applied calculus for the life and social sciences

4. Applied Calculus for the Life and Social Sciences R O N L A R S O N The Pennsylvania State University The Behrend College with the assistance of DAVID C. FALVO The Pennsylvania State University The Behrend College HOUGHTON MIFFLIN HARCOURT PUBLISHING COMPANY Boston New York Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

5. Publisher: Richard Stratton Senior Sponsoring Editor: Cathy Cantin Senior Marketing Manager: Jennifer Jones Development Editor: Peter Galuardi Associate Editor: Jeannine Lawless Project Editor: Margaret M. Kearney/Sarah S. Evans Senior Media Producer: Douglas Winicki Senior Content Manager: Maren Kunert Art and Design Manager: Jill Haber Cover Design Manager: Anne S. Katzeff Senior Photo Editor: Jennifer Meyer Dare Senior Composition Buyer: Chuck Dutton Manager of New Title Project Management: Pat ONeill Editorial Assistant: Amy Haines Cover photo: Multiple jellyfish. Copyright iStock International Inc. Copyright 2009 by Houghton Mifflin Harcourt Publishing 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 Harcourt Publishing Company unless such copying is expressly permitted by federal copyright law. Address inquiries to College Permissions, Houghton Mifflin Harcourt Publishing Company, 222 Berkeley Street, Boston, MA 02116-3764. Printed in the U.S.A. Library of Congress Control Number: 2008924205 Instructors examination copy ISBN-10: 0-547-20367-5 ISBN-13: 978-0-547-20367-6 For orders, use student text ISBNs ISBN-10: 0-618-96259-X ISBN-13: 978-0-618-96259-4 123456789DOW12 11 10 09 08 Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

6. Contents v Contents From the Desk of Ron Larson ix Goals for this Text x A Plan for You as a Student xi Supplements xii Acknowledgments xiii Features xiv 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 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 Life Science Capsule: Nursing Careers 81 1.5 Limits 82 1.6 Continuity 94 Chapter 1 Algebra Review 104 Chapter Summary and Study Strategies 106 Review Exercises 108 Chapter Test 112 Differentiation 113 2.1 The Derivative and the Slope of a Graph 114 2.2 Some Rules for Differentiation 125 2.3 Rates of Change 137 Mid-Chapter Quiz 146 2.4 The Product and Quotient Rules 147 Social Science Capsule: Social Work 157 2.5 The Chain Rule 158 2.6 Higher-Order Derivatives 167 Chapter 2 Algebra Review 174 Chapter Summary and Study Strategies 176 Review Exercises 178 Chapter Test 182 0 1 2 Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

7. vi Contents Applications of the Derivative 183 3.1 Increasing and Decreasing Functions 184 3.2 Extrema and the First-Derivative Test 193 3.3 Concavity and the Second-Derivative Test 203 Life Science Capsule: Biomedical Science 212 3.4 Optimization Problems 213 Mid-Chapter Quiz 222 3.5 Asymptotes 223 3.6 Curve Sketching: A Summary 233 3.7 Differentials: Linear Approximation 242 Chapter 3 Algebra Review 249 Chapter Summary and Study Strategies 251 Review Exercises 253 Chapter Test 257 Exponential and Logarithmic Functions 258 4.1 Exponential Functions 259 4.2 Natural Exponential Functions 265 4.3 Derivatives of Exponential Functions 271 Mid-Chapter Quiz 279 4.4 Logarithmic Functions 280 4.5 Derivatives of Logarithmic Functions 289 Life Science Capsule: Agriculture and Food Science 297 4.6 Exponential Growth and Decay 298 Chapter 4 Algebra Review 306 Chapter Summary and Study Strategies 308 Review Exercises 310 Chapter Test 314 Trigonometric Functions 315 5.1 Radian Measure of Angles 316 5.2 The Trigonometric Functions 324 Mid-Chapter Quiz 335 5.3 Graphs of Trigonometric Functions 336 5.4 Derivatives of Trigonometric Functions 346 Social Science Capsule: Landscape Architecture 354 Chapter 5 Algebra Review 355 Chapter Summary and Study Strategies 357 Review Exercises 359 Chapter Test 362 Integration and Its Applications 363 6.1 Antiderivatives and Indefinite Integrals 364 6.2 Integration by Substitution and The General Power Rule 375 6.3 Exponential and Logarithmic Integrals 384 Mid-Chapter Quiz 391 6.4 Area and the Fundamental Theorem of Calculus 392 6.5 The Area of a Region Bounded by Two Graphs 403 Social Science Capsule: USDA 410 6.6 Volumes of Solids of Revolution 411 Chapter 6 Algebra Review 418 Chapter Summary and Study Strategies 420 Review Exercises 422 Chapter Test 426 3 4 5 6 Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

8. Contents vii Techniques of Integration 427 7.1 Integration by Parts 428 7.2 Partial Fractions and Logistic Growth 436 7.3 Integrals of Trigonometric Functions 446 Mid-Chapter Quiz 455 7.4 The Definite Integral as the Limit of a Sum 456 7.5 Numerical Integration 462 7.6 Improper Integrals 471 Life Science Capsule: Orthodontics 480 Chapter 7 Algebra Review 481 Chapter Summary and Study Strategies 483 Review Exercises 485 Chapter Test 488 Matrices 489 8.1 Systems of Linear Equations in Two Variables 490 8.2 Systems of Linear Equations in More Than Two Variables 498 Life Science Capsule: Chemical and Material Science 506 8.3 Matrices and Systems of Linear Equations 507 Mid-Chapter Quiz 517 8.4 Operations with Matrices 518 8.5 The Inverse of a Matrix 529 Chapter 8 Algebra Review 538 Chapter Summary and Study Strategies 540 Review Exercises 542 Chapter Test 546 Functions of Several Variables 547 9.1 The Three-Dimensional Coordinate System 548 9.2 Surfaces in Space 556 9.3 Functions of Several Variables 565 9.4 Partial Derivatives 573 Social Science Capsule: Empirisoft 583 Mid-Chapter Quiz 584 9.5 Extrema of Functions of Two Variables 585 9.6 Least Squares Regression Analysis 594 9.7 Double Integrals and Area in the Plane 604 9.8 Applications of Double Integrals 612 Chapter 9 Algebra Review 620 Chapter Summary and Study Strategies 622 Review Exercises 624 Chapter Test 628 7 8 9 Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

9. viii Contents Differential Equations 629 10.1 Solutions of Differential Equations 630 10.2 Separation of Variables 636 Life Science Capsule: Radiation Oncology 642 Mid-Chapter Quiz 643 10.3 First-Order Linear Differential Equations 644 10.4 Applications of Differential Equations 649 Chapter 10 Algebra Review 657 Chapter Summary and Study Strategies 659 Review Exercises 660 Chapter Test 663 Probability and Calculus 664 11.1 Discrete Probability 665 11.2 Continuous Random Variables 674 Life Science Capsule: Occupational Health and Safety 680 11.3 Expected Value and Variance 681 Chapter 11 Algebra Review 691 Chapter Summary and Study Strategies 693 Review Exercises 694 Chapter Test 697 Appendices Appendix A: Differentiation and Integration Formulas A2 Appendix B: Additional Topics in Differentiation A3 B.1 Implicit Differentiation A3 B.2 Related Rates A10 Appendix C: Probability and Probability Distributions (web only)* C.1 Probability C.2 Probability Computations C.3 Conditional Probability C.4 Tree Diagrams and Bayes Theorem C.5 Probability Distributions C.6 Normal Distribution C.7 Binomial Distribution 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)* Answers to Selected Exercises A17 Answers to Checkpoints A109 Index A125 *Available at the text-specific website: college.hmco.com/pic/larsonACLS 10 11 Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

10. A Word from the Author ix From the Desk of Ron Larson This text was written in response to requests from several users of my Calculus: An Applied Approach book to write a book intended for students majoring in the Life and Social Sciences. Enrollments in environmental science, biology, nursing, and social science programs are increasing each year. Most students in these programs are required to take one or more applied calculus courses. However, most traditional applied calculus texts have a strong business focus, which many students may find dull or irrelevant. It was my goal to write a textbook that would teach calculus concepts within a context best suited for these students. Im excited about this new textbook because it acknowledges where students are when they enter the courseand where they should be when they complete it. We review the precalculus that students have studied previously (in Chapter 0, skills reviews, notes, study tips, and algebra reviews throughout the text), and present a solid applied calculus course that balances understanding of concepts with the practicality of skills. I am also excited about this textbook program because it is being published as part of a whole series of textbooks tailored to the needs of college algebra and applied calculus students majoring in business, biology, and related courses: College Algebra with Applications for Business and the Life Sciences College Algebra and Calculus: An Applied Approach (2010) Calculus: An Applied Approach, 8th edition Brief Calculus: An Applied Approach, 8th edition Applied Calculus for the Life and Social Sciences This new textbook program helps students learn the math in the ways I have found most effective for my students, by practicing their problem-solving skills and reinforcing their understanding in the context of actual problems they may encounter in their lives and careers. I hope you and your students enjoy Applied Calculus for the Life and Social Sciences. Feel free to tell me what you think about it. Over the years, I have received many useful comments from both instructors and students, and I value these comments very much. Ron Larson Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

