains the Indian team for participation in the International Mathematical Olympiad every year. While NBHM coordinates this Maths Olympiad at the national level, regional bodies (mostly voluntary) play an important role at different stages. Eligibility Only students selected on the basis of the RMO (Regional Mathematical Olympiad) from different regions are eligible to appear for INMO. All high school students up to class XII are eligible to appear for RMO How To Prepare The syllabus for Mathematics Olympiads (regional, national and international) is pre-degree college mathematics. The topics to be covered are: number systems, arithmetic of integers, geometry, quadratic equations and expressions, trigonometry, co-ordinate geometry, systems of linear equations, permutations and combinations, factorisation of polynomials, inequalities, elementary combinatorics, probability theory, number theory, infinite series, complex numbers and elementary graph theory. Calculus and statistics are not included. The typical areas for problems are: number theory, geometry, algebra and combinatorics. Even though the syllabus is spread over class 9-12 levels, the questions are exceptionally high level in difficulty and sophistication. The difficulty level increases from RMO to INMO to IMO. Books like Mathematics Olympiad Primer, by V.Krishnamurthy, C.R.Pranesachar, K.N. Ranganathan and B.J. Venkatachala (Interline Publishing Pvt. Ltd., Bangalore).Challenge and Thrill of Pre-College Mathematics, by V.Krishnamurthy, C.R.Pranesachar, K.N.Ranganathan and B.J.Venkatachala (New Age International Publishers, New Delhi - 1996) will help you get a feel of the kind of questions that you will face, Important Dates The Indian National Mathematical Olympiad (INMO) examination is conducted in February every year for students of all classes. The Regional Mathematical Olympiad (RMO) is held in each region normally between September and the first Sunday of December each year.
Computational science is an exciting new field at the intersection of the sciences, computer science, and mathematics because much scientific investigation now involves computing as well as theory and experiment. This textbook provides students with a versatile and accessible introduction to the subject. It assumes only a background in high school algebra, enables instructors to follow tailored pathways through the material, and is the only textbook of its kind designed specifically for an introductory course in the computational science and engineering curriculum. While the text itself is generic, an accompanying website offers tutorials and files in a variety of software packages. This fully updated and expanded edition features two new chapters on agent-based simulations and modeling with matrices, ten new project modules, and an additional module on diffusion. Besides increased treatment of high-performance computing and its applications, the book also includes additional quick review questions with answers, exercises, and individual and team projects. The only introductory textbook of its kind--now fully updated and expanded Features two new chapters on agent-based simulations and modeling with matrices An online instructor's manual with exercise answers, selected project solutions, and a test bank and solutions (available only to professors) An online illustration package is available to professors Angela B. Shiflet is the Larry Hearn McCalla Professor of Mathematics and Computer Science and director of computational science at Wofford College. George W. Shiflet is the Larry Hearn McCalla Professor of Biology at Wofford College. Review: Praise for the previous edition: "The heart of Introduction to Computational Science is a collection of modules. Each module is either a discussion of a general computational issue or an investigation of an application. . . . [This book] has been carefully and thoughtfully written with students clearly in mind."--William J. Satzer, MAA Reviews Praise for the previous edition: "Introduction to Computational Science is useful for students and others who want to obtain some of the basic skills of the field. Its impressive collection of projects allows readers to quickly enjoy the power of modern computing as an essential tool in building scientific understanding."--Wouter van Joolingen, Physics Today Praise for the previous edition: "A masterpiece. I know of nothing comparable. I give it five stars."--James M. Cargal, UMAP Journal Praise for the previous edition: "This is an important book with a wonderful collection of examples, models, and references."--Robert M. Panoff, Shodor Education Foundation Praise for the previous edition: "This is a very good introduction to the field of computational science."--Peter Turner, Clarkson University
Chapter Summary Image Attributions Description In this chapter, students will plot points in a polar coordinate system, graph and recognize limaçons and cardiods, and work with real-world applications involving polar coordinates and polar equations.
Get ready to master the principles and operations of algebra! Master Math: Algebra is a comprehensive reference guide that explains and clarifies algebraic principles in a simple, easy-to-follow style and format. Beginning with the most basic fundamental topics and progressing through to the more advanced topics that will help prepare you for pre-calculus... more... Get ready to master the principles and operations of calculus! Master Math: Calculus is a comprehensive reference guide that explains and clarifies the principles of calculus in a simple, easy-to-follow style and format. Beginning with the most basic fundamental topics and progressing through to the more advanced, the book helps clarify calculus using... more...... more...... more... Get ready to master the fundamentals of trigonometry! Master Math: Trigonometry is a comprehensive reference guide that explains and clarifies the principles of trigonometry in a simple, easy-to-follow style and format. Beginning with the most basic fundamental topics, including a review of basic geometry, and progressing through to the more advanced... more...
eh, for me i didn't really like math until calculus 2. That class is what made me want to be a math major. Algebra can be really boring because it's basically factoring and multiplying polynomials (usually binomials) and the quadratic equation. simple topics that they cover for an entire semester or longer. math is fun once you learn new concepts each week to week and a half. We learn new things every hour and do not get enough time to master them, so to speak. I need something that can motivate me to study for two math tests next week. With that said, I mean motivation that doesn't count the grade in. This is a great time to expand your mathematical knowledge. Go look for layman or introductory math books (not textbooks) that look interesting. At that age I would go to a Barnes and Noble (the actual physical store) and just browse the math and sciences isles for whatever caught my eye. Does your teacher have a bookcase in class with additional materials? Ask her/him what's a good or interesting read. If you want particular topics, some introductory number theory, combinatorics, linear algebra are all approachable either without or with minor algebra skills. Grab some books on Escher. You'll learn about symmetry, translations, scalings, projections and more without actually having the math of it forced at you. Not sure how helpful this will be, but if you think of math like an art (say painting) then you're sort of at the point of learning what brushes to use and how to choose the right colors. It's not exactly "fun" but it's a necessary step to move onto the point of being able to paint. Even when you start to paint you'll begin with simple shapes and pictures.

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