Maple by Example

556 Pages·2005·16.646 MB·English

by  M. Abell, J. Braselton

Maple by Example
Third Edition
Martha L. Abell and James P. Braselton
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05 06 07 08 09 10 9 8 7 6 5 4 3 2 1
Contents
Preface ix
1 GettingStarted 1
1.1 IntroductiontoMaple . . . . . . . . . . . . . . . . . . . . . . . . 1
ANoteRegardingDifferentVersionsofMaple . . . . . . . . . . . 2
1.1.1 GettingStartedwithMaple . . . . . . . . . . . . . . . . . 3
Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 LoadingPackages . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 GettingHelpfromMaple . . . . . . . . . . . . . . . . . . . . . . . 11
MapleHelp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
TheMapleMenu . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 BasicOperationsonNumbers,Expressions,andFunctions 19
2.1 NumericalCalculationsandBuilt-InFunctions . . . . . . . . . . . 19
2.1.1 NumericalCalculations . . . . . . . . . . . . . . . . . . . 19
2.1.2 Built-InConstants . . . . . . . . . . . . . . . . . . . . . . 22
2.1.3 Built-InFunctions . . . . . . . . . . . . . . . . . . . . . . 23
AWordofCaution . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2 ExpressionsandFunctions:ElementaryAlgebra . . . . . . . . . . 27
2.2.1 BasicAlgebraicOperationsonExpressions . . . . . . . . . 27
2.2.2 NamingandEvaluatingExpressions . . . . . . . . . . . . 31
TwoWordsofCaution . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.3 DefiningandEvaluatingFunctions . . . . . . . . . . . . . 33
2.3 GraphingFunctions,Expressions,andEquations . . . . . . . . . . 40
2.3.1 FunctionsofaSingleVariable . . . . . . . . . . . . . . . . 40
2.3.2 ParametricandPolarPlotsinTwoDimensions . . . . . . 51
v
vi Contents
2.3.3 Three-DimensionalandContourPlots;Graphing
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.3.4 ParametricCurvesandSurfacesinSpace . . . . . . . . . . 66
2.4 SolvingEquationsandInequalities . . . . . . . . . . . . . . . . . 73
2.4.1 ExactSolutionsofEquations . . . . . . . . . . . . . . . . 73
2.4.2 SolvingInequalities . . . . . . . . . . . . . . . . . . . . . 82
2.4.3 ApproximateSolutionsofEquations . . . . . . . . . . . . 84
3 Calculus 91
3.1 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.1.1 UsingGraphsandTablestoPredictLimits . . . . . . . . . 91
3.1.2 ComputingLimits . . . . . . . . . . . . . . . . . . . . . . 93
3.1.3 One-SidedLimits . . . . . . . . . . . . . . . . . . . . . . . 96
3.2 DifferentialCalculus . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.2.1 DefinitionoftheDerivative . . . . . . . . . . . . . . . . . 98
3.2.2 CalculatingDerivatives . . . . . . . . . . . . . . . . . . . 102
3.2.3 ImplicitDifferentiation . . . . . . . . . . . . . . . . . . . 105
3.2.4 TangentLines . . . . . . . . . . . . . . . . . . . . . . . . 105
3.2.5 TheFirstDerivativeTestandSecondDerivativeTest . . . 116
3.2.6 AppliedMax/MinProblems . . . . . . . . . . . . . . . . 121
3.2.7 Antidifferentiation . . . . . . . . . . . . . . . . . . . . . . 131
3.3 IntegralCalculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
3.3.1 Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
3.3.2 TheDefiniteIntegral . . . . . . . . . . . . . . . . . . . . . 139
3.3.3 ApproximatingDefiniteIntegrals . . . . . . . . . . . . . . 144