11. x Goals for This Text Goals for This Text Establish a Solid Foundation in Calculus Many effective tools are incorporated throughout the text to help students master calculus concepts. These features help students evaluate and reinforce their understanding of the math. After each worked Example, a Checkpoint offers the opportunity for immediate practice. At the end of each section before the Section Exercises, a Concept Check poses noncomputa- tional questions designed to test students basic understanding of that sections concepts. Each exercise set begins with a Skills Review of cumulative exercises that test prerequisite skills from earlier sections. The Mid-Chapter Quiz offers frequent opportunities for self-assessment so students can discov- er any topics they might need to study further before they progress too far into the chapter. The Chapter Summary summarizes skills presented in the chapter and correlates each skill to Review Exercises for extra practice. The Study Strategies provide invaluable tips for overcoming common study obstacles. The Chapter Test enables students to identify and strengthen any weaknesses before their exam. Present Real-World Problems to Motivate Interest and Understanding Great care was taken in choosing the applications for this text, drawing them from news sources, current events, industry data, world events, and government data. If students can see calculus as it applies to the world around them, they will see calculus relevance. Ample Review Students coming into the course approach from many different paths. Students who have not taken a math course recently may have difficulty at first. This book contains a number of features designed to get students up to speed quickly and keep them there, including: Algebra Reviews appear in each chapter to review algebra concepts relevant to the calculus being studied. Prerequisite Skills Reviews appear before each exercise set and review important material students will need to use in order to successfully complete the exercises. Enhance Understanding Using Technology Students can visualize the math by using powerful technology, such as graphing calculators and spreadsheet software, and so develop a deeper comprehension of mathematical concepts. Optional Technology boxes feature exercises that offer students opportunities to practice using these tools. The icon in the exercises suggests when a graphing calculator or other technology tool can be used. The icon appears when spreadsheet software, such as Microsofts Excel, can be incorporated. Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

12. A Plan for You as a Student xi 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 are located 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 beforethis 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/larsonACLS. A Plan for You as a Student Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

13. xii Supplements Supplements for the Instructor Online Complete Solutions Manual Found on the instructor website, this manual contains the complete, worked-out solutions for all the exercises in the text. Supplements for the Student Student Solutions Manual This manual contains worked-out solutions to all odd-numbered exercises in the text. Excel Made Easy CD This CD offers easy-to-follow videos to help students use Microsoft Excel to 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/larsonACLS contain an abundance of resources for teaching and learning, such as Note Taking Guides, Digital Art, ACE Practice Quizzes and a graphing calculator simulator. Instructional 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 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 testsincluding 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. Diploma is currently in use at thousands of college and university campuses throughout the United States and Canada. Houghton Mifflins unique partnership with SMARTHINKING brings students live, online tutorial support when they need it most. SMARTHINKING provides an easy-to-use and effective online, text-specific tutoring service. A dynamic Whiteboard and Graphing Calculator function enables students and e-structors to collaborate easily. Visit www.smarthinking.com for more information. Online Course Content for Blackboard/WebCT Houghton Mifflin can provide you with valuable content to include in your existing Blackboard and WebCT systems. This text-specific content enables instructors to teach all or part of their course online. Contact your Houghton Mifflin sales rep for cartridge availability. WebAssign Developed by teachers for teachers, WebAssign allows instructors to focus on teaching rather than grading. Instructors can create assignments from a ready-to-use database of algorithmic questions based on end-of-section exercises or write and customize their own. With WebAssign, students can access homework, quizzes, and tests anytime of day or night. Get more value from your textbook! Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

14. Acknowledgments xiii I would like to thank the many people who have helped at the various stages of this project. Their encouragement, criticisms, and suggestions have been invaluable. Thank you to all of the instructors who took the time to provide feedback on the material for this book. I would also like to thank the reviewers of the other books in this series. Reviewers Dr. Mohammad Z. Abu-Sbeih, King Fahd University of Petroleum and Minerals; Lateef Adelani, HarrisStowe State University; Frederick Adkins, Indiana University of Pennsylvania; Polly Amstutz, University of Nebraska at Kearney; Judy Barclay, Cuesta College, Saint Louis; Jean Michelle Benedict, Augusta State University; Ben Brink, Wharton County Junior College; Jimmy Chang, St. Petersburg College; Derron Coles, Oregon State University; Steve Comer, The CitadelMilitary College of SC; 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; Dr. Eric Marland, Appalachain State University; John Nardo, Oglethorpe University; Darla Ottman, Elizabeth- town Community & Technical College; William Parzynski, Montclair State University; Laurie Poe, Santa Clara University; Adelaida Quesada, Miami Dade CollegeKendall; 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 MassachusettsLowell; Devki Talwar, Indiana University of Pennsylvania; Linda Taylor, Northern Virginia Community College; Stephen Tillman, Wilkes University; Blake Thornton, Washington University in St. Louis; Donna Tupper, CCBC-Essex; Jay Wiestling, Palomar College; John Williams, St. Petersburg College; Ted Williamson, Montclair State University. 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. 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 ONeil. If you have suggestions for improving this text, please feel free to write to me. Over the years I have received many useful comments from both instructors and students, and I value these comments highly. Ron Larson Acknowledgments Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

15. xiv Features How to get the most out of your textbook . . . 114 CHAPTER 2 Differentiation Identify tangent lines to a graph at a point. Approximate the slopes of tangent lines to graphs at points. Use the limit definition to find the slopes of graphs at points. Use the limit definition to find the derivatives of functions. Describe the relationship between differentiability and continuity. Tangent Line to a Graph Calculus is a branch of mathematics that studies rates of change of functions. In 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 line rises or falls. For a line, this rate (or slope) is the same at every point on the 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 quickly at the point than it is at the point At the vertex the graph levels off, and at the point 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 the graph of a function f at a point is the line that best approximates the graph at that point, as shown in Figure 2.1. Figure 2.2 shows other examples of tangent lines. Px1, y1 x4, y4, x3, y3,x2, y2 .x1, y1 Section 2.1 The Derivative and the Slope of a Graph x y (x1 , y1 ) (x2, y2) (x3 , y3 ) (x4 , y4 ) FIGURE 2.1 The slope of a nonlinear graph changes from one point to another. 113 2Differentiation Demography, a field within sociology, studies changes in population. Derivatives are used to determine the rates of change of populations over time. Recently, growth of Japans population has been slowing down. (See Section 2.3, Exercise 23.) Differentiation has many real-life applications. The applications listed below represent a sample of the applications in this chapter. Psychology: Migraine Prevalence, Exercise 64, page 136 Center of Population, Exercise 28, page 145 Biology: Wildlife Management, Exercise 60, page 156 Make a Decision: Environment, Exercise 63, page 157 Fruit and Vegetable Consumption, Exercise 73, page 166 Applications IainMasterton/Alamy 2.1 The Derivative and the Slope of a Graph 2.2 Some Rules for Differentiation 2.3 Rates of Change 2.4 The Product and Quotient Rules 2.5 The Chain Rule 2.6 Higher-Order Derivatives CHAPTER OPENERS Each opener has an applied example of a core topic from the chapter. The section outline provides a comprehensive overview of the material being presented. SECTION OBJECTIVES A bulleted list of learning objectives allows you the opportunity to preview what will be presented in the upcoming section. Establish a Solid Foundation in Applied Calculus Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

16. Features xv The Sum and Difference Rules The derivative of the sum or difference of two differentiable functions is the sum or difference of their derivatives. Sum Rule Difference Rule d dx fx gx fx gx d dx fx) gx fx gx Definition of Instantaneous Rate of Change The instantaneous rate of change (or simply rate of change) of at is the limit of the average rate of change on the interval as approaches 0. If is a distance and is time, then the rate of change is a velocity.xy lim x0 y x lim x0 fx x fx x x x, x x,x y fxDEFINITIONS AND THEOREMS All definitions and theorems are highlighted for emphasis and easy reference. 154 CHAPTER 2 Differentiation Application Example 9 Rate of Change of Systolic Blood Pressure As blood moves from the heart through the major arteries out to the capillaries and back through the veins, the systolic blood pressure continuously drops. Consider a person whose systolic blood pressure P (in millimeters of mercury) is given by where t is measured in seconds. At what rate is the blood pressure changing 5 seconds after blood leaves the heart? SOLUTION Begin by applying the Quotient Rule. Quotient Rule Simplify. When the rate of change is millimeters per second. So, the pressure is dropping at a rate of 1.48 millimeters per second when seconds. CHECKPOINT 9 In Example 9, find the rate at which systolic blood pressure is changing at each time shown in the table below. Describe the changes in blood pressure as the blood moves away from the heart. t 5 2005 262 1.48 t 5, 200t t2 12 50t3 50t 50t3 250t t2 12 dP dt t2 150t 25t2 1252t t2 12 0 t 10P 25t2 125 t2 1 , t 0 1 2 3 4 5 6 7 dP dt aorta vein vein artery artery EXAMPLES There are a wide variety of relevant examples in the text, each titled for easy reference. Many of the solutions are presented graphically, analytically, and/or numerically to provide further insight into mathematical concepts. Examples using a real-life situation are identified with an icon. CHECKPOINT After each example, a similar problem is presented to allow for immediate practice, and to further reinforce your understanding of the concepts just learned. Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

