Brief Calculus: An Applied Approach, 8th Edition PDF

 Brief Calculus An Applied Approach RON LARSON The Pennsylvania State University The Behrend College Eighth Edition with the assistance of D AV I D C. FA LV O...

 Brief Calculus An Applied Approach RON LARSON The Pennsylvania State University The Behrend College Eighth Edition with the assistance of D AV I D C. FA LV O HOUGHTON MIFFLIN The Pennsylvania State University C O M PA N Y The Behrend College Boston New York Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Publisher: Richard Stratton Sponsoring Editor: Cathy Cantin Senior Marketing Manager: Jennifer Jones Development Editor: Peter Galuardi Art and Design Manager: Jill Haber Cover Design Manager: Anne S. Katzeff Senior Photo Editor: Jennifer Meyer Dare Senior Composition Buyer: Chuck Dutton Senior New Title Project Manager: Pat O’Neill Editorial Associate: Jeannine Lawless Marketing Associate: Mary Legere Editorial Assistant: Jill Clark Cover photo © Torsten Andreas Hoffmann/Getty Images Copyright © 2009 by Houghton Mifflin Company. All rights reserved. No part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system, without the prior written permission of Houghton Mifflin Company unless such copying is expressly permitted by federal copyright law. Address inquiries to College Permissions, Houghton Mifflin Company, 222 Berkeley Street, Boston, MA 02116-3764. Printed in the U.S.A. Library of Congress Control Number: 2007925316 Instructor’s examination copy ISBN-10: 0-547-00480-X ISBN-13: 978-0-547-00480-8 For orders, use student text ISBNs ISBN-10: 0-618-95847-9 ISBN-13: 978-0-618-95847-4 123456789–DOW– 11 10 09 08 07 Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.  Contents iii Contents A Word from the Author (Preface) vii A Plan for You as a Student ix Features xiii 0 A Precalculus Review 1 0.1 The Real Number Line and Order 2 0.2 Absolute Value and Distance on the Real Number Line 8 0.3 Exponents and Radicals 13 0.4 Factoring Polynomials 19 0.5 Fractions and Rationalization 25 1 Functions, Graphs, and Limits 33 1.1 The Cartesian Plane and the Distance Formula 34 1.2 Graphs of Equations 43 1.3 Lines in the Plane and Slope 56 Mid-Chapter Quiz 68 1.4 Functions 69 1.5 Limits 82 1.6 Continuity 94 Chapter 1 Algebra Review 105 Chapter Summary and Study Strategies 107 Review Exercises 109 Chapter Test 113 2 Differentiation 114 2.1 The Derivative and the Slope of a Graph 115 2.2 Some Rules for Differentiation 126 2.3 Rates of Change: Velocity and Marginals 138 2.4 The Product and Quotient Rules 153 Mid-Chapter Quiz 164 2.5 The Chain Rule 165 2.6 Higher-Order Derivatives 174 2.7 Implicit Differentiation 181 2.8 Related Rates 188 Chapter 2 Algebra Review 196 Chapter Summary and Study Strategies 198 Review Exercises 200 Chapter Test 204 Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. iv Contents 3 Applications of the Derivative 205 3.1 Increasing and Decreasing Functions 206 3.2 Extrema and the First-Derivative Test 215 3.3 Concavity and the Second-Derivative Test 225 3.4 Optimization Problems 235 Mid-Chapter Quiz 244 3.5 Business and Economics Applications 245 3.6 Asymptotes 255 3.7 Curve Sketching: A Summary 266 3.8 Differentials and Marginal Analysis 275 Chapter 3 Algebra Review 283 Chapter Summary and Study Strategies 285 Review Exercises 287 Chapter Test 291 4 Exponential and Logarithmic Functions 292 4.1 Exponential Functions 293 4.2 Natural Exponential Functions 299 4.3 Derivatives of Exponential Functions 308 Mid-Chapter Quiz 316 4.4 Logarithmic Functions 317 4.5 Derivatives of Logarithmic Functions 326 4.6 Exponential Growth and Decay 335 Chapter 4 Algebra Review 344 Chapter Summary and Study Strategies 346 Review Exercises 348 Chapter Test 352 5 Integration and Its Applications 353 5.1 Antiderivatives and Indefinite Integrals 354 5.2 Integration by Substitution and the General Power Rule 365 5.3 Exponential and Logarithmic Integrals 374 Mid-Chapter Quiz 381 5.4 Area and the Fundamental Theorem of Calculus 382 5.5 The Area of a Region Bounded by Two Graphs 394 5.6 The Definite Integral as the Limit of a Sum 403 Chapter 5 Algebra Review 409 Chapter Summary and Study Strategies 411 Review Exercises 413 Chapter Test 417 Copyright 2009 Cengage Learning, Inc. 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May not be copied, scanned, or duplicated, in whole or in part.  