Hungerford T.W., Shaw D.J. Contemporary Precalculus: A Graphing Approach

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vii

Contents

Preface xi

To the Instructor xiv

Ancillaries xvii

To the Student xix

chapter

Basics 1

1.1 The Real Number System 2

1.1.A Special Topics: Decimal Representation of Real Numbers 16

1.2 Solving Equations Algebraically 19

1.2.A Special Topics: Absolute Value Equations 32

1.2.B Special Topics: Variation 33

1.3 The Coordinate Plane 39

1.4 Lines 53

CHAPTER 1 REVIEW 69

■

DISCOVERY PROJECT 1 75

chapter

Graphs and Technology 77

2.1 Graphs 78

2.2 Solving Equations Graphically and Numerically 92

2.3 Applications of Equations 101

2.4 Optimization Applications 114

2.5 Linear Models 120

CHAPTER 2 REVIEW 131

■

DISCOVERY PROJECT 2 137

chapter

Functions and Graphs 141

3.1 Functions 142

3.2 Functional Notation 151

3.3 Graphs of Functions 161

3.3.A Special Topics: Parametric Graphing 175

3.4 Graphs and Transformations 179

3.4.A Special Topics: Symmetry 189

3.5 Operations on Functions 195

3.6 Rates of Change 204

3.7 Inverse Functions 217

CHAPTER 3 REVIEW 228 DISCOVERY PROJECT 3 236

(3, −1)

(2, 4)

−6

−33

⫺5

⫺3

viii Contents

chapter

Polynomial and Rational Functions 239

4.1 Quadratic Functions 240

4.2 Polynomial Functions 250

4.2.A Special Topics: Synthetic Division 259

4.3 Real Roots of Polynomials 262

4.4 Graphs of Polynomial Functions 270

4.4.A Special Topics: Polynomial Models 283

4.5 Rational Functions 288

4.5.A Special Topics: Other Rational Functions 304

4.6 Polynomial and Rational Inequalities 308

4.6.A Special Topics: Absolute Value Inequalities 317

4.7 Complex Numbers 321

4.8 Theory of Equations 328

CHAPTER 4 REVIEW 334

■

DISCOVERY PROJECT 4 340

chapter

Exponential and Logarithmic Functions 341

5.1 Radicals and Rational Exponents 342

5.1.A Special Topics: Radical Equations 350

5.2 Exponential Functions 357

5.2.A Special Topics: Compound Interest and the Number e 369

5.3 Common and Natural Logarithmic Functions 375

5.4 Properties of Logarithms 385

5.4.A Special Topics: Logarithmic Functions to Other Bases 392

5.5 Algebraic Solutions of Exponential and Logarithmic Equations 399

5.6 Exponential, Logarithmic, and Other Models 409

CHAPTER 5 REVIEW 419

■

DISCOVERY PROJECT 5 425

chapter

Trigonometric Functions 427

6.1 Angles and Their Measurement 428

6.1.A Special Topics: Arc Length and Angular Speed 435

6.2 The Sine, Cosine, and Tangent Functions 442

6.2 Alternate: The Sine, Cosine, and Tangent Functions 452

6.3 Algebra and Identities 457

6.4 Basic Graphs 466

6.5 Periodic Graphs and Simple Harmonic Motion 477

6.5.A Special Topics: Other Trigonometric Graphs 490

6.6 Other Trigonometric Functions 496

CHAPTER 6 REVIEW 505

■

DISCOVERY PROJECT 6 512

(0, 0)

(3, 1)

y = x

g(x)

f(x)