17. xvi Features 174 CHAPTER 2 Differentiation Algebra Review Simplifying Algebraic Expressions To be successful in using derivatives, you must be good at simplifying algebraic expressions. Here are some helpful simplification techniques. 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. 3. Factor an expression. 4. Rationalize a denominator. 5. Add, subtract, multiply, or divide fractions. Example 1 Simplifying Fractional Expressions a. Expand expression. Combine like terms. Factor. Divide out like factors. b. Expand expression. Remove parentheses. Combine like terms. c. Multiply factors. Divide out like factors. 22x 1 9x3 Combine like terms and factor. 22x 13 39x3 Multiply fractions and remove parentheses. 22x 16x 6x 3 3x3 22x 1 3x 6x 6x 3 3x2 22x 1 3x 3x2 2x 13 3x2 2x3 6x 4 x2 12 2x2 2x3 2 2x 6 4x 2x2 x2 12 2x2 2x3 2 2x 6 4x 2x2 x2 12 x2 12 2x 3 2x x22 x2 12 x 0 2x x, x2x x x 2xx x2 x x x2 x2 x x2 2xx x2 x2 x Symbolic algebra systems can simplify algebraic expressions. If you have access to such a system, try using it to simplify the expressions in this Algebra Review. T E C H N O L O G Y When applying the Quotient Rule, it is suggested that you enclose all factors and derivatives in symbols of grouping, such as parentheses. Also, pay special attention to the subtraction required in the numerator. For help in evaluating expressions such as the one in Example 4, see the Chapter 2 Algebra Review on page 175, Example 2(d). Algebra Review ALGEBRA REVIEWS These appear throughout each chapter and offer algebraic support at point of use. Many of the reviews are then revisited in the Algebra Review at the end of the chapter, where additional details of examples with solutions and explanations are provided. S T U D Y T I P In real-life problems, it is important to list the units of measure for a rate of change. The units for are -units per -units. For example, if is measured in miles and is measured in hours, then is measured in miles per hour. yxx yxyyx S T U D Y T I P An interpretation of the Constant Rule says that the tangent line to a constant function is the function itself. Find an equation of the tangent line to at x 3.fx 4 STUDY TIPS Scattered throughout the text, study tips address special cases, expand on concepts, and help you to avoid common errors. Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

18. Features xvii S o c i a l S c i e n c e C a p s u l e Social work is a profession for those with a strong desire to help improve peoples lives. About 90% of social workers are in health care and social assistance industries, as well as state and local government agencies. Most states require 3000 hours of supervised clinical experi- ence for licensure. Employment of social workers is expected to increase faster than the average for all occupations through 2014. 70. Research Project Use your schools library, the Internet, or some other reference source to find information about the many career paths possible for someone with a degree in social work. Choose one career path and write a short paper about it. Bill Aron/PhotoEdit LIFE AND SOCIAL SCIENCE CAPSULES Life and Social Science Capsules appear at the ends of several sections. These capsules and their accompanying exercises discuss life and social science careers and companies that are related to the mathematical concepts covered in the chapter. 1. Complete the following: When determining the rate of change in the concentration of a drug over a 2-minute time interval, you are finding an rate of change. 2. Complete the following: When determining the rate of change in the concentration of a drug exactly 1 hour after the drug was administered, you are finding an rate of change. 3. You are asked to find the rate of change of a function over a certain interval. Should you find the average rate of change or the instantaneous rate of change? 4. You are asked to find the rate of change of a function at a certain instant. Should you find the average rate of change or the instantaneous rate of change? C O N C E P T C H E C K 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. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

19. xviii Features g g DISCOVERY Use a graphing utility to graph On the same screen, sketch the graphs of and Which of these lines, if any, appears to be tangent to the graph of at the point Explain your reasoning.0, 5? f y 3x 5.y 2x 5,y x 5, fx 2x3 4x2 3x 5.DISCOVERY These projects appear before selected topics and allow you to explore concepts on your own. These boxed features are optional, so they can be omitted with no loss of continuity in the coverage of material. Enhance Your Understanding Using Technology There are several ways to use technology to find relative extrema of a function. One way is to use a graphing utility to graph the function, and then use the zoom and trace features to find the relative minimum and relative maximum points. For instance, consider the graph of as shown below. From the graph, you can see that the function has one relative maximum and one relative minimum. You can approximate these values by zooming in and using the trace feature, as shown below. A second way to use technology to find relative extrema is to perform the First-Derivative Test with a symbolic differentiation utility. You can use the utility to differentiate the function, set the derivative equal to zero, and then solve the resulting equation. After obtaining the critical numbers, 1.48288 and 0.0870148, you can graph the function and observe that the first yields a relative minimum and the second yields a relative maximum. Compare the two ways shown above with doing the calculations by hand, as shown below. Write original function. First derivative Set derivative equal to 0. Solve for x. Approximate.x 1.48288, x 0.0870148 x 73 4213 93 9.3x2 14.6x 1.2 0 fx 9.3x2 14.6x 1.2 fx d dx 3.1x3 7.3x2 1.2x 2.5 fx 3.1x3 7.3x2 1.2x 2.5 when x 0.09 Relative 0.5 2 0.5 3 maximum when x 1.48 Relative minimum 1 2 2 0 2 4 3 4 fx 3.1x3 7.3x2 1.2x2.5 T E C H N O L O G Y Differentiate with respect to x. You can use a graphing utility to confirm the result given in Example 7. One way to do this is to choose a point on the graph of such as and find the equation of the tangent line at that point. Using the derivative found in the example, you know that the slope of the tangent line when is This means that the tangent line at the point is or By graphing and in the same viewing window, as shown below, you can confirm that the line is tangent to the graph at the point * 6 4 4 6 1, 2. 2t 4 y y 2t y 2t 4. y 2 2t 1 y y1 mt t1 1, 2 m 2. t 1 1, 2, y 2t, T E C H N O L O G Y TECHNOLOGY BOXES These boxes appear throughout the text and provide guidance on using technology to ease lengthy calculations, present a graphical solution, or discuss where using technology can lead to misleading or wrong solutions. Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

