Trace the conic x2 2xy y2 3x 2y 3 0

Cool Polynomial Project

GEOMETRIC FORMULAS

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

Rectangle Box

A � l„ V � l„ h

P � 2l � 2„

Triangle Pyramid

A � �1 2

� bh V � �1 3

�ha2

Circle Sphere

A � �r 2 V � �4 3

��r 3

C � 2�r A � 4�r 2

Cylinder Cone

V � �r 2h V � �1 3

��r 2h

HERON’S FORMULA

Area � �s 1�s��� a�2 1�s��� b�2 1�s��� c�2� where s � �

a � 2 b � c �

hh

r

r

r r

h

b a a

h

„

l

h

l „

EXPONENTS AND RADICALS

xmxn � xm�n � x x

m

n� � x m�n

1xm2n � xm n x�n � � x 1

n�

1xy2n � xnyn a�xy�b n

� � x y

n

n�

x1�n � �n x� xm�n � �n xm�� � Q�n x�Rm

�n xy� � �n x� �n y� �n xy��� � �

n x�

�n y� ��

�m �n��x� ��n �m��x� � �mnx�

SPECIAL PRODUCTS

1x � y22 � x 2 � 2xy � y 2 1x � y22 � x 2 � 2xy � y 2 1x � y23 � x 3 � 3x 2y � 3xy 2 � y 3 1x � y23 � x 3 � 3x 2y � 3xy 2 � y 3

FACTORING FORMULAS

x 2 � y 2 � 1x � y2 1x � y2 x 2 � 2xy � y 2 � 1x � y22 x 2 � 2xy � y 2 � 1x � y22 x 3 � y 3 � 1x � y2 1x 2 � xy � y 22 x 3 � y 3 � 1x � y2 1x 2 � xy � y 22

QUADRATIC FORMULA

If ax 2 � bx � c � 0, then

x �

INEQUALITIES AND ABSOLUTE VALUE

If a � b and b � c, then a � c.

If a � b, then a � c � b � c.

If a � b and c � 0, then ca � cb.

If a � b and c � 0, then ca � cb.

If a � 0, then

⏐x⏐ � a means x � a or x � �a.

⏐x⏐ � a means �a � x � a.

⏐x⏐ � a means x � a or x � �a.

�b �b�2��� 4�a�c� ��

2a

b

B

CA

ac

DISTANCE AND MIDPOINT FORMULAS

Distance between P11x1, y12 and P21x2, y2 2 : d � � 1x�2��� x�12�2��� 1�y2� �� y�12�2�

Midpoint of P1P2: a�x1 � 2

x2. �, �

y1 �

2

y2. �b

LINES

Slope of line through P11x1, y12 and P21x2, y2 2 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 a loga x � x

