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Summary: Focusing on the important ideas of geometry, this book shows how to investigate two- and three-dimensional shapes with very young students. It introduces methods to describe location and position, explores simple transformations, and addresses visualization, spatial reasoning, and the building and drawing of constructions. Activities in each chapter pose questions that stimulate students to think more deeply about mathematical ideas. The CD-ROM features fourteen arti...show morecles from NCTM publications. The supplemental CD-ROM also features interactive electronic activities, master copies of activity pages for students, and additional readings for teachers. ...show less
Edition/Copyright: 01 Cover: Paperback Publisher: National Council of Teachers of Mathematics Published: 01/28/2001 International: No
Shows definite wear, and perhaps considerable marking on inside. 100% Money Back Guarantee. Shipped to over one million happy customers. Your purchase benefits world literacy!
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Math is tough but Microsoft has just given us a free graphical symbolic calculator to download. It can solve equations, do calculus, work with matrices and plot the results. As they say, why pay more?.
Microsoft Mathematics is a interactive tool for doing mathematics and it can do both numeric and symbolic calculations. Give it an equation and it can solve it. Give it a function and it will integrate or differentiate it for you and offer to graph the result. It is an amazing tool and it's easy to use.
It is intended for use by students and promoted by Microsoft Education and previously sold for around $20. Now the new Version 4 is available free to download. There hasn't really been much information forthcoming from Microsoft about it and it is almost as if it has been slipped out in the hope that no one will notice.
After you have downloaded it you can use it like a simple calculator to do arithmetic, advanced arithmetic even, but you can also use it to do symbolic maths. You can type in an equation and ask it for the solution and in many cases it makes a sensible attempt at an answer. It can do matrix calculations, algebra and calculus. You need the derivative or integral of some function - just type it in and the program will perform the symbolic manipulation for you. It also draws graphs of functions and data and ... well the list goes on.
It also comes with a good help file, tutorials and it supports ink input so you can write equations into the edit box.
It isn't as good as Mathematica or Maple but it does enough for many users not to need to go beyond it. Given that it is free it also represents a bargain. And yes it has a multibase conversion function so you can use it to do programming calculations. It's a great educational tool but it is also suitable for serious calculations.
The only problem is that there is no scripting language for it and no API specification, so it looks as if it can't be easily extended. As it is a .NET WPF application the usual techniques for taking control of it are unlikely to work. This is a shame because with a scripting language it could do so much more.
As it stands Microsoft has just given us a free graphical symbolic calculator - why not download it and give it a try.
32- and 64-bit versions are available and it runs under just about everything from Windows XP SP3 up.
More Information
JPEG is well known, well used and well understood. Surely there cannot be anything left to squeeze out of this old compression algorithm? Mozilla seems to think that we can get more if we are careful. [ ... ]
As programmers we often think that users are overly sensitive about their data. What could it hurt to allow the collection of location data, for example. Here is a short video from the ACLU that might [ ... ] |
Joy of Mathematics & Change and Motion: Calculus Made Clear (Set)
COURSE DESCRIPTION
Exercise your brain cells with two fascinating courses that make mathematics both accessible and enjoyable. In The
Joy of Mathematics, you discover the amazing utility of this exciting field in everything from science and
engineering to finance, games of chance, and many other aspects of life. Taught by award-winning Professor Arthur T.
Benjamin—a literal magician with numbers—these 24 lectures reveal how the beautiful and often imposing edifice
that has given us algebra, trigonometry, geometry, probability, and so much else is based on nothing more than fooling around
with numbers.
Then, focus on calculus and the ways it appears around you. Change and Motion: Calculus Made Clear, 2nd
Edition is crafted to make the key concepts and triumphs of this mathematical field accessible to nonmathematicians.
Over the course of 24 lectures, award-winning Professor Michael Starbird teaches you how to grasp the power and beauty of
calculus without the technical background traditionally required in calculus courses. As you follow the intellectual
development of calculus, your appreciation of its inner workings will deepen, and your skill in seeing how it can solve a
variety of problems will increase.
Course 1 of 2:
Calculus has had a notorious reputation for being difficult to understand, but the 24 lectures of Change and Motion: Calculus Made Clear are crafted to make the key concepts and triumphs of this field accessible to non-mathematicians. This course teaches you how to grasp the power and beauty of calculus without the technical background traditionally required in calculus courses. Follow award-winning Professor Michael Starbird as he takes you through derivatives and integrals—the two concepts that serve as the foundation for all of calculus. As you investigate the field's intellectual development, your appreciation of its inner workings and your skill in seeing how it can solve a variety of problems will deepenTwo Ideas, Vast Implications
Calculus is a subject of enormous importance and historical impact. It provides a dynamic view of the world and is an invaluable tool for measuring change. Calculus is applicable in many situations, from the trajectory of a baseball to changes in the Dow Jones average or elephant populations. Yet, at its core, calculus is the study of two ideas about motion and change.
13.
Achilles, Tortoises, Limits, and Continuity
The integral's strategy of adding up little pieces solves a variety of problems, such as finding the volume of a pyramid or the total pressure on the face of a dam.
2.
Stop Sign Crime—The First Idea of Calculus—The Derivative
The example of a car moving down a straight road is a simple and effective way to study motion. An everyday scenario that involves running a stop sign and the use of a camera illustrates the first fundamental idea of calculus: the derivative.
3.
Another Car, Another Crime—The Second Idea of Calculus—The Integral
You are kidnapped and driven away in a car. You can't see out the window, but you are able to shoot a videotape of the speedometer. The process by which you can use information about speed to compute the exact location of the car at the end of one hour is the second idea of calculus: the integral.
15.
The Best of All Possible Worlds—Optimization
Calculus is useful in many branches of mathematics. The 18th-century French scientist Georges Louis Leclerc Compte de Buffon used calculus and breadsticks to perform an experiment in probability. His experiment showed how random events can ultimately lead to an exact number.
4.
The Fundamental Theorem of Calculus
The moving car scenario illustrates the Fundamental Theorem of Calculus. This states that the derivative and the integral are two sides of the same coin. The insight of calculus, the Fundamental Theorem creates a method for finding a value that would otherwise be hard or impossible to get, even with a computer.
16.
Economics and Architecture
Zeno's Arrow Paradox concerns itself with the fact that an arrow traveling to a target must cover half the total distance, then half the remaining distance, etc. How does it ever get there? The concept of limit solves the problem.
5.
Visualizing the Derivative—Slopes
Change is so fundamental to our vision of the world that we view it as the driving force in our understanding of physics, biology, economics—virtually anything. Graphs are a way to visualize the derivative's ability to analyze and quantify change.
17.
Galileo, Newton, and Baseball
The real numbers in toto constitute a smooth, seamless continuum. Viewing the world as continuous in time and space allows us to make mathematical models that are helpful and predictive.
6.
Derivatives the Easy Way—Symbol Pushing
The derivative lets us understand how a change in one variable affects a dependent quantity. We have studied this relationship with respect to time. But the derivative can be abstracted to many other dependencies, such as that of the area of a circle on the length of its radius, or supply or demand on price.
18.
Getting off the Line—Motion in Space
Zeno's Arrow Paradox shows us that an infinite addition problem (1/2 + 1/4 + 1/8 + . . .) can result in a single number: 1. Similarly, it is possible to approximate values such as π or the square root of 2 by adding up the first few hundred terms of infinite sum. Calculators use this method when we push the "sin" or square root keys.
7.
Abstracting the Derivative—Circles and Belts
One of the most useful ways to consider derivatives is to view them algebraically. We can find the derivative of a function expressed algebraically by using a mechanical process, bypassing the infinite process of taking derivatives at each point.
19.
Mountain Slopes and Tangent Planes
We have seen how to analyze change and dependency according to one varying quantity. But many processes and things in nature vary according to several features. The steepness of a mountain slope is one example. To describe these real-world situations, we must use planes instead of lines to capture the philosophy of the derivative.
8.
Circles, Pyramids, Cones, and Spheres
The description of moving objects is one of the most direct applications of calculus. Analyzing the trajectories and speeds of projectiles has an illustrious history. This includes Galileo's famous experiments in Pisa and Newton's theories that allow us to compute the path and speed of projectiles, from baseballs to planets.
20.
Several Variables—Volumes Galore
After developing the ideas of calculus for cars moving in a straight line, we have gained enough expertise to apply the same reasoning to anything moving in space—from mosquitoes to planets.
9.
Archimedes and the Tractrix
Optimization problems—for example, maximizing the area that can be enclosed by a certain amount of fencing—often bring students to tears. But they illustrate questions of enormous importance in the real world. The strategy for solving these problems involves an intriguing application of derivatives.
21.
The Fundamental Theorem Extended
Calculus plays a central role in describing much of physics. It is integral to the description of planetary motion, mechanics, fluid dynamics, waves, thermodynamics, electricity, optics, and more. It can describe the physics of sound, but can't explain why we enjoy Bach.
10.
The Integral and the Fundamental Theorem
Formulas for areas and volumes can be deduced by dividing such objects as cones and spheres into thin pieces. Ancient examples of this method were precursors to the modern idea of the integral.
22.
Fields of Arrows—Differential Equations
Many money matters are prime examples of rates of change. The difference between getting rich and going broke is often determined by our ability to predict future trends. The perspective and methods of calculus are helpful tools in attempts to decide such questions as what production levels of a good will maximize profit.
11.
Abstracting the Integral—Pyramids and Dams
Archimedes devised an ingenious method that foreshadowed the idea of the integral in that it involved slicing a sphere into thin sections. Integrals provide effective techniques for computing volumes of solids and areas of surfaces. The image of an onion is useful in investigating how a solid ball can be viewed as layers of surfaces.
23.
Owls, Rats, Waves, and Guitars
Whether looking at people or pachyderms, the models for predicting future populations all involve the rates of population change. Calculus is well suited to this task. However, the discrete version of the Verhulst Model is an example of chaotic behavior—an application for which calculus may not be appropriate.
12.
Buffon's Needle or π from Breadsticks
The integral involves breaking intervals of change into small pieces and then adding them up. We use Leibniz's notation for the integral because the long S shape reminds us that the definition of the integral involves sums.
24.
Calculus Everywhere
There are limits to the realms of applicability of calculus, but it would be difficult to exaggerate its importance and influence in our lives. When considered in all of its aspects, calculus truly has been—and will continue to be—one of the most effective and influential strategies for analyzing our world that has ever been devised.
Course 2 of 2:
Humans have been having fun and games with mathematics for thousands of years. Along the way, they've discovered the amazing utility of this field—in science, engineering, finance, games of chance, and many other aspects of life. This course of 24 half-hour lectures celebrates the sheer joy of math, taught by a mathematician who is literally a magician with numbers. Professor Arthur T. Benjamin shows how everything in mathematics is connected—how the beautiful and often imposing edifice that has given us algebra, geometry, trigonometry, calculus, probability, and so much else is based on nothing more than fooling around with numbersThe Joy of Math—The Big Picture
Professor Benjamin introduces the ABCs of math appreciation: The field can be loved for its applications, its beauty and structure, and its certainty. Most of all, mathematics is a source of endless delight through creative play with numbers.
13.
The Joy of TrigonometryTrigonometry deals with the sides and angles of triangles. This lecture defines sine, cosine, and tangent, along with their reciprocals, the cosecant, secant, and cotangent. Extending these definitions to the unit circle allows a handy measure of angle: the radian.
2.
The Joy of Numbers
How do you add all the numbers from 1 to 100—instantly? What makes a square number square and a triangular number triangular? Why do the rules of arithmetic really work, and how do you calculate in bases other than 10?
14.
The Joy of the Imaginary Number i
Could the apparently nonsensical number the square root of –1 be of any use? Very much so, as this lecture shows. Such imaginary and complex numbers play an indispensable role in physics and other fields, and are easier to understand than they appear.
3.
The Joy of Primes
A number is prime if it is evenly divisible by only itself and one: for example, 2, 3, 5, 7, 11. Professor Benjamin proves that there are an infinite number of primes and shows how they are the building blocks of our number system.
15.
The Joy of the Number e
Another indispensable number to learn is e = 2.71828 ... Defined as the base of the natural logarithm, e plays a central role in calculus, and it arises naturally in many spheres of mathematics, including calculations of compound interest.
4.
The Joy of CountingCombinatorics is the study of counting questions such as: How many outfits are possible if you own 8 shirts, 5 pairs of pants, and 10 ties? A trickier question: How many ways are there to arrange 10 books on a shelf? Combinatorics can also be used to analyze numbering systems, such as ZIP Codes or license plates, as well as games of chance.
16.
The Joy of Infinity
What is the meaning of infinity? Are some infinite sets "more" infinite than others? Could there possibly be an infinite number of levels of infinity? This lecture explores some of the strange ideas associated with mathematical infinity.
5.
The Joy of Fibonacci Numbers
The Fibonacci numbers follow the simple pattern 1, 1, 2, 3, 5, 8, etc., in which each number is the sum of the two preceding numbers. Fibonacci numbers have many beautiful and unexpected properties, and show up in nature, art, and poetry.
17.
The Joy of Infinite Series
Starting with the analysis of the proposition 0.999999999 ... = 1, this lecture explores what it means to add up an infinite series of numbers. Some infinite series converge on a definite value, while others grow arbitrarily large.
6.
The Joy of Algebra
Arguably the most important area of mathematics, algebra introduces the powerful idea of using an abstract variable to represent an unknown quantity. This lecture demonstrates algebra's golden rule: Do unto one side of an equation as you do unto the other.
18.
The Joy of Differential Calculus
Calculus is the mathematics of change, and answers questions such as: How fast is a function growing? This lecture introduces the concepts of limits and derivatives, which allow the slope of a curve to be measured at any point.
7.
The Joy of Higher Algebra
This lecture shows how to solve quadratic (second-degree) equations from the technique of completing the square and the quadratic formula. The quadratic formula reveals the connection between Fibonacci numbers and the golden ratio.
19.
The Joy of Approximating with Calculus
Exploiting the idea of the derivative, we can approximate just about any function using simple polynomials. This lecture also shows why a formula sometimes known as "God's equation" (involving e, i, p, 1, and 0) is true, and how to calculate square roots in your head.
8.
The Joy of Algebra Made Visual
Algebra can be used to solve geometrical problems, such as finding where two lines cross. The technique is useful in real-life problems, for example, in choosing a telephone plan. Graphs help us better understand everything from lines to equations with negative or fractional exponents.
20.
The Joy of Integral Calculus
Geometry and trigonometry are used to determine the areas of simple figures such as triangles and circles. But how are more complex shapes measured? Calculus comes to the rescue with a technique called integration, which adds the simple areas of many tiny quantities.
9.
The Joy of 9
Adding the digits of a multiple of 9 always gives a multiple of 9. For example: 9 x 4 = 36, and 3 + 6 = 9. In modular arithmetic, this property allows checking answers by "casting out nines." A related trick: mentally computing the day of the week for any date in history.
21.
The Joy of Pascal's Triangle
A geometric arrangement of binomial coefficients called Pascal's triangle is a treasure trove of beautiful number patterns. It even provides an answer to the song "The Twelve Days of Christmas": Exactly how many gifts did my true love give to me?
10.
The Joy of Proofs
Professor Benjamin begins his discussion of mathematical proofs with intuitive cases like "even plus even is even" and "odd times odd is odd." He builds to more complex proofs by existence and induction, and ends with a checkerboard challenge.
22.
The Joy of Probability
Mathematics can draw detailed inferences about random events. This lecture covers major concepts in probability, such as the law of large numbers, the central limit theorem, and how to measure variance.
11.
The Joy of Geometry
Geometry is based on a handful of definitions and axioms involving points, lines, and angles. These lead to important conclusions about the properties of polygons. This lecture uses geometric reasoning to derive the Pythagorean theorem and other interesting results.
23.
The Joy of Mathematical Games
This lecture applies the law of total probability and other concepts from the course to predict the long-term losses to be expected from playing games such as roulette and craps and understand what is known as the "Gambler's Ruin Problem."
12.
The Joy of Pi
Pi is the ratio of the circumference of a circle to its diameter. It starts 3.14 and continues in an infinite nonrepeating sequence. Professor Benjamin shows how to learn the first hundred digits of this celebrated number, making it look as easy as pie.
24.
The Joy of Mathematical Magic
Closing the course with a magician's flair, Professor Benjamin shows a trick for producing anyone's phone number, how to create a magic square based on your birthday, how to play "mathematical survivor," a technique for computing cube roots in your head, and a card trick to ponder. |
Calculus,
one of the most useful areas of mathematics, is the study of continuous
change. It provides the language and concepts used by modern science to
quantify the laws of nature and the numerical techniques through which
this knowledge is applied to enrich daily life. Using the mathematics
computer laboratory, students gain a clear understanding of the
fundamental principles of calculus and how they are applied in
real-world situations. Topics: infinite series, functions of several
variables and their derivatives, gradient, directional derivatives,
vector-valued functions and their derivatives, the Jacobian matrix, and
chain rule. (4 credits) Prerequisite: MATH 286 |
1 Introduction About the Student Version The Student Version of MATLAB® & Simulink® is the premier software package for technical computation, data analysis, and visualization in education and industry. The Student Version of MATLAB & Simulink provides all of the features of professional MATLAB, with no limitations, and the full functionality of professional Simulink, with model sizes up to 300 blocks. The Student Version gives you immediate access to the high-performance numeric computing power you need. MATLAB allows you to focus on your course work and applications rather than on programming details. It enables you to solve many numerical problems in a fraction of the time it would take you to write a program in a lower level language. MATLAB helps you better understand and apply concepts in applications ranging from engineering and mathematics to chemistry, biology, and economics. Simulink, included with the Student Version, provides a block diagram tool for modeling and simulating dynamical systems, including signal processing, controls, communications, and other complex systems. The Symbolic Math Toolbox, also included with the Student Version, is based on the Maple® V symbolic kernel and lets you perform symbolic computations and variable-precision arithmetic. MATLAB products are used in a broad range of industries, including automotive, aerospace, electronics, environmental, telecommunications, computer peripherals, finance, and medical. More than 400,000 technical professionals at the world's most innovative technology companies, government research labs, financial institutions, and at more than 2,000 universities rely on MATLAB and Simulink as the fundamental tools for their engineering and scientific work. Student Use Policy This Student License is for use in conjunction with courses offered at a degree-granting institution. The MathWorks offers this license as a special service to the student community and asks your help in seeing that its terms are not abused. To use this Student License, you must be a student using the software in conjunction with courses offered at degree-granting institutions.1-2
About the Student VersionYou may not use this Student License at a company or government lab, or ifyou are an instructor at a university. Also, you may not use it for research orfor commercial or industrial purposes. In these cases, you can acquire theappropriate professional or academic version of the software by contacting TheMathWorks.Differences Between the Student Version and theProfessional VersionMATLABThis version of MATLAB provides full support for all language features as wellas graphics, external (Application Program Interface) support, and access toevery other feature of the professional version of MATLAB.Note MATLAB does not have a matrix size limitation in this Student Version.MATLAB Differences. There are a few small differences between the StudentVersion and the professional version of MATLAB:1 The MATLAB prompt in the Student Version is EDU>>2 The window title bars include the words <Student Version>3 All printouts contain the footer Student Version of MATLAB This footer is not an option that can be turned off; it will always appear in your printouts. 1-3
1 Introduction Simulink This Student Version contains the complete Simulink product, which is used with MATLAB to model, simulate, and analyze dynamical systems. Simulink Differences. 1 Models are limited to 300 blocks. 2 The window title bars include the words <Student Version> 3 All printouts contain the footer Student Version of MATLAB This footer is not an option that can be turned off; it will always appear in your printouts.1-4
Obtaining Additional MathWorks ProductsObtaining Additional MathWorks Products Many college courses recommend MATLAB as their standard instructional software. In some cases, the courses may require particular toolboxes, blocksets, or other products. Many of these products are available for student use. You may purchase and download these additional products at special student prices from the MathWorks Store at Although many professional toolboxes are available at student prices from the MathWorks Store, not every one is available for student use. Some of the toolboxes you can purchase include: Communications Neural Network Control System Optimization Fuzzy Logic Signal Processing Image Processing Statistics For an up-to-date list of which toolboxes are available, visit the MathWorks Store. Note The toolboxes that are available for the Student Version of MATLAB & Simulink have the same functionality as the full, professional versions. However, these student versions will only work with the Student Version. Likewise, the professional versions of the toolboxes will not work with the Student Version. Patches and Updates From time to time, the MathWorks makes changes to some of its products between scheduled releases. When this happens, these updates are made available from our Web site. As a registered user of the Student Version, you will be notified by e-mail of the availability of product updates. Note To register your product, see "Product Registration" in "Troubleshooting and Other Resources" in this chapter. 1-5
1 Introduction Getting Started with MATLAB What I Want What I Should Do I need to install MATLAB. See Chapter 2, "Installation," in this book. I'm new to MATLAB and Start by reading Chapters 1 through 5 of Learning MATLAB. want to learn it quickly. The most important things to learn are how to enter matrices, how to use the : (colon) operator, and how to invoke functions. You will also get a brief overview of graphics and programming in MATLAB. After you master the basics, you can access the rest of the documentation through the online help (Help Desk) facility. I want to look at some There are numerous demonstrations included with MATLAB. samples of what you can do You can see the demos by selecting Examples and Demos from with MATLAB. the Help menu. (Linux users type demo at the MATLAB prompt.) There are demos in mathematics, graphics, visualization, and much more. You also will find a large selection of demos at
Finding Reference InformationFinding Reference InformationWhat I Want What I Should DoI want to know how to use a Use the online help (Help Desk) facility, or, use the M-file helpspecific function. window to get brief online help. These are available using the command helpdesk or from the Help menu on the PC. The MATLAB Function Reference is also available on the Help Desk in PDF format (under Online Manuals) if you want to print out any of the function descriptions in high-quality form.I want to find a function for There are several choices:a specific purpose but I don'tknow its name. • Use lookfor (e.g., lookfor inverse) from the command line. • See Appendix A, "MATLAB Quick Reference," in this book for a list of MATLAB functions. • From the Help Desk peruse the MATLAB functions by Subject or by Index. • Use the full text search from the Help Desk.I want to learn about a Use the Help Desk facility to locate the appropriate chapter inspecific topic like sparse Using MATLAB.matrices, ordinarydifferential equations, or cellarrays.I want to know what Use the Help Desk facility to see the Function Referencefunctions are available in a grouped by subject, or see Appendix A, "MATLAB Quickgeneral area. Reference," in this book for a list of MATLAB functions. The Help Desk provides access to the reference pages for the hundreds of functions included with MATLAB.I want to learn about the See Chapter 6, "Symbolic Math Toolbox," and Appendix B,Symbolic Math Toolbox. "Symbolic Math Toolbox Quick Reference," in this book. For complete descriptions of the Symbolic Math Toolbox functions, use the Help Desk and select Symbolic Math Toolbox functions. 1-7
1 Introduction Troubleshooting and Other Resources What I Want What I Should Do I have a MATLAB specific Visit the Technical Support section problem I want help with. ( of the MathWorks Web site and use the Solution Support Engine to search the Knowledge Base of problem solutions. I want to report a bug or Use the Help Desk or send e-mail to bugs@mathworks.com or make a suggestion. suggest@mathworks.com. Documentation Library Your Student Version of MATLAB & Simulink contains much more documentation than the two printed books, Learning MATLAB and Learning Simulink. On your CD is a personal reference library of every book and reference page distributed by The MathWorks. Access this documentation library from the Help Desk. Note Even though you have the documentation set for the MathWorks family of products, not every product is available for the Student Version of MATLAB & Simulink. For an up-to-date list of available products, visit the MathWorks Store. At the store you can also purchase printed manuals for the MATLAB family of products. Accessing the Online Documentation Access the online documentation (Help Desk) directly from your product CD. (Linux users should refer to Chapter 2, "Installation," for specific information on configuring and accessing the Help Desk from the CD.) 1 Place the CD in your CD-ROM drive. 2 Select Documentation (Help Desk) from the Help menu. The Help Desk appears in a Web browser.1-8
Troubleshooting and Other ResourcesUsenet NewsgroupIf you have access to Usenet newsgroups, you can join the active community ofparticipants in the MATLAB specific group, comp.soft-sys.matlab. Thisforum is a gathering of professionals and students who use MATLAB and havequestions or comments about it and its associated products. This is a greatresource for posing questions and answering those of others. MathWorks staffalso participates actively in this newsgroup.MathWorks Web SiteUse your browser to visit the MathWorks Web site, You'llfind lots of information about MathWorks products and how they are used ineducation and industry, product demos, and MATLAB based books. From theWeb site you will also be able to access our technical support resources, view alibrary of user and company supplied M-files, and get information aboutproducts and upcoming events.MathWorks Education Web SiteThis education-specific Web site, containsmany resources for various branches of mathematics and science. Many ofthese include teaching examples, books, and other related products. You willalso find a comprehensive list of links to Web sites where MATLAB is used forteaching and research at universities.MATLAB Related BooksHundreds of MATLAB related books are available from many differentpublishers. An up-to-date list is available at StoreThe MathWorks Store ( gives you an easy way topurchase products, upgrades, and documentation. 1-9
1 Introduction MathWorks Knowledge Base You can access the MathWorks Knowledge Base from the Support link on our Web site. Our Technical Support group maintains this database of frequently asked questions (FAQ). You can peruse the Knowledge Base by topics, categories, or use the Solution Search Engine to quickly locate relevant data. You can answer many of your questions by spending a few minutes with this around-the-clock resource. Also, Technical Notes, which is accessible from our Technical Support Web site ( contains numerous examples on graphics, mathematics, API, Simulink, and others. Technical Support Registered users of the Student Version of MATLAB & Simulink can use our electronic technical support services to answer product questions. Visit our Technical Support Web site at Student Version Support Policy The MathWorks does not provide telephone technical support to users of the Student Version of MATLAB & Simulink. There are numerous other vehicles of technical support that you can use. The Sources of Information card included with the Student Version identifies the ways to obtain support. After checking the available MathWorks sources for help, if you still cannot resolve your problem, you should contact your instructor. Your instructor should be able to help you, but if not, there is telephone technical support for registered instructors who have adopted the Student Version of MATLAB & Simulink in their courses. Product Registration Visit the MathWorks Web site ( and register your Student Version.1-10
About MATLAB and SimulinkAbout MATLAB and Simulink What Is MATLAB? MATLAB is a high-performance language for technical computing. It integrates computation, visualization, and programming in an easy-to-use environment where problems and solutions are expressed in familiar mathematical notation. Typical uses include: • Math and computation • Algorithm development • Modeling, simulation, and prototyping • Data analysis, exploration, and visualization • Scientific and engineering graphics • Application development, including graphical user interface building MATLAB is an interactive system whose basic data element is an array that does not require dimensioning. This allows you to solve many technical computing problems, especially those with matrix and vector formulations, in a fraction of the time it would take to write a program in a scalar noninteractive language such as C or Fortran. The name MATLAB stands for matrix laboratory. MATLAB was originally written to provide easy access to matrix software developed by the LINPACK and EISPACK projects, which together represent the state-of-the-art in software for matrix computation. MATLAB has evolved over a period of years with input from many users. In university environments, it is the standard instructional tool for introductory and advanced courses in mathematics, engineering, and science. In industry, MATLAB is the tool of choice for high-productivity research, development, and analysis. Toolboxes MATLAB features a family of application-specific solutions called toolboxes. Very important to most users of MATLAB, toolboxes allow you to learn and apply specialized technology. Toolboxes are comprehensive collections of MATLAB functions (M-files) that extend the MATLAB environment to solve particular classes of problems. Areas in which toolboxes are available include 1-11
1 Introduction signal processing, control systems, neural networks, fuzzy logic, wavelets, simulation, and many others. The MATLAB System The MATLAB system consists of five main parts: The MATLAB language. This is a high-level matrix/array language with control flow statements, functions, data structures, input/output, and object-oriented programming features. It allows both "programming in the small" to rapidly create quick and dirty throw-away programs, and "programming in the large" to create complete large and complex application programs. The MATLAB working environment. This is the set of tools and facilities that you work with as the MATLAB user or programmer. It includes facilities for managing the variables in your workspace and importing and exporting data. It also includes tools for developing, managing, debugging, and profiling M-files, MATLAB's applications. Handle Graphics®. This is the MATLAB graphics system. It includes high-level commands for two-dimensional and three-dimensional data visualization, image processing, animation, and presentation graphics. It also includes low-level commands that allow you to fully customize the appearance of graphics as well as to build complete graphical user interfaces on your MATLAB applications. The MATLAB mathematical function library. This is a vast collection of computational algorithms ranging from elementary functions like sum, sine, cosine, and complex arithmetic, to more sophisticated functions like matrix inverse, matrix eigenvalues, Bessel functions, and fast Fourier transforms. The MATLAB Application Program Interface (API). This is a library that allows you to write C and Fortran programs that interact with MATLAB. It include facilities for calling routines from MATLAB (dynamic linking), calling MATLAB as a computational engine, and for reading and writing MAT-files.1-12
About MATLAB and SimulinkWhat Is Simulink?Simulink, a companion program to MATLAB, is an interactive system forsimulating nonlinear dynamic systems. It is a graphical mouse-driven programthat allows you to model a system by drawing a block diagram on the screenand manipulating it dynamically. It can work with linear, nonlinear,continuous-time, discrete-time, multirate, and hybrid systems.Blocksets are add-ons to Simulink that provide additional libraries of blocks forspecialized applications like communications, signal processing, and powersystems.Real-Time Workshop® is a program that allows you to generate C code fromyour block diagrams and to run it on a variety of real-time systems. 1-13
Installing on a PCAdobe Acrobat Reader is required to view and print the MATLAB onlinedocumentation that is in PDF format. Adobe Acrobat Reader is available on theMATLAB CD.MEX-FilesMEX-files are dynamically linked subroutines that MATLAB canautomatically load and execute. They provide a mechanism by which you cancall your own C and Fortran subroutines from MATLAB as if they were built-infunctions.For More Information The Application Program Interface Guide describeshow to write MEX-files and the Application Program Interface Referencedescribes the collection of API functions. Both of these are available from theHelp Desk.If you plan to build your own MEX-files, one of the following is required:• DEC Visual Fortran version 5.0 or 6.0• Microsoft Visual C/C++ version 4.2, 5.0, or 6.0• Borland C++ version 5.0, 5.2, or 5.3• Watcom C/C++ version 10.6 or 11Note For an up-to-date list of all the compilers supported by MATLAB, seethe MathWorks Technical Support Department's Technical Notes at 2-3
2 Installation Installing MATLAB This list summarizes the steps in the standard installation procedure. You can perform the installation by simply following the instructions in the dialog boxes presented by the installation program; it walks you through this process. 1 Stop any virus protection software you have running. 2 Insert the MathWorks CD into your CD-ROM drive. The installation program starts automatically when the CD-ROM drive is ready. You can also run setup.exe from the CD. View the Welcome screen. 3 Review the Student Use Policy. 4 Review the Software License Agreement. 5 Enter your name and school name. 6 To install the complete set of software (MATLAB, Simulink, and the Symbolic Math Toolbox), make sure all of the components are selected in the Select MATLAB Components dialog box. 7 Specify the destination directory, that is, the directory where you want to save the files on your hard drive. To change directories, use the Browse button. 8 When the installation is complete, verify the installation by starting MATLAB and running one of the demo programs. 9 Customize any MATLAB environment options, if desired. For example, to include default definitions or any MATLAB expressions that you want executed every time MATLAB is invoked, create a file named startup.m in the $MATLABtoolboxlocal directory. MATLAB executes this file each time MATLAB is invoked. 1 Perform any additional necessary configuration by typing the appropriate 0 command at the MATLAB command prompt. For example, to configure the MATLAB Notebook, type notebook -setup. To configure a compiler to work with the MATLAB Application Program Interface, type mex -setup.2-4
Installing on a PCFor More Information The MATLAB Installation Guide for PC providesadditional installation information. This manual is available in PDF formfrom Online Manuals on the Help Desk.Installing Additional ToolboxesTo purchase additional toolboxes, visit the MathWorks Store at( Once you purchase a toolbox, it is downloaded toyour computer.When you download a toolbox, you receive an installation program for thetoolbox. To install the toolbox, run the installation program by double-clickingon its icon. After you successfully install the toolbox, all of its functionality willbe available to you when you start MATLAB.Note Some toolboxes have ReadMe files associated with them. When youdownload the toolbox, check to see if there is a ReadMe file. These files containimportant information about the toolbox and possibly installation andconfiguration notes. To view the ReadMe file for a toolbox, use the whatsnewcommand.Accessing the Online Documentation (Help Desk)Access the online documentation (Help Desk) directly from your product CD:1 Place the CD in your CD-ROM drive.2 Select Documentation (Help Desk) from the Help menu in the MATLAB command window. You can also type helpdesk at the MATLAB prompt. 2-5
