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Unformatted text preview: PRECALCULUS Mathematics for Calculus FIFTH EDITION This page intentionally left blank PRECALCULUS Mathematics for Calculus FIFTH EDITION James Stewart McMaster University Lothar Redlin The Pennsylvania State University Saleem Watson California State University, Long Beach Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States Precalculus: Mathematics for Calculus, Fifth Edition, Enhanced WebAssign Edition James Stewart, Lothar Redlin, Saleem Watson Acquisitions Editor: Gary Whalen Assistant Editor: Natasha Coats ALL RIGHTS RESERVED. 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Further permissions questions can be emailed to [email protected] Project Manager, Editorial Production: Jennifer Risden Creative Director: Rob Hugel Art Director: Vernon Boes Print Buyer: Judy Inouye Student Edition: Permissions Editor: Bob Kauser Production Service: Martha Emry Text Designer: John Edeen Art Editor: Martha Emry Brooks/Cole Cengage Learning Photo Researcher: Stephen Forsling Copy Editor: Luana Richards USA Illustrator: Jade Myers, Matrix Cover Designer: Roy E. Neuhaus Cover Image: Bill Ralph Compositor: Newgen–India Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local office at international.cengage.com/region. Cengage Learning products are represented in Canada by Nelson Education, Ltd. For your course and learning solutions, visit academic.cengage.com. Purchase any of our products at your local college store or at our preferred online store . Printed in the United States of America 1 2 3 4 5 6 7 12 11 10 09 08 07 To our students, from whom we have learned so much. About the Cover The art on the cover was created by Bill Ralph, a mathematician who uses modern mathematics to produce visual representations of “dynamical systems.” Examples of dynamical systems in nature include the weather, blood pressure, the motions of the planets, and other phenomena that involve continual change. Such systems, which tend to be unpredictable and even chaotic at times, are modeled mathematically using the concepts of composition and iteration of functions (see Section 2.7 and the Discovery Project on pages 223–224). The basic idea is to start with a particular function and evaluate it at some point in its domain, yielding a new number. The function is then evaluated at the new number. Repeating this process produces a sequence of numbers called iterates of the function. The original domain is “painted” by assigning a color to each starting point; the color is determined by certain properties of its sequence of iterates and the mathematical concept of “dimension.” The result is a picture that reveals the complex patterns of the dynamical system. In a sense, these pictures allow us to look, through the lens of mathematics, at exotic little universes that have never been seen before. Professor Ralph teaches at Brock University in Canada. He can be contacted by e-mail at [email protected] About the Authors James Stewart was educated at the University of Toronto and Stanford University, did research at the University of London, and now teaches at McMaster University. His research field is harmonic analysis. He is the author of a best-selling calculus textbook series published by Brooks/Cole, including Calculus, 5th Ed., Calculus: Early Transcendentals, 5th Ed., and Calculus: Concepts and Contexts, 3rd Ed., as well as a series of high-school mathematics textbooks. Lothar Redlin grew up on Vancouver Island, received a Bachelor of Science degree from the University of Victoria, and a Ph.D. from McMaster University in 1978. He subsequently did research and taught at the University of Washington, the University of Waterloo, and California State University, Long Beach. He is currently Professor of Mathematics at The Pennsylvania State University, Abington College. His research field is topology. Saleem Watson received his Bachelor of Science degree from Andrews University in Michigan. He did graduate studies at Dalhousie University and McMaster University, where he received his Ph.D. in 1978. He subsequently did research at the Mathematics Institute of the University of Warsaw in Poland. He also taught at The Pennsylvania State University. He is currently Professor of Mathematics at California State University, Long Beach. His research field is functional analysis. The authors have also published College Algebra, Fourth Edition (Brooks/Cole, 2004), Algebra and Trigonometry, Second Edition (Brooks/Cole, 2007), and Trigonometry (Brooks/Cole, 2003). Contents Preface xiii To the Student xxi Calculators and Calculations 1 Fundamentals ■ 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 2 2.1 2.2 1 Chapter Overview 1 Real Numbers 2 Exponents and Radicals 12 Algebraic Expressions 24 ● DISCOVERY PROJECT Visualizing a Formula 34 Rational Expressions 35 Equations 44 Modeling with Equations 58 ● DISCOVERY PROJECT Equations through the Ages 75 Inequalities 76 Coordinate Geometry 87 Graphing Calculators; Solving Equations and Inequalities Graphically 101 Lines 111 Modeling Variation 123 Chapter 1 Review 130 Chapter 1 Test 135 ■ FOCUS ON PROBLEM SOLVING General Principles 138 Functions ■ xxii 146 Chapter Overview 147 What is a Function? 