Cengage learning precalculus mathematics for calculus 6th edition 0840068077

■Book Companion Website A new website www.stewartmath.com contains Dis-covery Projects for each chapter and Focus on Problem Solving sections that high-light different problem-solving

  x m n 1x m2n  x m n x n 



 n x

n y

ac

DISTANCE AND MIDPOINT FORMULAS

Distance between P11x1, y12 and P21x2, y22:

Point-slope equation of line y  y1 m 1x  x12

through P11x1, y12 with slope m

Slope-intercept equation of y  mx  b

line with slope m and y-intercept b

Two-intercept equation of line

with x-intercept a and y-intercept b

LOGARITHMS

y logax means a y  x

loga a x  x alogax  x

log x log10x ln x logex

loga x y logax  logay logaax yb  loga x  logay

loga x b  b loga x logb x

EXPONENTIAL AND LOGARITHMIC FUNCTIONS

01

10

1 1

1

xy

Ï=| x |

xy

≈xy

xxy

Ï=£œ∑x

xy

Ï=œ∑x

xy

Ï=x£

xy

Ï=b

b

xy

f1a2  limx a f1x2  f1a2 x  a

f1a2  limh0 f1a  h2  f1a2

Polar form of a complex number

For z a  bi, the polar form is

z r where r

Angle-of-rotation formula for conic sections

To eliminate the xy-term in the equation

Re

Imbi

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S I X T H E D I T I O N

MATHEMATICS FOR CALCULUS

A B O U T T H E AU T H O R S

J AMES S TEWART received his MS

from Stanford University and his PhD

from the University of Toronto He did

research at the University of London

and was influenced by the famous

mathematician George Polya at

Stan-ford University Stewart is Professor

Emeritus at McMaster University and is

currently Professor of Mathematics at

the University of Toronto His research

field is harmonic analysis and the

con-nections between mathematics and

music James Stewart is the author of a

bestselling calculus textbook series

published by Brooks/Cole, Cengage

Learning, including Calculus, Calculus:

Early Transcendentals, and Calculus:

Concepts and Contexts; a series of

pre-calculus texts; and a series of

high-school mathematics textbooks.

L OTHAR R EDLIN grew up on couver Island, received a Bachelor of Science degree from the University of Victoria, and received a PhD from McMaster University in 1978 He sub- sequently did research and taught at the University of Washington, the Uni- versity of Waterloo, and California State University, Long Beach He is currently Professor of Mathematics at The Pennsylvania State University, Abington Campus His research field is topology.

Van-S ALEEM W ATSON 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 PhD 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.

Stewart, Redlin, and Watson have also published College Algebra, Trigonometry, Algebra and Trigonometry, and (with Phyllis Panman) College Algebra: Concepts and Contexts.

The cover photograph shows the Science Museum in the City of

Arts and Sciences in Valencia, Spain, with a planetarium in the

dis-tance Built from 1991 to 1996, it was designed by Santiago

Cala-trava, a Spanish architect Calatrava has always been very

inter-ested in how mathematics can help him realize the buildings he

imagines As a young student, he taught himself descriptive

geom-etry from books in order to represent three-dimensional objects intwo dimensions Trained as both an engineer and an architect, hewrote a doctoral thesis in 1981 entitled “On the Foldability ofSpace Frames,” which is filled with mathematics, especially geo-metric transformations His strength as an engineer enables him to

be daring in his architecture

A BOUT THE C OVER

Precalculus: Mathematics for Calculus,

Sixth Edition

James Stewart, Lothar Redlin, Saleem Watson

Acquisitions Editor: Gary Whalen

Developmental Editor: Stacy Green

Assistant Editor: Cynthia Ashton

Editorial Assistant: Naomi Dreyer

Media Editor: Lynh Pham

Marketing Manager: Myriah Fitzgibbon

Marketing Assistant: Shannon Myers

Marketing Communications Manager:

Darlene Macanan

Content Project Manager: Jennifer Risden

Design Director: Rob Hugel

Art Director: Vernon Boes

Print Buyer: Karen Hunt

Rights Acquisitions Specialist:

Dean Dauphinais

Production Service: Martha Emry

Text Designer: Lisa Henry

Photo Researcher: Bill Smith Group

Copy Editor: Barbara Willette

Illustrators: Matrix Art Services,

Precision Graphics

Cover Designer: Lisa Henry

Cover Image: © Jose Fuste Raga/CORBIS

Compositor: Graphic World, Inc

© 2012, 2006 Brooks/Cole, Cengage LearningALL RIGHTS RESERVED No part of this work covered by the copyright hereinmay be reproduced, transmitted, stored, or used in any form or by any meansgraphic, electronic, or mechanical, including but not limited to photocopying,recording, scanning, digitizing, taping, Web distribution, information networks,

or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the priorwritten permission of the publisher

Library of Congress Control Number: 2010935410ISBN-13: 978-0-8400-6807-1

ISBN-10: 0-8400-6807-7

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Printed in the United States of America

1 2 3 4 5 6 7 14 13 12 11

2.3 Getting Information from the Graph of a Function 163

2.4 Average Rate of Change of a Function 172

2.5 Transformations of Functions 179

2.6 Combining Functions 190

2.7 One-to-One Functions and Their Inverses 199

v

Chapter 2 Review 207 Chapter 2 Test 211

■ FOCUS ON MODELING Modeling with Functions 213

Chapter Overview 223 3.1 Quadratic Functions and Models 224 3.2 Polynomial Functions and Their Graphs 232 3.3 Dividing Polynomials 246

3.4 Real Zeros of Polynomials 253 3.5 Complex Numbers 264 3.6 Complex Zeros and the Fundamental Theorem of Algebra 269 3.7 Rational Functions 277

