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Contents
1 Functions 2
The Concept of a Function . . . . . . . . . . . . . . . . . . . . 2
Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . 12
Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . 19
Logarithmic, Exponential and Hyperbolic Functions . . . . . . 26
2 Limits and Continuity 35
Intuitive treatment and definitions . . . . . . . . . . . . . . . 35
Introductory Examples . . . . . . . . . . . . . . . . . . 35
Limit: Formal Definitions . . . . . . . . . . . . . . . . 41
Continuity: Formal Definitions . . . . . . . . . . . . . 43
Continuity Examples . . . . . . . . . . . . . . . . . . . 48
Linear Function Approximations . . . . . . . . . . . . . . . . . 61
Limits and Sequences . . . . . . . . . . . . . . . . . . . . . . . 72
Properties of Continuous Functions . . . . . . . . . . . . . . . 84
Limits and Infinity . . . . . . . . . . . . . . . . . . . . . . . . 94
3 Differentiation 99
The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 99
The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Differentiation of Inverse Functions . . . . . . . . . . . . . . . 118
Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . 130
Higher Order Derivatives . . . . . . . . . . . . . . . . . . . . . 137
4 Applications of Differentiation 146
Mathematical Applications . . . . . . . . . . . . . . . . . . . . 146
Antidifferentiation . . . . . . . . . . . . . . . . . . . . . . . . 157
Linear First Order Differential Equations . . . . . . . . . . . . 164
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ii CONTENTS
Linear Second Order Homogeneous Differential Equations . . . 169
Linear Non-Homogeneous Second Order Differential Equations 179
5 The Definite Integral 183
Area Approximation . . . . . . . . . . . . . . . .