loga 1 � 0 loga a � 1

log x � log10 x ln x � loge x

loga xy � loga x � loga y loga a�xy�b � loga x � loga y loga x

b � b loga x logb x �

EXPONENTIAL AND LOGARITHMIC FUNCTIONS

0

1

y=a˛ 0

0

1

y=a˛ a>1

1

y=loga x a>1

0

y=loga x 0

10

y

x

y

x

y

x

y

x

loga x

log a b

� a x

� � � b y

� � 1

m � � x

y2

2

�

�

y

x 1

1 �

GRAPHS OF FUNCTIONS

Linear functions: f 1x2 � mx � b

Power functions: f 1x2 � xn

Root functions: f 1x2 � �n x�

Reciprocal functions: f 1x2 � 1/xn

Absolute value function Greatest integer function

Ï=“x ‘

1

1

x

y

Ï=|x |

x

y

Ï= 1≈

x

y

Ï= 1x

x

y

Ï=£œ∑x

x

y

Ï=œ∑x

x

y

Ï=x£

x

y

Ï=≈ x

y

Ï=mx+b

b

x

y

Ï=b

b

x

y

COMPLEX NUMBERS

For the complex number z � a � bi the conjugate is

the modulus is ⏐z⏐ � �a2 � b2���� the argument is , where tan � b/a

Polar form of a complex number

For z � a � bi, the polar form is

z � r 1cos � i sin 2 where r � ⏐z⏐ is the modulus of z and is the argument of z

De Moivre’s Theorem

z n � �r 1cos � i sin 2 n � rn 1cos n � i sin n 2 �n z� � �r 1cos � i sin 2 1�n

� r 1�n acos � � n 2k� � � i sin �

�

n 2k� �b

where k � 0, 1, 2, . . . , n � 1

ROTATION OF AXES

0

P(x, y) P(X, Y)

Y

X

ƒ x

y

Re

Im

bi

0

| z| a+bi

¨ a

z � a � bi

Angle-of-rotation formula for conic sections

To eliminate the xy-term in the equation

Ax2 � Bxy � Cy2 � Dx � Ey � F � 0

rotate the axis by the angle � that satisfies

cot 2� � � A �

B C

�

POLAR COORDINATES

x � r cos

y � r sin

r2 � x2 � y 2

tan � � y

x �

x

y

0

r

¨ x

y

P (x, y) P (r, ¨)

Rotation of axes formulas

x � X cos � � Y sin �

y � X sin � � Y cos �

CONIC SECTIONS

Circles

1x � h2 2 � 1y � k2 2 � r 2

Parabolas x 2 � 4py y 2 � 4px

Focus 10, p2 , directrix y � �p Focus 1p, 02 , directrix x � �p

y � a 1x � h22 � k, y � a 1x � h2 2 � k, a � 0, h � 0, k � 0 a � 0, h � 0, k � 0

Ellipses

� a

x2

2� � �b

y2

2� � 1 x2

b2 �� �

y2

a2 �� � 1

Foci 1 c, 02 , c2 � a2 � b2 Foci 10, c2 , c2 � a2 � b2

Hyperbolas

� a

x2

2� � �b

y2

2� � 1 � x2

b2 �� �

y2

a2 �� � 1

Foci 1 c, 02 , c2 � a2 � b2 Foci 10, c2 , c2 � a2 � b2

a

b

_a

_b

b

a

_b

_a

_c c

c

_c

x

y

x

y

a>b

b

a

_b

_a

c

_c

a>b

a

b

_a

_b

c_c x

y

x

y

0

y

x

(h, k)

0

y

x

(h, k)

y

x

p>0

p<0

y

x

p>0p<0

p

p

0

C(h, k)

r

x

y

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

ALGEBRA AND TRIGONOMETRY

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

JAMES STEWART 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.

LOTHAR REDLIN grew up on Van- 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.

SALEEM WATSON received his Bachelor of Science degree from

Andrews University in Michigan. He

did graduate studies at Dalhousie

University and McMaster University,

where he received his 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 an engineer and an architect, he wrote a doctoral thesis in 1981 entitled “On the Foldability of Space Frames,” which is filled with mathematics, especially geometric transformations. His strength as an engineer enables him to be daring in his architecture.

ABOUT THE COVER

ALGEBRA AND TRIGONOMETRY JAMES STEWART M C M A S T E R U N I V E R S I T Y A N D U N I V E R S I T Y O F TO R O N TO

LOTHAR REDLIN T H E P E N N S Y LVA N I A S TAT E U N I V E R S I T Y

SALEEM WATSON C A L I F O R N I A S TAT E U N I V E R S I T Y, LO N G B E A C H

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

Algebra and Trigonometry, Third Edition James Stewart, Lothar Redlin, Saleem Watson

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CO N T E N T S

PREFACE xi

TO THE STUDENT xix

PROLOGUE: PRINCIPLES OF PROBLEM SOLVING P1

C H A P T E R P PREREQUISITES 1 Chapter Overview 1

P.1 Real Numbers and Their Properties 2 P.2 The Real Number Line and Order 8 P.3 Integer Exponents 14 P.4 Rational Exponents and Radicals 22 P.5 Algebraic Expressions 28 P.6 Factoring 33 P.7 Rational Expressions 40