Chapter 3 Review 292 Chapter 3 Test 295

■ FOCUS ON MODELING Fitting Polynomial Curves to Data 296

C H A P T E R 4 E XPONENTIAL AND L OGARITHMIC F UNCTIONS 301

4.5 Exponential and Logarithmic Equations 331

4.6 Modeling with Exponential and Logarithmic Functions 340 Chapter 4 Review 353

Chapter 4 Test 356

■ FOCUS ON MODELING Fitting Exponential and Power Curves to Data 357

Cumulative Review Test: Chapters 2, 3, and 4 367

C H A P T E R 5 T RIGONOMETRIC F UNCTIONS : U NIT C IRCLE A PPROACH 369

Chapter Overview 369

5.1 The Unit Circle 370

5.2 Trigonometric Functions of Real Numbers 377

5.3 Trigonometric Graphs 386

5.4 More Trigonometric Graphs 399

5.5 Inverse Trigonometric Functions and Their Graphs 406

5.6 Modeling Harmonic Motion 412 Chapter 5 Review 423

Chapter 5 Test 426

■ FOCUS ON MODELING Fitting Sinusoidal Curves to Data 427

vi Contents

C H A P T E R 6 T RIGONOMETRIC F UNCTIONS : R IGHT T RIANGLE A PPROACH 433

Chapter Overview 433

6.1 Angle Measure 434

6.2 Trigonometry of Right Triangles 443

6.3 Trigonometric Functions of Angles 451

6.4 Inverse Trigonometric Functions and Right Triangles 462

6.5 The Law of Sines 469

6.6 The Law of Cosines 476 Chapter 6 Review 483 Chapter 6 Test 487

■ FOCUS ON MODELING Surveying 489

Chapter Overview 493

7.1 Trigonometric Identities 494

7.2 Addition and Subtraction Formulas 500

7.3 Double-Angle, Half-Angle, and Product-Sum Formulas 507

7.4 Basic Trigonometric Equations 517

7.5 More Trigonometric Equations 524 Chapter 7 Review 530

Chapter 7 Test 532

■ FOCUS ON MODELING Traveling and Standing Waves 533

Cumulative Review Test: Chapters 5, 6, and 7 538

C H A P T E R 8 P OLAR C OORDINATES AND P ARAMETRIC E QUATIONS 541

Chapter Overview 541

8.1 Polar Coordinates 542

8.2 Graphs of Polar Equations 547

8.3 Polar Form of Complex Numbers; De Moivre's Theorem 555

8.4 Plane Curves and Parametric Equations 564 Chapter 8 Review 572

Chapter 8 Test 574

■ FOCUS ON MODELING The Path of a Projectile 575

C H A P T E R 9 V ECTORS IN T WO AND T HREE D IMENSIONS 579

Chapter Overview 579

9.1 Vectors in Two Dimensions 580

9.2 The Dot Product 589

9.3 Three-Dimensional Coordinate Geometry 597

9.4 Vectors in Three Dimensions 603

9.5 The Cross Product 610

Contents vii

9.6 Equations of Lines and Planes 616 Chapter 9 Review 620

Chapter 9 Test 623

■ FOCUS ON MODELING Vector Fields 624

Cumulative Review Test: Chapters 8 and 9 628

C H A P T E R 10 S YSTEMS OF E QUATIONS AND I NEQUALITIES 629

Chapter Overview 629

10.1 Systems of Linear Equations in Two Variables 630

10.2 Systems of Linear Equations in Several Variables 640

10.3 Matrices and Systems of Linear Equations 649

10.4 The Algebra of Matrices 661

10.5 Inverses of Matrices and Matrix Equations 672

10.6 Determinants and Cramer's Rule 682

10.7 Partial Fractions 693

10.8 Systems of Nonlinear Equations 698

10.9 Systems of Inequalities 703 Chapter 10 Review 710 Chapter 10 Test 714

■ FOCUS ON MODELING Linear Programming 716

■ FOCUS ON MODELING Conics in Architecture 776

Cumulative Review Test: Chapters 10 and 11 780

12.6 The Binomial Theorem 820 Chapter 12 Review 829 Chapter 12 Test 832

■ FOCUS ON MODELING Modeling with Recursive Sequences 833

Chapter Overview 839

13.1 Finding Limits Numerically and Graphically 840

13.2 Finding Limits Algebraically 848

13.3 Tangent Lines and Derivatives 856

13.4 Limits at Infinity; Limits of Sequences 865

Chapter 13 Review 881 Chapter 13 Test 883

■ FOCUS ON MODELING Interpretations of Area 884

Cumulative Review Test: Chapters 12 and 13 888

APPENDIX: Calculations and Signific ant Figures 889 ANSWERS A1

INDEX I1

Contents ix

P R E FAC E

What do students really need to know to be prepared for calculus? What tools do tors really need to assist their students in preparing for calculus? These two questions havemotivated the writing of this book

instruc-To be prepared for calculus a student needs not only technical skill but also a clear

un-derstanding of concepts Indeed, conceptual unun-derstanding 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

de-voted to fostering these goals

In writing this Sixth Edition our purpose is to further enhance the utility of the book as aninstructional tool for teachers and as a learning tool for students There are several majorchanges in this edition including a restructuring of each exercise set to better align the exer-

cises with the examples of each section In this edition each exercise set begins with Concepts

Exercises, which encourage students to work with basic concepts and to use mathematical

vo-cabulary appropriately Several chapters have been reorganized and rewritten (as describedbelow) to further focus the exposition on the main concepts; we have added a new chapter onvectors in two and three dimensions In all these changes and numerous others (small andlarge) we have retained the main features that have contributed to the success of this book

New to the Sixth Edition

■Exercises More than 20% of the exercises are new This includes new Concept ercises and new Cumulative Review exercises Key exercises are now linked to ex-

Ex-amples in the text

■Book Companion Website A new website www.stewartmath.com contains

Dis-covery Projects for each chapter and Focus on Problem Solving sections that

high-light different problem-solving principles outlined in the Prologue

■CHAPTER 2 Functions This chapter has been completely rewritten to focus more

sharply on the fundamental and crucial concept of function The material on quadratic

functions, formerly in this chapter, is now part of the chapter on polynomial functions

