Calculus, once again

194 Pages·2007·1.43 MB·English

by  Santos D.

CalcCualagoluanacsi,cCgunleaaoliunacns,cCguleaaoliuCnacnsa,cCgulleaacoliuCunacnsla,cCgululeaacosliu,Cunacnsla,cCgululeaacosliu,Cunacnsla,cgululeacosiu,Cunansla,cglueacosi,Cunanlacgolueancosi,cCunanleacgolueancsi,cCuanleagoluanacsi,cCguanleaagoliuanacnsi,cCguanleaagoliuanacnsi,cCgunleaaoliunacns,cCguleaaoliuCnacnsa,cCgulleaacoliuCunacnsla,cCgululeaacosliu,Cunacnsla,cCgululeaacosliu,Cunacnsla,cgululeacosiu,Cunansla,cglueacosi,Cunanlacgolueancosi,cCunanleacgolueancsi,cCuanleagoluanacsi,cCguanleaagoliuanacnsi,cCgunleaaoliunacns,cCguleaaoliunacns,cCguleaaoliuCnacnsa,cCgulleaacoliuCunacnsla,cCgululeaacosliu,Cunacnsla,cgululeacosiu,Cunansla,cglueacosi,Cunanlacgolueancosi,cCunanleacgolueancsi,cCuanleagoluancsi,cCuanleagoluanacsi,cCguanleaagoliuanacnsi,cCgunleaaoliunacns,cCguleaaoliuCnacnsa,cCgulleaacoliuCunacnsla,cCgululeaacosliu,Cunacnsla,cCgululeaacosliu,Cunacnsla,cgululeacosiu,Cunansla,cglueacosi,Cunanlacgolueancosi,cCunanleacgolueancsi,cCuanleagoluanacsi,cCguanleaagoliuanacnsi,cCguanleaagoliuanacnsi,cCgunleaaoliunacns,cCguleaaoliuCnacnsa,cCgulleaacoliuCunacnsla,cCgululeaacosliu,Cunacnsla,cgululeacosiu,Cunansla,cglueacosi,Cunanlacgolueancosi,cCunanleacgolueancosi,cCunanleacgolueancsi,cCuanleagoluanacsi,cCguanleaagoliuanacnsi,cCgunleaaoliunacns,cCguleaaoliunacns,cCguleaaoliuCnacnsa,cCgulleaacoliuCunacnsla,cCgululeaacosliu,Cunacnsla,cgululeacosiu,Cunansla,cglueacosi,Cunanlacgolueancosi,cCunanleacgolueancsi,cCuanleagoluanacsi,cCguanleaagoliuanacnsi,cCguanleaagoliuanacnsi,cCgunleaaoliunacns,cCguleaaoliuCnacnsa,cCgulleaacoliuCunacnsla,cCgululeaacosliu,Cunacnsla,cCgululeaacosliu,Cunacnsla,cgululeacosiu,Cunansla,cglueacosi,Cunanlacgolueancosi,cCunanleacgolueancsi,cCuanleagoluanacsi,cCgunleaaoliunacns,cCguleaaoliunacns,cguleaoiunans,cgeaoiCnanacgleaciCuanlagluacsi,Cunlalucs,Culalucs,ulouns,ceonce
DavidA.SANTOS
[email protected]
July 17, 2008 Version
ii
Copyright©2007DavidAnthonySANTOS.Permissionisgrantedtocopy,distributeand/ormodifythisdocu-
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Que a quien robe este libro, o lo tome prestado y no lo devuelva, se le convierta en una serpiente en las
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Contents
GNUFreeDocumentationLicense iii
1.APPLICABILITYANDDEFINITIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
2.VERBATIMCOPYING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
3.COPYINGINQUANTITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
4.MODIFICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
5.COMBININGDOCUMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
6.COLLECTIONSOFDOCUMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
7.AGGREGATIONWITHINDEPENDENTWORKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
8.TRANSLATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
9.TERMINATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
10.FUTUREREVISIONSOFTHISLICENSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Preface viii
1 Preliminaries 1
1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 NumericalFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 InjectiveandSurjectiveFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 AlgebraofFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.3 InverseImage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.4 InverseFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Countability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 GroupsandFields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 AdditionandMultiplicationinR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.6 OrderAxioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.6.1 AbsoluteValue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.7 ClassicalInequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.7.1 TriangleInequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.7.2 Bernoulli’sInequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.7.3 RearrangementInequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.7.4 ArithmeticMean-GeometricMeanInequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.7.5 Cauchy-Bunyakovsky-SchwarzInequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.7.6 Minkowski’sInequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.8 CompletenessAxiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.8.1 GreatestIntegerFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
v
vi CONTENTS
2 TopologyofR 34
2.1 Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2 DenseSets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3 OpenandClosedSets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4 Interior,Boundary,andClosureofaSet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.5 ConnectedSets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.6 CompactSets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.7 R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.8 LebesgueMeasure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.9 TheCantorSet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3 Sequences 47
