calculus - Single Variable Calculus Early Transcendentals...

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Single Variable Calculus Early Transcendentals
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This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike License. To view a copy of this license, visit or send a letter to Creative Commons, 543 Howard Street, 5th Floor, San Francisco, California, 94105, USA. If you distribute this work or a derivative, include the history of the document. This text was initially written by David Guichard. The single variable material in chapters 1–9 is a mod- ification and expansion of notes written by Neal Koblitz at the University of Washington, who generously gave permission to use, modify, and distribute his work. New material has been added, and old material has been modified, so some portions now bear little resemblance to the original. The book includes some exercises and examples from Elementary Calculus: An Approach Using Infinitesi- mals , by H. Jerome Keisler, available at under a Creative Commons license. In addition, the chapter on differential equations (in the multivariable version) and the section on numerical integration are largely derived from the corresponding portions of Keisler’s book. Albert Schueller, Barry Balof, and Mike Wills have contributed additional material. This copy of the text was compiled from source at 14:13 on 4/29/2016. I will be glad to receive corrections and suggestions for improvement at [email protected] .
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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
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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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