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Special Edition Chapters 1-5 Seventh Edition one-time online - access code included Karl J. Smith Monty J. Strauss Magdalena D. Toda ⅛⅛ / ∙ ⅛ ' ■ __ _ LOG IN INSTRUCTIONS Browser minimum: Internet Explorer 11, Firefox 3.6 +, Safari 4.0 +, and Chrome 9.0 +. JavaScript must also be enabled in your browser. Some online publications may require additional free browser plug-ins (e.g., Flash, Adobe Acrobat). https://www.grtep.com You will use your one-time-use online access code to set up your account the first time that you access this page. After you set up your account and create a password, you will be able to enter the Login portion of this page. TO SET UP A NEW ACCOUNT 1. Go to https://www.grtep.com. In the First Time User (with access code) section, enter the Access Code provided below. Click the Next button. 2. Confirm your institution and your publication. Click Next. 3. If your publication has multiple sections, you must select the section you are enrolled in. 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If you purchase this book and the scratch off material over the access code has been removed, the code is no good. You will need to contact Kendall Hunt to purchase a new code for your course. Contact: Kendall Hunt Customer Service: 800-228-0810 Or go to www.kendallhunt.com Kendall Hunt p u b lis h in g com pany Calculus Seventh Edition Special Edition, Chapters 1-5 Karl J. Smith Santa Rosa Junior College Monty J. Strauss Texas Tech University Magdalena D. Toda Texas Tech University Kendall Hunt p u b lis h in g com pany Cover image Copyright © Shutterstock.com The cover image of the Louvre was selected to highlight the melding of the old and the new. The study of calculus has a rich heritage, full of anecdotes and mountaintop ideas, symbolized by the majestic old building in Paris known as the Louvre. It was constructed as a fortress in 1190 and went thorough many changes until becoming the palace of the Sun King, Louis XIV, only to be transformed into a museum by Louis XVI. The building has constantly undergone change, as does mathematics, and in 1989 a modern Pyramid was added to the museum. At about that time, calculus was “reinvented” using new technologies to capture the ideas of calculus. Just as the Louvre blends the old and the new in its structure, this book also blends the old and the new in its presentation. To symbolize this symbiosis, we use the logo: THINKING IN <⅛→NE1U DIRECTION All images, unless otherwise noted, are © Kendall Hunt Publishing Company. Kendall Hunt publishing company www.kendallhunt.com Send all inquiries to: 4050 Westmark Drive Dubuque, IA 52004-1840 Copyright © 1995, 1999, 2002 by Prentice-Hall, Inc. Special Edition © 2006, 2007 by Prentice-Hall, Inc. Copyright © 2014, 2018 by Kendall Hunt Publishing Company PAK ISBN 978-1-5249-7135-9 Text ISBN 978-1-5249-7137-3 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the copyright owner. Published in the United States of America C O N T E N T S Preface v 1 Functions and Graphs 1 1.1 What Is Calculus? 2 1.2 Preliminaries 15 1.3 Lines in the Plane; Parametric Equations 27 1.4 Functions and Graphs 36 1.5 Inverse Functions; Inverse Trigonometric Functions 51 Chapter 1 Review 62 Ethnomathematics by Marcia Ascher 71 Chapter 1 Group Research Project 72 2 Limits and Continuity 73 2.1 The Limit of a Function 74 2.2 Algebraic Computation of Limits 85 2.3 Continuity 95 2.4 Exponential and Logarithmic Functions 107 Chapter 2 Review 122 Chapter 2 Group Research Project 129 3 Differentiation 131 3.1 An Introduction to the Derivative: Tangents 132 3.2 Techniques of Differentiation 145 3.3 Derivatives of Trigonometric, Exponential, and Logarithmic Functions 154 3.4 Rates of Change: Modeling Rectilinear Motion 161 3.5 The Chain Rule 173 3.6 Implicit Differentiation 181 3.7 Related Rates and Applications 193 3.8 Linear Approximation and Differentials 202 Chapter 3 Review 215 [>233EEED3 Fermat’s Enigma by Simon Singh 217 Chapter 3 Group Research Project 223 4 Additional Applications of the Derivative 225 4.1 Extreme Values of a Continuous Function 226 4.2 The Mean Value Theorem 237 4.3 Using Derivatives to Sketch the Graph of a Function 244 4.4 Curve Sketching with Asymptotes: Limits Involving Infinity 261 4.5 l,H6pitaΓs Rule 273 4.6 Optimization in the Physical Sciences and Engineering 282 4.7 Optimization in Business, Economics, and the Life Sciences 298 Chapter 4 Review 313 Chapter 4 Group Research Project 322 iii iv Contents 5 Integration 323 5.1 Antidifferentiation 324 5.2 Area as the Limit of a Sum 336 5.3 Riemann Sums and the Definite Integral 344 5.4 The Fundamental Theorems of Calculus 357 5.5 Integration by Substitution 364 5.6 Introduction to Differential Equations 372 5.7 The Mean Value Theorem for Integrals; Average Value 385 5.8 Numerical Integration: The Trapezoidal Rule and Simpson’s Rule 392 5.9 An Alternative Approach: The Logarithm as an Integral 402 Chapter 5 Review 409 Chapter 5 Group Research Project 417 Cumulative Review Problems—Chapters 1-5 418 Appendices 421 A: Introduction to