Algebra and trigonometry

P.1 Real Numbers and Their Properties 2P.2 The Real Number Line and Order 8 ■ FOCUS ON MODELING Modeling the Real World with Algebra 53... ■Book Companion Website A new website www.stewa

Formulas for area A, perimeter P, circumference C, volume V :

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 loga x means a y  x

loga a x  x aloga x  x

log x log10x ln x loge x

loga x y loga x  loga y logaax yb  loga x  loga y

loga x b  b log a x logb x

EXPONENTIAL AND LOGARITHMIC FUNCTIONS

01

10

1 1

Ï=| x |

xy

≈

xy

x

xy

xy

xy

Ï=x£

xy

Ï=b

b

xy

For the complex number z a  bi

the conjugate is

the modulus is ⏐z⏐  a2 b2

the argument is

Polar form of a complex number

For z a  bi, the polar form is

Re

Imbi

0

| z|

a+bi

¨a

z a  bi

Angle-of-rotation formula for conic sections

To eliminate the xy-term in the equation

  

b

y2 2



a

x2 2

  

b

y2 2

y

xy

c

y

xy

0

y

x(h, k)

0y

x(h, k)

y

xp>0

x

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T H I R D E D I T I O N

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 Precalculus: Mathematics for Calculus, College Algebra, 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 Built from 1991 to 1996, it was

designed by Santiago Calatrava, a Spanish architect Calatrava has

always been very interested in how mathematics can help him

realize the buildings he imagines As a young student, he taught

himself descriptive geometry from books in order to represent

three-dimensional objects in two dimensions Trained as both anengineer and an architect, he wrote a doctoral thesis in 1981 entitled “On the Foldability of Space Frames,” which is filled withmathematics, especially geometric transformations His strength