Contents v 6 Techniques of Integration 418 6.1 Integration by Parts and Present Value 419 6.2 Partial Fractions and Logistic Growth 429 6.3 Integration Tables 439 Mid-Chapter Quiz 449 6.4 Numerical Integration 450 6.5 Improper Integrals 459 Chapter 6 Algebra Review 470 Chapter Summary and Study Strategies 472 Review Exercises 474 Chapter Test 477 7 Functions of Several Variables 478 7.1 The Three-Dimensional Coordinate System 479 7.2 Surfaces in Space 487 7.3 Functions of Several Variables 496 7.4 Partial Derivatives 505 7.5 Extrema of Functions of Two Variables 516 Mid-Chapter Quiz 525 7.6 Lagrange Multipliers 526 7.7 Least Squares Regression Analysis 535 7.8 Double Integrals and Area in the Plane 545 7.9 Applications of Double Integrals 553 Chapter 7 Algebra Review 561 Chapter Summary and Study Strategies 563 Review Exercises 565 Chapter Test 569 Appendices Appendix A: Alternative Introduction to the Fundamental Theorem of Calculus A1 Appendix B: Formulas A10 B.1 Differentiation and Integration Formulas A10 B.2 Formulas from Business and Finance A14 Appendix C: Differential Equations (web only)* C.1 Solutions of Differential Equations C.2 Separation of Variables C.3 First-Order Linear Differential Equations C.4 Applications of Differential Equations Appendix D: Properties and Measurement (web only)* D.1 Review of Algebra, Geometry, and Trigonometry D.2 Units of Measurements Appendix E: Graphing Utility Programs (web only)* E.1 Graphing Utility Programs Answers to Selected Exercises A17 Answers to Checkpoints A91 Index A103 *Available at the text-specific website at college.hmco.com/pic/larsonBrief8e Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.  A Word from the Author vii A Word from the Author Welcome to Brief Calculus: An Applied Approach, Eighth Edition. In this revision, I focused not only on providing a meaningful revision to the text, but also a completely integrated learning program. Applied calculus students are a diverse group with varied interests and backgrounds. The revision strives to address the diversity and the different learning styles of students. I also aimed to alleviate and remove obstacles that prevent students from mastering the material. An Enhanced Text The table of contents was streamlined to enable instructors to spend more time on each topic. This added time will give students a better understanding of the concepts and help them to master the material. Real data and applications were updated, rewritten, and added to address more modern topics, and data was gathered from news sources, current events, industry, world events, and government. Exercises derived from other disciplines’ textbooks are included to show the relevance of the calculus to students’ majors. I hope these changes will give students a clear picture that the math they are learning exists beyond the classroom. Two new chapter tests were added: a Mid-Chapter Quiz and a Chapter Test. The Mid-Chapter quiz gives students the opportunity to discover any topics they might need to study further before they progress too far into the chapter. The Chapter Test allows students to identify and strengthen any weaknesses in advance of an exam. Several new section-level features were added to promote further mastery of the concepts. Concept Checks appear at the end of each section, immediately before the exercise sets. They ask non-computational questions designed to test students’ basic understanding of that sections’ concepts. Make a Decision exercises and examples ask open-ended questions that force students to apply concepts to real-world situations. Extended Applications are more in-depth, applied exercises requiring students to work with large data sets and often involve work in creating or analyzing models. I hope the combination of these new features with the existing features will promote a deeper understanding of the mathematics. Enhanced Resources Although the textbook often forms the basis of the course, today’s students often find greater value in an integrated text and technology program. With that in mind, I worked with the publisher to enhance the online and media resources available to students, to provide them with a complete learning program. Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. viii A Word from the Author An HM MathSPACE course has been developed with dynamic, algorithmic exercises tied to exercises within the text. These exercises provide students with unlimited practice for complete mastery of the topics. An additional resource for the 8th edition is a Multimedia Online eBook. This eBook breaks the physical constraints of a traditional text and binds a number of multimedia assets and features to the text itself. Based in Flash, students can read the text, watch the videos when they need extra explanation, view enlarged math graphs, and more. The eBook promotes multiple learning styles and provides students with an engaging learning experience. For students who work best in groups or whose schedules don’t allow them to come to office hours, Calc Chat is now available with this edition. Calc Chat (located at www.CalcChat.com) provides solutions to exercises. Calc Chat also has a moderated online forum for students to discuss any issues they may be having with their calculus work. I hope you enjoy the enhancements made to the eighth edition. I believe the whole suite of learning options available to students will enable any student to master applied calculus. Ron Larson Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.  