the Theory of Limits 421 B: Selected Proofs 427 C: Significant Digits 429 D: Short Table of Integrals 434 E: Trigonometry 444 F: Parabolas 452 G: Ellipses 462 H: Hyperbolas 473 I: Determinants 484 J: Answers to Selected Problems 489 Index 515 P R E F A C E “M 1 rZ athematics is the instrument by which the engineer tunnels our mountains, bridges our rivers, constructs our aqueducts, erects our factories and makes them musical by the busy hum of spindles. Take away the results of the rea soning of mathematics, and there would go with it nearly all the material achievements which give convenience and glory to modem civilization. ” Edward Brooks. Mental Science and Culture, Philadelphia: Normal Publishing, 1891, p. 255. THINKING IN &. FOR THE STU D EN T C∣→NEUJ DIRECTION You have enrolled in a calculus course, and you might be thinking, “I’ve finally made it ... All my life I’ve studied mathematics, and now Γm enrolled in ‘the big one.’ Certainly, calculus is the pinnacle of my studies in mathematics!” But wait! So you are here,... but do you know why? Can you answer the question, “What is calculus?” In the first section of this text, we begin by answering this question, and if you look at college catalogues, you will see that calculus is the first course in college mathematics, not the last one. This course is a prerequisite for almost all the other nonremedial mathematics courses. OK, so now you are here in a calculus course and you are asked to pay a great deal of money for this book. “Why does this book cost so much, and how could it possibly be worth all that money?” The value of any purchase is relative to many aspects which differ from person to person, but ultimately, value must be measured by the extent to which it changes our lives. The intent and goal of this book is to change your life by giving you life skills which enhance your problem-solving ability. By the time you finish this course you will have a book, which at different times, you will have loved, hated, cursed, and respected. You will have a reference book that should last you a lifetime. Calculus is a difficult subject, and there is no magic key to success; it will require hard work. This book should make your calculus journey easier because it builds your problem-solving skills and helps you form some good study habits. You will need to read the book, work the examples in the book using your own pencil and paper, and make a commitment to do your mathematics homework on a daily basis. & Read that last sentence once again. It is the best hint you will see about building success in mathematics. £ We have written this book so that it will be easy for you to know what is important. Each chapter begins with a preview and ends with a proficiency examination (generally 30 problems) consisting of concept problems (to help you know what is important) and practice problems (to help you with a self-test). Important terms are presented in boldface type. Important ideas, definitions, and procedures are enclosed in screened boxes: Definitions , Properties , and Procedures Common pitfalls, helpful hints and explanations are shown using this font. & WARNINGS are given to call your attention to common mistakes. V vi Preface • Color is used in a functional way to help you “see” what to do next. • INTERACTIVE figures allow you to “see” a figure in a dynamic way—essential since calculus discusses dynamic, not static, processes. Success in this course is a joint effort by the student, the instructor, and the author. The student must be willing to attend class and devote time to the course on a daily basis. There is no substitute for working problems in mathematics. FOR THE INSTRUCTOR As the instructor of a calculus course, you must select materials which are relevant to your students at your school or university. Some aspects of your course are necessitated by your environment, and others are selected by you or a committee in your department. We have put together a calculus textbook which we believe will meet your needs, as well as those of your department. Most importantly, our book will meet the needs of your students in the twenty-first century. One of the first questions asked by most instructors: “Does this calculus book do early or late transcendentals?” The quick answer is that this is an early transcendental book, but we understand the mathematical reasons why one might want a late transcendental development. For this reason, we have offered an optional Section 5.9 which defines the logarithm as an integral. The writing of a calculus textbook is a monumental task, one that grows out of a love of teaching and of helping students struggle until we see “the light” in their eyes as he or she has an “ah-hah” moment. The creation of this text has been a life-long journey for its authors, and to understand the nature of this book we travel back to 1930 to tell you of the books which have influenced us as we have developed the materials in this text. At one time, the defining and almost universally used calculus textbook was Granville, Smith, and Longley’s calculus book published in 1929.