as an engineer enables him to be daring in his architecture

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1 2 3 4 5 6 7 14 13 12 11

P.1 Real Numbers and Their Properties 2

P.2 The Real Number Line and Order 8

■ FOCUS ON MODELING Modeling the Real World with Algebra 53

C H A P T E R 2 C OORDINATES AND G RAPHS 125

Chapter Overview 125

2.1 The Coordinate Plane 126

2.2 Graphs of Equations in Two Variables 132

2.3 Graphing Calculators: Solving Equations and Inequalities Graphically 140

2.4 Lines 149

2.5 Making Models Using Variations 162 Chapter 2 Review 167

Chapter 2 Test 170

■ FOCUS ON MODELING Fitting Lines to Data 171

Cumulative Review Test: Chapters 1 and 2 181

Chapter Overview 183

3.1 What Is a Function? 184

3.2 Graphs of Functions 194

3.3 Getting Information from the Graph of a Function 205

3.4 Average Rate of Change of a Function 214

■ FOCUS ON MODELING Modeling with Functions 255

Chapter Overview 265

4.1 Quadratic Functions and Models 266

4.2 Polynomial Functions and Their Graphs 274

4.3 Dividing Polynomials 288

4.4 Real Zeros of Polynomials 295

4.5 Complex Zeros and the Fundamental Theorem of Algebra 306

4.6 Rational Functions 314 Chapter 4 Review 329 Chapter 4 Test 332

■ FOCUS ON MODELING Fitting Polynomial Curves to Data 333

Chapter Overview 339

5.1 Exponential Functions 340

5.2 The Natural Exponential Function 348

5.3 Logarithmic Functions 353

5.4 Laws of Logarithms 363

5.5 Exponential and Logarithmic Equations 369

5.6 Modeling with Exponential and Logarithmic Functions 378

Chapter 5 Review 391 Chapter 5 Test 394

■ FOCUS ON MODELING Fitting Exponential and Power Curves to Data 395

Cumulative Review Test: Chapters 3, 4, and 5 405

Chapter Overview 407

6.1 Angle Measure 408

6.2 Trigonometry of Right Triangles 417

6.3 Trigonometric Functions of Angles 425

6.4 Inverse Trigonometric Functions and Right Triangles 436

6.5 The Law of Sines 443

6.6 The Law of Cosines 450

Chapter 6 Review 457 Chapter 6 Test 461

■ FOCUS ON MODELING Surveying 463

Chapter Overview 467

7.1 The Unit Circle 468

7.2 Trigonometric Functions of Real Numbers 475

7.3 Trigonometric Graphs 484

7.4 More Trigonometric Graphs 497

7.5 Inverse Trigonometric Functions and Their Graphs 504

7.6 Modeling Harmonic Motion 510

Chapter 7 Review 521 Chapter 7 Test 524

■ FOCUS ON MODELING Fitting Sinusoidal Curves to Data 525

Chapter Overview 531

8.1 Trigonometric Identities 532

8.2 Addition and Subtraction Formulas 538

8.3 Double-Angle, Half-Angle, and Product-Sum Formulas 545

8.4 Basic Trigonometric Equations 555

8.5 More Trigonometric Equations 562

Chapter 8 Review 568 Chapter 8 Test 570

■ FOCUS ON MODELING Traveling and Standing Waves 571

Cumulative Review Test: Chapters 6, 7, and 8 576

C H A P T E R 9 P OLAR C OORDINATES AND P ARAMETRIC E QUATIONS 579

Chapter Overview 579

9.1 Polar Coordinates 580

9.2 Graphs of Polar Equations 585

9.3 Polar Form of Complex Numbers; De Moivre's Theorem 593

9.4 Plane Curves and Parametric Equations 602 Chapter 9 Review 610

Chapter 9 Test 612

■ FOCUS ON MODELING The Path of a Projectile 613

Chapter Overview 617

10.1 Vectors in Two Dimensions 618

10.2 The Dot Product 627

10.3 Three-Dimensional Coordinate Geometry 635

10.4 Vectors in Three Dimensions 641

10.5 The Cross Product 648

10.6 Equations of Lines and Planes 654 Chapter 10 Review 658

Chapter 10 Test 661

■ FOCUS ON MODELING Vector Fields 662

Cumulative Review Test: Chapters 9 and 10 666

Chapter Overview 667

11.1 Systems of Linear Equations in Two Variables 668

11.2 Systems of Linear Equations in Several Variables 678

11.3 Matrices and Systems of Linear Equations 687

11.4 The Algebra of Matrices 699

11.5 Inverses of Matrices and Matrix Equations 710

11.6 Determinants and Cramer's Rule 720

11.7 Partial Fractions 731

11.8 Systems of Nonlinear Equations 736

11.9 Systems of Inequalities 741 Chapter 11 Review 748 Chapter 11 Test 752

■ FOCUS ON MODELING Linear Programming 754

12.4 Shifted Conics 788

12.5 Rotation of Axes 795

12.6 Polar Equations of Conics 803

Chapter 12 Review 810 Chapter 12 Test 813

■ FOCUS ON MODELING Conics in Architecture 814

Cumulative Review Test: Chapters 11 and 12 818

■ FOCUS ON MODELING Modeling with Recursive Sequences 871

■ FOCUS ON MODELING The Monte Carlo Method 913

Cumulative Review Test: Chapters 13 and 14 917

APPENDIX: Calculations and Signific ant Figures 919 ANSWERS A1

INDEX I1

P R E FAC E

For many students an Algebra and Trigonometry course represents the first opportunity todiscover the beauty and practical power of mathematics Thus instructors are faced withthe challenge of teaching the concepts and skills of the subject while at the same time im-parting an appreciation for its effectiveness in modeling the real world This book aims tohelp instructors meet this challenge

In writing this Third Edition, our purpose is to further enhance the usefulness of thebook as an instructional tool for teachers and as a learning tool for students There are sev-eral major changes in this edition including a restructuring of each exercise set to better alignthe exercises 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

math-ematical vocabulary appropriately Several chapters have been reorganized and rewritten (asdescribed below) to further focus the exposition on the main concepts; we have added a newchapter on vectors in two and three dimensions In all these changes and numerous others(small and large) we have retained the main features that have contributed to the success ofthis book

New to the Third Edition

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

Ex-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 3 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 4 Polynomial and Rational Functions This chapter now begins with asection on quadratic functions, leading to higher degree polynomial functions

■CHAPTER 5 Exponential and Logarithmic Functions The material on the naturalexponential function is now in a separate section

■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 8

in-■CHAPTER 7 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 8 Analytic Trigonometry This chapter has been completely revised.There are two new sections on trigonometric equations (Sections 8.4 and 8.5) Thematerial on this topic (formerly in Section 8.5) has been expanded and revised

■CHAPTER 9 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 10 Vectors in Two and Three Dimensions This is a new chapter with a

new Focus on Modeling section.

