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Calculus II PDF

339 Pages·2016·14.18 MB·english
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Calculus II by Chris Monahan A member of Penguin Random House LLC First American Edition, 2016 Published in the United States by DK Publishing 6081 E. 82nd Street, Indianapolis, Indiana 46250 Copyright © 2016 Dorling Kindersley Limited A Penguin Random House Company 16 17 18 19 10 9 8 7 6 5 4 3 2 1 001-295791-DECEMBER2016 Published in the United States by Dorling Kindersley Limited. IDIOT’S GUIDES and Design are trademarks of Penguin Random House LLC ISBN: 9781465454409 Library of Congress Catalog Card Number: 2016941029 Printed and bound in the United States of America idiotsguides.com Contents Part 1: Review of Pre-Calculus and Calculus I ............................. 1 1 Pre-Calculus Topics Used in Calculus II .............................3 Trigonometry .............................................................................................................................................4 Exponents and Logarithms ................................................................................................................6 Parametric Equations ............................................................................................................................8 Polar Coordinates .................................................................................................................................10 Geometric Sequences and Series .................................................................................................14 Partial Fractions .....................................................................................................................................15 2 Limits, Derivatives, and Basic Integration .......................19 Limits ............................................................................................................................................................20 Derivatives .................................................................................................................................................23 Implicit Differentiation .....................................................................................................................28 Mean Value Theorem ........................................................................................................................29 Relative Extremes and Concavity ..............................................................................................30 Applications of the Derivative ......................................................................................................33 Related Rates ...........................................................................................................................................36 3 Definite and Indefinite Integrals ........................................41 Indefinite Integrals...............................................................................................................................42 u-Substitutions ........................................................................................................................................46 Fundamental Theorem of Calculus ..........................................................................................50 Numerical Approximation ..............................................................................................................54 Part 2: Length, Area, and Volumes ..............................................61 4 Areas and Approximations ..................................................63 Riemann Sums ........................................................................................................................................64 From Numerical Approximations to the True Area ...............................................................................................................................................76 Average Value of a Function ..........................................................................................................80 Area Between Two Curves .............................................................................................................81 Simpson’s Rule ........................................................................................................................................86 5 Volumes and Areas of Solids of Revolutions ..................91 Volumes of Solids with Defined Cross Sections ...............................................................92 Disks and Washers ................................................................................................................................97 Cylindrical Shell Method ..............................................................................................................106 Arc Length ..............................................................................................................................................110 Surface Area ...........................................................................................................................................113 iv Calculus II Part 3: More Definite and Indefinite Integrals ...........................117 6 More Integration Techniques .............................................119 Integration by Parts ...........................................................................................................................120 Polynomials and Transcendentals ............................................................................................123 Two Transcendentals ........................................................................................................................129 7 Integration with Trigonometric Functions ....................133 Trigonometric Substitutions ........................................................................................................134 Case I: a2+x2 ..................................................................................................................................134 Case II: x2−a2 ................................................................................................................................136 Case III: a2−x2 ..............................................................................................................................138 Integrals of the Form sinn(x) cosm(x) (When Either m or n Is Odd) ...................139 Case I: Both m and n Are Odd.....................................................................................................139 Case II: Both m and n Are Even .................................................................................................140 Case III: Either m or n Is Odd ....................................................................................................141 Integrals with Integrands of the Form tann(x) secm(x) (m Is Even) ......................142 Integrals with Integrands of the Form tann(x) secm(x) (m Is Odd, n Is Even) .....................................................................................................................143 8 Integration with Fractions .................................................147 Completing the Square ...................................................................................................................148 Integration by Partial Fractions .................................................................................................152 Nonrepeating Linear Factors ........................................................................................................152 Repeated Linear Factors .................................................................................................................154 