3.3.4 Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
3.3.5 ArcLength . . . . . . . . . . . . . . . . . . . . . . . . . . 154
3.3.6 SolidsofRevolution . . . . . . . . . . . . . . . . . . . . . 158
3.4 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
3.4.1 IntroductiontoSequencesandSeries . . . . . . . . . . . . 164
3.4.2 ConvergenceTests . . . . . . . . . . . . . . . . . . . . . . 170
3.4.3 AlternatingSeries . . . . . . . . . . . . . . . . . . . . . . 174
3.4.4 PowerSeries . . . . . . . . . . . . . . . . . . . . . . . . . 176
3.4.5 TaylorandMaclaurinSeries . . . . . . . . . . . . . . . . . 179
3.4.6 Taylor’sTheorem . . . . . . . . . . . . . . . . . . . . . . 185
3.4.7 OtherSeries . . . . . . . . . . . . . . . . . . . . . . . . . 188
3.5 Multi-VariableCalculus . . . . . . . . . . . . . . . . . . . . . . . 190
3.5.1 LimitsofFunctionsofTwoVariables . . . . . . . . . . . . 190
3.5.2 PartialandDirectionalDerivatives . . . . . . . . . . . . . 193
3.5.3 IteratedIntegrals . . . . . . . . . . . . . . . . . . . . . . . 212
4 IntroductiontoListsandTables 223
4.1 ListsandListOperations . . . . . . . . . . . . . . . . . . . . . . . 223
Contents vii
4.1.1 DefiningLists. . . . . . . . . . . . . . . . . . . . . . . . . 223
4.1.2 PlottingListsofPoints . . . . . . . . . . . . . . . . . . . . 227
4.2 ManipulatingLists:Moreonopandmap . . . . . . . . . . . . . . 238
4.2.1 MoreonGraphingLists . . . . . . . . . . . . . . . . . . . 247
4.3 MathematicsofFinance . . . . . . . . . . . . . . . . . . . . . . . 253
4.3.1 CompoundInterest . . . . . . . . . . . . . . . . . . . . . 254
4.3.2 FutureValue . . . . . . . . . . . . . . . . . . . . . . . . . 256
4.3.3 AnnuityDue . . . . . . . . . . . . . . . . . . . . . . . . . 257
4.3.4 PresentValue . . . . . . . . . . . . . . . . . . . . . . . . . 259
4.3.5 DeferredAnnuities . . . . . . . . . . . . . . . . . . . . . . 260
4.3.6 Amortization . . . . . . . . . . . . . . . . . . . . . . . . . 262
4.3.7 MoreonFinancialPlanning . . . . . . . . . . . . . . . . . 267
4.4 OtherApplications . . . . . . . . . . . . . . . . . . . . . . . . . . 274
4.4.1 ApproximatingListswithFunctions . . . . . . . . . . . . 274
4.4.2 IntroductiontoFourierSeries . . . . . . . . . . . . . . . . 281
4.4.3 TheMandelbrotSetandJuliaSets . . . . . . . . . . . . . . 294
5 MatricesandVectors:TopicsfromLinearAlgebraandVector
Calculus 311
5.1 NestedLists:IntroductiontoMatrices,Vectors,and
MatrixOperations . . . . . . . . . . . . . . . . . . . . . . . . . . 312
5.1.1 DefiningNestedLists,Matrices,andVectors . . . . . . . . 312
5.1.2 ExtractingElementsofMatrices . . . . . . . . . . . . . . . 320
5.1.3 BasicComputationswithMatrices . . . . . . . . . . . . . 322
5.1.4 BasicComputationswithVectors . . . . . . . . . . . . . . 328
5.2 LinearSystemsofEquations . . . . . . . . . . . . . . . . . . . . . 336
5.2.1 CalculatingSolutionsofLinearSystemsofEquations . . . 336
5.2.2 Gauss-JordanElimination . . . . . . . . . . . . . . . . . . 342
5.3 SelectedTopicsfromLinearAlgebra . . . . . . . . . . . . . . . . 349
5.3.1 FundamentalSubspacesAssociatedwithMatrices . . . . . 349
5.3.2 TheGram-SchmidtProcess . . . . . . . . . . . . . . . . . 352
5.3.3 LinearTransformations . . . . . . . . . . . . . . . . . . . 355
5.3.4 EigenvaluesandEigenvectors . . . . . . . . . . . . . . . . 360