■CHAPTER 11 Systems of Equations and Inequalities The material on systems ofnonlinear equations is now in a separate section

■CHAPTER 12 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 9

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

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 3)

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 61).

been 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 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-■Chapter 7 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-Another way to teach trigonometry is to intertwine the two approaches Some tors teach this material in the following order: Sections 7.1, 7.2, 6.1, 6.2, 6.3, 7.3, 7.4, 7.5,7.6, 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

computers in examples and exercises throughout the book Our calculator-oriented ples are always preceded by examples in which students must graph or calculate by hand,

exam-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,

and 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 825 and 829.)

clarify 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 256 and 674)

■Constructing Models There are numerous applied problems throughout the bookwhere students are given a model to analyze (see, for instance, page 270) 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 255, 378, and 525)

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

The first such section, after Chapter P, introduces the basic idea of modeling a life situation by using algebra Other sections present ways in which linear, polyno-mial, exponential, logarithmic, and trigonometric functions, and systems of inequal-ities can all be used to model familiar phenomena from the sciences and fromeveryday life (see for example pages 333, 395, and 525)

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

aspect 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.)

pro-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

■ 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 82; Salt Lake City, page 127; and radiocarbon dating, page 371)

■ 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 321, 738, and 797, for example)

section 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 2, 5,

8, 10, 12, and 14 combine skills and concepts from the preceding chapters and aredesigned to highlight the connections between the topics in these related chapters

■ 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

Hadavas, Armstrong Atlantic University; and Gary Lippman, California State UniversityEast Bay

County 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 intoteaching mathematics We especially thank Andrew Bulman-Fleming for writing theStudy Guide and the Solutions Manual and Doug Shaw at the University of Northern Iowafor writing the Instructor Guide.

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

Instructor's Guide ISBN-10: 1-111-56813-8; ISBN-13: 978-1-111-56813-9

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

program 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

tests 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: 1-111-56815-4; ISBN-13: 978-1-111-56815-3

This CD-ROM provides the instructor with dynamic media tools for teaching Create,

Computer-ized Testing Featuring Algorithmic Equations Easily build solution sets for homework or

slides and figures from the book are also included on this CD-ROM

giv-Study Guide ISBN-10: 1-111-56810-3; ISBN-13: 978-1-111-56810-8

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

program 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.com

Visit 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: 1-111-57275-5; ISBN-13: 978-1-111-57275-4

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

TO THE STUDENT

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

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

problem-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 the concept cises and to each chapter test, appear at the back of the book If your answer differs from the one given, don’t immediately assume that you are wrong There may be a calculation that connects the two answers 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 numer-

exer-ator and denominexer-ator of your answer by  1 to change it to the given answer In

round-ing approximate answers, follow the guidelines in the Appendix: Calculations and cant Figures.

Signifi-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

Standing Room Only 381

Half-Lives of Radioactive Elements 383

AM and FM Radio 493Root-Mean Square 515Jean Baptiste Joseph Fourier 539Maria Gaetana Agnesi 603Galileo Galilei 614William Rowan Hamilton 649Julia Robinson 701

Olga Taussky-Todd 706Arthur Cayley 712David Hilbert 721Emmy Noether 724The Rhind Papyrus 732Linear Programming 755Archimedes 767Eccentricities of the Orbits

of the Planets 776Paths of Comets 783Johannes Kepler 792Large Prime Numbers 824Eratosthenes 825

Fibonacci 825The Golden Ratio 829Srinavasa Ramanujan 840

Blaise Pascal 856Pascal’s Triangle 860Sir Isaac Newton 863Persi Diaconis 879Ronald Graham 886Probability Theory 892The Contestant’s Dilemma 916