■CHAPTER 3 Polynomial and Rational Functions This chapter now begins with asection on quadratic functions, leading to higher degree polynomial functions

■CHAPTER 4 Exponential and Logarithmic Functions The material on the naturalexponential function is now in a separate section

■CHAPTER 5 Trigonometric Functions: Unit Circle Approach This chapter cludes a new section on inverse trigonometric functions and their graphs Introduc-ing this topic here reinforces the function concept in the context of trigonometry

in-■CHAPTER 6 Trigonometric Functions: Right Triangle ApproachThis chapter cludes a new section on inverse trigonometric functions and right triangles (Section6.4) which is needed in applying the Laws of Sines and Cosines in the followingsection, as well as for solving trigonometric equations in Chapter 7.

in-■CHAPTER 7 Analytic Trigonometry This chapter has been completely revised.There are two new sections on trigonometric equations (Sections 7.4 and 7.5) Thematerial on this topic (formerly in Section 7.5) has been expanded and revised

■CHAPTER 8 Polar Coordinates and Parametric Equations This chapter is now moresharply focused on the concept of a coordinate system The section on parametricequations is new to this chapter The material on vectors is now in its own chapter

■CHAPTER 9 Vectors in Two and Three Dimensions This is a new chapter with a

new Focus on Modeling section.

■CHAPTER 10 Systems of Equations and Inequalities The material on systems ofnonlinear equations is now in a separate section

■CHAPTER 11 Conic Sections This chapter is now more closely devoted to thetopic of analytic geometry, especially the conic sections; the section on parametricequations has been moved to Chapter 8

Teaching with the Help of This Book

We are keenly aware that good teaching comes in many forms, and that there are manydifferent approaches to teaching the concepts and skills of precalculus The organization

of the topics in this book is designed to accommodate different teaching styles For ample, the trigonometry chapters have been organized so that either the unit circle ap-proach or the right triangle approach can be taught first Here are other special featuresthat can be used to complement different teaching styles:

ex-EXERCISESETS The most important way to foster conceptual understanding and honetechnical skill is through the problems that the instructor assigns To that end we haveprovided a wide selection of exercises

■Concept Exercises These exercises ask students to use mathematical language tostate fundamental facts about the topics of each section

■Skills Exercises Each exercise set is carefully graded, progressing from basic development exercises to more challenging problems requiring synthesis of previ-ously learned material with new concepts

skill-■Applications Exercises We have included substantial applied problems that we lieve will capture the interest of students

be-■Discovery, Writing, and Group Learning Each exercise set ends with a block of

exercises labeled Discovery ■Discussion■Writing These exercises are designed to

encourage students to experiment, preferably in groups, with the concepts oped in the section, and then to write about what they have learned, rather than sim-ply look for the answer

devel-■Now Try Exercise At the end of each example in the text the student is directed

to a similar exercise in the section that helps reinforce the concepts and skills oped in that example (see, for instance, page 4)

devel-■Check Your Answer Students are encouraged to check whether an answer they

ob-tained is reasonable This is emphasized throughout the text in numerous Check

Your Answer sidebars that accompany the examples (See, for instance, page 52).

A COMPLETEREVIEWCHAPTER We have included an extensive review chapter ily as a handy reference for the basic concepts that are preliminary to this course

primar-■Chapter 1This is the review chapter; it contains the fundamental concepts from gebra and analytic geometry that a student needs in order to begin a precalculus

al-xii Preface

course As much or as little of this chapter can be covered in class as needed, pending on the background of the students.

de-■Chapter 1 Test The test at the end of Chapter 1 is designed as a diagnostic test fordetermining what parts of this review chapter need to be taught It also serves tohelp students gauge exactly what topics they need to review

FLEXIBLEAPPROACH TOTRIGONOMETRY The trigonometry chapters of this text havebeen written so that either the right triangle approach or the unit circle approach may betaught first Putting these two approaches in different chapters, each with its relevant ap-plications, helps to clarify the purpose of each approach The chapters introducingtrigonometry are as follows:

■Chapter 5 Trigonometric Functions: Unit Circle Approach This chapter duces trigonometry through the unit circle approach This approach emphasizes thatthe trigonometric functions are functions of real numbers, just like the polynomialand exponential functions with which students are already familiar

intro-■Chapter 6 Trigonometric Functions: Right Triangle Approach This chapter troduces trigonometry through the right triangle approach This approach builds onthe foundation of a conventional high-school course in trigonometry

in-Another way to teach trigonometry is to intertwine the two approaches Some tors teach this material in the following order: Sections 5.1, 5.2, 6.1, 6.2, 6.3, 5.3, 5.4, 5.5,5.6, 6.4, 6.5, and 6.6 Our organization makes it easy to do this without obscuring the factthat the two approaches involve distinct representations of the same functions

instruc-GRAPHINGCALCULATORS ANDCOMPUTERS We make use of graphing calculators andcomputers in examples and exercises throughout the book Our calculator-oriented exam-ples are always preceded by examples 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 optionaland may be omitted without loss of continuity We use the following capabilities of thecalculator

■Graphing, Regression, Matrix Algebra The capabilities of the graphing calculatorare used 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 tor to simulate real-life situations, to sum series, or to compute the terms of a recur-sive sequence (See, for instance, pages 787 and 791.)

calcula-FOCUS ON MODELING The “modeling” theme has been used throughout to unify andclarify the many applications of precalculus We have made a special effort to clarify theessential process of translating problems from English into the language of mathematics(see pages 214 and 636)

■Constructing Models There are numerous applied problems throughout the bookwhere students are given a model to analyze (see, for instance, page 228) But the

material on modeling, in which students are required to construct mathematical

models, has been organized into clearly defined sections and subsections (see forexample, pages 213, 340, and 427)

■Focus on Modeling Each chapter concludes with a Focus on Modeling section.