148 Graphs of Functions 158 ● DISCOVERY PROJECT Relations and Functions 171 vii viii Contents 2.3 2.4 2.5 2.6 2.7 2.8 3 Polynomial and Rational Functions ■ 3.1 3.2 3.3 3.4 3.5 3.6 4 Increasing and Decreasing Functions; Average Rate of Change 173 Transformations of Functions 182 Quadratic Functions; Maxima and Minima 193 Modeling with Functions 203 Combining Functions 214 ● DISCOVERY PROJECT Iteration and Chaos 223 One-to-One Functions and Their Inverses 225 Chapter 2 Review 233 Chapter 2 Test 237 ■ FOCUS ON MODELING Fitting Lines to Data 239 248 Chapter Overview 249 Polynomial Functions and Their Graphs 250 Dividing Polynomials 265 Real Zeros of Polynomials 272 ● DISCOVERY PROJECT Zeroing in on a Zero 283 Complex Numbers 285 Complex Zeros and the Fundamental Theorem of Algebra 291 Rational Functions 299 Chapter 3 Review 316 Chapter 3 Test 319 ■ FOCUS ON MODELING Fitting Polynomial Curves to Data 320 Exponential and Logarithmic 326 Functions ■ 4.1 4.2 4.3 4.4 4.5 Chapter Overview 327 Exponential Functions 328 ● DISCOVERY PROJECT Exponential Explosion 341 Logarithmic Functions 342 Laws of Logarithms 352 Exponential and Logarithmic Equations 358 Modeling with Exponential and Logarithmic Functions 369 Chapter 4 Review 382 Chapter 4 Test 385 ■ FOCUS ON MODELING Fitting Exponential and Power Curves to Data 386 Contents 5 Trigonometric Functions of Real 398 Numbers ■ 5.1 5.2 5.3 5.4 5.5 6 Trigonometric Functions of Angles ■ 6.1 6.2 6.3 6.4 6.5 7 Chapter Overview 399 The Unit Circle 400 Trigonometric Functions of Real Numbers 408 Trigonometric Graphs 418 ● DISCOVERY PROJECT Predator/Prey Models 432 More Trigonometric Graphs 434 Modeling Harmonic Motion 442 Chapter 5 Review 454 Chapter 5 Test 458 ■ FOCUS ON MODELING Fitting Sinusoidal Curves to Data 459 Chapter Overview 467 Angle Measure 468 Trigonometry of Right Triangles 478 Trigonometric Functions of Angles 488 ● DISCOVERY PROJECT Similarity 499 The Law of Sines 501 The Law of Cosines 508 Chapter 6 Review 516 Chapter 6 Test 520 ■ FOCUS ON MODELING Surveying 522 Analytic Trigonometry ■ 7.1 7.2 7.3 7.4 7.5 466 526 Chapter Overview 527 Trigonometric Identities 528 Addition and Subtraction Formulas 535 Double-Angle, Half-Angle, and Sum-Product Formulas 541 Inverse Trigonometric Functions 550 ● DISCOVERY PROJECT Where to Sit at the Movies Trigonometric Equations 561 Chapter 7 Review 571 Chapter 7 Test 574 ■ FOCUS ON MODELING Traveling and Standing Waves 575 560 ix x Contents 8 Polar Coordinates and Vectors ■ 8.1 8.2 8.3 8.4 8.5 9 580 Chapter Overview 581 Polar Coordinates 582 Graphs of Polar Equations 587 Polar Form of Complex Numbers; DeMoivre’s Theorem 596 ● DISCOVERY PROJECT Fractals 605 Vectors 607 The Dot Product 617 ● DISCOVERY PROJECT Sailing Against the Wind 626 Chapter 8 Review 627 Chapter 8 Test 629 ■ FOCUS ON MODELING Mapping the World 630 Systems of Equations and 634 Inequalities ■ 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 Chapter Overview 635 Systems of Equations 636 Systems of Linear Equations in Two Variables 644 Systems of Linear Equations in Several Variables 651 ● DISCOVERY PROJECT Best Fit versus Exact Fit 660 Systems of Linear Equations: Matrices 662 The Algebra of Matrices 675 ● DISCOVERY PROJECT Will the Species Survive? 688 Inverses of Matrices and Matrix Equations 689 ● DISCOVERY PROJECT Computer Graphics I 700 Determinants and Cramer’s Rule 704 Partial Fractions 715 Systems of Inequalities 721 Chapter 9 Review 728 Chapter 9 Test 733 ■ FOCUS ON MODELING Linear Programming 735 10 Analytic Geometry ■ 10.1 10.2 10.3 Chapter Overview Parabolas 744 Ellipses 753 Hyperbolas 762 743 742 Contents ● DISCOVERY PROJECT 10.4 10.5 10.6 10.7 Conics in Architecture 771 Shifted Conics 775 Rotation of Axes 783 ● DISCOVERY PROJECT Computer Graphics II 792 Polar Equations of Conics 795 Plane Curves and Parametric Equations 801 Chapter 10 Review 810 Chapter 10 Test 814 ■ FOCUS ON MODELING The Path of a Projectile 816 11 Sequences and Series ■ 11.1 11.2 11.3 11.4 11.5 11.6 820 Chapter Overview 821 Sequences and Summation Notation 822 Arithmetic Sequences 833 Geometric Sequences 838 ● DISCOVERY PROJECT Finding Patterns 847 Mathematics of Finance 848 Mathematical Induction 854 The Binomial Theorem 860 Chapter 11 Review 870 Chapter 11 Test 873 ■ FOCUS ON MODELING Modeling with Recursive Sequences 874 12 