2 Installation The Help Desk, similar to this figure, appears in your Web browser.2-6
Installing on LinuxInstalling on Linux System Requirements Note For the most up-to-date information about system requirements, see the system requirements page, available in the products area at the MathWorks Web site ( MATLAB and Simulink • Intel-based Pentium, Pentium Pro, or Pentium II personal computer • Linux 2.0.34 kernel (Red Hat 4.2, 5.1, Debian 2.0) • X Windows (X11R6) • 60 MB free disk space for MATLAB & Simulink • 64 MB memory, additional memory strongly recommended • 64 MB swap space (recommended) • CD-ROM drive (for installation and online documentation) • 8-bit graphics adapter and display (for 256 simultaneous colors) • Netscape Navigator 3.0 or higher (to view the online documentation) Adobe Acrobat Reader is required to view and print the MATLAB online documentation that is in PDF format. Adobe Acrobat Reader is available on the MATLAB CD. MEX-Files MEX-files are dynamically linked subroutines that MATLAB can automatically load and execute. They provide a mechanism by which you can call your own C and Fortran subroutines from MATLAB as if they were built-in functions. 2-7
2 Installation For More Information The Application Program Interface Guide describes how to write MEX-files and the Application Program Interface Reference describes the collection of API functions. Both of these are available from the Help Desk. If you plan to build your own MEX-files, you need an ANSIC C compiler (e.g., the GNU C compiler, gcc). Note For an up-to-date list of all the compilers supported by MATLAB, see the MathWorks Technical Support Department's Technical Notes at Installing MATLAB The following instructions describe how to install the Student Version of MATLAB & Simulink on your computer. Note It is recommended that you log in as root to perform your installation. Installing the Software To install the Student Version: 1 If your CD-ROM drive is not accessible to your operating system, you will need to create a directory to be the mount point for it. mkdir /cdrom 2 Place the CD into the CD-ROM drive.2-8
Installing on Linux3 Execute the command to mount the CD-ROM drive on your system. For example, # mount -t iso9660 /dev/cdrom /cdrom should work on most systems. If your /etc/fstab file has a line similar to /dev/cdrom /cdrom iso9660 noauto,ro,user,exec 0 0 then nonroot users can mount the CD-ROM using the simplified command $ mount /cdromNote If the exec option is missing (as it often is by default, for securityreasons), you will receive a "Permission denied" error when attempting to runthe install script. To remedy this, either use the full mount command shownabove (as root) or add the exec option to the file /etc/fstab.4 Move to the installation location using the cd command. For example, if you are going to install into the location /usr/local/matlab5, use the commands cd /usr/local mkdir matlab5 cd matlab5 Subsequent instructions in this section refer to this directory as $MATLAB.5 Copy the license file, license.dat, from the CD to $MATLAB.6 Run the CD install script. /cdrom/install_lnx86.sh The welcome screen appears. Select OK to proceed with the installation.Note If you need additional help on any step during this installation process,click the Help button at the bottom of the dialog box. 2-9
2 Installation 7 Accept or reject the software licensing agreement displayed. If you accept the terms of the agreement, you may proceed with the installation. 8 The MATLAB Root Directory screen is displayed. Select OK if the pathname for the MATLAB root directory is correct; otherwise, change it to the desired location. 9 The system displays your license file. Press OK.2-10
Installing on Linux10 The installation program displays the Product Installation Options screen, which is similar to this. The products you are licensed to install are listed in the Items to install list box. The right list box displays the products that you do not want to install. To install the complete Student Version of MATLAB & Simulink, you must install all the products for which you are licensed (MATLAB, MATLAB Toolbox, MATLAB Kernel, Simulink, Symbolic Math, Symbolic Math Library, and GhostScript). Select OK. 2-11
2 Installation 1 The installation program displays the Installation Data screen. 1 Specify the directory location in your file system for symbolic links to the matlab, matlabdoc, and mex scripts. Choose a directory such as /usr/local/bin. You must be logged in as root to do this. In the MATLAB License No. field, enter student. Select OK to continue. 1 The Begin Installation screen is displayed. Select OK to start the 2 installation. After the installation is complete, the Installation Complete screen is displayed, assuming your installation is successful. Select Exit to exit from the setup program. 1 If desired, customize any MATLAB environment options. For example, to 3 include default definitions or any MATLAB expressions that you want executed every time MATLAB is invoked, create a file named startup.m in the $MATLAB/toolbox/local directory. MATLAB executes this file each time MATLAB is invoked. 1 You must edit the docopt.m M-file located in the $MATLAB/toolbox/local 4 directory to specify the path to the online documentation (Help Desk). For example, if /cdrom is the path to your CD-ROM drive, then you would use2-12
Installing on Linux /cdrom/help. To set the path using this example, change the lines in the if isunix block in the docopt.m file to if isunix % UNIX % doccmd = ; % options = ; docpath = /cdrom/help; The docopt.m file also allows you to specify an alternative Web browser or additional initial browser options. It is configured for Netscape Navigator.15 Start MATLAB by entering the matlab command. If you did not set up symbolic links in a directory on your path, type $MATLAB/bin/matlab.Post Installation ProceduresSuccessful InstallationIf you want to use the MATLAB Application Program Interface, you mustconfigure the mex script to work with your compiler. Also, some toolboxes mayrequire some additional configuration. For more information, see "InstallingAdditional Toolboxes" later in this section.Unsuccessful InstallationIf MATLAB does not execute correctly after installation:1 Check the MATLAB Known Software and Documentation Problems document for the latest information concerning installation. This document is accessible from the Help Desk.2 Repeat the installation procedure from the beginning but run the CD install script using the -t option. /cdrom/install_lnx86.sh -tFor More Information The MATLAB Installation Guide for UNIX providesadditional installation information. This manual is available in PDF formfrom Online Manuals on the Help Desk. 2-13
2 Installation Installing Additional Toolboxes To purchase additional toolboxes, visit the MathWorks Store at ( Once you purchase a toolbox, it is downloaded to your computer. When you download a toolbox on Linux, you receive a tar file (a standard, compressed formatted file). To install the toolbox, you must: 1 Place the tar file in $MATLAB and un-tar it. tar -xf filename 2 Run install_matlab. After you successfully install the toolbox, all of its functionality will be available to you when you start MATLAB. Note Some toolboxes have ReadMe files associated with them. When you download the toolbox, check to see if there is a ReadMe file. These files contain important information about the toolbox and possibly installation and configuration notes. To view the ReadMe file for a toolbox, use the whatsnew command. Accessing the Online Documentation (Help Desk) Access the online documentation (Help Desk) directly from your product CD: 1 Place the CD in your CD-ROM drive and mount it. 2 Type helpdesk at the MATLAB prompt.2-14
Installing on LinuxThe Help Desk, similar to this figure, appears in your Web browser. 2-15
3 Getting Started Starting MATLAB This book is intended to help you start learning MATLAB. It contains a number of examples, so you should run MATLAB and follow along. To run MATLAB on a PC, double-click on the MATLAB icon. To run MATLAB on a Linux system, type matlab at the operating system prompt. To quit MATLAB at any time, type quit at the MATLAB prompt. If you feel you need more assistance, you can: • Access the Help Desk by typing helpdesk at the MATLAB prompt. • Type help at the MATLAB prompt. • Pull down the Help menu on a PC. For more information about help and online documentation, see "Help and Online Documentation" later in this chapter. Also, Chapter 1 provides additional help resources.3-2
Matrices and Magic SquaresMatrices and Magic Squares The best way for you to get started with MATLAB is to learn how to handle matrices. This section shows you how to do that. In MATLAB, a matrix is a rectangular array of numbers. Special meaning is sometimes attached to 1-by-1 matrices, which are scalars, and to matrices with only one row or column, which are vectors. MATLAB has other ways of storing both numeric and nonnumeric data, but in the beginning, it is usually best to think of everything as a matrix. The operations in MATLAB are designed to be as natural as possible. Where other programming languages work with numbers one at a time, MATLAB allows you to work with entire matrices quickly and easily. 3-3
3 Getting Started A good example matrix, used throughout this book, appears in the Renaissance engraving Melancholia I by the German artist and amateur mathematician Albrecht Dürer. This image is filled with mathematical symbolism, and if you look carefully, you will see a matrix in the upper right corner. This matrix is known as a magic square and was believed by many in Dürer's time to have genuinely magical properties. It does turn out to have some fascinating characteristics worth exploring. Entering Matrices You can enter matrices into MATLAB in several different ways: • Enter an explicit list of elements. • Load matrices from external data files. • Generate matrices using built-in functions. • Create matrices with your own functions in M-files. Start by entering Dürer's matrix as a list of its elements. You have only to follow a few basic conventions: • Separate the elements of a row with blanks or commas. • Use a semicolon, ; , to indicate the end of each row. • Surround the entire list of elements with square brackets, [ ]. To enter Dürer's matrix, simply type A = [16 3 2 13; 5 10 11 8; 9 6 7 12; 4 15 14 1]3-4
Matrices and Magic SquaresMATLAB displays the matrix you just entered, A = 16 3 2 13 5 10 11 8 9 6 7 12 4 15 14 1This exactly matches the numbers in the engraving. Once you have entered thematrix, it is automatically remembered in the MATLAB workspace. You canrefer to it simply as A. Now that you have A in the workspace, take a look atwhat makes it so interesting. Why is it magic?sum, transpose, and diagYou're probably already aware that the special properties of a magic squarehave to do with the various ways of summing its elements. If you take the sumalong any row or column, or along either of the two main diagonals, you willalways get the same number. Let's verify that using MATLAB. The firststatement to try is sum(A)MATLAB replies with ans = 34 34 34 34When you don't specify an output variable, MATLAB uses the variable ans,short for answer, to store the results of a calculation. You have computed a rowvector containing the sums of the columns of A. Sure enough, each of thecolumns has the same sum, the magic sum, 34.How about the row sums? MATLAB has a preference for working with thecolumns of a matrix, so the easiest way to get the row sums is to transpose thematrix, compute the column sums of the transpose, and then transpose theresult. The transpose operation is denoted by an apostrophe or single quote, .It flips a matrix about its main diagonal and it turns a row vector into a columnvector. So A 3-5
Matrices and Magic SquaresThe other diagonal, the so-called antidiagonal, is not so importantmathematically, so MATLAB does not have a ready-made function for it. But afunction originally intended for use in graphics, fliplr, flips a matrix from leftto right. sum(diag(fliplr(A))) ans = 34You have verified that the matrix in Dürer's engraving is indeed a magicsquare and, in the process, have sampled a few MATLAB matrix operations.The following sections continue to use this matrix to illustrate additionalMATLAB capabilities.SubscriptsThe element in row i and column j of A is denoted by A(i,j). For example,A(4,2) is the number in the fourth row and second column. For our magicsquare, A(4,2) is 15. So it is possible to compute the sum of the elements in thefourth column of A by typing A(1,4) + A(2,4) + A(3,4) + A(4,4)This produces ans = 34but is not the most elegant way of summing a single column.It is also possible to refer to the elements of a matrix with a single subscript,A(k). This is the usual way of referencing row and column vectors. But it canalso apply to a fully two-dimensional matrix, in which case the array isregarded as one long column vector formed from the columns of the originalmatrix. So, for our magic square, A(8) is another way of referring to the value15 stored in A(4,2).If you try to use the value of an element outside of the matrix, it is an error. t = A(4,5) Index exceeds matrix dimensions. 3-7
3 Getting Started On the other hand, if you store a value in an element outside of the matrix, the size increases to accommodate the newcomer. X = A; X(4,5) = 17 X = 16 3 2 13 0 5 10 11 8 0 9 6 7 12 0 4 15 14 1 17 The Colon Operator The colon, :, is one of MATLAB's most important operators. It occurs in several different forms. The expression 1:10 is a row vector containing the integers from 1 to 10 1 2 3 4 5 6 7 8 9 10 To obtain nonunit spacing, specify an increment. For example, 100:-7:50 is 100 93 86 79 72 65 58 51 and 0:pi/4:pi is 0 0.7854 1.5708 2.3562 3.1416 Subscript expressions involving colons refer to portions of a matrix. A(1:k,j) is the first k elements of the jth column of A. So sum(A(1:4,4))3-8
Matrices and Magic Squares computes the sum of the fourth column. But there is a better way. The colon by itself refers to all the elements in a row or column of a matrix and the keyword end refers to the last row or column. So sum(A(:,end)) computes the sum of the elements in the last column of A. ans = 34 Why is the magic sum for a 4-by-4 square equal to 34? If the integers from 1 to 16 are sorted into four groups with equal sums, that sum must be sum(1:16)/4 which, of course, is ans = 34Using the Symbolic Math The magic FunctionToolbox, you can discover MATLAB actually has a built-in function that creates magic squares of almostthat the magic sum for an any size. Not surprisingly, this function is named magic.n-by-n magic square is(n 3 + n )/2. B = magic(4) B = 16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1 This matrix is almost the same as the one in the Dürer engraving and has all the same "magic" properties; the only difference is that the two middle columns are exchanged. To make this B into Dürer's A, swap the two middle columns. A = B(:,[1 3 2 4]) 3-9
3 Getting Started This says "for each of the rows of matrix B, reorder the elements in the order 1, 3, 2, 4." It produces A = 16 3 2 13 5 10 11 8 9 6 7 12 4 15 14 1 Why would Dürer go to the trouble of rearranging the columns when he could have used MATLAB's ordering? No doubt he wanted to include the date of the engraving, 1514, at the bottom of his magic square. For More Information Using MATLAB provides comprehensive material on the MATLAB language, environment, mathematical topics, and programming in MATLAB. Access Using MATLAB from the Help Desk.3-10
ExpressionsExpressions Like most other programming languages, MATLAB provides mathematical expressions, but unlike most programming languages, these expressions involve entire matrices. The building blocks of expressions are: • Variables • Numbers • Operators • Functions Variables MATLAB does not require any type declarations or dimension statements. When MATLAB encounters a new variable name, it automatically creates the variable and allocates the appropriate amount of storage. If the variable already exists, MATLAB changes its contents and, if necessary, allocates new storage. For example, num_students = 25 creates a 1-by-1 matrix named num_students and stores the value 25 in its single element. Variable names consist of a letter, followed by any number of letters, digits, or underscores. MATLAB uses only the first 31 characters of a variable name. MATLAB is case sensitive; it distinguishes between uppercase and lowercase letters. A and a are not the same variable. To view the matrix assigned to any variable, simply enter the variable name. Numbers MATLAB uses conventional decimal notation, with an optional decimal point and leading plus or minus sign, for numbers. Scientific notation uses the letter e to specify a power-of-ten scale factor. Imaginary numbers use either i or j as a suffix. Some examples of legal numbers are 3 -99 0.0001 9.6397238 1.60210e-20 6.02252e23 1i -3.14159j 3e5i 3-11
3 Getting Started All numbers are stored internally using the long format specified by the IEEE floating-point standard. Floating-point numbers have a finite precision of roughly 16 significant decimal digits and a finite range of roughly 10-308 to 10+308. Operators Expressions use familiar arithmetic operators and precedence rules. + Addition - Subtraction * Multiplication / Division Left division (described in "Matrices and Linear Algebra" in Using MATLAB) ^ Power Complex conjugate transpose ( ) Specify evaluation order Functions MATLAB provides a large number of standard elementary mathematical functions, including abs, sqrt, exp, and sin. Taking the square root or logarithm of a negative number is not an error; the appropriate complex result is produced automatically. MATLAB also provides many more advanced mathematical functions, including Bessel and gamma functions. Most of these functions accept complex arguments. For a list of the elementary mathematical functions, type help elfun3-12
ExpressionsFor a list of more advanced mathematical and matrix functions, type help specfun help elmatFor More Information Appendix A, "MATLAB Quick Reference," containsbrief descriptions of the MATLAB functions. Use the Help Desk to accesscomplete descriptions of all the MATLAB functions by Subject or by Index.Some of the functions, like sqrt and sin, are built-in. They are part of theMATLAB core so they are very efficient, but the computational details are notreadily accessible. Other functions, like gamma and sinh, are implemented inM-files. You can see the code and even modify it if you want.Several special functions provide values of useful constants. pi 3.14159265… i Imaginary unit, √-1 j Same as i eps Floating-point relative precision, 2-52 realmin Smallest floating-point number, 2-1022 realmax Largest floating-point number, (2-ε)21023 Inf Infinity NaN Not-a-numberInfinity is generated by dividing a nonzero value by zero, or by evaluating welldefined mathematical expressions that overflow, i.e., exceed realmax.Not-a-number is generated by trying to evaluate expressions like 0/0 orInf-Inf that do not have well defined mathematical values.The function names are not reserved. It is possible to overwrite any of themwith a new variable, such as eps = 1.e-6 3-13
3 Getting Started and then use that value in subsequent calculations. The original function can be restored with clear eps Expressions You have already seen several examples of MATLAB expressions. Here are a few more examples, and the resulting values. rho = (1+sqrt(5))/2 rho = 1.6180 a = abs(3+4i) a = 5 z = sqrt(besselk(4/3,rho-i)) z = 0.3730+ 0.3214i huge = exp(log(realmax)) huge = 1.7977e+308 toobig = pi*huge toobig = Inf3-14
3 Getting Started The load Command The load command reads binary files containing matrices generated by earlier MATLAB sessions, or reads text files containing numeric data. The text file should be organized as a rectangular table of numbers, separated by blanks, with one row per line, and an equal number of elements in each row. For example, outside of MATLAB, create a text file containing these four lines. 16.0 3.0 2.0 13.0 5.0 10.0 11.0 8.0 9.0 6.0 7.0 12.0 4.0 15.0 14.0 1.0 Store the file under the name magik.dat. Then the command load magik.dat reads the file and creates a variable, magik, containing our example matrix. M-Files You can create your own matrices using M-files, which are text files containing MATLAB code. Just create a file containing the same statements you would type at the MATLAB command line. Save the file under a name that ends in .m. Note To access a text editor on a PC, choose Open or New from the File menu or press the appropriate button on the toolbar. To access a text editor under Linux, use the ! symbol followed by whatever command you would ordinarily use at your operating system prompt. For example, create a file containing these five lines. A = [ ... 16.0 3.0 2.0 13.0 5.0 10.0 11.0 8.0 9.0 6.0 7.0 12.0 4.0 15.0 14.0 1.0 ];3-16
3 Getting Started Deleting Rows and Columns You can delete rows and columns from a matrix using just a pair of square brackets. Start with X = A; Then, to delete the second column of X, use X(:,2) = [] This changes X to X = 16 2 13 5 11 8 9 7 12 4 14 1 If you delete a single element from a matrix, the result isn't a matrix anymore. So, expressions like X(1,2) = [] result in an error. However, using a single subscript deletes a single element, or sequence of elements, and reshapes the remaining elements into a row vector. So X(2:2:10) = [] results in X = 16 9 2 7 13 12 13-18
More About Matrices and ArraysMore About Matrices and Arrays This sections shows you more about working with matrices and arrays, focusing on: • Linear algebra • Arrays • Multivariate data Linear Algebra Informally, the terms matrix and array are often used interchangeably. More precisely, a matrix is a two-dimensional numeric array that represents a linear transformation. The mathematical operations defined on matrices are the subject of linear algebra. Dürer's magic square A = 16 3 2 13 5 10 11 8 9 6 7 12 4 15 14 1 provides several examples that give a taste of MATLAB matrix operations. You've already seen the matrix transpose, A. Adding a matrix to its transpose produces a symmetric matrix. A + A ans = 32 8 11 17 8 20 17 23 11 17 14 26 17 23 26 2 For More Information All of the MATLAB math functions are described in the MATLAB Function Reference, which is accessible from the Help Desk. 3-19
3 Getting Started The multiplication symbol, *, denotes the matrix multiplication involving inner products between rows and columns. Multiplying the transpose of a matrix by the original matrix also produces a symmetric matrix. A*A ans = 378 212 206 360 212 370 368 206 206 368 370 212 360 206 212 378 The determinant of this particular matrix happens to be zero, indicating that the matrix is singular. d = det(A) d = 0 The reduced row echelon form of A is not the identity. R = rref(A) R = 1 0 0 1 0 1 0 -3 0 0 1 3 0 0 0 0 Since the matrix is singular, it does not have an inverse. If you try to compute the inverse with X = inv(A) you will get a warning message Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 1.175530e-017. Roundoff error has prevented the matrix inversion algorithm from detecting exact singularity. But the value of rcond, which stands for reciprocal condition estimate, is on the order of eps, the floating-point relative precision, so the computed inverse is unlikely to be of much use.3-20
3 Getting Started Such matrices represent the transition probabilities in a Markov process. Repeated powers of the matrix represent repeated steps of the process. For our example, the fifth power P^5 is 0.2507 0.2495 0.2494 0.2504 0.2497 0.2501 0.2502 0.2500 0.2500 0.2498 0.2499 0.2503 0.2496 0.2506 0.2505 0.2493 This shows that as k approaches infinity, all the elements in the kth power, Pk, approach 1/4. Finally, the coefficients in the characteristic polynomial poly(A) are 1 -34 -64 2176 0 This indicates that the characteristic polynomial det( A - λI ) is λ4 - 34λ3 - 64λ2 + 2176λ The constant term is zero, because the matrix is singular, and the coefficient of the cubic term is -34, because the matrix is magic! Arrays When they are taken away from the world of linear algebra, matrices become two dimensional numeric arrays. Arithmetic operations on arrays are done element-by-element. This means that addition and subtraction are the same for arrays and matrices, but that multiplicative operations are different. MATLAB uses a dot, or decimal point, as part of the notation for multiplicative array operations.3-22
More About Matrices and ArraysAs an example, consider a data set with three variables:• Heart rate• Weight• Hours of exercise per weekFor five observations, the resulting array might look like D = 72 134 3.2 81 201 3.5 69 156 7.1 82 148 2.4 75 170 1.2The first row contains the heart rate, weight, and exercise hours for patient 1,the second row contains the data for patient 2, and so on. Now you can applymany of MATLAB's data analysis functions to this data set. For example, toobtain the mean and standard deviation of each column: mu = mean(D), sigma = std(D) mu = 75.8 161.8 3.48 sigma = 5.6303 25.499 2.2107For a list of the data analysis functions available in MATLAB, type help datafunIf you have access to the Statistics Toolbox, type help statsScalar ExpansionMatrices and scalars can be combined in several different ways. For example,a scalar is subtracted from a matrix by subtracting it from each element. Theaverage value of the elements in our magic square is 8.5, so B = A - 8.5 3-25
3 Getting Started forms a matrix whose column sums are zero. B = 7.5 -5.5 -6.5 4.5 -3.5 1.5 2.5 -0.5 0.5 -2.5 -1.5 3.5 -4.5 6.5 5.5 -7.5 sum(B) ans = 0 0 0 0 With scalar expansion, MATLAB assigns a specified scalar to all indices in a range. For example, B(1:2,2:3) = 0 zeros out a portion of B B = 7.5 0 0 4.5 -3.5 0 0 -0.5 0.5 -2.5 -1.5 3.5 -4.5 6.5 5.5 -7.5 Logical Subscripting The logical vectors created from logical and relational operations can be used to reference subarrays. Suppose X is an ordinary matrix and L is a matrix of the same size that is the result of some logical operation. Then X(L) specifies the elements of X where the elements of L are nonzero. This kind of subscripting can be done in one step by specifying the logical operation as the subscripting expression. Suppose you have the following set of data. x = 2.1 1.7 1.6 1.5 NaN 1.9 1.8 1.5 5.1 1.8 1.4 2.2 1.6 1.8 The NaN is a marker for a missing observation, such as a failure to respond to an item on a questionnaire. To remove the missing data with logical indexing,3-26
More About Matrices and Arraysuse finite(x), which is true for all finite numerical values and false for NaNand Inf. x = x(finite(x)) x = 2.1 1.7 1.6 1.5 1.9 1.8 1.5 5.1 1.8 1.4 2.2 1.6 1.8Now there is one observation, 5.1, which seems to be very different from theothers. It is an outlier. The following statement removes outliers, in this casethose elements more than three standard deviations from the mean. x = x(abs(x-mean(x)) <= 3*std(x)) x = 2.1 1.7 1.6 1.5 1.9 1.8 1.5 1.8 1.4 2.2 1.6 1.8For another example, highlight the location of the prime numbers in Dürer'smagic square by using logical indexing and scalar expansion to set thenonprimes to 0. A(~isprime(A)) = 0 A = 0 3 2 13 5 0 11 0 0 0 7 0 0 0 0 0The find FunctionThe find function determines the indices of array elements that meet a givenlogical condition. In its simplest form, find returns a column vector of indices.Transpose that vector to obtain a row vector of indices. For example, k = find(isprime(A))picks out the locations, using one-dimensional indexing, of the primes in themagic square. k = 2 5 9 10 11 13 3-27
The Command WindowThe Command Window So far, you have been using the MATLAB command line, typing commands and expressions, and seeing the results printed in the command window. This section describes a few ways of altering the appearance of the command window. If your system allows you to select the command window font or typeface, we recommend you use a fixed width font, such as Fixedsys or Courier, to provide proper spacing. The format Command The format command controls the numeric format of the values displayed by MATLAB. The command affects only how numbers are displayed, not how MATLAB computes or saves them. Here are the different formats, together with the resulting output produced from a vector x with components of different magnitudes. x = [4/3 1.2345e-6] format short 1.3333 0.0000 format short e 1.3333e+000 1.2345e-006 format short g 1.3333 1.2345e-006 format long 1.33333333333333 0.00000123450000 format long e 1.333333333333333e+000 1.234500000000000e-006 3-29
3 Getting Started format long g 1.33333333333333 1.2345e-006 format bank 1.33 0.00 format rat 4/3 1/810045 format hex 3ff5555555555555 3eb4b6231abfd271 If the largest element of a matrix is larger than 103 or smaller than 10-3, MATLAB applies a common scale factor for the short and long formats. In addition to the format commands shown above format compact suppresses many of the blank lines that appear in the output. This lets you view more information on a screen or window. If you want more control over the output format, use the sprintf and fprintf functions. Suppressing Output If you simply type a statement and press Return or Enter, MATLAB automatically displays the results on screen. However, if you end the line with a semicolon, MATLAB performs the computation but does not display any output. This is particularly useful when you generate large matrices. For example, A = magic(100);3-30
The Command WindowLong Command LinesIf a statement does not fit on one line, use three periods, ..., followed byReturn or Enter to indicate that the statement continues on the next line. Forexample, s = 1 -1/2 + 1/3 -1/4 + 1/5 - 1/6 + 1/7 ... - 1/8 + 1/9 - 1/10 + 1/11 - 1/12;Blank spaces around the =, +, and - signs are optional, but they improvereadability.Command Line EditingVarious arrow and control keys on your keyboard allow you to recall, edit, andreuse commands you have typed earlier. For example, suppose you mistakenlyenter rho = (1 + sqt(5))/2You have misspelled sqrt. MATLAB responds with Undefined function or variable sqt.Instead of retyping the entire line, simply press the ↑ key. The misspelledcommand is redisplayed. Use the ← key to move the cursor over and insert themissing r. Repeated use of the ↑ key recalls earlier lines. Typing a fewcharacters and then the ↑ key finds a previous line that begins with thosecharacters.The list of available command line editing keys is different on differentcomputers. Experiment to see which of the following keys is available on yourmachine. (Many of these keys will be familiar to users of the EMACS editor.) ↑ Ctrl-p Recall previous line ↓ Ctrl-n Recall next line ← Ctrl-b Move back one character → Ctrl-f Move forward one character → Ctrl-→ Ctrl-r Move right one word 3-31
3 Getting Started ← Ctrl-← Ctrl-l Move left one word Home Ctrl-a Move to beginning of line End Ctrl-e Move to end of line Esc Ctrl-u Clear line Del Ctrl-d Delete character at cursor Backspace Ctrl-h Delete character before cursor Ctrl-k Delete to end of line3-32
The MATLAB EnvironmentThe MATLAB Environment The MATLAB environment includes both the set of variables built up during a MATLAB session and the set of disk files containing programs and data that persist between sessions. The Workspace The workspace is the area of memory accessible from the MATLAB command line. Two commands, who and whos, show the current contents of the workspace. The who command gives a short list, while whos also gives size and storage information. Here is the output produced by whos on a workspace containing results from some of the examples in this book. It shows several different MATLAB data structures. As an exercise, you might see if you can match each of the variables with the code segment in this book that generates it. whos Name Size Bytes Class A 4x4 128 double array D 5x3 120 double array M 10x1 3816 cell array S 1x3 442 struct array h 1x11 22 char array n 1x1 8 double array s 1x5 10 char array v 2x5 20 char array Grand total is 471 elements using 4566 bytes. To delete all the existing variables from the workspace, enter clear 3-33
3 Getting Started save Commands The save commands preserve the contents of the workspace in a MAT-file that can be read with the load command in a later MATLAB session. For example, save August17th saves the entire workspace contents in the file August17th.mat. If desired, you can save only certain variables by specifying the variable names after the filename. Ordinarily, the variables are saved in a binary format that can be read quickly (and accurately) by MATLAB. If you want to access these files outside of MATLAB, you may want to specify an alternative format. -ascii Use 8-digit text format. -ascii -double Use 16-digit text format. -ascii -double -tabs Delimit array elements with tabs. -v4 Create a file for MATLAB version 4. -append Append data to an existing MAT-file. When you save workspace contents in text format, you should save only one variable at a time. If you save more than one variable, MATLAB will create the text file, but you will be unable to load it easily back into MATLAB. The Search Path MATLAB uses a search path, an ordered list of directories, to determine how to execute the functions you call. When you call a standard function, MATLAB executes the first M-file function on the path that has the specified name. You can override this behavior using special private directories and subfunctions. The command path shows the search path on any platform. On PCs, choose Set Path from the File menu to view or modify the path.3-34
The MATLAB EnvironmentDisk File ManipulationThe commands dir, type, delete, and cd implement a set of generic operatingsystem commands for manipulating files. This table indicates how thesecommands map to other operating systems. MATLAB MS-DOS Linux dir dir ls type type cat delete del or erase rm cd chdir cdFor most of these commands, you can use pathnames, wildcards, and drivedesignators in the usual way.The diary CommandThe diary command creates a diary of your MATLAB session in a disk file. Youcan view and edit the resulting text file using any word processor. To create afile called diary that contains all the commands you enter, as well asMATLAB's printed output (but not the graphics output), enter diaryTo save the MATLAB session in a file with a particular name, use diary filenameTo stop recording the session, use diary offRunning External ProgramsThe exclamation point character ! is a shell escape and indicates that the restof the input line is a command to the operating system. This is quite useful forinvoking utilities or running other programs without quitting MATLAB. OnLinux, for example, !emacs magik.m 3-35
3 Getting Started invokes an editor called emacs for a file named magik.m. When you quit the external program, the operating system returns control to MATLAB.3-36
Help and Online DocumentationHelp and Online Documentation There are several different ways to access online information about MATLAB functions: • The MATLAB Help Desk • Online reference pages • The help command • Link to The MathWorks, Inc. The Help Desk The MATLAB Help Desk provides access to a wide range of help and reference information stored on CD. Many of the underlying documents use HyperText Markup Language (HTML) and are accessed with an Internet Web browser such as Netscape or Microsoft Explorer. The Help Desk process can be started on PCs by selecting the Help Desk option under the Help menu, or, on all computers, by typing helpdesk All of MATLAB's operators and functions have online reference pages in HTML format, which you can reach from the Help Desk. These pages provide more details and examples than the basic help entries. HTML versions of other documents, including this manual, are also available. A search engine, running on your own machine, can query all the online reference material. 3-37
3 Getting Started Using the Help Desk When you access the Help Desk, you see its entry screen. MATLAB Function Simulink instruction In-depth instruction Reference pages and reference pages on Simulink blocks Symbolic Math Toolbox Introduction to reference pages MATLAB Access all toolbox In-depth instruction documentation on MATLAB In-depth instruction on MATLAB graphics Access other product documentation Find answers to your questions (WWW) A particular MATLAB Contact the Function Reference MathWorks (WWW) page Search all documents Access all documents for particular text in PDF format Online Reference Pages The doc Command If you know the name of a specific function, you can view its reference page directly. For example, to get the reference page for the eval function, type doc eval3-38
Help and Online DocumentationThe doc command starts your Web browser, if it is not already running.Printing Online Reference PagesVersions of the online reference pages, as well as the rest of the MATLABdocumentation set, are also available in Portable Document Format (PDF)through the Help Desk. These pages are processed by Adobe's Acrobat reader.They reproduce the look and feel of the printed page, complete with fonts,graphics, formatting, and images. This is the best way to get printed copies ofreference material. To access the PDF versions of the books, select OnlineManuals from the Help Desk and then choose the desired book.The help CommandThe help command is the most basic way to determine the syntax and behaviorof a particular function. Information is displayed directly in the commandwindow. For example, help magicprints MAGIC Magic square. MAGIC(N) is an N-by-N matrix constructed from the integers 1 through N^2 with equal row, column, and diagonal sums. Produces valid magic squares for N = 1,3,4,5....Note MATLAB online help entries use uppercase characters for the functionand variable names to make them stand out from the rest of the text. Whentyping function names, however, always use the corresponding lowercasecharacters because MATLAB is case sensitive and all function names areactually in lowercase.All the MATLAB functions are organized into logical groups, and MATLAB'sdirectory structure is based on this grouping. For example, all the linear 3-39
3 Getting Started algebra functions reside in the matfun directory. To list the names of all the functions in that directory, with a brief description of each help matfun Matrix functions - numerical linear algebra. Matrix analysis. norm - Matrix or vector norm. normest - Estimate the matrix 2-norm ... The command help by itself lists all the directories, with a description of the function category each represents. matlab/general matlab/ops ... The lookfor Command The lookfor command allows you to search for functions based on a keyword. It searches through the first line of help text, which is known as the H1 line, for each MATLAB function, and returns the H1 lines containing a specified keyword. For example, MATLAB does not have a function named inverse. So the response from help inverse is inverse.m not found. But lookfor inverse3-40
Help and Online Documentationfinds over a dozen matches. Depending on which toolboxes you have installed,you will find entries like INVHILB Inverse Hilbert matrix. ACOSH Inverse hyperbolic cosine. ERFINV Inverse of the error function. INV Matrix inverse. PINV Pseudoinverse. IFFT Inverse discrete Fourier transform. IFFT2 Two-dimensional inverse discrete Fourier transform. ICCEPS Inverse complex cepstrum. IDCT Inverse discrete cosine transform.Adding -all to the lookfor command, as in lookfor -allsearches the entire help entry, not just the H1 line.Link to the MathWorksIf your computer is connected to the Internet, the Help Desk provides aconnection to The MathWorks, the home of MATLAB. You can also use theSolution Search Engine at The MathWorks Web site to query an up-to-datedata base of technical support information. 3-41
4 Graphics Basic Plotting MATLAB has extensive facilities for displaying vectors and matrices as graphs, as well as annotating and printing these graphs. This section describes a few of the most important graphics functions and provides examples of some typical applications. For More Information Using MATLAB Graphics provides in-depth coverage of MATLAB graphics and visualization tools. Access Using MATLAB Graphics from the Help Desk. Creating a Plot The plot function has different forms, depending on the input arguments. If y is a vector, plot(y) produces a piecewise linear graph of the elements of y versus the index of the elements of y. If you specify two vectors as arguments, plot(x,y) produces a graph of y versus x. For example, these statements use the colon operator to create a vector of x values ranging from zero to 2π, compute the sine of these values, and plot the result. x = 0:pi/100:2*pi; y = sin(x); plot(x,y) Now label the axes and add a title. The characters pi create the symbol π. xlabel(x = 0:2pi) ylabel(Sine of x) title(Plot of the Sine Function,FontSize,12)4-2