The first such section, after Chapter 1, introduces the basic idea of modeling a life situation by fitting lines to data (linear regression) Other sections present ways

real-in which polynomial, exponential, logarithmic, and trigonometric functions, andsystems of inequalities can all be used to model familiar phenomena from the sci-ences and from everyday life (see for example pages 296, 357, and 427)

Preface xiii

BOOK COMPANION WEBSITE A website that accompanies this book is located at

www stewartmath.com The site includes many useful resources for teaching

precalcu-lus, including the following:

■ Discovery Projects Discovery Projects for each chapter are available on the

web-site Each project provides a challenging but accessible set of activities that enablestudents (perhaps working in groups) to explore in greater depth an interesting as-pect of the topic they have just learned (See for instance the Discovery Projects

Visualizing a Formula, Relations and Functions, Will the Species Survive?, and Computer Graphics I and II.)

■ Focus on Problem Solving Several Focus on Problem Solving sections are

avail-able on the website Each such section highlights one of the problem-solving ciples introduced in the Prologue and includes several challenging problems (See

prin-for instance Recognizing Patterns, Using Analogy, Introducing Something Extra,

Taking Cases, and Working Backward.)

MATHEMATICALVIGNETTES Throughout the book we make use of the margins to vide historical notes, key insights, or applications of mathematics in the modern world.These serve to enliven the material and show that mathematics is an important, vital ac-tivity, and that even at this elementary level it is fundamental to everyday life

pro-■ Mathematical Vignettes These vignettes include biographies of interestingmathematicians and often include a key insight that the mathematician discoveredand which is relevant to precalculus (See, for instance, the vignettes on Viète,page 49; Salt Lake City, page 84; and radiocarbon dating, page 333)

■ Mathematics in the Modern World This is a series of vignettes that emphasizesthe central role of mathematics in current advances in technology and the sciences(see pages 283, 700, and 759, for example)

REVIEW SECTIONS AND CHAPTER TESTS Each chapter ends with an extensive reviewsection including the following

■ Concept Check The Concept Check at the end of each chapter is designed to get

the students to think about and explain in their own words the ideas presented inthe chapter These can be used as writing exercises, in a classroom discussion set-ting, or for personal study

■ Review Exercises The Review Exercises at the end of each chapter recapitulate

the basic concepts and skills of the chapter and include exercises that combine thedifferent ideas learned in the chapter

■ Chapter Test The review sections conclude with a Chapter Test designed to help

students gauge their progress

■ Cumulative Review Tests The Cumulative Review Tests following Chapters 4, 7,

9, 11, and 13 combine skills and concepts from the preceding chapters and are signed to highlight the connections between the topics in these related chapters

de-■ Answers Brief answers to odd-numbered exercises in each section (including thereview exercises), and to all questions in the Concepts Exercises and ChapterTests, are given in the back of the book

Acknowledgments

We thank the following reviewers for their thoughtful and constructive comments

REVIEWERS FOR THEFIFTHEDITION Kenneth Berg, University of Maryland; ElizabethBowman, University of Alabama at Huntsville; William Cherry, University of NorthTexas; Barbara Cortzen, DePaul University; Gerry Fitch, Louisiana State University;Lana Grishchenko, Cal Poly State University, San Luis Obispo; Bryce Jenkins, Cal PolyState University, San Luis Obispo; Margaret Mary Jones, Rutgers University; Victoria

xiv Preface

Kauffman, University of New Mexico; Sharon Keener, Georgia Perimeter College;YongHee Kim-Park, California State University Long Beach; Mangala Kothari, RutgersUniversity; Andre Mathurin, Bellarmine College Prep; Donald Robertson, Olympic Col-lege; Jude Socrates, Pasadena City College; Enefiok Umana, Georgia Perimeter College;Michele Wallace, Washington State University; and Linda Waymire, Daytona BeachCommunity College.

REVIEWERS FOR THE SIXTH EDITION Raji Baradwaj, UMBC; Chris Herman, LorainCounty Community College; Irina Kloumova, Sacramento City College; Jim McCleery,Skagit Valley College, Whidbey Island Campus; Sally S Shao, Cleveland State Univer-sity; David Slutzky, Gainesville State College; Edward Stumpf, Central Carolina Com-munity College; Ricardo Teixeira, University of Texas at Austin; Taixi Xu, Southern Poly-technic State University; and Anna Wlodarczyk, Florida International University

We are grateful to our colleagues who continually share with us their insights into ing mathematics We especially thank Andrew Bulman-Fleming for writing the StudyGuide and the Solutions Manual and Doug Shaw at the University of Northern Iowa forwriting the Instructor Guide

teach-We thank Martha Emry, our production service and art editor; her energy, devotion, perience, and intelligence were essential components in the creation of this book Wethank Barbara Willette, our copy editor, for her attention to every detail in the manuscript

ex-We thank Jade Myers and his staff at Matrix Art Services for their attractive and accurategraphs and Precision Graphics for bringing many of our illustrations to life We thank ourdesigner Lisa Henry for the elegant and appropriate design for the interior of the book

At Brooks/Cole we especially thank Stacy Green, developmental editor, for guidingand facilitating every aspect of the production of this book Of the many Brooks/Cole staffinvolved in this project we particularly thank the following: Jennifer Risden, content proj-ect manager, Cynthia Ashton, assistant editor; Lynh Pham, media editor; Vernon Boes, artdirector; and Myriah Fitzgibbon, marketing manager They have all done an outstandingjob

Numerous other people were involved in the production of this book—including missions editors, photo researchers, text designers, typesetters, compositors, proof read-ers, printers, and many more We thank them all

per-Above all, we thank our editor Gary Whalen His vast editorial experience, his sive knowledge of current issues in the teaching of mathematics, and especially his deepinterest in mathematics textbooks, have been invaluable resources in the writing of thisbook

exten-Preface xv

Instructor's Guide ISBN-10: 0-8400-6883-2; ISBN-13: 978-0-8400-6883-5

Doug Shaw, author of the Instructor Guides for the widely used Stewart calculus texts,wrote this helpful teaching companion It contains points to stress, suggested time to al-lot, text discussion topics, core materials for lectures, workshop/discussion suggestions,group work exercises in a form suitable for handout, solutions to group work exercises,and suggested homework problems

