Calculus Demystified

416 Pages·2011·3.245 MB·English

by  Steven G. Krantz

Calculus
DeMYSTiFieD®
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Calculus
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StevenG.Krantz
SecondEdition
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Copyright © 2011, 2003 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act
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ISBN: 978-0-07-174364-8
MHID: 0-07-174364-2
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ToArchimedes,PierredeFermat,IsaacNewton,andGottfriedWilhelmvonLeibniz,
thefathersofcalculus.
About the Author
Steven G. Krantz, Ph.D., is a professor of mathematics at Washington Univer-
sity in St. Louis, Missouri. He is an award-winning teacher, and the author of
How to Teach Mathematics, Discrete Mathematics Demystified, and Differential
EquationsDemystified,amongotherbooks.
p¨
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Contents
Preface xi
HowtoUseThisBook xiii
......................................................................................................................................
CHAPTER1 Basics 1
1.0 IntroductoryRemarks 2
1.1 NumberSystems 2
1.2 CoordinatesinOneDimension 4
1.3 CoordinatesinTwoDimensions 6
1.4 TheSlopeofaLineinthePlane 10
1.5 TheEquationofaLine 15
1.6 LociinthePlane 18
1.7 Trigonometry 20
1.8 SetsandFunctions 35
1.8.1 ExamplesofFunctionsofaRealVariable 36
1.8.2 GraphsofFunctions 39
1.8.3 PlottingtheGraphofaFunction 41
1.8.4 CompositionofFunctions 46
1.8.5 TheInverseofaFunction 48
1.9 AFewWordsaboutLogarithmsandExponentials 57
Quiz 60
......................................................................................................................................
CHAPTER2 FoundationsofCalculus 65
2.1 Limits 66
2.1.1 One-SidedLimits 70
2.2 PropertiesofLimits 71
vii
viii CALCULUS DeMYSTiFieD
2.3 Continuity 74
2.4 TheDerivative 76
2.5 RulesforCalculatingDerivatives 83
2.5.1 TheDerivativeofanInverse 90
2.6 TheDerivativeasaRateofChange 91
Quiz 94
......................................................................................................................................
CHAPTER3 ApplicationsoftheDerivative 97
3.1 GraphingofFunctions 98
3.2 Maximum/MinimumProblems 103
3.3 RelatedRates 109
3.4 FallingBodies 113
Quiz 117
......................................................................................................................................
CHAPTER4 TheIntegral 119
4.0 Introduction 120
4.1 AntiderivativesandIndefiniteIntegrals 120
4.1.1 TheConceptofAntiderivative 120
4.1.2 TheIndefiniteIntegral 121
4.2 Area 124
4.3 SignedArea 134
4.4 TheAreaBetweenTwoCurves 140
4.5 RulesofIntegration 144
4.5.1 LinearProperties 144
4.5.2 Additivity 144
Quiz 145
......................................................................................................................................
CHAPTER5 IndeterminateForms 147
5.1 l'Hôpital'sRule 148
5.1.1 Introduction 148
5.1.2 l'Hôpital'sRule 148
5.2 OtherIndeterminateForms 154
5.2.1 Introduction 154
5.2.2 WritingaProductasaQuotient 154
5.2.3 TheUseoftheLogarithm 155
5.2.4 PuttingTermsoveraCommonDenominator 156
5.2.5 OtherAlgebraicManipulations 158
5.3 ImproperIntegrals:AFirstLook 160
5.3.1 Introduction 160
Contents ix
5.3.2 IntegralswithInfiniteIntegrands 160
5.3.3 AnApplicationtoArea 168
5.4 MoreonImproperIntegrals 170
5.4.1 Introduction 170
5.4.2 TheIntegralonanInfiniteInterval 170
5.4.3 SomeApplications 173
Quiz 176
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CHAPTER6 TranscendentalFunctions 179
6.0 IntroductoryRemarks 180
6.1 LogarithmBasics 180
6.1.1 ANewApproachtoLogarithms 181
6.1.2 TheLogarithmFunctionandtheDerivative 183
6.2 ExponentialBasics 188
6.2.1 FactsabouttheExponentialFunction 189
6.2.2 CalculusPropertiesoftheExponential 191
6.2.3 TheNumbere 193
6.3 ExponentialswithArbitraryBases 195
6.3.1 ArbitraryPowers 195
6.3.2 LogarithmswithArbitraryBases 199
6.4 CalculuswithLogsandExponentials
toArbitraryBases 203
6.4.1 DifferentiationandIntegration
oflog xandax 203
a
6.4.2 GraphingofLogarithmic
andExponentialFunctions 206
6.4.3 LogarithmicDifferentiation 208
6.5 ExponentialGrowthandDecay 210
6.5.1 ADifferentialEquation 211
6.5.2 BacterialGrowth 213
6.5.3 RadioactiveDecay 215
6.5.4 CompoundInterest 218
6.6 InverseTrigonometricFunctions 221
6.6.1 IntroductoryRemarks 221
6.6.2 InverseSineandCosine 222
6.6.3 TheInverseTangentFunction 227
6.6.4 IntegralsinWhichInverseTrigonometric
FunctionsArise 229
6.6.5 OtherInverseTrigonometricFunctions 231