Calculus demystified

356 Pages·2002·1.76 MB·English

by  Steven Krantz

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CALCULUS DEMYSTIFIED
OtherTitlesintheMcGraw-HillDemystifiedSeries
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CALCULUS DEMYSTIFIED
STEVEN G. KRANTZ
McGRAW-HILL
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DOI: 10.1036/0071412115
To Archimedes, Pierre de Fermat, Isaac Newton, and Gottfried Wilhelm
vonLeibniz,thefathersofcalculus
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CONTENTS
Preface xi
CHAPTER1 Basics 1
1.0 IntroductoryRemarks 1
1.1 NumberSystems 1
1.2 CoordinatesinOneDimension 3
1.3 CoordinatesinTwoDimensions 5
1.4 TheSlopeofaLineinthePlane 8
1.5 TheEquationofaLine 13
1.6 LociinthePlane 15
1.7 Trigonometry 19
1.8 SetsandFunctions 30
1.8.1 ExamplesofFunctionsofaRealVariable 31
1.8.2 GraphsofFunctions 33
1.8.3 PlottingtheGraphofaFunction 35
1.8.4 CompositionofFunctions 40
1.8.5 TheInverseofaFunction 42
1.9 AFewWordsAboutLogarithmsandExponentials 49
CHAPTER2 Foundations of Calculus 57
2.1 Limits 57
2.1.1 One-SidedLimits 60
2.2 PropertiesofLimits 61
2.3 Continuity 64
2.4 TheDerivative 66
2.5 RulesforCalculatingDerivatives 71
2.5.1 TheDerivativeofanInverse 76
2.6 TheDerivativeasaRateofChange 76
vii
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viii Contents
CHAPTER3 Applications of the Derivative 81
3.1 GraphingofFunctions 81
3.2 Maximum/MinimumProblems 86
3.3 RelatedRates 91
3.4 FallingBodies 94
CHAPTER4 The Integral 99
4.0 Introduction 99
4.1 AntiderivativesandIndefiniteIntegrals 99
4.1.1 TheConceptofAntiderivative 99
4.1.2 TheIndefiniteIntegral 100
4.2 Area 103
4.3 SignedArea 111
4.4 TheAreaBetweenTwoCurves 116
4.5 RulesofIntegration 120
4.5.1 LinearProperties 120
4.5.2 Additivity 120
CHAPTER5 Indeterminate Forms 123
5.1 l’Hôpital’sRule 123
5.1.1 Introduction 123
5.1.2 l’Hôpital’sRule 124
5.2 OtherIndeterminateForms 128
5.2.1 Introduction 128
5.2.2 WritingaProductasaQuotient 128
5.2.3 TheUseoftheLogarithm 128
5.2.4 PuttingTermsOveraCommonDenominator 130
5.2.5 OtherAlgebraicManipulations 131
5.3 ImproperIntegrals:AFirstLook 132
5.3.1 Introduction 132
5.3.2 IntegralswithInfiniteIntegrands 133
5.3.3 AnApplicationtoArea 139
5.4 MoreonImproperIntegrals 140
5.4.1 Introduction 140
5.4.2 TheIntegralonanInfiniteInterval 141
5.4.3 SomeApplications 143
Contents ix
CHAPTER6 Transcendental Functions 147
6.0 IntroductoryRemarks 147
6.1 LogarithmBasics 147
6.1.1 ANewApproachtoLogarithms 148
6.1.2 TheLogarithmFunctionandtheDerivative 150
6.2 ExponentialBasics 154
6.2.1 FactsAbouttheExponentialFunction 155
6.2.2 CalculusPropertiesoftheExponential 156
6.2.3 TheNumbere 158
6.3 ExponentialswithArbitraryBases 160
6.3.1 ArbitraryPowers 160
6.3.2 LogarithmswithArbitraryBases 163
6.4 CalculuswithLogsandExponentialstoArbitraryBases 166
6.4.1 DifferentiationandIntegrationoflog x andax 166
a
6.4.2 GraphingofLogarithmicandExponential
Functions 168
6.4.3 LogarithmicDifferentiation 170
6.5 ExponentialGrowthandDecay 172
6.5.1 ADifferentialEquation 173
6.5.2 BacterialGrowth 174
6.5.3 RadioactiveDecay 176
6.5.4 CompoundInterest 178
6.6 InverseTrigonometricFunctions 180
6.6.1 IntroductoryRemarks 180
6.6.2 InverseSineandCosine 180
6.6.3 TheInverseTangentFunction 185
6.6.4 IntegralsinWhichInverseTrigonometricFunctions
Arise 187
6.6.5 OtherInverseTrigonometricFunctions 189
6.6.6 AnExampleInvolvingInverseTrigonometric
Functions 193
CHAPTER7 Methods of Integration 197
7.1 IntegrationbyParts 197
7.2 PartialFractions 202
7.2.1 IntroductoryRemarks 202
7.2.2 ProductsofLinearFactors 203
7.2.3 QuadraticFactors 206