4 Graphics 1 sin(x) sin(x−.25) 0.8 sin(x−.5) 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0 1 2 3 4 5 6 7 For More Information See "Defining the Color of Lines for Plotting" in the Axes Properties chapter of Using MATLAB Graphics. Access Using MATLAB Graphics from the Help Desk. Specifying Line Styles and Colors It is possible to specify color, line styles, and markers (such as plus signs or circles) with the syntax plot(x,y,color_style_marker) color_style_marker is a string containing from one to four characters (enclosed in single quotation marks) constructed from a color, a line style, and a marker type:4-4
Basic Plotting• Color strings are c, m, y, r, g, b, w, and k. These correspond to cyan, magenta, yellow, red, green, blue, white, and black.• Linestyle strings are - for solid, -- for dashed, : for dotted, -. for dash-dot, and none for no line.• The marker types are +, o, *, and x and the filled marker types s for square, d for diamond, ^ for up triangle, v for down triangle, > for right triangle, < for left triangle, p for pentagram, h for hexagram, and none for no marker.Plotting Lines and MarkersIf you specify a marker type but not a linestyle, MATLAB draws only themarker. For example, plot(x,y,ks)plots black squares at each data point, but does not connect the markers witha line.The statement plot(x,y,r:+)plots a red dotted line and places plus sign markers at each data point. Youmay want to use fewer data points to plot the markers than you use to plot thelines. This example plots the data twice using a different number of points forthe dotted line and marker plots. x1 = 0:pi/100:2*pi; x2 = 0:pi/10:2*pi; plot(x1,sin(x1),r:,x2,sin(x2),r+) 4-5
4 Graphics 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0 1 2 3 4 5 6 7 For More Information See the "Basic Plotting" chapter of Using MATLAB Graphics for more examples of plotting options. Access Using MATLAB Graphics from the Help Desk. Imaginary and Complex Data When the arguments to plot are complex, the imaginary part is ignored except when plot is given a single complex argument. For this special case, the command is a shortcut for a plot of the real part versus the imaginary part. Therefore, plot(Z) where Z is a complex vector or matrix, is equivalent to plot(real(Z),imag(Z))4-6
Basic PlottingFor example, t = 0:pi/10:2*pi; plot(exp(i*t),-o) axis equaldraws a 20-sided polygon with little circles at the vertices. The command,axis equal, makes the individual tick mark increments on the x- and y-axesthe same length, which makes this plot more circular in appearance. 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1 −0.5 0 0.5 1Adding Plots to an Existing GraphThe hold command enables you to add plots to an existing graph. When youtype hold onMATLAB does not replace the existing graph when you issue another plottingcommand; it adds the new data to the current graph, rescaling the axes ifnecessary. 4-7
4 Graphics For example, these statements first create a contour plot of the peaks function, then superimpose a pseudocolor plot of the same function. [x,y,z] = peaks; contour(x,y,z,20,k) hold on pcolor(x,y,z) shading interp hold off The hold on command causes the pcolor plot to be combined with the contour plot in one figure. For More Information See the "Specialized Graphs" chapter in Using MATLAB Graphics for information on a variety of graph types. Access Using MATLAB Graphics from the Help Desk.4-8
Basic PlottingFigure WindowsGraphing functions automatically open a new figure window if there are nofigure windows already on the screen. If a figure window exists, MATLAB usesthat window for graphics output. If there are multiple figure windows open,MATLAB targets the one that is designated the "current figure" (the last figureused or clicked in).To make an existing figure window the current figure, you can click the mousewhile the pointer is in that window or you can type figure(n)where n is the number in the figure title bar. The results of subsequentgraphics commands are displayed in this window.To open a new figure window and make it the current figure, type figureFor More Information See the "Figure Properties" chapter in UsingMATLAB Graphics and the reference page for the figure command. AccessUsing MATLAB Graphics and the figure reference page from the Help Desk.Multiple Plots in One FigureThe subplot command enables you to display multiple plots in the samewindow or print them on the same piece of paper. Typing subplot(m,n,p)partitions the figure window into an m-by-n matrix of small subplots and selectsthe pth subplot for the current plot. The plots are numbered along first the top 4-9
Basic PlottingControlling the AxesThe axis command supports a number of options for setting the scaling,orientation, and aspect ratio of plots.Setting Axis LimitsBy default, MATLAB finds the maxima and minima of the data to choose theaxis limits to span this range. The axis command enables you to specify yourown limits axis([xmin xmax ymin ymax])or for three-dimensional graphs, axis([xmin xmax ymin ymax zmin zmax])Use the command axis autoto re-enable MATLAB's automatic limit selection.Setting Axis Aspect Ratioaxis also enables you to specify a number of predefined modes. For example, axis squaremakes the x-axes and y-axes the same length. axis equalmakes the individual tick mark increments on the x- and y-axes the samelength. This means plot(exp(i*[0:pi/10:2*pi]))followed by either axis square or axis equal turns the oval into a propercircle. axis auto normalreturns the axis scaling to its default, automatic mode. 4-11
4 Graphics Setting Axis Visibility You can use the axis command to make the axis visible or invisible. axis on makes the axis visible. This is the default. axis off makes the axis invisible. Setting Grid Lines The grid command toggles grid lines on and off. The statement grid on turns the grid lines on and grid off turns them back off again. For More Information See the axis and axes reference pages and the "Axes Properties" chapter in Using MATLAB Graphics. Access these reference pages and Using MATLAB Graphics from the Help Desk. Axis Labels and Titles The xlabel, ylabel, and zlabel commands add x-, y-, and z-axis labels. The title command adds a title at the top of the figure and the text function inserts text anywhere in the figure. A subset of TeX notation produces Greek letters. t = -pi:pi/100:pi; y = sin(t); plot(t,y) axis([-pi pi -1 1]) xlabel(-pi leq {itt} leq pi) ylabel(sin(t)) title(Graph of the sine function) text(1,-1/3,{itNote the odd symmetry.})4-12
Basic Plotting Graph of the sine function 1 0.8 0.6 0.4 0.2 sin(t) 0 −0.2 Note the odd symmetry. −0.4 −0.6 −0.8 −1 −3 −2 −1 0 1 2 3 −π ≤ t ≤ πFor More Information See the "Labeling Graphs" chapter in UsingMATLAB Graphics for more information. Access Using MATLAB Graphicsfrom the Help Desk.Annotating Plots Using the Plot EditorAfter creating a plot, you can make changes to it and annotate it with the PlotEditor, which is an easy-to-use graphical interface. The following illustration 4-13
4 Graphics shows the plot in a figure window and labels the main features of the figure window and the Plot Editor. Use the Tools Click this button to Get help for Annotate Zoom and rotate the Menu to access Plot start Plot Editor the Plot the plot plot Editor features mode Editor To save a figure, select Save from the File menu. To save it using a graphics format, such as TIFF, for use with other applications, select Export from the File menu. You can also save from the command line – use the saveas command, including any options to save the figure in a different format.4-14
Mesh and Surface PlotsMesh and Surface Plots MATLAB defines a surface by the z-coordinates of points above a grid in the x-y plane, using straight lines to connect adjacent points. The mesh and surf plotting functions display surfaces in three dimensions. mesh produces wireframe surfaces that color only the lines connecting the defining points. surf displays both the connecting lines and the faces of the surface in color. Visualizing Functions of Two Variables To display a function of two variables, z = f (x,y): • Generate X and Y matrices consisting of repeated rows and columns, respectively, over the domain of the function. • Use X and Y to evaluate and graph the function. The meshgrid function transforms the domain specified by a single vector or two vectors x and y into matrices X and Y for use in evaluating functions of two variables. The rows of X are copies of the vector x and the columns of Y are copies of the vector y. Example – Graphing the sinc Function This example evaluates and graphs the two-dimensional sinc function, sin(r)/r, between the x and y directions. R is the distance from origin, which is at the center of the matrix. Adding eps (a MATLAB command that returns the smallest floating-point number on your system) avoids the indeterminate 0/0 at the origin. [X,Y] = meshgrid(-8:.5:8); R = sqrt(X.^2 + Y.^2) + eps; Z = sin(R)./R; mesh(X,Y,Z,EdgeColor,black) 4-15
4 Graphics 1 0.5 0 −0.5 10 5 10 0 5 0 −5 −5 −10 −10 By default, MATLAB colors the mesh using the current colormap. However, this example uses a single-colored mesh by specifying the EdgeColor surface property. See the surface reference page for a list of all surface properties. You can create a transparent mesh by disabling hidden line removal. hidden off See the hidden reference page for more information on this option. Example – Colored Surface Plots A surface plot is similar to a mesh plot except the rectangular faces of the surface are colored. The color of the faces is determined by the values of Z and the colormap (a colormap is an ordered list of colors). These statements graph the sinc function as a surface plot, select a colormap, and add a color bar to show the mapping of data to color. surf(X,Y,Z) colormap hsv colorbar4-16
Mesh and Surface Plots 1 1 0.8 0.6 0.5 0.4 0 0.2−0.5 10 0 5 10 0 5 0 −5 −0.2 −5 −10 −10See the colormap reference page for information on colormaps.For More Information See the "Creating 3-D Graphs" chapter in UsingMATLAB Graphics for more information on surface plots. Access UsingMATLAB Graphics from the Help Desk.Surface Plots with LightingLighting is the technique of illuminating an object with a directional lightsource. In certain cases, this technique can make subtle differences in surfaceshape easier to see. Lighting can also be used to add realism tothree-dimensional graphs.This example uses the same surface as the previous examples, but colors it redand removes the mesh lines. A light object is then added to the left of the"camera" (that is the location in space from where you are viewing the surface). 4-17
4 Graphics After adding the light and setting the lighting method to phong, use the view command to change the view point so you are looking at the surface from a different point in space (an azimuth of -15 and an elevation of 65 degrees). Finally, zoom in on the surface using the toolbar zoom mode. surf(X,Y,Z,FaceColor,red,EdgeColor,none); camlight left; lighting phong view(-15,65) For More Information See the "Lighting as a Visualization Tool" and "Defining the View" chapters in Using MATLAB Graphics for information on these techniques. Access Using MATLAB Graphics from the Help Desk.4-18
ImagesImages Two-dimensional arrays can be displayed as images, where the array elements determine brightness or color of the images. For example, the statements load durer whos Name Size Bytes Class X 648x509 2638656 double array caption 2x28 112 char array map 128x3 3072 double array load the file durer.mat, adding three variables to the workspace. The matrix X is a 648-by-509 matrix and map is a 128-by-3 matrix that is the colormap for this image. Note MAT-files, such as durer.mat, are binary files that can be created on one platform and later read by MATLAB on a different platform. The elements of X are integers between 1 and 128, which serve as indices into the colormap, map. Then image(X) colormap(map) axis image reproduces Dürer's etching shown at the beginning of this book. A high resolution scan of the magic square in the upper right corner is available in another file. Type load detail and then use the uparrow key on your keyboard to reexecute the image, colormap, and axis commands. The statement colormap(hot) adds some twentieth century colorization to the sixteenth century etching. The function hot generates a colormap containing shades of reds, oranges, and 4-19
4 Graphics yellows. Typically a given image matrix has a specific colormap associated with it. See the colormap reference page for a list of other predefined colormaps. For More Information See the "Displaying Bit-Mapped Images" chapter in Using MATLAB Graphics for information the image processing capabilities of MATLAB. Access Using MATLAB Graphics from the Help Desk.4-20
Printing GraphicsPrinting Graphics You can print a MATLAB figure directly on a printer connected to your computer or you can export the figure to one of the standard graphic file formats supported by MATLAB. There are two ways to print and export figures: • Using the Print option under the File menu • Using the print command Printing from the Menu There are four menu options under the File menu that pertain to printing: • The Page Setup option displays a dialog box that enables you to adjust characteristics of the figure on the printed page. • The Print Setup option displays a dialog box that sets printing defaults, but does not actually print the figure. • The Print Preview option enables you to view the figure the way it will look on the printed page. • The Print option displays a dialog box that lets you select standard printing options and print the figure. Generally, use Print Preview to determine whether the printed output is what you want. If not, use the Page Setup dialog box to change the output settings. The Page Setup dialog box Help button displays information on how to set up the page. Exporting Figure to Graphics Files The Export option under the File menu enables you to export the figure to a variety of standard graphics file formats. Using the Print Command The print command provides more flexibility in the type of output sent to the printer and allows you to control printing from M-files. The result can be sent directly to your default printer or stored in a specified file. A wide variety of output formats, including TIFF, JPEG, and PostScript, is available. For example, this statement saves the contents of the current figure window as color Encapsulated Level 2 PostScript in the file called magicsquare.eps. It 4-21
4 Graphics also includes a TIFF preview, which enables most word processors to display the picture print -depsc2 -tiff magicsquare.eps To save the same figure as a TIFF file with a resolution of 200 dpi, use the command print -dtiff -r200 magicsquare.tiff If you type print on the command line, print MATLAB prints the current figure on your default printer. For More Information See the print command reference page and the "Printing MATLAB Graphics" chapter in Using MATLAB Graphics for more information on printing. Access this information from the Help Desk.4-22
Handle GraphicsHandle Graphics When you use a plotting command, MATLAB creates the graph using various graphics objects, such as lines, text, and surfaces (see Table 4-1 for a complete list). All graphics objects have properties that control the appearance and behavior of the object. MATLAB enables you to query the value of each property and set the value of most properties. Whenever MATLAB creates a graphics object, it assigns an identifier (called a handle) to the object. You can use this handle to access the object's properties. Handle Graphics is useful if you want to: • Modify the appearance of graphs. • Create custom plotting commands by writing M-files that create and manipulate objects directly. The material in this manual concentrates on modifying the appearance of graphs. See the "Handle Graphics" chapter in Using MATLAB Graphics for more information on programming with Handle Graphics. Graphics Objects Graphics objects are the basic elements used to display graphics and user interface elements. Table 4-1 lists the graphics objects. Table 4-1: Handle Graphics Objects Object Description Root Top of the hierarchy corresponding to the computer screen Figure Window used to display graphics and user interfaces Uicontrol User interface control that executes a function in response to user interaction Uimenu User-defined figure window menu Uicontextmenu Pop-up menu invoked by right clicking on a graphics object 4-23
4 Graphics Table 4-1: Handle Graphics Objects (Continued) Object Description Axes Axes for displaying graphs in a figure Image Two-dimensional pixel-based picture Light Light sources that affect the coloring of patch and surface objects Line Line used by functions such as plot, plot3, semilogx Patch Filled polygon with edges Rectangle Two-dimensional shape varying from rectangles to ovals Surface Three-dimensional representation of matrix data created by plotting the value of the data as heights above the x-y plane Text Character string Object Hierarchy The objects are organized in a tree structured hierarchy reflecting their interdependence. For example, line objects require axes objects as a frame of reference. In turn, axes objects exist only within figure objects. This diagram illustrates the tree structure. Root Figure Axes Uicontrol Uimenu Uicontextmenu Image Light Line Patch Rectangle Surface Text4-24
Handle GraphicsCreating ObjectsEach object has an associated function that creates the object. These functionshave the same name as the objects they create. For example, the text functioncreates text objects, the figure function creates figure objects, and so on.MATLAB's high-level graphics functions (like plot and surf) call theappropriate low-level function to draw their respective graphics.For More Information See the object creation function reference page formore information about the object and a description of the object's properties.Commands for Working with ObjectsThis table lists commands commonly used when working with objects. Function Purpose copyobj Copy graphics object delete Delete an object findobj Find the handle of objects having specified property values gca Return the handle of the current axes gcf Return the handle of the current figure gco Return the handle of the current object get Query the value of an objects properties set Set the value of an objects propertiesFor More Information See MATLAB Functions in the Help Desk for adescription of each of these functions. 4-25
4 Graphics Setting Object Properties All object properties have default values. However, you may find it useful to change the settings of some properties to customize your graph. There are two ways to set object properties: • Specify values for properties when you create the object. • Set the property value on an object that already exists. You can specify object property values as arguments to object creation functions as well as with plotting function, such as plot, mesh, and surf. You can use the set command to modify the property values of existing objects. For More Information See Handle Graphics Properties in the Help Desk for a description of all object properties. Setting Properties from Plotting Commands Plotting commands that create lines or surfaces enable you to specify property name/property value pairs as arguments. For example, the command plot(x,y,LineWidth,1.5) plots the data in the variables x and y using lines having a LineWidth property set to 1.5 points (one point = 1/72 inch). You can set any line object property this way. Setting Properties of Existing Objects Many plotting commands can also return the handles of the objects created so you can modify the objects using the set command. For example, these statements plot a five-by-five matrix (creating five lines, one per column) and then set the Marker to a square and the MarkerFaceColor to green. h = plot(magic(5)); set(h,Marker,s,MarkerFaceColor,g) In this case, h is a vector containing five handles, one for each of the five lines in the plot. The set statement sets the Marker and MarkerFaceColor properties of all lines to the same values.4-26
Handle GraphicsSetting Multiple Property ValuesIf you want to set the properties of each line to a different value, you can usecell arrays to store all the data and pass it to the set command. For example,create a plot and save the line handles. h = plot(magic(5));Suppose you want to add different markers to each line and color the marker'sface color to the same color as the line. You need to define two cell arrays – onecontaining the property names and the other containing the desired values ofthe properties.The prop_name cell array contains two elements. prop_name(1) = {Marker}; prop_name(2) = {MarkerFaceColor};The prop_values cell array contains 10 values – five values for the Markerproperty and five values for the MarkerFaceColor property. Notice thatprop_values is a two-dimensional cell array. The first dimension indicateswhich handle in h the values apply to and the second dimension indicateswhich property the value is assigned to. prop_values(1,1) = {s}; prop_values(1,2) = {get(h(1),Color)}; prop_values(2,1) = {d}; prop_values(2,2) = {get(h(2),Color)}; prop_values(3,1) = {o}; prop_values(3,2) = {get(h(3),Color)}; prop_values(4,1) = {p}; prop_values(4,2) = {get(h(4),Color)}; prop_values(5,1) = {h}; prop_values(5,2) = {get(h(5),Color)};The MarkerFaceColor is always assigned the value of the corresponding line'scolor (obtained by getting the line's Color property with the get command).After defining the cell arrays, call set to specify the new property values. set(h,prop_name,prop_values) 4-27
4 Graphics 25 20 15 10 5 0 1 1.5 2 2.5 3 3.5 4 4.5 5 For More Information See the "Structures and Cell Arrays" chapter in Using MATLAB for information on cell arrays. Access Using MATLAB from the Help Desk. Finding the Handles of Existing Objects The findobj command enables you to obtain the handles of graphics objects by searching for objects with particular property values. With findobj you can specify the value of any combination of properties, which makes it easy to pick one object out of many. For example, you may want to find the blue line with square marker having blue face color. You can also specify which figures or axes to search, if there is more than one. The following sections provide examples illustrating how to use findobj.4-28
Handle GraphicsFinding All Objects of a Certain TypeSince all objects have a Type property that identifies the type of object, you canfind the handles of all occurrences of a particular type of object. For example, h = findobj(Type,line);finds the handles of all line objects.Finding Objects with a Particular PropertyYou can specify multiple properties to narrow the search. For example, h = findobj(Type,line,Color,r,LineStyle,:);finds the handles of all red, dotted lines.Limiting the Scope of the SearchYou can specify the starting point in the object hierarchy by passing the handleof the starting figure or axes as the first argument. For example, h = findobj(gca,Type,text,String,pi/2);finds the string π/2 only within the current axes.Using findobj as an ArgumentSince findobj returns the handles it finds, you can use it in place of the handleargument. For example, set(findobj(Type,line,Color,red),LineStyle,:)finds all red lines and sets their line style to dotted.For More Information See the "Accessing Object Handles" section of theHandle Graphics chapter in Using MATLAB Graphics for more information.Access Using MATLAB Graphics from the Help Desk. 4-29
4 Graphics Graphics User Interfaces Here is a simple example illustrating how to use Handle Graphics to build user interfaces. The statement b = uicontrol(Style,pushbutton, ... Units,normalized, ... Position,[.5 .5 .2 .1], ... String,click here); creates a pushbutton in the center of a figure window and returns a handle to the new object. But, so far, clicking on the button does nothing. The statement s = set(b,Position,[.8*rand .9*rand .2 .1]); creates a string containing a command that alters the pushbutton's position. Repeated execution of eval(s) moves the button to random positions. Finally, set(b,Callback,s) installs s as the button's callback action, so every time you click on the button, it moves to a new position. Graphical User Interface Design Tools MATLAB provides GUI Design Environment (GUIDE) tools that simplify the creation of graphical user interfaces. To display the GUIDE control panel, issue the guide command. For More Information Type help guide at the MATLAB command line.4-30
AnimationsAnimations MATLAB provides two ways of generating moving, animated graphics: • Continually erase and then redraw the objects on the screen, making incremental changes with each redraw. • Save a number of different pictures and then play them back as a movie. Erase Mode Method Using the EraseMode property is appropriate for long sequences of simple plots where the change from frame to frame is minimal. Here is an example showing simulated Brownian motion. Specify a number of points, such as n = 20 and a temperature or velocity, such as s = .02 The best values for these two parameters depend upon the speed of your particular computer. Generate n random points with (x,y) coordinates between -1/2 and +1/2. x = rand(n,1)-0.5; y = rand(n,1)-0.5; Plot the points in a square with sides at -1 and +1. Save the handle for the vector of points and set its EraseMode to xor. This tells the MATLAB graphics system not to redraw the entire plot when the coordinates of one point are changed, but to restore the background color in the vicinity of the point using an "exclusive or" operation. h = plot(x,y,.); axis([-1 1 -1 1]) axis square grid off set(h,EraseMode,xor,MarkerSize,18) Now begin the animation. Here is an infinite while loop, which you can eventually exit by typing Ctrl-c. Each time through the loop, add a small amount of normally distributed random noise to the coordinates of the points. 4-31
4 Graphics Then, instead of creating an entirely new plot, simply change the XData and YData properties of the original plot. while 1 drawnow x = x + s*randn(n,1); y = y + s*randn(n,1); set(h,XData,x,YData,y) end How long does it take for one of the points to get outside of the square? How long before all of the points are outside the square? 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1 −0.5 0 0.5 1 Creating Movies If you increase the number of points in the Brownian motion example to something like n = 300 and s = .02, the motion is no longer very fluid; it takes too much time to draw each time step. It becomes more effective to save a predetermined number of frames as bitmaps and to play them back as a movie.4-32
5 Programming with MATLAB Flow Control MATLAB has several flow control constructs: • if statements • switch statements • for loops • while loops • break statements For More Information Using MATLAB discusses programming in MATLAB. Access Using MATLAB from the Help Desk. if The if statement evaluates a logical expression and executes a group of statements when the expression is true. The optional elseif and else keywords provide for the execution of alternate groups of statements. An end keyword, which matches the if, terminates the last group of statements. The groups of statements are delineated by the four keywords – no braces or brackets are involved. MATLAB's algorithm for generating a magic square of order n involves three different cases: when n is odd, when n is even but not divisible by 4, or when n is divisible by 4. This is described by if rem(n,2) ~= 0 M = odd_magic(n) elseif rem(n,4) ~= 0 M = single_even_magic(n) else M = double_even_magic(n) end In this example, the three cases are mutually exclusive, but if they weren't, the first true condition would be executed.5-2
Flow ControlIt is important to understand how relational operators and if statements workwith matrices. When you want to check for equality between two variables, youmight use if A == B, ...This is legal MATLAB code, and does what you expect when A and B are scalars.But when A and B are matrices, A == B does not test if they are equal, it testswhere they are equal; the result is another matrix of 0's and 1's showingelement-by-element equality. In fact, if A and B are not the same size, thenA == B is an error.The proper way to check for equality between two variables is to use theisequal function, if isequal(A,B), ...Here is another example to emphasize this point. If A and B are scalars, thefollowing program will never reach the unexpected situation. But for mostpairs of matrices, including our magic squares with interchanged columns,none of the matrix conditions A > B, A < B or A == B is true for all elementsand so the else clause is executed. if A > B greater elseif A < B less elseif A == B equal else error(Unexpected situation) endSeveral functions are helpful for reducing the results of matrix comparisons toscalar conditions for use with if, including isequal isempty all any 5-3
5 Programming with MATLAB switch and case The switch statement executes groups of statements based on the value of a variable or expression. The keywords case and otherwise delineate the groups. Only the first matching case is executed. There must always be an end to match the switch. The logic of the magic squares algorithm can also be described by switch (rem(n,4)==0) + (rem(n,2)==0) case 0 M = odd_magic(n) case 1 M = single_even_magic(n) case 2 M = double_even_magic(n) otherwise error(This is impossible) end Note for C Programmers Unlike the C language switch statement, MATLAB's switch does not fall through. If the first case statement is true, the other case statements do not execute. So, break statements are not required. for The for loop repeats a group of statements a fixed, predetermined number of times. A matching end delineates the statements. for n = 3:32 r(n) = rank(magic(n)); end r The semicolon terminating the inner statement suppresses repeated printing, and the r after the loop displays the final result.5-4
Flow ControlIt is a good idea to indent the loops for readability, especially when they arenested. for i = 1:m for j = 1:n H(i,j) = 1/(i+j); end endwhileThe while loop repeats a group of statements an indefinite number of timesunder control of a logical condition. A matching end delineates the statements.Here is a complete program, illustrating while, if, else, and end, that usesinterval bisection to find a zero of a polynomial. a = 0; fa = -Inf; b = 3; fb = Inf; while b-a > eps*b x = (a+b)/2; fx = x^3-2*x-5; if sign(fx) == sign(fa) a = x; fa = fx; else b = x; fb = fx; end end xThe result is a root of the polynomial x3 - 2x - 5, namely x = 2.09455148154233The cautions involving matrix comparisons that are discussed in the section onthe if statement also apply to the while statement.breakThe break statement lets you exit early from a for or while loop. In nestedloops, break exits from the innermost loop only. 5-5
Other Data StructuresOther Data Structures This section introduces you to some other data structures in MATLAB, including: • Multidimensional arrays • Cell arrays • Characters and text • Structures For More Information For a complete discussion of MATLAB's data structures, see Using MATLAB, which is accessible from the Help Desk. Multidimensional Arrays Multidimensional arrays in MATLAB are arrays with more than two subscripts. They can be created by calling zeros, ones, rand, or randn with more than two arguments. For example, R = randn(3,4,5); creates a 3-by-4-by-5 array with a total of 3x4x5 = 60 normally distributed random elements. A three-dimensional array might represent three-dimensional physical data, say the temperature in a room, sampled on a rectangular grid. Or, it might represent a sequence of matrices, A(k), or samples of a time-dependent matrix, A(t). In these latter cases, the (i, j)th element of the kth matrix, or the tkth matrix, is denoted by A(i,j,k). MATLAB's and Dürer's versions of the magic square of order 4 differ by an interchange of two columns. Many different magic squares can be generated by interchanging columns. The statement p = perms(1:4); 5-7
5 Programming with MATLAB produces a 1-by-3 cell array. The three cells contain the magic square, the row vector of column sums, and the product of all its elements. When C is displayed, you see C = [4x4 double] [1x4 double] [20922789888000] This is because the first two cells are too large to print in this limited space, but the third cell contains only a single number, 16!, so there is room to print it. Here are two important points to remember. First, to retrieve the contents of one of the cells, use subscripts in curly braces. For example, C{1} retrieves the magic square and C{3} is 16!. Second, cell arrays contain copies of other arrays, not pointers to those arrays. If you subsequently change A, nothing happens to C. Three-dimensional arrays can be used to store a sequence of matrices of the same size. Cell arrays can be used to store a sequence of matrices of different sizes. For example, M = cell(8,1); for n = 1:8 M{n} = magic(n); end M produces a sequence of magic squares of different order, M = [ 1] [ 2x2 double] [ 3x3 double] [ 4x4 double] [ 5x5 double] [ 6x6 double] [ 7x7 double] [ 8x8 double]5-10
5 Programming with MATLAB Internally, the characters are stored as numbers, but not in floating-point format. The statement a = double(s) converts the character array to a numeric matrix containing floating-point representations of the ASCII codes for each character. The result is a = 72 101 108 108 111 The statement s = char(a) reverses the conversion. Converting numbers to characters makes it possible to investigate the various fonts available on your computer. The printable characters in the basic ASCII character set are represented by the integers 32:127. (The integers less than 32 represent nonprintable control characters.) These integers are arranged in an appropriate 6-by-16 array with F = reshape(32:127,16,6); The printable characters in the extended ASCII character set are represented by F+128. When these integers are interpreted as characters, the result depends on the font currently being used. Type the statements char(F) char(F+128) and then vary the font being used for the MATLAB command window. On a PC, select Preferences under the File menu. Be sure to try the Symbol and5-12
5 Programming with MATLAB same length, and forms a character array with each line in a separate row. For example, S = char(A,rolling,stone,gathers,momentum.) produces a 5-by-9 character array S = A rolling stone gathers momentum. There are enough blanks in each of the first four rows of S to make all the rows the same length. Alternatively, you can store the text in a cell array. For example, C = {A;rolling;stone;gathers;momentum.} is a 5-by-1 cell array C = A rolling stone gathers momentum. You can convert a padded character array to a cell array of strings with C = cellstr(S) and reverse the process with S = char(C) Structures Structures are multidimensional MATLAB arrays with elements accessed by textual field designators. For example, S.name = Ed Plum; S.score = 83; S.grade = B+5-14
Other Data Structurescreates a scalar structure with three fields. S = name: Ed Plum score: 83 grade: B+Like everything else in MATLAB, structures are arrays, so you can insertadditional elements. In this case, each element of the array is a structure withseveral fields. The fields can be added one at a time, S(2).name = Toni Miller; S(2).score = 91; S(2).grade = A-;or, an entire element can be added with a single statement. S(3) = struct(name,Jerry Garcia,... score,70,grade,C)Now the structure is large enough that only a summary is printed. S = 1x3 struct array with fields: name score gradeThere are several ways to reassemble the various fields into other MATLABarrays. They are all based on the notation of a comma separated list. If you type S.scoreit is the same as typing S(1).score, S(2).score, S(3).scoreThis is a comma separated list. Without any other punctuation, it is not veryuseful. It assigns the three scores, one at a time, to the default variable ans anddutifully prints out the result of each assignment. But when you enclose theexpression in square brackets, [S.score] 5-15
5 Programming with MATLAB it is the same as [S(1).score, S(2).score, S(3).score] which produces a numeric row vector containing all of the scores. ans = 83 91 70 Similarly, typing S.name just assigns the names, one at time, to ans. But enclosing the expression in curly braces, {S.name} creates a 1-by-3 cell array containing the three names. ans = Ed Plum Toni Miller Jerry Garcia And char(S.name) calls the char function with three arguments to create a character array from the name fields, ans = Ed Plum Toni Miller Jerry Garcia5-16
Scripts and FunctionsScripts and Functions MATLAB is a powerful programming language as well as an interactive computational environment. Files that contain code in the MATLAB language are called M-files. You create M-files using a text editor, then use them as you would any other MATLAB function or command. There are two kinds of M-files: • Scripts, which do not accept input arguments or return output arguments. They operate on data in the workspace. • Functions, which can accept input arguments and return output arguments. Internal variables are local to the function. If you're a new MATLAB programmer, just create the M-files that you want to try out in the current directory. As you develop more of your own M-files, you will want to organize them into other directories and personal toolboxes that you can add to MATLAB's search path. If you duplicate function names, MATLAB executes the one that occurs first in the search path. To view the contents of an M-file, for example, myfunction.m, use type myfunction Scripts When you invoke a script, MATLAB simply executes the commands found in the file. Scripts can operate on existing data in the workspace, or they can create new data on which to operate. Although scripts do not return output arguments, any variables that they create remain in the workspace, to be used in subsequent computations. In addition, scripts can produce graphical output using functions like plot. 5-17
5 Programming with MATLAB For example, create a file called magicrank.m that contains these MATLAB commands. % Investigate the rank of magic squares r = zeros(1,32); for n = 3:32 r(n) = rank(magic(n)); end r bar(r) Typing the statement magicrank causes MATLAB to execute the commands, compute the rank of the first 30 magic squares, and plot a bar graph of the result. After execution of the file is complete, the variables n and r remain in the workspace. 35 30 25 20 15 10 5 0 0 5 10 15 20 25 30 355-18
Scripts and FunctionsFunctionsFunctions are M-files that can accept input arguments and return outputarguments. The name of the M-file and of the function should be the same.Functions operate on variables within their own workspace, separate from theworkspace you access at the MATLAB command prompt.A good example is provided by rank. The M-file rank.m is available in thedirectory toolbox/matlab/matfunYou can see the file with type rankHere is the file. function r = rank(A,tol) % RANK Matrix rank. % RANK(A) provides an estimate of the number of linearly % independent rows or columns of a matrix A. % RANK(A,tol) is the number of singular values of A % that are larger than tol. % RANK(A) uses the default tol = max(size(A)) * norm(A) * eps. s = svd(A); if nargin==1 tol = max(size(A)) * max(s) * eps; end r = sum(s > tol);The first line of a function M-file starts with the keyword function. It gives thefunction name and order of arguments. In this case, there are up to two inputarguments and one output argument.The next several lines, up to the first blank or executable line, are commentlines that provide the help text. These lines are printed when you type help rankThe first line of the help text is the H1 line, which MATLAB displays when youuse the lookfor command or request help on a directory. 5-19