* This 516 page book contained no color, few illustrations, and even fewer problems. At that time, there were about a million students enrolled in the American university system, and education was designed to be for the privileged few. In 1957, Sputnik created a revolution in American education. Rigor was added to the teaching of calculus. The most influential calculus textbook of this era was a book by Thomas' which was initially published in 1952 and became the standard text in the 1960s. This book added rigor and problems, but still offered little pedagogy. By the end of 1968, Thomas was in its fourth edition, and had grown to 818 pages, which matched the growth of college enrollments at that time to almost four million students. Textbooks were still relatively inexpensive, but were colorless and without learning aids or technology. By 1970, Calculus by Apostol÷ was being used as a rigid and rigorous alternative to calculus,... correct, but lifeless. Many instructors of that time adopted a “sink-or-swim” attitude as there seemed to be an unlimited supply of students, each year growing larger and larger. By the 1980s, college enrollments had passed the ten million mark and calculus needed to change to keep up with the times. In 1987 the next generation of calcu lus books was bom. Stewart’s Calculus^ changed both the content and pedagogy of calculus. During this period of time, there was a movement to “reform” calculus. In 1986 and 1987, Steve Maurer and Ronald Douglas held two conferences, “Toward a Lean and Lively Calculus” and “Calculus for a New Century: A Pump not a Filter.” This move ment gave rise to what has become known as the Harvard Calculus approach and the *W. A. Granville, P. F. Smith, and W. R. Longley, Elements of the Differential and Integral Calculus. Boston: Ginn and Company, 1929. i G. B. Thomas, Calculus and Analytic Geometry, Classic Edition (Reprint of the 1952 edition). Reading: Addison Wesley, 1983. $Tom Apostle, Calculus (2 volums): Hoboken, Wiley, 1967 $James Stewart, Calculus. Belmont: Brooks/Cole Publishing Company, a division of Wadsworth, Inc., 1987 Preface vii publication of a book commonly known as the Hughes-Hallett Calculus^- The reform calculus movement divided the mathematics community into two “camps” and the teach ing of calculus went through great changes, most of which have had lasting benefits to students. With these books (and we have either learned or taught from all of them) and others we found those features which are essential and those which are superficial. Over the years, we have class tested our ideas and with this edition we have brought to you a book which is as fresh as this century, but preserves all the tried-and-true techniques which have been found to provide the necessary foundation for future work in mathematics. As college enrollments soar past the twenty million mark, education must now be designed for the masses. Regardless of the equipment offered at school, individual students have access to more and more technology (computers, iPads, smartphones, calculators, and Facebook, etc.). They use that technology not only to learn, but to communicate with each other in a cyber community. The calculus book for next generation needs to “work” in today’s world and with today’s changing technology, but at the same time, not compro mise the student’s ability to work complicated mathematical problems without relying on technology. These issues were our guiding principle as we wrote this edition of Calculus. FEATURES OF THIS BOOK Some of the distinguishing characteristics of the earlier editions are continued with this edition: • It is possible to begin the course with either Chapter 1 or Chapter 2 (where the calculus topics begin). • This edition offers an early presentation of transcendental functions: logarithms, expo nential functions, and trigonometric functions are heavily integrated into all chapters of the book (especially Chapters 1-5). • Because of user feedback, this edition has expanded drawing lessons for the student. We offer practical solutions to students on using engineering paper to draw curves in two and three dimensions. These drawing lessons include hints on plotting points in three dimensions, drawing circles, ellipses, hyperbolas, trigonometric curves, and polar-form curves, as well as planes and lines in three dimensions. • We have taken the introduction of differential equations seriously. Students in many allied disciplines need to use differential equations early in their studies and conse quently cannot wait for a postcalculus course. In this edition, we introduce differential equations in a natural and reasonable way. Slope fields are introduced as a geometric view of antidifferentiation in Section 5.1, and then are used to introduce a graphi cal solution to differential equations in Section 5.6. We consider separable differential equations in Chapter 5 and first-order linear differential equations in Chapter 7, and demonstrate the use of both, modeling a variety of applied situations. Exact and homo geneous differential equations appear in