multivariable_00_intro

multivariable_00_intro - Calculus Early Transcendentals...

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Unformatted text preview: Calculus Early Transcendentals This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ 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 19 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 http://www.math.wisc.edu/~keisler/calc.html under a Creative Commons license. In addition, the chapter on differential equations is largely derived from the correspond- ing chapter in Keislers book. Albert Schueller, Barry Balof, and Mike Wills have contributed additional material. This copy of the text was compiled from source at 11:16 on 9/22/2011. I will be glad to receive corrections and suggestions for improvement at guichard@whitman.edu . For Kathleen, without whose encouragement this book would not have been written. Contents 1 Analytic Geometry 1 1.1 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Distance Between Two Points; Circles . . . . . . . . . . . . . . . . 7 1.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Shifts and Dilations . . . . . . . . . . . . . . . . . . . . . . . . 13 2 Instantaneous Rate of Change: The Derivative 17 2.1 The slope of a function . . . . . . . . . . . . . . . . . . . . . . 17 2.2 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4 The Derivative Function . . . . . . . . . . . . . . . . . . . . . 34 2.5 Adjectives For Functions . . . . . . . . . . . . . . . . . . . . . 39 v vi Contents 3 Rules for Finding Derivatives 43 3.1 The Power Rule . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2 Linearity of the Derivative . . . . . . . . . . . . . . . . . . . . 46 3.3 The Product Rule . . . . . . . . . . . . . . . . . . . . . . . . 48 3.4 The Quotient Rule . . . . . . . . . . . . . . . . . . . . . . . . 50 3.5 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4 Transcendental Functions 59 4.1 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . 59 4.2 The Derivative of sin x . . . . . . . . . . . . . . . . . . . . . . 62 4.3 A hard limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.4 The Derivative of sin x , continued . . . . . . . . . . . . . . . . . 65 4.5 Derivatives of the Trigonometric Functions . . . . . . . . . . . . 66Derivatives of the Trigonometric Functions ....
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multivariable_00_intro - Calculus Early Transcendentals...

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