Calculus

1180 Pages·2015·80.944 MB·English

by  Jon Ragawski, Colin Adams

TOJULIE–Jon
TOALEXAANDCOLTON–Colin
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ABOUT THE AUTHORS
COLIN ADAMS
ColinAdams is the Thomas T. Read professor of Mathematics at Williams College,
wherehehastaughtsince1985.ColinreceivedhisundergraduatedegreefromMIT
andhisPhDfromtheUniversityofWisconsin.Hisresearchisintheareaofknottheory
and low-dimensional topology. He has held various grants to support his research, and
writtennumerousresearcharticles.
Colinistheauthororco-authorofTheKnotBook,HowtoAceCalculus:TheStreetwise
Guide, How to Ace the Rest of Calculus: The Streetwise Guide, Riot at the Calc Exam
andOtherMathematicallyBentStories,WhyKnot?,IntroductiontoTopology:Pureand
Applied,andZombies&Calculus.Heco-wroteandappearsinthevideos“TheGreatPi
vs.EDebate”and“Derivativevs.Integral:theFinalSmackdown.”
He is a recipient of the Haimo National Distinguished Teaching Award from the
MathematicalAssociationofAmerica(MAA)in1998,anMAAPolyaLecturerfor1998-
2000,aSigmaXiDistinguishedLecturerfor2000-2002,andtherecipientoftheRobert
FosterCherryTeachingAwardin2003.
Colin has two children and one slightly crazy dog, who is great at providing the
entertainment.
JON ROGAWSKI
As a successful teacher for more than 30 years, Jon Rogawski listened and learned
muchfromhisownstudents.Thesevaluablelessonsmadeanimpactonhisthinking,
hiswriting,andhisshapingofacalculustext.
Jon Rogawski received his undergraduate and master’s degrees in mathematics si-
multaneouslyfromYaleUniversity,andheearnedhisPhDinmathematicsfromPrinceton
University,wherehestudiedunderRobertLanglands.BeforejoiningtheDepartmentof
MathematicsatUCLAin1986,wherehewasafullprofessor,heheldteachingandvisiting
positionsattheInstituteforAdvancedStudy,theUniversityofBonn,andtheUniversity
ofParisatJussieuandOrsay.
Jon’sareasofinterestwerenumbertheory,automorphicforms,andharmonicanaly-
sisonsemisimplegroups.Hepublishednumerousresearcharticlesinleadingmathemat-
icsjournals,includingtheresearchmonographAutomorphicRepresentationsofUnitary
GroupsinThreeVariables(PrincetonUniversityPress).HewastherecipientofaSloan
Fellowship and an editor of the Pacific Journal of Mathematics and theTransactions of
theAMS.
Sadly,JonRogawskipassedawayinSeptember2011.Jon’scommitmenttopresent-
ingthebeautyofcalculusandtheimportantroleitplaysinstudents’understandingofthe
widerworldisthelegacythatlivesonineachneweditionofCalculus.
CONTENTS CALCULUS
Chapter 1 PRECALCULUS REVIEW 1 5.4 TheFundamentalTheoremofCalculus,PartI 254
5.5 TheFundamentalTheoremofCalculus,PartII 259
1.1 RealNumbers,Functions,andGraphs 1 5.6 NetChangeastheIntegralofaRateofChange 264
1.2 LinearandQuadraticFunctions 12 5.7 SubstitutionMethod 270
1.3 TheBasicClassesofFunctions 19 ChapterReviewExercises 277
1.4 TrigonometricFunctions 23
1.5 Technology:CalculatorsandComputers 32 Chapter 6 APPLICATIONS OF THE INTEGRAL 281
ChapterReviewExercises 36
6.1 AreaBetweenTwoCurves 281
Chapter 2 LIMITS 39 6.2 SettingUpIntegrals:Volume,Density,AverageValue 289
6.3 VolumesofRevolution 299
2.1 Limits,RatesofChange,andTangentLines 39 6.4 TheMethodofCylindricalShells 307
2.2 Limits:ANumericalandGraphicalApproach 47 6.5 WorkandEnergy 313
2.3 BasicLimitLaws 56 ChapterReviewExercises 319
2.4 LimitsandContinuity 59
2.5 EvaluatingLimitsAlgebraically 68 Chapter 7 EXPONENTIAL FUNCTIONS 321
2.6 TrigonometricLimits 73
2.7 LimitsatInfinity 78 7.1 Derivativeoff(x) bx andtheNumbere 321
=
2.8 IntermediateValueTheorem 84 7.2 InverseFunctions 329
2.9 TheFormalDefinitionofaLimit 87 7.3 LogarithmsandTheirDerivatives 336
ChapterReviewExercises 94 7.4 ExponentialGrowthandDecay 345
7.5 CompoundInterestandPresentValue 351
Chapter 3 DIFFERENTIATION 97 7.6 ModelsInvolvingy′ k(y b) 356
= −
7.7 L’Hôpital’sRule 361
3.1 DefinitionoftheDerivative 97 7.8 InverseTrigonometricFunctions 368
3.2 TheDerivativeasaFunction 105 7.9 HyperbolicFunctions 376
3.3 ProductandQuotientRules 117 ChapterReviewExercises 386
3.4 RatesofChange 123
3.5 HigherDerivatives 132 Chapter 8 TECHNIQUES OF INTEGRATION 391
3.6 TrigonometricFunctions 137
