5 Reasons to Use for Applied Calculus ❶ Thousands of high-quality exercises. Algorithmic exercises of all types and difficulty levels are available to meet the needs of students with diverse mathematical backgrounds. We’ve also added even more conceptual exercises to the already abundant skill and application exercises. ➋ Helps students help themselves. Homework isn’t effective if students don’t do it. MyMathLab not only grades homework, but it also does the more subtle task of providing specific feedback and guidance along the way. As an instructor, you can control the amount of guidance students receive. Breaks the problem into manageable steps. Students enter answers along the way. Reviews a problem like the one assigned. Links to the appropriate section in the textbook. Features an instructor explaining the concept. ➌ Addresses gaps in prerequisite skills. Our “Getting Ready for Applied Calculus” content addresses gaps in prerequisite skills that can impede student success. MyMathLab identifies precise areas of weakness, then automatically provides remediation for those skills. ➍ Adaptive Study Plan. MyMathLab’s Adaptive Study Plan makes studying more efficient and effective. Each student’s work and activity are assessed continually in real time. The data and analytics are used to provide personalized content to remediate any gaps in understanding. ➎ Ready-to-Go Courses. To make it even easier for first-time users to start using MyMathLab, we have enlisted experienced instructors to create premade assignments for the Ready-to-Go Courses. You can alter these assignments at any time, but they provide a terrific starting point, right out of the box. Since 2001, more than 15 million students at more than 1,950 colleges have used MyMathLab. Users have reported significant increases in pass rates and retention. Why? Students do more work and get targeted help when they need it. See www.mymathlab.com/success_report.html for the latest information on how schools are successfully using MyMathLab. Learn more at www.mymathlab.com CALCULUS FOR BUSINESS, ECONOMICS, LIFE SCIENCES, AND SOCIAL SCIENCES Thirteenth Edition RAYMOND A. BARNETT Merritt College MICHAEL R. ZIEGLER Marquette University KARL E. BYLEEN Marquette University Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montréal Toronto Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo Editor in Chief: Deirdre Lynch Executive Editor: Jennifer Crum Project Manager: Kerri Consalvo Editorial Assistant: Joanne Wendelken Senior Managing Editor: Karen Wernholm Senior Production Supervisor: Ron Hampton Associate Design Director: Andrea Nix Interior and Cover Design: Beth Paquin Executive Manager, Course Production: Peter Silvia Associate Media Producer: Christina Maestri Digital Assets Manager: Marianne Groth Executive Marketing Manager: Jeff Weidenaar Marketing Assistant: Brooke Smith Rights and Permissions Advisor: Joseph Croscup Senior Manufacturing Buyer: Carol Melville Production Coordination and Composition: Integra Cover photo: Leigh Prather/Shutterstock; Dmitriy Raykin/Shutterstock; Image Source/Getty Images Photo credits: Page 2, iStockphoto/Thinkstock; Page 94, Purestock/Thinkstock; Page 180, Vario Images/Alamy; Page 237, P. Amedzro/Alamy; Page 319, Anonymous Donor/ Alamy; Page 381, Shime/Fotolia; Page 424, Aurora Photos/Alamy; Page 494, Gary Whitton/Fotolia Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book, and Pearson was aware of a trademark claim, the designations have been printed in initial caps or all caps. Library of Congress Cataloging-in-Publication Data Calculus for business, economics, life sciences, and social sciences / Raymond A. Barnett … [et al.].—13th ed. p. cm. Includes index. ISBN-13: 978-0-321-86983-8 ISBN-10: 0-321-86983-4 1. Calculus—Textbooks I. Ziegler, Michael R. II. Byleen, Karl E. III. Title QA303.2.B285 2015 515—dc23 2013023206 Copyright © 2015, 2011, 2008, Pearson Education, Inc. All rights reserved. No part of this publica- tion 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 publisher. Printed in the United States of America. For information on obtaining permission for use of material in this work, please submit a written request to Pearson Education, Inc., Rights and Contracts Department, 501 Boylston Street, Suite 900, Boston, MA 02116. 