5 Programming with MATLAB The rest of the file is the executable MATLAB code defining the function. The variable s introduced in the body of the function, as well as the variables on the first line, r, A and tol, are all local to the function; they are separate from any variables in the MATLAB workspace. This example illustrates one aspect of MATLAB functions that is not ordinarily found in other programming languages – a variable number of arguments. The rank function can be used in several different ways. rank(A) r = rank(A) r = rank(A,1.e-6) Many M-files work this way. If no output argument is supplied, the result is stored in ans. If the second input argument is not supplied, the function computes a default value. Within the body of the function, two quantities named nargin and nargout are available which tell you the number of input and output arguments involved in each particular use of the function. The rank function uses nargin, but does not need to use nargout. Global Variables If you want more than one function to share a single copy of a variable, simply declare the variable as global in all the functions. Do the same thing at the command line if you want the base workspace to access the variable. The global declaration must occur before the variable is actually used in a function. Although it is not required, using capital letters for the names of global variables helps distinguish them from other variables. For example, create an M-file called falling.m. function h = falling(t) global GRAVITY h = 1/2*GRAVITY*t.^2; Then interactively enter the statements global GRAVITY GRAVITY = 32; y = falling((0:.1:5)); The two global statements make the value assigned to GRAVITY at the command prompt available inside the function. You can then modify GRAVITY interactively and obtain new solutions without editing any files.5-20
Scripts and FunctionsPassing String Arguments to FunctionsYou can write MATLAB functions that accept string arguments without theparentheses and quotes. That is, MATLAB interprets foo a b cas foo(a,b,c)However, when using the unquoted form, MATLAB cannot return outputarguments. For example, legend apples orangescreates a legend on a plot using the strings apples and oranges as labels. If youwant the legend command to return its output arguments, then you must usethe quoted form. [legh,objh] = legend(apples,oranges);In addition, you cannot use the unquoted form if any of the arguments are notstrings.Building Strings on the FlyThe quoted form enables you to construct string arguments within the code.The following example processes multiple data files, August1.dat,August2.dat, and so on. It uses the function int2str, which converts aninteger to a character, to build the filename. for d = 1:31 s = [August int2str(d) .dat]; load(s) % Code to process the contents of the d-th file end 5-21
5 Programming with MATLAB A Cautionary Note While the unquoted syntax is convenient, it can be used incorrectly without causing MATLAB to generate an error. For example, given a matrix A, A = 0 -6 -1 6 2 -16 -5 20 -10 The eig command returns the eigenvalues of A. eig(A) ans = -3.0710 -2.4645+17.6008i -2.4645-17.6008i The following statement is not allowed because A is not a string, however MATLAB does not generate an error. eig A ans = 65 MATLAB actually takes the eigenvalues of ASCII numeric equivalent of the letter A (which is the number 65). The eval Function The eval function works with text variables to implement a powerful text macro facility. The expression or statement eval(s) uses the MATLAB interpreter to evaluate the expression or execute the statement contained in the text string s.5-22
Scripts and Functions The example of the previous section could also be done with the following code, although this would be somewhat less efficient because it involves the full interpreter, not just a function call. for d = 1:31 s = [load August int2str(d) .dat]; eval(s) % Process the contents of the d-th file end Vectorization To obtain the most speed out of MATLAB, it's important to vectorize the algorithms in your M-files. Where other programming languages might use for or DO loops, MATLAB can use vector or matrix operations. A simple example involves creating a table of logarithms. x = 0; for k = 1:1001 y(k) = log10(x); x = x + .01; endExperienced MATLAB users A vectorized version of the same code islike to say "Life is too short x = 0:.01:10;to spend writing for loops." y = log10(x); For more complicated code, vectorization options are not always so obvious. When speed is important, however, you should always look for ways to vectorize your algorithms. Preallocation If you can't vectorize a piece of code, you can make your for loops go faster by preallocating any vectors or arrays in which output results are stored. For example, this code uses the function zeros to preallocate the vector created in the for loop. This makes the for loop execute significantly faster. r = zeros(32,1); for n = 1:32 r(n) = rank(magic(n)); end 5-23
5 Programming with MATLAB Without the preallocation in the previous example, the MATLAB interpreter enlarges the r vector by one element each time through the loop. Vector preallocation eliminates this step and results in faster execution. Function Functions A class of functions, called "function functions," works with nonlinear functions of a scalar variable. That is, one function works on another function. The function functions include: • Zero finding • Optimization • Quadrature • Ordinary differential equations MATLAB represents the nonlinear function by a function M-file. For example, here is a simplified version of the function humps from the matlab/demos directory. function y = humps(x) y = 1./((x-.3).^2 + .01) + 1./((x-.9).^2 + .04) - 6; Evaluate this function at a set of points in the interval 0 ≤ x ≤ 1 with x = 0:.002:1; y = humps(x); Then plot the function with plot(x,y)5-24
Scripts and Functions 100 90 80 70 60 50 40 30 20 10 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1The graph shows that the function has a local minimum near x = 0.6. Thefunction fmins finds the minimizer, the value of x where the function takes onthis minimum. The first argument to fmins is the name of the function beingminimized and the second argument is a rough guess at the location of theminimum. p = fmins(humps,.5) p = 0.6370To evaluate the function at the minimizer, humps(p) ans = 11.2528Numerical analysts use the terms quadrature and integration to distinguishbetween numerical approximation of definite integrals and numerical 5-25
5 Programming with MATLAB integration of ordinary differential equations. MATLAB's quadrature routines are quad and quad8. The statement Q = quad8(humps,0,1) computes the area under the curve in the graph and produces Q = 29.8583 Finally, the graph shows that the function is never zero on this interval. So, if you search for a zero with z = fzero(humps,.5) you will find one outside of the interval z = -0.13165-26
Demonstration Programs Included with MATLABDemonstration Programs Included with MATLAB This section includes information on many of the demonstration programs that are included with MATLAB. For More Information The MathWorks Web site ( contains numerous M-files that have been written by users and MathWorks staff. These are accessible by selecting Download M-Files. Also, Technical Notes, which is accessible from our Technical Support Web site ( contains numerous examples on graphics, mathematics, API, Simulink, and others. There are many programs included with MATLAB that highlight various features and functions. For a complete list of the demos, at the command prompt type help demos To view a specific file, for example, airfoil, type edit airfoil To run a demonstration, type the filename at the command prompt. For example, to run the airfoil demonstration, type airfoil Note Many of the demonstrations use multiple windows and require you to press a key in the MATLAB command window to continue through the demonstration. 5-27
6 Symbolic Math Toolbox Introduction The Symbolic Math Toolbox incorporates symbolic computation into MATLAB's numeric environment. This toolbox supplements MATLAB's numeric and graphical facilities with several other types of mathematical computation. Facility Covers Calculus Differentiation, integration, limits, summation, and Taylor series Linear Algebra Inverses, determinants, eigenvalues, singular value decomposition, and canonical forms of symbolic matrices Simplification Methods of simplifying algebraic expressions Solution of Symbolic and numerical solutions to algebraic and Equations differential equations Transforms Fourier, Laplace, z-transform, and corresponding inverse transforms Variable-Precision Numerical evaluation of mathematical expressions Arithmetic to any specified accuracy The computational engine underlying the toolboxes is the kernel of Maple, a system developed primarily at the University of Waterloo, Canada, and, more recently, at the Eidgenössiche Technische Hochschule, Zürich, Switzerland. Maple is marketed and supported by Waterloo Maple, Inc. This version of the Symbolic Math Toolbox is designed to work with MATLAB 5.3 and Maple V Release 5. The Symbolic Math Toolbox is a collection of more than one-hundred MATLAB functions that provide access to the Maple kernel using a syntax and style that is a natural extension of the MATLAB language. The toolbox also allows you to access functions in Maple's linear algebra package. With this toolbox, you can write your own M-files to access Maple functions and the Maple workspace.6-2
IntroductionThe following sections of this tutorial provide explanation and examples onhow to use the toolbox. Section Covers "Getting Help" How to get online help for Symbolic Math Toolbox functions "Getting Started" Basic symbolic math operations "Calculus" How to differentiate and integrate symbolic expressions "Simplifications and How to simplify and substitute values into Substitutions" expressions "Variable-Precision How to control the precision of Arithmetic" computations "Linear Algebra" Examples using the toolbox functions "Solving Equations" How to solve symbolic equationsFor More Information You can access complete reference information for theSymbolic Math Toolbox functions from the Help Desk. Also, you can print thePDF version of the Symbolic Math Toolbox User's Guide (tutorial andreference information) by selecting Symbolic Math Toolbox User's Guidefrom Online Manuals on the Help Desk. 6-3
6 Symbolic Math Toolbox Getting Help There are several ways to find information on using Symbolic Math Toolbox functions. One, of course, is to read this chapter! Another is to use the Help Desk, which contains reference information for all the functions. You can also use MATLAB's command line help system. Generally, you can obtain help on MATLAB functions simply by typing help function where function is the name of the MATLAB function for which you need help. This is not sufficient, however, for some Symbolic Math Toolbox functions. The reason? The Symbolic Math Toolbox "overloads" many of MATLAB's numeric functions. That is, it provides symbolic-specific implementations of the functions, using the same function name. To obtain help for the symbolic version of an overloaded function, type help sym/function where function is the overloaded function's name. For example, to obtain help on the symbolic version of the overloaded function, diff, type help sym/diff To obtain information on the numeric version, on the other hand, simply type help diff How can you tell whether a function is overloaded? The help for the numeric version tells you so. For example, the help for the diff function contains the section Overloaded methods help char/diff.m help sym/diff.m This tells you that there are two other diff commands that operate on expressions of class char and class sym, respectively. See the next section for information on class sym. For more information on overloaded commands, see the Using MATLAB guide, which is accessible from the Help Desk.6-4
Getting StartedGetting Started This section describes how to create and use symbolic objects. It also describes the default symbolic variable. If you are familiar with version 1 of the Symbolic Math Toolbox, please note that version 2 uses substantially different and simpler syntax. To get a quick online introduction to the Symbolic Math Toolbox, type demos at the MATLAB command line. MATLAB displays the MATLAB Demos dialog box. Select Symbolic Math (in the left list box) and then Introduction (in the right list box). Symbolic Objects The Symbolic Math Toolbox defines a new MATLAB data type called a symbolic object or sym (see Using MATLAB for an introduction to MATLAB classes and objects). Internally, a symbolic object is a data structure that stores a string representation of the symbol. The Symbolic Math Toolbox uses symbolic objects to represent symbolic variables, expressions, and matrices. 6-5
Getting Startedoperations (e.g., integration, differentiation, substitution, etc.) on f, you needto create the variables explicitly. You can do this by typing a = sym(a) b = sym(b) c = sym(c) x = sym(x)or simply syms a b c xIn general, you can use sym or syms to create symbolic variables. Werecommend you use syms because it requires less typing.Symbolic and Numeric ConversionsConsider the ordinary MATLAB quantity t = 0.1The sym function has four options for returning a symbolic representation ofthe numeric value stored in t. The f option sym(t,f)returns a symbolic floating-point representation 1.999999999999a*2^(-4)The r option sym(t,r)returns the rational form 1/10This is the default setting for sym. That is, calling sym without a secondargument is the same as using sym with the r option. sym(t) ans = 1/10 6-7
6 Symbolic Math Toolbox The third option e returns the rational form of t plus the difference between the theoretical rational expression for t and its actual (machine) floating-point value in terms of eps (the floating-point relative accuracy). sym(t,e) ans = 1/10+eps/40 The fourth option d returns the decimal expansion of t up to the number of significant digits specified by digits. sym(t,d) ans = .10000000000000000555111512312578 The default value of digits is 32 (hence, sym(t,d) returns a number with 32 significant digits), but if you prefer a shorter representation, use the digits command as follows. digits(7) sym(t,d) ans = .1000000 A particularly effective use of sym is to convert a matrix from numeric to symbolic form. The command A = hilb(3) generates the 3-by-3 Hilbert matrix. A = 1.0000 0.5000 0.3333 0.5000 0.3333 0.2500 0.3333 0.2500 0.2000 By applying sym to A A = sym(A)6-8
6 Symbolic Math Toolbox The command clear x does not make x a nonreal variable. Creating Abstract Functions If you want to create an abstract (i.e., indeterminant) function f(x), type f = sym(f(x)) Then f acts like f(x) and can be manipulated by the toolbox commands. To construct the first difference ratio, for example, type df = (subs(f,x,x+h) - f)/h or syms x h df = (subs(f,x,x+h)-f)/h which returns df = (f(x+h)-f(x))/h This application of sym is useful when computing Fourier, Laplace, and z-transforms. Example: Creating a Symbolic Matrix A circulant matrix has the property that each row is obtained from the previous one by cyclically permuting the entries one step forward. We create the circulant matrix A whose elements are a, b, and c, using the commands syms a b c A = [a b c; b c a; c a b] which return A = [ a, b, c ] [ b, c, a ] [ c, a, b ]6-10
Getting StartedSince A is circulant, the sum over each row and column is the same. Let's checkthis for the first row and second column. The command sum(A(1,:))returns ans = a+b+cThe command sum(A(1,:)) == sum(A(:,2)) % This is a logical test.returns ans = 1Now replace the (2,3) entry of A with beta and the variable b with alpha. Thecommands syms alpha beta; A(2,3) = beta; A = subs(A,b,alpha)return A = [ a, alpha, c] [ alpha, c, beta] [ c, a, alpha]From this example, you can see that using symbolic objects is very similar tousing regular MATLAB numeric objects. 6-11
6 Symbolic Math Toolbox The Default Symbolic Variable When manipulating mathematical functions, the choice of the independent variable is often clear from context. For example, consider the expressions in the table below. Mathematical Function MATLAB Command f = xn f = x^n g = sin(at+b) g = sin(a*t + b) h = Jν(z) h = besselj(nu,z) If we ask for the derivatives of these expressions, without specifying the independent variable, then by mathematical convention we obtain f = nxn, g = a cos(at + b), and h = Jv(z)(v/z) - Jv+1(z). Let's assume that the independent variables in these three expressions are x, t, and z, respectively. The other symbols, n, a, b, and v, are usually regarded as "constants" or "parameters." If, however, we wanted to differentiate the first expression with respect to n, for example, we could write d d n ------ ( x ) or ------ -f -x dn dn to get xn ln x. By mathematical convention, independent variables are often lower-case letters found near the end of the Latin alphabet (e.g., x, y, or z). This is the idea behind findsym, a utility function in the toolbox used to determine default symbolic variables. Default symbolic variables are utilized by the calculus, simplification, equation-solving, and transform functions. To apply this utility to the example discussed above, type syms a b n nu t x z f = x^n; g = sin(a*t + b); h = besselj(nu,z); This creates the symbolic expressions f, g, and h to match the example. To differentiate these expressions, we use diff. diff(f)6-12
Getting Startedreturns ans = x^n*n/xSee the section "Differentiation" for a more detailed discussion ofdifferentiation and the diff command.Here, as above, we did not specify the variable with respect to differentiation.How did the toolbox determine that we wanted to differentiate with respect tox? The answer is the findsym command findsym(f,1)which returns ans = xSimilarly, findsym(g,1) and findsym(h,1) return t and z, respectively. Herethe second argument of findsym denotes the number of symbolic variables wewant to find in the symbolic object f, using the findsym rule (see below). Theabsence of a second argument in findsym results in a list of all symbolicvariables in a given symbolic expression. We see this demonstrated below. Thecommand findsym(g)returns the result ans = a, b, tfindsym Rule The default symbolic variable in a symbolic expression is theletter that is closest to x alphabetically. If there are two equally close, theletter later in the alphabet is chosen. 6-13
Getting StartedCreating an M-FileM-files permit a more general use of functions. Suppose, for example, you wantto create the sinc function sin(x)/x. To do this, create an M-file in the @symdirectory. function z = sinc(x) %SINC The symbolic sinc function % sin(x)/x. This function % accepts a sym as the input argument. if isequal(x,sym(0)) z = 1; else z = sin(x)/x; endYou can extend such examples to functions of several variables. For a moredetailed discussion on object-oriented programming, see the Using MATLABguide. 6-15
6 Symbolic Math Toolbox Calculus The Symbolic Math Toolbox provides functions to do the basic operations of calculus; differentiation, limits, integration, summation, and Taylor series expansion. The following sections outline these functions. Differentiation Let's create a symbolic expression. syms a x f = sin(a*x) Then diff(f) differentiates f with respect to its symbolic variable (in this case x), as determined by findsym. ans = cos(a*x)*a To differentiate with respect to the variable a, type diff(f,a) which returns df/da ans = cos(a*x)*x To calculate the second derivatives with respect to x and a, respectively, type diff(f,2) or diff(f,x,2) which return ans = -sin(a*x)*a^26-16
6 Symbolic Math Toolbox Limits The fundamental idea in calculus is to make calculations on functions as a variable "gets close to" or approaches a certain value. Recall that the definition of the derivative is given by a limit f(x + h) – f(x) f′(x) = lim --------------------------------- - h→0 h provided this limit exists. The Symbolic Math Toolbox allows you to compute the limits of functions in a direct manner. The commands syms h n x limit( (cos(x+h) - cos(x))/h,h,0 ) which return ans = -sin(x) and limit( (1 + x/n)^n,n,inf ) which returns ans = exp(x) illustrate two of the most important limits in mathematics: the derivative (in this case of cos x) and the exponential function. While many limits lim f ( x ) x→a are "two sided" (that is, the result is the same whether the approach is from the right or left of a), limits at the singularities of f(x) are not. Hence, the three limits, 1 1 1 lim -- , lim -- , and lim -- - - - x→0 x x → 0- x x → 0+ x yield the three distinct results: undefined, - ∞ , and + ∞ , respectively.6-20
CalculusIn contrast to differentiation, symbolic integration is a more complicated task.A number of difficulties can arise in computing the integral. Theantiderivative, F, may not exist in closed form; it may define an unfamiliarfunction; it may exist, but the software can't find the antiderivative; thesoftware could find it on a larger computer, but runs out of time or memory onthe available machine. Nevertheless, in many cases, MATLAB can performsymbolic integration successfully. For example, create the symbolic variables syms a b theta x y n x1 uThis table illustrates integration of expressions containing those variables. f int(f) x^n x^(n+1)/(n+1) y^(-1) log(y) n^x 1/log(n)*n^x sin(a*theta+b) -cos(a*theta+b)/a exp(-x1^2) 1/2*pi^(1/2)*erf(x1) 1/(1+u^2) atan(u)The last example shows what happens if the toolbox can't find theantiderivative; it simply returns the command, including the variable ofintegration, unevaluated.Definite integration is also possible. The commands int(f,a,b)and int(f,v,a,b)are used to find a symbolic expression for b b ∫a f ( x ) dx and ∫a f ( v ) dvrespectively. 6-23
6 Symbolic Math Toolbox Here are some additional examples. f a, b int(f,a,b) x^7 0, 1 1/8 1/x 1, 2 log(2) log(x)*sqrt(x) 0, 1 -4/9 exp(-x^2) 0, inf 1/2*pi^(1/2) bessel(1,z) 0, 1 -besselj(0,1)+1 For the Bessel function (besselj) example, it is possible to compute a numerical approximation to the value of the integral, using the double function. The command a = int(besselj(1,z),0,1) returns a = -besselj(0,1)+1 and the command a = double(a) returns a = 0.23480231344203 Integration with Real Constants One of the subtleties involved in symbolic integration is the "value" of various parameters. For example, the expression – ( kx ) 2 e is the positive, bell shaped curve that tends to 0 as x tends to ±∞ for any real number k. An example of this curve is depicted below with6-24
6 Symbolic Math Toolbox ∞ – ( kx ) 2 ∫ e dx –∞ in the Symbolic Math Toolbox, using the commands syms x k; f = exp(-(k*x)^2); int(f,x,-inf,inf) results in the output Definite integration: Cant determine if the integral is convergent. Need to know the sign of --> k^2 Will now try indefinite integration and then take limits. Warning: Explicit integral could not be found. ans = int(exp(-k^2*x^2),x= -inf..inf) In the next section, you well see how to make k a real variable and therefore k2 positive. Real Variables via sym Notice that Maple is not able to determine the sign of the expression k^2. How does one surmount this obstacle? The answer is to make k a real variable, using the sym command. One particularly useful feature of sym, namely the real option, allows you to declare k to be a real variable. Consequently, the integral above is computed, in the toolbox, using the sequence syms k real int(f,x,-inf,inf) which returns ans = signum(k)/k*pi^(1/2) Notice that k is now a symbolic object in the MATLAB workspace and a real variable in the Maple kernel workspace. By typing clear k6-26
Calculusyou only clear k in the MATLAB workspace. To ensure that k has no formalproperties (that is, to ensure k is a purely formal variable), type syms k unrealThis variation of the syms command clears k in the Maple workspace. You canalso declare a sequence of symbolic variables w, y, x, z to be real, using syms w x y z realIn this case, all of the variables in between the words syms and real areassigned the property real. That is, they are real variables in the Mapleworkspace.Symbolic SummationYou can compute symbolic summations, when they exist, by using the symsumcommand. For example, the p-series 1 1 1 + ----- + ----- + … - - 2 2 2 3adds to π2/6, while the geometric series 1 + x + x2 + ... adds to 1/(1-x), provided|x| < 1. Three summations are demonstrated below. syms x k s1 = symsum(1/k^2,1,inf) s2 = symsum(x^k,k,0,inf) s1 = 1/6*pi^2 s2 = -1/(x-1) 6-27
6 Symbolic Math Toolbox Extended Calculus Example The function 1 f ( x ) = ----------------------------- - 5 + 4 cos ( x ) provides a starting point for illustrating several calculus operations in the toolbox. It is also an interesting function in its own right. The statements syms x f = 1/(5+4*cos(x)) store the symbolic expression defining the function in f. The function ezplot(f) produces the plot of f(x) as shown below. 1/(5+4*cos(x)) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 −6 −4 −2 0 2 4 6 x The ezplot function tries to make reasonable choices for the range of the x-axis and for the resulting scale of the y-axis. Its choices can be overridden by an additional input argument, or by subsequent axis commands. The default6-30
Calculusdomain for a function displayed by ezplot is -2π ≤ x ≤ 2π. To produce a graphof f(x) for a ≤ x ≤ b, type ezplot(f,[a b])Let's now look at the second derivative of the function f. f2 = diff(f,2) f2 = 32/(5+4*cos(x))^3*sin(x)^2+4/(5+4*cos(x))^2*cos(x)Equivalently, we can type f2 = diff(f,x,2). The default scaling in ezplotcuts off part of f2's graph. Set the axes limits manually to see the entirefunction. ezplot(f2) axis([-2*pi 2*pi -5 2]) 32/(5+4*cos(x))^3*sin(x)^2+4/(5+4*cos(x))^2*cos(x) 2 1 0−1−2−3−4−5 −6 −4 −2 0 2 4 6 x 6-31
Calculuseach of whose entries is a zero of f(x). The command format; % Default format of 5 digits zr = double(z)converts the zeros to double form. zr = 0 0+ 2.4381i 0- 2.4381i 2.4483 -2.4483So far, we have found three real zeros and two complex zeros. However, a graphof f3 shows that we have not yet found all its zeros. ezplot(f3) hold on; plot(zr,0*zr,ro) plot([-2*pi,2*pi], [0,0],g-.); title(Zeros of f3) 6-33
6 Symbolic Math Toolbox This is certainly not the original expression for f(x). Let's look at the difference f(x) - g(x). d = f - g pretty(d) 1 8 –––––––––––– + ––––––––––––––– 5 + 4 cos(x) 2 tan(1/2 x) + 9 We can simplify this using simple(d) or simplify(d). Either command produces ans = 1 This illustrates the concept that differentiating f(x) twice, then integrating the result twice, produces a function that may differ from f(x) by a linear function of x. Finally, integrate f(x) once more. F = int(f) The result F = 2/3*atan(1/3*tan(1/2*x)) involves the arctangent function.6-40
CalculusThough F(x) is the antiderivative of a continuous function, it is itselfdiscontinuous as the following plot shows. ezplot(F) 2/3*atan(1/3*tan(1/2*x)) 10.80.60.40.2 00.20.40.60.8−1 −6 −4 −2 0 2 4 6 xNote that F(x) has jumps at x = ± π. This occurs because tan x is singular atx = ± π. 6-41
6 Symbolic Math Toolbox In fact, as ezplot(atan(tan(x))) shows, the numerical value of atan(tan(x))differs from x by a piecewise constant function that has jumps at odd multiples of π/2. atan(tan(x)) 1.5 1 0.5 0 −0.5 −1 −1.5 −6 −4 −2 0 2 4 6 x To obtain a representation of F(x) that does not have jumps at these points, we must introduce a second function, J(x), that compensates for the discontinuities. Then we add the appropriate multiple of J(x) to F(x) J = sym(round(x/(2*pi))); c = sym(2/3*pi); F1 = F+c*J F1 = 2/3*atan(1/3*tan(1/2*x))+2/3*pi*round(1/2*x/pi)6-42
Calculusand plot the result. ezplot(F1,[-6.28,6.28])This representation does have a continuous graph. 2/3*atan(1/3*tan(1/2*x))+2/3*pi*round(1/2*x/pi) 2.5 2 1.5 1 0.5 0−0.5 −1−1.5 −2−2.5 −6 −4 −2 0 2 4 6 xNotice that we use the domain [-6.28, 6.28] in ezplot rather than the defaultdomain [-2π, 2π]. The reason for this is to prevent an evaluation ofF1 = 2/3 atan(1/3 tan 1/2 x) at the singular points x = -π and x = π where thejumps in F and J do not cancel out one another. The proper handling of branchcut discontinuities in multivalued functions like arctan x is a deep and difficultproblem in symbolic computation. Although MATLAB and Maple cannot dothis entirely automatically, they do provide the tools for investigating suchquestions. 6-43
6 Symbolic Math Toolbox Simplifications and Substitutions There are several functions that simplify symbolic expressions and are used to perform symbolic substitutions. Simplifications Here are three different symbolic expressions. syms x f = x^3-6*x^2+11*x-6 g = (x-1)*(x-2)*(x-3) h = x*(x*(x-6)+11)-6 Here are their prettyprinted forms, generated by pretty(f), pretty(g), pretty(h) 3 2 x - 6 x + 11 x - 6 (x - 1) (x - 2) (x - 3) x (x (x - 6) + 11) - 6 These expressions are three different representations of the same mathematical function, a cubic polynomial in x. Each of the three forms is preferable to the others in different situations. The first form, f, is the most commonly used representation of a polynomial. It is simply a linear combination of the powers of x. The second form, g, is the factored form. It displays the roots of the polynomial and is the most accurate for numerical evaluation near the roots. But, if a polynomial does not have such simple roots, its factored form may not be so convenient. The third form, h, is the Horner, or nested, representation. For numerical evaluation, it involves the fewest arithmetic operations and is the most accurate for some other ranges of x. The symbolic simplification problem involves the verification that these three expressions represent the same function. It also involves a less clearly defined objective — which of these representations is "the simplest"?6-44
Simplifications and SubstitutionsThis toolbox provides several functions that apply various algebraic andtrigonometric identities to transform one representation of a function intoanother, possibly simpler, representation. These functions are collect,expand, horner, factor, simplify, and simple.collectThe statement collect(f)views f as a polynomial in its symbolic variable, say x, and collects all thecoefficients with the same power of x. A second argument can specify thevariable in which to collect terms if there is more than one candidate. Here area few examples. f collect(f) (x-1)*(x-2)*(x-3) x^3-6*x^2+11*x-6 x*(x*(x-6)+11)-6 x^3-6*x^2+11*x-6 (1+x)*t + x*t 2*x*t+t 6-45
Simplifications and SubstitutionssimplifyThe simplify function is a powerful, general purpose tool that applies anumber of algebraic identities involving sums, integral powers, square rootsand other fractional powers, as well as a number of functional identitiesinvolving trig functions, exponential and log functions, Bessel functions,hypergeometric functions, and the gamma function. Here are some examples. f simplify(f) x∗(x∗(x-6)+11)-6 x^3-6∗x^2+11∗x-6 (1-x^2)/(1-x) x+1 (1/a^3+6/a^2+12/a+8)^(1/3) ((2*a+1)^3/a^3)^(1/3) syms x y positive log(x∗y) log(x)+log(y) exp(x) ∗ exp(y) exp(x+y) besselj(2,x) + besselj(0,x) 2/x*besselj(1,x) gamma(x+1)-x*gamma(x) 0 cos(x)^2 + sin(x)^2 1simpleThe simple function has the unorthodox mathematical goal of finding asimplification of an expression that has the fewest number of characters. Ofcourse, there is little mathematical justification for claiming that oneexpression is "simpler" than another just because its ASCII representation isshorter, but this often proves satisfactory in practice.The simple function achieves its goal by independently applying simplify,collect, factor, and other simplification functions to an expression andkeeping track of the lengths of the results. The simple function then returnsthe shortest result.The simple function has several forms, each returning different output. Theform simple(f) 6-49
Simplifications and SubstitutionsThis form is useful when you want to check, for example, whether the shortestform is indeed the simplest. If you are not interested in how simple achievesits result, use the form f = simple(f)This form simply returns the shortest expression found. For example, thestatement f = simple(cos(x)^2+sin(x)^2)returns f = 1If you want to know which simplification returned the shortest result, use themultiple output form. [F, how] = simple(f)This form returns the shortest result in the first variable and the simplificationmethod used to achieve the result in the second variable. For example, thestatement [f, how] = simple(cos(x)^2+sin(x)^2)returns f = 1 how = combineThe simple function sometimes improves on the result returned by simplify,one of the simplifications that it tries. For example, when applied to the 6-51
Simplifications and SubstitutionsNext, substitute the symbol S into E with E = subs(E,S,S) E = [ S, 0, 0] [ 0, -S, 0] [ 0, 0, b+c+a]Now suppose we want to evaluate v at a = 10. We can do this using the subscommand. subs(v,a,10)This replaces all occurrences of a in v with 10. [ -(10+S-b)/(10-c), -(10-S-b)/(10-c), 1] [ -(b-c-S)/(10-c), -(b-c+S)/(10-c), 1] [ 1, 1, 1]Notice, however, that the symbolic expression represented by S is unaffected bythis substitution. That is, the symbol a in S is not replaced by 10. The subscommand is also a useful function for substituting in a variety of values forseveral variables in a particular expression. Let's look at S. Suppose that inaddition to substituting a = 10, we also want to substitute the values for 2 and10 for b and c, respectively. The way to do this is to set values for a, b, and c inthe workspace. Then subs evaluates its input using the existing symbolic anddouble variables in the current workspace. In our example, we first set a = 10; b = 2; c = 10; subs(S) ans = 8 6-57
Variable-Precision ArithmeticVariable-Precision Arithmetic Overview There are three different kinds of arithmetic operations in this toolbox. • Numeric MATLAB's floating-point arithmetic • Rational Maple's exact symbolic arithmetic • VPA Maple's variable-precision arithmetic For example, the MATLAB statements format long 1/2+1/3 use numeric computation to produce 0.83333333333333 With the Symbolic Math Toolbox, the statement sym(1/2)+1/3 uses symbolic computation to yield 5/6 And, also with the toolbox, the statements digits(25) vpa(1/2+1/3) use variable-precision arithmetic to return .8333333333333333333333333 The floating-point operations used by numeric arithmetic are the fastest of the three, and require the least computer memory, but the results are not exact. The number of digits in the printed output of MATLAB's double quantities is controlled by the format statement, but the internal representation is always the eight-byte floating-point representation provided by the particular computer hardware. In the computation of the numeric result above, there are actually three roundoff errors, one in the division of 1 by 3, one in the addition of 1/2 to the 6-61
6 Symbolic Math Toolbox result of the division, and one in the binary to decimal conversion for the printed output. On computers that use IEEE floating-point standard arithmetic, the resulting internal value is the binary expansion of 5/6, truncated to 53 bits. This is approximately 16 decimal digits. But, in this particular case, the printed output shows only 15 digits. The symbolic operations used by rational arithmetic are potentially the most expensive of the three, in terms of both computer time and memory. The results are exact, as long as enough time and memory are available to complete the computations. Variable-precision arithmetic falls in between the other two in terms of both cost and accuracy. A global parameter, set by the function digits, controls the number of significant decimal digits. Increasing the number of digits increases the accuracy, but also increases both the time and memory requirements. The default value of digits is 32, corresponding roughly to floating-point accuracy. The Maple documentation uses the term "hardware floating-point" for what we are calling "numeric" or "floating-point" and uses the term "floating-point arithmetic" for what we are calling "variable-precision arithmetic." Example: Using the Different Kinds of Arithmetic Rational Arithmetic By default, the Symbolic Math Toolbox uses rational arithmetic operations, i.e., Maple's exact symbolic arithmetic. Rational arithmetic is invoked when you create symbolic variables using the sym function. The sym function converts a double matrix to its symbolic form. For example, if the double matrix is A = 1.1000 1.2000 1.3000 2.1000 2.2000 2.3000 3.1000 3.2000 3.3000 its symbolic form, S = sym(A), is S = [11/10, 6/5, 13/10] [21/10, 11/5, 23/10] [31/10, 16/5, 33/10]6-62
Variable-Precision ArithmeticFor this matrix A, it is possible to discover that the elements are the ratios ofsmall integers, so the symbolic representation is formed from those integers.On the other hand, the statement E = [exp(1) sqrt(2); log(3) rand]returns a matrix E = 2.71828182845905 1.41421356237310 1.09861228866811 0.21895918632809whose elements are not the ratios of small integers, so sym(E) reproduces thefloating-point representation in a symbolic form. [3060513257434037*2^(-50), 3184525836262886*2^(-51)] [2473854946935174*2^(-51), 3944418039826132*2^(-54)]Variable-Precision NumbersVariable-precision numbers are distinguished from the exact rationalrepresentation by the presence of a decimal point. A power of 10 scale factor,denoted by e, is allowed. To use variable-precision instead of rationalarithmetic, create your variables using the vpa function.For matrices with purely double entries, the vpa function generates therepresentation that is used with variable-precision arithmetic. Continuing onwith our example, and using digits(4), applying vpa to the matrix S vpa(S)generates the output S = [1.100, 1.200, 1.300] [2.100, 2.200, 2.300] [3.100, 3.200, 3.300]and with digits(25) F = vpa(E) 6-63
6 Symbolic Math Toolbox generates F = [2.718281828459045534884808, 1.414213562373094923430017] [1.098612288668110004152823, .2189591863280899719512718] Converting to Floating-Point To convert a rational or variable-precision number to its MATLAB floating-point representation, use the double function. In our example, both double(sym(E)) and double(vpa(E)) return E. Another Example The next example is perhaps more interesting. Start with the symbolic expression f = sym(exp(pi*sqrt(163))) The statement double(f) produces the printed floating-point value 2.625374126407687e+17 Using the second argument of vpa to specify the number of digits, vpa(f,18) returns 262537412640768744. whereas vpa(f,25) returns 262537412640768744.0000000 We suspect that f might actually have an integer value. This suspicion is reinforced by the 30 digit value, vpa(f,30) 262537412640768743.9999999999996-64
Variable-Precision ArithmeticFinally, the 40 digit value, vpa(f,40) 262537412640768743.9999999999992500725944shows that f is very close to, but not exactly equal to, an integer. 6-65
6 Symbolic Math Toolbox Linear Algebra Basic Algebraic Operations Basic algebraic operations on symbolic objects are the same as operations on MATLAB objects of class double. This is illustrated in the following example. The Givens transformation produces a plane rotation through the angle t. The statements syms t; G = [cos(t) sin(t); -sin(t) cos(t)] create this transformation matrix. G = [ cos(t), sin(t) ] [ -sin(t), cos(t) ] Applying the Givens transformation twice should simply be a rotation through twice the angle. The corresponding matrix can be computed by multiplying G by itself or by raising G to the second power. Both A = G*G and A = G^2 produce A = [cos(t)^2-sin(t)^2, 2*cos(t)*sin(t)] [ -2*cos(t)*sin(t), cos(t)^2-sin(t)^2] The simple function A = simple(A)6-66
Linear AlgebraAll three of these results, the inverse, the determinant, and the solution to thelinear system, are the exact results corresponding to the infinitely precise,rational, Hilbert matrix. On the other hand, using digits(16), the command V = vpa(hilb(3))returns [ 1., .5000000000000000, .3333333333333333] [.5000000000000000, .3333333333333333, .2500000000000000] [.3333333333333333, .2500000000000000, .2000000000000000]The decimal points in the representation of the individual elements are thesignal to use variable-precision arithmetic. The result of each arithmeticoperation is rounded to 16 significant decimal digits. When inverting thematrix, these errors are magnified by the matrix condition number, which forhilb(3) is about 500. Consequently, inv(V)which returns [ 9.000000000000082, -36.00000000000039, 30.00000000000035] [-36.00000000000039, 192.0000000000021, -180.0000000000019] [ 30.00000000000035, -180.0000000000019, 180.0000000000019]shows the loss of two digits. So does det(V)which gives .462962962962958e-3and Vbwhich is [ 3.000000000000041] [-24.00000000000021] [ 30.00000000000019]Since H is nonsingular, the null space of H null(H) 6-69
Linear Algebraand inv(H)produces an error message ??? error using ==> inv Error, (in inverse) singular matrixbecause H is singular. For this matrix, Z = null(H) and C = colspace(H) arenontrivial. Z = [ 1] [ -4] [10/3] C = [ 0, 1] [ 1, 0] [6/5, -3/10]It should be pointed out that even though H is singular, vpa(H) is not. For anyinteger value d, setting digits(d)and then computing det(vpa(H)) inv(vpa(H))results in a determinant of size 10^(-d) and an inverse with elements on theorder of 10^d.EigenvaluesThe symbolic eigenvalues of a square matrix A or the symbolic eigenvalues andeigenvectors of A are computed, respectively, using the commands E = eig(A) [V,E] = eig(A) 6-71