3.7 TheChainRule 141 8.1 IntegrationbyParts 391
3.8 ImplicitDifferentiation 148 8.2 TrigonometricIntegrals 397
3.9 RelatedRates 155 8.3 TrigonometricSubstitution 405
8.4 IntegralsInvolvingHyperbolicandInverseHyperbolic
ChapterReviewExercises 162
Functions 411
Chapter 4 APPLICATIONS OF THE DERIVATIVE 167 8.5 TheMethodofPartialFractions 416
8.6 StrategiesforIntegration 424
4.1 LinearApproximationandApplications 167 8.7 ImproperIntegrals 432
4.2 ExtremeValues 174 8.8 ProbabilityandIntegration 443
4.3 TheMeanValueTheoremandMonotonicity 184 8.9 NumericalIntegration 450
4.4 TheShapeofaGraph 190 ChapterReviewExercises 459
4.5 GraphSketchingandAsymptotes 197
4.6 AppliedOptimization 204 Chapter 9 FURTHER APPLICATIONS OF THE
4.7 Newton’sMethod 216 INTEGRAL AND TAYLOR
ChapterReviewExercises 221 POLYNOMIALS 463
Chapter 5 THE INTEGRAL 225 9.1 ArcLengthandSurfaceArea 463
9.2 FluidPressureandForce 470
5.1 ApproximatingandComputingArea 225 9.3 CenterofMass 476
5.2 TheDefiniteIntegral 237 9.4 TaylorPolynomials 485
5.3 TheIndefiniteIntegral 247 ChapterReviewExercises 496
iv
CONTENTS v
Chapter 10 INTRODUCTION TO DIFFERENTIAL 15.3 PartialDerivatives 773
EQUATIONS 499 15.4 DifferentiabilityandTangentPlanes 783
15.5 TheGradientandDirectionalDerivatives 790
10.1 SolvingDifferentialEquations 499 15.6 TheChainRule 803
10.2 GraphicalandNumericalMethods 507 15.7 OptimizationinSeveralVariables 811
10.3 TheLogisticEquation 515 15.8 LagrangeMultipliers:OptimizingwithaConstraint 825
10.4 First-OrderLinearEquations 519 ChapterReviewExercises 834
ChapterReviewExercises 525
Chapter 16 MULTIPLE INTEGRATION 837
Chapter 11 INFINITE SERIES 529
16.1 IntegrationinTwoVariables 837
11.1 Sequences 529 16.2 DoubleIntegralsoverMoreGeneralRegions 848
11.2 SumminganInfiniteSeries 539 16.3 TripleIntegrals 861
11.3 ConvergenceofSerieswithPositiveTerms 550 16.4 IntegrationinPolar,Cylindrical,andSpherical
11.4 AbsoluteandConditionalConvergence 559 Coordinates 872
11.5 TheRatioandRootTestsandStrategiesforChoosing 16.5 ApplicationsofMultipleIntegrals 882
Tests 564 16.6 ChangeofVariables 894
11.6 PowerSeries 569 ChapterReviewExercises 907
11.7 TaylorSeries 579
ChapterReviewExercises 591 Chapter 17 LINE AND SURFACE INTEGRALS 911
Chapter 12 PARAMETRIC EQUATIONS, POLAR 17.1 VectorFields 911
COORDINATES, AND CONIC 17.2 LineIntegrals 921
SECTIONS 595 17.3 ConservativeVectorFields 935
17.4 ParametrizedSurfacesandSurfaceIntegrals 946
12.1 ParametricEquations 595 17.5 SurfaceIntegralsofVectorFields 960
12.2 ArcLengthandSpeed 606 ChapterReviewExercises 970
12.3 PolarCoordinates 612
12.4 AreaandArcLengthinPolarCoordinates 620 Chapter 18 FUNDAMENTAL THEOREMS OF
12.5 ConicSections 625 VECTOR ANALYSIS 973
ChapterReviewExercises 638
18.1 Green’sTheorem 973
Chapter 13 VECTOR GEOMETRY 641
18.2 Stokes’Theorem 987
18.3 DivergenceTheorem 997
13.1 VectorsinthePlane 641
ChapterReviewExercises 1009
13.2 VectorsinThreeDimensions 651
13.3 DotProductandtheAngleBetweenTwoVectors 661 APPENDICES A1
13.4 TheCrossProduct 669 A. TheLanguageofMathematics A1
13.5 Planesin3-Space 680 B. PropertiesofRealNumbers A7
13.6 ASurveyofQuadricSurfaces 686 C. InductionandtheBinomialTheorem A12
13.7 CylindricalandSphericalCoordinates 694 D. AdditionalProofs A16
ChapterReviewExercises 701
ANSWERSTOODD-NUMBEREDEXERCISES ANS1
Chapter 14 CALCULUS OF VECTOR-VALUED
REFERENCES R1
FUNCTIONS 705
INDEX I1
14.1 Vector-ValuedFunctions 705
14.2 CalculusofVector-ValuedFunctions 713
Additional content can be accessed online via LaunchPad:
14.3 ArcLengthandSpeed 722
14.4 Curvature 727
ADDITIONALPROOFS
14.5 Motionin3-Space 738
14.6 PlanetaryMotionAccordingtoKeplerandNewton 747 • L’Hôpital’sRule
ChapterReviewExercises 753 ErrorBoundsforNumericalIntegration
•
ComparisonTestforImproperIntegrals
•
Chapter 15 DIFFERENTIATION IN SEVERAL
VARIABLES 755
ADDITIONALCONTENT
15.1 FunctionsofTwoorMoreVariables 755 SecondOrderDifferentialEquations
•
15.2 LimitsandContinuityinSeveralVariables 766 ComplexNumbers
•
PREFACE
ABOUT CALCULUS
On Teaching Mathematics
Iconsidermyselfveryluckytohaveacareerasateacherandpractitionerofmathematics.