1 2 3 4 5 6 7 8 9 10—V011—18 17 16 15 14 ISBN-10: 0-321-86983-4 www.pearsonhighered.com ISBN-13: 978-0-321-86983-8 CONTENTS Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Diagnostic Prerequisite Test. . . . . . . . . . . . . . . . . . . . xvi PART 1 A LIBRARY OF ELEMENTARY FUNCTIONS Chapter 1 Functions and Graphs . . . . . . . . . . . . . . . . . . . . 2 1.1 Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Elementary Functions: Graphs and Transformations . . . . . . . . 18 1.3 Linear and Quadratic Functions . . . . . . . . . . . . . . . . . 30 1.4 Polynomial and Rational Functions . . . . . . . . . . . . . . . . 52 1.5 Exponential Functions. . . . . . . . . . . . . . . . . . . . . . 62 1.6 Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . 73 Chapter 1 Summary and Review . . . . . . . . . . . . . . . . . . . 84 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 87 PART 2 CALCULUS Chapter 2 Limits and the Derivative . . . . . . . . . . . . . . . . . . 94 2.1 Introduction to Limits . . . . . . . . . . . . . . . . . . . . . . 95 2.2 Infinite Limits and Limits at Infinity . . . . . . . . . . . . . . . 109 2.3 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 121 2.4 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . 132 2.5 Basic Differentiation Properties. . . . . . . . . . . . . . . . . 147 2.6 Differentials . . . . . . . . . . . . . . . . . . . . . . . . . 156 2.7 Marginal Analysis in Business and Economics. . . . . . . . . . 163 Chapter 2 Summary and Review . . . . . . . . . . . . . . . . . . 174 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 175 Chapter 3 Additional Derivative Topics . . . . . . . . . . . . . . . . 180 3.1 The Constant e and Continuous Compound Interest . . . . . . . 181 3.2 Derivatives of Exponential and Logarithmic Functions . . . . . . 187 3.3 Derivatives of Products and Quotients . . . . . . . . . . . . . 196 3.4 The Chain Rule. . . . . . . . . . . . . . . . . . . . . . . . 204 3.5 Implicit Differentiation. . . . . . . . . . . . . . . . . . . . . 214 3.6 Related Rates. . . . . . . . . . . . . . . . . . . . . . . . . 220 3.7 Elasticity of Demand . . . . . . . . . . . . . . . . . . . . . 226 Chapter 3 Summary and Review . . . . . . . . . . . . . . . . . . 233 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 235 Chapter 4 Graphing and Optimization . . . . . . . . . . . . . . . . 237 4.1 First Derivative and Graphs . . . . . . . . . . . . . . . . . . 238 4.2 Second Derivative and Graphs . . . . . . . . . . . . . . . . 254 4.3 L’Hôpital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . 271 4.4 Curve-Sketching Techniques . . . . . . . . . . . . . . . . . . 280 iii iv CONTENTS 4.5 Absolute Maxima and Minima. . . . . . . . . . . . . . . . . 293 4.6 Optimization. . . . . . . . . . . . . . . . . . . . . . . . . 301 Chapter 4 Summary and Review . . . . . . . . . . . . . . . . . . 314 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 315 Chapter 5 Integration. . . . . . . . . . . . . . . . . . . . . . . . . 319 5.1 Antiderivatives and Indefinite Integrals . . . . . . . . . . . . . 320 5.2 Integration by Substitution. . . . . . . . . . . . . . . . . . . 331 5.3 Differential Equations; Growth and Decay . . . . . . . . . . . 342 5.4 The Definite Integral. . . . . . . . . . . . . . . . . . . . . . 353 5.5 The Fundamental Theorem of Calculus . . . . . . . . . . . . . 363 Chapter 5 Summary and Review . . . . . . . . . . . . . . . . . . 375 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 377 Chapter 6 Additional Integration Topics . . . . . . . . . . . . . . . . 381 6.1 Area Between Curves. . . . . . . . . . . . . . . . . . . . . 382 6.2 Applications in Business and Economics . . . . . . . . . . . . 391 6.3 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . 403 6.4 Other Integration Methods. . . . . . . . . . . . . . . . . . . 409 Chapter 6 Summary and Review . . . . . . . . . . . . . . . . . . 420 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 421 Chapter 7 Multivariable Calculus . . . . . . . . . . . . . . . . . . . 