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CALCULUS
Simplified
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Other Books by Oscar Fernandez
EverydayCalculus:DiscoveringtheHiddenMathAllAroundUs
TheCalculusofHappiness:HowaMathematicalApproachtoLifeAddsUpto
Health,Wealth,andLove
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CALCULUS
Simplified
Oscar E. Fernandez
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
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Copyright(cid:2)c 2019byPrincetonUniversityPress
PublishedbyPrincetonUniversityPress
41WilliamStreet,Princeton,NewJersey08540
6OxfordStreet,Woodstock,OxfordshireOX201TR
press.princeton.edu
AllRightsReserved
LCCN2019936027
ISBN978-0-691-17539-3
BritishLibraryCataloging-in-PublicationDataisavailable
Editorial:VickieKearn,SusannahShoemaker,andLaurenBucca
ProductionEditorial:KathleenCioffi
TextDesign:LorraineDoneker
Coverdesign:LaylaMacRory
Production:ErinSuydam
Publicity:MatthewTaylorandKathrynStevens
Copyeditor:TheresaKornak
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Printedonacid-freepaper.∞
PrintedintheUnitedStatesofAmerica
1 3 5 7 9 10 8 6 4 2
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To Emilia and Alicia
Manyyearsfromnowwhenyouconfrontcalculus,
Comebacktothisbookandgiveitaread.
Comebacktome,too,andgivemeahug.
AsmuchasIlovemath,Iloveyoubothmuch,muchmore.
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Contents
Preface xi
TotheStudent xvii
TotheInstructor xix
BeforeYouBegin... xxi
1 The Fast Track Introduction to Calculus 1
1.1 WhatIsCalculus? 1
CalculusasaWayofThinking 1
WhatDoes“InfinitesimalChange”Mean? 2
1.2 Limits:TheFoundationofCalculus 3
1.3 TheThreeDifficultProblemsThatLedtotheInvention
ofCalculus 5
2 Limits: How to Approach Indefinitely (and Thus Never Arrive) 8
2.1 One-SidedLimits:AGraphicalApproach 8
2.2 ExistenceofOne-SidedLimits 11
2.3 Two-SidedLimits 13
2.4 ContinuityataPoint 15
2.5 ContinuityonanInterval 17
2.6 TheLimitLaws 21
2.7 CalculatingLimits—AlgebraicTechniques 25
2.8 LimitsApproachingInfinity 30
2.9 LimitsYieldingInfinity 33
2.10 PartingThoughts 37
Chapter2Exercises 37
3 Derivatives: Change, Quantified 43
3.1 SolvingtheInstantaneousSpeedProblem 43
3.2 SolvingtheTangentLineProblem—TheDerivative
ataPoint 47
3.3 TheInstantaneousRateofChangeInterpretation
oftheDerivative 50
3.4 Differentiability:WhenDerivativesDo(andDon’t)Exist 51
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viii • Contents
3.5 TheDerivative,aGraphicalApproach 53
3.6 TheDerivative,anAlgebraicApproach 55
LeibnizNotation 59
3.7 DifferentiationShortcuts:TheBasicRules 60
3.8 DifferentiationShortcuts:ThePowerRule 61
3.9 DifferentiationShortcuts:TheProductRule 64
3.10 DifferentiationShortcuts:TheChainRule 65
3.11 DifferentiationShortcuts:TheQuotientRule 68
3.12 (Optional)DerivativesofTranscendentalFunctions 69
3.13 Higher-OrderDerivatives 74
3.14 PartingThoughts 75
Chapter3Exercises 76
4 Applications of Differentiation 82
4.1 RelatedRates 82
4.2 Linearization 89
4.3 TheIncreasing/DecreasingTest 93
4.4 OptimizationTheory:LocalExtrema 98
4.5 OptimizationTheory:AbsoluteExtrema 101
4.6 ApplicationsofOptimization 106
4.7 WhattheSecondDerivativeTellsUsAbouttheFunction 112
4.8 PartingThoughts 117
Chapter4Exercises 118
5 Integration: Adding Up Change 125
5.1 DistanceasArea 125
5.2 Leibniz’sNotationfortheIntegral 128
5.3 TheFundamentalTheoremofCalculus 130
5.4 AntiderivativesandtheEvaluationTheorem 133
5.5 IndefiniteIntegrals 135
5.6 PropertiesofIntegrals 138
5.7 NetSignedArea 139
5.8 (Optional)IntegratingTranscendentalFunctions 141
5.9 TheSubstitutionRule 143
5.10 ApplicationsofIntegration 148
5.11 PartingThoughts 152
Chapter5Exercises 153
Epilogue 159
Acknowledgments 161
AppendixA:ReviewofAlgebraandGeometry 163
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Contents • ix
AppendixB:ReviewofFunctions 177
AppendixC:AdditionalAppliedExamples 215
AnswerstoAppendixandChapterExercises 227
Bibliography 239
IndexofApplications 241
IndexofSubjects 243
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