Media

Enhanced WebAssign ISBN-10: 0-538-73810-3; ISBN-13: 978-0-538-73810-1

Exclusively from Cengage Learning, Enhanced WebAssign®offers an extensive onlineprogram for Precalculus to encourage the practice that's so critical for concept mastery.The meticulously crafted pedagogy and exercises in this text become even more effective

in Enhanced WebAssign, supplemented by multimedia tutorial support and immediatefeedback as students complete their assignments Algorithmic problems allow you to as-sign unique versions to each student The Practice Another Version feature (activated atyour discretion) allows students to attempt the questions with new sets of values until theyfeel confident enough to work the original problem Students benefit from a new PremiumeBook with highlighting and search features; Personal Study Plans (based on diagnosticquizzing) that identify chapter topics they still need to master; and links to video solutions,interactive tutorials, and even live online help

ExamView Computerized Testing

ExamView®testing software allows instructors to quickly create, deliver, and customizetests for class in print and online formats, and features automatic grading Includes a testbank with hundreds of questions customized directly to the text ExamView is availablewithin the PowerLecture CD-ROM

Solution Builder www.cengage.com/solutionbuilder

This online instructor database offers complete worked solutions to all exercises in thetext, allowing you to create customized, secure solutions printouts (in PDF format)matched exactly to the problems you assign in class

A N C I L L A R I E S

xvii

PowerLecture with ExamView

ISBN-10: 0-8400-6901-4; ISBN-13: 978-0-8400-6901-6

This CD-ROM provides the instructor with dynamic media tools for teaching Create, liver, and customize tests (both print and online) in minutes with ExamView®Computer-ized Testing Featuring Algorithmic Equations Easily build solution sets for homework orexams using Solution Builder's online solutions manual Microsoft®PowerPoint®lectureslides and figures from the book are also included on this CD-ROM

giv-Study Guide ISBN-10: 0-8400-6917-0; ISBN-13: 978-0-8400-6917-7

This carefully crafted learning resource helps students develop their problem-solvingskills while reinforcing their understanding with detailed explanations, worked-out ex-amples, and practice problems Students will also find listings of key ideas to master Eachsection of the main text has a corresponding section in the Study Guide

Media

Enhanced WebAssign ISBN-10: 0-538-73810-3; ISBN-13: 978-0-538-73810-1

Exclusively from Cengage Learning, Enhanced WebAssign® offers an extensive onlineprogram for Precalculus to encourage the practice that's so critical for concept mastery.You'll receive multimedia tutorial support as you complete your assignments You'll alsobenefit from a new Premium eBook with highlighting and search features; Personal StudyPlans (based on diagnostic quizzing) that identify chapter topics you still need to master;and links to video solutions, interactive tutorials, and even live online help

Book Companion Website

A new website www.stewartmath.com contains Discovery Projects for each chapter and

Focus on Problem Solving sections that highlight different problem-solving principles

outlined in the Prologue

CengageBrain.comVisit www.cengagebrain.com to access additional course materials and companion re-

sources At the CengageBrain.com home page, search for the ISBN of your title (from theback cover of your book) using the search box at the top of the page This will take you

to the product page where free companion resources can be found

Text-Specific DVDs ISBN-10: 0-8400-6882-4; ISBN-13: 978-0-8400-6882-8

The Text-Specific DVDs include new learning objective based lecture videos TheseDVDs provide comprehensive coverage of the course—along with additional explana-tions of concepts, sample problems, and applications—to help students review essentialtopics

xviii Ancillaries

TO THE STUDENT

This textbook was written for you to use as a guide to mastering precalculus ics Here are some suggestions to help you get the most out of your course.

mathemat-First of all, you should read the appropriate section of text before you attempt your

homework problems Reading a mathematics text is quite different from reading a novel,

a newspaper, or even another textbook You may find that you have to reread a passage several times before you understand it Pay special attention to the examples, and work them out yourself with pencil and paper as you read Then do the linked exercises referred

to in “Now Try Exercise ” at the end of each example With this kind of preparation

you will be able to do your homework much more quickly and with more understanding Don’t make the mistake of trying to memorize every single rule or fact you may come

across Mathematics doesn’t consist simply of memorization Mathematics is a

problem-solving art, not just a collection of facts To master the subject you must solve problems—

lots of problems Do as many of the exercises as you can Be sure to write your solutions

in a logical, step-by-step fashion Don’t give up on a problem if you can’t solve it right away Try to understand the problem more clearly—reread it thoughtfully and relate it to what you have learned from your teacher and from the examples in the text Struggle with

it until you solve it Once you have done this a few times you will begin to understand what mathematics is really all about.

Answers to the odd-numbered exercises, as well as all the answers to each chapter test, appear at the back of the book If your answer differs from the one given, don’t immedi- ately assume that you are wrong There may be a calculation that connects the two an- swers and makes both correct For example, if you get 1/( ) but the answer given

is 1  , your answer is correct, because you can multiply both numerator and

de-nominator of your answer by  1 to change it to the given answer In rounding

ap-proximate answers, follow the guidelines in the Appendix: Calculations and Significant

Figures.

The symbol is used to warn against committing an error We have placed this bol in the margin to point out situations where we have found that many of our students make the same mistake.

sym-12

xix

per liter of solution ⇔ is equivalent to

Standing Room Only 343

Half-Lives of Radioactive Elements 345

Olga Taussky-Todd 668Arthur Cayley 674David Hilbert 683Emmy Noether 686The Rhind Papyrus 694Linear Programming 717Archimedes 729Eccentricities of the Orbits

of the Planets 738Paths of Comets 745Johannes Kepler 754Large Prime Numbers 786Eratosthenes 787

Fibonacci 787The Golden Ratio 791

Srinavasa Ramanujan 802Blaise Pascal 818Pascal’s Triangle 822Sir Isaac Newton 852Newton and Limits 859

Mathematics in the Modern World 16Changing Words, Sound, and Picturesinto Numbers 30

Error Correcting Codes 38Computers 182

Splines 234Automotive Design 238Unbreakable Codes 284Law Enforcement 318Evaluating Functions on a Calculator 400Weather Prediction 632

Mathematical Ecology 679Global Positioning System 700Looking Inside Your Head 759Fair Division of Assets 796Fractals 804

Mathematical Economics 810

M AT H E M AT I C S I N

T H E M O D E R N W O R L D

The ability to solve problems is a highly prized skill in many aspects of our lives; it is tainly an important part of any mathematics course There are no hard and fast rules that will ensure success in solving problems However, in this Prologue we outline some gen- eral steps in the problem-solving process and we give principles that are useful in solv- ing certain problems These steps and principles are just common sense made explicit.

cer-They have been adapted from George Polya’s insightful book How To Solve It.

