MATH_137P_Notes - Math 137P Course Notes Physics-Based Calculus University of Waterloo By Francis J Poulin ii Contents Preface Part I 0 INTRODUCTION 1

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Math 137P Course Notes Physics-Based Calculus University of Waterloo By Francis J. Poulin
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Contents Preface vii Part I INTRODUCTION 1 0 Introduction 3 0.1 Mathematical Models 4 0.2 Historical Note 5 0.3 The Fundamentals of Calculus 5 0.4 Numbers 6 0.4.1 Set Theoretic Notation 6 0.4.2 Types of Numbers 6 0.4.3 Intervals 7 0.4.4 Inequalities 7 0.4.5 Distance and the Absolute Value 8 0.4.6 Triangle Inequality 10 0.5 Solving inequalities with absolute values 10 1 Functions 13 1.1 Four Ways to Represent a Function 13 1.1.1 Representations of Functions 14 1.1.2 Piecewise Defined Functions 16 1.1.3 Symmetry 17 1.1.4 Increasing and Decreasing Function 17 1.1.5 Conic Sections 18 1.2 Mathematical Models: A Catalog of Essential Functions 19 1.3 New Functions from Old Functions 21 iii
iv Contents 1.3.1 Transformation of Functions 22 1.3.2 Combinations of functions 23 1.4 Exponential Functions 24 1.4.1 Applications of the Exponential 26 1.4.2 The Number e 26 1.5 Trigonometric Functions 26 1.5.1 Angles 27 1.5.2 The Trigonometric Functions 27 1.5.3 Trigonometric Identities 28 1.5.4 Graphs of the Trigonometric Functions 29 1.6 Hyperbolic functions 29 1.7 Inverse Functions and Logarithms 32 1.7.1 Inverse Functions 33 1.7.2 Logarithmic Functions 35 1.7.3 Natural Logarithms 36 1.7.4 Graph and Growth of the Natural Logarithm 39 1.7.5 Inverse Trigonometric Functions 39 1.7.6 Inverse Hyperbolic Functions 40 2 Limits and Rates of Change 43 2.1 The Tangent and Velocity Problems 43 2.1.1 The Tangent Problem 43 2.2 The Limit of a Function 43 2.3 The Precise Definition of the Limit 49 2.4 Calculating Limits using Limit Laws 56 2.5 Continuity 61 2.5.1 Application of the IVP: Estimating zeros of functions 67 2.6 Limits at Infinity 69 2.6.1 Rational Functions 70 2.6.2 Infinite Limits at Infinity 72 2.7 Derivatives and Rates of Change 73 2.7.1 Tangents 73 2.7.2 Velocity 74 2.7.3 Rates of Change 74 2.7.4 Derivatives 74 2.8 The derivative of a function 75 2.8.1 Other Notation 83 2.8.2 Differentiability 83 2.8.3 Higher Derivatives 84 Part II DIFFERENTIAL CALCULUS 87 3 Differentiation Rules 89 3.1 Derivatives of Polynomials and Exponential Functions 89 3.1.1 Power Functions 89
Contents v 3.1.2 New Derivatives from Old 89 3.1.3 Exponential Functions 90 3.2 Product and Quotient Rules 92 3.3 Trigonometric Functions 94 3.4 Chain Rule 98 3.4.1 Differentiation of Hyperbolic Trigonometric Functions 101 3.5 Implicit Differentiation 102 3.5.1 Derivatives of inverse trigonometric functions 104 3.5.2 An application of the inverse sine function 105 3.6 Derivatives of Exponential Logarithmic Functions 107 3.6.1 Logarithmic differentiation 109 3.6.2 The number e as a limit 111 3.7 Derivatives and Rates of Change 113 3.7.1 Rates of change in the physical sciences 113 3.7.2 Biology 117 3.8 Linear Approximations and Differentiations 120 3.8.1 Differentials 125 3.9 Related Rates 127 3.10 Antiderivatives and Applications to Classical Mechanics 128 3.10.1 Applications to Classical Mechanics 128 3.11 Antiderivatives 131 3.11.1 Oscillations in Potential Well 142 4 Applications of the Derivative 145 4.1 Maximum and Minimum Values 145 4.2 Mean Value Theorem 148 4.2.1 Important results due to the MTV 150 4.3 How Derivatives Affect the Shape of a Graph 152
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