20. Features xix In Exercises 18, find the open intervals on which the graph is concave upward and those on which it is concave downward. 1. 2. 3. 4. 5. 6. 7. 8. In Exercises 922, find all relative extrema of the function. Use the Second-Derivative Test when applicable. 9. 10. 11. 12. 13. 14. 15. 16. fx 2x2 6fx x2 1 fx x 4 x fx x23 3 fx x4 4x3 2fx x3 5x2 7x fx x 52 fx 6x x2 y x 8 8 4 4 8 8 y x 2 4 62 2 2 4 6 y 3x5 40x3 135x 270 y x3 6x2 9x 1 y x 4 2 42 2 4 2 4 y x 4 2 42 2 4 4 fx x2 x2 1 fx 24 x2 12 y x 4 2 42 2 2 4 4 y x 4 42 2 4 fx x2 4 4 x2 fx x2 1 2x 1 y x 2 4 2 4 4 y x 2 4 2 4 2 4 y x3 3x2 2y x2 x 2 210 CHAPTER 3 Applications of the Derivative The following warm-up exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review Sections 2.4, 2.5, 2.6, and 3.1. In Exercises 16, find the second derivative of the function. 1. 2. 3. 4. 5. 6. In Exercises 710, find the critical numbers of the function. 7. 8. 9. 10. hx x4 50x2 8 gt 16 t2 t fx x4 4x3 10fx 5x3 5x 11 fx 2x 1 3x 2 hx 4x 3 5x 1 fx x 343gx x2 14 gs s2 1s2 3s 2fx 4x4 9x3 5x 1 Skills Review 3.3 Exercises 3.3 See www.CalcChat.com for worked-out solutions to odd-numbered exercises. SECTION 3.3 Concavity and the Second-Derivative Test 211 17. 18. 19. 20. 21. 22. In Exercises 2326, use a graphing utility to estimate graphically all relative extrema of the function. 23. 24. 25. 26. In Exercises 2730, state the signs of and on the interval 27. 28. 29. 30. In Exercises 3138, find the point(s) of inflection of the graph of the function. 31. 32. 33. 34. 35. 36. 37. 38. In Exercises 3950, use a graphing utility to graph the function and identify all relative extrema and points of inflection. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. In Exercises 5154, sketch a graph of a function having the given characteristics. 51. 52. 53. 54. In Exercises 55 and 56, use the graph to sketch the graph of Find the intervals on which (a) is positive, (b) is negative, (c) is increasing, and (d) is decreasing. For each of these intervals, describe the corresponding behavior of 55. 56. In Exercises 5760, you are given Find the intervals on which (a) is increasing or decreasing and (b) the graph of is concave upward or concave downward. (c) Find the relative extrema and inflection points of (d) Then sketch a graph of 57. 58. 59. 60. Productivity In Exercises 61 and 62, consider a college student who works from 7 P.M. to 11 P.M. assembling mechanical components. The number of components assembled after hours is given by the function. At what time is the student assembling components at the greatest rate? 61. 62. 0 t 4N 20t2 4 t2 , 0 t 4N 0.12t3 0.54t2 8.22t, t N fx x2 x 6fx x2 2x 1 fx 3x2 2fx 2x 5 f.f. f fx f. 422 1 1 1 2 3 2 3 x y 1 3 2 1 1 x y f. f ffx fxf. fx > 0fx < 0 fx > 0 if x > 1fx < 0 if x > 1 f1 0f1 0 fx < 0 if x < 1fx > 0 if x < 1 f0 f2 0f0 f2 0 f x > 0, x 3fx > 0 fx < 0 if x > 3fx > 0 if x > 3 f3 is undefined.f3 0 fx > 0 if x < 3fx < 0 if x < 3 f2 f4 0f2 f4 0 f fx 2 x2 1 fx 4 1 x2 gx x9 xgx xx 3 gx x 6x 23 gx x 2x 12 fx 2x4 8x 3fx 1 4x4 2x2 fx x3 3 2x2 6xfx x3 6x2 12x fx x3 3xfx x3 12x ft 1 tt 4t2 4 hx x 23 x 1 fx 4x3 8x2 32 gx 2x4 8x3 12x2 12x fx x4 18x2 5fx x 13 x 5 fx x6 x2 fx x3 9x2 24x 18 21 x ))xy f y 21 x ))f xy y 21 x ))fy x y 21 x ))xfy y 0, 2. fxfx fx 3x3 5x2 2fx 5 3x2 x3 fx 1 3x5 1 2x4 xfx 1 2x4 1 3x3 1 2x2 fx x x2 1 fx x x 1 fx 18 x2 3 fx 8 x2 2 fx 4 x2 fx 9 x2 212 CHAPTER 3 Applications of the Derivative In Exercises 6366, use a graphing utility to graph and in the same viewing window. Graphically locate the relative extrema and points of inflection of the graph of State the relationship between the behavior of and the signs of and 63. 64. 65. 66. 67. Phishing Phishing is a criminal activity used by an individual or group to fraudulently acquire information by masquerading as a trustworthy person or business in an electronic communication. Criminals create spoof sites on the Internet to trick victims into giving them information. A model for the number of reported spoof sites from November 2005 through October 2006 is where represents the number of months since November 2005. (Source: Anti-Phishing Working Group) (a) Use a graphing utility to graph the model on the interval (b) Use the graph in part (a) to estimate the month corresponding to the absolute minimum number of spoof sites. (c) Use the graph in part (a) to estimate the month corresponding to the absolute maximum number of spoof sites. (d) During approximately which month was the rate of increase of the number of spoof sites the greatest? the least? 68. Medicine The spread of a virus can be modeled by where is the number of people infected (in hundreds), and is the time (in weeks). (a) What is the maximum number of people projected to be infected? (b) When will the virus be spreading most rapidly? (c) Use a graphing utility to graph the model and to verify your results. 69. High School Dropouts From 2000 through 2005, the number of high school dropouts not in the labor force (in thousands) can be modeled by where is the year, with corresponding to 2000. (Source: U.S. Bureau of Labor Statistics) (a) Use a graphing utility to graph the model. (b) Use the second derivative to determine the concavity of (c) Find the point(s) of inflection of the graph of (d) Interpret the meaning of the inflection point(s) of the graph of 70. Veteran Benefits From 1995 through 2005, the number of veterans (in thousands) receiving compensa- tion and pension benefits for service in the armed forces can be modeled by where is the year, with corresponding to 1995. (Source: U.S. Department of Veterans Affairs) (a) Use a graphing utility to graph the model. (b) Use the second derivative to determine the concavity of (c) Find the point(s) of inflection of the graph of (d) Interpret the meaning of the inflection point(s) of the graph of v. v. v. t 5t v 0.1731t4 7.405t3 106.50t2 607.2t 2133 v d. d. d. t 0t d 20.444t3 152.33t2 266.6t 1162 d t N 0 t 12N t3 12t2 , 0, 11. t ft 88.253t3 1116.16t2 4541.4t 4161, 0 t 11 3, 3fx x2 x2 1 ,3, 3fx 2 x2 1 , 2, 2fx 1 20 x5 1 12 x2 1 3 x 1, 0, 3fx 1 2 x3 x2 3x 5, f.ff f. f f,f, L i f e S c i e n c e C a p s u l e Biomedical scientists have made worthwhile contributions to the benefit of humankind. For example, James D. Watson is best known for his discovery of the structure of DNA. Also, Fredrick Sanger determined the complete amino acid sequence of insulin. According to a recent survey from ASEE (American Society of Engi- neering Education), the number of biomedical bachelors degrees has doubled since 2001. 71. Research Project Use your schools library, the Internet, or some other reference source to research the number of students earning bachelors, masters, or doctoral degrees like the one above. Gather the data for the number of degrees earned in a particular field over a period of time, and use a graphing utility to graph a scatter plot of the data. Fit models to the data. Do the models appear to be concave upward or downward? Do they appear to be increasing or decreasing? Discuss the implications of your answers. Owen Franken/CORBIS SKILLS REVIEW These exercises at the beginning of each exercise set help you review skills covered in previous sections. The answers are provided at the back of the text to reinforce your understanding of the skill sets learned. EXERCISE SETS These exercises offer opportunities for practice and review. They progress in difficulty from skill-development problems to more challenging problems, to build confidence and understanding. Practice Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

21. xx Features 78. U.S. AIDS Epidemic The number of cases (in thousands) of AIDS reported in the years 1995 through 2005 can be modeled by where represents the year, with corresponding to 1995. (Source: U.S. Centers for Disease Control and Prevention) (a) Find the derivative of the model. Which differentiation rule(s) did you use? (b) Use a graphing utility to graph the derivative on the interval (c) Use the trace feature to find the years during which the number of AIDS cases was changing the most. (d) Use the trace feature to find the years during which the number of AIDS cases was changing the least. 5 t 15. t 5t f 1.8840t4 66.325t3 739.92t2 2307.0t 5125 f 68. Human Memory Model Consider the learning curve modeled by where is the percent of the correct responses after trials. (a) Use a spreadsheet software program to complete the table for the model. (b) Find the limit of P as n approaches infinity. nP n > 0P 0.55 0.87n 1 1 0.87n 1 , n 1 2 3 4 5 6 7 8 9 10 P 83. Modeling Data The table shows the numbers (in thousands) of students enrolled in public schools in Puerto Rico for the years 2000 through 2005, where is the year, with corresponding to 2000. (Source: Puerto Rico Planning Board) t 0 t n t 0 1 2 3 4 5 n 612.3 610.8 603.5 596.3 584.9 575.9 TECHNOLOGY EXERCISES Many exercises in the text can be solved with or without technology. The symbol identifies exercises for which you are specifically instructed to use a graphing calculator or a computer algebra system to solve the problem. Additionally, the symbol denotes exercises best solved by using a spreadsheet. p 63. MAKE A DECISION: ENVIRONMENT The predicted cost (in hundreds of thousands of dollars) for a company to remove of a chemical from its waste water can be modeled by (a) Find the rates of change in the cost when and (b) Use a graphing utility to graph (c) What happens to and as approaches 100? (d) What would you say is a reasonable percent of the chemical for the company to remove? Explain. 64. Consumer Awareness The price per pound of lean d l d b f i h i d f P pCC C. p 75%. p 25% 0 p < 100.C 124p 10 p100 p , p% C 55. MAKE A DECISION: SOCIAL SECURITY The table lists the average monthly Social Security benefits (in dollars) for retired workers aged 62 and over from 1998 through 2005. A model for the data is where is the year, with corresponding to 1998. (Source: U.S. Social Security Administration) t 8t B 582.6 38.38t 1 0.025t 0.0009t2 , 8 t 15 B t 8 9 10 11 12 13 14 15 B 780 804 844 874 895 922 955 1002 MAKE A DECISION Multi-step exercises reinforce your problem-solving skills and mastery of concepts, as well as taking a real-life application further by testing what you know about a given problem to make a decision within the context of the problem. Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

