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# M408C - MATHEMATICS 408C Written by Aaron Gladish...

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MATHEMATICS 408C Differential and Integral Calculus M 408C Written by Aaron Gladish P.O. Box 7637, Austin, Texas 78713-7637 http://www.utextension.org ©2009 by The University of Texas at Austin. All rights reserved. Published 2009. Printed in the United States of America.

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Table of Contents Course Introduction . .............................................................................................. 7 The Course . ........................................................................................................................................... 8 How to Succeed in this Course. ......................................................................................................... 8 Coursework and Grades. .................................................................................................................. 12 Textbook. ............................................................................................................................................ 12 1: Precalculus Review. ......................................................................................... 13 Reading Assignment. ......................................................................................................................... 14 Overview. ............................................................................................................................................ 14 Discussion. .......................................................................................................................................... 14 Assignment. ........................................................................................................................................ 22 2: Overview of Calculus; Functions. ................................................................... 23 Reading Assignment. ......................................................................................................................... 24 Overview. ............................................................................................................................................ 24 Discussion. .......................................................................................................................................... 24 Assignment. ........................................................................................................................................ 31 3: Limits of Functions; Graphing Functions; Trigonometry. ............................ 32 Reading Assignment. ......................................................................................................................... 33 Overview. ............................................................................................................................................ 33 Discussion. .......................................................................................................................................... 33 Assignment. ........................................................................................................................................ 41 4: Continuity; Precise Definitions of Limits . ...................................................... 42 Reading Assignment. ......................................................................................................................... 43 2 MATHEMATICS 408C
Overview. ............................................................................................................................................ 43 Discussion. .......................................................................................................................................... 43 Assignment. ........................................................................................................................................ 46 5: The Derivative. .................................................................................................. 47 Reading Assignment. ......................................................................................................................... 48 Overview. ............................................................................................................................................ 48 Discussion. .......................................................................................................................................... 48 Assignment. ........................................................................................................................................ 51 6: Theorems for Calculating Derivatives; Derivatives of Trig Functions . ....... 52 Reading Assignment. ......................................................................................................................... 53 Overview. ............................................................................................................................................ 53 Discussion. .......................................................................................................................................... 53 Assignment. ........................................................................................................................................ 56 7: Graphing . .......................................................................................................... 57 Reading Assignment. ......................................................................................................................... 58 Overview. ............................................................................................................................................ 58 Discussion. .......................................................................................................................................... 58 Assignment. ........................................................................................................................................ 59 8: Second Derivative; Related Rates; More Graphing. ...................................... 60 Reading Assignment. ......................................................................................................................... 61 Overview. ............................................................................................................................................ 61 Discussion. .......................................................................................................................................... 61 Assignment. ........................................................................................................................................ 63 3 MATHEMATICS 408C

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9: Max/Min Problems; Implicit Differentiation. ................................................... 64 Reading Assignment. ......................................................................................................................... 65 Overview. ............................................................................................................................................ 65 Discussion. .......................................................................................................................................... 65 Assignment. ........................................................................................................................................ 67 10: Midterm Examination. .................................................................................... 68 The Content of the Midterm Exam . .............................................................................................. 69 Studying for the Midterm. ................................................................................................................ 70 Applying to Take the Midterm Exam . ........................................................................................... 71 11: The Definite Integral. ...................................................................................... 72 Reading Assignment. ......................................................................................................................... 73 Overview. ............................................................................................................................................ 73 Discussion. .......................................................................................................................................... 73 Assignment. ........................................................................................................................................ 74 12: The Fundamental Theorems . ........................................................................ 75 Reading Assignment. ......................................................................................................................... 76 Overview. ............................................................................................................................................ 76 Discussion. .......................................................................................................................................... 76 Assignment. ........................................................................................................................................ 77 13: Logarithmic and Exponential Functions. ..................................................... 78 Reading Assignment. ......................................................................................................................... 79 Overview. ............................................................................................................................................ 79 Discussion. .......................................................................................................................................... 79 4 MATHEMATICS 408C
Assignment. ........................................................................................................................................ 84 14: Inverse Trig Functions; L’Hôpital’s Rule. ..................................................... 85 Reading Assignment. ......................................................................................................................... 86 Overview. ............................................................................................................................................ 86 Discussion. .......................................................................................................................................... 86 Assignment. ........................................................................................................................................ 88 15: Substitution; Integration by Parts. ................................................................ 89 Reading Assignment. ......................................................................................................................... 90 Overview. ............................................................................................................................................ 90 Review . ................................................................................................................................................ 90 Discussion. .......................................................................................................................................... 91 Assignment. ........................................................................................................................................ 94 16: Partial Fractions . ............................................................................................ 95 Reading Assignment. ......................................................................................................................... 96 Overview. ............................................................................................................................................ 96 Discussion. .......................................................................................................................................... 96 Assignment. ........................................................................................................................................ 98 17: Special Integration Techniques . ................................................................... 99 Reading Assignment. ....................................................................................................................... 100 Overview. .......................................................................................................................................... 100 Discussion. ........................................................................................................................................ 100 Assignment. ...................................................................................................................................... 102 18: Geometric Applications. .............................................................................. 103 5 MATHEMATICS 408C

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6 MATHEMATICS 408C Reading Assignment. ....................................................................................................................... 104 Overview. .......................................................................................................................................... 104 Discussion. ........................................................................................................................................ 104 Assignment. ...................................................................................................................................... 105 19: Physics Applications; Improper Integrals. ................................................. 106 Reading Assignment. ....................................................................................................................... 107 Overview. .......................................................................................................................................... 107 Discussion. ........................................................................................................................................ 107 Assignment. ...................................................................................................................................... 108 Final Exam Content. ....................................................................................................................... 109 Applying to Take the Final Exam. ................................................................................................ 110
COURSE INTRODUCTION

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Course Introduction THE COURSE Welcome to the first half of engineering/science calculus. The textbook for this course is Calculus and Analytic Geometry by Sherman Stein and Anthony Barcellos, 5th edition. The text is the same for the second half of the course, and is currently being used in the classroom courses taught at The University of Texas at Austin.
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M408C - MATHEMATICS 408C Written by Aaron Gladish...

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