6 Symbolic Math Toolbox The variable-precision counterparts are E = eig(vpa(A)) [V,E] = eig(vpa(A)) The eigenvalues of A are the zeros of the characteristic polynomial of A, det(A-x*I), which is computed by poly(A) The matrix H from the last section provides our first example. H = [8/9, 1/2, 1/3] [1/2, 1/3, 1/4] [1/3, 1/4, 1/5] The matrix is singular, so one of its eigenvalues must be zero. The statement [T,E] = eig(H) produces the matrices T and E. The columns of T are the eigenvectors of H. T = [ 1, 28/153+2/153*12589^(1/2), 28/153-2/153*12589^(12)] [ -4, 1, 1] [ 10/3, 92/255-1/255*12589^(1/2), 292/255+1/255*12589^(12)] Similarly, the diagonal elements of E are the eigenvalues of H. E = [0, 0, 0] [0, 32/45+1/180*12589^(1/2), 0] [0, 0, 32/45-1/180*12589^(1/2)] It may be easier to understand the structure of the matrices of eigenvectors, T, and eigenvalues, E, if we convert T and E to decimal notation. We proceed as follows. The commands Td = double(T) Ed = double(E)6-72
6 Symbolic Math Toolbox The commands p = poly(R); pretty(factor(p)) produce 2 2 2 x (x - 1020) (x - 1020 x + 100)(x - 1040500) (x - 1000) The characteristic polynomial (of degree 8) factors nicely into the product of two linear terms and three quadratic terms. We can see immediately that four of the eigenvalues are 0, 1020, and a double root at 1000. The other four roots are obtained from the remaining quadratics. Use eig(R) to find all these values [ 0] [ 1020] [510+100*26^(1/2)] [510-100*26^(1/2)] [ 10*10405^(1/2)] [ -10*10405^(1/2)] [ 1000] [ 1000] The Rosser matrix is not a typical example; it is rare for a full 8-by-8 matrix to have a characteristic polynomial that factors into such simple form. If we change the two "corner" elements of R from 29 to 30 with the commands S = R; S(1,8) = 30; S(8,1) = 30; and then try p = poly(S) we find p = 40250968213600000+51264008540948000*x- 1082699388411166000*x^2+4287832912719760*x^-3- 5327831918568*x^4+82706090*x^5+5079941*x^6- 4040*x^7+x^86-74
Linear AlgebraWe also find that factor(p) is p itself. That is, the characteristic polynomialcannot be factored over the rationals.For this modified Rosser matrix F = eig(S)returns F = [ -1020.0532142558915165931894252600] [ -.17053529728768998575200874607757] [ .21803980548301606860857564424981] [ 999.94691786044276755320289228602] [ 1000.1206982933841335712817075454] [ 1019.5243552632016358324933278291] [ 1019.9935501291629257348091808173] [ 1020.4201882015047278185457498840]Notice that these values are close to the eigenvalues of the original Rossermatrix. Further, the numerical values of F are a result of Maple's floating-pointarithmetic. Consequently, different settings of digits do not alter the numberof digits to the right of the decimal place.It is also possible to try to compute eigenvalues of symbolic matrices, but closedform solutions are rare. The Givens transformation is generated as the matrixexponential of the elementary matrix A = 0 1 –1 0The Symbolic Math Toolbox commands syms t A = sym([0 1; -1 0]); G = expm(t*A)return [ cos(t), sin(t)] [ -sin(t), cos(t)] 6-75
Linear Algebra how = combineNotice the first application of simple uses simplify to produce a sum of sinesand cosines. Next, simple invokes radsimp to produce cos(t) + i*sin(t) forthe first eigenvector. The third application of simple uses convert(exp) tochange the sines and cosines to complex exponentials. The last application ofsimple uses simplify to obtain the final form.Jordan Canonical FormThe Jordan canonical form results from attempts to diagonalize a matrix by asimilarity transformation. For a given matrix A, find a nonsingular matrix V,so that inv(V)*A*V, or, more succinctly, J = VA*V, is "as close to diagonal aspossible." For almost all matrices, the Jordan canonical form is the diagonalmatrix of eigenvalues and the columns of the transformation matrix are theeigenvectors. This always happens if the matrix is symmetric or if it hasdistinct eigenvalues. Some nonsymmetric matrices with multiple eigenvaluescannot be diagonalized. The Jordan form has the eigenvalues on its diagonal,but some of the superdiagonal elements are one, instead of zero. The statement J = jordan(A)computes the Jordan canonical form of A. The statement [V,J] = jordan(A)also computes the similarity transformation. The columns of V are thegeneralized eigenvectors of A.The Jordan form is extremely sensitive to perturbations. Almost any change inA causes its Jordan form to be diagonal. This makes it very difficult to computethe Jordan form reliably with floating-point arithmetic. It also implies that Amust be known exactly (i.e., without round-off error, etc.). Its elements must beintegers, or ratios of small integers. In particular, the variable-precisioncalculation, jordan(vpa(A)), is not allowed. 6-77
Linear Algebra ( A – λ 2 I )v 4 = v 3 ( A – λ 1 I )v 2 = v 1Singular Value DecompositionOnly the variable-precision numeric computation of the singular valuedecomposition is available in the toolbox. One reason for this is that theformulas that result from symbolic computation are usually too long andcomplicated to be of much use. If A is a symbolic matrix of floating-point orvariable-precision numbers, then S = svd(A)computes the singular values of A to an accuracy determined by the currentsetting of digits. And [U,S,V] = svd(A);produces two orthogonal matrices, U and V, and a diagonal matrix, S, so that A = U*S*V;Let's look at the n-by-n matrix A with elements defined by A(i,j) = 1/(i-j+1/2)For n = 5, the matrix is [ 2 -2 -2/3 -2/5 -2/7] [2/3 2 -2 -2/3 -2/5] [2/5 2/3 2 -2 -2/3] [2/7 2/5 2/3 2 -2] [2/9 2/7 2/5 2/3 2]It turns out many of the singular values of these matrices are close to π.The most obvious way of generating this matrix is for i=1:n for j=1:n A(i,j) = sym(1/(i-j+1/2)); end end 6-79
6 Symbolic Math Toolbox The most efficient way to generate the matrix is [J,I] = meshgrid(1:n); A = sym(1./(I - J+1/2)); Since the elements of A are the ratios of small integers, vpa(A) produces a variable-precision representation, which is accurate to digits precision. Hence S = svd(vpa(A)) computes the desired singular values to full accuracy. With n = 16 and digits(30), the result is S = [ 1.20968137605668985332455685357 ] [ 2.69162158686066606774782763594 ] [ 3.07790297231119748658424727354 ] [ 3.13504054399744654843898901261 ] [ 3.14106044663470063805218371924 ] [ 3.14155754359918083691050658260 ] [ 3.14159075458605848728982577119 ] [ 3.14159256925492306470284863102 ] [ 3.14159265052654880815569479613 ] [ 3.14159265349961053143856838564 ] [ 3.14159265358767361712392612384 ] [ 3.14159265358975439206849907220 ] [ 3.14159265358979270342635559051 ] [ 3.14159265358979323325290142781 ] [ 3.14159265358979323843066846712 ] [ 3.14159265358979323846255035974 ] There are two ways to compare S with pi, the floating-point representation of π. In the vector below, the first element is computed by subtraction with variable-precision arithmetic and then converted to a double. The second element is computed with floating-point arithmetic. format short e [double(pi*ones(16,1)-S) pi-double(S)]6-80
Linear AlgebraThe results are 1.9319e+00 1.9319e+00 4.4997e-01 4.4997e-01 6.3690e-02 6.3690e-02 6.5521e-03 6.5521e-03 5.3221e-04 5.3221e-04 3.5110e-05 3.5110e-05 1.8990e-06 1.8990e-06 8.4335e-08 8.4335e-08 3.0632e-09 3.0632e-09 9.0183e-11 9.0183e-11 2.1196e-12 2.1196e-12 3.8846e-14 3.8636e-14 5.3504e-16 4.4409e-16 5.2097e-18 0 3.1975e-20 0 9.3024e-23 0Since the relative accuracy of pi is pi*eps, which is 6.9757e-16, either columnconfirms our suspicion that four of the singular values of the 16-by-16 exampleequal π to floating-point accuracy.Eigenvalue TrajectoriesThis example applies several numeric, symbolic, and graphic techniques tostudy the behavior of matrix eigenvalues as a parameter in the matrix isvaried. This particular setting involves numerical analysis and perturbationtheory, but the techniques illustrated are more widely applicable.In this example, we consider a 3-by-3 matrix A whose eigenvalues are 1, 2, 3.First, we perturb A by another matrix E and parameter t: A → A + tE. As t 6-81
Linear Algebraeigenvalues may vary from one machine to another, but on a typicalworkstation, the statements format long e = eig(A)produce e = 0.99999999999642 2.00000000000579 2.99999999999780Of course, the example was created so that its eigenvalues are actually 1, 2, and3. Note that three or four digits have been lost to roundoff. This can be easilyverified with the toolbox. The statements B = sym(A); e = eig(B) p = poly(B) f = factor(p)produce e = [1, 2, 3] p = x^3-6*x^2+11*x-6 f = (x-1)*(x-2)*(x-3)Are the eigenvalues sensitive to the perturbations caused by roundoff errorbecause they are "close together"? Ordinarily, the values 1, 2, and 3 would beregarded as "well separated." But, in this case, the separation should be viewedon the scale of the original matrix. If A were replaced by A/1000, theeigenvalues, which would be .001, .002, .003, would "seem" to be closertogether.But eigenvalue sensitivity is more subtle than just "closeness." With a carefullychosen perturbation of the matrix, it is possible to make two of its eigenvalues 6-83
6 Symbolic Math Toolbox coalesce into an actual double root that is extremely sensitive to roundoff and other errors. One good perturbation direction can be obtained from the outer product of the left and right eigenvectors associated with the most sensitive eigenvalue. The following statement creates E = [130,-390,0;43,-129,0;133,-399,0] the perturbation matrix E = 130 -390 0 43 -129 0 133 -399 0 The perturbation can now be expressed in terms of a single, scalar parameter t. The statements syms x t A = A+t*E replace A with the symbolic representation of its perturbation. A = [-149+130*t, -50-390*t, -154] [ 537+43*t, 180-129*t, 546] [ -27+133*t, -9-399*t, -25] Computing the characteristic polynomial of this new A p = poly(A) gives p = x^3-6*x^2+11*x-t*x^2+492512*t*x-6-1221271*t Prettyprinting pretty(collect(p,x)) shows more clearly that p is a cubic in x whose coefficients vary linearly with t. 3 2 x + (- t - 6) x + (492512 t + 11) x - 6 - 1221271 t6-84
6 Symbolic Math Toolbox One way to find τ is based on the fact that, at a double root, both the function and its derivative must vanish. This results in two polynomial equations to be solved for two unknowns. The statement sol = solve(p,diff(p,x)) solves the pair of algebraic equations p = 0 and dp/dx = 0 and produces sol = t: [4x1 sym] x: [4x1 sym] Find τ now by tau = double(sol.t(2)) which reveals that the second element of sol.t is the desired value of τ. format short tau = 7.8379e-07 Therefore, the second element of sol.x sigma = double(sol.x(2)) is the double eigenvalue sigma = 1.5476 Let's verify that this value of τ does indeed produce a double eigenvalue at σ = 1.5476. To achieve this, substitute τ for t in the perturbed matrix A(t) = A + tE and find the eigenvalues of A(t). That is, e = eig(double(subs(A,t,tau))) e = 1.5476 1.5476 2.9047 confirms that σ = 1.5476 is a double eigenvalue of A(t) for t = 7.8379e-07.6-88
Solving EquationsSolving Equations Solving Algebraic Equations If S is a symbolic expression, solve(S) attempts to find values of the symbolic variable in S (as determined by findsym) for which S is zero. For example, syms a b c x S = a*x^2 + b*x + c; solve(S) uses the familiar quadratic formula to produce ans = [1/2/a*(-b+(b^2-4*a*c)^(1/2))] [1/2/a*(-b-(b^2-4*a*c)^(1/2))] This is a symbolic vector whose elements are the two solutions. If you want to solve for a specific variable, you must specify that variable as an additional argument. For example, if you want to solve S for b, use the command b = solve(S,b) which returns b = -(a*x^2+c)/x Note that these examples assume equations of the form f(x) = 0. If you need to solve equations of the form f(x) = q(x), you must use quoted strings. In particular, the command s = solve(cos(2*x)+sin(x)=1) 6-89
6 Symbolic Math Toolbox The solutions for a reside in the "a-field" of S. That is, S.a produces ans = [ -1] [ 3] Similar comments apply to the solutions for u and v. The structure S can now be manipulated by field and index to access a particular portion of the solution. For example, if we want to examine the second solution, we can use the following statement s2 = [S.a(2), S.u(2), S.v(2)] to extract the second component of each field. s2 = [ 3, 5, -4] The following statement M = [S.a, S.u, S.v] creates the solution matrix M M = [ -1, 1, 0] [ 3, 5, -4] whose rows comprise the distinct solutions of the system. Linear systems of simultaneous equations can also be solved using matrix division. For example, clear u v x y syms u v x y S = solve(x+2*y-u, 4*x+5*y-v); sol = [S.x;S.y]6-92
Solving Equationsand A = [1 2; 4 5]; b = [u; v]; z = Abresult in sol = [ -5/3*u+2/3*v] [ 4/3*u-1/3*v] z = [ -5/3*u+2/3*v] [ 4/3*u-1/3*v]Thus s and z produce the same solution, although the results are assigned todifferent variables.Single Differential EquationThe function dsolve computes symbolic solutions to ordinary differentialequations. The equations are specified by symbolic expressions containing theletter D to denote differentiation. The symbols D2, D3, ... DN, correspond to thesecond, third, ..., Nth derivative, respectively. Thus, D2y is the Symbolic MathToolbox equivalent of d2y/dt2. The dependent variables are those preceded by Dand the default independent variable is t. Note that names of symbolicvariables should not contain D. The independent variable can be changed fromt to some other symbolic variable by including that variable as the last inputargument.Initial conditions can be specified by additional equations. If initial conditionsare not specified, the solutions contain constants of integration, C1, C2, etc.The output from dsolve parallels the output from solve. That is, you can calldsolve with the number of output variables equal to the number of dependentvariables or place the output in a structure whose fields contain the solutionsof the differential equations. 6-93
6 Symbolic Math Toolbox Example 1 The following call to dsolve dsolve(Dy=1+y^2) uses y as the dependent variable and t as the default independent variable. The output of this command is ans = tan(t+C1) To specify an initial condition, use y = dsolve(Dy=1+y^2,y(0)=1) This produces y = tan(t+1/4*pi) Notice that y is in the MATLAB workspace, but the independent variable t is not. Thus, the command diff(y,t) returns an error. To place t in the workspace, type syms t. Example 2 Nonlinear equations may have multiple solutions, even when initial conditions are given. x = dsolve((Dx)^2+x^2=1,x(0)=0) results in x = [-sin(t)] [ sin(t)] Example 3 Here is a second order differential equation with two initial conditions. The commands y = dsolve(D2y=cos(2*x)-y,y(0)=1,Dy(0)=0, x) simplify(y)6-94
A MATLAB Quick Reference Introduction This appendix lists the MATLAB functions as they are grouped in the Help Desk by subject. Each table contains the function names and brief descriptions. For complete information about any of these functions, refer to the Help Desk and either: • Select the function from the MATLAB Functions list (By Subject or By Index), or • Type the function name in the Go to MATLAB function field and click Go. Note If you are viewing this book from the Help Desk, you can click on any function name and jump directly to the corresponding MATLAB function page.A-2
A MATLAB Quick ReferenceCharacter String Functions String to Number ConversionThis set of functions lets you manipulate strings char Create character array (string)such as comparison, concatenation, search, and int2str Integer to string conversionconversion. mat2str Convert a matrix into a string num2str Number to string conversionGeneral sprintf Write formatted data to a stringabs Absolute value and complex sscanf Read string under format magnitude controleval Interpret strings containing str2double Convert string to MATLAB expressions double-precision valuereal Real part of complex number str2num String to number conversionstrings MATLAB string handling Radix ConversionString Manipulation bin2dec Binary to decimal numberdeblank Strip trailing blanks from the conversion end of a string dec2bin Decimal to binary numberfindstr Find one string within another conversionlower Convert string to lower case dec2hex Decimal to hexadecimal numberstrcat String concatenation conversionstrcmp Compare strings hex2dec IEEE hexadecimal to decimal number conversionstrcmpi Compare strings ignoring case hex2num Hexadecimal to double numberstrjust Justify a character array conversionstrmatch Find possible matches for a string Low-Level File I/O Functionsstrncmp Compare the first n characters of two strings The low-level file I/O functions allow you to openstrrep String search and replace and close files, read and write formatted andstrtok First token in string unformatted data, operate on files, and perform other specialized file I/O such as reading andstrvcat Vertical concatenation of strings writing images and spreadsheets.symvar Determine symbolic variables in an expression File Opening and Closingtexlabel Produce the TeX format from a character string fclose Close one or more open filesupper Convert string to upper case fopen Open a file or obtain information about open filesA-14
Bitwise FunctionsUnformatted I/O Specialized File I/O (Continued)fread Read binary data from file wk1read Read a Lotus123 WK1 spreadsheet file into a matrixfwrite Write binary data to a file wk1write Write a matrix to a Lotus123 WK1 spreadsheet fileFormatted I/Ofgetl Return the next line of a file as a Bitwise Functions string without line terminator(s)fgets Return the next line of a file as a These functions let you operate at the bit level string with line terminator(s) such as shifting and complementing.fprintf Write formatted data to filefscanf Read formatted data from file Bitwise Functions bitand Bit-wise ANDFile Positioning bitcmp Complement bitsfeof Test for end-of-file bitor Bit-wise ORferror Query MATLAB about errors in bitmax Maximum floating-point integer file input or output bitset Set bitfrewind Rewind an open file bitshift Bit-wise shiftfseek Set file position indicator bitget Get bitftell Get file position indicator bitxor Bit-wise XORString Conversion Structure Functionssprintf Write formatted data to a string Structures are arrays whose elements can holdsscanf Read string under format any MATLAB data type such as text, numeric control arrays, or other structures. You access structure elements by name. Use the structure functions toSpecialized File I/O create and operate on this array type.dlmread Read an ASCII delimited file into a matrix Structure Functionsdlmwrite Write a matrix to an ASCII deal Deal inputs to outputs delimited file fieldnames Field names of a structurehdf HDF interface getfield Get field of structure arrayimfinfo Return information about a rmfield Remove structure fields graphics file setfield Set field of structure arrayimread Read image from graphics file struct Create structure arrayimwrite Write an image to a graphics file struct2cell Structure to cell arraytextread Read formatted data from text conversion file A-15
A MATLAB Quick ReferenceObject Functions Multidimensional Array FunctionsUsing the object functions you can create objects, cat Concatenate arraysdetect objects of a given class, and return the class flipdim Flip array along a specifiedof an object. dimension ind2sub Subscripts from linear indexObject Functions ipermute Inverse permute the dimensionsclass Create object or return class of of a multidimensional array object ndgrid Generate arrays forisa Detect an object of a given class multidimensional functions and interpolationCell Array Functions ndims Number of array dimensions permute Rearrange the dimensions of aCell arrays are arrays comprised of cells, which can multidimensional arrayhold any MATLAB data type such as text, numeric reshape Reshape arrayarrays, or other cell arrays. Unlike structures, you shiftdim Shift dimensionsaccess these cells by number. Use the cell arrayfunctions to create and operate on these arrays. squeeze Remove singleton dimensions sub2ind Single index from subscriptsCell Array Functionscell Create cell array Plotting and Data Visualizationcellfun Apply a function to each element This extensive set of functions gives you the ability in a cell array to create basic graphs such as bar, pie, polar, andcellstr Create cell array of strings from three-dimensional plots, and advanced graphs character array such as surface, mesh, contour, and volumecell2struct Cell array to structure array visualization plots. In addition, you can use these conversion functions to control lighting, color, view, and manycelldisp Display cell array contents other fine manipulations.cellplot Graphically display the structure of cell arrays Basic Plots and Graphsnum2cell Convert a numeric array into a bar Vertical bar chart cell array barh Horizontal bar chart hist Plot histogramsMultidimensional Array Functions hold Hold current graphThese functions provide a mechanism for working loglog Plot using log-log scaleswith arrays of dimension greater than 2. pie Pie plot plot Plot vectors or matrices. polar Polar coordinate plot semilogx Semi-log scale plotA-16
B Symbolic Math Toolbox Quick Reference Introduction This appendix lists the Symbolic Math Toolbox functions that are available in the Student Version of MATLAB & Simulink. For complete information about any of these functions, use the Help Desk and either: • Select the function from the Symbolic Math Toolbox Functions, or • Select Online Manuals and view the Symbolic Math Toolbox User's Guide. Note All of the functions listed in Symbolic Math Toolbox Functions are available in the Student Version of MATLAB & Simulink except maple, mapleinit, mfun, mfunlist, and mhelp.B-2 |
What kinds of curriculum materials do mathematics teachers select and use, and how? This question is complex, in a period of deep evolutions of teaching resources, with the proficiency of online resources in particular. How do teachers learn from these materials, and in which ways do they 'tailor' them for their use and pupil learning? Teachers... more...Preempt your anxiety about PRE-ALGEBRA!
Ready to learn math fundamentals but can't seem to get your brain to function? No problem! Add Pre-Algebra Demystified , Second Edition, to the equation and you'll solve your dilemma in no time.
Written in a step-by-step format, this practical guide begins by covering whole numbers, integers, fractions,... more...
Lesson study is a professional development process that teachers engage in to systematically examine their practice, with the goal of becoming more effective. Originating in Japan, lesson study has gained significant momentum in the mathematics education community in recent years. As a process for professional development, lesson study became highly... more...
Build student success in math with the only comprehensive guide for developing math talent among advanced learners. The authors, nationally recognized math education experts, offer a focused look at educating gifted and talented students for success in math. More than just a guidebook for educators, this book offers a comprehensive approach to mathematics... more...
Mathematical sciences have been playing an increasingly important role in modern society. They are in high demand for investigating complex problems in physical science, environmental and geophysical sciences, materials science, life science and chemical sciences. This is a review volume on some timely and interesting topics in applied mathematical... more... |
Summary: The Third Edition of the Bittinger Graphs and Models series helps students succeed in algebra by emphasizing a visual understanding of concepts. This latest edition incorporates a new Visualizing the Graph feature that helps students make intuitive connections between graphs and functions without the aid of a graphing calculator.
In addition, students learn problem-solving skills from the Bittinger hallmark five-step problem-solving process coupled with Co...show morennecting the Concepts and Aha! Exercises. As you have come to expect with any Bittinger text, we bring you a complete supplements package including MyMathLab® and the new Instructor and Adjunct Support Manual. ...show less
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Sheldon Axler
For the review of Precalculus, the recent recommendations from the Committee on the Undergraduate Program in Mathematics (CUPM) are used as an outline. The headings below are the CUPM recommendations; each is followed by my attempt to assess the extent to which the book conforms to them.
Present key ideas and concepts from a variety of perspectives
Chapter 4 (randomly selected), Area, e, and the Natural Logarithm, includes five sections and 66 pages. Of the 66 pages, twenty-seven pages contain graphs. The following are the section number and the amount of examples in each sections: {(1, 7), (2, 5), (3, 2), (4, 0), (5, 4)}. Definitions, properties and concepts are boxed in and colored to highlight their importance throughout the text.
Promote awareness of connections to other subjects and strengthen each student's ability to apply the course material to these subjects
Looking through the chapter review questions for exercises that include context, the following amounts are per chapter (including chapter 0) out of the total problems in the review were found: {0 out of 20, 0 out of 30, 0 out of 30, 7 out of 36, 5 out of 30, 0 out of 50, 1 out of 41 and 0 out of 14}.
Employ a broad range of instructional techniques, and require students to confront, explore, and communicate important ideas of modern mathematics and the uses of mathematics in society. Students need more classroom experiences in which they learn to think, to do, to analyze – not just memorize and reproduce theories or algorithms
Included in each section of every chapter are exercises (and worked solutions) and problems. The problems tend to be open-ended and need explanations, not just computation.
Understand and respond to the impact of computer technology on course content and instructional techniques
A calculator icon is used throughout the exercises and problems to delineate the questions that need a calculator to work through or to estimate an answer.
At first glance, I was really excited by this text. I like the idea of having open-ended problems for the students to struggle with. But very few of the open-ended problems incorporate real-world applications. So, in order to utilize this text and teach with connections to other subjects, I would need to supplement it. Furthermore, when teaching precalculus I emphasize the use of multiple representations of functions (graphic, numeric, symbolic, and verbal) by using a graphing calculator. Therefore, more supplementation would be needed.
Jane Ries Cushman currently works at Buffalo State College in Buffalo, NY as an assistant professor. She received her doctorate at The University of Texas at Austin in August 2006. She is editor of the Association of Mathematics Teachers of New York State Newsletter and she is the chair of the Association of Mathematics Teacher Educators Affiliate's Connection Committee. Her research interests include Inquiry-Based Learning, Problem-Solving and Functions-Based Approach to Algebra. |
This site is a selection of mathlets designed for "geometry classes to review basic algebra skills in such a way that they...
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This site contains an online library with datafiles and stories that demonstrate the use of statistical methods. Instructors...
see more
This site contains an online library with datafiles and stories that demonstrate the use of statistical methods. Instructors can use the "stories" and datafiles to create real world examples that will interest students. The list here is organized by Method, but one can search the material by Topic or Data Subject as wellDASL is designed to help instructors locate and identify datafiles for teaching. DASL also serves as an archive for datasets...
see more
DASL is designed to help instructors locate and identify datafiles for teaching. DASL also serves as an archive for datasets from the statistics literature. The archive contains two types of files, stories and datafiles. Each story applies a particular statistical method to aset of data. Each datafile has one or more associated stories. The data can be downloaded as a space- or tab-delimited table of text, easily read by most statistics programs. The archive is searchable by discipline as well as statistical method.
'Over 100 math formulas at high school level. The covered areas include algebra, geometry, calculus, trigonometry,...
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'Over 100 math formulas at high school level. The covered areas include algebra, geometry, calculus, trigonometry, probability and statistics. Most of the formulas come with examples for better understanding. Use the powerful search function to find what you are looking for and mark your favorites for easier access. A convenient tool for students and teachers and a handy reference for anyone interested in math!'This app costs $0.99
This website was created to "use Web 2.0 tools to collect, organize, and redistribute free online educational resources for...
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This website was created to "use Web 2.0 tools to collect, organize, and redistribute free online educational resources for the students and families to use.״ The site has learning activities that students can utilize during the summer so they don't fall behind in classes. There are activities for Language Arts, Math, and ESL for grades K-2, 3-5, and 6-8. These resources have been collected by students, parents, and educators and organized by students in a USC writing course.
Keith Devlin is NPR's Math Guy who appears on NPR's Weekend Edition and Morning Edition programs. Over the years he has...
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Keith Devlin is NPR's Math Guy who appears on NPR's Weekend Edition and Morning Edition programs. Over the years he has contributed 78 segments that explain the mathematics of items in the news. The Math Guy archives could be a great place to find some "real world" mathematics examples and problems for your students to tackle. The episodes provide a good cross-over between mathematics and current events, perfect for teachers trying to show the relevance of mathematics to their students. |
Ruskeepaa gives a general introduction to the most recent versions of Mathematica, the symbolic computation software from Wolfram. The book emphasizes graphics, methods of applied mathematics and statistics, and programming. Mathematica Navigator can be used both as a tutorial and as a handbook. While no previous experience with Mathematica is required, most chapters also include advanced material, so that the book will be a valuable resource for both beginners and experienced users.Covers both Mathematica v6 and Mathematica v7Fully-revised and updated, based on Mathematica v6CD-ROM contains material about the new properties of Mathematica v7 and can be installed into the help system of MathematicaComprehensive coverage from basic, introductory information through to more advanced topicsStudies several real data sets (included in the CD-ROM) and many classical mathematical models |
Staten Island Algebra 2 lot of models and analogies that help explain some abstract concepts and help visualize tiniest atoms and molecules and their interactions. Finally, I encourage students to work on additional assignments to enhance their experience and strengthen their problem-solving skills; learn to as... |
Mathematics for Elementary Teachers, New York Correlation Guide Book: A Contemporary Approach
Book Description: This leading mathematics text for elementary and middle school educators helps you quickly develop a true understanding of mathematical concepts. It integrates rich problem-solving strategies with relevant topics and extensive opportunities for hands-on experience. By progressing from the concrete to the pictorial to the abstract, Musser captures the way math is generally taught in elementary schools.This title will give you all the essentials mathematics teachers need for teaching at the elementary and middle school levels:Highlights algebraic concepts throughout the text and includes additional supporting information. Provides enhanced coverage of order of operations, Z-scores, union of two events, Least Common Multiple, and Greatest Common Factor. Focuses on solid mathematical content in an accessible and appealing way. Offers the largest collection of problems (over 3,000!), worked examples, and problem-solving strategies in any text of its kind.Includes a comprehensive, five-chapter treatment of geometry based on the van Hiele |
This is a free, online textbook offered by the CK-12 Foundation. Although designed for high schools, it could also be used...
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This is a free, online textbook offered by the CK-12 Foundation. Although designed for high schools, it could also be used for college freshmen. Chapters include the following topics: 1. Basics of Geometry2. Reasoning and Proof3. Parallel and Perpendicular Lines4. Congruent Triangles5. Relationships Within Triangles6. Quadrilaterals7. Similarity8. Right Triangle Trigonometry9. Circles10. Perimeter and Area11. Surface Area and Volume12. Transformations Geometry - Second Edition is a clear presentation of the essentials of geometry for the high school student. Topics include: Proofs, Triangles, Quadrilaterals, Similarity, Perimeter & Area, Volume, and Transformations. Volume 2 includes the last 6 chapters: Similarity, Right Triangle Trigonometry, Circles, Perimeter and Area, Surface Area and Volume, and Rigid Transformations.'
This is a free, online textbook that is comprised of articles from a variety of authors. "Comparison Geometry asks: What can...
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This is a free, online textbook that is comprised of articles from a variety of authors. "Comparison Geometry asks: What can we say about a Riemannian manifold if we know a (lower or upper) bound for its curvature, and perhaps something about its topology? Powerful results that allow the exploration of this question were first obtained in the 1950s by Rauch, Alexandrov, Toponogov, and Bishop, with some ideas going back to Hopf, Morse, Schoenberg, Myers, and Synge in the 1930s.״
This is a free, online textbook that is designed "for the basic course of differential geometry. It is recommended as an...
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This is a free, online textbook that is designed "for the basic course of differential geometry. It is recommended as an introductory material for this subject. The PDF file with 15 color pictures is designed for double-side printing on the standard Letter size paper.״ |
9:00-9:50, M W F, DuSable 348
COURSE DESCRIPTION:Algebra I (3).
Introduction to group theory.
Properties of the integers, functions, and equivalence relations.
A concrete approach to cyclic groups and permutation groups;
isomorphisms and the theorems of Lagrange and Cayley.
COURSE OBJECTIVES:
The student is expected to acquire an understanding
of the elementary theory of groups, together with the
necessary number theoretic prerequisites. There will
be some discussion of the computational aspects of
these topics, but the main thrust of the course will
be theoretical. The student will be expected not
only to follow the proofs presented in class and in
the text, but also to learn to construct new proofs.
Proofs must be logically correct and care must be
taken to write them precisely and in grammatically
correct English.
COURSE PREREQUISITE:
MATH 240, Linear Algebra.
We will use matrices in some important examples, but
the main reason for the requirement is to attempt to
guarantee a certain level of "mathematical maturity."
GRADING:
Final grades will be based on 600 points:
3 hour tests (300), homework (100), and final exam (200).
The homework problems are extremely important.
In many ways the course is like an English composition
course, since it requires you to write out very carefully
the reasons for each step in your solutions of problems.
CAAR STATEMENT:
Students who request accommodation due to a physical or learning disability
must contact their instructor at the beginning of the semester. The
instructor has the right to see documentation of the student's condition
from the CAAR office.
HONORS:
The honors section meets an additional hour each week.
We will do some extra problems from the text,
and then study symmetry groups.
Reference: Goodman,
Algebra, abstract and concrete, stressing symmetry,
Sections 1.1-1.4, 1.12.
Meeting time and place: 9:30-10:20 Thursday mornings in DU 464
ASSIGNMENTS:
Assignment 1: from the text, 1.2 #16,18,25
Assignment 2: from the text, Appendix 4, #4,5,7,9 (pp 443-444)
Assignment 3: from the text, 1.3 #25,61 and 1.4 #29,55,56
Assignment 4: from the text, 2.1 #16,17,18
Assignment 5: from the text, 2.1 #7, 3.6 #12
Assignment 6: find references for a 4 to 5 page paper on Rubik's cube,
which will be due the last week of the semester.
Some possibilities are listed below. |
Synopses & Reviews
Publisher Comments:
Master the fundamentals of computer graphics with Schaum's -the high-performance study guide. It will help you cut study time, home, but want to excel in class, this book helps you: *Use detailed examples to solve problems *Brush up before tests *Find answers fast *Study quickly and more effectively *Get the big picture without spending hours poring over lengthy textbooks Schaum's Outlines give you the information your teachers expect youto know in a handy and succinct format - without - fast! And Schaum's are so complete, they're perfect for preparing for graduate or professional exams. Inside, you will find: *Full coverage of Computer Graphics, from the traditional 2D to the recent 3D advances *Simplified explanations of the algorithmic aspects of image synthesis *Hundreds of solved problems in computer graphics, including step-by-step annotations *Examples and worked problems that help you master computer graphics If you want top grades and a thorough understanding of computer graphics, this powerful study tool is the best tutor you can have! Chapters include: Introduction, Image Representation, Scan Conversion, Two-Dimensional Transformations, Two-Dimensional Viewing and Clipping, Three-Dimensional Transformations, Mathematics of Projection, Three-Dimensional Viewing and Clipping, Geometric Represenation, Hidden Surfaces, Color and Shading Models, Ray Tracing, Appendixes include: Mathematics for Two-Dimensional computer Graphics, Mathematics for Three-Dimensional Computer Graphics.
Synopsis:
Synopsis:
"Synopsis"
by Gardners,"Synopsis"
by McGraw, |
Content MathML
Summary: A short introduction to writing Content MathML by hand. It covers
tokens, prefix notation, and applying functions and operators. In
addition it introduces writing derivatives, integrals, vectors, and
matrices.
The authoritative reference for Content MathML is Section 4
of the MathML 2.0 Specification. The World Wide Web
Consortium (W3C) is the body that wrote the specification for
MathML. The text is very readable and it is easy to find what
you are looking for. Look there for answers to questions that
are not answered in this tutorial or when you need more
elaboration. This tutorial is based on MathML 2.0.
In this document, the m prefix is used to
denote tags in the MathML namespace. Thus the
<apply> tag is referred to as
<m:apply>. Remember all markup in
the MathML namespace must be surrounded by
<m:math> tags.
The Fundamentals of Content MathML: Applying Functions and
Operators
The fundamental concept to grasp about Content MathML is that
it consists of applying a series of functions and operators to
other elements. To do this, Content MathML uses prefix
notation. Prefix notation is when the operator
comes first and is followed by the operands. Here is how to
write "2 plus 3".
There are three types of elements in the Content MathML
example shown above. First, there is the apply tag, which
indicates that an operator (or function) is about to be
applied to the operands. Second, there is the function or
operator to be applied. In this case the operator, plus, is
being applied. Third, the operands follow the operator. In
this case the operands are the numbers being added. In
summary, the apply tag applies the function (which could be
sin or ff, etc.) or operator
(which could be plus or minus, etc.) to the elements that
follow it.
Tokens
Content MathML has three tokens: ci, cn, and csymbol. A
token is basically the lowest level element.
The tokens denote what kind of element you are acting on.
The cn tag indicates that the content of the
tag is a number. The ci tag indicates that the
content of the tag is an identifier. An
identifier could be any variable or function;
xx,
yy, and
ff are examples of identifiers.
In addition, ci elements can contain Presentation MathML.
Tokens, especially ci and cn, are used profusely in Content
MathML. Every number, variable, or function is marked by a
token.
csymbol is a different type of token from ci
and cn. It is used to create a new object whose semantics
is defined externally. It can contain plain text or
Presentation MathML. If you find that you need something,
such as an operator or function, that is not defined in
Content MathML, then you can use csymbol to create it.