Limits: A Preview of Calculus ■ 12.1 12.2 12.3 12.4 12.5 880 Chapter Overview 881 Finding Limits Numerically and Graphically 882 Finding Limits Algebraically 890 Tangent Lines and Derivatives 898 ● DISCOVERY PROJECT Designing a Roller Coaster 908 Limits at Infinity: Limits of Sequences 908 Areas 916 Chapter 12 Review 925 Chapter 12 Test 928 ■ FOCUS ON MODELING Interpretations of Area 929 Cumulative Review Answers A1 Index I1 Photo Credits C1 CR1 xi This page intentionally left blank Preface The art of teaching is the art of assisting discovery. MARK VAN DOREN What do students really need to know to be prepared for calculus? What tools do instructors really need to assist their students in preparing for calculus? These two questions have motivated the writing of this book. To be prepared for calculus a student needs not only technical skill but also a clear understanding of concepts. Indeed, conceptual understanding and technical skill go hand in hand, each reinforcing the other. A student also needs to gain an appreciation for the power and utility of mathematics in modeling the real-world. Every feature of this textbook is devoted to fostering these goals. We are keenly aware that good teaching comes in many different forms, and that each instructor brings unique strengths and imagination to the classroom. Some instructors use technology to help students become active learners; others use the rule of four, “topics should be presented geometrically, numerically, algebraically, and verbally,” to promote conceptual reasoning; some use an expanded emphasis on applications to promote an appreciation for mathematics in everyday life; still others use group learning, extended projects, or writing exercises as a way of encouraging students to explore their own understanding of a given concept; and all present mathematics as a problem-solving endeavor. In this book we have included all these methods of teaching precalculus as enhancements to a central core of fundamental skills. These methods are tools to be utilized by instructors and their students to navigate their own course of action in preparing for calculus. In writing this fifth edition our purpose was to further enhance the utility of the book as an instructional tool. The main change in this edition is an expanded emphasis on modeling and applications: In each section the applications exercises have been expanded and are grouped together under the heading Applications, and each chapter (except Chapter 1) now ends with a Focus on Modeling section. We have also made some organizational changes, including dividing the chapter on analytic trigonometry into two chapters, each of more manageable size. There are numerous other smaller changes—as we worked through the book we sometimes realized that an additional example was needed, or an explanation could be clarified, or a section could benefit from different types of exercises. Throughout these changes, however, we have retained the overall structure and the main features that have contributed to the success of this book. xiii xiv Preface Many of the changes in this edition have been drawn from our own experience in teaching, but, more importantly, we have listened carefully to the users of the current edition, including many of our closest colleagues. We are also grateful to the many letters and e-mails we have received from users of this book, instructors as well as students, recommending changes and suggesting additions. Many of these have helped tremendously in making this edition even more user-friendly. Special Features The most important way to foster conceptual understanding and hone technical skill is through the problems that the instructor assigns. To that end we have provided a wide selection of exercises. EXERCISE SETS ■ ■ ■ Exercises Each exercise set is carefully graded, progressing from basic conceptual exercises and skill-development problems to more challenging problems requiring synthesis of previously learned material with new concepts. Applications Exercises We have included substantial applied problems that we believe will capture the interest of students. These are integrated throughout the text in both examples and exercises. In the exercise sets, applied problems are grouped together under the heading, Applications. (See, for example, pages 127, 156, 314, and 451.) Discovery, Writing, and Group Learning Each exercise set ends with a block of exercises called Discovery•Discussion. These exercises are designed to encourage students to experiment, preferably in groups, with the concepts developed in the section, and then to write out what they have learned, rather than simply look for “the answer.” (See, for example, pages 232 and 369.) A COMPLETE REVIEW CHAPTER We have included an extensive review chapter