Mathematics in the Modern World 18Error-Correcting Codes 98

Changing Words, Sound, and Picturesinto Numbers 158

Computers 224Splines 276Automotive Design 280Unbreakable Codes 321Law Enforcement 356Evaluating Functions on a Calculator 498Weather Prediction 670

Mathematical Ecology 717Global Positioning System 738Looking Inside Your Head 797Fair Division of Assets 834Fractals 842

Mathematical Economics 848Fair Voting Methods 893

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

P R O LO G U E PRINCIPLES OF PROBLEM SOLVING

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 92) was unique among

great mathematicians because he

ex-plained how he found his results Polya

often said to his students and

col-leagues,“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 13.5.

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.

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

P R O B L E M | Average Speed

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

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 1990,

We use numbers every day Sometimes we notice patterns in numbers, and that’s where algebra comes in By using letters to stand for numbers, we write formulas that help us to predict properties of real-world objects or processes For example, an ocean diver knows that the deeper he dives, the higher the water pressure The pattern of how pressure changes with depth can be

expressed as an algebra formula (or model) If we let P stand for pressure (in

lb/in2) and d for depth (in ft) we can express the relationship by the formula

This formula can be used to predict water pressure at great depths (without having to dive to those depths) To get more information from this model, such

as the depth at a given pressure, we need to know the rules of algebra, that is, the rules for working with numbers In this chapter we review these rules.

P.2 The Real Number

Line and Order

▼ Types of Real Numbers

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

(Recall that division by 0 is always ruled out, so expressions such as and are

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

de-grees 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:

12  1.414213562373095 p  3.141592653589793 .

9

7 1.285714285714  1.285714157

495 0.3171717  0.317

2

3 0.66666  0.61

P.1 R EAL N UMBERS AND T HEIR P ROPERTIES

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 such as “half a gallon of

milk,” and irrational numbers for

mea-suring certain distances, such as 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

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

re-tain, the better our approximation

E X A M P L E 1 Classifying Real Numbers

Determine whether each given real number is a natural number, an integer, a rational ber, or an irrational number

S O L U T I O N

(a) 999 is a positive whole number, so it is a natural number.

(b) is a ratio of two integers, so it is a rational number.

(c) equals 2, so it is an integer.

(d) equals 5, so it is a natural number.

(e) is a nonrepeating decimal (approximately 1.7320508075689), so it is an tional number.

Real numbers can be combined by using the familiar operations of addition, subtraction,multiplication, and division When evaluating arithmetic expressions that contain several

of these operations, we use the following conventions to determine the order in which theoperations are performed:

In dividing two expressions, the numerator and denominator of the quotient aretreated as if they are within parentheses

E X A M P L E 2 Evaluating an Arithmetic Expression

Find the value of the expression

S O L U T I O N First we evaluate the numerator and denominator of the quotient, since these are treated as if they are inside parentheses:

Evaluate quotientEvaluate parenthesesEvaluate productsEvaluate difference

6 3

6 5

13 125

6 3

6 5

p  3.14159265

BHASKARA (born 1114) was an

In-dian mathematician, astronomer, and

astrologer Among his many

accom-plishments was an ingenious proof of

the Pythagorean Theorem (See Focus

on Problem Solving 5, Problem 12,

at the book companion website

www.stewartmath.com.) His

impor-tant mathematical book Lilavati [The

Beautiful] consists of algebra problems

posed in the form of stories to his

daughter Lilavati Many of the

prob-lems begin “Oh beautiful maiden,

sup-pose ” The story is told that using

as-trology, Bhaskara had determined that

great misfortune would befall his

daughter if she married at any time

other than at a certain hour of a certain

day On her wedding day, as she was

anxiously watching the water clock, a

pearl fell unnoticed from her

head-dress It stopped the flow of water in

the clock, causing her to miss the

opportune moment for marriage.

Bhaskara’s Lilavati was written to

console her.