424 7.1 Functions of Several Variables . . . . . . . . . . . . . . . . . 425 7.2 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . 434 7.3 Maxima and Minima . . . . . . . . . . . . . . . . . . . . . 443 7.4 Maxima and Minima Using Lagrange Multipliers . . . . . . . . 451 7.5 Method of Least Squares . . . . . . . . . . . . . . . . . . . 460 7.6 Double Integrals over Rectangular Regions . . . . . . . . . . . 470 7.7 Double Integrals over More General Regions . . . . . . . . . . 480 Chapter 7 Summary and Review . . . . . . . . . . . . . . . . . . 488 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 491 Chapter 8 Trigonometric Functions . . . . . . . . . . . . . . . . . . 494 8.1 Trigonometric Functions Review . . . . . . . . . . . . . . . . 495 8.2 Derivatives of Trigonometric Functions . . . . . . . . . . . . . 502 8.3 Integration of Trigonometric Functions . . . . . . . . . . . . . 507 Chapter 8 Summary and Review . . . . . . . . . . . . . . . . . . 512 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 513 Appendix A Basic Algebra Review . . . . . . . . . . . . . . . . . . . 514 A.1 Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . 514 A.2 Operations on Polynomials . . . . . . . . . . . . . . . . . . 520 A.3 Factoring Polynomials. . . . . . . . . . . . . . . . . . . . . 526 A.4 Operations on Rational Expressions . . . . . . . . . . . . . . 532 A.5 Integer Exponents and Scientific Notation . . . . . . . . . . . 538 A.6 Rational Exponents and Radicals. . . . . . . . . . . . . . . . 542 A.7 Quadratic Equations . . . . . . . . . . . . . . . . . . . . . 548 CONTENTS v Appendix B Special Topics . . . . . . . . . . . . . . . . . . . . . . . 557 B.1 Sequences, Series, and Summation Notation . . . . . . . . . . 557 B.2 Arithmetic and Geometric Sequences. . . . . . . . . . . . . . 563 B.3 Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . 569 Appendix C Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 Answers. . . . . . . . . . . . . . . . . . . . . . . . . . A-1 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . I-1 Index of Applications. . . . . . . . . . . . . . . . . . . . I-9 Available separately: Calculus Topics to Accompany Calculus, 13e, and College Mathematics, 13e Chapter 1 Differential Equations 1.1 Basic Concepts 1.2 Separation of Variables 1.3 First-Order Linear Differential Equations Chapter 1 Review Review Exercises Chapter 2 Taylor Polynomials and Infinite Series 2.1 Taylor Polynomials 2.2 Taylor Series 2.3 Operations on Taylor Series 2.4 Approximations Using Taylor Series Chapter 2 Review Review Exercises Chapter 3 Probability and Calculus 3.1 Improper Integrals 3.2 Continuous Random Variables 3.3 Expected Value, Standard Deviation, and Median 3.4 Special Probability Distributions Chapter 3 Review Review Exercises Appendixes A and B ( Refer to back of Calculus for Business, Economics, Life Sciences and Social Sciences, 13e) Appendix C Tables Table III Area Under the Standard Normal Curve Appendix D Special Calculus Topic D.1 Interpolating Polynomials and Divided Differences Answers Solutions to Odd-Numbered Exercises Index Applications Index PREFACE The thirteenth edition of Calculus for Business, Economics, Life Sciences, and Social Sci- ences is designed for a one-term course in Calculus for students who have had one to two years of high school algebra or the equivalent. The book’s overall approach, refined by the authors’ experience with large sections of college freshmen, addresses the challenges of teaching and learning when prerequisite knowledge varies greatly from student to student. The authors had three main goals when writing this text: ▶ To write a text that students can easily comprehend ▶ To make connections between what students are learning and how they may apply that knowledge ▶ To give flexibility to instructors to tailor a course to the needs of their students. Many elements play a role in determining a book’s effectiveness for students. Not only is it critical that the text be accurate and readable, but also, in order for a book to be effective, aspects such as the page design, the interactive nature of the presentation, and the ability to support