1 Understand the Problem

The first step is to read the problem and make sure that you understand it Ask yourself the following questions:

What is the unknown?

What are the given quantities?

What are the given conditions?

For many problems it is useful to

draw a diagram

and identify the given and required quantities on the diagram Usually, it is necessary to

introduce suitable notation

In choosing symbols for the unknown quantities, we often use letters such as a, b, c, m,

n, x, and y, but in some cases it helps to use initials as suggestive symbols, for instance,

V for volume or t for time.

2 Think of a Plan

Find a connection between the given information and the unknown that enables you to calculate the unknown It often helps to ask yourself explicitly: “How can I relate the given to the unknown?” If you don’t see a connection immediately, the following ideas may be helpful in devising a plan.

Tr y t o R e c o g n i z e S o m e t h i n g Fa m i l i a r

Relate the given situation to previous knowledge Look at the unknown and try to recall

a more familiar problem that has a similar unknown.

GEORGE POLYA (1887–1985) is famous

among mathematicians for his ideas on

problem solving His lectures on

prob-lem solving at Stanford University

at-tracted overflow crowds whom he held

on the edges of their seats, leading

them to discover solutions for

them-selves He was able to do this because

of his deep insight into the psychology

of problem solving His well-known

book How To Solve It has been

trans-lated into 15 languages He said that

Euler (see page 266) was unique

among great mathematicians because

he explained how he found his results.

Polya often said to his students and

colleagues,“Yes, I see that your proof is

correct, but how did you discover it?” In

the preface to How To Solve It, Polya

writes,“A great discovery solves a great

problem but there is a grain of

discov-ery in the solution of any problem Your

problem may be modest; but if it

chal-lenges your curiosity and brings into

play your inventive faculties, and if you

solve it by your own means, you may

experience the tension and enjoy the

triumph of discovery.”

ar-Wo r k B a c k w a r d

Sometimes it is useful to imagine that your problem is solved and work backward, step

by step, until you arrive at the given data Then you might be able to reverse your steps and thereby construct a solution to the original problem This procedure is commonly

used in solving equations For instance, in solving the equation 3x  5  7, we suppose

that x is a number that satisfies 3x  5  7 and work backward We add 5 to each side of

the equation and then divide each side by 3 to get x  4 Since each of these steps can be reversed, we have solved the problem.

 E s t a b l i s h S u b g o a l s

In a complex problem it is often useful to set subgoals (in which the desired situation is only partially fulfilled) If you can attain or accomplish these subgoals, then you might be able to build on them to reach your final goal.

 I n d i r e c t R e a s o n i n g

Sometimes it is appropriate to attack a problem indirectly In using proof by

contradic-tion to prove that P implies Q, we assume that P is true and Q is false and try to see why

this cannot happen Somehow we have to use this information and arrive at a tion to what we absolutely know is true.

contradic- M a t h e m a t i c a l I n d u c t i o n

In proving statements that involve a positive integer n, it is frequently helpful to use the

Principle of Mathematical Induction, which is discussed in Section 12.5.

3 Carry Out the Plan

In Step 2, a plan was devised In carrying out that plan, you must check each stage of the plan and write the details that prove that each stage is correct.

P2 Prologue

4 Look Back

Having completed your solution, it is wise to look back over it, partly to see whether any errors have been made and partly to see whether you can discover an easier way to solve the problem Looking back also familiarizes you with the method of solution, which may

be useful for solving a future problem Descartes said, “Every problem that I solved came a rule which served afterwards to solve other problems.”

be-We illustrate some of these principles of problem solving with an example

A driver sets out on a journey For the first half of the distance, she drives at the leisurely pace of 30 mi/h; during the second half she drives 60 mi/h What is her average speed on this trip?

THINKING ABOUT THE PROBLEM

It is tempting to take the average of the speeds and say that the average speed for the entire trip is

But is this simple-minded approach really correct?

Let’s look at an easily calculated special case Suppose that the total distance traveled is 120 mi Since the first 60 mi is traveled at 30 mi/h, it takes 2 h The second 60 mi is traveled at 60 mi/h, so it takes one hour Thus, the total time is

2  1  3 hours and the average speed is

So our guess of 45 mi/h was wrong.

S O L U T I O N

We need to look more carefully at the meaning of average speed It is defined as

Let d be the distance traveled on each half of the trip Let t1and t2be the times taken for the first and second halves of the trip Now we can write down the information we have been given For the first half of the trip we have

and for the second half we have

Now we identify the quantity that we are asked to find:

To calculate this quantity, we need to know t1and t2, so we solve the above equations for these times:

Try a special case 

Understand the problem 

State what is given 

Identify the unknown 

Connect the given

with the unknown 

Now we have the ingredients needed to calculate the desired quantity:

P R O B L E M S

1 Distance, Time, and Speed An old car has to travel a 2-mile route, uphill and down.Because it is so old, the car can climb the first mile—the ascent—no faster than an averagespeed of 15 mi/h How fast does the car have to travel the second mile—on the descent it can

go faster, of course—to achieve an average speed of 30 mi/h for the trip?