▼ Properties of Real Numbers

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

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

fact is called the Commutative Property of 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 2explains why this property works for the case in which all the numbers are positive inte-

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

E X A M P L E 3 Using the Properties of Real Numbers

Associative Property of Addition

PROPERTIES OF REAL NUMBERS

Property Example Description

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

(c) Distributive Property

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

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 5 is often used with more than two terms:

E X A M P L E 4 Using Properties of Negatives

Let x, y, and z be real numbers.

(a) 13  22  3  2 Property 5: –(a + b) = –a – b

(b) 1x  22  x  2 Property 5: –(a + b) = –a – b

Property 2: –(–a) = a

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

Don’t assume that a is a negative

positive depends on the value of a For

(Property 2), a itive number

pos-a  152  5

PROPERTIES OF NEGATIVESProperty Example 1.

satisfies Division is the operation that undoes multiplication; to divide by a

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 stead we rewrite the fractions so that they have the smallest possible common denominator(often smaller than the product of the denominators), and then we use Property 3 This de-

In-nominator is the Least Common DeIn-nominator (LCD) described in the next example.

E X A M P L E 5 Using the LCD to Add Fractions 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 Thus the LCD

Use common denominator

Property 3: Adding fractions with the same denominator

 50

360  21

360  71 360

ad  bc a

b  c

d

Cancel numbers that are common factors in the

numerator and denominator

2 # 5

3 # 5  2 3

ac

bc  a

b

When adding fractions with different

denomina-tors, find a common denominator Then add the

When adding fractions with the same

denomi-nator, add the numerators.

# 7

5  14 15

When multiplying fractions, multiply numerators

and denominators

2 3

# 5

7  2 # 5

3 # 7  10 21

The word algebra comes from the

9th-century Arabic book Hisâb

al-Jabr w’al-Muqabala, written by

al-Khowarizmi The title refers to

trans-posing and combining terms, two

processes that are used in solving

equations In Latin translations the title

was shortened to Aljabr, from which we

get the word algebra The author’s

name itself made its way into the

English language in the form of our

word algorithm.

S E C T I O N P 1 | Real Numbers and Their Properties 7

C O N C E P T S

1 Give an example of each of the following:

(a) A natural number

(b) An integer that is not a natural number

(c) A rational number that is not an integer

(d) An irrational number

2 Complete each statement, and name the property of real

numbers you have used

3 To add two fractions, you must first express them so that they

4. To divide two fractions, you the divisor and then

multiply

S K I L L S

5–6 ■ List the elements of the given set that are

(a) natural numbers

19 Commutative Property of Addition,

20 Associative Property of Multiplication, 713x2 

A  600  20x?

44 Sums and Products of Rational and Irrational Numbers Explain why the sum, the difference, and theproduct of two rational numbers are rational numbers

Is the product of two irrational numbers necessarily irrational? What about the sum?

5.23

0.570.28

0.7

2

51 2 1

10 3 15

23 4 1

1

1 12

3

1

41 5 3

10 4 15

(e) Are the actions of washing laundry and drying it

commutative?

(f) Give an example of a pair of actions that is commutative (g) Give an example of a pair of actions that is not

commutative

45 Combining Rational Numbers with Irrational

Numbers Is rational or irrational? Is

rational or irrational? In general, what can you say about

the sum of a rational and an irrational number? What about the

product?

46 Commutative and Noncommutative Operations

We have seen that addition and multiplication are both

commutative operations

(a) Is subtraction commutative?

(b) Is division of nonzero real numbers commutative?

(c) Are the actions of putting on your socks and putting on

your shoes commutative?

(d) Are the actions of putting on your hat and putting on your

P.2 T HE R EAL N UMBER L INE AND O RDER

Value and Distance

The real numbers can be represented by points on a line, as shown in Figure 1 The positive direction (toward the right) is indicated by an arrow We choose an arbitrary reference point

O, called the origin, which corresponds to the real number 0 Given any convenient 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 represented by the point x

units to the left of the origin Thus every real number is represented by a point on the line,

and every point P on the line corresponds to exactly one real number The number

associ-ated 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

co-ordinate and think of a number as being a point on the real line.

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

means that either a

instance, the following are true inequalities (see Figure 2):

F I G U R E 2

_1_2_3_4

1 2

1 4 1 8

4.3

1 16

2_2.63

_3.1725_4.7