# calculus - Single Variable Calculus Early Transcendentals...

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Single Variable Calculus Early Transcendentals

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For Kathleen, without whose encouragement this book would not have been written.

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Contents 1 Analytic Geometry 13 1.1 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2 Distance Between Two Points; Circles . . . . . . . . . . . . . . . 19 1.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.4 Shifts and Dilations . . . . . . . . . . . . . . . . . . . . . . . . 25 2 Instantaneous Rate of Change: The Derivative 29 2.1 The slope of a function . . . . . . . . . . . . . . . . . . . . . . 29 2.2 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.3 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.4 The Derivative Function . . . . . . . . . . . . . . . . . . . . . 46 2.5 Adjectives For Functions . . . . . . . . . . . . . . . . . . . . . 51 5

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6 Contents 3 Rules for Finding Derivatives 55 3.1 The Power Rule . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2 Linearity of the Derivative . . . . . . . . . . . . . . . . . . . . 58 3.3 The Product Rule . . . . . . . . . . . . . . . . . . . . . . . . 60 3.4 The Quotient Rule . . . . . . . . . . . . . . . . . . . . . . . . 62 3.5 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4 Transcendental Functions 71 4.1 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . 71 4.2 The Derivative of sin x . . . . . . . . . . . . . . . . . . . . . . 74 4.3 A hard limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.4 The Derivative of sin x , continued . . . . . . . . . . . . . . . . . 78 4.5 Derivatives of the Trigonometric Functions . . . . . . . . . . . . 79 4.6 Exponential and Logarithmic functions . . . . . . . . . . . . . . 80 4.7 Derivatives of the exponential and logarithmic functions . . . . . 82 4.8 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . 87 4.9 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . 92 4.10 Limits revisited . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.11 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . 100 5 Curve Sketching 105 5.1 Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . 105 5.2 The first derivative test . . . . . . . . . . . . . . . . . . . . . 109 5.3 The second derivative test . . . . . . . . . . . . . . . . . . . 110 5.4 Concavity and inflection points . . . . . . . . . . . . . . . . . 111 5.5 Asymptotes and Other Things to Look For . . . . . . . . . . . 113
Contents 7 6 Applications of the Derivative 117 6.1 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.2 Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.3 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . 137 6.4 Linear Approximations . . . . . . . . . . . . . . . . . . . . . 141 6.5 The Mean Value Theorem . . . . . . . . . . . . . . . . . . . 143 7 Integration 147 7.1 Two examples . . . . . . . . . . . . . . . . . . . . . . . . . 147 7.2 The Fundamental Theorem of Calculus . . . . . . . . . . . . . 151 7.3 Some Properties of Integrals . . . . . . . . . . . . . . . . . . 158 8 Techniques of Integration 163 8.1 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . 164 8.2 Powers of sine and cosine . . . . . . . . . . . . . . . . . . . . 169 8.3 Trigonometric Substitutions . . . . . . . . . . . . . . . . . . . 171 8.4 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . 174 8.5 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . 178 8.6 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . 182 8.7 Additional exercises . . . . . . . . . . . . . . . . . . . . . . . 187

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