Introduction to differential calculus : systematic studies with engineering applications for beginners

779 Pages·2012·4.75 MB·English

by  Rohde

INTRODUCTION TO
DIFFERENTIAL
CALCULUS

INTRODUCTION TO
DIFFERENTIAL
CALCULUS
Systematic Studies with
Engineering Applications
for Beginners
Ulrich L. Rohde
Prof. Dr.-Ing.Dr. h.c. mult.
BTUCottbus, Germany
Synergy MicrowaveCorporation
Paterson, NJ, USA
G. C. Jain
(Retd. Scientist)Defense Research andDevelopmentOrganization
Maharashtra,India
Ajay K. Poddar
Chief Scientist,Synergy MicrowaveCorporation,
Paterson, NJ, USA
A. K. Ghosh
Professor, Department ofAerospace Engineering
Indian Institute ofTechnology – Kanpur
Kanpur, India
Copyright(cid:1)2012byJohnWiley&Sons.Allrightsreserved
PublishedbyJohnWiley&Sons,Inc.,Hoboken,NewJersey
PublishedsimultaneouslyinCanada
Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedinanyformor
byanymeans,electronic,mechanical,photocopying,recording,scanning,orotherwise,exceptas
permittedunderSection107or108ofthe1976UnitedStatesCopyrightAct,withouteithertheprior
writtenpermissionofthePublisher,orauthorizationthroughpaymentoftheappropriateper-copyfeeto
theCopyrightClearanceCenter,Inc.,222RosewoodDrive,Danvers,MA01923,[phone],
fax [phone], or on the web at www.copyright.com. Requests to the Publisher for permission
shouldbeaddressedtothePermissionsDepartment,JohnWiley&Sons,Inc.,111RiverStreet,Hoboken,
NJ 07030, [phone], fax [phone], or online at http://www.wiley.com/go/permission.
LimitofLiability/DisclaimerofWarranty:Whilethepublisherandauthorhaveusedtheirbestefforts
inpreparingthisbook,theymakenorepresentationsorwarrantieswithrespecttotheaccuracyor
completenessofthecontentsofthisbookandspecificallydisclaimanyimpliedwarrantiesof
merchantabilityorfitnessforaparticularpurpose.Nowarrantymaybecreatedorextendedbysales
representativesorwrittensalesmaterials.Theadviceandstrategiescontainedhereinmaynotbesuitable
foryoursituation.Youshouldconsultwithaprofessionalwhereappropriate.Neitherthepublisher
norauthorshallbeliableforanylossofprofitoranyothercommercialdamages,includingbutnot
limitedtospecial,incidental,consequential,orotherdamages.
Forgeneralinformationonourotherproductsandservicesorfortechnicalsupport,pleasecontact
ourCustomerCareDepartmentwithintheUnitedStatesat(800)762-2974,outsidetheUnitedStates
at(317)572-3993orfax(317)572-4002.
Wileyalsopublishesitsbooksinavarietyofelectronicformats.Somecontentthatappearsinprint
maynotbeavailableinelectronicformats.FormoreinformationaboutWileyproducts,visitourwebsite
at www.wiley.com.
LibraryofCongressCataloging-in-PublicationData:
Introductiontodifferentialcalculus:systematicstudieswithengineering
applicationsforbeginners/UlrichL.Rohde...[etal.].– 1sted.
p.cm.
Includesbibliographicalreferencesandindex.
ISBN978-1-118-11775-0(hardback)
