Problem solving with Maple, a handbook (1998), Notas de estudo de Matemática

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Problem Solving with Maple

A Handb o ok

Carl Eb erhart Department of Mathematics University of Kentucky

June  

CONTENTS

 Water tank problem                                   A ladder problem                                   Another Ladder Problem                                Variation on the last ladder problem                         

I I Calculus  

 Di erentiation and its uses  Dening Derivatives                                   The student package                                   Problems                                         Newtons Metho d                                     Use of the derivatives in plotting                             Implicit Dierentiation                                  Maxmin Problems                                    A Pap er folding problem                           

More Max min Problems   Stumbling onto maxmin Problems                            Problems                                         Solutions                                       

 Early Integration  Learning to use the Maple words Sum and sum                      Riemann Sums with the student package                         Learning to use Int and int                               Average value                                      Mo deling the ow of air in lungs                             Two Area problems                                  

Moments and Center of Mass   Center of mass of a Wire                                 Center of mass of a solid of revolution                        

 Denitions and Theorems of Calculus I 

I I I Calculus I I 

 Inverse functions    A Useful Function  The natural logarithm                        Inverse Functions  The exp onential function                        Inverse Functions The inverse trig functions                    

CONTENTS

Integration Techniques and Applications   Symb olic Integration Problems                             A Substitution Problem                             An Integration by Parts Problem                        A Trig Substitution                                A Partial Fractions Problem                            Problems                                     Numerical Integration Problems                             The Trap ezoid Rule                                Problems                                      Simpsons Rule                                  More problems with Trap ezoid and Simpson                 

 Taylors Theorem   Taylor p olynomials                                    Taylor remainder theorem                                 Problems                                        

 Sequences and Series    Sequences                                          Perio dic Points of functions                            Problems                                        Series                                             Problems                                       Two interesting curves                                     The Snowake Curve                                  A Spaceller                                  

 Di erential equations    Terminology                                          Problems leading to rst order equations                          Logistic Growth                                    

Part I

Sans Calculus

 CHAPTER  RAISON DMAPLE

The existing vo cabulary is large enough to carry out the solution to many problems After awhile it b ecomes very useful to b e able to add new words to the vo cabulary If you develop some words to work on a sp ecialized class of problems these can b e put into a package of words for easy access b etween worksheets Maple comes equipp ed with several such packages already including plots a package of drawing words linalg  a very useful package of vector and matrix manipulation words combinat  a package of words from combinatorics and networks  a graph theory package

It comes with a worksheet environment  see b elow We want to learn to use Maple to solve problems In order make go o d use of the language for this purp ose we need to b ecome familiar with the worksheet environnment get to know the language and foster our exp erimental urges

 The Worksheet A handy place to solve problems

When you click on the Maple Icon in Windows an untitled worksheet op ens up Think of it as a clean sheet of pap er Typically after awhile the worksheet will contain a record of the work done to date on the problem or problems you have b een working on Very often you might b e in a problemsolving team working on the problems The worksheet can b e given a name and saved onto a disk for later working or for handing in The worksheet le consists of a numb er of cells of three dierent typ es Input cells Output cells and Text cells

Input Cells These are started with a right angle bracket   Here are a couple An input cell is the place where you put the commands or statements you want Maple to execute The cell can contain one or more statements each ending with a semicolon The nice thing ab out these cells in a worksheet is that they can b e mo died and reexecuted over and over again This enables you to correct typing mistakes with relative ease For example supp ose I wanted Maple to calculate      but left out a parenthesis     Syntax error  unexpected An error message is generated which may help you nd your mistake So you can make a change in that input cell and reexecute it Use the mouse to put the cursor at the sp ot where the error o ccurs and make the correction You can also use this ability to change and reexecute input cells to change the numb ers in whatever problem you have worked out a solution to and see how the solution is aected

Output Cells Almost every input cell when executed gives an output cell containing the results of the calculations It is app ended to the input cell which pro duced it For example lets add to the  rd and to the th  then lets factor the result into prime factors s     ifactors s         

 THE WORKSHEET A HANDY PLACE TO SOLVE PROBLEMS 

You insp ect the output cell to see if it is what you want If it is not then go back and change the input cell and reexecute it

Certain Maple words such as plot generate a separate window containing a picture or a page of text see factor in the next section You can copy and paste these items into an output cell of your worksheet if the need arises plotx x  x

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–2 –1 (^1) x 2

 Exercise Execute the fol lowing plot command Then copy the graph from the plot window use the Edit Menu in that window and paste it into your worksheet

plotx x x

 GET TO KNOW THE LANGUAGE 

 Get to know the language

Maple has a large built in vo cabulary of words esp ecially dened to carry out many of the algorithms you have learned in your previous math classes There are Maple words like factor  expand  simplify  etc You can learn ab out them with online Help There is a Browser available in b oth X and Windows Maple which has the Maple vo cabulary nicely indexed by category Alternatively you can ask for help in an input cell For example to nd out ab out factor just typ e factor Knowing the word is one thing but you also need to know the syntax of the word What is syntax Every algorithm requires certain input information in order to b e p erformed After it has b een p erformed certain output information is pro duced To know the syntax of a Maple word is to know the input information needed and the output information pro duced by the word The help screen gives you the syntax of the word and thus tells you how to use it For example the help screen for factor tells us in CALLING SEQUENCES that there are one and p ossibly two inputs needed and one output pro duced by factor The most useful part of the help screen is the b ottom part where there are examples of the usage of the word in question These examples can b e copied into a worksheet and tested out which gives you a chance to develop a feel for the word by exp erimenting with its use In fact there is a Maple word example which brings up a page of examples of the usage of the word in many cases