and challenge all students have an incredible impact on how easily students com- prehend the material. Here are some of the ways this text addresses the needs of students at all levels: ▶ Page layout is clean and free of potentially distracting elements. ▶ Matched Problems that accompany each of the completely worked examples help students gain solid knowledge of the basic topics and assess their own level of under- standing before moving on. ▶ Review material (Appendix A and Chapter 1) can be used judiciously to help remedy gaps in prerequisite knowledge. ▶ A Diagnostic Prerequisite Test prior to Chapter 1 helps students assess their skills, while the Basic Algebra Review in Appendix A provides students with the content they need to remediate those skills. ▶ Explore and Discuss problems lead the discussion into new concepts or build upon a current topic. They help students of all levels gain better insight into the mathemati- cal concepts through thought-provoking questions that are effective in both small and large classroom settings. ▶ Instructors are able to easily craft homework assignments that best meet the needs of their students by taking advantage of the variety of types and difficulty levels of the exercises. Exercise sets at the end of each section consist of a Skills Warm-up (four to eight problems that review prerequisite knowledge specific to that section) followed by problems divided into categories A, B, and C by level of difficulty, with level-C exercises being the most challenging. ▶ The MyMathLab course for this text is designed to help students help themselves and provide instructors with actionable information about their progress. The immedi- ate feedback students receive when doing homework and practice in MyMathLab is invaluable, and the easily accessible e-book enhances student learning in a way that the printed page sometimes cannot. Most important, all students get substantial experience in modeling and solving real-world problems through application examples and exercises chosen from business and econom- ics, life sciences, and social sciences. Great care has been taken to write a book that is mathematically correct, with its emphasis on computational skills, ideas, and problem solving rather than mathematical theory. vi PREFACE vii Finally, the choice and independence of topics make the text readily adaptable to a variety of courses (see the chapter dependencies chart on page xi). This text is one of three books in the authors’ college mathematics series. The others are Finite Mathematics for Business, Economics, Life Sciences, and Social Sciences, and College Mathematics for Business, Economics, Life Sciences, and Social Sciences; the latter contains selected con- tent from the other two books. Additional Calculus Topics, a supplement written to accom- pany the Barnett/Ziegler/Byleen series, can be used in conjunction with any of these books. New to This Edition Fundamental to a book’s effectiveness is classroom use and feedback. Now in its thir- teenth edition, Calculus for Business, Economics, Life Sciences, and Social Sciences has had the benefit of a substantial amount of both. Improvements in this edition evolved out of the generous response from a large number of users of the last and previous editions as well as survey results from instructors, mathematics departments, course outlines, and college catalogs. In this edition, ▶ The Diagnostic Prerequisite Test has been revised to identify the specific deficiencies in prerequisite knowledge that cause students the most difficulty with calculus. ▶ Chapters 1 and 2 of the previous edition have been revised and combined to create a single introductory chapter (Chapter 1) on functions and graphs. ▶ Most exercise sets now begin with a Skills Warm-up—four to eight problems that review prerequisite knowledge specific to that section in a just-in-time approach. References to review material are given in the answer section of the text for the benefit of students who struggle with the warm-up problems and need a refresher. ▶ Section 6.4 has been rewritten to cover the trapezoidal rule and Simpson’s rule. ▶ Examples and exercises