A Plan for You as a Student ix A Plan for You as a Student Study Strategies Your success in mathematics depends on your active participation both in class and outside of class. Because the material you learn each day builds on the material you have learned previously, it is important that you keep up with your course work every day and develop a clear plan of study. This set of guidelines highlights key study strategies to help you learn how to study mathematics. Preparing for Class The syllabus your instructor provides is an invaluable resource that outlines the major topics to be covered in the course. Use it to help you prepare. As a general rule, you should set aside two to four hours of study time for each hour spent in class. Being prepared is the first step toward success. Before class: Review your notes from the previous class. Read the portion of the text that will be covered in class. Keeping Up Another important step toward success in mathematics involves your ability to keep up with the work. It is very easy to fall behind, especially if you miss a class. To keep up with the course work, be sure to: Attend every class. Bring your text, a notebook, a pen or pencil, and a calculator (scientific or graphing). If you miss a class, get the notes from a classmate as soon as possible and review them carefully. Participate in class. As mentioned above, if there is a topic you do not understand, ask about it before the instructor moves on to a new topic. Take notes in class. After class, read through your notes and add explanations so that your notes make sense to you. Fill in any gaps and note any questions you might have. Getting Extra Help It can be very frustrating when you do not understand concepts and are unable to complete homework assignments. However, there are many resources available to help you with your studies. Your instructor may have office hours. If you are feeling overwhelmed and need help, make an appointment to discuss your difficulties with your instructor. Find a study partner or a study group. Sometimes it helps to work through problems with another person. Special assistance with algebra appears in the Algebra Reviews, which appear throughout each chapter. These short reviews are tied together in the larger Algebra Review section at the end of each chapter. Preparing for an Exam The last step toward success in mathematics lies in how you prepare for and complete exams. If you have followed the suggestions given above, then you are almost ready for exams. Do not assume that you can cram for the exam the night before—this seldom works. As a final preparation for the exam: When you study for an exam, first look at all definitions, properties, and formulas until you know them. Review your notes and the portion of the text that will be covered on the exam. Then work as many exercises as you can, especially any kinds of exercises that have given you trouble in the past, reworking homework problems as necessary. Start studying for your exam well in advance (at least a week). The first day or two, study only about two hours. Gradually increase your study time each day. Be completely prepared for the exam two days in advance. Spend the final day just building confidence so you can be relaxed during the exam. For a more comprehensive list of study strategies, please visit college.hmco.com/pic/larsonBrief8e. Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. x Supplements Get more value from your textbook! Supplements for the Instructor Supplements for the Student Digital Instructor’s Solution Manual Student Solutions Guide Found on the instructor website, this manual contains This guide contains complete solutions to all the complete, worked-out solutions for all the exercises odd-numbered exercises in the text. in the text. Excel Made Easy CD This CD uses easy-to-follow videos to help students master mathematical concepts introduced in class. Electronic spreadsheets and detailed tutorials are included. Instructor and Student Websites The Instructor and Student websites at college.hmco.com/pic/larsonBrief8e contain an abundance of resources for teaching and learning, such as Note Taking Guides, a Graphing Calculator Guide, Digital Lessons, ACE Practice Tests, and a graphing calculator simulator. Instruction DVDs Hosted by Dana Mosely and captioned for the hearing-impaired, these DVDs cover all sections in the text. Ideal for promoting individual study and review, these comprehensive DVDs also support students in online courses or those who have missed a lecture. HM MathSPACE® HM MathSPACE encompasses the interactive online products and services integrated with Houghton Mifflin mathematics programs. Students and instructors can access HM MathSPACE content through text-specific Student and Instructor websites and via online learning platforms including WebAssign as well as Blackboard®, WebCT®, and other course management systems. HM Testing™ (powered by Diploma™) HM Testing (powered by Diploma) provides instructors with a wide array of algorithmic items along