5.3.5 JordanCanonicalForm . . . . . . . . . . . . . . . . . . . 365
5.3.6 TheQRMethod . . . . . . . . . . . . . . . . . . . . . . . 369
5.4 MaximaandMinimaUsingLinearProgramming . . . . . . . . . 372
5.4.1 TheStandardFormofaLinearProgrammingProblem . . 372
5.4.2 TheDualProblem . . . . . . . . . . . . . . . . . . . . . . 375
5.5 SelectedTopicsfromVectorCalculus . . . . . . . . . . . . . . . . 384
5.5.1 Vector-ValuedFunctions . . . . . . . . . . . . . . . . . . 384
5.5.2 LineIntegrals . . . . . . . . . . . . . . . . . . . . . . . . . 397
5.5.3 SurfaceIntegrals . . . . . . . . . . . . . . . . . . . . . . . 401
5.5.4 ANoteonNonorientability . . . . . . . . . . . . . . . . . 406
viii Contents
6 ApplicationsRelatedtoOrdinaryandPartialDifferentialEquations 417
6.1 First-OrderDifferentialEquations . . . . . . . . . . . . . . . . . . 417
6.1.1 SeparableEquations . . . . . . . . . . . . . . . . . . . . . 417
6.1.2 LinearEquations . . . . . . . . . . . . . . . . . . . . . . . 422
6.1.3 NonlinearEquations . . . . . . . . . . . . . . . . . . . . . 433
6.1.4 NumericalMethods . . . . . . . . . . . . . . . . . . . . . 437
6.2 Second-OrderLinearEquations . . . . . . . . . . . . . . . . . . . 443
6.2.1 BasicTheory . . . . . . . . . . . . . . . . . . . . . . . . . 443
6.2.2 ConstantCoefficients . . . . . . . . . . . . . . . . . . . . 444
6.2.3 UndeterminedCoefficients . . . . . . . . . . . . . . . . . 452
6.2.4 VariationofParameters . . . . . . . . . . . . . . . . . . . 457
6.3 Higher-OrderLinearEquations . . . . . . . . . . . . . . . . . . . 460
6.3.1 BasicTheory . . . . . . . . . . . . . . . . . . . . . . . . . 460
6.3.2 ConstantCoefficients . . . . . . . . . . . . . . . . . . . . 460
6.3.3 UndeterminedCoefficients . . . . . . . . . . . . . . . . . 463
6.3.4 LaplaceTransformMethods . . . . . . . . . . . . . . . . . 473
6.3.5 NonlinearHigher-OrderEquations . . . . . . . . . . . . . 486
6.4 SystemsofEquations . . . . . . . . . . . . . . . . . . . . . . . . . 487
6.4.1 LinearSystems . . . . . . . . . . . . . . . . . . . . . . . . 487
6.4.2 NonhomogeneousLinearSystems . . . . . . . . . . . . . 498
6.4.3 NonlinearSystems . . . . . . . . . . . . . . . . . . . . . . 502
6.5 SomePartialDifferentialEquations . . . . . . . . . . . . . . . . . 518
6.5.1 TheOne-DimensionalWaveEquation . . . . . . . . . . . 519
6.5.2 TheTwo-DimensionalWaveEquation . . . . . . . . . . . 524
6.5.3 OtherPartialDifferentialEquations . . . . . . . . . . . . . 534
Bibliography 539
SubjectIndex 541
Preface
Maple by Example bridges the gap that exists between the very elementary
handbooksavailableonMapleandthosereferencebookswrittenfortheadvanced
Maple users. Maple by Example is an appropriate reference for all users of Maple
and,inparticular,forbeginninguserslikestudents,instructors,engineers,business
people,andotherprofessionalsfirstlearningtouseMaple.MaplebyExampleintro-
ducestheverybasiccommandsandincludestypicalexamplesofapplicationsof
thesecommands.Inaddition,thetextalsoincludescommandsusefulinareassuch
ascalculus,linearalgebra,businessmathematics,ordinaryandpartialdifferential
equations, andgraphics. Inallcases, however, examplesfollowtheintroduction
ofnewcommands.Readersfromthemostelementarytoadvancedlevelswillfind
thattherangeoftopicscoveredaddressestheirneeds.