have been given up-to-date contexts and data. ▶ Exposition has been simplified and clarified throughout the book. ▶ An Annotated Instructor’s Edition is now available, providing answers to exercises directly on the page (whenever possible). Teaching Tips provide less-experienced instructors with insight on common student pitfalls, suggestions for how to approach a topic, or reminders of which prerequisite skills students will need. Lastly, the difficulty level of exercises is indicated only in the instructor’s edition so as not to discourage students from attempting the most challenging “C” level exercises. ▶ MyMathLab for this text has been enhanced greatly in this revision. Most notably, a “Getting Ready for Chapter X” has been added to each chapter as an optional resource for instructors and students as a way to address the prerequisite skills that students need, and are often missing, for each chapter. Many more improvements have been made. See the detailed description on pages xiv and xv for more information. Trusted Features Emphasis and Style As was stated earlier, this text is written for student comprehension. To that end, the focus has been on making the book both mathematically correct and accessible to students. Most derivations and proofs are omitted, except where their inclusion adds significant insight into a particular concept as the emphasis is on computational skills, ideas, and problem solving rather than mathematical theory. General concepts and results are typically pre- sented only after particular cases have been discussed. Design One of the hallmark features of this text is the clean, straightforward design of its pages. Navigation is made simple with an obvious hierarchy of key topics and a judicious use of call-outs and pedagogical features. We made the decision to maintain a two-color design to viii PREFACE help students stay focused on the mathematics and applications. Whether students start in the chapter opener or in the exercise sets, they can easily reference the content, examples, and Conceptual Insights they need to understand the topic at hand. Finally, a functional use of color improves the clarity of many illustrations, graphs, and explanations, and guides students through critical steps (see pages 22, 75, and 306). Examples and Matched Problems More than 300 completely worked examples are used to introduce concepts and to dem- onstrate problem-solving techniques. Many examples have multiple parts, significantly increasing the total number of worked examples. The examples are annotated using blue text to the right of each step, and the problem-solving steps are clearly identified. To give students extra help in working through examples, dashed boxes are used to e nclose steps that are usually performed mentally and rarely mentioned in other books (see Example 4 on page 9). Though some students may not need these additional steps, many will a ppreciate the fact that the authors do not assume too much in the way of prior knowledge. EXAMPLE 2 Tangent Lines Let f1x2 = 12x - 921x2 + 62. (A) Find the equation of the line tangent to the graph of f(x) at x = 3. (B) Find the value(s) of x where the tangent line is horizontal. SOLUTION (A) First, find f′1x2: f′1x2 = 12x - 921x2 + 62′ + 1x2 + 6212x - 92′ = 12x - 9212x2 + 1x2 + 62122 Then, find f132 and f′132: f132 = 32132 - 94132 + 62 = 1-321152 = -45 f′132 = 32132 - 942132 + 132 + 62122 = -18 + 30 = 12 Now, find the equation of the tangent line at x = 3: y - y = m1x - x 2 y = f1x 2 = f132 = -45 1 1 1 1 y - 1-452 = 121x - 32 m = f′1x 2 = f′132 = 12 1 y = 12x - 81 Tangent line at x = 3 (B) The tangent line is horizontal at any value of x such that f′1x2 = 0, so f′1x2 = 12x - 922x + 1x2 + 622 = 0 6x2 - 18x + 12 = 0 x2 - 3x + 2 = 0 1x - 121x - 22 = 0 x = 1, 2 The tangent line is horizontal at x = 1 and at x = 2. Matched Problem 2 Repeat Example 2 for f1x2 = 12x + 921x2 - 122. Each example is followed by a similar Matched Problem for the student to work while reading the material. This actively involves the student in the learning process. The answers to these matched problems are included at the end of each section for easy reference.