22. Features xxi 146 CHAPTER 2 Differentiation Mid-Chapter Quiz Take this test as you would take a test in class. When you are done, check your work against the answers given in the back of the book. In Exercises 13, use the limit definition to find the derivative of the function. Then find the slope of the tangent line to the graph of at the given point. 1. 2. 3. In Exercises 49, find the derivative of the function. 4. 5. 6. 7. 8. 9. In Exercises 1013, use a graphing utility to graph the function and find its average rate of change on the interval. Compare this rate with the instantaneous rates of change at the endpoints of the interval. 10. 11. 12. 13. 14. The graph represents the average salary (in dollars) of classroom teachers in public schools for the years 2000 through 2005, where represents the year, with corresponding to 2000. Estimate the slopes of the graph for the years 2002 and 2004. (Source: Educational Research Service) In Exercises 15 and 16, find an equation of the tangent line to the graph of f at the given point. Then use a graphing utility to graph the function and the tangent line in the same viewing window. 15. 16. 17. At time a diver jumps from a diving board that is 25 feet high. Because the divers initial velocity is 15 feet per second, her position function is (a) When does the diver hit the water? (b) What is the divers velocity at impact? h 16t2 15t 25. t 0, 0, 1fx x 1x 1, 1, 2fx 5x2 6x 1, t 0t S 8, 27fx 3x, 2, 5fx 1 2x , 1, 1fx 2x3 x2 x 4, 0, 3fx x2 3x 1, fx 2xfx 4x2 fx 12x14 f x 5 3x2 fx 19x 9fx 12 1, 4fx 4 x , 1, 2fx x 3, 2, 0fx x 2, f t 41,000 S 40,000 43,000 42,000 45,000 44,000 46,000 47,000 48,000 1 2 3 4 5 Salary(indollars) Average Salary of Public School Teachers Year (0 2000) Figure for 14 MID-CHAPTER QUIZ Appearing in the middle of each chapter, this one- page test allows you to practice skills and concepts learned in the chapter. This opportunity for self-assessment will uncover any potential weak areas that might require further review of the material. 182 CHAPTER 2 Differentiation Chapter Test See www.CalcChat.com for worked-out solutions to odd-numbered exercises. Take this test as you would take a test in class. When you are done, check your work against the answers given in the back of the book. In Exercises 1 and 2, use the limit definition to find the derivative of the function. Then find the slope of the tangent line to the graph of at the given point. 1. 2. In Exercises 311, find the derivative of the function. Simplify your result. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. Find an equation of the tangent line to the graph of at the point Then use a graphing utility to graph the function and the tangent line in the same viewing window. 13. The number of bone graft procedures (in thousands) in the United States for the years 2000 through 2005 can be modeled by where represents the year, with corresponding to 2000. (Source: U.S. Department of Health and Human Services) (a) Find the average rate of change from 2001 through 2004. (b) Find the instantaneous rates of change of the model for 2001 and 2004. (c) Interpret the results of parts (a) and (b) in the context of the problem. In Exercises 1416, find the third derivative of the function. Simplify your result. 14. 15. 16. 17. The table shows the numbers of bald eagle breeding pairs in the lower 48 states for selected years from 1996 through 2006. (Source: U.S. Fish & Wildlife Service) A model for the data is where is the year, with corresponding to 1996. (a) Use a graphing utility to graph the model and the data in the same viewing window. How well does the model fit the data? (b) Find the first and second derivatives of the function. (c) Use the model to show that the number of breeding pairs was decreasing in the years 2000 through 2002. (d) Find the year in which the number of breeding pairs was decreasing at the greatest rate by solving Nt 0. t 6 tN 0.00263t6 1.035t4 124.5t2 1590, N fx 2x 1 2x 1 fx 3 xfx 2x2 3x 1 t 0 tN 12.95t2 110.3t 780, N 1, 0.fx x 1 x fx 5x 13 x fx 1 2xfx 3x2 42 fx x5 xfx 3x3 fx x 3x 3 fx x32fx 4x2 8x 1ft t3 2t 4, 0fx x 2,2, 5f x x2 1, f t 6 7 8 9 10 15 16 N 5094 5295 5748 6404 6471 7066 9789 CHAPTER TEST Appearing at the end of the chapter, this test is designed to simulate an in-class exam. Taking these tests will help you to determine what concepts require further study and review. Study and Review Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

23. xxii Features 176 CHAPTER 2 Differentiation Chapter Summary and Study Strategies After studying this chapter, you should have acquired the following skills. The exercise numbers are keyed to the Review Exercises that begin on page 178. Answers to odd-numbered Review Exercises are given in the back of the text.* * Use a wide range of valuable study aids to help you master the material in this chapter. The Student Solutions Guide includes step-by-step solutions to all odd-numbered exercises to help you review and prepare. The student website at college.hmco.com/info/larsonapplied offers algebra help and a Graphing Technology Guide. The Graphing Technology Guide contains step-by-step commands and instructions for a wide variety of graphing calculators, including the most recent models. Section 2.1 Review Exercises Approximate the slope of the tangent line to a graph at a point. 14 Interpret the slope of a graph in a real-life setting. 58 Use the limit definition to find the derivative of a function and the slope of a graph 916 at a point. Use the derivative to find the slope of a graph at a point. 1724 Use the graph of a function to recognize points at which the function is not 2528 differentiable. fx lim x0 f x x f x x Section 2.2 Use the Constant Multiple Rule for differentiation. 29, 30 Use the Sum and Difference Rules for differentiation. 3138 d dx fx gx fx gx d dx cfx cfx Section 2.3 Find the average rate of change of a function over an interval and the instantaneous 39, 40 rate of change at a point. Find the average and instantaneous rates of change of a quantity in a real-life problem. 41, 42 Use rates of change to solve real-life problems. 4348 Instantaneous rate of change lim x0 fx x fx x Average rate of change fb fa b a CHAPTER SUMMARY AND STUDY STRATEGIES The Chapter Summary reviews the skills covered in the chapter and correlates each skill to the Review Exercises that test the skill. Following each Chapter Summary is a short list of Study Strategies for addressing topics or situations in the chapter. Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

26. 1 0A Precalculus Review One way erosion occurs is by running water. Running water transports particles such as soil or fragments of rock. These particles strike against the bedrock of the stream channel, grind it away, and eventually settle in the channel or find their way to the sea. (See Section 0.3, Exercise 58.) Topics in precalculus have many real-life applications. The applications listed below represent a sample of the applications in this chapter. Biology: pH Values, Exercise 33, page 7 Life Expectancy, Exercise 46, page 12 Chemistry: Finding Concentrations, Exercise 77, page 24 Plants, Exercise 46, page 32 Applications LarryMiller/PhotoResearchers,Inc. 0.1 The Real Number Line and Order 0.2 Absolute Value and Distance on the Real Number Line 0.3 Exponents and Radicals 0.4 Factoring Polynomials 0.5 Fractions and Rationalization Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

27. 2 CHAPTER 0 A Precalculus Review Section 0.1 The Real Number Line and Order Represent, classify, and order real numbers. Use inequalities to represent sets of real numbers. Solve inequalities. Use inequalities to model and solve real-life problems. The Real Number Line Real numbers can be represented with a coordinate system called the real number line (or x-axis), as shown in Figure 0.1. The positive direction (to the right) is denoted by an arrowhead and indicates the direction of increasing values of x. The real number corresponding to a particular point on the real number line is called the coordinate of the point. As shown in Figure 0.1, it is customary to label those points whose coordinates are integers. The point on the real number line corresponding to zero is called the origin. Numbers to the right of the origin are positive, and numbers to the left of the origin are negative. The term nonnegative describes a number that is either positive or zero. The importance of the real number line is that it provides you with a conceptually perfect picture of the real numbers. That is, each point on the real number line corresponds to one and only one real number, and each real number corresponds to one and only one point on the real number line. This type of relationship is called a one-to-one correspondence and is illustrated in Figure 0.2. Each of the four points in Figure 0.2 corresponds to a real number that can be expressed as the ratio of two integers. Such numbers are called rational. Rational numbers have either terminating or infinitely repeating decimal representations. Terminating Decimals Infinitely Repeating Decimals * Real numbers that are not rational are called irrational, and they cannot be represented as the ratio of two integers (or as terminating or infinitely repeating decimals). So, a decimal approximation is used to represent an irrational number. Some irrational numbers occur so frequently in applications that mathematicians have invented special symbols to represent them. For example, the symbols and e represent irrational numbers whose decimal approximations are as shown. (See Figure 0.3.) *The bar indicates which digit or digits repeat infinitely. e 2.7182818284 3.14159265352 1.4142135623 , 2, 12 7 1.714285714285 . . . 1.714285 7 8 0.875 1 3 0.333 . . . 0.3 2 5 0.4 1.85 37 20 7 3 5 42.6 13 5 x 4 3 2 1 0 4321 Positive direction (x increases) Negative direction (x decreases) FIGURE 0.1 The Real Number Line 0 211 3 x e 2 FIGURE 0.3 Every real number corresponds to one and only one point on the real number line. Every point on the real number line corresponds to one and only one real number. 3 2 1 1 30 2 x 1.85 7 3 3 2 1 1 30 2 x 5 42.6 FIGURE 0.2 Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

28. Order and Intervals on the Real Number Line One important property of the real numbers is that they are ordered: 0 is less than 1, is less than is less than and so on. You can visualize this property on the real number line by observing that a is less than b if and only if a lies to the left of b on the real number line. Symbolically, a is less than b is denoted by the inequality For example, the inequality follows from the fact that lies to the left of 1 on the real number line, as shown in Figure 0.4. FIGURE 0.4 When three real numbers a, x, and b are ordered such that and we say that x is between a and b and write x is between a and b. The set of all real numbers between a and b is called the open interval between a and b and is denoted by An interval of the form does not contain the endpoints a and b. Intervals that include their endpoints are called closed and are denoted by Intervals of the form and are neither open nor closed. Figure 0.5 shows the nine types of intervals on the real number line. a, ba, ba, b. a, ba, b. a < x < b. x < b,a < x 211 0 x 1 3 4 lies to the left of 1, so < 1.3 4 3 4 3 4 3 4 < 1a < b. 22 7 ,2.5, 3 SECTION 0.1 The Real Number Line and Order 3 a a < x < b b (a, b) ba ba a < x b (a, b] a x < b [a, b) a b a b a b baba x < a x > b x a x b (, ) (, a] [b, ) (, a) (b, ) ba a x b [a, b] FIGURE 0.5 Intervals on the Real Number Line Open interval Intervals that are neither open nor closed Infinite intervals Closed interval S T U D Y T I P Note that a square bracket is used to denote less than or equal to or greater than or equal to Furthermore, the symbols and denote positive and negative infinity. These symbols do not denote real numbers; they merely let you describe unbounded conditions more concisely. For instance, the interval is unbounded to the right because it includes all real numbers that are greater than or equal to b. b, . Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