Both ci and csymbol can use Presentation MathML to determine
how an identifier or a new symbol will be rendered. To
learn more about Presentation MathML see Section
3 of the MathML 2.0 Specification. For example, to
denote "xx with a subscript 2",
where the 2 does not have a more semantic meaning, you would
use the following code.
The ci elements have a type attribute which can be used to
provide more information about the content of the element.
For example, you can declare the contents of a ci tag to be
a function (type='fn'), or a vector (type='vector'), or a
complex number (type='complex'), as well as any number of
other things. Using the type attribute helps encode the
meaning of the math that you are writing.
Functions and Operators
In order to apply a function to a variable, make the
function the first argument of an apply. The second
argument will be the variable. For example, you would use
the following code to encode the meaning, "the function
ff of
xx". (Note that you have to
include the attribute type='fn' on the ci tag denoting
ff.)
There are also pre-defined functions and operators in
Content MathML. For example, sine and cosine are
predefined. These predefined functions and operators are
all empty tags and they directly follow the apply tag. "The
sine of xx" is similar to the
example above.
<m:math>
<m:apply>
<m:sin/>
<m:ci>x</m:ci>
</m:apply>
</m:math>
You can find a more thorough description of the different
predefined functions in the MathML specification.
In addition to the predefined functions, there are also many
predefined operators. A few of these are plus (for
addition), minus (for subtraction), times (for
multiplication), divide (for division), power (for taking
the nn-power of something), and
root (for taking the nn-root of
something).
Most operators expect a specific number of child tags. For
example, the power operator expects two children. The first
child is the base and the second is the value in the
exponent. However, there are other tags which can take many
children. For example, the plus operator merely expects one
or more children. It will add together all of its children
whether there are two or five.
Representing "the negative of a variable" and explicitly
representing "the positive of a variable or number" has
slightly unusual syntax. In this case you apply the plus or
minus operator to the variable or number, etc., in question.
The following is the code for "negative
xx."
<m:math>
<m:apply>
<m:minus/>
<m:ci>x</m:ci>
</m:apply>
</m:math>
In contrast to representing the negative of a variable, the
negative of a number may be coded as follows:
<m:math><m:cn>-1</m:cn></m:math>
To create more complicated expressions, you can nest these
bits of apply code within each other. You can create
arbitrarily complex expressions this way.
"aa times the quantity
bb plus
cc" would be written as
follows.
The eq operator is used to write equations. It is used in
the same way as any other operator. That is, it is the
first child of an apply. It takes two children which are
the two quantities that are equal to each other. For
example, "aa times
bb plus
aa times
cc equals
aa times the quantity
bb plus
cc" would be written as shown.
Integrals
The operator for an integral is int. However, unlike the
operators and functions discussed above, it has children that
define the independent variable that you integrate with
respect to (bvar) and the interval over which the integral is
taken (use either lowlimit and uplimit, or interval, or
condition). lowlimit and uplimit (which go together),
interval, and condition are just three different ways of
denoting the integrands. Don't forget that the bvar,
lowlimit, uplimit, interval, and condition children take token
elements as well. The following is "the integral of
ff of
xx with respect to
xx from 0 to
bb."
Derivatives
The derivative operator is diff. The derivative is done in
much the same way as the integral. That is, you need to
define a base variable (using bvar). The following is "the
derivative of the function ff of
xx, with respect to
xx."
To apply a higher level derivative to a function, add a degree
tag inside of the bvar tag. The degree tag will contain the
order of the derivative. The following shows "the second
derivative of the function ff of
xx, with respect to
xx."
There are also operators to take the determinant and the
transpose of a matrix as well as to select elements from
within the matrix.
Entities
MathML defines its own entities for many characters that you
might need to use (Greek letters for example). They are also
very useful when you need to embed Presentation MathML within
Content MathML. A list of these
entities is found in the MathML 2.0 specification. It
is better to use these entities than the Unicode character
that they stand for, because these entities can be redefined
as necessary.
Other Resources
There is a lot more that can be done with Content MathML.
Especially if you are planning on writing a lot of Content
MathML, it is well worth your time to take a look at the MathML
specification.
Content actions
Share content
Share module:
Give feedback:
Download module as:
Add module to:
'My Favorites' is a special kind |
Contemporary's Number Power 4: Geometry: a real world approach to math (The Number Power Series)
Book Description: Number Power is the first choice for those who want to develop and improve their math skills! 4: Geometry introduces lines, angles, triangles, other plane figures, and solid figures |
MATH 3110 - A careful study, with emphasis on proofs, of the following topics associated with the set of integers: divisibility, congruences, arithmetic functions, sums of squares, quadratic residues and reciprocity, and elementary results on distributions of primes.
Prerequisites: MATH 2150 (Discrete Mathematics), and either MATH 1360 (Calculus 2) or score of 4 or 5 on the AP Calculus AB or BC exam.
Note: 100-level courses are courses that students would under normal circumstances take during their freshman year at the university, 200-level courses during their sophomore year, and 300-level courses during their junior year. |
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$35.40 by two former instructors at The Culinary Institute of America, this revised and updated guide is an indispensable math resource for foodservice professionals everywhere. Covering topics such as calculating yield percent, determining portion costs, changing recipe yields, and converting between metric and U.S. measures, it offers a review of math basics, easy-to-follow lessons, detailed examples, and newly revised practice problems in every chapter.
Author Biography
Linda Blocker taught middle and high school mathematics for ten years before joining the faculty of The Culinary Institute of America. During her three years at the CIA, she taught culinary math. She spent many years involved in her family's gourmet food business, Meredith Mountain Farms.
Julia Hill taught cost control and culinary math at The Culinary Institute of America for fifteen years. Prior to her teaching position at the CIA, she was a public accountant and restaurant manager.
Founded in 1946, The Culinary Institute of America is an independent, not-for-profit college offering bachelor's and associate New York, and at The Culinary Institute of America at Greystone, in St. Helena, California. Greystone also offers baking and pastry, advanced culinary arts, and wine certifications.
Table of Contents
Acknowledgments
p. vii
Preface
p. ix
Math Basics
p. 1
Customary Units of Measure
p. 25
Metric Measures
p. 40
Basic Conversion of Units of Measure within Volume or Weight
p. 49
Converting Weight and Volume Mixed Measures
p. 60
Advanced Conversions between Weight and Volume
p. 71
Yield Percent
p. 87
Applying Yield Percent
p. 106
Finding Cost
p. 120
Edible Portion Cost
p. 138
Recipe Costing
p. 154
Yield Percent: When to Ignore It
p. 176
Beverage Costing
p. 186
Recipe Size Conversion
p. 200
Kitchen Ratios
p. 222
Formula Reference Review
p. 237
Units of Measure and Equivalency Charts
p. 241
Approximate Volume to Weight Chart and Approximate Yield of Fruits and Vegetables Chart |
Calculus,
one of the most useful areas of mathematics, is the study of continuous
change. It provides the language and concepts used by modern science to
quantify the laws of nature and the numerical techniques through which
this knowledge is applied to enrich daily life. Using the mathematics
computer laboratory, students gain a clear understanding of the
fundamental principles of calculus and how they are applied in
real-world situations. Topics: techniques of integration, further
applications of derivatives, and applications of integration. (4
credits) Prerequisite: MATH 281 |
With this second volume, we enter the intriguing world of complex analysis. From the first theorems on, the elegance and sweep of the results is evident. The starting point is the simple idea of extending a function initially given for real values of the argument to one that is defined when the argument is complex. From there, one proceeds to the main properties of holomorphic functions, whose proofs are generally short and quite illuminating: the Cauchy theorems, residues, analytic continuation, the argument principle.
With this background, the reader is ready to learn a wealth of additional material connecting the subject with other areas of mathematics: the Fourier transform treated by contour integration, the zeta function and the prime number theorem, and an introduction to elliptic functions culminating in their application to combinatorics and number theory.
Thoroughly developing a subject with many ramifications, while striking a careful balance between conceptual insights and the technical underpinnings of rigorous analysis, Complex Analysis will be welcomed by students of mathematics, physics, engineering and other sciences.
The Princeton Lectures in Analysis represents a sustained effort to introduce the core areas of mathematical analysis while also illustrating the organic unity between them. Numerous examples and applications throughout its four planned volumes, of which Complex Analysis is the second, highlight the far-reaching consequences of certain ideas in analysis to other fields of mathematics and a variety of sciences. Stein and Shakarchi move from an introduction addressing Fourier series and integrals to in-depth considerations of complex analysis; measure and integration theory, and Hilbert spaces; and, finally, further topics such as functional analysis, distributions and elements of probability theory |
Technical Math courses that do not include calculus. This text addresses curriculum and pedagogy standards that are initiatives of the American Mathematical Association of two-year Colleges (AMATYC), the National Council of Teachers of Mathematics (NCTM), and the Mathematics Association of America (MAA). It uses simplified language that appeals to a variety of student learning styles, promotes active and independent learning, and strengthens critical thinking and writing skills. A six-step approach to problem solving, numerous tips, and clear, concise explanations throughout the text enable students to understand the concepts underlying mathematical processes. |
Introductory and Intermediate Algebra - 07 edition
ISBN13:978-0073298078 ISBN10: 0073298077 This edition has also been released as: ISBN13: 978-0073298146 ISBN10: 007329814X
Summary: Miller/O'Neill/Hyde's Introductory and Intermediate Algebra is an insightful and engaging textbook written for teachers by teachers. Through strong pedagogical features, conceptual learning methodologies, student friendly writing, and a wide-variety of exercise sets, Introductory and Intermediate Algebra is a book committed to student success in mathematics.
1.1 Sets of Numbers and the Real Number Line 1.2 Order of Operations 1.3 Addition of Real Numbers 1.4 Subtraction of Real Numbers 1.5 Multiplication and Division of Real Numbers 1.6 Properties of Real Numbers and Simplifying Expressions
7.1 Solving Systems of Linear Equations by Graphing 7.2 Solving Systems of Equations by Using the Substitution Method 7.3 Solving Systems of Equations by Using the Addition Method 7.4 Applications of Systems of Linear Equations in Two Variables 7.5 Systems of Linear Equations in Three Variables and Applications 7.6 Solving Systems of Linear Equations by Using Matrices |
What knowledge of mathematics do secondary school math teachers need to facilitate understanding, competency, and interest in mathematics for all of their students? This unique text and resource bridges the gap between the mathematics learned in college and the mathematics taught in secondary schools. Written in an informal, clear, and interactive... more...
Mathematics education in the United States can reproduce social inequalities whether schools use either "basic-skills" curricula to prepare mainly low-income students of color for low-skilled service jobs or "standards-based" curricula to ready students for knowledge-intensive positions. And working for fundamental social change and rectifying injustice... more...
Signal processing is the discipline of extracting information from collections of measurements. To be effective, the measurements must be organized and then filtered, detected, or transformed to expose the desired information. Distortions caused by uncertainty, noise, and clutter degrade the performance of practical signal processing systems.In aggressively... more... |
Course:
For finite mathematics students
majoring in Business, Management, or Economics /Social & Life
sciences
Created by Kameswari Tekumalla for Synergy Project
(Feb 2002)
Uses websites about Venn diagrams produced by A.J.Ronald, Campbell, and NY
State Exam Prep. Center.
Background: In the mid-nineteenth
century, John Venn (1834-1923), a Fellow of Cambridge University, devised
a scheme for visualizing logical relationships. His single contribution to
the field of Mathematics made him immortal. That contribution is the Venn
diagram. Venn wrote Logic of Chance in 1866 which his colleague
Keynes
described as "strikingly original and considerably influenced
the development of the theory of Statistics. " It is a technique
for analyzing visually and solving many problems and logical relationships.
A Venn diagram is simply a field within which circular areas represent
groups of items sharing common properties. The Venn diagram is made up of
two or more overlapping circles and a rectangle is used to represent the
Universal Set (Universal Set includes all objects being analyzed). Venn Diagrams
are often used in mathematics to show relationships between sets.Consider
a Universal set with two subsets A and B. The union of A and B is everything
which is in either A or B. We may represent this as
AÈB. The intersection of
two sets is that which is in both sets, as represented as
AÇB. The complement of a
set A is everything that is not in A; it is represented by A' or
Ac. To know more about them, visit the
website.
Venn
Diagrams . . A set is a list of objects in no particular
order; they could be numbers, letters, or even words. A Venn
diagram is a way of representing sets visually. To learn more, visit
the site Sets
and Venn diagrams. Venn Diagrams are useful to solve many problems that
involve critical thinking. To see examples visit the website
Practice
Problems . If you like to create a Venn diagram on web,
Click here.
Question:
Using the data from a survey of critical care
hospital patients, complete the Venn diagram
and answer the questions below.
80 patients were surveyed with the following results;
40 patients had Diabetes (D)
30 Patients had Heart Disease
(H)
30 Patients had Cancer (C)
15 Patients had Diabetes and Heart Disease
14 Patients had Diabetes and Cancer
18 Patients had Heart Disease and Cancer
10 Patients had all three
a) How many patients had exactly two of the
ailments?
_________
b) How many patients suffered from none
of the three ailments? ________ |
Summary: These popular and proven workbooks help students build confidence before attempting end-of-chapter problems. They provide short problems and exercises that focus on developing a particular skill, often requiring students to draw or interpret sketches and graphs, or reason with math relationships. New to the Second Edition are exercises that provide guided practice for the textbook's Problem-Solving Strategies, focusing in particular on working symbolically.
Very Good Text may contain some highlighting. Order shipped same day if if rec'd by 1PM CST, otherwise ships the next business day. Great Customer Service. Upgrade shipping available.
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2009-10-30 Paperback Good Expedited shipping is available for this item!
$19.00 +$3.99 s/h
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One Stop Text Books Store Sherman Oaks, CA
2009-10 |
Calculus and Analytic Geometry 2
Credits: 5Catalog #20804232
Calculus and Analytic Geometry 2 is designed for students of mathematics, science, and engineering. Topics covered include the techniques of integration, numerical approximation of definite integrals, applications of integration and an introduction to first order differential equations, analysis of infinite sequences and series, parametric equations and derivatives of parametric curves, polar coordinates in the plane and integrals using polar coordinates, the analytic geometry of the conic sections, an introduction to vectors in two and three dimensions, scalar and vector cross products, graphs of quadratic surfaces.
Course Offerings
last updated: 09:02:12 course may make use of an online homework system. Your instructor may choose to use a variety of teaching methods and tools including 'online math software, active learning principles, supplemental video lessons, etc..' Please contact your instructor prior to the first day of class for details about specific approaches to be used in this class sectionAn online subscription to MyMathLab ( is required for this course. Online interactive educational systems provide additional homework support, immediate feedback, and automated grading on many required assignments. Publishers often include an access code in the price of a new textbook, but these can also be purchased separately. In most cases, the subscription will include electronic access to the textbook, so there is no need to purchase a paper copy of the textbook for this class. Please consult your instructor for more information.
This section of Calc 2 uses primarily the "inverted classroom" model. Students are responsible for watching roughly three video lectures on-line (via Blackboard or YouTube) per week before class. Class time is spent doing example problems, group work, applications and skill-building. There will be in-class quizzes most Fridays and three to four in-class exams |
To take a practice quiz before a test:
1. Go to
2. Choose your state – Louisiana
3. Choose student/parent
4. Subject - choose mathematics
5. Click enter
6. Louisiana programs – choose Mathematics: Applications &
Concepts, Course I, 2005
7. Click self check quiz Leap 21
8. Click by lesson.
9. Click on the chapter you need.
10. Click on the lesson you need.
11. Click check it at the bottom when you are finished.
To access your math book online:
1. Go to
2. Choose your state – Louisiana
3. Choose student/parent.
4. Subject – choose mathematics
5. Click enter.
6. Louisiana programs – choose Mathematics: Applications
and Concepts: Course I 2005
7. Click on online student edition.
8. User name: MAC1LA05
Password: qL5toAch
9. Click submit.
9. Click on contents in the bottom right hand corner on the
red math book.
10. Click Table of Contents on the left side of the screen.
11. Click on the unit you need on the left side of the screen.
12. Click on the lesson you need on the left side of the screen.
13. You can enlarge the page with the + button towards the top
of the screen |
co... read more
Our Editors also recommend:Another Fine Math You've Got Me Into. . . by Ian Stewart Sixteen columns from the French edition of Scientific American feature oddball characters and wacky wordplay in a mathematical wonderland of puzzles and games that also imparts significant mathematical ideas. 1992Game, Set and Math: Enigmas and Conundrums by Ian Stewart Twelve essays take a playful approach to mathematics, investigating the topology of a blanket, the odds of beating a superior tennis player, and how to distinguish between fact and fallacy.Game Theory: A Nontechnical Introduction by Morton D. Davis This fascinating, newly revised edition offers an overview of game theory, plus lucid coverage of two-person zero-sum game with equilibrium points; general, two-person zero-sum game; utility theory; and other topics.
Arithmetic Refresher by A. A. Klaf These 937 most-asked questions deal with tax problems, interest and discount, time-payment, etc. Features 809 problems and answers. "More than just a refresher . . . contains a great number of items that are not just reminders but entirely new ideas. — Bookmarks.
Trigonometry Refresher by A. Albert Klaf Covers the most important aspects of plane and spherical trigonometry. Discusses special problems in navigation, surveying, elasticity, architecture, and various fields of engineering. Includes 1,738 problems, many with solutions. 1946 edition. Features 494 figures.
Product Description:
concepts of applied mathematics useful for solving problems that arise in business, industry, science, and technology. Contents include examinations of the theory of sets, numbers and groups; matrices and determinants; probability, statistics, and quality control; and game theory. Additional subjects include inequalities, linear programming, and the transportation problem; combinatorial mathematics; transformations and transforms; and numerical analysis. Accessible explanations of important concepts feature a total of more than 150 diagrams and graphs, in addition to worked-out examples with step-by-step explanations of methods. Answers to exercises and problems appear at the end |
Exercises
I've
discovered that simple exercises are exceptionally useful during a seminar to
complete a student's understanding, so you'll find a set at the end
of each chapter.
Most
exercises are designed to be easy enough that they can be finished in a
reasonable amount of time in a classroom situation while the instructor
observes, making sure that all the students are absorbing the material. Some
exercises are more advanced to prevent boredom for experienced students. The
majority are designed to be solved in a short time and test and polish your
knowledge. Some are more challenging, but none present major challenges.
(Presumably, you'll find those on your own – or more likely
they'll find you |
The following placement guide may be used to estimate which book a student should use. A placement test, to be used for initial placement only, will provide a more accurate measure.
Grade Level
Accelerated Student
Average Student
Slower Student
12th
Calculus (A.P. Calculus*)
Advanced Math (Geo., Trig., Alg. 4**)
Advanced Math***
11th
Advanced Math (Pre-Calculus II*) Calculus
Advanced Math (Geo., Trig., Alg. 3**)
Algebra 2
10th
Advanced Math (Pre-Calculus I*)
Algebra 2
Algebra 1
9th
Algebra 2
Algebra 1
Algebra 1/2
8th
Algebra 1
Algebra 1/2
Math 87
7th
Algebra 1/2
Math 87
Math 76
6th
Math 76
Math 76
Math 65
5th
Math 65
Math 65
Math 54
4th
Math 54
Math 54
*
Suggested course titles. Accelerated students will finish Advanced Mathematics in the first semester of eleventh grade and begin Calculus in the second semester. Pre-Calculus I consists of the first 70-90 lessons in Advanced Mathematics. Pre-Calculus II comprises the remaining lessons.
**
Suggested course titles. The content in Geometry-Trigonometry-Algebra 3 is identical to that in Pre-Calculus I but is presented at a slower pace. Likewise, Geometry-Trigonometry-Algebra 4 is the same as Pre-Calculus II. Using these course titles allows accelerated students who are not successful in Advanced Mathematics during the tenth grade to try again and still receive credit. Students who complete Advanced Mathematics will have taken the equivalent of two semesters of geometry, one semester of trigonometry, and one semester of advanced algebra. ("Geometry" in the course title ensures credit for geometry.)
***
These students may find the material in Advanced Mathematics difficult. Consider placing them in a less rigorous mathematics course.
Notes:
Math 87 covers a broad spectrum of topics that are required by the mathematical standards of many states. Considerable pre-algebra content is included, and students who complete the text successfully (80% or higher test scores) will be prepared to take Algebra 1 as their next mathematics course.
Algebra 1/2 focuses on developing skills that prepare students for algebra. The pre-algebra content is more extensive than the pre-algebra content of Math 87; however, the scope is narrower. |
Essential Calculus
9780495014423
ISBN:
0495014427
Edition: 1 Pub Date: 2006 Publisher: Thomson Learning
Summary: This book is a response to those instructors who feel that calculus textbooks are too big. In writing the book James Stewart asked himself: What is essential for a three-semester calculus course for scientists and engineers? Stewart's ESSENTIAL CALCULUS offers a concise approach to teaching calculus, focusing on major concepts and supporting those with precise definitions, patient explanations, and carefully graded p...roblems. ESSENTIAL CALCULUS is about two-thirds the size of Stewart's other calculus texts (Calculus, Fifth Edition and Calculus, Early Transcendentals, Fifth Edition) and yet it contains almost all of the same topics. The author achieved this relative brevity mainly by condensing the exposition and by putting some of the features on the web site Despite the reduced size of the book, there is still a modern flavor: Conceptual understanding and technology are not neglected, though they are not as prominent as in Stewart's other books. ESSENTIAL CALCULUS has been written with the same attention to detail, eye for innovation, and meticulous accuracy that have made Stewart's textbooks the best-selling calculus texts in the world.
Stewart, James is the author of Essential Calculus, published 2006 under ISBN 9780495014423 and 0495014427. Three hundred sixty nine Essential Calculus textbooks are available for sale on ValoreBooks.com, two hundred two used from the cheapest price of $0.01, or buy new starting at $530495014427 New, Unread Copy with school stamp on 3 sides. This is Student US Edition. All reference pages included. May be publisher overstock. Might have minor shelf wear on [more]
0495014427 New, Unread Copy with school stamp on 3 sides. This is Student US Edition. All reference pages included. May be publisher overstock. Might have minor shelf wear on covers. Same day shipping with free tracking number. Expedited shipping available. A+ Customer Service![ |
What kinds of curriculum materials do mathematics teachers select and use, and how? This question is complex, in a period of deep evolutions of teaching resources, with the proficiency of online resources in particular. How do teachers learn from these materials, and in which ways do they 'tailor' them for their use and pupil learning? Teachers... more...
Helping teachers prepare elementary students to master the common core math standards With the common core math curriculum being adopted by forty-three states, it is imperative that students learn to master those key math standards. Teaching the Common Core Math Standards with Hands-On Activities, Grades 3-5 is the only book currently available... more...
Develops the statistical approach to inverse problems with an emphasis on modeling and computations. The book discusses the measurement noise modeling and Bayesian estimation, and uses Markov Chain Monte Carlo methods to explore the probability distributions. It is for researchers and advanced students in applied mathematics. more... |
Summer 2013 Math 45 – Pre-Algebra (4 units)
Section # 8120 Begins 06/10/13 Ends 08/01/13
Course Description
This course will be a review of arithmetic involving whole numbers, fractions, decimals, and signed numbers; and a study of basic algebra concepts and techniques, such as variables, distributive property, combining like terms, and solving equations. Students will also solve problems involving ratios, proportions, percents and geometry.
Estimated Time per Week: Students can expect to spend approximately 10 to 16 hours per week reading, writing, and taking quizzes and participating in online class discussions. Remember this is a 17 week class condensed into 8 weeks!
Special Requirements: Log into Etudes the first day of class. Complete the online orientation for math classes. Access the class following the instructions in the orientation for Course Compass.
Assignments & Tests: Assignments will consist of textbook exercises (provided online), journal responses, and discussion board activities. Exams will be conducted online. You will be required to submit your written work for the final exam within one week of the course ending date.
Additional Comments: The entire course will be conducted online through the Course Compass program. Etudes will only be used for orientation to the Course Compass program. Students are required to have Internet access, an active email account, the ability to use word processing, conduct Internet searches, attach files, send emails, and work independently.
New to Etudes: Here is an Online Orientation (Flash presentation opens in a new window) that will show you the basics of how to use Etudes. Here is a flash tutorial (Flash presentation opens in a new window) that demonstrates the log in protocol. Be sure to check System Requirements before getting started with Etudes. You need to do this on each computer you use while taking a class through Etudes.
Etudes Course: You will log into the Etudes classroom with the log-in information provided below.
Login ID
Password
First 2 letters of first name +
First 2 letters of last name +
Last 5 digits of Student COLLEAGUE ID
(Type using all lower case letters)
Example: Jose A. Garcia
Student ID: 1021945
Username = joga21945 |
MTH110
Math
Perspectives
TEXT:
For All Practical Purposes,COMAP,
8th
edition
OBJECTIVE:
The objective of this course is to introduce you
to the ways
in which mathematics impinges upon your daily lives. You will be
introduced
to quantitative concepts and skills which will enable you to interpret
and
reason with quantitative information. In addition you will be
able to see
how mathematics can be used to model many different types of real-world
situations, from presidential elections to garbage collection.
GOALS:
learn the
mathematical principals behind the decision making processes within
groups (social choice).
EXAM REVIEWS:
HOMEWORK ASSIGNMENTS:
Find an Euler Circuit (starting at
A) for the graph shown in Figure 1.10, pg. 10.
Find an Euler Path for the graph in
Figure 1.15a, pg. 14. To answer this question, label the
vertices with the letters A, B, C, etc. and then list the letters in
the order that they lie on the path.
In the graph shown below, color in the odd-valence vertices. Draw in
dashed edges to show an optimal Eulerization of this graph
A complete n-graph is a
graph with n vertices where each vertex is connected to every other
vertex. The figure below shows complete graphs for n = 3, 4 and 5.
Which of these graphs have Euler Circuits? What conditions on n will guarantee that a
complete n-graph will or will not have an Euler Circuit? Explain your answer.
DUE: Sept. 6
Homework 2: Chapter 2, Exercises 2 (list the vertices for each circuit
that exists), 4a, 8, 16b (just answer the question above the 3x4 grid), 18, 26,
38 (show your work using the method of trees), 42a,b, 44, 46. For Exercises 42, 44 and 46 be sure to specify both the Hamilton circuits and the cost of those circuits. Also, do
the following problems:
Read Exercise 24 and then answer the following question: If Jill always insists on wearing her green boots whenever she wears her green scarf, how many outfits might her friends see her in? HINT: Determine how many outfits she has when she wears a green scarf and how many outfits she has when she doesn't wear a green scarf. Then add these two values together to get your answer.
Using the method of trees and starting at vertex A, determine the number of Hamilton Circuits in the graph below. Show your work.
We discussed two measures of optimality for the list scheduling problem:
the length of the critical path, and the total time of all tasks divided by the number
of processors. Apply these measures to the order-requirement digraphs in Exercise
70 in Chapter 2 (pg. 63) for a) 2 processors and b) 3 processors. For each, specify which measure is
a more accurate estimate of the true optimal time.
Use the list processing algorithm and the order-requirement digraph shown below to schedule the tasks
with the priority list T1, T2, ..., T10 using a) two processors, b)
three processors and c) four processors. Which, if any, of these three
schedules can you be sure is optimal?
Determine the critical path lengths for the vertices in the order-requirement digraph used in Problem B and create a priority list using those values. Repeat part a) of Problem B with this new list.
Assume we have to cut 30 pipes of the
following lengths: 22, 1, 24, 26, 27, 14, 8, 27, 7, 17, 16,
11, 16, 23, 11, 21, 27, 29, 14, 3, 8, 14, 12, 16, 26,
6, 15, 30, 11, and 29. If pipes only come in lengths
of 40 determine how many pipes would be needed if we used each
of the following algorithms: next fit, first fit, worst fit, best
fit, next-fit decreasing, first-fit decreasing, worst-fit decreasing
and best-fit decreasing.
For all problems no credit
will be given for a simple numerical answer (min number
of bins, min time to schedule a set of tasks, etc.); you must
show your work (for example, show how the bins are filled,
or how the tasks are scheduled).
Determine if there are any Condorcet
Winners in Exercises 10, 12, 14 and 16.
Read the description of the Coombs
procedure in Exercise 17, and apply it to the preference lists in
Exercises 10, 12, 14 and 16.
Apply the Plurality Runoff method to
the preference lists in Exercises 10, 12, 14 and 16.
Assume we use the preference lists
in Exercise 10 and are using approval voting. Who wins if
everyone votes for their top two favorite candidates? Who wins if
everyone votes for their top three favorite candidates?
Assume we have an election with 5
candidates and 11 voters with the preference lists shown below.
Determine the winner if the following methods are used: plurality,
plurality runoff, Borda count, Hare and sequential pairwise
with the agenda AEDBC.
DUE: Oct. 25
Weighted Voting Systems:
Chapter 11,
Exercises 4, 8, 14, and 18.Also do the following problems (be sure to show your work for all of these problems):
Determine the Shapley-Shubik indices
for the weighted voting systems in Exercise 14.
The current # of representatives for
the New England states are as follows: Connecticut - 5, Maine - 2, Massachusetts - 9, New Hampshire - 2, Rhode Island - 2, and Vermont - 1. Suppose
they all got together to vote on new-englandy
type matters, and decided that the quota to pass any measure would be
18 (we assume all voters from any state always vote together). Determine the BPIs for this weighted voting
system. Which states would you say are the
biggest winners in this system? Which are
the biggest losers?
Suppose we wanted to create a
5-person basketball team using members of the class (which currently has 23 students). Determine the
number of ways we could do this if
we pick a particular person for
each position (point guard, center, etc);
we just pick five people who can play any of the
positions.
Euchre is a card game played with a
deck of 24 cards divided into 4 suits (clubs, diamonds, hearts and
spades) each containing 6 cards (the A, K, Q, J, 10 and 9).
Determine the number of possible 5
cards Euchre hands.
Determine the number of possible
full houses in a 5-card hand.
Determine the number of possible
flushes in a 5-card hand.
Use the techniques discussed in
class to determine the Banzhaf Power Indices for the following weighted
voting system: [9: 3, 2, 2, 2, 2, 2, 2].
Repeat with a quota of 10 and 11. For
which of these quotas does the voter with weight 3 have the most power?
Determine the Banzhaf Power Indices
for the following weighted voting system: [16:5, 5, 5, 4, 4, 4, 4]. Repeat with a quota of 17.
Solve problem 6 in the Skills Check section (pg. 426)
completely (i.e., show the steps and the final results for a
complete division of the items listed).
Solve problem 8 in the Skills Check section (pg. 426)
completely (i.e., show the steps and the final results for a
complete division of the items listed).
Four co-workers - Tessa, Alex, Abby and Sarah - have
decided to use the Knaster Inheritance method to decide who gets two
highly prized items at
the office: an electric foot massager and a mousepad with a picture of
Elvis on it. Tessa values the foot massager at
$24 and the mousepad at $10; Alex values them at $36 and $12;
Abby values them at $24 and $18; and Sarah values them
at $28 and $8. Determine who gets which item, and how much money each
co-worker either makes or spends.
Read the description of the lone-chooser method in Exercise 29,
then
apply it to the cake shown in Exercise 27, where
Player 1 is Bob, Player 2 is Carol and Player 3 is Ted, and Bob divides
the cake initially. Determine if any of the
three are envious after the cake has been divided up.
Five people have pooled their financial resources to start a small company. The amount of money put in by each person is $56,000, $41,000, $25,000, $15,000 and $4,000. In order to decide on company policies they decide to create a weighted voting system by dividing 100 votes between them based on the amount that they originally contributed. Determine how the 100 votes should be divided using the Hamilton, Jefferson and Webster methods.
A country has four states, A, B, C and D. Its house of
representatives has 100 members, apportioned by the Hamilton
method. A new census is taken, and the house is
reapportioned. Here are the data:
State
Old
Census
New
Census
A
6390
6395
B
5890
5890
C
2920
3015
D
1389
1389
Totals
16589
16689
Apportion the house using both the old and new census. Explain
how this is an example of the population paradox described on pages 440-441 in the textbook.
The ThreeBears Corporation has four stock holders: Mama Bear, who owns 971 shares;
Papa Bear, who owns 807 shares; Baby Bear, who owns 510 shares; and Goldilocks, who
owns 315 shares. At the end of year they have 50 units of porridge as dividends to
be distributed to the four stock holders. Use the Webster method to come up with
a fair apportionment of the porridge. Repeat this procedure for 51 units and 52 units.
(Can you figure out why the Three Bears scenario was used for this problem?) |
Math for Life
Synopses & Reviews
Publisher Comments:
How can we solve the national debt crisis? Should you or your child take on a student loan? Is it safe to talk on a cell phone while driving? Are there viable energy alternatives to fossil fuels? What could you do with a billion dollars? Could simple policy changes reduce political polarization? These questions may all seem very different, but they share two things in common. First, they are all questions with important implications for either personal success or our success as a nation. Second, they all concern topics that we can fully understand only with the aid of clear quantitative or mathematical thinking. In other words, they are topics for which we need math for life—a kind of math that looks quite different from most of the math that we learn in school, but that is just as (and often more) important. In Math for Life, award-winning author Jeffrey Bennett simply and clearly explains the key ideas of quantitative reasoning and applies them to all the above questions and many more. He also uses these questions to analyze our current education system, identifying both shortfalls in the teaching of mathematics and solutions for our educational future. No matter what your own level of mathematical ability, and no matter whether you approach the book as an educator, student, or interested adult, you are sure to find something new and thought-provoking in Math for Life.
About college textbooks in four subjects—astronomy, astrobiology, mathematics, and statistics—and has written critically acclaimed books for the general public including Beyond UFOs and On the Cosmic Horizon. He is also the author of childrens books, including those in the Science Adventures with Max the Dog series and The Wizard Who Saved the World |
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This math unit from Illuminations introduces students to the concepts of cryptology and coding. It includes two lessons, which cover the Caesar Cipher and the Vignere Cipher. Students will learn to encode and decode...
This lesson from Illuminations asks students to look at different classes of polynomial functions by exploring the graphs of the functions. Students should already have a grasp of linear functions, quadratic functions,... |
Graphs and Digraphs - 5th edition
Summary: Written for advanced undergraduate and beginning graduate students, the fifth edition of this best-selling book provides a wide range of new examples along with historical discussions of mathematicians, problems, and conjectures. It features new and expanded coverage of such topics as toughness, graph minors, perfect graphs, list colorings, nowhere zero flows, list edge colorings, the road coloring problem, and the rainbow number of a graph. Additional applications, exercises, and ex...show moreamples illustrate the concepts and theorems. A solutions manual is available for qualifying instructors. ...show less
2010 Hardback NEAR FINE Hardback, 59879 |
0764142062
Edition Description:
Revised
ISBN-13:
9780764142062
Publication Year:
2009
Author:
Johanna Holm
Language:
English
Format:
Trade Paper
ISBN:
9780764142062
Detailed item info
SynopsisProduct DescriptionFrom the Inside Flap
(back cover)
A solid review of math basics emphasizes topics that appear most frequently on the GED: number operations and number sense; measurement; geometry; algebra, functions and patterns; data analysis; statistics and probability
Hundreds of exercises with answers
A diagnostic test and four practice tests with answers
Questions reflect math questions on the actual GED in format and degree of difficulty
Paperback: 240 pages
Publisher: Barron's Educational Series; 3.0 edition (August 1, 2009)
Language: English
ISBN-10: 0764142062
ISBN-13: 978-0764142062
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Find a Lonetree, CO Algebra 1Elementary math includes number theory, which is the study of whole numbers and relations between them. Things like factors, multiples, primes, composites, divisibility tests, and exponents provide a critical basis for later mathematical understanding. This may be the most important subject for |
MathGrapher is a stand-out graphing tool designed for students, scientists and engineers. Visitors can read the Introduction to get started, as it contains information about the various functions that the tool can...