TakingadvantageofVersion9ofMaple,MaplebyExample,ThirdEdition,intro-
ducesthefundamentalconceptsofMapletosolvetypicalproblemsofinterestto
students,instructors,andscientists.OtherfeaturestohelpmakeMaplebyExample,
ThirdEdition,aseasytouseandasusefulaspossibleincludethefollowing.
1. Version 9 Compatibility. All examples illustrated in Maple by Example, Third
Edition,werecompletedusingVersion9ofMaple.Althoughmostcomputations
cancontinuetobecarriedoutwithearlierversionsofMaple,likeVersions5–8,
wehavetakenadvantageofthenewfeaturesinVersion9asmuchaspossible.
2. Applications. New applications, many of which are documented by refer-
ences, from a variety of fields, especially biology, physics, and engineering,
areincludedthroughoutthetext.
3. DetailedTableofContents.Thetableofcontentsincludesallchapter,section,
and subsection headings. Along with the comprehensive index, we hope that
userswillbeabletolocateinformationquicklyandeasily.
ix
x Preface
4. Additional Examples. We have considerably expanded the topics in Chap-
ters 1 through 6. The results should be more useful to instructors, students,
businesspeople,engineers,andotherprofessionalsusingMapleonavarietyof
platforms.Inaddition,severalsectionshavebeenaddedtohelpmakelocating
informationeasierfortheuser.
5. ComprehensiveIndex. Intheindex, mathematicalexamplesandapplications
arelistedbytopic, orname, aswellascommandsalongwithfrequentlyused
options:particularmathematicalexamplesaswellasexamplesillustratinghow
tousefrequentlyusedcommandsareeasytolocate.Inaddition,commandsin
theindexarecross-referencedwithfrequentlyusedoptions.Functionsavailable
inthevariouspackagesarecross-referencedbothbypackageandalphabetically.
6. IncludedCD.AllMaplecodethatappearsinMaplebyExample,ThirdEdition,
isincludedontheCDpackagedwiththetext.
WebeganMaplebyExamplein1991andthefirsteditionwaspublishedin1992.
Backthen,wewereontopoftheworldusingMacintoshIIcx’swith8megsofRAM
and40megharddrives. Wetriedtochooseexamplesthatwethoughtwouldbe
relevanttobeginningusers–typicallyinthecontextofmathematicsencountered
in the undergraduate curriculum. Those examples could also be carried out by
MapleinatimelymanneronacomputeraspowerfulasaMacintoshIIcx.
Now,weareontopoftheworldwithPowerMacintoshG4’swith768megsof
RAMand50gigharddrives,whichwillalmostcertainlybeobsoletebythetime
youarereadingthis. TheexamplespresentedinMaplebyExample continuetobe
theonesthatwethinkaremostsimilartotheproblemsencounteredbybeginning
usersandarepresentedinthecontextofsomeonefamiliarwithmathematicstyp-
ically encountered by undergraduates. However, for this third edition of Maple
by Example we have taken the opportunity to expand on several of our favorite
examplesbecausethemachinesnowhavethespeedandpowertoexplorethemin
greaterdetail.