29. Solving Inequalities In calculus, you are frequently required to solve inequalities involving variable expressions such as The number a is a solution of an inequality if the inequality is true when a is substituted for x. The set of all values of x that satisfy an inequality is called the solution set of the inequality. The following properties are useful for solving inequalities. (Similar properties are obtained if is replaced by and is replaced by ) Note that you reverse the inequality when you multiply by a negative number. For example, if then This principle also applies to division by a negative number. So, if then Example 1 Solving an Inequality an Inequality Find the solution set of the inequality SOLUTION Write original inequality. Add 4 to each side. Simplify. Multiply each side by Simplify. So, the solution set is the interval as shown in Figure 0.6. CHECKPOINT 1 Find the solution set of the inequality In Example 1, all five inequalities listed as steps in the solution have the same solution set, and they are called equivalent inequalities. 2x 3 < 7. , 3, x < 3 1 3. 1 3 3x < 1 3 9 3x < 9 3x 4 4 < 5 4 3x 4 < 5 3x 4 < 5. x < 2.2x > 4, 4x > 12.x < 3, .> < 3x 4 < 5. 4 CHAPTER 0 A Precalculus Review S T U D Y T I P Notice the differences between Properties 3 and 4. For example, and 3 < 432 > 42. 3 < 4 32 < 42 S T U D Y T I P Once you have solved an inequality, it is a good idea to check some -values in your solution set to see whether they satisfy the original inequality. You might also check some values outside your solution set to verify that they do not satisfy the inequality. For example, Figure 0.6 shows that when or the inequality is satisfied, but when the inequality is not satisfied. x 4 x 2x 0 x 876543210 x Solution set for 3x 4 < 5 r x = 2, 3(2) 4 = 2.oF For x = 4, 3(4) 4 = 8. 1 For x = 0, 3(0) 4 = 4. FIGURE 0.6 Properties of Inequalities Let and be real numbers. 1. Transitive property: and 2. Adding inequalities: and 3. Multiplying by a (positive) constant: 4. Multiplying by a (negative) constant: 5. Adding a constant: 6. Subtracting a constant: a c < b ca < b a c < b ca < b c < 0ac > bc,a < b c > 0ac < bc,a < b a c < b dc < da < b a < cb < ca < b dc,b,a, Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

30. The inequality in Example 1 involves a first-degree polynomial. To solve inequalities involving polynomials of higher degree, you can use the fact that a polynomial can change signs only at its real zeros (the real numbers that make the polynomial zero). Between two consecutive real zeros, a polynomial must be entirely positive or entirely negative. This means that when the real zeros of a polynomial are put in order, they divide the real number line into test intervals in which the polynomial has no sign changes. That is, if a polynomial has the factored form then the test intervals are and For example, the polynomial can change signs only at and To determine the sign of the polynomial in the intervals and you need to test only one value from each interval. Example 2 Solving a Polynomial Inequality Inequality Find the solution set of the inequality SOLUTION Write original inequality. Polynomial form Factor. So, the polynomial has and as its zeros. You can solve the inequality by testing the sign of the polynomial in each of the following intervals. To test an interval, choose a representative number in the interval and compute the sign of each factor. For example, for any both of the factors and are negative. Consequently, the product (of two negative numbers) is positive, and the inequality is not satisfied in the interval A convenient testing format is shown in Figure 0.7. Because the inequality is satisfied only by the center test interval, you can conclude that the solution set is given by the interval Solution set CHECKPOINT 2 Find the solution set of the inequality x2 > 3x 10. 2 < x < 3. x < 2. x 2 x 3x < 2, x > 32 < x < 3,x < 2, x 3x 2x2 x 6 x 3x 2 < 0 x2 x 6 < 0 x2 < x 6 x2 < x 6. 3, ,, 2, 2, 3, x 3.x 2 x2 x 6 x 3x 2 rn, .rn1, rn ,. . . ,r1, r2,, r1, r1 < r2 < r3 < . . . < rnx r1x r2, . . . , x rn , SECTION 0.1 The Real Number Line and Order 5 x 32 NoYesNo ()() > 0 (+)(+) > 0()(+) < 0 FIGURE 0.7 Is x 3x 2 < 0? Sign of x 3x 1 2 x Sign < 0? 3 No 2 0 No 1 Yes 0 Yes 1 Yes 2 Yes 3 0 No 4 No Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

31. Application Inequalities are frequently used to describe conditions that occur in business and science. For instance, the inequality describes the recommended weight W for a man whose height is 5 feet 10 inches. Example 3 shows how an inequality can be used to describe temperatures at which a lizard will be comfortable during the day. Example 3 MAKE A DECISION Temperature According to a pet care guide, your pet lizard should be kept in a room where the daytime temperature ranges from to Keeping the lizard in higher or lower temperatures could make the lizard ill. Find the high and low temperatures in degrees Celsius. Will a daytime temperature of be harmful to the lizard? SOLUTION The range of temperatures in degrees Fahrenheit is Use the temperature conversion formula to write the following. Write original inequality. Substitute for Subtract 32 from each part. Simplify. Divide each part by 1.8. C Simplify. So, the daytime temperature ranges from 22C to 25C. No, a daytime temperature of 24.5C will not be harmful to the lizard, as shown in Figure 0.8. FIGURE 0.8 CHECKPOINT 3 During nighttime, the lizard in Example 3 should be kept in a room where the temperature ranges from 68F to 75.2F. Find the high and low temperatures in degrees Celsius. C 262524 24.5 232221 Low daytime temperature High daytime temperature Each temperature that is safe for the lizard falls in this interval. 2522 45 1.8 1.8C 1.8 39.6 1.8 451.8C39.6 77 321.8C 32 3271.6 32 F.1.8C 32 771.8C 3271.6 77F71.6 F 1.8C 32 71.6 F 77.0. F 24.5C 77F.71.6F 144 W 180 6 CHAPTER 0 A Precalculus Review The symbol indicates an example that uses or is derived from real-life data. Crested geckos are native to Southern Grand Terre, New Caledonia and at least one nearby island (Isle of Pines). At temperatures of 85F or warmer, crested geckos will become stressed, possibly leading to illness or death. Danita Delimont/Alamy Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

32. SECTION 0.1 The Real Number Line and Order 7 In Exercises 110, determine whether the real number is rational or irrational. *1. 0.25 2. 3. 4. 5. 6. 7. 8. 9. 10. In Exercises 1114, determine whether each given value of x satisfies the inequality. 11. (a) (b) (c) 12. (a) (b) (c) 13. (a) (b) (c) 14. (a) (b) (c) In Exercises 1528, solve the inequality and sketch the graph of the solution on the real number line. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. In Exercises 2932, use inequality notation to describe the subset of real numbers. 29. A gas station expects the cost per gallon of 87-octane gasoline for the next week to be no less than $2.39 per gallon and no more than $3.25 per gallon. 30. The average length of a fish is no less than 7 inches and no more than 12 inches. 31. A person expects their weight loss for the next three months to be no less than 18 pounds. 32. To grow cabbage properly, gardeners keep the pH level of the soil at no more than 7.5. 33. Biology: pH Values The pH scale measures the concentration of hydrogen ions in a solution. Strong acids produce low pH values, while strong bases produce high pH values. Represent the following approximate pH values on a real number line: hydrochloric acid, 0.0; lemon juice, 2.0; oven cleaner, 13.0; baking soda, 9.0; pure water, 7.0; black coffee, 5.0. (Source: Adapted from Levine/Miller, Biology: Discovering Life, Second Edition) 34. Physiology The maximum heart rate of a person in normal health is related to the persons age by the equation where r is the maximum heart rate in beats per minute and A is the persons age in years. Some physiologists recom- mend that during physical activity a person should strive to increase his or her heart rate to at least 60% of the maximum heart rate for sedentary people and at most 90% of the maximum heart rate for highly fit people. Express as an interval the range of the target heart rate for a 20-year-old. 35. Terrarium Temperature Your pet frog should be kept in a terrarium where the temperature ranges from 68F to 75F. The current temperature of the terrarium is 18.3C. Does this temperature fall within the acceptable range? 36. Egg Production The number of eggs (in billions) produced in the United States from 2000 through 2005 can be modeled by where represents the year, with corresponding to 2000. According to this model, when was the annual egg production at least 86 billion, but no more than 88 billion? (Source: U.S. Department of Agriculture) In Exercises 37 and 38, determine whether each state- ment is true or false, given 37. (a) 38. (a) (b) (b) (c) (c) (d) (d) a 4 < b 4 1 a < 1 b 3b < 3a6a < 6b 4 a < 4 ba 2 < b 2 a 4 < b 42a < 2b a < b. t 0t 0 t 5E 1.02t 84.9, E r 220 A p w l g 2x2 1 < 9x 32x2 x < 6 x 2 x 3 > 5 x 2 x 3 > 5 1 < x 3 < 1 3 4 > x 1 > 1 4 0 x 3 < 54 < 2x 3 < 4 x 4 2x 14 2x < 3x 1 2x 7 < 34x 1 < 2x 2x > 3x 5 7 x 5x 1x 0 1 < 3 x 2 1 x 0x 10x 4 0 < x 2 4 < 2 x 4x 4x 0 x 1 < x 3 x 5 2x 3x 3 5x 12 > 0 2e360 0.8177364 22 7 4.3451 32 1 3 2 3678 Exercises 0.1 See www.CalcChat.com for worked-out solutions to odd-numbered exercises. *The answers to the odd-numbered and selected even-numbered exercises are given in the back of the text. Worked-out solutions to the odd-numbered exercises are given in the Student Solutions Guide. Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