Irreducible Quadratic Factors ......................................................................................................155 Part 4: The Infinite Series and More ..........................................159 9 To Infinity and Beyond ........................................................161 Improper Integrals ..............................................................................................................................162 Infinite Limits of Integration ......................................................................................................162 Discontinuities in the Integrand................................................................................................165 Comparison Test for Improper Integrals .............................................................................168 10 Parametric Equations ..........................................................171 First and Second Derivatives of Parametric Curves .....................................................172 Arc Length of a Parametric Curve ..........................................................................................178 11 Polar Coordinates .................................................................181 Slope of the Tangent Line .............................................................................................................182 Length of an Arc of a Polar Curve ...........................................................................................187 Area Under a Curve ..........................................................................................................................189 Contents v 12 Introduction to Vectors ......................................................195 Scalars and Vectors ............................................................................................................................196 Displacement, Velocity, and Acceleration ..........................................................................200 13 Differential Equations ........................................................205 Separable Differential Equations ..............................................................................................206 Linear Approximations....................................................................................................................212 Euler’s Method......................................................................................................................................214 Slope Fields .............................................................................................................................................217 First Order Linear Differential Equations ..........................................................................222 14 Infinite Sequences ...............................................................227 Convergence and Divergence of Sequences .......................................................................228 Squeeze Theorem ...............................................................................................................................231 Increasing, Decreasing, and Monotonic Sequences ......................................................233 15 Infinite Series ........................................................................237 Infinite Geometric Series ..............................................................................................................238 Tests of Convergence........................................................................................................................239 Alternating Series ...............................................................................................................................244 Estimating the Sum of Alternating Series ...........................................................................247 16 Power Series ........................................................................249 Power Series ...........................................................................................................................................250 MacLaurin Series ................................................................................................................................252 Taylor Series ..........................................................................................................................................258 Error Estimates for the MacLaurin and Taylor Series.......................................................................................................................................260 17 Calculus II Final Exam .......................................................263 Chapter 1 .................................................................................................................................................264 Chapter 2 .................................................................................................................................................264 Chapter 3 ..................................................................................................................................................265 Chapter 4 ..................................................................................................................................................266 Chapter 5 ..................................................................................................................................................266 Chapter 6 ..................................................................................................................................................267 Chapter 7 ..................................................................................................................................................267 Chapter 8 ..................................................................................................................................................268 Chapter 9 ..................................................................................................................................................268 Chapter 10 ................................................................................................................................................268 Chapter 11 ................................................................................................................................................269 Chapter 12 ...............................................................................................................................................269 Chapter 13 ................................................................................................................................................270 Chapter 14 ................................................................................................................................................270 Chapter 15................................................................................................................................................271 Chapter 16 ................................................................................................................................................271 Solutions ...................................................................................................................................................272 Chapter 1 .............................................................................................................................................272 Chapter 2 .............................................................................................................................................272 Chapter 3 .............................................................................................................................................273 