with improved functionality and ease of use. HM Testing offers all the tools needed to create, deliver, and customize multiple types of tests—including authoring and editing algorithmic questions. In addition to producing an unlimited number of tests for each chapter, including cumulative tests and final exams, HM Testing also offers instructors the ability to deliver tests online, or by paper and pencil. Online Course Content for Blackboard®, WebCT®, and eCollege® Deliver program or text-specific Houghton Mifflin content online using your institution’s local course management system. Houghton Mifflin offers homework, tutorials, videos, and other resources formatted for Blackboard®, WebCT®, eCollege®, and other course management systems. Add to an existing online course or create a new one by selecting from a wide range of powerful learning and instructional materials. Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.  Acknowledgments xi Acknowledgments I would like to thank the many people who have helped me at various stages of this project during the past 27 years. Their encouragement, criticisms, and suggestions have been invaluable. Thank you to all of the instructors who took the time to review the changes to this edition and provide suggestions for improving it. Without your help this book would not be possible. Reviewers of the Eighth Edition Lateef Adelani, Harris-Stowe State University, Saint Louis; Frederick Adkins, Indiana University of Pennsylvania; Polly Amstutz, University of Nebraska at Kearney; Judy Barclay, Cuesta College; Jean Michelle Benedict, Augusta State University; Ben Brink, Wharton County Junior College; Jimmy Chang, St. Petersburg College; Derron Coles, Oregon State University; David French, Tidewater Community College; Randy Gallaher, Lewis & Clark Community College; Perry Gillespie, Fayetteville State University; Walter J. Gleason, Bridgewater State College; Larry Hoehn, Austin Peay State University; Raja Khoury, Collin County Community College; Ivan Loy, Front Range Community College; Lewis D. Ludwig, Denison University; Augustine Maison, Eastern Kentucky University; John Nardo, Oglethorpe University; Darla Ottman, Elizabethtown Community & Technical College; William Parzynski, Montclair State University; Laurie Poe, Santa Clara University; Adelaida Quesada, Miami Dade College—Kendall; Brooke P. Quinlan, Hillsborough Community College; David Ray, University of Tennessee at Martin; Carol Rychly, Augusta State University; Mike Shirazi, Germanna Community College; Rick Simon, University of La Verne; Marvin Stick, University of Massachusetts—Lowell; Devki Talwar, Indiana University of Pennsylvania; Linda Taylor, Northern Virginia Community College; Stephen Tillman, Wilkes University; Jay Wiestling, Palomar College; John Williams, St. Petersburg College; Ted Williamson, Montclair State University Reviewers of the Seventh Edition George Anastassiou, University of Memphis; Keng Deng, University of Louisiana at Lafayette; Jose Gimenez, Temple University; Shane Goodwin, Brigham Young University of Idaho; Harvey Greenwald, California Polytechnic State University; Bernadette Kocyba, J. Sergeant Reynolds Community College; Peggy Luczak, Camden County College; Randall McNiece, San Jacinto College; Scott Perkins, Lake Sumter Community College Reviewers of Previous Editions Carol Achs, Mesa Community College; David Bregenzer, Utah State University; Mary Chabot, Mt. San Antonio College; Joseph Chance, University of Texas—Pan American; John Chuchel, University of California; Miriam E. Connellan, Marquette University; William Conway, University of Arizona; Karabi Datta, Northern Illinois University; Roger A. Engle, Clarion University of Pennsylvania; Betty Givan, Eastern Kentucky University; Mark Greenhalgh, Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. xii Acknowledgments Fullerton College; Karen Hay, Mesa Community College; Raymond Heitmann, University of Texas at Austin; William C. Huffman, Loyola University of Chicago; Arlene Jesky, Rose State College; Ronnie Khuri, University of Florida; Duane Kouba, University of California—Davis; James A. Kurre, The Pennsylvania State University; Melvin Lax, California State University—Long Beach; Norbert Lerner, State University of New York at Cortland; Yuhlong Lio, University of South Dakota; Peter J. Livorsi, Oakton Community College; Samuel A. Lynch, Southwest Missouri State University; Kevin McDonald, Mt. San Antonio College; Earl H. McKinney, Ball State University; Philip R. Montgomery, University of Kansas; Mike Nasab, Long Beach City College; Karla Neal, Louisiana State University; James Osterburg, University of Cincinnati; Rita Richards, Scottsdale Community College; Stephen B. Rodi, Austin Community College; Yvonne Sandoval-Brown, Pima Community College; Richard Semmler, Northern Virginia Community College—Annandale; Bernard Shapiro, University of Massachusetts—Lowell; Jane Y. Smith, University of Florida; DeWitt L. Sumners, Florida State