33. Find the absolute values of real numbers and understand the properties of absolute value. Find the distance between two numbers on the real number line. Define intervals on the real number line. Find the midpoint of an interval and use intervals to model and solve real-life problems. Absolute Value of a Real Number At first glance, it may appear from this definition that the absolute value of a real number can be negative, but this is not possible. For example, let Then, because you have The following properties are useful for working with absolute values. Be sure you understand the fourth property in this list. A common error in algebra is to imagine that by squaring a number and then taking the square root, you come back to the original number. But this is true only if the original number is nonnegative. For instance, if then but if then The reason for this is that (by definition) the square root symbol denotes only the nonnegative root. 22 4 2. a 2, 22 4 2 a 2, 3. 3a 3 3 < 0, a 3. 8 CHAPTER 0 A Precalculus Review Section 0.2 Absolute Value and Distance on the Real Number Line Definition of Absolute Value The absolute value of a real number is a a, a, if a 0 if a < 0. a Properties of Absolute Value 1. Multiplication: 2. Division: 3. Power: 4. Square root: a2 a an an b 0 a b a b , ab ab Absolute value expressions can be evaluated on a graphing utility. When an expression such as is evaluated, parentheses should surround the expression, as in abs3 8. 3 8 T E C H N O L O G Y Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

34. SECTION 0.2 Absolute Value and Distance on the Real Number Line 9 Distance on the Real Number Line Consider two distinct points on the real number line, as shown in Figure 0.9. 1. The directed distance from a to b is 2. The directed distance from b to a is 3. The distance between a and b is or In Figure 0.9, note that because b is to the right of a, the directed distance from a to b (moving to the right) is positive. Moreover, because a is to the left of b, the directed distance from b to a (moving to the left) is negative. The distance between two points on the real number line can never be negative. Note that the order of subtraction with and does not matter because and Example 1 Finding Distance on the Real Number Line Determine the distance between and 4 on the real number line. What is the directed distance from to 4? What is the directed distance from 4 to SOLUTION The distance between and 4 is given by or or as shown in Figure 0.10. FIGURE 0.10 The directed distance from to 4 is The directed distance from 4 to is CHECKPOINT 1 Determine the distance between and 6 on the real number line. What is the directed distance from to 6? What is the directed distance from 6 to 2?2 2 a b3 4 7. 3 b a4 3 7. 3 543214 3 2 1 0 x Distance = 7 b aa b4 3 7 73 4 7 7 3 3?3 3 x2 x12 x1 x22 .x2 x1 x1 x2 x2x1 b a.a b a b. b a.x x x Distance between a and b: Directed distance from b to a: b a a b a b a a b b ora b b a Directed distance from a to b: FIGURE 0.9 Distance Between Two Points on the Real Number Line The distance between points and on the real number line is given by d x2 x1 x2 x12 . x2x1d Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

35. 10 CHAPTER 0 A Precalculus Review Intervals Defined by Absolute Value Example 2 Defining an Interval on the Real Number Line Find the interval on the real number line that contains all numbers that lie no more than two units from 3. SOLUTION Let x be any point in this interval.You need to find all such that the distance between x and 3 is less than or equal to 2. This implies that Requiring the absolute value of to be less than or equal to 2 means that must lie between and 2. So, you can write Solving this pair of inequalities, you have x Solution set So, the interval is as shown in Figure 0.11. CHECKPOINT 2 Find the interval on the real number line that contains all numbers that lie no more than four units from 6. 1, 5, 5.1 2 3 x 3 3 2 3 2 x 3 2. 2x 3 x 3 x 3 2. x Two Basic Types of Inequalities Involving Absolute Value Let and be real numbers, where if and only if if and only if x a d or a d x.x a d a d x a d.x a d d > 0.da All numbers x whose distance from a is less than or equal to d. InterpretationInequality Graph All numbers x whose distance from a is greater than or equal to d. x a d x a d x x a d a d a a + d a + d a d d d d 6 x 54210 3 2 units 2 units x 3 2 FIGURE 0.11 S T U D Y T I P Be sure you see that inequalities of the form have solution sets consisting of two intervals. To describe the two intervals without using absolute values, you must use two separate inequalities, connected by an or to indicate union. x a d Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

36. SECTION 0.2 Absolute Value and Distance on the Real Number Line 11 Midpoint of an Interval The midpoint of the interval with endpoints and is found by taking the average of the endpoints. Midpoint a b 2 ba Application Example 3 Voter Poll In a poll for an upcoming mayoral election, 40% of likely voters said they would vote for Candidate A. The poll has a margin of error of a. Write and solve an inequality that represents the percent of likely voters who said they would vote for Candidate A. b. Assume there are 50,000 likely voters. Of these voters, how many does the poll predict would vote for Candidate A? SOLUTION a. Because the poll has a margin of error of Candidate A expects to get of likely voters. Let be the percent (written in decimal form) of likely voters for Candidate A.You know that will differ from 0.40 by at most 0.03. Figure 0.12(a) b. Letting be the number of people out of 50,000 who vote for Candidate A, it follows that So, the number of people who would vote for Candidate A would be given by V Figure 0.12(b) According to the poll, Candidate A can expect at least 18,500 votes and at most 21,500 votes. CHECKPOINT 3 Repeat Example 3 for Candidate B, where 37% of likely voters said they would vote for Candidate B with a margin of error of . In Example 3(b), the candidate should expect to receive between 18,500 and 21,500 votes. Of course, a conservative expectation would be the lower of these estimates. However, from a statistical point of view, the most representative estimate would be the average of these two extremes. Graphically, the average of two numbers is the midpoint of the interval with the two numbers as endpoints, as shown in Figure 0.13. 3% 21,500.18,500 0.3750,000 50,000A 0.4350,000 V 50,000A. V 0.37 A 0.43 0.40 0.03 A 0.40 0.03 A A40% 3% 3%, 3%. A 36 38 40 42 44 (b) Number of likely voters (a) Percent of likely voters 37 43 V 18,000 20,000 22,000 18,500 21,500 FIGURE 0.12 = 20,0002 Midpoint = 18,500 + 21,500 V 18,000 20,000 22,000 18,500 21,500 FIGURE 0.13 Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

37. In Exercises 16, find (a) the directed distance from a to b, (b) the directed distance from b to a, and (c) the distance between a and b. 1. 2. 3. 4. 5. 6. In Exercises 718, use absolute values to describe the given interval (or pair of intervals) on the real number line. 7. 8. 9. 10. 11. 12. 13. 14. 15. All numbers less than three units from 5 16. All numbers more than five units from 2 17. y is at most two units from a. 18. y is less than h units from c. In Exercises 1934, solve the inequality and sketch the graph of the solution on the real number line. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. In Exercises 3540, find the midpoint of the given interval. 35. 36. 37. 38. 39. 40. 41. Chemistry Copper has a melting point M within of Use absolute values to write the range as an inequality. 42. Body Temperature Normal body temperature in humans is within of Use absolute values to write the range as an inequality. 43. Heights of a Population The heights of two-thirds of the members of a population satisfy the inequality where is measured in inches. Determine the interval on the real number line in which these heights lie. 44. Biology The American Kennel Club has developed guidelines for judging the features of various breeds of dogs. For collies, the guidelines specify that the weights for males satisfy the inequality where is measured in pounds. Determine the interval on the real number line in which these weights lie. 45. Landscaping The amount of grass contained in one square foot of a typical lawn is given by where is measured in blades of grass. Determine the high and low amounts of grass per square foot. 46. Life Expectancy Estimates of the life expectancy of a loggerhead sea turtle are given by where is measured in years. Determine the high and low life expectancy estimates. 47. Voter Poll In a poll for an upcoming election, 45% of likely voters said they would vote for Candidate X. The poll has a margin of error of (a) Write and solve an inequality that represents the percent of likely voters who said they would vote for Candidate X. (b) Assume there are 80,000 likely voters. Of these voters, how many does the poll predict would vote for Candidate X? 48. Survey In a survey on health insurance, 92% of persons covered by a particular plan said they were satisfied with the coverage they were receiving. The survey has a margin of error of (a) Write and solve an inequality that represents the percent of persons who said they were satisfied with their coverage. (b) Assume there are 120,000 persons covered by the plan. According to the survey, how many said they were satisfied with their coverage? 5%. 3%. x x 67.5 7.5, n n 850 100 w w 67.5 7.5 1 h h 68.5 2.7 1 h 98.6F.1F T 1083.4C. 0.2C 5 6, 5 2 1 2, 3 4 4.6, 1.36.85, 9.35 7.3, 12.78, 24 b > 0 a 5x 2 > b,b > 0 3x a 4 < 2b, b > 02x a b,b > 0x a b, 1 2x 3 < 19 2x < 1 25 x 2010 x > 4 2x 1 < 5 x 3 2 5 3x 1 4x 5 < 2 3x > 12 x 2> 3 2x < 6x < 4 , 20 24, , 0 4, 7, 12, 8 , 3 3, , 2 2, 3, 32, 2 a 18 5 , b 61 15a 16 5 , b 112 75 a 2.05, b 4.25a 9.34, b 5.65 a 126, b 75a 126, b 75 12 CHAPTER 0 A Precalculus Review Exercises 0.2 See www.CalcChat.com for worked-out solutions to odd-numbered exercises. Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