1. Differentialcalculus–Textbooks.I. Rohde,UlrichL.
QA304.I592012
513’.33–dc23
2011018421
PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1
CONTENTS
Foreword xiii
Preface xvii
Biographies xxv
Introduction xxvii
Acknowledgments xxix
1 FromArithmetictoAlgebra
(WhatmustyouknowtolearnCalculus?) 1
1.1 Introduction 1
1.2 TheSetofWholeNumbers 1
1.3 TheSetofIntegers 1
1.4 TheSetofRationalNumbers 1
1.5 TheSetofIrrationalNumbers 2
1.6 TheSetofRealNumbers 2
1.7 EvenandOddNumbers 3
1.8 Factors 3
1.9 PrimeandCompositeNumbers 3
1.10 CoprimeNumbers 4
1.11 HighestCommonFactor(H.C.F.) 4
1.12 LeastCommonMultiple(L.C.M.) 4
1.13 TheLanguageofAlgebra 5
1.14 AlgebraasaLanguageforThinking 7
1.15 Induction 9
1.16 AnImportantResult:TheNumberofPrimesisInfinite 10
1.17 AlgebraastheShorthandofMathematics 10
1.18 NotationsinAlgebra 11
1.19 ExpressionsandIdentitiesinAlgebra 12
1.20 OperationsInvolvingNegativeNumbers 15
1.21 DivisionbyZero 16
2 TheConceptofaFunction
(WhatmustyouknowtolearnCalculus?) 19
2.1 Introduction 19
2.2 EqualityofOrderedPairs 20
2.3 RelationsandFunctions 20
2.4 Definition 21
v
vi CONTENTS
2.5 Domain,Codomain,Image,andRangeofaFunction 23
2.6 DistinctionBetween“f”and“f(x)” 23
2.7 DependentandIndependentVariables 24
2.8 FunctionsataGlance 24
2.9 ModesofExpressingaFunction 24
2.10 TypesofFunctions 25
2.11 InverseFunctionf(cid:1)1 29
2.12 ComparingSetswithoutCountingtheirElements 32
2.13 TheCardinalNumberofaSet 32
2.14 EquivalentSets(Definition) 33
2.15 FiniteSet(Definition) 33
2.16 InfiniteSet(Definition) 34
2.17 CountableandUncountableSets 36
2.18 CardinalityofCountableandUncountableSets 36
2.19 SecondDefinitionofanInfinitySet 37
2.20 TheNotionofInfinity 37
2.21 AnImportantNoteAbouttheSizeofInfinity 38
2.22 AlgebraofInfinity(1) 38
3 DiscoveryofRealNumbers:ThroughTraditionalAlgebra
(WhatmustyouknowtolearnCalculus?) 41
3.1 Introduction 41
3.2 PrimeandCompositeNumbers 42
3.3 TheSetofRationalNumbers 43
3.4 TheSetofIrrationalNumbers 43
3.5 TheSetofRealNumbers 43
3.6 DefinitionofaRealNumber 44
3.7 GeometricalPictureofRealNumbers 44
3.8 AlgebraicPropertiesofRealNumbers 44
3.9 Inequalities(OrderPropertiesinRealNumbers) 45
3.10 Intervals 46
3.11 PropertiesofAbsoluteValues 51
3.12 NeighborhoodofaPoint 54
3.13 PropertyofDenseness 55
3.14 CompletenessPropertyofRealNumbers 55
3.15 (Modified)DefinitionII(l.u.b.) 60
3.16 (Modified)DefinitionII(g.l.b.) 60
4 FromGeometrytoCoordinateGeometry
(WhatmustyouknowtolearnCalculus?) 63
4.1 Introduction 63
4.2 CoordinateGeometry(orAnalyticGeometry) 64
4.3 TheDistanceFormula 69
4.4 SectionFormula 70
4.5 TheAngleofInclinationofaLine 71
4.6 Solution(s)ofanEquationanditsGraph 76
4.7 EquationsofaLine 83
4.8 ParallelLines 89
CONTENTS vii
4.9 RelationBetweentheSlopesof(Nonvertical)Linesthatare
PerpendiculartoOneAnother 90
4.10 AngleBetweenTwoLines 92
4.11 PolarCoordinateSystem 93
5 TrigonometryandTrigonometricFunctions
(WhatmustyouknowtolearnCalculus?) 97
5.1 Introduction 97
5.2 (Directed)Angles 98
5.3 Rangesofsin(cid:1)andcos(cid:1) 109
5.4 UsefulConceptsandDefinitions 111