University; Jonathan Wilkin, Northern Virginia Community College; Carol G. Williams, Pepperdine University; Melvin R. Woodard, Indiana University of Pennsylvania; Carlton Woods, Auburn University at Montgomery; Jan E. Wynn, Brigham Young University; Robert A. Yawin, Springfield Technical Community College; Charles W. Zimmerman, Robert Morris College My thanks to David Falvo, The Behrend College, The Pennsylvania State University, for his contributions to this project. My thanks also to Robert Hostetler, The Behrend College, The Pennsylvania State University, and Bruce Edwards, University of Florida, for their significant contributions to previous edi- tions of this text. I would also like to thank the staff at Larson Texts, Inc. who assisted with proofreading the manuscript, preparing and proofreading the art package, and checking and typesetting the supplements. On a personal level, I am grateful to my spouse, Deanna Gilbert Larson, for her love, patience, and support. Also, a special thanks goes to R. Scott O’Neil. If you have suggestions for improving this text, please feel free to write to me. Over the past two decades I have received many useful comments from both instructors and students, and I value these comments very highly. Ron Larson Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.  Features xiii How to get the most out of your textbook... CHAPTER OPENERS Each opener has an applied example of a core topic from the chapter. The section outline 2 Differentiation provides a comprehensive overview of the material being presented. © Schlegelmilch/Corbis Higher-order derivatives are used to determine the acceleration function of a 2.1 The Derivative and sports car. The acceleration function shows the changes in the car’s velocity. As the Slope of a Graph the car reaches its “cruising”speed, is the acceleration increasing or decreasing? (See Section 2.6, Exercise 45.) 2.2 Some Rules for Differentiation 2.3 Rates of Change: Velocity and Applications Marginals Differentiation has many real-life applications. The applications 2.4 The Product and listed below represent a sample of the applications in this chapter. Quotient Rules 2.5 The Chain Rule Sales, Exercise 61, page 137 2.6 Higher-Order Political Fundraiser, Exercise 63, page 137 Derivatives Make a Decision: Inventory Replenishment, Exercise 65, 2.7 Implicit page 163 Differentiation Modeling Data, Exercise 51, page 180 2.8 Related Rates Health: U.S. HIV/AIDS Epidemic, Exercise 47, page 187 114 SECTION 2.1 The Derivative and the Slope of a Graph 115 SECTION Section 2.1 OBJECTIVES Identify tangent lines to a graph at a point. The Derivative Approximate the slopes of tangent lines to graphs at points. A bulleted list of learning Use the limit definition to find the slopes of graphs at points. and the Slope Use the limit definition to find the derivatives of functions. objectives allows you the of a Graph Describe the relationship between differentiability and continuity. opportunity to preview what will be presented in the upcoming section. Tangent Line to a Graph y Calculus is a branch of mathematics that studies rates of change of functions. In (x3, y3) this course, you will learn that rates of change have many applications in real life. In Section 1.3, you learned how the slope of a line indicates the rate at which the (x2, y2) line rises or falls. For a line, this rate (or slope) is the same at every point on the (x4, y4) line. For graphs other than lines, the rate at which the graph rises or falls changes from point to point. For instance, in Figure 2.1, the parabola is rising more x quickly at the point 共x1, y1兲 than it is at the point 共x2, y2 兲. At the vertex 共x3, y3兲, the (x1, y1) graph levels off, and at the point 共x4, y4兲, the graph is falling. To determine the rate at which a graph rises or falls at a single point, you can find the slope of the tangent line at the point. In simple terms, the tangent line to F I G U R E 2. 1 The slope of a the graph of a function f at a point P共x1, y1兲 is the line that best approximates the nonlinear graph changes from one graph at that point, as shown in Figure 2.1. Figure 2.2 shows other examples of point to another. tangent lines. Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. xiv Features 45. MAKE A DECISION: FUEL COST A car is driven 15,000 miles a year and gets x miles per gallon. Assume that the average fuel cost is $2.95 per gallon. Find the annual cost NEW! MAKE A DECISION of fuel C as a function of x and use this function to Multi-step exercises reinforce your problem-solving complete the table. skills and mastery of concepts, as well as taking a real-life application further by testing what you know x 10 15 20 25 30 35 40 about a given problem to make a decision within the C context of the problem. dC兾dx g Who would benefit more from a 1 mile per gallon increase 61. MAKE A DECISION: NEGOTIATING A PRICE You in fuel efficiency—the driver who gets 15 miles per gallon decide to form a partnership with another business. Your or the driver who gets 35 miles per gallon? Explain. business determines that the demand x for your product is inversely proportional to the square of the price for x ≥ 5. (a) The price is $1000 and the demand is 16 units. Find the demand function. (b) Your partner determines that the product costs $250 per unit and the fixed cost is $10,000. Find the cost function. (c) Find the profit function and use a graphing utility to graph it. From the graph, what price would you negotiate with your partner for this product? Explain your reasoning. CONCEPT CHECK 1. What is the name of the line that best approximates the slope of a graph at a point? 