net This page intentionally left blank www.net Calculus with Applications TENTH EDITION www.net This page intentionally left blank www.net Calculus with Applications TENTH EDITION Margaret L. Lial American River College Raymond N. Greenwell Hofstra University Nathan P. Ritchey Youngstown State University www.net Editor in Chief: Deirdre Lynch Executive Editor: Jennifer Crum NOTICE: Executive Content Editor: Christine O’Brien This work is Senior Project Editor: Rachel S.
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All recipi- ents of this work are Library of Congress Cataloging-in-Publication Data expected to abide by Lial, Margaret L. these restrictions Calculus with applications — 10th ed. and to honor the Greenwell, Nathan P. Includes bibliographical references and index.
purposes and the ISBN-13: 978-0-321-74900-0 (student ed.) needs of other ISBN-10: 0-321-74900-6 (student ed.) instructors who rely 1. on these materials.L53 2012 515— dc22 2010030933 Copyright © 2012, 2008, 2005, 2002 Pearson Education, Inc. 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 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, fax your request to 617-671-3447, or e-mail at http://www.com/legal/permissions. 1 2 3 4 5 6 7 8 9 10—QG—15 14 13 12 11 ISBN-10: 0-321-74900-6 www.com ISBN-13: 978-0-321-74900-0 www.net Contents Preface ix Dear Student xxi Prerequisite Skills Diagnostic Test xxii CHAPTER Algebra Reference R-1 R R.7 Radicals R-25 CHAPTER Linear Functions 1 1 1.2 Slopes and Equations of Lines Linear Functions and Applications 2 17 1.3 The Least Squares Line 25 CHAPTER 1 REVIEW 38 EXTENDED APPLICATION Using Extrapolation to Predict Life Expectancy 42 CHAPTER Nonlinear Functions 44 2 2.2 Properties of Functions 45 Quadratic Functions;Translation and Reflection 57 2.3 Polynomial and Rational Functions 67 2.6 Applications: Growth and Decay; Mathematics of Finance 102 CHAPTER 2 REVIEW 110 EXTENDED APPLICATION Power Functions 118 CHAPTER The Derivative 121 3 3.3 Rates of Change 149 3.4 Definition of the Derivative 162 3.5 Graphical Differentiation 180 CHAPTER 3 REVIEW 186 EXTENDED APPLICATION A Model for Drugs Administered Intravenously 193 v www.net vi CONTENTS CHAPTER Calculating the Derivative 196 4 4.2 Techniques for Finding Derivatives Derivatives of Products and Quotients 197 211 4.3 The Chain Rule 218 4.4 Derivatives of Exponential Functions 228 4.5 Derivatives of Logarithmic Functions 236 CHAPTER 4 REVIEW 243 EXTENDED APPLICATION Electric Potential and Electric Field 248 CHAPTER Graphs and the Derivative 251 5 5.2 Increasing and Decreasing Functions Relative Extrema 263 252 5.3 Higher Derivatives, Concavity, and the Second Derivative Test 274 5.4 Curve Sketching 287 CHAPTER 5 REVIEW 296 EXTENDED APPLICATION A Drug Concentration Model for Orally Administered Medications 300 CHAPTER Applications of the Derivative 303 6 6.2 Absolute Extrema Applications of Extrema 304 313 6.3 Further Business Applications: Economic Lot Size; Economic Order Quantity; Elasticity of Demand 322 6.6 Differentials: Linear Approximation 343 CHAPTER 6 REVIEW 349 EXTENDED APPLICATION A Total Cost Model for a Training Program 353 CHAPTER Integration 355 7 7.3 Area and the Definite Integral 376 7.4 The Fundamental Theorem of Calculus 388 7.5 The Area Between Two Curves 398 7.6 Numerical Integration 408 CHAPTER 7 REVIEW 416 EXTENDED APPLICATION Estimating Depletion Dates for Minerals 421 www.net CONTENTS vii CHAPTER Further Techniques and Applications of Integration 425 8 8.2 Integration by Parts Volume and Average Value 426 434 8.3 Continuous Money Flow 441 8.4 Improper Integrals 448 CHAPTER 8 REVIEW 454 EXTENDED APPLICATION Estimating Learning Curves in Manufacturing with Integrals 457 CHAPTER Multivariable Calculus 459 9 9.2 Functions of Several Variables Partial Derivatives 471 460 9.3 Maxima and Minima 482 9.5 Total Differentials and Approximations 499 9.6 Double Integrals 504 CHAPTER 9 REVIEW 515 EXTENDED APPLICATION Using Multivariable Fitting to Create a Response Surface Design 521 CHAPTER Differential Equations 525 10 10.1 Solutions of Elementary and Separable Differential Equations 10.2 Linear First-Order Differential Equations 539 526 10.4 Applications of Differential Equations 551 CHAPTER 10 REVIEW 559 EXTENDED APPLICATION Pollution of the Great Lakes 564 CHAPTER Probability and Calculus 567 11 11.1 Continuous Probability Models 568 11.2 Expected Value and Variance of Continuous Random Variables 579 11.3 Special Probability Density Functions 588 CHAPTER 11 REVIEW 600 EXTENDED APPLICATION Exponential Waiting Times 605 www.net viii CONTENTS CHAPTER Sequences and Series 608 12 12.2 Annuities: An Application of Sequences 613 12.3 Taylor Polynomials at 0 624 12.4 Infinite Series 633 12.7 l’Hôpital’s Rule 653 CHAPTER 12 REVIEW 660 EXTENDED APPLICATION Living Assistance and Subsidized Housing 663 CHAPTER The Trigonometric Functions 665 13 13.1 Definitions of the Trigonometric Functions 13.2 Derivatives of Trigonometric Functions 682 666 13.3 Integrals of Trigonometric Functions 692 CHAPTER 13 REVIEW 699 EXTENDED APPLICATION The Shortest Time and the Cheapest Path 704 Appendix A Solutions to Prerequisite Skills Diagnostic Test A-1 B Learning Objectives A-4 C MathPrint Operating System for TI-84 and TI-84 Plus Silver Edition A-8 D Tables A-10 1 Formulas of Geometry 2 Area Under a Normal Curve 3 Integrals 4 Integrals Involving Trigonometric Functions Answers to Selected Exercises A-15 Credits C-1 Index of Applications I-1 Index I-5 Sources S-1 www.net Preface Calculus with Applications is a thorough, application-oriented text for students majoring in business, management, economics, or the life or social sciences. In addition to its clear exposition, this text consistently connects the mathematics to career and everyday-life situations.