38. SECTION 0.3 Exponents and Radicals 13 Evaluate expressions involving exponents or radicals. Simplify expressions with exponents. Find the domains of algebraic expressions. Expressions Involving Exponents or Radicals Example 1 Evaluating Expressions Expression -Value Substitution a. b. c. d. Example 2 Evaluating Expressions Expression -Value Substitution a. b. y 823 813 2 22 4x 8y 3x2 y 24 22 4x 4y 2x12 x y 2 32 232 18x 3y 2 x2 y 1 2 2 1 4 x 1 2 y x2 y 313 3 13 3 1 3x 1y 3x3 y 242 216 32x 4y 2x2 x Section 0.3 Exponents and Radicals Properties of Exponents 1. Whole-number exponents: n factors 2. Zero exponent: 3. Negative exponents: 4. Radicals (principal nth root): 5. Rational exponents 6. Rational exponents 7. Special convention (square root): 2x x xmn xm 1n nxm xmn x1n m nxm mn: x1n nx1n: x annx a x 0xn 1 xn , x 0x0 1, xn x x x . . . x S T U D Y T I P If is even, then the principal th root is positive. For example, and 481 3.4 2 n n CHECKPOINT 1 Evaluate for x 3.y 4x2 CHECKPOINT 2 Evaluate for x 8.y 4x13 Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

39. 14 CHAPTER 0 A Precalculus Review Operations with Exponents Example 3 Simplifying Expressions with Exponents Simplify each expression. a. b. c. d. e. f. SOLUTION a. b. c. d. e. f. CHECKPOINT 3 Simplify each expression. a. b. c. *Specific calculator keystroke instructions for operations in this and other technology boxes can be found at college.hmco.com/info/larsonapplied. 4x2 x13 2 2x3x3x2 x4 xn xm xnm x 5x1 1 5x12 x1 1 5 x121 1 5 x32 xnxm xnmx1 2x2 2x1 x2 2x21 2x xn xm xnm xn m xnm , 5x4 x2 3 5x4 x6 5x46 5x2 5 x2 xn xm xnmxnm xnm, 3x2 x12 3 3 x2 x32 3x232 3x12 xn xm xnm 3x2 3x 9x2 x13 9x213 9x73 xn xm xnm 2x2 x3 2x23 2x5 x 5x1x1 2x2 5x4 x2 3 3x2 x12 33x2 3x2x2 x3 Graphing utilities perform the established order of operations when evaluating an expression. To see this, try entering the expressions and into your graphing utility to see that the expressions result in different values.* 1200 1 0.09 12 126 12001 0.09 12 126 T E C H N O L O G Y Operations with Exponents 1. Multiplying like bases: Add exponents. 2. Dividing like bases: Subtract exponents. 3. Removing parentheses: 4. Special conventions: xnm xn m xnm xnm , cxn cxn cxn cxn , xn xn xn xn , xn m xnm x y n xn yn xyn xn yn xn xm xnm xn xm xnm Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

40. Note in Example 3 that one characteristic of simplified expressions is the absence of negative exponents. Another characteristic of simplified expressions is that sums and differences are written in factored form. To do this, you can use the Distributive Property. Study the next example carefully to be sure that you understand the concepts involved in the factoring process. Example 4 Simplifying by Factoring Simplify each expression by factoring. a. b. c. d. SOLUTION a. b. c. d. Many algebraic expressions obtained in calculus occur in unsimplified form. For instance, the two expressions shown in the following example are the result of an operation in calculus called differentiation. The first is the derivative of and the second is the derivative of Example 5 Simplifying by Factoring Simplify each expression by factoring. a. b. 2x 332 12x 7 x 112 x 112 2x 332 12x 7 x 112 2x 332 2x 3 10x 10 x 112 2x 332 2x 3 10x 1 x 112 2x 352 10x 112 2x 332 x 112 2x 332 16x 1 x 112 2x 332 6x 9 10x 10 x 112 2x 332 32x 3 10x 1 3x 112 2x 352 10x 132 2x 332 2x 112 2x 352 . 2x 132 2x 352 , 2x12 3x52 x12 2 3x3 2 3x3 x 2x12 4x52 2x12 1 2x2 2x3 x2 x2 2x 1 2x2 x3 x2 2 x 2x12 3x52 2x12 4x52 2x3 x2 2x2 x3 abxn acxnm axn b cxm SECTION 0.3 Exponents and Radicals 15 CHECKPOINT 4 Simplify each expression by factoring. a. b. 2x12 8x32 x3 2x CHECKPOINT 5 Simplify the expression by factoring. 4x 2123x 152 x 2123x 132 S T U D Y T I P To check that the simplified expression is equivalent to the original expression, try substituting values for into each expression. x Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

41. 16 CHAPTER 0 A Precalculus Review Example 6 shows some additional types of expressions that can occur in calculus. The expression in Example 6(d) is an antiderivative of and the expression in Example 6(e) is the derivative of Example 6 Factors Involving Quotients Simplify each expression by factoring. a. b. c. d. e. SOLUTION a. b. c. d. e. 9x 22 x 14 3x 22 x 1 x 2 x 162 3x 22 x 12 x 1 x 2 x 16 3x 22 x 13 3x 23 x 12 x 13 2 3 20 x 153 5x 9 3 20 x 1534 5x 5 3 20 x 153 4 5x 1 3 5 x 153 3 4 x 183 12 20 x 153 15 20 x 183 1 189x 243 39x 2 9x 213 189x 2 9x 213 1 189x 243 x x32 x x12 1 x x 1 x x112 1 x x 3x2 x4 2x x2 3 x2 2x x21 3 x2 2 x3 x2 2 3x 22 x 13 3x 23 x 12 x 13 2 3 5 x 153 3 4 x 183 9x 213 189x 2 x x32 x 3x2 x4 2x x 23 x 13 . x 123 2x 3, A graphing utility offers several ways to calculate rational exponents and radicals. You should be familiar with the -squared key . This key squares the value of an expression. For rational exponents or exponents other than 2, use the key. For radical expressions, you can use the square root key , the cube root key , or the th root key . Consult your graphing utility users guide for specific keystrokes you can use to evaluate rational exponents and radical expressions. Use a graphing utility to evaluate each expression. a. b. c. d. e. 4163 3729576 16 54 823 x x T E C H N O L O G Y CHECKPOINT 6 Simplify the expression by factoring. 5x3 x6 3x x2 > 3 x Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

42. Domain of an Algebraic Expression When working with algebraic expressions involving x, you face the potential difficulty of substituting a value of x for which the expression is not defined (does not produce a real number). For example, the expression is not defined when because is not a real number. The set of all values for which an expression is defined is called its domain. So, the domain of is the set of all values of x such that is a real number. In order for to represent a real number, it is necessary that In other words, is defined only for those values of x that lie in the interval as shown in Figure 0.14. FIGURE 0.14 Example 7 Finding the Domain of an Expression Find the domain of each expression. a. b. c. SOLUTION a. The domain of consists of all x such that Expression must be nonnegative. which implies that So, the domain is b. The domain of is the same as the domain of except that is not defined when Because this occurs when the domain is c. Because is defined for all real numbers, its domain is CHECKPOINT 7 Find the domain of each expression. a. b. c. 3x 2 1 x 2 x 2 , .39x 1 2 3, . x 2 3,3x 2 0.13x 2 3x 2,13x 2 2 3, .x 2 3. 3x 2 0 3x 2 39x 1 1 3x 2 3x 2 321123 0 x 3 2 2x + 3 is not defined for these x. 2x + 3 is defined for these x. 3 2, , 2x 32x 3 0. 2x 3 2x 32x 3 22 3x 2 2x 3 SECTION 0.3 Exponents and Radicals 17 Copyright 2009 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

43. In Exercises 120, evaluate the expression for the given value of x. Expression x-Value Expression x-Value 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. In Exercises 2130, simplify the expression. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. In Exercises 3136, simplify by removing all possible factors from the radical. 31. 32. 33. 34. 35. 36. In Exercises 37 44, simplify each expression by factoring. 37. 38. 39. 40. 41. 42. 43. 44. In Exercises 4552, find the domain of the given expression. 45. 46. 47. 48. 49. 50. 51. 52. Body Surface Area In Exercises 5356, the body surface area (BSA) of a person is given by the formula where is the BSA (in square meters), is the height (in centimeters), and is the weight (in kilograms). Enter the formula into a graphing utility and use it to find the BSA for the given height and weight. 53. 54. 55. 56. 57. Population Growth The population of a species of mollusk is given by the formula where is the initial population, is the annual percentage growth rate of the population (expressed as a decimal), and is the time period (in years). Find the population in 4 years if the initial population i