2. What is the name of a line through the point of tangency and a second point on the graph? 3. Sketch a graph of a function whose derivative is always negative. 4. Sketch a graph of a function whose derivative is always positive. NEW! CONCEPT CHECK These non-computational questions appear at the end of each section and are designed to check your understanding of the concepts covered in that section. Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.  Features xv The Sum and Difference Rules DEFINITIONS AND THEOREMS The derivative of the sum or difference of two differentiable functions is the All definitions and theorems are highlighted sum or difference of their derivatives. for emphasis and easy recognition. d 关 f 共x) ⫹ g共x兲兴 ⫽ f⬘共x兲 ⫹ g⬘共x兲 Sum Rule dx d 关 f 共x兲 ⫺ g共x兲兴 ⫽ f⬘共x兲 ⫺ g⬘共x兲 Difference Rule dx Definition of Average Rate of Change If y ⫽ f 共x兲, then the average rate of change of y with respect to x on the interval 关a, b兴 is f 共b兲 ⫺ f 共a兲 Average rate of change ⫽ b⫺a ⌬y. ⫽ ⌬x x Note that f 共a兲 is the value of the function at the left endpoint of the interval, f 共b兲 is the value of the function at the right endpoint of the interval, and b ⫺ a is the width of the interval, as shown in Figure 2.18. 1 y g(x) = − 2 x 4 + 3x 3 − 2x Example 9 Using the Sum and Difference Rules 60 Find an equation of the tangent line to the graph of 50 1 40 g共x兲 ⫽ ⫺ x 4 ⫹ 3x 3 ⫺ 2x 30 2 at the point 共⫺1, ⫺ 2 兲. 3 20 Slope = 9 SOLUTION The derivative of g共x兲 is g⬘共x兲 ⫽ ⫺2x3 ⫹ 9x2 ⫺ 2, which implies x that the slope of the graph at the point 共⫺1, ⫺ 2 兲 is 3 −3 −2 1 2 3 4 5 7 − 10 − 20 (−1, − ) 3 2 Slope ⫽ g⬘共⫺1兲 ⫽ ⫺2共⫺1兲3 ⫹ 9共⫺1兲2 ⫺ 2 EXAMPLES ⫽2⫹9⫺2 FIGURE 2.16 ⫽9 There are a wide variety of as shown in Figure 2.16. Using the point-slope form, you can write the equation ✓CHECKPOINT 9 of the tangent line at 共⫺1, ⫺ 32 兲 as shown. relevant examples in the text, 134 C H A P T E R 2 Find an equation of the Differentiation y⫺ ⫺ 冢 32冣 ⫽ 9关x ⫺ 共⫺1兲兴 Point-slope form each titled for easy reference. tangent line to the graph of f 共x兲 ⫽ ⫺x2 ⫹ 3x ⫺ 2 at the Application y ⫽ 9x ⫹ 15 Equation of tangent line Many of the solutions are point 共2, 0兲. 2 presented graphically, analyti- Example 10 Modeling Revenue cally, and/or numerically to From 2000 through 2005, the revenue R (in millions of dollars per year) for provide further insight into Microsoft Corporation can be modeled by R ⫽ ⫺110.194t 3 ⫹ 993.98t2 ⫹ 1155.6t ⫹ 23,036, 0 ≤ t ≤ 5 mathematical concepts. where t represents the year, with t ⫽ 0 corresponding to 2000. At what rate was Examples using a real-life Microsoft’s revenue changing in 2001? (Source: Microsoft Corporation) Microsoft Revenue R SOLUTION One way to answer this question is to find the derivative of the situation are identified with 45,000 revenue model with respect to time. the symbol. 40,000 dR (in millions of dollars) 35,000 ⫽ ⫺330.582t 2 ⫹ 1987.96t ⫹ 1155.6, 0 ≤ t ≤ 5 30,000 dt Revenue 25,000 20,000 In 2001 (when t ⫽ 1), the rate of change of the revenue with respect to time is CHECKPOINT 15,000 Slope ≈ 2813 given by 10,000 5,000 ⫺330.582共1兲2 ⫹ 1987.96共1兲 ⫹ 1155.6 ⬇ 2813. After each example, a similar 1 2 3 4 5 t Because R is measured in millions of dollars and t is measured in years, it follows that the derivative dR兾dt is measured in millions of dollars per year. So, at the end problem is presented to Year (0 ↔ 2000) FIGURE 2.17 of 2001, Microsoft’s revenue was increasing at a rate of about $2813 million per year, as shown in Figure 2.17. allow for immediate practice, and to further reinforce your ✓CHECKPOINT 10 From 1998 through 2005, the revenue per share R (in dollars) for McDonald’s understanding of the concepts Corporation can be modeled by just learned. R ⫽ 0.0598t 2 ⫺ 0.379t ⫹ 8.44, 8 ≤ t ≤ 15 where t represents the year, with t ⫽ 8 corresponding to 1998. At what rate was McDonald’s revenue per share changing in 2003? (Source: McDonald’s Corporation) Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. xvi Features g g D I S C O V E RY D I S C O V E RY These projects appear before selected topics Use a graphing utility to graph f 共x兲 ⫽ 2x 3 ⫺ 4x 2 ⫹ 3x ⫺ 5. On the same and allow you to explore concepts on your screen, sketch the graphs of y ⫽ x ⫺ 5, y ⫽ 2x ⫺ 5, and y ⫽ 3x ⫺ 5. own. These boxed features are optional, so Which of these lines, if any, appears to be tangent to the graph of f at the point 共0, ⫺5兲? Explain your reasoning. they can be omitted with no loss of continuity in the coverage of material. 