TLFeBOOK WHAT READERS ARE SAYIN6 "I wish I had had this book when I needed it most, which was during my pre-med classes. It could have also been a great tool for me in a few medical school courses." Or. Kellie Aosley8 Recent Hedical school &a&ate "CALCULUS FOR THE UTTERLY CONFUSED has proven to be a wonderful review enabling me to move forward in application of calculus and advanced topics in mathematics. I found it easy to use and great as a reference for those darker aspects of calculus. I' Aaron Ladeville, Ekyiheeriky Student 'I1 am so thankful for CALCULUS FOR THE UTTERLY CONFUSED! I started out Clueless but ended with an All' Erika Dickstein8 0usihess school Student "As a non-traditional student one thing I have learned is the importance of material supplementary to texts. Especially in calculus it helps to have a second source, especially one as lucid and fun to read as CALCULUS FOR THE UTTERtY CONFUSED. Anyone, whether you are a math weenie or not, will get something out of this book. With this book, your chances of survival in the calculus jungle are greatly increased.'I Brad &3~ker,Physics Student Other books in the Utterly Conhrsed Series include: Financial Planning for the Utterly Confrcsed, Fifth Edition Job Hunting for the Utterly Confrcred Physics for the Utterly Confrred CALCULUS FOR THE UTTERLY CONFUSED Robert M. Oman Daniel M. Oman McGraw -Hill New York San Francisco Washington, D.C. Auckland Bogoth Caracas Lisbon London Madrid Mexico City Milan Montreal New Delhi San Juan Singapore Sydney Tokyo Toronto Library of Congress Cataloging-in-Publication Data Oman, Robert M. Calculus for the utterly confused / Robert M. Oman, Daniel M. Oman. p. cm. ISBN 0-07-048261-6 1. Calculus-Study and teaching. I. Oman, Daniel M. II. Title. QA303.3.043 1998 5 154~21 98-25802 CIP Copyright 0 1999 by The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher. 34567890 FGRFGR 9032109 ISBN 0-07-04826 1-6 The sponsoring editor for this book was Barbara Gilson, the editing supervisor was Stephen M. Smith, and the production supervisor was Pamela A. Pelton. Printed and bound by QuebecorlFairfield. McGraw-Hill books are available at special quantity discounts to use as premi- ums and sales promotions, or for use in corporate training programs. For more information, please write to the Director of Special Sales, McGraw-Hill, 11 West 19th Street, New York, NY 10011. Or contact your local bookstore. This book is printed on recycled, acid-free paper containing a minimum of 50% recycled, de-inked fiber. Information contained in this work has been obtained by The McGraw- Hill Companies, Inc. (“McGraw-Hill”) from sources believed to be reli- able. However, neither McGraw-Hill nor its authors guarantee the accu- racy or completeness of any information published herein and neither McGraw-Hill nor its authors shall be responsible for any errors, omis- sions, or damages arising out of use of this information. This work is published with the understanding that McCraw-Hill and its authors are supplying information but are not attempting to render engineering or other professional services. If such services are required, the assistance of an appropriate professional should be sought. CONTENTS A Special Message ...................... vi i How to Study Calculus .............. ix Preface ......................................... xi 1. Mathematical Background ......... 1 2. Limits and Continuity .................. 27 3. Derivatives ................................... 33 4. Graphing ........................................ 45 5. Max-Min Problems ...................... 57 6. Related Rate Problems ............... 65 7. Integration................................... 75 8. Trigonometric Functions ............ 105 9. Exponents and Logarithms......... 131 10. More Integrals ............................ 155 Mathematical Tab les .................. 181 Index ............................................ 187 V This page intentionally left blank A SPECIAL MESSAGE TO THE UTTERLY CONFUSED CALCULUS STUDENT Our message to the utterly conhsed calculus student is very simple: You don't have to be confused anvore. We were once conhsed calculus students. We aren't confbsed anymore. We have taught many utterly confused calculus students both in formal class settings and one-on-one. They aren't confbsed anymore. All this experience has taught us what causes utter confbsion in calculus and how to eliminate that confusion. The topics we discuss here are aimed right at the heart of those topics that we know cause the most trouble. Follow us through this book, and you won't be confused anymore either. Anyone who has taught calculus will tell you that there are two problem areas that prevent students from learning the subject. The frrst problem is a lack of algebra skills. Sometimes it's not a lack of algebra skills but a lack of confidence in applying recently learned algebra skills. We attack this problem two ways. One of the largest chapters in this book is the one devoted to a review of the algebra skills you need to be successfirl in working calculus problems. Don't pass by this chapter. Spend time there and refer back to is as needed. There are insights for even those who consider themselves good at algebra. When we do a problem we take you through the steps, the calculus steps and all those pesky little algebra steps, tricks some might call them. When we present a problem it is a complete presentation. Not only do we do the problem completely but also we explain along the way why things are done a certain way. The second problem of the utterly confused calculus student is the inability to set up the problems. In most problems the calculus is easy, the algebra possibly tedious, but writing the problem in mathematical statements the most difficult step of all. Translating a word problem into a math problem (words to equation) is not easy. We spend time in the problems showing you how to make word sentences into mathematical equations. Where there are patterns to problems we point them out so when you see similar problems, on tests perhaps, you will remember how to do them. To aid you in refdgback to important parts of the book we use a collection of icons as described on the next page. Our message to utterly conhsed calculus students is simple. You don't have to be confused anymore. We have been there, done that, know what it takes to remove the confusion, and have written it all down for you. vii This icon highlights things you should memorize. Right before a test, go over these items to keep them fresh in your mind. Ranember* $v=This icon appears next to the "deeper" insights into a problem. If you have trouble understanding the details of why a problem makes physical Insight sense, then this is the icon to follow. \& This icon highlights trouble spots and common traps that students fie encounter. If you are womed about making frustrating little mistakes or Watch feel you are loosing points on tests due to missing little "tricks" then this is the icon to follow. The intention of this icon is to help you identifL a pattern of solving one problem that works for a general category of problems. In many cases the pattern is reviewed in a step by step summary along examples of Pattern with similar problems. Items next to this icon can be skipped if you are really struggling. On a second pass through the book, or for the more advanced student, this icon is intended to show a few extra tricks that will allow you to do problems faster. These items are included since speed is many times important to 'pd success on calculus tests. viii How To Study Calculus Calculus courses are different from most courses in other disciplines. One big difference is in testing. There is 'very little writing for a calculus tests. There is a lot of mathematical manipulation. In many disciplines you learn the material by reading and listening and demonstrate mastery of that material by writing about it. In mathematics there is some reading, and some listening, but demonstrating mastery of the material is by doing problems. Another example of the difference between learning and demonstrating mastery of a subject is history. There is a great deal of reading in a history course, but mastery of the material is demonstrated by writing about history. If you are not already doing this you can improve your grades on history exams by practicing writing the answers to questions you expect to encounter on those exams. Guess the questions on the test, practice writing answers to those questions and watch your grades go up and your study time go down in your history course or any other read-to-learn, write-to-demonstrate-mastery course. In your calculus course practicing working potential problems as test preparation is even more important than practicing writing the answers to potential questions in a history course. Writing is more familiar to most people than pdorming mathematical manipulations. You can almost always say something about a topic, but it is not at all unusual to have no clue as to how to start a calculus problem. Practicing writing for a history test will improve your grades. Practicing problems, not just reading them but actually writing them down, may be the only way for you to achieve the most modest of success on a calculus test. To succeed on your calculus tests you need to do three things, PROBLEMS, PROBLEMS and PROBLEMS.