Table Of ContentCalculus
DeMYSTiFieD®
..................................................................................................................................................
http://avaxhome.ws/blogs/ChrisRedfield
DeMYSTiFieD® Series
AccountingDemystified LinearAlgebraDemystified
AdvancedCalculusDemystified LogicDemystified
AdvancedPhysicsDemystified MacroeconomicsDemystified
AdvancedStatisticsDemystified ManagementAccountingDemystified
AlgebraDemystified MathProofsDemystified
AlternativeEnergyDemystified MathWordProblemsDemystified
AnatomyDemystified MATLAB®Demystified
asp.net2.0Demystified MedicalBillingandCodingDemystified
AstronomyDemystified MedicalTerminologyDemystified
AudioDemystified MeteorologyDemystified
BiologyDemystified MicrobiologyDemystified
BiotechnologyDemystified MicroeconomicsDemystified
BusinessCalculusDemystified NanotechnologyDemystified
BusinessMathDemystified NurseManagementDemystified
BusinessStatisticsDemystified OOPDemystified
C++Demystified OptionsDemystified
CalculusDemystified OrganicChemistryDemystified
ChemistryDemystified PersonalComputingDemystified
CircuitAnalysisDemystified PharmacologyDemystified
CollegeAlgebraDemystified PhysicsDemystified
CorporateFinanceDemystified PhysiologyDemystified
DataStructuresDemystified Pre-AlgebraDemystified
DatabasesDemystified PrecalculusDemystified
DifferentialEquationsDemystified ProbabilityDemystified
DigitalElectronicsDemystified ProjectManagementDemystified
DiscreteMathematicsDemystified PsychologyDemystified
EarthScienceDemystified QualityManagementDemystified
ElectricityDemystified QuantumMechanicsDemystified
ElectronicsDemystified RealEstateMathDemystified
EngineeringStatisticsDemystified RelativityDemystified
EnvironmentalScienceDemystified RoboticsDemystified
EverydayMathDemystified SalesManagementDemystified
FertilityDemystified SignalsandSystemsDemystified
FinancialPlanningDemystified SixSigmaDemystified
ForensicsDemystified SpanishDemystified
FrenchDemystified SQLDemystified
GeneticsDemystified StaticsandDynamicsDemystified
GeometryDemystified StatisticsDemystified
GermanDemystified TechnicalAnalysisDemystified
HomeNetworkingDemystified TechnicalMathDemystified
InvestingDemystified TrigonometryDemystified
ItalianDemystified UMLDemystified
JavaDemystified VisualBasic2005Demystified
JavaScriptDemystified VisualC#2005Demystified
LeanSixSigmaDemystified XMLDemystified
Calculus
DeMYSTiFieD®
..................................................................................................................................................
StevenG.Krantz
SecondEdition
NewYork Chicago SanFrancisco Lisbon London Madrid MexicoCity
Milan NewDelhi SanJuan Seoul Singapore Sydney Toronto
Copyright © 2011, 2003 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act
of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system,
without the prior written permission of the publisher.
ISBN: 978-0-07-174364-8
MHID: 0-07-174364-2
The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-174363-1,
MHID: 0-07-174363-4.
All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name,
we use names in an editorial fashion only, and to the benefi t of the trademark owner, with no intention of infringement of the trademark. Where
such designations appear in this book, they have been printed with initial caps.
McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training
programs. To contact a representative please e-mail us at [email protected].
Information in this book has been obtained by The McGraw-Hill Companies, Inc. (‘‘McGraw-Hill”) from sources believed to be reliable.
However, neither McGraw-Hill nor its authors guarantee the accuracy or completeness of any information published herein, and neither
McGraw-Hill nor its authors shall be responsible for any errors, omissions, or damages arising out of use of this information. This work is
published with the understanding that McGraw-Hill and its authors are supplying information but are not attempting to render engineering or
other professional services. If such services are required, the assistance of an appropriate professional should be sought.
TERMS OF USE
This is a copyrighted work and The McGraw-Hill Companies, Inc. (“McGrawHill”) and its licensors reserve all rights in and to the work. Use
of this work is subject to these terms. Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the
work, you may not decompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute,
disseminate, sell, publish or sublicense the work or any part of it without McGraw-Hill’s prior consent. You may use the work for your own
noncommercial and personal use; any other use of the work is strictly prohibited. Your right to use the work may be terminated if you fail to
comply with these terms.
THE WORK IS PROVIDED “AS IS.” McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO
THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING
ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY
DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MER-
CHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. McGraw-Hill and its licensors do not warrant or guarantee that the
functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free. Neither McGraw-Hill nor
its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages
resulting therefrom. McGraw-Hill has no responsibility for the content of any information accessed through the work. Under no circumstances
shall McGraw-Hill and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from
the use of or inability to use the work, even if any of them has been advised of the possibility of such damages. This limitation of liability shall
apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise.
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¨
–
5
5
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
......................................................................................................................................
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