primarily as a handy reference for the student to revisit basic concepts in algebra and analytic geometry. ■ ■ Chapter 1 This is the review chapter; it contains the fundamental concepts a student needs to begin a precalculus course. As much or as little of this chapter can be covered in class as needed, depending on the background of the students. Chapter 1 Test The test at the end of Chapter 1 is intended as a diagnostic instrument for determining what parts of this review chapter need to be taught. It also serves to help students gauge exactly what topics they need to review. The trigonometry chapters of this text have been written so that either the right triangle approach or the unit circle approach may be taught first. Putting these two approaches in different chapters, each with its relevant applications, helps clarify the purpose of each approach. The chapters introducing trigonometry are as follows: FLEXIBLE APPROACH TO TRIGONOMETRY ■ ■ Chapter 5: Trigonometric Functions of Real Numbers This chapter introduces trigonometry through the unit circle approach. This approach emphasizes that the trigonometric functions are functions of real numbers, just like the polynomial and exponential functions with which students are already familiar. Chapter 6: Trigonometric Functions of Angles This chapter introduces trigonometry through the right triangle approach. This approach builds on the foundation of a conventional high-school course in trigonometry. Preface xv Another way to teach trigonometry is to intertwine the two approaches. Some instructors teach this material in the following order: Sections 5.1, 5.2, 6.1, 6.2, 6.3, 5.3, 5.4, 6.4, 6.5. Our organization makes it easy to do this without obscuring the fact that the two approaches involve distinct representations of the same functions. GRAPHING CALCULATORS AND COMPUTERS Calculator and computer technology extends in a powerful way our ability to calculate and visualize mathematics. The availability of graphing calculators makes it not less important, but far more important to understand the concepts that underlie what the calculator produces. Accordingly, all our calculator-oriented subsections are preceded by sections in which students must graph or calculate by hand, so that they can understand precisely what the calculator is doing when they later use it to simplify the routine, mechanical part of their work. The graphing calculator sections, subsections, examples, and exercises, all marked with the special symbol , are optional and may be omitted without loss of continuity. We use the following capabilities of the calculator: ■ ■ Graphing Calculators The use of the graphing calculator is integrated throughout the text to graph and analyze functions, families of functions, and sequences, to calculate and graph regression curves, to perform matrix algebra, to graph linear inequalities, and other powerful uses. Simple Programs We exploit the programming capabilities of a graphing calculator to simulate real-life situations, to sum series, or to compute the terms of a recursive sequence. (See, for instance, pages 702, 825, and 829.) FOCUS ON MODELING The “modeling” theme has been used throughout to unify and clarify the many applications of precalculus. We have made a special effort, in these modeling sections and subsections, to clarify the essential process of translating problems from English into the language of mathematics. (See pages 204 or 647.) ■ ■ Constructing Models There are numerous applied problems throughout the book where students are given a model to analyze (see, for instance, page 200). But the material on modeling, where students are required to construct mathematical models for themselves, has been organized into clearly defined sections and subsections (see, for example, pages 203, 369, 442, and 848). Focus on Modeling Each chapter concludes with a Focus on Modeling section. The first such section, after Chapter 2, introduces the basic idea of modeling a real-life situation by fitting lines to data (linear regression). Other sections present ways in which polynomial, exponential, logarithmic, and trigonometric functions, and systems of inequalities can all be used to model familiar phenomena from the sciences and from everyday life (see, for example, pages 320, 386, or 459). Chapter 1 concludes with a section entitled Focus on Problem Solving. One way to engage students and make them active learners is to have them work (perhaps in groups) on extended projects that give a feeling of substantial accomplishment when completed. Each chapter contains one or more Discovery Projects (see the table of contents); these provide a challenging but accessible set of activities that enable students to ...
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