3.1 LimitofaSequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 ConvergenceofSequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3 ClassicalLimitsofSequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.4 AveragesofSequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.5 OrdersofInfinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.6 CauchySequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.7 Topologyofsequences.LimitSuperiorandLimitInferior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4 Series 70
4.1 ConvergenceandDivergenceofSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2 ConvergenceandDivergenceofSeriesofPositiveTerms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3 SummationbyParts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.4 AbsoluteConvergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5 RealFunctionsofOneRealVariable 85
5.1 LimitsofFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.3 AlgebraicOperationswithContinuousFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.4 MonotonicityandInverseImage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.5 ConvexFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.5.1 GraphsofFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.6 ClassicalFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.6.1 AffineFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.6.2 QuadraticFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.6.3 PolynomialFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.6.4 ExponentialFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.6.5 LogarithmicFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.6.6 TrigonometricFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.6.7 InverseTrigonometricFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.7 ContinuityofSomeStandardFunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.7.1 ContinuityPolynomialFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
CONTENTS vii
5.7.2 ContinuityoftheExponentialandLogarithmicFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.7.3 ContinuityofthePowerFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.8 InequalitiesObtainedbyContinuityArguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.9 IntermediateValueProperty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.10 VariationofaFunctionandUniformContinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.11 ClassicalLimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6 DifferentiableFunctions 113
6.1 DerivativeataPoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2 DifferentiationRules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.3 Rolle’sTheoremandtheMeanValueTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.4 Extrema. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.5 ConvexFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.6 InequalitiesObtainedThroughDifferentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.7 AsymptoticPreponderance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.8 AsymptoticEquivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.9 AsymptoticExpansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7 IntegrableFunctions 143
7.1 TheAreaProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.2 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.3 Riemann-StieltjesIntegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.4 Euler’sSummationFormula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
8 SequencesandSeriesofFunctions 160
8.1 PointwiseConvergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
8.2 UniformConvergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
8.3 IntegralsandDerivativesofSequencesofFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.4 PowerSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.5 MaclaurinExpansionstoknowbyinspection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
8.6 ComparisonTests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
8.7 TaylorPolynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
8.8 Abel’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
A AnswersandHints 163
AnswersandHints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Preface
FormanyyearsIhavebeenluckyenoughtohavestudentsaskformore: morechallengingproblems,moreilluminating
proofstodifferenttheorems,adeeperlookatvarioustopics,etc. TothosestudentsInormallyrecommendthebooksin
thebibliography. Someofthesamestudentshavecomplainedofnotfindingthebooksorwantingtobuythem,butbeing
impecunious,notbeingabletoaffordtobuythem.HenceIhavedecidedtomakethiscompilation.
Herewetakeasemi-rigoroustourthroughCalculus. Wedon’tconstructtherealnumbers,butweexaminecloserthe
realnumberaxiomsandsomeofthebasictheoremsofCalculus. WealsoconsidersomeOlympiad-levelproblemswhose
solutioncanbeobtainedthroughCalculus.
Thereaderisassumedtobefamiliarwithproofsusingmathematicalinduction,proofsbycontradiction,andtheme-
chanicsofdifferentiationandintegration.
DavidA.SANTOS
[email protected]
viii
Chapter 1
Preliminaries
Whybother? Wewill usethe languageofset theory throughout these notes. There arevariouselementary
resultsthatpopupinlaterproofs,amongthem,theDeMorganLawsandtheMonotonicityReversingofCom-
plementationRule.
The conceptof a functionlies at thecoreofmathematics. Wewillgive abriefoverview hereof some basic
propertiesoffunctions.