146 CHAPTER 2 Differentiation TECHNOLOGY Modeling a Demand Function To model a demand function, you need data that indicate how many units of a product will sell at a given price. As you might imagine, such data are not easy to obtain for a new product. After a product has been on the market awhile, however, its sales history can provide the necessary data. As an example, consider the two bar graphs shown below. From these graphs, you can see that from 2001 through 2005, the number of prerecorded DVDs sold increased from about 300 million to about 1100 million. During that time, the price per unit dropped from an average price of about $18 to an average price of about $15. (Source: Kagan Research, LLC) Prerecorded DVDs Prerecorded DVDs x p 1200 20 Average price per unit 18 Number of units sold 1000 16 (in millions) 14 (in dollars) 800 12 600 10 8 400 6 200 4 2 t t 1 2 3 4 5 1 2 3 4 5 Year (1 ↔ 2001) Year (1 ↔ 2001) SECTION 2.1 The Derivative and the Slope of a Graph 121 In many applications, it is convenient to use a variable other than x as the The information in the two bar graphs is combined in the table, where x independent variable. Example 7 shows a function that uses t as the independent represents the units sold (in millions) and p represents the price (in dollars). variable. t 1 2 3 4 5 TECHNOLOGY Example 7 Finding a Derivative x 291.5 507.5 713.0 976.6 1072.4 You can use a graphing p Find the derivative of y with respect to t for the function 18.40 17.11 15.83 15.51 14.94 utility to confirm the result given in Example 7. One 2 y⫽. way to do this is to choose a t By entering the ordered pairs 共x, p兲 into a graphing utility, you can find point on the graph of y ⫽ 2兾t, that the power model for the demand for prerecorded DVDs is: SOLUTION Consider y ⫽ f 共t兲, and use the limit process as shown. such as 共1, 2兲, and find the p ⫽ 44.55x⫺0.155, 291.5 ≤ x ≤ 1072.4. A graph of this demand function equation of the tangent line at dy f 共t ⫹ ⌬t兲 ⫺ f 共t兲 and its data points is shown below ⫽ lim Set up difference quotient. that point. Using the derivative dt ⌬t→0 ⌬t found in the example, you 2 2 20 know that the slope of the ⫺ t ⫹ ⌬t t tangent line when t ⫽ 1 is ⫽ lim Use f 共t兲 ⫽ 2兾t. ⌬t→0 ⌬t m ⫽ ⫺2. This means that the 2t ⫺ 2t ⫺ 2 ⌬t tangent line at the point 共1, 2兲 is t共t ⫹ ⌬t兲 y ⫺ y1 ⫽ m共t ⫺ t1兲 ⫽ lim Expand terms. ⌬t→0 ⌬t 200 1100 ⫺2 ⌬t 5 y ⫺ 2 ⫽ ⫺2共t ⫺ 1兲 or ⫽ lim Factor and divide out. ⌬t→0 t共⌬t兲共t ⫹ ⌬t兲 y ⫽ ⫺2t ⫹ 4. ⫺2 By graphing y ⫽ 2兾t and y ⫽ ⫽ lim Simplify. ⌬t→0 t共t ⫹ ⌬t兲 ⫺2t ⫹ 4 in the same viewing 2 window, as shown below, you ⫽⫺2 Evaluate the limit. t can confirm that the line is tangent to the graph at the So, the derivative of y with respect to t is point 共1, 2兲.* dy 2 ⫽ ⫺ 2. 4 dt t TECHNOLOGY BOXES Remember that the derivative of a function gives you a formula for finding the −6 6 slope of the tangent line at any point on the graph of the function. For example, the slope of the tangent line to the graph of f at the point 共1, 2兲 is given by These boxes appear throughout the text and f⬘ 共1兲 ⫽ ⫺ 2 12 ⫽ ⫺2. provide guidance on using technology to −4 To find the slopes of the graph at other points, substitute the t-coordinate of the ease lengthy calculations, present a graphical point into the derivative, as shown below. Point t-Coordinate Slope solution, or discuss where using technology 2 1 ✓CHECKPOINT 7 共2, 1兲 t⫽2 m ⫽ f⬘ 共2兲 ⫽ ⫺ 22 ⫽⫺ 2 can lead to misleading or wrong solutions. Find the derivative of y with 2 1 respect to t for the function 共⫺2, ⫺1兲 t ⫽ ⫺2 m ⫽ f⬘ 共⫺2兲 ⫽ ⫺ ⫽⫺ 共⫺2兲2 2 y ⫽ 4兾t. *Specific calculator keystroke instructions for operations in this and other technology boxes can be found at college.hmco.com/info/larsonapplied. Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.  Features xvii 78. Credit Card Rate The average annual rate r (in percent form) for commercial bank credit cards from 2000 through 2005 can be modeled by TECHNOLOGY EXERCISES r ⫽ 冪⫺1.7409t4 ⫹ 18.070t3 ⫺ 52.68t2 ⫹ 10.9t ⫹ 249 Many exercises in the text can be solved with or where t represents the year, with t ⫽ 0 corresponding to without technology. The symbol identifies 2000. (Source: Federal Reserve Bulletin) exercises for which students are specifically (a) Find the derivative of this model. Which differentiation instructed to use a graphing calculator or rule(s) did you use? a computer algebra system to solve the (b) Use aGraphical, Numerical, graphing utility to graph the and Analytic derivative on theAnalysis In Exercises 63–66, interval 0 ≤ t ≤ 5. use a graphing utility to graph f on problem. Additionally, the symbol denotes the interval [ⴚ2, 2]. Complete the table by graphically exercises best solved by using a spreadsheet. (c) Use the trace feature estimating thetoslopes find theofyears theduring graphwhich at thethegiven points. finance