Contents 1 Functions 1.1 The Concept of a Function . . . . . . . . . . . . . . 1.2 Trigonometric Functions . . . . . . . . . . . . . . . 1.3 Inverse Trigonometric Functions . . . . . . . . . . . 1.4 Logarithmic, Exponential and Hyperbolic Functions 2 Limits and Continuity 2.1 Intuitive treatment and definitions . . 2.1.1 Introductory Examples . . . . . 2.1.2 Limit: Formal Definitions . . . 2.1.3 Continuity: Formal Definitions 2.1.4 Continuity Examples . . . . . . 2.2 Linear Function Approximations . . . . 2.3 Limits and Sequences . . . . . . . . . . 2.4 Properties of Continuous Functions . . 2.5 Limits and Infinity . . . . . . . . . . . 3 Differentiation 3.1 The Derivative . . . . . . . . . . . 3.2 The Chain Rule . . . . . . . . . . . 3.3 Differentiation of Inverse Functions 3.4 Implicit Differentiation . . . . . . . 3.5 Higher Order Derivatives . . . . . .
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35 35 35 41 43 48 61 72 84 94
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99 99 111 118 130 137
4 Applications of Differentiation 146 4.1 Mathematical Applications . . . . . . . . . . . . . . . . . . . . 146 4.2 Antidifferentiation . . . . . . . . . . . . . . . . . . . . . . . . 157 4.3 Linear First Order Differential Equations . . . . . . . . . . . . 164 i