1.1 Sets
Thissectioncontainssomeofthesetnotationtobeusedthroughoutthesenotes. Theone-directionalarrow reads
=⇒
“implies”andthetwo-directionalarrow reads“ifandonlyif.”
⇐⇒
1Definition Wewillacceptthenotionofsetasaprimitivenotion,thatis,anotionthatcannotbedefinedintermsofmore
elementarynotions. Byasetwewillunderstandawell-definedcollectionofobjects,whichwewillcalltheelementsofthe
set.IftheelementxbelongstothesetSwewillwritex S,andinthecontrarycasewewillwritex S.1 Thecardinalityof
∈ 6∈
asetisthenumberofelementsthesethas. Itcaneitherbefiniteorinfinite. WewilldenotethecardinalityofthesetS by
card(S).
+
Somesetsareusedsooftenthatmeritspecialnotation.Wewilldenoteby
N {0,1,2,3,...}
=
thesetofnaturalnumbers,by
Z {..., 3, 2, 1,0,1,2,3,...}2
= − − −
byQthesetofrationalnumbers3,byRtherealnumbers,andbyCthesetofcomplexnumbers.Wewilloccasionallyalsouse
αZ {..., 3α, 2α, α,0,α,2α,3α,...},etc.
= − − −
Wewillalsodenotetheemptyset,thatis,thesethavingnoelementsby∅.
2Definition TheunionoftwosetsAandB istheset
A B {x:(x A)or(x B)}.
∪ = ∈ ∈
Thisisread“AunionB.”Seefigure1.1.TheintersectionoftwosetsAandB is
A B {x:(x A)and(x B)}.
∩ = ∈ ∈
1GeorgCantor(1845-1918),thecreatorofsettheory,said“Asetisanycollectionintoawholeofdefinite,distinguishableobjects,calledelements,ofour
intuitionorthought.”
2ZfortheGermanwordZählenmeaning“integer.”
3Qfor“quotients.”
1
Sets
A B A B A B
Figure1.1:A B Figure1.2:A B Figure1.3:A\B
∪ ∩
Thisisread“AintersectionB.”Seefigure1.2.ThesetdifferenceoftwosetsAandB is
A\B {x:(x A)and(x B)}.
= ∈ 6∈
Thisisread“AsetminusB.”Seefigure1.3.
3Definition TwosetsAandB aredisjointifA B ∅.
∩ =
4Example WriteA B asthedisjointunionofthreesets.
∪
Solution: Observethat
A B (A\B) (A B) (B\A),
∪ = ∪ ∩ ∪
andthatthesetsonthedextralsidearedisjoint.
5Definition AsubsetB ofasetAisasubcollectionofA,andwedenotethisbyBjA.4 Thismeansthatx B x A.
∈ =⇒ ∈
+
∅andAarealwayssubsetsofanysetA.
Observethat
A B (A B) and (B A).
= ⇐⇒ ⊆ ⊆
Weusethisobservationonthenexttheorem.
6THEOREM(DeMorganLaws) LetA,B,C besets.Then
A\(B C) (A\B) (A\C), A\(B C) (A\B) (A\C).
∩ = ∪ ∪ = ∩
Proof: Wehave
x A\(B C) x A and x (B or C)
∈ ∪ ⇐⇒ ∈ 6∈
(x A) and ((x B) and (x C))
⇐⇒ ∈ 6∈ 6∈
(x A and x B) and (x A and x C)
⇐⇒ ∈ 6∈ ∈ 6∈
(x A\B) and (x A\C)
⇐⇒ ∈ ∈
x (A\B) (A\C).
⇐⇒ ∈ ∩
Also,
x A\(B C) x A and x (B and C)
∈ ∩ ⇐⇒ ∈ 6∈
(x A) and ((x B) or (x C))
⇐⇒ ∈ 6∈ 6∈
(x A and x B) or (x A and x C)
⇐⇒ ∈ 6∈ ∈ 6∈
(x A\B) or (x A\C)
⇐⇒ ∈ ∈
x (A\B) (A\C)
⇐⇒ ∈ ∪
q
4Thereseemsnottobeanagreementherebyauthors.Someusethenotation or insteadofj.Someseeinthenotation theexclusionofequality.
⊂ ⊆ ⊂
Inthesenotes,wewillalwaysusethenotationj,andifwewishedtoexcludeequalitywewillwrite .
á
2