This course, presented by MIT and taught by Professor Denis Auroux, presents multivariable calculus. It is intended for use in a freshman calculus course. It includes material relating to vectors and matrices, partial...
This is a basic course, produced by Gilbert Strang of the Massachusetts Institute of Technology, on matrix theory and linear algebra. Emphasis is given to topics that will be useful in other disciplines, including...
Created by Lewis Blake and David Smith for the Connected Curriculum Project, the purposes of this module are to experiment with matrix operations, espcially multiplication, inversion, and determinants, and to explore... |
Calculus : Single and Multivariable - 5th edition
Summary: Calculus teachers recognize Calculus as the leading resource among the ''reform'' projects that employ the rule of four and streamline the curriculum in order to deepen conceptual understanding. The fifth edition uses all strands of the ''Rule of Four'' - graphical, numeric, symbolic/algebraic, and verbal/applied presentations - to make concepts easier to understand. The book focuses on exploring fundamental ideas rather than comprehensive coverage of multiple similar cases that are ...show morenot fundamentally unique. Readers will also gain access to WileyPLUS, an online tool that allows for extensive drills and practice. Calculus teachers will build on their understanding in the field and discover new ways to present concepts to their students89148208 |
Text: R. Larson and R.P. Hostetler
Precalculus, Houghton Mifflin Company 7th Edition
Supplies: A spiral notebook of graph paper for notes.
A cheap scientific calculator with buttons for sin, cos, tan,
Ln, and Log.
Homework: Approximately 2 hours of homework will be assigned
each meeting and additional review assignments will be given on weekends.
Most faculty
will be checking whether the homework is completed while students do
classwork. Odd problems have answers in the
back so students can check their work before proceeding to the next problem.
Working with study partners can make homework easier and more fun.
Math Lab:
All students should plan on spending at least an hour a week at the Math Lab in
Gillet 222. Be sure to have someone look over your homework to see if you are doing it correctly as an answer which matches the one in the back of the book doesn't guarantee you are doing things correctly. |
1 Program Components Introduction ... The Honors Gold Series helps students develop a deep understanding of mathematics through thinking, reasoning, ... This workbook contains daily lesson support with Think About a Plan, Practice, and
Gatti Evaluation Inc. 4 Objective To further assess the effectiveness of the Prentice Hall Algebra online mathematics curriculum. Participants Two urban public school districts; one in western Pennsylvania to provide the below grade level students,
Homework Practice and Problem-Solving Practice Workbook. Pdf Pass Crxs Homework Practice and Problem-Solving ... Resource Masters for California Mathematics, Grade 2. The answers to these worksheets are available at the end of each Chapter Resource Masters booklet.
Reading to Learn MathematicsWorkbook 0-07-861058-3 Answers for Workbooks The answers for Chapter 1 of these workbooks can be found in the back of this Chapter Resource Masters booklet. Spanish Assessment Masters Spanish versions of forms 2A and 2C of
Grade 10 TAKS Mathematics—Objective 1 Understanding functional relationshipsis critical for algebra and geometry. Students need to understand that functions represent pairs of numbers in which the value of one number is dependent |
Mathematical Application in Agriculture - 2nd edition
Summary: Get the specialized math skills you need to be successful in today's farming industry with MATHEMATICAL APPLICATIONS IN AGRICULTURE, 2nd Edition. Invaluable in any area of agriculture-from livestock and dairy production to horticulture and agronomy--this easy to follow book gives you steps by step instructions on how to address problems in the field using math and logic skills. Clearly written and thoughtfully organized, the stand-alone chapters on mathematics involved in crop produc...show moretion, livestock production, and financial management allow you to focus on those topics specific to your area while useful graphics, case studies, examples, and sample problems to help you hone your critical thinking skills and master the93 |
Why is Eulers method to solving (numerically) DE's included on the new AP syllabus? It seems (with calculator in hand) that slope fields are numrically/graphically sufficient. What have I missed about Eulers method that should give it such "weighty" status on the new syllabus? Are there plans to review, evaluate and if needed modify the new syllabus by ETS? |
Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for). |
Experience mathematics--and develop problem-solving skills that will benefit you throughout your life--with THE NATURE OF MATHEMATICS. Karl Smith introduces you to proven problem-solving techniques and shows you how to use these techniques to solve unfamiliar problems that you encounter in your day-to-day world. You'll find coverage of interesting historical topics, and practical applications to real-world settings and situations, such as finance (amortization, installment buying, annuities) and voting. With Smith's guidance, you'll both understand mathematical concepts and master the techniques270.95
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Enhanced WebAssign - Start Smart Guide for Students
Nature of Mathematics, 12 are |
Precalculus - 7th edition
Summary: Get a good grade in your precalculus course with PRECALCULUS, Seventh Edition. Written in a clear, student-friendly style, the book also provides a graphical perspective so you can develop a visual understanding of college algebra and trigonometry. With great examples, exercises, applications, and real-life data--and a range of online study resources--this book provides you with the tools you need to be successful69421 -used book - book appears to be recovered - has some used book stickers - free tracking number with every order. book may have some writing or highlighting, or used book stickers on front ...show moreor back ...show less
$161 |
Publisher's Description
"Patterns" is a 2D graphing calculator to create an infinity of colorful image patterns from mathematical expressions. "Patterns" can help students improve their math skills by developing some visual intuition of 2D mathematical expressions as function of Cartesian (x,y) or polar coordinates (r, theta). Using "Patterns" graphic designers can create fancy icons or illustrations, and vision scientists can recreate their favorite visual stimuli for publication purposes or create new ones when on the road without any need for scientific software like Matlab or Mathematica- only black and white contrast patterns (z in [-1,1] range), - no favorites list, - no export to Photos Album or by email, - no support.
What's new in this version: Support for in-application email capability under iPhone 3.*, Fix issue with disclosure button in favorite list when running OS 3.1.1, Fix issue with saved preferences, Fix a rare computation bug |
Intended Learning Outcomes
The main intent of mathematics instruction at the secondary level is for students to develop mathematical proficiency that will enable them to efficiently use mathematics to make sense of and improve the world around them.
The Intended Learning Outcomes (ILOs) describe the skills and attitudes students should acquire as a result of successful mathematics instruction. They are an essential part of the Mathematics Core Curriculum and provide teachers with a standard for student learning in mathematics.
The ILOs for mathematics at the secondary level are:
Develop positive attitudes toward mathematics, including the confidence, creativity, enjoyment, and perseverance that come from achievement.
Course Description
The main goal of Geometry is for students to develop a Euclidean geometric structure and apply the resulting theorems and formulas to address meaningful problems. Students will use experimentation and inductive reasoning to construct geometric concepts, discover geometric relationships, and formulate conjectures. Students will employ deductive logic to prove theorems and justify conclusions. Students will extend their pre-existing experiences with algebra and geometry to trigonometry, coordinate geometry, and probability. Students will use dynamic geometry software, compass and straightedge, and other tools to investigate and explore mathematical ideas and relationships and develop multiple strategies for analyzing complex situations. Students will apply mathematical skills and make meaningful connections to life's experiences.
Prove lines parallel or perpendicular using slope or angle relationships.
Objective 3
Analyze characteristics and properties of triangles.
Prove congruency and similarity of triangles using postulates and theorems.
Prove the Pythagorean Theorem in multiple ways, find missing sides of right triangles using the Pythagorean Theorem, and determine whether a triangle is a right triangle using the converse of the Pythagorean Theorem.
Prove and apply theorems involving isosceles triangles.
Apply triangle inequality theorems.
Identify medians, altitudes, and angle bisectors of a triangle, and the perpendicular bisectors of the sides of a triangle, and justify the concurrency theorems.
Objective 4
Analyze characteristics and properties of polygons and circles.
Use examples and counterexamples to classify subsets of quadrilaterals. |
, Ninth Edition, by Howard Anton. The first ten chapters of this book are identical to the first ten chapters of that text; the eleventh chapter consists of 21 applications of linear algebradrawn from business, economics, engineering, physics, computer science, approximation theory, ecology, sociology,demography, and genetics. The applications are, with one exception, independent of one another and each comes with a list of mathematical prerequisites. Thus, each instructor has the flexibility to choose those applications that are suitable for his or herstudents and to incorporate each application anywhere in the course after the mathematical prerequisites have been satisfied.This edition of
Elementary Linear Algebra
, like those that have preceded it, gives an elementary treatment of linear algebra thatis suitable for students in their freshman or sophomore year. The aim is to present the fundamentals of linear algebra in theclearest possibleway; pedagogy is the main consideration. Calculus is not a prerequisite, but there are clearly labeled exercisesand examples for students who have studied calculus. Those exercises can be omitted without loss of continuity. Technology isalso not required, but for those who would like to use
MATLAB
, Maple,
Mathematica
, or calculators with linear algebracapabilities, exercises have been included at the ends of the chapters that allow for further exploration of that chapter'scontents.
SUMMARY OF CHANGESIN THIS EDITION
This edition contains organizational changes and additional material suggested by users of the text. Most of the text isunchanged. The entire text has been reviewed for accuracy, typographical errors, and areas where the exposition could beimproved or additional examples are needed. The following changes have been made:
Section 6.5 has been split into two sections: Section 6.5 Change of Basis and Section 6.6 Orthogonal Matrices. Thisallows for sharper focus on each topic.
A new Section 4.4 Spaces of Polynomials has been added to further smooth the transition to general lineartransformations, and a new Section 8.6 Isomorphisms has been added to provide explicit coverage of this topic.
Chapter 2 has been reorganized by switching Section 2.1 with Section 2.4. The cofactor expansion approach todeterminants is now covered first and the combinatorial approach is now at the end of the chapter.
Additional exercises, including Discussion and Discovery, Supplementary, and Technology exercises, have been addedthroughout the text.
In response to instructors' requests, the number of exercises that have answers in the back of the book has been reducedconsiderably.
The page design has been modified to enhance the readability of the text.
A new section on the earliest applications of linear algebra has been added to Chapter 11. This section shows how linearequations were used to solve practical problems in ancient Egypt, Babylonia, Greece, China, and India.
Hallmark Features
Relationships Between Concepts
One of the important goals of a course in linear algebra is to establish the intricatethread of relationships between systems of linear equations, matrices, determinants, vectors, linear transformations, andeigenvalues. That thread of relationships is developed through the following crescendo of theorems that link each newidea with ideas that preceded it: 1.5.3, 1.6.4, 2.3.6, 4.3.4, 5.6.9, 6.2.7, 6.4.5, 7.1.5. These theorems bring a coherence tothe linear algebra landscape and also serve as a constant source of review.
Smooth Transition to Abstraction
The transition from to general vector spaces is often difficult for students. Tosmooth out that transition, the underlying geometry of is emphasized and key ideas are developed in beforeproceeding to general vector spaces.
Early Exposure to Linear Transformations and Eigenvalues
To ensure that the material on linear transformationsand eigenvalues does not get lost at the end of the course, some of the basic concepts relating to those topics aredeveloped early in the text and then reviewed and expanded on when the topic is treated in more depth later in the text.For example, characteristic equations are discussed briefly in the chapter on determinants, and linear transformations fromto are discussed immediately after is introduced, then reviewed later in the context of general lineartransformations.
About the Exercises
Each section exercise set begins with routine drill problems, progresses to problems with more substance, and concludes withtheoretical problems. In most sections, the main part of the exercise set is followed by the
Discussion and Discovery
problemsdescribed above. Most chapters end with a set of supplementary exercises that tend to be more challenging and force thestudent to draw on ideas from the entire chapter rather than a specific section. The technology exercises follow thesupplementary exercises and are classified according to the section in which we suggest that they be assigned. Data for theseexercises in
MATLAB
, Maple, and
Mathematica
formats can be downloaded from
About Chapter 11
This chapter consists of 21 applications of linear algebra. With one clearly marked exception, each application is in its ownindependent section, so that sections can be deleted or permuted freely to fit individual needs and interests. Each topic beginswith a list of linear algebra prerequisites so that a reader can tell in advance if he or she has sufficient background to read thesection.Because the topics vary considerably in difficulty, we have included a subjective rating of each topic—easy, moderate, moredifficult. (See "A Guide for the Instructor" following this preface.) Our evaluation is based more on the intrinsic difficulty of the material rather than the number of prerequisites; thus, a topic requiring fewer mathematical prerequisites may be ratedharder than one requiring more prerequisites.Because our primary objective is to present applications of linear algebra, proofs are often omitted. We assume that the readerhas met the linear algebra prerequisites and whenever results from other fields are needed, they are stated precisely (withmotivation where possible), but usually without proof.Since there is more material in this book than can be covered in a one-semester or one-quarter course, the instructor will haveto make a selection of topics. Help in making this selection is provided in the Guide for the Instructor below.
Supplementary Materials for Students
Student Solutions Manual, Ninth Edition
—This supplement provides detailed solutions to most theoretical exercises and to atleast one nonroutine exercise of every type. (ISBN 0-471-43329-2) |
97805216504, aimed at advanced undergraduate or beginning graduate students in mathematics, introduces both the theory of Riemann surfaces, and of analytic functions between Riemann surfaces. The first half of the book describes the basic theory, the second half develops the theory of harmonic and subharmonic functions on a Riemann surface, and culminates with a detailed proof of the famous Uniformisation Theorem and some of its applications to Riemann surface theory. The book is a major revision of the author's earlier 'Primer', with new chapters and more exercises and |
SparkCharts-created by Harvard students for students everywhere-serve as study companions and reference tools that cover a wide range of high school, college, and gradu...show moreate school subjects, including math, business, history, computer programming, medicine, law, foreign language, humanities, and science. Titles like Spanish Vocabulary, Microsoft Excel, Study Tactics, the Bible, Algebra I, Chemistry, and Literary Terms give you what it takes to find success in school and beyond. Outlines and summaries cover key points, while diagrams and tables make difficult concepts easier to digest. ...show lessEdition/Copyright: 05 Cover: Paperback Publisher: Sparkchart |
Though I consider myself good at Math, I had a hard time understanding this application. It does the basic math that everyone uses calculators for, plus tons of extra functions that can be done if you take the time to read the directions and look at the examples on the homepage. |
Summary:This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses the least common multiple. By the end of the module students should be able to find the least common multiple of two or more whole numbers.
Summary:This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses nonterminating divisions. By the end of the module students should understand the meaning of a nonterminating division and be able to recognize a nonterminating number by its notation.
Summary:This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module reviews the key concepts from the chapter "Introduction to Fractions and Multiplication and Division of Fractions."
Summary:This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses multiplication of fractions. By the end of the module students should be able to understand the concept of multiplication of fractions, multiply one fraction by another, multiply mixed numbers and find powers and roots of various fractions.
Summary: ... in multiplication.[Expand Summary] by a power of 10 and understand how to use the word "of" in multiplication.[Collapse Summary]
Summary:This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to combine like terms using addition and subtraction. By the end of the module students should be able to combine like terms in an algebraic expression.
Summary:This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses applications involving fractions. By the end of the module students should be able to solve missing product statements and solve missing factor statements.
Summary: ... by another.[Expand Summary] zero is undefined, and use a calculator to divide one whole number by another.[Collapse Summary] |
... read more
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to Euclidean and non-Euclidean geometry, and then it addresses the specific differences that constitute elliptic and hyperbolic geometry. Major topics include hyperbolic geometry, single elliptic geometry, and analytic non-Euclidean geometry. 1901 edition.
Unabridged republication of Non-Euclidean Geometry, published by Ginn and Company, 1901 |
Best Mathematics Instructional Solution
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Literacy Advantage Courses: Mathematics - APEX Learning, Inc.
Literacy Advantage courses support academic success in standards-based high school courses for students who are reading below proficient. Literacy Advantage courses assist students in mastering required math to earn credits toward graduation, while simultaneously developing reading skills. Courses are based on the most current research in adolescent literacy and best practices for instruction and intervention.
Literacy Advantage is part of Apex Learning Foundations courses, which–8. Foundations courses have been designed to be age-appropriate with respect to content, illustrations, and examples for students ages 13 and older. Each semester course offers 60–80 hours of interactive direct instruction, guided practice, and integrated formative, summative, and diagnostic assessment.
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The power of the Trigsted eCourse series comes from the interactivity and interface of the eText, along with the 4-Step Learning Path and integration of MyMathLab. Recognizing that today's students start with the homework instead of reading the text, Kirk Trigsted created an online learning environment that is a seamless mix of exposition, videos, interactive animations, tutorials, and assessment. This approach leverages the power of MyMathLab and leads students to interact with course materials in a way that is proving to be more effective.
Trigsted continues to innovate with a design that improves navigation and usability, videos, and animation coverage with all multimedia elements, exercises, feedback, and content written by the author himself so the students have a consistent voice throughout. |
books.google.com - This book bridges the gap between the many elementary introductions to set theory that are available today and the more advanced, specialized monographs. The authors have taken great care to motivate concepts as they are introduced. The large number of exercises included make this book especially suitable... Modern Set Theory: The basics |
Precalculus Tutorial
This tutorial has been ported to a new domain. Please change your links to this resource from "jwbales.home.mindspring.com" to "jwbales.us" Direct inquiries to Prof. Bales
This course assumes the student has a good grounding in basic school algebra. Students who diligently follow the instructions and examples and work the exercises should have a good grounding for a first course in calculus.
Prof. Bales is beginning the process of updating this site, written in 2000, to modern technology. The site was moved several years ago to and the site here at will soon be shut down. Look for a much improved appearance of the jwbales.us site sometime in the Fall of 2012. |
Pre-Calculus Help
In this section you'll find study materials for pre-calculus help. Use the links below to find the area of pre-calculus you're looking for help with. Each study guide comes complete with an explanation, example problems, and practice problems with solutions to help you learn pre-calculus.
Study Guides
Introduction to Conversion from Degrees to Radians
Trigonometry has been used for over two thousand years to solve many real-world problems, among them surveying, navigating, and problems in engineering. Another important use is analytic—the trigonometric ... |
Closing at noon, Friday, March 14
and will remain closed during spring break.
Monday & Tuesday
9:00 am - 7:00 pm
Wednesday & Thursday
9:00 am - 5:00 pm
Friday
9:00 am - 3:00 pm
Library Commons Hours
Sunday, Monday, Tuesday, Wednesday
4:00 pm - 9:00 pm
Goal
Help students improve their grades in math classes. This also includes helping students with basic math in courses outside the Math Department.
Intended Audience
Students who need assistance in the following classes:
Mathematical Reasoning (Math 113)
Statistical Reasoning (Math 115)
College Algebra (Math 119)
Finite Math (Math 123)
Basic Calculus (Math 125)
Precalculus (Math 130)
Calculus I and II (Math 141, 142)
We will also help with second year courses such as Calculus III and Differential Equations to the best of the staff's ability.
Undergraduate classes outside the Math Department where the assistance needed is mathematical in nature.
Note: The Math Department has asked the we not help any students in math courses numbered 300 or above.
Staff
The Center is staffed primarily by undergraduates, sophomores through seniors, all of who have passed at least Calculus I and II here at UT. Almost all are Math or Engineering majors.
Format
The Tutorial Center is not a study lounge. Students should come in prepared to ask questions of the staff and should already know what question(s) they will ask. After the student checks in a Staff person will work with the student as soon as possible. If the Center is near capacity, a student may be asked to leave after having some of their questions answered so as to allow other students a chance to come in for help. Last year the Center was at its fullest between 11 and 3.
Disclaimer
Our staff are not geniuses and there will be questions that they cannot answer. This happens most often with material that an instructor assigns which is not part of the standard curriculum. In situations such as this the student is referred to their instructor and/or teaching assistant.
Contact
For more information call 974-4266 or 974-2461. You may also reach us by e-mail at mtc@math.utk.edu. |
Algebra1
Algebra1 is the basic algebra, or algebra learned in high school, till grade eight. Algebra is the study of unknown quantities which keep on changing. Algebra1 is the introduction to these changing quantities or variables around us.
There are hundreds of changing quantities (variables) around us. For example; temperature during the day keep on changing, our weight changes on a daily basis and the share prices on an exchange market keep on changing.
In mathematics, the changing quantities are called the variables or unknowns. Variables are represented using the letters (mainly lower case letters) from the alphabet. Algebra is all about the study of these variables.
This site can help you to understand algebra1, so that you can better focus on different sections of this math topic.
Main topics for algebra1 and covered in this site are given below:
1. Input and out put tables.
3. Algebraic expressions to polynomials.
5. Basic algebraic equations.
2. Introduction to algebraic expressions.
4. Types of polynominals.
6. Algebraic inequalities.
We'll discuss all the above topics one by one, on this page. Most often kids keep learning arithmetic till grade five in most schools around the world.
Also arithmetic is the base to learn algebra1 and hence, overall mathematics. In other words, if we say that basic arithmetic skills are the key to learn algebra1, there will be nothing wrong. We can list all the basic arithmetic skills as given below:
1. Number sense (counting numbers, place value and rounding numbers)
2. Four operations of math (addition, subtractions, multiplication and division)
3. Good and sharpen memory to do times tables in head for mental math.
4. Knowledge of greatest common factor and least common multiple.
5. Good understanding and knowledge of fractions and their applications.
6. Last but not least the ability to understand number and shape patterns.
All the above basic topics are the part of arithmetic and mostly, kids learn them till grade five in elementary school. All these topics are mandatory to learn higher math in grade six and beyond, which consists of 80% algebra.
Hence, encourage your kids or students to pay close attention to these key topics in lower grades. Specially, in grade three to grade five, students should switch their daily activities more towards math.
Parents should reward the kids if they make math learning as their daily routine. If kids in grade three to five have difficulty to sit everyday and learn and practice math skills, parents should talk to them and resolve any issues or challenges their kids face to learn the basic math in these key grades.
Once kids get used to sit for math every day, they start to love this subject. Otherwise math becomes very dry or boring if kids just want to do it occasionally. Basic arithmetic is not very hard and even most parents can help their kids to get better at it.
Again, make sure kids are good at basic arithmetic before starting algebra1 topics with them. For example; if kids are not good at patterns, it will be very hard for them to understand input and output tables and to make a relation between the independent and dependent variables. In other words, arithmetic is the key to higher math.
Algebra1 - The beginning of relationships between unknowns
Below are all the sections of algebra1 and a brief introduction about each of them:
1. Input and output tables:
Input and output tables are learned in grade 6. These tables can be very helpful to make a relation between two quantities. For example; the price of sugar in dollars and its amount which can be purchased with the money, can be the input and output respectively.
To learn this skill effectively, kids need a sound knowledge of basic operations of math and good pattern skills.
2. Algebraic expressions - An introduction:
In the input and output tables, there are two quantities discussed. One of the quantity vary according to the other. For example; the quantity of sugar you can buy is directly hit by price of per pound of sugar.
In other words, if sugar is expensive we have buy less with fixed amount of money. So there is a number relation between amount of sugar and it cost per pound, which is called an algebraic expression.
3. Polynominals:
Once kids get able to draw a relation between two quantities, then they begin to learn about the basic terminology about the algebra1. They start to know variables and how we denote them, coefficient and constants are some other basic terms in algebra 1 vocabulary.
So, there are algebraic relations between two quantities. For example; if we can buy 5 pounds of sugar with $6, then the cost of per pound of sugar is an algebraic relation which helps us to find any other cost of sugar for a given amount of it.
These relationship between two (or more) quantities are called algebraic relations and if these relations pass certain criterion; they are called polynominals
4. Types of polynominals:
Once kids get used to term polynominals, then they need to know their types. Polynomials can be classified many ways, but classification on the basis of number of terms is most important.
5. Basic algebraic equations:
When we equate an algebraic relation, equal to zero then it becomes an equations. The basic equations are an integral and most important part to learn in algebra1.
6. Algebraic inequalities:
Finally, algebra1 comes to an end with the knowledge of inequalities. Inequalities are the next stage to equations and they use inequality symbols along with equal sign.
Most kids finish all the above skills till grade eight or nine. Then students start to learn algebra2 in grade ten and beyond.
Finally, it can be said that arithmetic is the most basic stage to learn math and it gives a platform to learn the next stage called algebra1. Which in turn, builds the foundation to learn algebra2 and higher mathematics. |
Unavailable
Short Description for Schaum's Outline of Group Theory Schaum's Outlines present all the essential course information in an easy-to-follow, topic-by-topic format. You also get hundreds of examples, solved problems, and practice exercises to test your skills. Full description
Full description for Schaum's Outline of Group Theory
The theory of abstract groups comes into play in an astounding number of seemingly unconnected areas like crystallography and quantum mechanics, geometry and topology, analysis and algebra, physics, chemistry, and even biology. Readers need only know high school mathematics, much of which is reviewed here, to grasp this important subject. Hundreds of problems with detailed solutions illustrate the text, making important points easy to understand and remember. |
Algebra 1 - 03 edition
Summary: Motivation Rationale Statements at the beginning of every lesson make abstract concepts real to students with interesting, easily-identifiable examples. Real-world applications add depth and relevancy to daily instruction. And, segmented lessons, with examples broken down into small steps, increase student success. Accessibility This text is specifically designed to make algebra accessible to a variety of learners by providing hands-o...show moren activities and multiple representations. Small chunks of text and many examples make it easy for students to follow along. Practice Ongoing, skill-building practice can be found throughout Algebra 1. The Extra Practice section in the appendix, the Practice Workbook, and the Basic Skills Practice Masters afford students additional practice if needed. ...show less
CD Missing. A306605110030660513 |
Mathematics Curriculum Studies 3
10 Units
This course introduces students to the key concepts underlying a deep understanding of mathematical proof and topology. This course will consider the historical development of mathematical proof and topology and will examine current related pedagogical models within the field of secondary mathematics including catering for differentiated learning needs in the contemporary classroom.
On satisfactory completion of this course students should be able to:
- understand the key concepts related to various forms of mathematical proof and the field of topology
- appreciate the mathematical knowledge and beliefs that learners bring to a learning task
- apply a range of strategies for teaching secondary mathematics
- recognise the common misconceptions that students may have in regard to the mathematical content covered.
- recognise the benefits and issues associated with differentiated learning
Content
The historical development of mathematical proof and its relationship to other forms of proof commonly accepted in contemporary society |
Thinking Mathematically - With 2 CDs - 4th edition
Summary: This general survey of mathematical topics helps a diverse audience, with different backgrounds and career plans, to understand mathematics. Blitzer provides the applications and technology readers need to gain an appreciation of mathematics in everyday life. Demonstrates how mathematics can be applied to readers' lives in interesting, enjoyable, and meaningful ways. Features abundant, step-by-step, annotated Examplesthat provide a problem-solving approach to reach the ...show moresolution; annotations are conversational in tone, explaining key steps and ideas as the problem is solved. Begins each section with a compelling vignette highlighting an everyday scenario, posing a question about it, and exploring how the chapter section subject can be applied to answer the question. A highly readable reference for anyone who needs to brush up their mathematics skills. ...show less
Fourth Edition. CD-Rom Included. If applicable,CD-ROM,online access or codes are not guaranteed to work. Some shelf wear with scratches to board. There may be writing marks in the book but we have no...show moret seen any. Pages are clean and crisp and binding is tight. Solid Book. ...show less
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Cheryls-Books Vinemont, AL
2006-12-31 Hardcover 4th Good Hardback book in good condition, but missing dust jacket if issued one. Still has CD. Few dog-eared edges at the bottom of |
4th and 5th grade students.
Not a C Minus is a comprehensive study aid for senior high school Mathematics. It covers topics such as calculus, probability, finance and trigonometry, and uses a conversational, informal teaching style. Every topic is explained in detail, with sample questions and worked solutions.
Practice and hone important addition skills with this second bookThis book was written by an experienced maths tutor to help parents and carers to be able to tutor their child in general maths. This book contains 14 lesson plans for hourly tuition sessions (which would cost £20-£25 if you paid a tutor)on maths units ranging from basic addition to more advanced ratio questions. Also suitable for adults taking basic or functional skills exams in adult numeracy.
Practice and hone important subtraction skills with this second book subtraction skills addition skillsPractice and hone important multiplication skills. Select one of twenty math problems with complete solutions that instruct the student in the multiplication process. The book also includes four bonus word problems with complete explanations and answers. Easily navigate the links from the problem list to view the solution. Most appropriate for 4th and 5th grade studentsOne of the best ways to succeed in Geometry is to practice taking real test questions. This volume contains 133 problems on Three-Dimensional Figures divided into four chapters: Definitions and Shapes; Rectangular Solids; Cylinders, Cones, Spheres; and Prisms and Pyramids. Try the problems. With a little Practice, Practice, Practice, you'll be Perfect, Perfect, Perfect. Good Luck!! |
Product Description
This innovative teaching tool targets four areas of study: linear graphs, quadratic equations, conic sections, and trig functions. Topics are supported by card decks and a teacher's manual. Cards within each deck display graphs, equations, point pairs, or other topic-specific information that support notation of the equation. Whether studying algebra basics or advanced equation functions, students will benefit by recognizing relationships between cards, improving problem-solving skills, and developing proficiency in recognizing algebraic equations. Teacher manuals focus on ready-to-use cooperative learning activities. Blackline transparency masters and answer keys save prep time for teachers. All decks have 60 cards: 12 graphs with five matching components for each graph. For up to 12 students working together. Includes one each of Linear Graphs, Quadratic Equations, Conic Sections, and Trig Functions card sets.
Prices listed are U.S. Domestic prices only and apply to orders shipped within the United States. Orders from outside the
United States may be charged additional distributor, customs, and shipping charges. |
MA 125 Intermediate Algebra Green, Catherine M.D Algebra is NOT a spectator sport. It is also not a subject that should be feared. Success is dependent on involvement. Students need to do the readings and the homework and come to class prepared. I welcome all questions and will be available if you need extra help. Students will be evaluated on class participation, homework, quizzes, midterm and the final exam
Homework Folder - 10% (Homework folder will be turned in for evaluation during week 4 and week 8, 5% each time)
6 Quizzes - 20% (discard two lowest scores, no makeups if absent)
Midterm - 30%
Final - 30%
Class Participation - 10%
100% - 90% = A
89% - 80% = B
79% - 70% = C
69% - 60% = D
59% - 0 = F
Late Submission of Course Materials: Homework folders may be turned in the following class after an excused absence. There are no makeups for missed quizzes however it can count as one of the discarded scores. Midterm can be made up after coordination with the instructor and the Park-Scott AFB office. Final can be made up if student has no unexcused absences and after coordination with the instructor and the Park-Scott AFB office.
Classroom Rules of Conduct: Students will conduct themselves in a civilized manner. Discussion is encouraged however students will show respect for each other and the instructor. Cell phones will be set on vibrate, and any cell phone use will be conducted outside the classroom.
Course Topic/Dates/Assignments:
Homework will be assigned at the end of each class and kept, by the student in a homework folder separate from class notes.
Week 1
8/16 - Class 1
Chapter 1 - 1.1-1.2
8/18 - Class 2
Chapter 1 - 1.3-1.4
Week 2
8/23 - Class 3
Quiz 1
Chapter 2 - 2.1-2.3
8/25 - Class 4
Chapter 2 - 2.4-2.7
Week 3
8/30 - Class 5
Quiz 2
Chapter 3 - 3.1-3.3
9/1 - Class 6
Chapter 3 - 3.4-3.7
Week 4
9/6 - Class 7
Quiz 3
Midterm Review, Chapters 1-3
9/8 - Class 8
Midterm - Chapters 1-3 Homework Folders Due
Week 5
9/13 - Class 9
Chapter 4 - 4.1-4.2
9/15 - Class 10
Chapter 4 - 4.3-4.4
Week 6
9/20 - Class 11
Quiz 4
Chapter 5 - 5.1-5.2
9/22 - Class 12
Chapter 5 - 5.3-5.5
Week 7
9/27 - Class 13
Quiz 5
Chapter 6 - 6.1-6.2
9/29 - Class 14
Chapter 6 - 6.3-6.5
Week 8
10/4 - Class 15
Quiz 6
Final Review
10/6 - Class 16
Final Exam Homework Folders |
's one of those classes where you sit, watch the prof, and wonder how on earth he got his position. WORST PROF EVER. He cannot teach and he is VERY difficult to understand! He's just as disorganized as the textbook. Just another self-teaching course, like any other math course at York. Please try to avoid!
He is the worst prof everrrr!!! avoid him at any cost. he is not fair in his marking at all. and you can never make sense of what he says basically you have to study on your own. he is just the worst professor.
On the upside, he's nice and tries to answer questions that you ask. However, his lecture notes jump from concept to concept, and aren't indicative of the material that you need to know. I relied EXCLUSIVELY on the textbook for this course; I did not learn any new concepts from his class alone. But if you work hard, you can do well in any course.
He's a nice guy, but very difficult to understand, notes are hard to read. Doesn't use technology at all, writes all his notes in incomprehensible script on chalkboard. Tests are reasonable, but you're better off just reading the textbook and teaching yourself.
He's really nice, and his tests are reasonable. His accent is awful but it can be understood if you pay attention. Though we find it easier to read the textbook directly and attend classes as practice.
Please listen to everyone else who posted. This prof is the WORST prof ever. He's like Alip+Purzitsky but WORSE. He acts nice on the outside, but he's just pretending to get his promotion. Don't waste your time and money with him. He gives ORAL exams! Comes late to class, ends classes late and he's in the biology department but teaches math >=(
worst professor ever! very monotone, very unexpected exam worth 55%, very hard and full of industrial statistics. Other group in the same exam class had an exam that had inclusion-exclusion question, while we had calculus questions! only solves his handout questions, no instuctions or a learning process provided! GARBAGE!
This is bad professor. Dont risk your money or mark with him. Messy, non-sequential, presentation of course material during lectures. Impossible final exam. Does not answer questions, just repeats your question and continues on. Does not answer emails at all. Not a pleasant experience.