Chapter P Review 49

Chapter P Test 52

■ FOCUS ON MODELING Modeling the Real World with Algebra 53

C H A P T E R 1 EQUATIONS AND INEQUALITIES 59 Chapter Overview 59

1.1 Basic Equations 60 1.2 Modeling with Equations 68 1.3 Quadratic Equations 80 1.4 Complex Numbers 90 1.5 Other Types of Equations 95 1.6 Inequalities 104 1.7 Absolute Value Equations and Inequalities 113

Chapter 1 Review 117

Chapter 1 Test 119

■ FOCUS ON MODELING Making the Best Decisions 120

v

C H A P T E R 2 COORDINATES AND GRAPHS 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

C H A P T E R 3 FUNCTIONS 183 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

3.5 Transformations of Functions 221

3.6 Combining Functions 232

3.7 One-to-One Functions and Their Inverses 241

Chapter 3 Review 249

Chapter 3 Test 253

■ FOCUS ON MODELING Modeling with Functions 255

C H A P T E R 4 POLYNOMIAL AND RATIONAL FUNCTIONS 265 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

C H A P T E R 5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 339 Chapter Overview 339

5.1 Exponential Functions 340 5.2 The Natural Exponential Function 348 5.3 Logarithmic Functions 353

vi Contents

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

C H A P T E R 6 TRIGONOMETRIC FUNCTIONS: RIGHT TRIANGLE APPROACH 407 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

C H A P T E R 7 TRIGONOMETRIC FUNCTIONS: UNIT CIRCLE APPROACH 467 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

C H A P T E R 8 ANALYTIC TRIGONOMETRY 531 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

Contents vii

C H A P T E R 9 POLAR COORDINATES AND PARAMETRIC EQUATIONS 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

C H A P T E R 10 VECTORS IN TWO AND THREE DIMENSIONS 617 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

C H A P T E R 11 SYSTEMS OF EQUATIONS AND INEQUALITIES 667 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

C H A P T E R 12 CONIC SECTIONS 761 Chapter Overview 761

12.1 Parabolas 762 12.2 Ellipses 770 12.3 Hyperbolas 779

viii Contents

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

C H A P T E R 13 SEQUENCES AND SERIES 821 Chapter Overview 821

13.1 Sequences and Summation Notation 822 13.2 Arithmetic Sequences 832 13.3 Geometric Sequences 838 13.4 Mathematics of Finance 846 13.5 Mathematical Induction 852 13.6 The Binomial Theorem 858

Chapter 13 Review 867

Chapter 13 Test 870

■ FOCUS ON MODELING Modeling with Recursive Sequences 871

C H A P T E R 14 COUNTING AND PROBABILITY 877 Chapter Overview 877

14.1 Counting Principles 878 14.2 Permutations and Combinations 882 14.3 Probability 891 14.4 Binomial Probability 902 14.5 Expected Value 907

Chapter 14 Review 909

Chapter 14 Test 912

■ FOCUS ON MODELING The Monte Carlo Method 913

Cumulative Review Test: Chapters 13 and 14 917

APPENDIX: Calculations and S ignific ant Figures 919

ANSWERS A1

INDEX I1

Contents ix

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xi

P R E FA C E

For many students an Algebra and Trigonometry course represents the first opportunity to discover the beauty and practical power of mathematics. Thus instructors are faced with the 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 to help instructors meet this challenge.

In writing this Third Edition, our purpose is to further enhance the usefulness of the book 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 align the 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 (as described below) to further focus the exposition on the main concepts; we have added a new chapter 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 of this book.

New to the Third Edition ■ Exercises More than 20% of the exercises are new. This includes new Concept Ex-

ercises and new Cumulative Review Tests. Key exercises are now linked to examples 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 a section on quadratic functions, leading to higher degree polynomial functions.

