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(MATH100)calculus.pdf
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MATH 100 Spring 2006-07

Single Variable Calculus

Supplementary Review Notes
Dr. Tony Yee
Department of Mathematics The Hong Kong University of Science and Technology
January 27, 2007

Contents
Table of Contents iii
1 Functions 1
1.1 FundamentalsofFunctions ................................ 1

1.2 LinearFunctionsandLines ................................ 18

1.3 QuadraticFunctionsandParabolas............................ 21

1.4 TrigonometricFunctions .................................. 24

1.5 ExponentialFunctions ................................... 27

1.6 LogarithmicFunctions ................................... 29

2 Limits and Continuity 33
2.1 LimitsofFunctions..................................... 33

2.2 TechniquesforEvaluatingLimits ............................. 39

2.3 ExistenceofLimits ..................................... 49

2.4 Continuity.......................................... 55

2.5 SolvingInequalitiesbyContinuity ............................ 64

3 Differentiation 67
3.1 Derivatives ......................................... 67

3.2 TangentLinesandDi.erentials .............................. 75

3.3 BasicRulesforDi.erentiation............................... 81

3.4 ChainRule ......................................... 93

3.5 TranscendentalFunctions ................................. 97

3.6 HigherDerivatives ..................................... 107

3.7 ImplicitDi.erentiation................................... 109

4 Applications of Differentiation 113
4.1 IncreasingandDecreasingFunctions ........................... 113

4.2 RelativeExtrema ...................................... 114

4.3 ConcavityandIn.ectionPoints .............................. 121

4.4 AbsoluteExtremaonaClosedInterval.......................... 123

4.5 CurveSketching ...................................... 126

4.6 LH.opitalRule ....................................... 129

4.7 Optimization ........................................ 131

CONTENTS
5 Integration 133
5.1 Inde.niteIntegrals ..................................... 134

5.2 IntegrationbySubstitution ................................ 141

5.3 De.niteIntegrals ...................................... 147

5.4 TheFundamentalTheoremofCalculus.......................... 150

5.5 TechniquesofIntegration ................................. 154

5.6 ApplicationsofIntegration ................................ 161

5.7 ImproperIntegrals ..................................... 168

Chapter 1
Functions
This chapter reviews the basic ideas you need to start calculus. The topics include the real number sys-tem, Cartesian coordinates in the plane, straight lines, parabolas, and graphs of elementary transcendental functions.
1.1 Fundamentals of Functions
In mathematics, we often encounter the word functions. When we discuss functions, we cannot avoid some abstract mathematical concepts of which many students .nd hesitant.
We start wi