Chapter 14, along with an introduction to second-order linear equations. The “early and often” approach to differential equations is intended to illustrate their value in continuous modeling and to provide a solid foundation for further study. • We continue to utilize the humanness of mathematics. History is not presented as additional material to leam. Rather we have placed history into problems that lead the reader from the development of a concept to actually participating in the discovery process. The problems are designated as historical Quest problems. The problems are not designed to be “add-on or challenge problems,” but rather to become an integral part of the assignment. The level of difficulty of Qwe.s ∙ problems ranges from easy to difficult. • This edition correctly reflects the precalculus mathematics being taught at most colleges and universities. We assume knowledge of the trigonometric functions, and in this , Deborah Hughes-Hallett, Andrew M. Gleason, et. al., Calculus. Hoboken: John Wiley & Sons, 1994. viii Preface edition we introduce ex and lnx in Chapter 2 after we have defined the notion of a limit. We also assume a knowledge of the conic sections and their graphs. • Think Tank Problems Thinking about and doing mathematics is different than solving textbook problems. When thinking about and doing mathematics, a proposition may be true or false, whereas the typical textbook problem has a nice and concise answer. In the Think Tank problems our task is to prove the proposition true or to find a counterexample to disprove the proposition. We believe this form of problem to be important in preparing the student for future work in not only advanced mathematics courses, but also for analytically oriented courses. • EXPLORATION PROBLEMS It has been said that mathematical discovery is directed toward two major goals—the formulation of proofs and the construction of counterexamples. Most calculus books focus only on the first goal (the body of proofs and true statements), but we feel that some attention should be paid to the formulation, exploring concepts that may prove to be true or for which a counterexample is appro priate. These exploration problems go beyond the category of counterexample problem to provide opportunities for innovative thinking. • Journal Problems In an effort to show that “mathematicians work problems too,” we have reprinted problems from leading mathematics journals. We have chosen prob lems that are within reach of the intended audience of this book. If students need help or hints for these problems, they can search out the original presentation and solution in the cited journal. In addition, we have included problems from various Putnam examinations. These problems, which are more challenging, are offered in the supple mentary problems at the end of various chapters and are provided to give insight into the type of problems that are asked in mathematical competitions. The Putnam Exam ination is a national annual examination given under the auspices of the Mathematical Association of America and is designed to recognize mathematically talented college and university students. • Modeling continues as a major theme in this edition. Modeling is discussed in Section, and then appears in almost every section of the book. These applications are desig nated Modeling Problems or MODELING EXPERIMENTS. Some authors use the words “Modeling Problem” to refer to any applied problem. We make a distinction between modeling problems and application problems by defining a modeling prob lem as follows. A modeling problem is a problem that requires the reader make some assumptions about the real world in order to derive or come up with the neces sary mathematical formula or mathematical information to answer the question. These problems also include real-world examples of modeling by citing the source of the book or journal that shows the modeling process. Much of the difficulty students encounter learning the ideas of calculus can be attributed to ways students study and learn mathematics in high school, which often involves stressing rote memorization over insight and understanding. On the other hand, some reform texts are perceived as spending so much time with the development of insight and understanding that students are not given enough exposure to important computational and problem-solving skills in order to perform well in more advanced courses. This text aims at a middle ground by providing sound development, simulating problems, and well-developed pedagogy within a framework of a traditional tropic structure. “Think, then do” is a fair summary of our approach. TEXT CONTENT The content of this text adapts itself to either semester or quarter systems, and both differentiation and integration can be introduced in the first course. We begin calculus with a minimum of review. Cumulative reviews are offered at locations that fit the way calculus is taught at most colleges and universities. The first one includes Chapters 1-5, the second includes Chapters 6-8. Furthermore there is a cumulative review for