2 Comparing Discounts Which price is better for the buyer, a 40% discount or two cessive discounts of 20%?

suc-3 Cutting up a Wire A piece of wire is bent as shown in the figure You can see that onecut through the wire produces four pieces and two parallel cuts produce seven pieces Howmany pieces will be produced by 142 parallel cuts? Write a formula for the number of pieces

produced by n parallel cuts.

4 Amoeba Propagation An amoeba propagates by simple division; each split takes

3 minutes to complete When such an amoeba is put into a glass container with a nutrientfluid, the container is full of amoebas in one hour How long would it take for the container

to be filled if we start with not one amoeba, but two?

5 Batting Averages Player A has a higher batting average than player B for the first half

of the baseball season Player A also has a higher batting average than player B for the ond half of the season Is it necessarily true that player A has a higher batting average than player B for the entire season?

sec-6 Coffee and Cream A spoonful of cream is taken from a pitcher of cream and put into acup of coffee The coffee is stirred Then a spoonful of this mixture is put into the pitcher ofcream Is there now more cream in the coffee cup or more coffee in the pitcher of cream?

7 Wrapping the World A ribbon is tied tightly around the earth at the equator Howmuch more ribbon would you need if you raised the ribbon 1 ft above the equator every-where? (You don’t need to know the radius of the earth to solve this problem.)

8 Ending Up Where You Started A woman starts at a point P on the earth’s surface and walks 1 mi south, then 1 mi east, then 1 mi north, and finds herself back at P, the start- ing point Describe all points P for which this is possible [Hint: There are infinitely many

such points, all but one of which lie in Antarctica.]

 120d

2d  d 

120d 3d  40

Multiply numerator anddenominator by 60

P4 Prologue

Don’t feel bad if you can’t solve these

problems right away Problems 1 and 4

were sent to Albert Einstein by his

friend Wertheimer Einstein (and his

friend Bucky) enjoyed the problems

and wrote back to Wertheimer Here is

part of his reply:

Your letter gave us a lot of

amusement The first

intelli-gence test fooled both of us

(Bucky and me) Only on

work-ing it out did I notice that no

time is available for the

down-hill run! Mr Bucky was also

taken in by the second example,

but I was not Such drolleries

show us how stupid we are!

(See Mathematical Intelligencer, Spring

In this first chapter we review the real numbers, equations, and the coordinate plane You are probably already familiar with these concepts, but it is help- ful to get a fresh look at how these ideas work together to solve problems and model (or describe) real-world situations.

Let's see how all these ideas are used in a real-life situation: Suppose

you get paid $9 an hour at your part-time job We can model your pay y for working x hours by the equation To find out how many hours you need to work to get paid 200 dollars, we solve the equation Graphing the equation in a coordinate plane helps us “see” how

pay increases with hours worked.

1.9 Graphing Calculators; Solving

Equations and Inequalities

Fitting Lines to Data

Image copyright Monkey Business Images 2010 Used under license from Shutterstock.com

Let’s review the types of numbers that make up the real number system We start with the

natural numbers:

The integers consist of the natural numbers together with their negatives and 0:

We construct the rational numbers by taking ratios of integers Thus, any rational

num-ber r can be expressed as

where m and n are integers and n 0 Examples are

(Recall that division by 0 is always ruled out, so expressions like and are undefined.)There are also real numbers, such as , that cannot be expressed as a ratio of integers

and are therefore called irrational numbers It can be shown, with varying degrees of

difficulty, that these numbers are also irrational:

The set of all real numbers is usually denoted by the symbol  When we use the word

number without qualification, we will mean “real number.” Figure 1 is a diagram of the

types of real numbers that we work with in this book

F I G U R E 1 The real number system

Every real number has a decimal representation If the number is rational, then its responding decimal is repeating For example,

cor-(The bar indicates that the sequence of digits repeats forever.) If the number is irrational,the decimal representation is nonrepeating:

The different types of real numbers

were invented to meet specific needs

For example, natural numbers are

needed for counting, negative

num-bers for describing debt or below-zero

temperatures, rational numbers for

concepts like “half a gallon of milk,”

and irrational numbers for measuring

certain distances, like the diagonal

of a square

A repeating decimal such as

is a rational number To convert it to a

ratio of two integers, we write

x by appropriate powers of 10 and then

subtract to eliminate the repeating part.)

If we stop the decimal expansion of any number at a certain place, we get an tion to the number For instance, we can write

approxima-where the symbol  is read “is approximately equal to.” The more decimal places we tain, the better our approximation

We all know that 2 3  3  2, and 5  7  7  5, and 513  87  87  513, and so

on In algebra, we express all these (infinitely many) facts by writing

where a and b stand for any two numbers In other words, “a  b  b  a” is a concise

way of saying that “when we add two numbers, the order of addition doesn’t matter.” This

fact is called the Commutative Property for addition From our experience with numbers

we know that the properties in the following box are also valid

The Distributive Property applies whenever we multiply a number by a sum Figure 2 explains why this property works for the case in which all the numbers are pos-

itive integers, but the property is true for any real numbers a, b, and c.

a  b  b  a

p  3.14159265

S E C T I O N 1 1 | Real Numbers 3

PROPERTIES OF REAL NUMBERS

Commutative Properties

When we add two numbers, order doesn’t matter.

When we multiply two numbers, order doesn’t matter.

F I G U R E 2 The Distributive Property

The Distributive Property is crucial

because it describes the way addition

and multiplication interact with each

other

E X A M P L E 1 Using the Distributive Property

Simplify

Distributive PropertyAssociative Property of Addition

In the last step we removed the parentheses because, according to the Associative Property, the order of addition doesn’t matter.

The number 0 is special for addition; it is called the additive identity because

a  0  a for any real number a Every real number a has a negative, a, that satisfies

Subtraction is the operation that undoes addition; to subtract a number

from another, we simply add the negative of that number By definition

To combine real numbers involving negatives, we use the following properties

Property 6 states the intuitive fact that a  b and b  a are negatives of each other.