A prerequisite of two years of high school algebra is assumed. A renewed focus on quick and effective assessments, new applications and exercises, as well as other new learning tools make this 10th edition an even richer learning resource for students. Our Approach Our main goal is to present applied calculus in a concise and meaningful way so that students can understand the full picture of the concepts they are learning and apply it to real-life situations. This is done through a variety of ways.
Focus on Applications Making this course meaningful to students is critical to their suc- cess. Applications of the mathematics are integrated throughout the text in the exposition, the examples, the exercise sets, and the supplementary resources. Calculus with Applications presents students with a myriad of opportunities to relate what they’re learning to career situ- ations through the Apply It questions, the applied examples, and the Extended Applications. To get a sense of the breadth of applications presented, look at the Index of Applications in the back of the book or the extended list of sources of real-world data on www.
Pedagogy to Support Students Students need careful explanations of the mathematics along with examples presented in a clear and consistent manner. Additionally students and instructors should have a means to assess the basic prerequisite skills. This can now be done with the Prerequisite Skills Diagnostic Test located just before Chapter R. In addition, the stu- dents need a mechanism to check their understanding as they go and resources to help them remediate if necessary.
Calculus with Applications has this support built into the pedagogy of the text through fully developed and annotated examples, Your Turn exercises, For Review references, and supplementary material. Beyond the Textbook Students today take advantage of a variety of resources and delivery methods for instruction. As such, we have developed a robust MyMathLab course for Calcu- lus with Applications. MyMathLab has a well-established and well-documented track record of helping students succeed in mathematics.
The MyMathLab online course for Calculus with Applications contains over 2000 exercises to challenge students and provides help when they need it. Students who learn best by seeing and hearing can view section- and example-level videos within MyMathLab or on the book-specific DVD-Rom. These and other resources are available to students as a unified and reliable tool for their success. New to the Tenth Edition Based on the authors’ experience in the classroom along with feedback from many instructors across the country, the focus of this revision is to improve the clarity of the presentation and provide students with more opportunities to learn, practice, and apply what they’ve learned on their own.
This is done in both the presentation of the content and in new features added to the text.net x PREFACE New and Revised Content • Chapter R The flow of the material was improved by reordering some exercises and examples. Exercises were added to Section R. • Chapter 1 Changes in the presentation were made throughout to increase clarity, includ- ing adding some examples and rewriting others. Terminology in Section 1.2 was adjusted to be more consistent with usage in economics.
• Chapter 2 The material in Section 2.1 on the Dow Jones Average was updated. Material on even and odd functions was added. Material on identifying the degree of a polynomial has been rewritten as an example to better highlight the concept. The discussion of the Rule of 70 and the Rule of 72 was improved.
A new Extended Application on Power Functions has been added. • Chapter 3 In Section 3.1, the introduction of limits was completely revised. The opening discussion and example were transformed into a series of examples that progress through different limit scenarios: a function defined at the limit, a function undefined at the limit (a hole in the graph), a function defined at the limit but with a different value than the limit (a piecewise function), and then finally, finding a limit when one does not exist. New figures were added to illustrate the different scenarios.2 the definition and example of continuity has been revised using a simple process to test for continuity.
The opening dis- cussion of Section 3.5, showing how to sketch the graph of the derivative given the graph of the original function, was rewritten as an example. • Chapter 4 The introduction to the chain rule was rewritten as an example in Section 4. Exercise topics were revised to cover subjects such as worldwide Internet users, online learning, and the Gateway arch. • Chapter 5 In Section 5.1 the definition of increasing/decreasing functions has been moved to the beginning of the chapter, followed by the discussion of using derivatives to determine where the function increases and decreases.
The determination of where a function is increasing or decreasing is divided into three examples: when the critical numbers are found by setting the derivative equal to zero, when the critical numbers are found by determining where the derivative is undefined, and when the function has no critical numbers. • Chapter 6 Changes in the presentation were made throughout to increase clarity and exercise sets were rearranged to improve progression and parity.