 Problems

 Exercise Discover the dierence between factor and ifactor using ifactor 

  Exercise Use ifactor to factor your social security number

  Exercise Use nextprime to nd the rst  primes after 

  Exercise How do col lect coe and expand work Use them to expand x   y ^ col lect the result into a polynomial in x and nd the coe cient of the x  term of that result

 Exp eriment

Exp erimentation is a prime source of learning We are constantly conducting little exp eriments learning from them and using that knowledge in some way The same holds true when you are learning Maple andor working on some math problem The help facility is a great aid to exp erimentation Say you are working on a math problem and you need to carry out some algorithm like solve an equation you have set up You know the Maple word solve will carry out the algorithm but you have forgotten the syntax You can use the word example to list some examples of its usage rather than bring up the entire help sheet on solve  examplesolve Keep in mind that the help sheets are written for general use by b oth novices and exp erts so dont b e intimidated by unfamiliar terminology Often the examples at the b ottom of the sheet suce to tell you what you need to know Keep using help

 CHAPTER  RAISON DMAPLE

  Problems

 Exercise Spend a few minutes with the online help deciding what the terms list and set mean in Maple In particular what is the main dierence between a list and a set

examplelist exampleset

  Exercise If you havent already nd out the usage of the terms seq NULL subsop as used in the example sheet for set and list

Exp eriment is used at the b eginning of solving a problem to generate data to conjecture a solution Towards the end of the pro cess exp eriment is used to test the solution Maple makes the act of exp erimentation easier to carry out and easier to mo dify A Maple Worksheet makes it easier to carry on work from one session to the next and to prepare public do cuments ie homework assignments reseach pap ers etc for public consumption

 CHAPTER  A SHORT INTRODUCTION TO THE MAPLE LANGUAGE

will pro duce an output of I  The name for pi the area of the circle of radius  in Maplese is Pi So to calculate the area of a circle of radius  you would enter Pi  

 Expressions Names Statements and Assignments

Quantities to b e computed like  are called expressions A name is a string of characters which can b e used to store the result of a computation A statement in Maple is a string of names and expressions terminated with a semicolon or a colon if you dont want to see the output which when entered will pro duce some action The assignment statement is one of the most common statements It is of the form name  expression For example the assignment area  Pi  area   stores Pi in a lo cation marked by the name area A more useful assignment for the area of a circle is area  Pir area   r In this case the expression Pir is stored in area and with this assignment the area of a circle of any given radius can b e computed using the Maple word subs So to calculate the area when r is  we enter subsr area  Here it is convenient to think of the assignment as dening area as a function of the radius r

 Functions

A function is a rule f p ossibly very complicated for assigning to each argument x in a given set a unique value fx in a set In calculus the arguments and values of a function are always real numb ers but the notion of function is much more exible than that Functions can b e dened in several useful ways in Maple As an expression The assignment

area  Pir area   r denes the area of a circle as a function of its radius The area function dened as an expression is evaluated with subs Since this function assigns real numb ers to real numb ers its values can b e plotted on a graph with the Maple word plot So the statement plotarear

 FUNCTIONS 

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(^1 2) r 3 4

will pro duce in a separate plot window the graph of the area function over the interval from r to r 

With the arrow op erator the assignment If you have a simple function you can often use the arrow op erator For example area  r  Pir

area  r  r

denes the area function also Now to nd the area of a circle of radius  we simply enter the statement

area



To plot this function over the domain r   typ e

plotarea

 FUNCTIONS 

Note the ifthen control statement here You can learn more ab out the word if by typing if in an input cell and entering it Functions of two variables can b e dened and plotted just as easily in Maple as functions of one variable For example the volume V of a cylinder of height h and radius r is dened by V  rh  Pirh

V  r h  r ^ h

To see what the graph of V lo oks like use plot d  plot dV  axesboxed

0 1 2 3 4

0 1 2 3 4

0 20 40 60 80 100 120 140 160 180 200

Which way of dening a function is the preferred way That really dep ends on the situation The expression metho d works well for functions which have only one rule of evaluation but eventually you cannot avoid using an  or pro c denition You will nd yourself using arrow or pro c denitions more and more as time go es by

Piecewise dened functions Many functions can only b e describ ed by stating various rules for various parts of the domain The Maple word piecewise will help with dening such functions Here is an example to show usage fx piecewisex  x  x    x x     cosx  x

fx 

x^   x  

  x x  

 cosx x 

 x otherwise f f As it stands f is not really a function We need to use unapply to make it into a funtion g unapplyfx x

g  x piecewisex   x^   x     x x   cosx  x

g

 CHAPTER  A SHORT INTRODUCTION TO THE MAPLE LANGUAGE

When plotting piecewise dened functions sometimes style  p oint is b etter plotg  style point

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 Built in Maple functions and Op erations with Functions

All of the standard scientic functions are built into Maple For example sqrt is the square ro ot function abs is the absolute value function the trig and inverse trig functions are sin  arcsin  cos  etc the natural logarithm and exp onential functions are ln and exp  For a complete list of built in functions typ e inifcns New functions can b e obtained from old functions by use of the arithmetic op erations of addi tion subtraction multiplication and division together with the op eration of comp osition which is denoted by!  Thus the function dened by the assignment y  sincosx y  sincosx^   and evaluated at x by subsx y sincos  could also b e dened by the assignment y  sincosxx y  sincosx x^   and evaluated at x by y