Train of thought is very difficult to follow, he jumps all over the place, notes on the board are very scattered. Does not answer questions asked, just explains something else. Sometimes stops partway through an explanation. Seems like a nice guy, but just does not know how to teach at all. THIS WAS MY WORST EXPERIENCE AT YORK UNIVERSITY. |
Integrated Arithmetic and Basic Algebra - 4th edition
Summary: KEY MESSAGE:Integrated Arithmetic and Basic Algebra, Fourth Edition, integrates arithmetic and algebra to allow students to see the big picture of math. Rather than separating these two subjects, this text helps students recognize algebra as a natural extension of arithmetic. As a result, students see how concepts are interrelated and are better prepared for future courses. KEY TOPICS: Adding and Subtracting Integers and Polynomials; Laws of Exponents, Products and Quotients of ...show moreIntegers and Polynomials; Linear Equations and Inequalities; Graphing Linear Equations and Inequalities; Factors, Divisors, and Factoring; Multiplication and Division of Rational Numbers and Expressions; Addition and Subtraction of Rational Numbers and Expressions; Ratios, Percents, and Applications; Systems of Linear Equations; Roots and Radicals; Solving Quadratic Equations MARKET: For all readers interested in algebra and basic algebra. ...show less
Ships same or next business day with delivery confirmation. Good condition. May or may not contain highlighting. Expedited shipping availablePAPERBACK New 0321442555 New Copy with minor shelf wear. This is Student US Edition. May be publisher overstock. Same day shipping with free tracking number. Expedited shipping available. A+ Custom...show moreer Service! ...show less
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0321442555 |
Mathomatic is a portable, general-purpose computer algebra system (CAS) that can solve, differentiate, simplify, combine, and compare algebraic equations, perform standard, complex number, modular, and polynomial arithmetic, etc. It does some calculus and is very easy to compile/install, learn, and use. The symbolic math application with a simple command-line interface is designed to be a colorful algebra calculator that is reliable, responsive, and convenient to use. The symbolic math library is lightweight and easy to include in other software, due to being written entirely in C with no additional dependencies. |
Mathematics
The Mathematics Program in the ParkwaySchool District is designed to ensure students demonstrate application of the Show-Me Standards through the integration of content and processes related to equity, use of technology, research strategies, and workplace readiness skills.
Equity
The ParkwaySchool District is committed to the belief that all students are able to learn mathematics regardless of gender, race, culture, or physical challenges. Therefore teachers integrate appropriate content and process related to equity, including gender equity, racial/ethnic equity, and disability awareness throughout the K-12 Mathematics Program. The integration of equity ensures that graduates of the ParkwaySchool District successfully demonstrate understanding and application of the Show-Me Standards and the ParkwaySchool District mission to prepare students to be effective problem solvers in an increasingly changing world. All Parkway mathematics courses integrate equity issues, including gender, multicultural, and disability awareness content into specific, appropriate lessons. In addition, all Parkway mathematics textbooks are reviewed for inclusion, equity, and diversity. Mathematics courses at all levels address valuable contributions made by mathematicians from men and women from a variety of cultures and countries around the world.
Technology
The design and inner workings of electronic tools are built upon a foundation of mathematical logic, reasoning, and processing. In order to ensure students are capable of understanding how to create and utilize these tools to help in problem solving, the ParkwaySchool District's K-12 Mathematics Program integrates appropriate content and processes related to the application of technology in the teaching and learning process. In addition to the Missouri Show-Me Standards related to technology integration, the ParkwaySchool District's Mathematics Program also utilizes the National Educational Technology Standards ( to ensure successful technology integration. As an example, our students in Algebra courses utilize graphing calculators as an effective tool for exploring, characterizing, and using functions in problem solving. In addition, the ParkwaySchool District's textbook selection process includes evaluation of the textbook for technology integration.
Research
An essential component of mathematical study is gathering data and using both deductive reasoning and critical thinking to draw conclusions. Thus teachers in the ParkwaySchool District integrate content and processes related to research throughout the K-12 Mathematics Program. Many of the mathematics courses provide meaningful opportunities for students to gather and analyze data. Students use a variety of available resources including the library, Internet, and community resources. The elementary curriculum emphasizes simple research techniques, such as periodically recording the total number of pockets that students have in their classroom. After charting the data and recognizing that more total pockets appear in winter, students realize this trend is a result of students wearing coats to school.
Workplace Readiness Skills
The ability to define and solve problems is perhaps the most important skill needed in the twenty-first century workplace. Therefore all Parkway mathematics courses integrate meaningful workplace readiness skills such as problem solving, organizing information, meeting deadlines, and cooperatively working in small groups. Courses at all levels provide opportunities for students to actively discuss mathematics problems with their peers as they work toward a solution. |
for Physicists
This best-selling title provides in one handy volume the essential mathematical tools and techniques used to solve problems in physics. It is a vital ...Show synopsisThis best-selling title provides in one handy volume the essential mathematical tools and techniques used to solve problems in physics. It is a vital addition to the bookshelf of any serious student of physics or research professional in the field. The authors have put considerable effort into revamping this new edition. * Updates the leading graduate-level text in mathematical physics * Provides comprehensive coverage of the mathematics necessary for advanced study in physics and engineering * Focuses on problem-solving skills and offers a vast array of exercises * Clearly illustrates and proves mathematical relations New in the Sixth Edition: * Updated content throughout, based on users' feedback * More advanced sections, including differential forms and the elegant forms of Maxwell's equations * A new chapter on probability and statistics * More elementary sections have been deleted Mathematical Methods for Physicists
The book covers a very large range of mathematical issues. Some topics are well developed, like the ones covering complex analysis, while others, like the group theory, are much concise (in my opinion). In general, the book offers a good introduction to several topics, not only for the physicists |
This complete guide to numerical methods in chemical engineering is the first to take full advantage of MATLAB's powerful calculation environment. Every chapter contains several examples using general MATLAB functions that implement the method and can also be applied to many other problems in the same category.
The authors begin by introducing the solution of nonlinear equations using several standard approaches, including methods of successive substitution and linear interpolation; the Wegstein method, the Newton-Raphson method; the Eigenvalue method; and synthetic division algorithms. With these fundamentals in hand, they move on to simultaneous linear algebraic equations, covering matrix and vector operations; Cramer's rule; Gauss methods; the Jacobi method; and the characteristic-value problem. Additional coverage includes:
The numerical methods covered here represent virtually all of those commonly used by practicing chemical engineers. The focus on MATLAB enables readers to accomplish more, with less complexity, than was possible with traditional FORTRAN. For those unfamiliar with MATLAB, a brief introduction is provided as an Appendix.
The accompanying website contains MATLAB 5.0 (and higher) source code for more than 60 examples, methods, and function scripts covered in the book. These programs are compatible with all three operating systems: Windows(r), MacOS(r), and UNIX(r).
Description:
Numerical Methods for Linear Control Systems Design and Analysis is
an interdisciplinary textbook aimed at systematic descriptions and implementations of numerically viable algorithms based on well established, efficient and stable modern numerical linear techniques for mathematical problems arising in ...
Description:
Written by an internationally respected field expert from Rutgers, this
second edition enables a basic understanding of mathematical methods involved in numerical computer analysis and highlights issues of particular importance to the field of engineering. |
mat... read more
Numerical Methods by Germund Dahlquist, Åke Björck Practical text strikes balance between students' requirements for theoretical treatment and the needs of practitioners, with best methods for both large- and small-scale computing. Many worked examples and problems. 1974 edition.
Mathematical Tools for Physics by James Nearing Encouraging students' development of intuition, this original work begins with a review of basic mathematics and advances to infinite series, complex algebra, differential equations, Fourier series, and more. 2010 editionA First Course in Numerical Analysis: Second Edition by Anthony Ralston, Philip Rabinowitz Outstanding text, oriented toward computer solutions, stresses errors in methods and computational efficiency. Problems — some strictly mathematical, others requiring a computer — appear at the end of each chapter.
Product Description:
Bonus Editorial Feature:
Richard W. Hamming: The Computer Icon
Richard W. Hamming (1915–1998) was first a programmer of one of the earliest digital computers while assigned to the Manhattan Project in 1945, then for many years he worked at Bell Labs, and later at the Naval Postgraduate School in Monterey, California. He was a witty and iconoclastic mathematician and computer scientist whose work and influence still reverberates through the areas he was interested in and passionate about. Three of his long-lived books have been reprinted by Dover: Numerical Methods for Scientists and Engineers, 1987; Digital Filters, 1997; and Methods of Mathematics Applied to Calculus, Probability and Statistics, 2004.
In the Author's Own Words: "The purpose of computing is insight, not numbers."
"There are wavelengths that people cannot see, there are sounds that people cannot hear, and maybe computers have thoughts that people cannot think."
"Whereas Newton could say, 'If I have seen a little farther than others, it is because I have stood on the shoulders of giants, I am forced to say, 'Today we stand on each other's feet.' Perhaps the central problem we face in all of computer science is how we are to get to the situation where we build on top of the work of others rather than redoing so much of it in a trivially different way."
"If you don't work on important problems, it's not likely that you'll do important work." — Richard W. Hamming |
TI-89 WORKSHEET 1: Basic Algebra Concepts i TI-89 BASICS To begin, you need to be able to find the ´˜button. It is in the lower left hand corner of the calculator.. Jhildreth ti89 ws 1 basics algebra concepts pdf.
Shape Of Polynomial Functions Worksheet.
Shape Of Polynomial Functions Worksheet. Newark k12 ny us 74420819213937360 lib 74420819213937360 files basic skills ws packet pdf.
Basic Skills Worksheets To jump to a location in this book 1. Click a bookmark on the left. To print a part of the book 1. Click the Print button.. |
in the Homework Helpers series tackles the one of the most advanced mathematical course in most high schools: Trigonometry. The concepts are explained in everyday language before the examples are worked. Good habits, such as checking your answers after every problem, are reinforced. There are practice problems throughout the book, and the answers to all of the practice problems are included. The problems are solved clearly and systematically, with step-by-step instructions provided. Particular attention is placed on topics that students traditionally struggle with the most. While this book could be used to supplement a standard calculus textbook, it could also be used by college students or adult learners to refresh long-forgotten concepts and skills. Trigonometry includes concepts that have both a geometric and an algebraic component. Homework Helpers: Trigonometry covers all of the topics in a typical Trigonometry class, including: • The unit circle • Trigonometric functions • Inverse trigonometric functions • Identities • Graphical analysis • Applications This book also contains a review of the algebraic and geometric ideas that are the foundation of trigonometry. |
Perfect for students of all backgrounds and interest levels, Pride and Ferrellsll need to succeed in todaysGoing beyond standard mathematical physics textbooks by integrating the mathematics with the associated physical content, this book presents mathematical topics with their applications to physics as well as basic physics topics linked to mathematical techniques. It is aimed at first–year graduate students, it is much more concise and discusses selected topics in full without omitting any steps. It covers the mathematical skills needed throughout common graduate level courses in physics and features around 450 end–of–chapter problems, with solutions available to lecturers from the Wiley website. |
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Starting at $33Loose Leaf Intermediate Algebra
LOOSE LEAF VERSION FOR INTERMEDIATE ALGEBRA
Student Solutions Manual for Intermediate Algebra
STUDENT SOLUTIONS MANUAL FOR INTERMEDIATE ALGEBRA
Summary
The revised exercise sets now include even more core exercises than were present in the second edition. This permits instructors to choose from a greater pool of problems, allowing ample opportunity for students to practice what they learn in lecture to hone their skills and develop the knowledge to make a successful transition to the next level course.
Table of Contents
Linear Equations and Inequalities in One Variable
Linear Equations in One Variable
Problem Recognition Exercises: Equations versus Expressions
Applications of Linear Equations in One Variable
Applications to Geometry and Literal Equations
Linear Inequalities in One Variable
Compound Inequalities
Absolute Value Equations
Absolute Value Inequalities
Problem Recognition Exercises: Identifying Equations and Inequalities
Group Activity: Understanding the Symbolism of Mathematics
Summary
Review Exercises
Test
Linear Equations in Two Variables and Functions
Linear Equations in Two Variables
Slope of a Line and Rate of Change
Equations of a Line
Problem Recognition Exercises: Characteristics of Linear Equations
Applications of Linear Equations and Modeling
Introduction to Relations
Introduction to Functions
Graphs of Basic Functions
Problem Recognition Exercises: Characteristics of Relations
Group Activity: Deciphering a Coded Message
Summary
Review Exercises
Test
Cumulative Review Exercises
Systems of Linear Equations and Inequalities
Solving Systems of Linear Equations by the Graphing Method
Solving Systems of Linear Equations by the Substitution Method
Solving Systems of Linear Equations by the Addition Method
Problem Recognition Exercises: Solving Systems of Linear Equations
Applications of Systems of Linear Equations in Two Variables
Linear Inequalities and Systems of Linear Inequalities in Two Variables
Systems of Linear Equations in Three Variables and Applications
Solving Systems of Linear Equations by Using Matrices
Group Activity: Creating a Quadratic Model of the Form y = at2 + bt + c
Summary
Review Exercises
Test
Cumulative Review Exercises
Polynomials
Properties of Integer Exponents and Scientific Notation
Addition and Subtraction of Polynomials and Polynomial Functions
Multiplication of Polynomials
Division of Polynomials
Problem Recognition Exercises: Operations on Polynomials
Greatest Common Factor and Factoring by Grouping
Factoring Trinomials
Factoring Binomials
Problem Recognition Exercises: Factoring Summary
Solving Equations by Using the Zero Product Rule
Group Activity: Investigating Pascal's Triangle
Summary
Review Exercises
Test
Cumulative Review Exercises
Rational Expressions and Rational Equations
Rational Expressions and Rational Functions
Multiplication and Division of Rational Expressions
Addition and Subtraction of Rational Expressions
Complex Fractions
Problem Recognition Exercises: Operations on Rational Expressions
Solving Rational Equationsv
Problem Recognition Exercises: Rational Equations vs. Expressions
Applications of Rational Equations and Proportions
Variation
Group Activity: Computing the Future Value of an Investment
Summary
Review Exercises
Test
Cumulative Review Exercises
Radicals and Complex Numbers
Definition of an nth Root
Rational Exponents
Simplifying Radical Expressions
Addition and Subtraction of Radicals
Multiplication of Radicals
Problem Recognition Exercises: Simplifying Radical Expressions
Division of Radicals and Rationalization
Solving Radical Equations
Complex Numbers
Group Activity: Margin of Error of Survey Results
Summary
Review Exercises
Test
Cumulative Review Exercises
Quadratic Equations and Functions
Square Root Property and Completing the Square
Quadratic Formula
Equations in Quadratic Form
Problem Recognition Exercises: Quadratic and Quadratic Type Equations
Graphs of Quadratic Functions
Vertex of a Parabola: Applications and Modeling
Nonlinear Inequalities
Problem Recognition Exercises: Recognizing Equations and Inequalities
Group Activity: Creating a Quadratic Model of the Form y = a(x - h)2 + k
Summary
Review Exercises
Test
Cumulative Review Exercises
Exponential and Logarithmic Functions and Applications
Algebra and Composition of Functions
Inverse Functions
Exponential Functions
Logarithmic Functions
Problem Recognition Exercises: Identifying Graphs of Functions
Properties of Logarithms
The Irrational Number and Change of Base
Problem Recognition Exercises: Logarithmic and Exponential Forms
Logarithmic and Exponential Equations and Applications
Group Activity: Creating a Population Model
Summary
Review Exercises
Test
Cumulative Review Exercises
Conic Sections
Distance Formula, Midpoint Formula, and Circles
More on the Parabola
The Ellipse and Hyperbola
Problem Recognition Exercises: Formulas and Conic Sections
Nonlinear Systems of Equations in Two Variables
Nonlinear Inequalities and Systems of Inequalities
Group Activity: Investigating the Graphs of Conic Sections on a Calculator |
Pre-Algebra
Description
This pre-algebra work-text gives a brief but complete review of all arithmetic topics, broadening many topics to include more than one approach to the correct solution. Much of the text is devoted to algebra and related topics, scientific notation, geometry, statistics, and trigonometry. Problem-solving strategies help students apply mathematical skills to word problems. Students build confidence in their mathematical potential as they successfully work in advanced topics that are presented in an understandable and interesting style |
Linear Algebra
9780135367971
ISBN:
0135367972
Edition: 2 Pub Date: 1971 Publisher: Prentice Hall
Summary: This introduction to linear algebra features intuitive introductions and examples to motivate important ideas and to illustrate the use of results of theorems. Linear Equations; Vector Spaces; Linear Transformations; Polynomials; Determinants; Elementary canonical Forms; Rational and Jordan Forms; Inner Product Spaces; Operators on Inner Product Spaces; Bilinear Forms For all readers interested in linear algebra. ...> Hoffman, Kenneth is the author of Linear Algebra, published 1971 under ISBN 9780135367971 and 0135367972. Three hundred sixty Linear Algebra textbooks are available for sale on ValoreBooks.com, twenty used from the cheapest price of $62.19, or buy new starting at $80.71 |
Book is Water Damaged, but it is still fully readable. ACCEPTABLE with noticeable wear to cover and pages. Binding intact. We offer a no hassle guarantee on all our items. Orders ...are generally shipped no later than next business day. We offer a no hassle guarantee on all our items Textbook
Overview
Be prepared when you get to the word-problem section of your test! With this easy-to-use pocket guide,solving word problems in algebra becomes almost fun. Written in an anxiety-quelling format,this guide prepares you You'll say "No problem" to word problems with this handy and helpful guide!
WORD PROBLEMS?? NO PROBLEM!! Be prepared when you get to the word-problem section of your test! With this easy-to-use pocket guide,solving word problems in algebra becomes almost fun. This anxiety-quelling guide helps you get ready Sanity-saving features include: Step-by-step approach to word problems; Complete explanations of every step; Fully explained answers Dozens of sample problems; Problems of every type; Skill-verifying practice drill. If you don't have a lot of time but want to excel in class,this book helps you: Brush up before tests; Locate answers quickly; Understand the material; Master word problemswithout spending hours with lengthy textbooks |
More About
This Textbook
Overview
Mathematical concepts and theories underpin engineering and many of the physical sciences. Yet many engineering and science students find math challenging and even intimidating.
The fourth edition of Mathematical Techniques provides a complete course in mathematics, covering all the essential topics with which a physical sciences or engineering student should be familiar.
By breaking the subject into small, modular chapters, the book introduces and builds on concepts in a progressive, carefully-layered way - always with an emphasis on using math to the best effect, rather than relying on theoretical proofs.
With a huge array of end of chapter problems and new self-check questions, the fourth edition of Mathematical Techniques provides extensive opportunities for students to exercise and enhance their mathematical knowledge and skills.
Distinctive Features
- Over 500 worked out examples offer the reader valuable guidance when tackling problems.
- Self-check questions and over 2,000 end of chapter problems provide extensive opportunities for students to actively master the concepts presented.
- A series of projects at the end of the book encourage students to use mathematical software to further their understanding.
- An Online Resource Centre features additional resources for lecturers and students, including figures from the book in electronic format, ready to download; a downloadable solutions manual featuring worked solutions to all end of chapter problems (password protected); and mathematical-based programs relating to the projects featured at the end of the book |
Tutorials on Mathematics to MATLAB
Written for science and engineering students, this book provides an introduction to basic mathematics problems using MATLAB. Topics covered include programming in MATLAB, matrix fundamentals, statistics, and differential and integral calculus.
MATLAB is introduced and used to solve numerous examples throughout the book. |
This text is intended to introduce freshman engineering students to problem solving using an m-file environment. Most of the information in this text applies to any m-file environment (MATLAB, LabVIEW MathScript, Octave, etc.). There are some differences between environments, and occasionally some material will be specific to a given environment. This material is offset from the surrounding text and labeled with the appropriate environment. For example: |
Arithmetic the Easy Way - 4th edition
Summary: Everybody uses arithmetic on virtually a daily basis, and this book serves as a handy brush-up for general readers while it also helps students master basic skills that they need before moving up to high-school-level math and beyond. It reviews addition, subtraction, multiplication, and division, then moves on to calculating with fractions, decimals, and percentages. A concluding chapter reviews units of measurement and word problems. Chapters are filled with short p...show moreractice exercises, all of which are answered at the back of the book. The book features many tables, charts, and line illustrations. Barron's Easy Way books focus on both practical and academic topics, presenting fundamental subject matter in clear, understandable language. Equally popular as self-teaching manuals and supplementary texts for classroom use, they are written to help students improve their grades and review subject matter before tests. They are also useful for introducing general readers to a new career-related skill. Easy Way titles cover virtually all subjects that are taught on advanced high school and college-101 levels. New subjects are periodically added, and existing titles are frequently updated to keep them timely and relevant to students' needs. Subject heads and key phrases are set in a second color |
This is a traditional A level Mathematics text book in use in the '70s and '80s in UK Schools. It covers all of the material needed for the Pure Mathematics part of the curriculum namely Equiations and Inequalities, Graph sketching,Partial Fractions, Logaritmic and Trigonometric functions, Calculus (Differentiation, Integration and Applications), and the basic conic section functions. Loads of worked examples and exercises mean that this book can be used as a complement to contemporary texts which tend to be a little thin in this area. The text does not cover "modern" topics such as matrices, groups, etc. but is nevertheless a good book to dip into. |
Intermediate Algebra (with CengageNOW Printed Access Card)
9780495389736
ISBN:
0495389730
Edition: 4 Pub Date: 2008 Publisher: Brooks Cole
Summary: Building a conceptual foundation in the 'language of algebra', this text provides an integrated learning process that will help readers expand their reasoning abilities as it teaches them how to read, write and think mathematically.
Tussy, Alan S. is the author of Intermediate Algebra (with CengageNOW Printed Access Card), published 2008 under ISBN 9780495389736 and 0495389730. Two hundred eighty two Interme...diate Algebra (with CengageNOW Printed Access Card) textbooks are available for sale on ValoreBooks.com, one hundred thirty one used from the cheapest price of $0.71, or buy new starting at $14 |
Excel HSC General Mathematics Quick Study is the perfect tool for studying and revising on the go! This app is designed specifically for the HSC General Mathematics course. There are two parts to the app: 1. HSC study cards • There are 134 study cards to revise. • All the Core topics (Measurement; Algebraic Modelling; Probability; Financial Mathematics; Data Analysis) are covered. • Special revision features include: - bookmarks—you can bookmark each card with a green, yellow or red bookmark depending on how well you know each card - revision notes—you can type in your own revision notes to customise your revision for each card or topic. 2. Quick Quiz • There are 553 questions in total. • You can take a randomly generated quick quiz of 10 questions from the topic of your choice, or from all topics combined. • Each question in the quiz is marked instantly for you, with the correct answer highlighted so you can learn from your mistakes. • You are given a score out of ten and your percentage mark at the end of each quiz. • You are also given a comprehensive summary of all your results for each topic including your percentage improvement, which helps you keep track of your progress. • You can take the quizzes as many times as you like until you get the consistently high percentage score you want. Excel Quick Study apps are a convenient and efficient way to revise—any time, anywhere!TechnicalPlease note: due to the large number of different devices supporting Android 2.2 and up, not all brands/models have been tested. Your feedback is appreciated: please email us to report any errors or bugs and we will do our utmost to fix them and to support other |
Avalon, CA Algebra experience working with children younger than myself was through an outreach program that was organized by my high school, which offered free basic-level theater classes to under-served elementary school students from the areas surrounding the Main Street campus, taught entirely by stude...
...In the upper grades of high school and at the college level, strong note-taking skills and time management strategies are especially important. Students at these levels are expected to learn enormous amounts of complex information in a short period of time. It is crucial, therefore, that their ...
...By stu...
...Knowing the basic properties of common will save you a lot of time in your calculus studies. Basic functions include trigonometry functions, exponential function, polynomials, and many more. Each set of functions has unique properties that make them useful in different ways.
3. of Biola University's Wri... |
This. I took a course that was an introduction to Boolean Algebra, Set Theory, Combinatorics, Probability, proofs, logic, some basic graph theory, and a bunch of other stuff. It kicked my ass, but now Math feels totally different to me. |
Electronics and Computer Math - 8th edition
The Eighth Edition continues the tradition of providing the most complete, thorough, and practical coverage of mathematics and its application in the world of electronics. This market leading text has been used in hundreds of classrooms by thousands of students who have benefited from the clear explanations and large practice sets that develop both computational and quantitative thinking skills.
Features <...show moreBR> Core Features Carried Over From the Seventh Edition:
Key Points highlight key concepts in each section.
Chapter topics are broken into small testable sections that move students carefully from concept to concept.
Over 300 worked examples model problem solving steps for students.
Over 1400 practice problems within the chapters provide a wealth of practice problems.
Self-tests at the end of each chapter sub-topic allow students to check their understanding upon completion of a topic.
Over 3500 end-of-chapter problems including over 500 new word problems that help students think and apply concepts quantitatively.
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Instructions: Read pages 307-317 of Chapter 5 to learn about angles in trigonometry. Pay particular attention to the new form of angle measure, the radian. A complete grasp of this concept will serve you well through the remainder of the course. Also note that this reading covers the material in subunits 1.2.1 through 1.2.5.
Instructions: Read pages 321-330 of Chapter 5 to learn points on circles using sine and cosine. The unit circle is one of the key concepts in trigonometry, and a complete understanding how the coordinates from the equation of the circle are used to create the trig functions is fundamental to understanding the derivations of the graphs of the functions and all the useful identities we will study in later sections. Committing the unit circle to memory is a useful skill. This reading covers the material in subunits 1.3.1 through 1.3.4.
Instructions: Read pages 333-338 of Chapter 5 to learn about the other trigonometric functions and some important identities, establishing some relationships between all six of the trigonometric functions. This reading covers the material in subunits 1.4.1 through 1.4.3.
Instructions: The graphs of the sinusoidal functions have some important features that help us construct them, and make them useful for modeling. Read pages 353-365 to gain an understanding of the properties of these graphs. This reading also covers the topics outlined in subunits 2.1.1 through 2.1.5.
Instructions: Much like the sinusoidal functions, the remaining trig function graphs have some key features that are important to understand. Read pages 369- 374 to understand these. This reading selection covers the topics outlined in subunits 2.2.1 through 2.2.4.
Instructions: The functions give us some powerful tools for equation solving. Read pages 379–384 to begin to understand them, their graphs, and their relationship to the trig functions. This reading covers the topics outlined in subunits 2.3.1 through 2.3.3.
Instructions: Now that you have an understanding of the inverse trig functions and the domains and ranges of both the trig and inverse trig functions, you can begin solving more complicated equations. Read pages 387-394 to understand how. This reading covers the topics outlined in subunits 2.4.1 and 2.4.2.
Instructions: Trigonometry is very useful for modeling real world data. Read the selection on pages 397–403 to develop some modeling techniques. Note that this reading covers the topics outlined in subunits 2.5.1 and 2.5.2.
Instructions: Because real world phenomena are often modeled with trig functions, it is important to understand how changes to the functions affect the resulting graphs and the phenomena being modeled. To increase your understanding of this, read pages 442–448 of Chapter 7. This selection also covers the topics outlined in subunits 3.4.1 through 3.4.3.
Instructions: Pages 451–466 introduce the idea of using trigonometric functions in triangles other than right triangles. Read this selection carefully. This selection also covers the topics outlined in subunits 4.1.1 and 4.1.2.
Instructions: Read the selection from pages 467–475. The selection defines a new system for graphing points and curves based on distances and angles rather than the horizontal and vertical distances used in the Cartesian Coordinate system. This reading covers the topics outlined in subunits 4.2.1 through 4.2.3.
Instructions: Vectors are geometric objects with both distance and direction, and they have numerous applications. Read pages 491-502 from Chapter 8 carefully to understand these applications. This reading selection also covers the topics outlined in subunits 4.4.1 through 4.4.3.
Instructions: Up until this point in the course, we have been defining functions in terms of two variables: a dependent and an independent variable. Parametric equations give us a new way to define functions, determining the coordinates of a point based on functions of a third variable, often time. Read pages 504–512 to learn about these concepts. This reading also covers the topics outlined for subunits 4.5.1 through 4.5.3. |
An Introduction to Grids, Graphs, and Networks aims to provide a concise introduction to graphs and networks at a level that is accessible to scientists, engineers, and students. In a practical approach, the book presents only the necessary theoretical concepts from mathematics and considers a variety of physical and conceptual configurations as prototypes... more...
The diversity of research domains and theories in the field of mathematics education has been a permanent subject of discussions from the origins of the discipline up to the present. On the one hand the diversity is regarded as a resource for rich scientific development on the other hand it gives rise to the often repeated criticism of the discipline?s... more...
Do you believe we should bomb our economy back to the dark ages? Carpet our beautiful countryside with bat-chomping, bird-slicing eco-crucifixes? Indoctrinate our kids with scary North Korea-style propaganda nonsense in order to deal with the alleged perils of 'climate change?? Neither does James Delingpole, author, polemicist, drowner of baby polar... more...
This book is the fruit of a symposium in honor of Ted Eisenberg concerning the growing divide between the mathematics community and the mathematics education community, a divide that is clearly unhealthy for both. The work confronts this disturbing gap by considering the nature of the relationship between mathematics education and mathematics, and... more...
Galileo Galilei said he was ?reading the book of nature? as he observed pendulums swinging, but he might also simply have tried to draw the numbers themselves as they fall into networks of permutations or form loops that synchronize at different speeds, or attach themselves to balls passing in and out of the hands of good jugglers. Numbers are, after... more...
In the ten years since the publication of the best-selling first edition, more than 1,000 graph theory papers have been published each year . Reflecting these advances, Handbook of Graph Theory, Second Edition provides comprehensive coverage of the main topics in pure and applied graph theory. This second edition?over 400 pages longer than its... more...
It describes each strategy and clarifies its advantages and drawbacks. Also included is a large sample of classroom-tested examples along with sample student responses. These examples can be used "as is" - or you can customize them for your own class. This book will help prepare your students for standardized tests that include items requiring evidence... more...
Topics in Matroid Theory provides a brief introduction to matroid theory with an emphasis on algorithmic consequences. Matroid theory is at the heart of combinatorial optimization and has attracted various pioneers such as Edmonds, Tutte, Cunningham and Lawler among others. Matroid theory encompasses matrices, graphs and other combinatorial entities... more... |
Calculus at RIT
Placement
One of the most important factors in student success in mathematics is correct placement, so calculus at RIT begins with the Math Placement Exam (MPE). Based on the results of the MPE, students in Science, Engineering, Mathematics and Computer Science are directed to a sequence that matches their academic needs, shown in the flow chart below (students in other majors are directed to other courses, pursuant to the requirements of their degree program).
Workshops
Each of the courses in the flow chart (above) has two hours of workshop per week. The academic content of a workshop depends on the particular educational objectives of the course to which it's attached; all workshops, regardless of the course they support, are organized around cooperative study, interaction, and participation in the problem-solving process. They are not traditional recitations nor are they a time for students to do or discuss homework from lecture.
Worksheets
As mentioned above, the particular content of a workshop is designed by the primary instructor to support the educational objectives of the course to which the workshop is attached. The worksheets that instructors design for the project-based calculus sequence are intended to stretch students' abilities and deepen their understanding by tapping into their imagination and sparking their creativity. Such exercises are very different than the standard "kill-and-drill" exercises that many students see in high school. Here are some examples of topics from worksheets in the project-based calculus sequence.
(Using calculus to demonstrate that the "focus" of a parabolic dish is really where the incoming rays are focused)
(Designing ellipitcal couplers for optical fiber)
53(Mathematically modeling wave fronts)
Worksheets are written to be relevant to students' lives (either personally or professionally) and often introduce students to "real" problems. Of course, "real" problems are "real" hard. To help students make the transition to collegiate level thinking and ability, each workshop is supported by both a faculty member and a Teaching Assistant (TA). The TAs attend workshop to help facilitate student group discussions.
Projects
Each course in the Project Based Calculus sequence has, as you might expect, a term project. These projects vary from quarter-to-quarter, and from instructor-to-instructor. Students are expected to solve the given problem, and to write a clear, concise, technical report in which they delineate the process by which they found the solution. Some recent topics for projects are given below.
Common Core Exam
The final exam for each section of each calculus course is given in two parts:
• a multiple-choice "common core" in which students are asked to demonstrate skills and knowledge that are fundamental to the subject
• a free-response part written by the individual instructor in which students demonstrate skills and knowledge particular to that section and instructor
This helps to ensure that students can "leap-frog" between professors if they need to, and also helps maintain a nominal degree of uniformity in grading criteria across all sections of a course.
The School of Mathematical Sciences prohibits calculators on the final exam of calculus (and other first-year) courses. Many professors prepare students for this by prohibiting calculators on exams during the term, or by giving exams in two parts (one with, and one without calculators).
"C"-or-Better Policy
Common sense points to adequate preparation as an important element in student success. Particularly when courses are in sequence, demonstrated competence in one course provides the best foundation for success in the next. For this reason, students in calculus must earn a letter grade of at least "C" before continuing on to subsequent courses.
Science and Engineering Sequence
Course
Grade Earned
Course Placement for Following Term
Project-Based Calculus I: (1016-281)
"C" or better
Project-Based Calculus II: (1016-282)
Project-Based Calculus I: (1016-281)
"D" or "F"
Project-Based Calculus I (1016-281) OR Calculus A and Calculus B: (1016-271) and (1016-272) |
Mercer Island Algebra it involves recognizing one's strengths and using them to create knowledge structures that can support effective learning |
Mathematica Teacher's Edition
Volume 8, Issue 4
Wolfram Research, Inc. is pleased to announce the release
of the all new Mathematica Teacher's Edition. Designed exclusively
for secondary-level math teachers, this software product combines powerful
computation, time-saving tools, and ready-to-use courseware with an
intuitive user interface. Mathematica Teacher's Edition allows
teachers to focus on the important aspects of teaching instead of on
mundane, routine tasks.
Mathematica Teacher's Edition provides a wealth of tools to help
teachers put their students on the fast track to learning. The unique
point-and-click problem generator enables math educators to quickly and
easily generate original assignments, quizzes, and answer keys. Instead
of pulling predefined problems out of a database, Mathematica Teacher's
Edition creates them on the fly--so teachers can compile new
assignments each time, or even make them unique for each student.
The underlying Mathematica computational engine in Mathematica
Teacher's Edition easily tackles everything from simple calculator
problems to
advanced symbolic manipulations, while the complete presentation
environment helps teachers create their own classroom demonstrations and
handouts. Mathematica Teacher's Edition also comes with built-in
courseware and classroom demos, ranging from prealgebra to calculus, to
enhance students' understanding and visualization capabilities. Teachers
can use these resources directly or customize them for their particular
curriculum.
Mathematica Teacher's Edition is the ideal tool to help teachers
succeed in the classroom and bring mathematics alive for their students.
With
Mathematica Teacher's Edition, teachers can:
Create classroom demos featuring live 2D and 3D graphics, animations,
and sounds--all designed to help augment students' intuition and
interest
Produce handouts with a full technical word-processing environment,
including spell checking and mathematical typesetting
Show students multiple problem-solving approaches to any given
problem
Mathematica Teacher's Edition works seamlessly with
Mathematica
for the Classroom to
give students a head start by exposing them to the computing
tools and methods they will use throughout their academic and professional
careers. Like all Wolfram Research products, Mathematica Teacher's
Edition
supports the platform-independent notebook (.nb) format. You can
easily
send notebooks by email or save them as HTML and post them on the
web--making them a convenient way to share reports and documents between
students and teachers.
Mathematica Teacher's Edition is available for Windows
95/98/Me/NT/2000/XP and Mac OS platforms.
More
information is available. |
History of Mathematics An Introduction
9780073051895
ISBN:
0073051896
Edition: 6 Pub Date: 2005 Publisher: McGraw-Hill College
Summary: David Burton covers the history behind the topics typically covered in an undergraduate maths curriculum or in elementary or high schools. He illuminates the people, stories, and social context behind mathematics' greatest historical advances, while maintaining appropriate focus on the mathematical concepts themselves.
Burton, David M. is the author of History of Mathematics An Introduction, published 2005 u...nder ISBN 9780073051895 and 0073051896. One hundred twenty six History of Mathematics An Introduction textbooks are available for sale on ValoreBooks.com, twenty two used from the cheapest price of $17.30, or buy new starting at $1890073051895-3-0-3 Orders ship the same or next business day... [more] |
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