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CHAPTER 1 Functions,Limits and Continuity 1

1.1 Mathematical Sign Language 1

1.1.1 Sets 2

1.1.2 Numbers 3

1.1.3 Intervals 4

1.1.4 Implication and Equivalence 5

1.1.5 Inequalities and Numbers 6

1.1.6 Absolute Value of a Number 7

1.1.7 Summation Notation 9

1.1.8 Factorial Notation 10

1.1.9 Binomial Coefficients 10

1.2 Functions 13

1.2.1 Definition of a Function 13

1.2.2 Properties of Functions 18

1.2.3 Inverse and Composite Functions 21

1.2.4 Combining Functions 26

1.2.5 Elementary Functions 26

1.3 Limits 27

1.3.1 The Limit of a Sequence 27

1.3.2 The Limits of a Function 30

1.3.3 One-sided Limits 33

1.3.4 Limits Involving the Infinity Symbol 35

1.3.5 Properties of Limits of Functions 36

1.3.6 Calculating Limits Using Limit Laws 37

1.3.7 Two Important Limit Results 41

1.3.8 Asymptotic Functions and Small o Notation 46

1.4 Continuous and Discontinuous Functions 49

1.4.1 Definitions 49

1.4.2 Building Continuous Functions 52

1.4.3 Theorems on Continuous Functions 55

1.5 Further Results on Limits 58

1.5.1 The Precise Definition of a Limit 58

1.5.2 Limits at Infinity and Infinite Limits 61

1.5.3 Real Numbers and Limits 64

1.5.4 Asymptotes 65

1.5.5 Uniform Continuity 68

1.6 Additional Material 69

1.6.1 Cauchy 69

1.6.2 Heine 70

1.6.3 Weierstrass 70

1.7 Exercises 71

1.7.1 Evaluating Limits 71

1.7.2 Continuous Functions 73

1.7.3 Questions to Guide Your Revision 74

CHAPTER 2 Differential Calculus 75

2.1 The Derivative 75

2.1.1 The Tangent to a Curve 75

2.1.2 Instantaneous Velocity 76

2.1.3 The Definition of a Derivative 77

2.1.4 Notations for the Derivative 80

2.1.5 The Derivative as a Function 80

2.1.6 One-sided Derivatives 83

2.1.7 Continuity of Differentiable Functions 83

2.1.8 Functions with no Derivative 84

2.2 Finding the Derivatives 86

2.2.1 Derivative Laws 86

2.2.2 Derivative of an Inverse Function 89

2.2.3 Differentiating a Composite Function—The Chain Rule 91

2.3 Derivatives of Higher Orders 93

2.4 Implicit Differentiation 96

2.4.1 Implicitly Defined Functions 96

2.4.2 Finding the Derivative of an Implicitly Defined Function 97

2.4.3 Logarithmic Differentiation 100

2.4.4 Functions Defined by Parametric Equations 100

2.5 Related Rates of Change 102

2.6 The Tangent Line Approximation and the Differential 104

2.7 Additional Material 107

2.7.1 Preliminary result needed to prove the Chain Rule 107

2.7.2 Proof of the Chain Rule 108

2.7.3 Leibnitz 109

2.7.4 Newton 109

2.8 Exercises 110

2.8.1 Finding Derivatives 110

2.8.2 Differentials 112

2.8.3 Questions to Guide Your Revision 113

CHAPTER 3 The Mean Value Theorem and Applications of the Derivative 114

3.1 The Mean Value Theorem 114

3.2 L'Hospital's Rule and Indeterminate Forms 122

3.2.1 The Indeterminate Forms 0/0,∞/∞,∞-∞ and ∞·0 122

3.2.2 The Indeterminate Forms 00,∞0, 0∞ and 1∞ 126

3.3 Taylor Series 128

3.4 Monotonic and Concave Functions and Graphs 131

3.4.1 Monotonic Functions 131

3.4.2 Concave Functions 133

3.5 Maximum and Minimum Values of Functions 137

3.5.1 Global Maximum and Global Minimum 143

3.5.2 Curve Sketching 145

3.6 Solving Equations Numerically 149

3.6.1 Decimal Search 149

3.6.2 Newton's Method 151

3.7 Additional Material 53

3.7.1 Fermat 153

3.7.2 L'Hospital 154

3.8 Exercises 155

3.8.1 The Mean Value Theorem 155

3.8.2 L'Hospital's Rules 156

3.8.3 Taylor's Theorem 156

3.8.4 Applications of the Derivative 156

3.8.5 Questions to Guide Your Revision 157

CHAPTER 4 Integral Calculus 158

4.1 The Indefinite Integral 159

4.1.1 Definitions and Properties of Indefinite Integrals 159

4.1.2 Basic Antiderivatives 161

4.1.3 Properties of Indefinite Integrals 163

4.1.4 Integration By Substitution 165

4.1.5 Further Results Using Integration by Substitution 169

4.1.6 Integration by Parts 172

4.1.7 Partial Fractions in Integration 175

4.1.8 Rationalizing Substitutions 182

4.2 Definite Integrals and the Fundamental Theorem of Calculus 183

4.2.1 Introduction 183

4.2.2 The Definite Integral 184

4.2.3 Interpreting ∫b a f(x) dx as an Area 187

4.2.4 Interpreting ∫b a f(t) dt as a Distance 190

4.2.5 Properties of the Definite Integral 191

4.2.6 The Fundamental Theorem of Calculus 192

4.2.7 Integration by Substitution 197

4.2.8 Integration by Parts 199

4.2.9 Numerical Integration 200

4.2.10 Improper Integrals 204

4.3 Applications of the Definite Integral 208

4.3.1 The Area of the Region Between Two Curves 208

4.3.2 Volumes of Solids of Revolution 211

4.3.3 Arc Length 213

4.4 Additional Material 215

4.4.1 Riemann 216

4.4.2 Lagrange 216

4.5 Exercises 217

4.5.1 Indefinite Integrals 217

4.5.2 Definite Integrals 219

4.5.3 Questions to Guide Your Revision 220

Answers 221

Reference Books 227