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

■ CHAPTER 6 Trigonometric Functions: Right Triangle Approach This chapter in- cludes a new section on inverse trigonometric functions and right triangles (Section 6.4) which is needed in applying the Laws of Sines and Cosines in the following section, as well as for solving trigonometric equations in Chapter 8.

www.stewartmath.com
■ CHAPTER 7 Trigonometric Functions: Unit Circle Approach This chapter in- 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.

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

■ CHAPTER 9 Polar Coordinates and Parametric Equations This chapter is now more sharply focused on the concept of a coordinate system. The section on parametric equations 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 of nonlinear equations is now in a separate section.

■ CHAPTER 12 Conic Sections This chapter is now more closely devoted to the topic of analytic geometry, especially the conic sections; the section on parametric equations 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 many different 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 ex- 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 features that can be used to complement different teaching styles:

EXERCISE SETS The most important way to foster conceptual understanding and hone technical skill is through the problems that the instructor assigns. To that end we have provided a wide selection of exercises.

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

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

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

■ 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 devel- oped in the section, and then to write about what they have learned, rather than sim- ply look for the answer.

■ 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 devel- oped in that example (see, for instance, page 3).

■ 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).

FLEXIBLE APPROACH TO TRIGONOMETRY The trigonometry chapters of this text have been written so that either the right triangle approach or the unit circle approach may be taught first. Putting these two approaches in different chapters, each with its relevant ap-

xii Preface

plications, helps to clarify the purpose of each approach. The chapters introducing trigonometry are as follows:

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

■ Chapter 7 Trigonometric Functions: Unit Circle Approach This chapter intro- duces trigonometry through the unit circle approach. This approach emphasizes that the trigonometric functions are functions of real numbers, just like the polynomial and exponential functions with which students are already familiar.

Another way to teach trigonometry is to intertwine the two approaches. Some instruc- 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 fact that the two approaches involve distinct representations of the same functions.

GRAPHING CALCULATORS AND COMPUTERS We make use of graphing calculators and computers 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 optional and may be omitted without loss of continuity. We use the following capabilities of the calculator.

■ Graphing, Regression, Matrix Algebra The capabilities of the graphing calculator are 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 calcula- 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.)

FOCUS ON MODELING The “modeling” theme has been used throughout to unify and clarify the many applications of precalculus. We have made a special effort to clarify the essential 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 book where 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 for example, 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 real- 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 from everyday life (see for example pages 333, 395, and 525).

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 enable students (perhaps working in groups) to explore in greater depth an interesting

Preface xiii

www.stewartmath.com
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 prin- ciples introduced in the Prologue and includes several challenging problems. (See for instance Recognizing Patterns, Using Analogy, Introducing Something Extra, Taking Cases, and Working Backward.)

MATHEMATICAL VIGNETTES Throughout the book we make use of the margins to 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 interesting mathematicians and often include a key insight that the mathematician discovered and 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 emphasizes the central role of mathematics in current advances in technology and the sciences (see pages 321, 738, and 797, for example).

REVIEW SECTIONS AND CHAPTER TESTS Each chapter ends with an extensive review 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 in the 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 the different 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 are designed to highlight the connections between the topics in these related chapters.

■ Answers Brief answers to odd-numbered exercises in each section (including the review exercises), and to all questions in the Concepts Exercises and Chapter Tests, are given in the back of the book.

Acknowledgments We thank the following reviewers for their thoughtful and constructive comments.

REVIEWERS FOR THE SECOND EDITION Heather Beck, Old Dominion University; Paul Hadavas, Armstrong Atlantic University; and Gary Lippman, California State University East Bay.

REVIEWERS FOR THE THIRD EDITION Raji Baradwaj, UMBC; Chris Herman, Lorain 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.

xiv Preface

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

We thank Martha Emry, our production service and art editor; her energy, devotion, ex- perience, and intelligence were essential components in the creation of this book. We thank Barbara Willette, our copy editor, for her attention to every detail in the manuscript. We thank Jade Myers and his staff at Matrix Art Services for their attractive and accurate graphs and Precision Graphics for bringing many of our illustrations to life. We thank our designer 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 guiding and facilitating every aspect of the production of this book. Of the many Brooks/Cole staff involved in this project we particularly thank the following: Jennifer Risden, content proj- ect manager, Cynthia Ashton, assistant editor; Lynh Pham, media editor; Vernon Boes, art director; and Myriah Fitzgibbon, marketing manager. They have all done an outstanding job.