WhenIwasyoung,IdecidedIwantedtobeawriter.Ilovedtellingstories.ButIwasalso
goodatmath,and,onceincollege,itdidn’ttakemelongtobecomeenamoredwithit.I
lovedthefactthatsuccessinmathematicsdoesnotdependonyourpresentationskillsor
yourinterpersonalrelationships.Youareeitherrightoryouarewrongandthereislittle
subjectiveevaluationinvolved.AndIlovedthesatisfactionofcomingupwithasolution.
ThatintensifiedwhenIstartedsolvingproblemsthatwereopenresearchquestionsthat
hadpreviouslyremainedunsolved.
So,Ibecameaprofessorofmathematics.AndIsoonrealizedthatteachingmathe-
maticsisabouttellingastory.Thegoalistoexplaintostudentsinanintriguingmanner,at
therightpace,andinasclearawayaspossible,howmathematicsworksandwhatitcan
doforyou.Ifindmathematicsimmenselybeautiful.Iwantstudentstofeelthatway,too.
On Writing a Calculus Text
IhadalwaysthoughtImightwriteacalculustext.Butthatisadauntingtask.Thesedays,
calculus books average over a thousand pages. And I would need to convince myself
thatIhadsomethingtoofferthatwasdifferentenoughfromwhatalreadyappearsinthe
existingbooks.Then,IwasapproachedaboutwritingthethirdeditionofJonRogawski’s
calculus book. Here was a book for which I already had great respect. Jon’s vision of
whatacalculusbookshouldbefitverycloselywithmyown.Jonbelievedthatasmath
teachers,howwesayitisasimportantaswhatwesay.Althoughheinsistedonrigorat
alltimes,healsowantedabookthatwaswritteninplainEnglish,abookthatcouldbe
readandthatwouldenticestudentstoreadfurtherandlearnmore.Moreover,Jonstrived
tocreateatextinwhichexposition,graphics,andlayoutwouldworktogethertoenhance
allfacetsofastudent’scalculusexperience.
Inwritinghisbook,Jonpaidspecialattentiontocertainaspectsofthetext:
1. Clear,accessibleexpositionthatanticipatesandaddressesstudentdifficulties.
2. Layoutandfiguresthatcommunicatetheflowofideas.
3. Highlightedfeaturesthatemphasizeconceptsandmathematicalreasoning:Conceptual
Insight,GraphicalInsight,AssumptionsMatter,Reminder,andHistoricalPerspective.
4. Arich collection of examples and exercises of graduated difficulty that teach basic
skills,problem-solvingtechniques,reinforceconceptualunderstanding,andmotivatecal-
culusthroughinterestingapplications.Eachsectionalsocontainsexercisesthatdevelop
additionalinsightsandchallengestudentstofurtherdeveloptheirskills.
Comingintotheprojectofcreatingthethirdedition,Iwassomewhatapprehensive.
Herewasanalreadyexcellentbookthathadattainedthegoalssetforitbyitsauthor.First
and foremost, I wanted to be sure that I did it no harm. On the other hand, I have been
teaching calculus now for 30 years, and in that time, I have come to some conclusions
aboutwhatdoesanddoesnotworkwellforstudents.
As a mathematician, I want to make sure that the theorems, proofs, arguments and
development are correct. There is no place in mathematics for sloppiness of any kind.
Asateacher,Iwantthematerialtobeaccessible.Thebookshouldnotbewrittenatthe
mathematicalleveloftheinstructor.Studentsshouldbeabletousethebooktolearnthe
material,withthehelpoftheirinstructor.WorkingfromthehighstandardthatJonset,I
havetriedhardtomaintainthelevelofqualityofthepreviouseditionwhilemakingthe
changesthatIbelievewillbringthebooktothenextlevel.
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