Property 5 is often used with more than two terms:

Let x, y, and z be real numbers.

number Whether a is negative or

positive depends on the value of a For

(Property 2), a itive number

pos-a  152  5

▼ Multiplication and Division

The number 1 is special for multiplication; it is called the multiplicative identity because

a1  a for any real number a Every nonzero real number a has an inverse, 1/a, thatsatisfies Division is the operation that undoes multiplication; to divide by a

number, we multiply by the inverse of that number If b 0, then, by definition,

We write as simply a/b We refer to a/b as the quotient of a and b or as the

frac-tion a over b; a is the numerator and b is the denominator (or divisor) To combine real

numbers using the operation of division, we use the following properties

When adding fractions with different denominators, we don’t usually use Property 4.Instead we rewrite the fractions so that they have the smallest possible common de-nominator (often smaller than the product of the denominators), and then we use Prop-

erty 3 This denominator is the Least Common Denominator (LCD) described in the

next example

Evaluate:

S O L U T I O N Factoring each denominator into prime factors gives

We find the least common denominator (LCD) by forming the product of all the factors that occur in these factorizations, using the highest power of each factor.

36  22# 32 and 120  23# 3 # 5

5

36  7 120

a

d

Cancel numbers that are common factors in

numer-ator and denominnumer-ator

2 # 5

3 # 5  2 3

# 7

5  14 15

When multiplying fractions, multiply numerators

and denominators

2 3

# 5

7  2 # 5

3 # 7  10 21

Thus the LCD is So

Use common denominator

Property 3: Adding fractions with the same denominator

The real numbers can be represented by points on a line, as shown in Figure 3 The itive direction (toward the right) is indicated by an arrow We choose an arbitrary refer-

pos-ence point O, called the origin, which corresponds to the real number 0 Given any

con-venient unit of measurement, each positive number x is represented by the point on the line a distance of x units to the right of the origin, and each negative number x is repre- sented by the point x units to the left of the origin The number associated with the point

P is called the coordinate of P, and the line is then called a coordinate line, or a real

number line, or simply a real line Often we identify the point with its coordinate and

think of a number as being a point on the real line

The real numbers are ordered We say that a is less than b and write if

b  a is a positive number Geometrically, this means that a lies to the left of b on

the number line Equivalently, we can say that b is greater than a and write b  a The

symbol means that either a b or a  b and is read “a is less than or equal to b.” For instance, the following are true inequalities (see Figure 4):

F I G U R E 4

A set is a collection of objects, and these objects are called the elements of the set If S

is a set, the notation a  S means that a is an element of S, and b  S means that b is not

an element of S For example, if Z represents the set of integers, then 3  Z but p  Z.

Some sets can be described by listing their elements within braces For instance, the

set A that consists of all positive integers less than 7 can be written as

We could also write A in set-builder notation as

which is read “A is the set of all x such that x is an integer and 0 x 7.”

A  51, 2, 3, 4, 5, 66

_1_2_3_4

1 2

1 4 1 8

4.3

1 16 2_2.63

_3.1725_4.7

If S and T are sets, then their union S  T is the set that consists of all elements that

are in S or T (or in both) The intersection of S and T is the set S  T consisting of all ements that are in both S and T In other words, S  T is the common part of S and T The

el-empty set, denoted by , is the set that contains no element

If S  {1, 2, 3, 4, 5}, T  {4, 5, 6, 7}, and V  {6, 7, 8}, find the sets S  T, S  T, and S  V.

S O L U T I O N

All elements in S or T

Elements common to both S and T

S and V have no element in common

Certain sets of real numbers, called intervals, occur frequently in calculus and

corre-spond geometrically to line segments If a b, then the open interval from a to b

con-sists of all numbers between a and b and is denoted The closed interval from a to

Note that parentheses in the interval notation and open circles on the graph in

Figure 5 indicate that endpoints are excluded from the interval, whereas square brackets and solid circles in Figure 6 indicate that the endpoints are included Intervals may also

include one endpoint but not the other, or they may extend infinitely far in one direction

or both The following table lists the possible types of intervals

F I G U R E 6 The closed interval 3a, b4

 (set of all real numbers)1q, q2

5x0x 61q, b4

bb

5x0a x b6 1a, b2

The symbol q (“infinity”) does not

for instance, simply indicates that the

interval has no endpoint on the right

but extends infinitely far in the positive

E X A M P L E 6 Finding Unions and Intersections of Intervals

Graph each set.

S O L U T I O N

(a) The intersection of two intervals consists of the numbers that are in both

intervals Therefore

This set is illustrated in Figure 7.

(b) The union of two intervals consists of the numbers that are in either one

interval or the other (or both) Therefore

This set is illustrated in Figure 8.

The absolute value of a number a, denoted by , is the distance from a to 0 on

the real number line (see Figure 9) Distance is always positive or zero, so we have

for every number a Remembering that a is positive when a is negative, we

have the following definition

(a) (b) (c) (d)

Any interval contains infinitely many

numbers—every point on the graph of

an interval corresponds to a real

num-ber In the closed interval , the

smallest number is 0 and the largest is

1, but the open interval contains

no smallest or largest number To see

this, note that 0.01 is close to zero, but

0.001 is closer, 0.0001 is closer yet, and

so on We can always find a number in

the interval closer to zero than

any given number Since 0 itself is not

in the interval, the interval contains no

smallest number Similarly, 0.99 is close

to 1, but 0.999 is closer, 0.9999 closer

yet, and so on Since 1 itself is not in

the interval, the interval has no largest

DEFINITION OF ABSOLUTE VALUE

If a is a real number, then the absolute value of a is

0 a 0  e a if a 0 a if a 0

50

_3

| 5 |=5

| _3 |=3

F I G U R E 9

Certain sets of real numbers, called intervals, occur frequently in calculus and

corre-spond geometrically to line segments If a b, then the open...

5x0a x b6 1a, b2

The symbol q (“infinity”) does not

for instance, simply indicates that the

interval has no endpoint on the right

but... intersection of two intervals consists of the numbers that are in both

intervals Therefore

This set is illustrated in Figure 7.

(b) The union of two