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

Above all, we thank our editor Gary Whalen. His vast editorial experience, his exten- sive knowledge of current issues in the teaching of mathematics, and especially his deep interest in mathematics textbooks, have been invaluable resources in the writing of this book.

Preface xv

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INSTRUCTOR RESOURCES Printed Complete Solution Manual ISBN-10: 1-111-56811-1; ISBN-13: 978-1-111-56811-5 The complete solutions manual provides worked-out solutions to all of the problems in the text.

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 Exclusively from Cengage Learning, Enhanced WebAssign® offers an extensive online 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 immediate feedback as students complete their assignments. Algorithmic problems allow you to as- sign unique versions to each student. The Practice Another Version feature (activated at your discretion) allows students to attempt the questions with new sets of values until they feel confident enough to work the original problem. Students benefit from a new Premium eBook with highlighting and search features; Personal Study Plans (based on diagnostic quizzing) 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 customize tests for class in print and online formats, and features automatic grading. Includes a test bank with hundreds of questions customized directly to the text. ExamView is available within the PowerLecture CD-ROM.

Solution Builder www.cengage.com/solutionbuilder This online instructor database offers complete worked solutions to all exercises in the text, 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

www.cengage.com/solutionbuilder
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, de- liver, and customize tests (both print and online) in minutes with ExamView® Computer- ized Testing Featuring Algorithmic Equations. Easily build solution sets for homework or exams using Solution Builder's online solutions manual. Microsoft® PowerPoint® lecture slides and figures from the book are also included on this CD-ROM.

STUDENT RESOURCES Printed Student Solution Manual ISBN-10: 0-8400-6923-5; ISBN-13: 978-0-8400-6923-8 Contains fully worked-out solutions to all of the odd-numbered exercises in the text, giv- ing students a way to check their answers and ensure that they took the correct steps to ar- rive at an answer.

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-solving skills while reinforcing their understanding with detailed explanations, worked-out ex- amples, and practice problems. Students will also find listings of key ideas to master. Each section 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 online 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 also benefit from a new Premium eBook with highlighting and search features; Personal Study Plans (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 the back 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. These DVDs provide comprehensive coverage of the course—along with additional explana- tions of concepts, sample problems, and applications—to help students review essential topics.

xviii Ancillaries

www.stewartmath.com
www.cengagebrain.com
TO T H E S T U D E N T

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

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 the concept exer- 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- ator and denominator 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 Signifi- cant Figures.

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

1212 12 � 1

xix

xx

A B B R E V I AT I O N S

cm centimeter mg milligram dB decibel MHz megahertz F farad mi mile ft foot min minute g gram mL milliliter gal gallon mm millimeter h hour N Newton H henry qt quart Hz Hertz oz ounce in. inch s second J Joule � ohm kcal kilocalorie V volt kg kilogram W watt km kilometer yd yard kPa kilopascal yr year L liter °C degree Celsius lb pound °F degree Fahrenheit lm lumen K Kelvin M mole of solute ⇒ implies

per liter of solution ⇔ is equivalent to m meter

xxi

M AT H E M AT I C A L V I G N E T T E S

George Polya P1 Einstein’s Letter P4 Bhaskara 3 Algebra 6 No Smallest or Largest Number

in an Open Interval 11 Diophantus 43 Euclid 63 François Viète 82 Leonhard Euler 92 Coordinates as Addresses 127 Pierre de Fermat 143 Alan Turing 157 Donald Knuth 200 René Descartes 223 Sonya Kovalevsky 227 Pythagoras 261 Evariste Galois 296 Carl Friedrich Gauss 309 Gerolamo Cardano 311 The Gateway Arch 348 John Napier 357 Radiocarbon Dating 371 Standing Room Only 381 Half-Lives of Radioactive Elements 383 Radioactive Waste 384 pH for Some Common Substances 386 Largest Earthquakes 386 Intensity Levels of Sounds 388