Chapter 4 .............................................................................................................................................273 Chapter 5 .............................................................................................................................................274 Chapter 6 .............................................................................................................................................274 Chapter 7 .............................................................................................................................................274 Chapter 8 .............................................................................................................................................275 Chapter 9 .............................................................................................................................................275 Chapter 10 ...........................................................................................................................................275 Chapter 11 ...........................................................................................................................................275 Chapter 12 ...........................................................................................................................................276 Chapter 13 ...........................................................................................................................................276 Chapter 14 ...........................................................................................................................................277 Chapter 15 ...........................................................................................................................................277 Chapter 16 ...........................................................................................................................................277 Appendixes A Solutions to “You’ve Got Problems” ..............................279 B Integration Practice Problems and Solutions ..............297 C Glossary ................................................................................307 Index .......................................................................................313 Introduction If you look up the word calculus in the dictionary, one of the first definitions given is that it is a hard object, like a kidney stone. People used to joke that it was no wonder that the subject was so difficult to pass. This book is intended to “soften the calculus” so you are able to better understand Calculus II. I have tried to avoid using the technical language of mathematics whenever I could. When I abso- lutely had to use more technical language, I have translated those terms into everyday language you can comprehend. Also, you’ll find plenty of examples in each chapter to help you understand the solutions to each problem. These examples include easy-to-understand explanations as well as the requisite mathematical notations. However, I warn you not to be a “mathematical voyeur,” one who likes to watch the math being done but who does not do it himself. As you are working through this book, be sure to have paper, pencil, and your graphing calculator by your side—and use them. Read the example, do the problem yourself, then look at my solution. You might want to hide the example solutions before you do the problem so you cannot cheat. People who have taken calculus will tell you that the first step in the problem is calculus, but the rest of the steps needed to solve the problem are algebra. Be careful as you work. Not every example problem included is easy. If I took that approach to teaching Calculus II, I would be giving you a false sense of what to expect. Making things too easy also would deny you the satisfaction of taking on a challenge and succeeding. Finally, I used to tell my students to have fun when I handed them their exams. Your first reac- tion to that might be similar to their reactions—“Wise a**!” However, I hope that after reading this book, you will come to understand—as my students did—learning Calculus II is a chance to show yourself what you can do when you set your mind to it and have fun with this book. You never know what you can do until you try. How This Book Is Organized This book is presented in four parts: In Part 1, Review of Pre-Calculus and Calculus I, you review a few key topics that you covered in Pre-Calculus and come into play in Calculus II. You’ll also do a quick review of limits and derivatives from Calculus I. In Part 2, Length, Area, and Volumes, you learn about the applications of integration to compute one-, two-, and three-dimensional measurements. In Part 3, More Definite and Indefinite Integrals, you study integration techniques beyond the notion of the simple antiderivative. viii Calculus II In Part 4, The Infinite Series and More, you look at topics that extend the notions of the Rectangular Coordinate System as well as topics to extend what you have learned about integra- tion. Finally, you end with a study of topics that are very applicable to how your calculator does math. I’ve also included a final exam as the last chapter so you can assess your understanding of what you’ve learned. At the very end of the book, I’ve included solutions to the You’ve Got Problems sidebars through- out the book. Also, you’ll find an appendix that provides you more integration practice as well as a glossary of helpful terms. Extras Throughout the book, you’ll see helpful sidebars that reinforce what you’re learning. Here’s what to look for: CRITICAL POINT These sidebars are meant to draw your attention to key issues in calculus and key people who have been very influential in the development of the topic. DEFINITION These sidebars help you break down the terms used in calculus so you can better understand what is presented to you. BE AWARE Although I warn you about common pitfalls and dangers throughout the book, these sidebars deserve special attention. They are also meant to draw a special light on critical errors students often make. YOU’VE GOT PROBLEMS After I discuss a topic, I explain how to work out a certain type of problem and then you get to try it on your own. These problems are very similar to those I walk you through in the chapters, but now it’s your turn to shine. Even though all the answers appear in Appendix A, you should only look there to check your work. PART 1 Review of Pre-Calculus and Calculus I The first part of this book is intended to refresh the work you did in Calculus I and review some topics you might have studied in your Pre-Calculus class that are applied in Calculus II. In addition, we review trigonometry and logarithms—topics you covered in Calculus I. I also include material on parametric equations, the polar coordinate system, and partial fractions. CHAPTER 1 Pre-Calculus Topics Used in Calculus II Pre-Calculus is usually the last class a student takes before In This Chapter entering Calculus I. The topics taught in Pre-Calculus vary • Reviewing key from state to state, school to school, and sometimes, from trigonometric relationships teacher to teacher. There are a few topics which often form used in calculus the mainstay of the course simply because they are key build- • ing blocks to one’s ability to answer topics in Calculus (as Euler’s number and his opposed to “understanding” calculus). The topics included in logarithms this chapter are the ones my students needed to review as we • The other coordinate were doing the calculus. system for the plane • Understanding x does not always have to be the independent variable • Learning still more about fractions

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