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

757 Pages·2012·3.35 MB·English

by  Ulrich L 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.
BTU Cottbus, 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:2)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],oronthewebatwww.copyright.com.RequeststothePublisherforpermission
shouldbeaddressedtothePermissionsDepartment,JohnWiley&Sons,Inc.,111RiverStreet,Hoboken,
NJ07030,[phone],fax[phone],oronlineathttp://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
atwww.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:2)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:2)andcos(cid:2) 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
CONTENTS ix
11b.3 LimitsoftheType(II)[lim f(x),wherea6¼0] 332
x!a
11b.4 LimitsofExponentialandLogarithmicFunctions 335
12 ExponentialForm(s)ofaPositiveRealNumberandits
Logarithm(s):Pre-RequisiteforUnderstandingExponential
andLogarithmicFunctions
(WhatmustyouknowtolearnCalculus?) 339
12.1 Introduction 339
12.2 ConceptofLogarithmic 339
12.3 TheLawsofExponent 340
12.4 LawsofExponents(orLawsofIndices) 341
12.5 TwoImportantBases:“10”and“e” 343
12.6 Definition:Logarithm 344
12.7 AdvantagesofCommonLogarithms 346
12.8 ChangeofBase 348
12.9 WhywereLogarithmsInvented? 351
12.10 FindingaCommonLogarithmofa(Positive)Number 351
12.11 Antilogarithm 353
12.12 MethodofCalculationinUsingLogarithm 355
13a ExponentialandLogarithmicFunctionsandTheirDerivatives
(WhatmustyouknowtolearnCalculus?) 359
13a.1 Introduction 359
13a.2 Originofe 360
13a.3 DistinctionBetweenExponentialandPowerFunctions 362
13a.4 TheValueofe 362
13a.5 TheExponentialSeries 364
13a.6 PropertiesofeandThoseofRelatedFunctions 365
13a.7 ComparisonofPropertiesofLogarithm(s)totheBases10ande 369
13a.8 ALittleMoreAboute 371
13a.9 GraphsofExponentialFunction(s) 373
13a.10 GeneralLogarithmicFunction 375
13a.11 DerivativesofExponentialandLogarithmicFunctions 378
13a.12 ExponentialRateofGrowth 383
13a.13 HigherExponentialRatesofGrowth 383
13a.14 AnImportantStandardLimit 385
13a.15 ApplicationsoftheFunctionex:ExponentialGrowthandDecay 390
13b MethodsforComputingLimitsofExponentialand
LogarithmicFunctions 401
13b.1 Introduction 401
13b.2 ReviewofLogarithms 401
13b.3 SomeBasicLimits 403
13b.4 EvaluationofLimitsBasedontheStandardLimit 410
14 InverseTrigonometricFunctionsandTheirDerivatives 417
14.1 Introduction 417
14.2 TrigonometricFunctions(WithRestrictedDomains)and
TheirInverses 420
x CONTENTS
14.3 TheInverseCosineFunction 425
14.4 TheInverseTangentFunction 428
14.5 DefinitionoftheInverseCotangentFunction 431
14.6 FormulafortheDerivativeofInverseSecantFunction 433
14.7 FormulafortheDerivativeofInverseCosecantFunction 436
14.8 ImportantSetsofResultsandtheirApplications 437
14.9 ApplicationofTrigonometricIdentitiesinSimplificationof
FunctionsandEvaluationofDerivativesofFunctionsInvolving
InverseTrigonometricFunctions 441
15a ImplicitFunctionsandTheirDifferentiation 453
15a.1 Introduction 453
15a.2 CloserLookattheDifficultiesInvolved 455
15a.3 TheMethodofLogarithmicDifferentiation 463
15a.4 ProcedureofLogarithmicDifferentiation 464
15b ParametricFunctionsandTheirDifferentiation 473
15b.1 Introduction 473
15b.2 TheDerivativeofaFunctionRepresentedParametrically 477
15b.3 LineofApproachforComputingtheSpeedofaMovingParticle 480
15b.4 Meaningofdy/dxwithReferencetotheCartesian
Formy¼f(x)andParametricFormsx¼f(t),y¼g(t)
oftheFunction 481
15b.5 DerivativeofOneFunctionwithRespecttotheOther 483
16 Differentials“dy”and“dx”:MeaningsandApplications 487
16.1 Introduction 487
16.2 ApplyingDifferentialstoApproximateCalculations 492
16.3 DifferentialsofBasicElementaryFunctions 494
16.4 TwoInterpretationsoftheNotationdy/dx 498
16.5 IntegralsinDifferentialNotation 499
16.6 ToCompute(Approximate)SmallChangesand
SmallErrorsCausedinVariousSituations 503
17 DerivativesandDifferentialsofHigherOrder 511
17.1 Introduction 511
17.2 DerivativesofHigherOrders:ImplicitFunctions 516
17.3 DerivativesofHigherOrders:ParametricFunctions 516
17.4 DerivativesofHigherOrders:ProductofTwoFunctions
(LeibnizFormula) 517
17.5 DifferentialsofHigherOrders 521
17.6 RateofChangeofaFunctionandRelatedRates 523
18 ApplicationsofDerivativesinStudyingMotioninaStraightLine 535
18.1 Introduction 535
18.2 MotioninaStraightLine 535