5.5 TwoImportantPropertiesofTrigonometricFunctions 114
5.6 GraphsofTrigonometricFunctions 115
5.7 TrigonometricIdentitiesandTrigonometricEquations 115
5.8 RevisionofCertainIdeasinTrigonometry 120
6 MoreAboutFunctions
(WhatmustyouknowtolearnCalculus?) 129
6.1 Introduction 129
6.2 FunctionasaMachine 129
6.3 DomainandRange 130
6.4 DependentandIndependentVariables 130
6.5 TwoSpecialFunctions 132
6.6 CombiningFunctions 132
6.7 RaisingaFunctiontoaPower 137
6.8 CompositionofFunctions 137
6.9 EqualityofFunctions 142
6.10 ImportantObservations 142
6.11 EvenandOddFunctions 143
6.12 IncreasingandDecreasingFunctions 144
6.13 ElementaryandNonelementaryFunctions 147
7a TheConceptofLimitofaFunction
(WhatmustyouknowtolearnCalculus?) 149
7a.1 Introduction 149
7a.2 UsefulNotations 149
7a.3 TheConceptofLimitofaFunction:InformalDiscussion 151
7a.4 IntuitiveMeaningofLimitofaFunction 153
7a.5 TestingtheDefinition[Applicationsofthe«,
dDefinitionofLimit] 163
7a.6 Theorem(B):SubstitutionTheorem 174
7a.7 Theorem(C):SqueezeTheoremorSandwichTheorem 175
7a.8 One-SidedLimits(ExtensiontotheConceptofLimit) 175
7b MethodsforComputingLimitsofAlgebraicFunctions
(WhatmustyouknowtolearnCalculus?) 177
7b.1 Introduction 177
7b.2 MethodsforEvaluatingLimitsofVariousAlgebraicFunctions 178
viii CONTENTS
7b.3 LimitatInfinity 187
7b.4 InfiniteLimits 190
7b.5 Asymptotes 192
8 TheConceptofContinuityofaFunction,andPointsofDiscontinuity
(WhatmustyouknowtolearnCalculus?) 197
8.1 Introduction 197
8.2 DevelopingtheDefinitionofContinuity“AtaPoint” 204
8.3 ClassificationofthePointsofDiscontinuity:TypesofDiscontinuities 214
8.4 CheckingContinuityofFunctionsInvolvingTrigonometric,
Exponential,andLogarithmicFunctions 215
8.5 FromOne-SidedLimittoOne-SidedContinuityanditsApplications 224
8.6 ContinuityonanInterval 224
8.7 PropertiesofContinuousFunctions 225
9 TheIdeaofaDerivativeofaFunction 235
9.1 Introduction 235
9.2 DefinitionoftheDerivativeasaRateFunction 239
9.3 InstantaneousRateofChangeofy[¼f(x)]atx¼x andthe
1
SlopeofitsGraphatx¼x 239
1
9.4 ANotationforIncrement(s) 246
9.5 TheProblemofInstantaneousVelocity 246
9.6 DerivativeofSimpleAlgebraicFunctions 259
9.7 DerivativesofTrigonometricFunctions 263
9.8 DerivativesofExponentialandLogarithmicFunctions 264
9.9 DifferentiabilityandContinuity 264
9.10 PhysicalMeaningofDerivative 270
9.11 SomeInterestingObservations 271
9.12 HistoricalNotes 273
10 AlgebraofDerivatives:RulesforComputingDerivativesof
VariousCombinationsofDifferentiableFunctions 275
10.1 Introduction 275
10.2 RecallingtheOperatorofDifferentiation 277
10.3 TheDerivativeofaCompositeFunction 290
10.4 UsefulnessofTrigonometricIdentitiesinComputingDerivatives 300
10.5 DerivativesofInverseFunctions 302
11a BasicTrigonometricLimitsandTheirApplications
inComputingDerivativesofTrigonometricFunctions 307
11a.1 Introduction 307
11a.2 BasicTrigonometricLimits 308
11a.3 DerivativesofTrigonometricFunctions 314
11b MethodsofComputingLimitsofTrigonometricFunctions 325
11b.1 Introduction 325
11b.2 LimitsoftheType(I) 328