rate was Then changing evaluate thetheslopes most. analytically and compare (d) Use the trace your results featurewith those to find obtained the years duringgraphically. which the finance rate was changing the least. x ⫺2 ⫺2 3 ⫺1 ⫺2 1 0 12 1 32 2 f 共x 兲 f⬘ 共x兲 63. f 共x兲 ⫽ 14x 3 64. f 共x兲 ⫽ 12x 2 ( ) 1 3 65. f 共x兲 ⫽ ⫺ 2x 66. f 共x兲 ⫽ ⫺ 32x 2 57. Income Distribution Using the Lorenz curve in Exercise 56 and a spreadsheet, complete the table, which lists the percent of total income earned by each quintile in the United States in 2005. Quintile Lowest 2nd 3rd 4th Highest Percent Business Capsule BUSINESS CAPSULES Business Capsules appear at the ends of numerous sections. These capsules and their accompanying exercises deal with business situations that are related to the mathematical concepts covered in AP/Wide World Photos the chapter. n 1978 Ben Cohen and Jerry Greenfield used I their combined life savings of $8000 to convert an abandoned gas station in Burlington, Vermont into their first ice cream shop. Today, Ben & Jerry’s Homemade Holdings, Inc. has over 600 scoop shops in 16 countries. The company’s three-part mission statement emphasizes product quality, economic reward, and a commitment to the community. Ben & Jerry’s contributes a minimum of $1.1 million annually through corporate philanthropy that is primarily employee led. 73. Research Project Use your school’s library, the Internet, or some other reference source to find information on a company that is noted for its philanthropy and community commitment. (One such business is described above.) Write a short paper about the company. Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. xviii Features 196 CHAPTER 2 Differentiation Algebra Review Simplifying Algebraic Expressions To be successful in using derivatives, you must be good at simplifying algebraic expres- sions. Here are some helpful simplification techniques. ALGEBRA REVIEWS TECHNOLOGY 1. Combine like terms. This may involve expanding an expression by multiplying factors. 2. Divide out like factors in the numerator and denominator of an expression. These appear throughout each chapter and offer Symbolic algebra systems can simplify 3. Factor an expression. algebraic expressions. If you algebraic support at point of use. Many of the have access to such a system, 4. Rationalize a denominator. try using it to simplify the 5. Add, subtract, multiply, or divide fractions. reviews are then revisited in the Algebra Review expressions in this Algebra Review. at the end of the chapter, where additional details Example 1 Simplifying a Fractional Expression of examples with solutions and explanations are a. 共x ⫹ ⌬x兲2 ⫺ x 2 x 2 ⫹ 2x共⌬x兲 ⫹ 共⌬x兲2 ⫺ x2 ⌬x ⫽ ⌬x Expand expression. provided. ⫽ 2x共⌬x兲 ⫹ 共⌬x兲2 ⌬x Combine like terms. ⌬x共2x ⫹ ⌬x兲 ⫽ Factor. ⌬x ⫽ 2x ⫹ ⌬x, ⌬x ⫽ 0 Divide out like factors. 共x 2 ⫺ 1兲共⫺2 ⫺ 2x兲 ⫺ 共3 ⫺ 2x ⫺ x 2兲共2兲 b. 共x 2 ⫺ 1兲2 共⫺2x 2 ⫺ 2x 3 ⫹ 2 ⫹ 2x兲 ⫺ 共6 ⫺ 4x ⫺ 2x 2兲 ⫽ Expand expression. 共x 2 ⫺ 1兲2 ⫺2x 2 ⫺ 2x 3 ⫹ 2 ⫹ 2x ⫺ 6 ⫹ 4x ⫹ 2x 2 ⫽ Remove parentheses. Algebra Review ⫺2x 3 ⫹ 6x ⫺ 4 共x 2 ⫺ 1兲2 ⫽ Combine like terms. 共x 2 ⫺ 1兲2 For help in evaluating the 2x ⫹ 1 3x共2兲 ⫺ 共2x ⫹ 1兲共3兲 c. 2 冢 冣冤 冥 expressions in Examples 3–6, 3x 共3x兲2 see the review of simplifying ⫽2 冢2x3x⫹ 1冣冤 6x ⫺共3x共6x兲 ⫹ 3兲冥 2 Multiply factors. 2共2x ⫹ 1兲共6x ⫺ 6x ⫺ 3兲 Multiply fractions and fractional expressions on page ⫽ 共3x兲3 2共2x ⫹ 1兲共⫺3兲 remove parentheses. Combine like terms 196. ⫽ 3共9兲x 3 and factor. ⫺2共2x ⫹ 1兲 ⫽ Divide out like factors. 9x 3 STUDY TIP When differentiating functions involving radicals, you should STUDY TIPS rewrite the function with rational Scattered throughout the text, study tips address exponents. For instance, you special cases, expand on concepts, and help you should rewrite y ⫽ 冪 3 x as to avoid common errors. y ⫽ x , and you should rewrite 1兾3 1 y⫽ 3 x4 as y ⫽ x⫺4兾3. 冪 STUDY TIP In real-life problems, it is important to list the units of measure for a rate of change. The units for ⌬y兾⌬x are “y-units” per “x-units.” For example, if y is measured in miles and x is measured in hours, then ⌬y兾⌬x is measured in miles per hour. Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.  Features xix SECTION 2.3 Rates of Change: Velocity and Marginals 149 The following warm-up exercises involve skills that were covered in earlier sections. You will use Skills Review 2.3 SKILLS REVIEW these skills in the exercise set for this section. For additional help, review Sections 2.1 and 2.2. In Exercises 1 and 2, evaluate the expression. These exercises at the beginning of each exercise 1. ⫺63 ⫺ 共⫺105兲 2. ⫺37 ⫺ 54 set help students review skills covered in previous 21 ⫺ 7 16 ⫺ 3 sections. The answers are provided at the back In Exercises 3–10, find the derivative of the function. 3. y ⫽ 4x 2 ⫺ 2x ⫹ 7 4. y ⫽ ⫺3t 3 ⫹ 2t 2 ⫺ 8 of the text to reinforce understanding of the skill 5. s ⫽ ⫺16t 2 ⫹ 24t ⫹ 30 1 6. y ⫽ ⫺16x 2 ⫹ 54x ⫹ 70 1 7. A ⫽ 10共⫺2r3 ⫹ 3r 2 ⫹ 5r兲 8. y ⫽ 9共6x 3 ⫺ 18x 2 ⫹ 63x ⫺ 15兲 sets learned.