Hipparchus 418 Aristarchus of Samos 420 Thales of Miletus 421 Surveying 446 The Value of p 481 Periodic Functions 492 AM and FM Radio 493 Root-Mean Square 515 Jean Baptiste Joseph Fourier 539 Maria Gaetana Agnesi 603 Galileo Galilei 614 William Rowan Hamilton 649 Julia Robinson 701 Olga Taussky-Todd 706 Arthur Cayley 712 David Hilbert 721 Emmy Noether 724 The Rhind Papyrus 732 Linear Programming 755 Archimedes 767 Eccentricities of the Orbits

of the Planets 776 Paths of Comets 783 Johannes Kepler 792 Large Prime Numbers 824 Eratosthenes 825 Fibonacci 825 The Golden Ratio 829 Srinavasa Ramanujan 840

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

Mathematics in the Modern World 18 Error-Correcting Codes 98 Changing Words, Sound, and Pictures

into Numbers 158 Computers 224 Splines 276 Automotive Design 280 Unbreakable Codes 321 Law Enforcement 356 Evaluating Functions on a Calculator 498 Weather Prediction 670 Mathematical Ecology 717 Global Positioning System 738 Looking Inside Your Head 797 Fair Division of Assets 834 Fractals 842 Mathematical Economics 848 Fair 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

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P1

P R O L O 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 cer- 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. 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.”

Ch uc

k Pa

in te

r/ St

an fo

rd N

ew s

Se rv

ic e

� Tr y t o R e c o g n i z e P a t t e r n s

Certain problems are solved by recognizing that some kind of pattern is occurring. The pattern could be geometric, numerical, or algebraic. If you can see regularity or repetition in a problem, then you might be able to guess what the pattern is and then prove it.

� U s e A n a l o g y

Try to think of an analogous problem, that is, a similar or related problem but one that is easier than the original. If you can solve the similar, simpler problem, then it might give you the clues you need to solve the original, more difficult one. For instance, if a prob- lem involves very large numbers, you could first try a similar problem with smaller num- bers. Or if the problem is in three-dimensional geometry, you could look for something similar in two-dimensional geometry. Or if the problem you start with is a general one, you could first try a special case.

� I n t r o d u c e S o m e t h i n g E x t r a

You might sometimes need to introduce something new—an auxiliary aid—to make the connection between the given and the unknown. For instance, in a problem for which a diagram is useful, the auxiliary aid could be a new line drawn in the diagram. In a more algebraic problem the aid could be a new unknown that relates to the original unknown.

� Ta k e C a s e s

You might sometimes have to split a problem into several cases and give a different ar- gument for each case. For instance, we often have to use this strategy in dealing with ab- solute value.

� 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 contradic- tion to what we absolutely know is true.

� 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.

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 be- came a rule which served afterwards to solve other problems.”

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 t1 and t2 be 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 t1 and t2, so we solve the above equations for these times:

t1 � d

30 t2 �

d

60

average speed for entire trip � total distance

total time �

2d

t1 � t2

60 � d

t2

30 � d

t1

average speed � distance traveled

time elapsed

120

3 � 40 mi/h

30 � 60

2 � 45 mi/h

Prologue P3

Introduce notation �

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:

So the average speed for the entire trip is 40 mi/h. ■

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 average speed 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 suc- cessive discounts of 20%?

3. Cutting up a Wire A piece of wire is bent as shown in the figure. You can see that one cut through the wire produces four pieces and two parallel cuts produce seven pieces. How many 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 nutrient fluid, 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 sec- ond half of the season. Is it necessarily true that player A has a higher batting average than player B for the entire season?