Otherimprovementstothethirdeditioninclude:
1. Throughoutthetext,wehaveattemptedtoeliminateredundantexamplesand
added several interesting ones. The following changes are especially worth
noting.
(a) In Chapter 2, we have increased the number of parametric and polar plots
intwoandthreedimensions.Forasample,seeExamples2.3.8,2.3.9,2.3.10,
2.3.11,2.3.17,and2.3.18.
(b) In Chapter 3, Calculus, we have added examples dealing with parametric
andpolarcoordinatestoeverysection. Examples3.2.9, 3.3.9, and3.3.10are
newexamplesworthnoting.
Preface xi
(c) Chapter4, IntroductiontoListsandTables, containsseveralnewexamples
illustrating various techniques of how to quickly create plots of bifurcation
diagrams,Juliasets,andtheMandelbrotset.SeeExamples4.1.7,4.2.5,4.2.7,
4.4.6,4.4.7,4.4.8,4.4.9,4.4.10,4.4.11,4.4.12,and4.4.13.
(d) Several examples illustrating how to determine graphically if a surface is
nonorientable have been added to Chapter 5, Matrices and Vectors. See
especiallyExamples5.5.8and5.5.9.
(e) Chapter 6, Differential Equations, has been completely reorganized. More
basic–andmoredifficult–exampleshavebeenaddedthroughout.
2. WehaveincludedreferencesthatwefindparticularlyinterestingintheBibli-
ography,eveniftheyarenotspecificMaple-relatedtexts.Acomprehensivelist
ofMaple-relatedpublicationscanbefoundattheMaplewebsite.
http://www.maplesoft.com/publications/
Finally,wemustexpressourappreciationtothosewhoassistedinthisproject.
We would like to express appreciation to our editors, Tom Singer and Barbara
Holland,andourproductioneditor,BrandyLilly,atAcademicPressforproviding
apleasantenvironmentinwhichtowork.Inaddition,FrancesMorgan,ourproject
manageratKeywordTypesettingServices,deservesthanksformakingtheproduc-
tionprocessrunsmoothly.Finally,wethankthoseclosetous,especiallyImogene
Abell,LoriBraselton,AdaBraselton,andMattieBraseltonforenduringwithusthe
pressuresofmeetingadeadlineandforgraciouslyacceptingourdemandingwork
schedules.Wecertainlycouldnothavecompletedthistaskwithouttheircareand
understanding.
MarthaAbell (email:[email protected])
JamesBraselton (email:[email protected])
Statesboro,Georgia
June,2004
1
Getting Started
1.1 Introduction to Maple
Maple,firstreleasedin1981byWaterlooMaple,Inc.,
http://www.maplesoft.com/,
is a system for doing mathematics on a computer. Maple combines symbolic
manipulation, numerical mathematics, outstanding graphics, and a sophisti-
cated programming language. Because of its versatility, Maple has established
itself as the computer algebra system of choice for many computer users includ-
ing commercial and government scientists and engineers, mathematics, science,
andengineeringteachersandresearchers,andstudentsenrolledinmathematics,
science, andengineeringcourses. However, duetoitsspecialnatureandsophis-
tication,beginningusersneedtobeawareofthespecialsyntaxrequiredtomake
Mapleperforminthewayintended.Youwillfindthatcalculationsandsequences
of calculations most frequently used by beginning users are discussed in detail
alongwithmanytypicalexamples.Inaddition,thecomprehensiveindexnotonly
listsavarietyoftopicsbutalsocross-referencescommandswithfrequentlyused
options.MaplebyExampleservesasavaluabletoolandreferencetothebeginning
userofMapleaswellastothemoresophisticateduser,withspecializedneeds.
Forinformation,includingpurchasinginformation,aboutMaplecontact:
CorporateHeadquarters:
Maplesoft
615KumpfDrive,Waterloo
Ontario,CanadaN2V1K8
telephone:[phone]
fax:[phone]
1