AM signal

Contents ix

chapter

Trigonometric Identities and Equations 513

7.1 Basic Identities and Proofs 514

7.2 Addition and Subtraction Identities 523

7.2.A Special Topics: Lines and Angles 532

7.3 Other Identities 535

7.4 Inverse Trigonometric Functions 545

7.5 Trigonometric Equations 555

CHAPTER 7 REVIEW 567

■

DISCOVERY PROJECT 7 571

chapter

Triangle Trigonometry 573

8.1 Trigonometric Functions of Angles 574

8.1 Alternate: Trigonometric Functions of Angles 584

8.2 Applications of Right Triangle Trigonometry 588

8.3 The Law of Cosines 597

8.4 The Law of Sines 606

8.4.A Special Topics: The Area of a Triangle 617

CHAPTER 8 REVIEW 620

■

DISCOVERY PROJECT 8 624

chapter

Applications of Trigonometry 625

9.1 The Complex Plane and Polar Form for Complex Numbers 626

9.2 DeMoivre’s Theorem and nth Roots

of Complex Numbers 632

9.3 Vectors in the Plane 639

9.4 The Dot Product 653

CHAPTER 9 REVIEW 662

■

DISCOVERY PROJECT 9 666

chapter

Analytic Geometry 669

10.1 Circles and Ellipses 671

10.2 Hyperbolas 686

10.3 Parabolas 700

10.3.A Special Topics: Parametric Equations for Conic Sections 713

10.4 Rotations and Second-Degree Equations 718

10.4.A Special Topics: Rotation of Axes 722

10.5 Plane Curves and Parametric Equations 727

10.6 Polar Coordinates 743

10.7 Polar Equations of Conics 753

CHAPTER 10 REVIEW 763

■

DISCOVERY PROJECT 10 768

−4

−66

46°34°

36°

(c, d)

(a, b)

(a – b, c – d)

u – v

–v

−10

−88

x Contents

chapter

Systems of Equations 771

11.1 Systems of Linear Equations in Two Variables 772

11.1.A Special Topics: Systems of Nonlinear Equations 784

11.2 Large Systems of Linear Equations 792

11.3 Matrix Methods for Square Systems 806

CHAPTER 11 REVIEW 819

■

DISCOVERY PROJECT 11 823

chapter

Discrete Algebra 825

12.1 Sequences and Sums 826

12.2 Arithmetic Sequences 837

12.3 Geometric Sequences 844

12.3.A Special Topics: Infinite Series 852

12.4 The Binomial Theorem 857

12.5 Mathematical Induction 864

CHAPTER 12 REVIEW 873

■

DISCOVERY PROJECT 12 877

chapter

Limits and Continuity 879

13.1 Limits of Functions 880

13.2 Properties of Limits 890

13.2.A Special Topics: The Formal Definition of Limit 897

13.3 Continuity 903

13.4 Limits Involving Infinity 914

CHAPTER 13 REVIEW 924

■

DISCOVERY PROJECT 13 927

APPENDIX 1: Algebra Review 931

APPENDIX 2: Geometry Review 953

APPENDIX 3: Programs 964

Selected Answers 967

Index of Applications I-1

Subject Index I-4

1−1

−1

−2

Preface

Content Changes in the Fifth Edition

The major changes in this edition include the following.

Chapter Tests In addition to the Chapter Review exercises, each chapter now

has a chapter test (in two parts for longer chapters), with answers for both odd

and even questions in the answer section.

Section Objectives The goals for student learning are summarized at the

beginning of each section.

Discovery Projects A new Discovery Project has been added to Chapter 1.

Exercises Every exercise in the previous edition was examined and a signifi-

cant number of them were updated, revised, or replaced. All in all, more than

20% of the exercises in this edition are new.

A number of smaller changes and additions—ranging from a few lines to a page

or so—have been made throughout the book to improve coverage and clarity.

Organizational Changes

Several changes in the order and arrangement of topics have been made to provide

more flexibility for the instructor.

Parametric Graphing The brief introduction in Chapter 3 is now in a section

by itself and graph reading is covered in Section 3.3 (Graphs of Functions).

Parametric equations for conic sections are now in a separate (optional) sec-

tion of Chapter 10, rather than being included as parts of other sections.

Arc Length and Angular Speed These topics are now in a separate (optional)

section instead of being included in Section 6.1.

Calculator Investigations These formerly appeared at the end of selected sec-

tions at the beginning of the book and are now incorporated into the exercise

sets for these sections.

This book provides the mathematical background needed for calculus for students

who have had two or three years of high school mathematics. Topics that are

essential for success in calculus are thoroughly covered, including functional

notation, graph reading, the natural exponential and logarithmic functions, average

rates of change, trigonometric functions of a real variable, and limits.

The mathematics is presented in an informal manner that stresses meaningful

motivation, careful explanations, and numerous examples, with an ongoing focus

on real-world problem solving. Technology is integrated into the presentation and

students are expected to use it to participate actively in exploring topics from

algebraic, graphical, and numerical perspectives.

xii Preface

Ongoing Features

All the helpful pedagogical features of earlier editions are retained here, such as:

A Flexible Approach to Trigonometry that allows instructors to use the order of

topics they prefer: Trig functions of a real variable can be covered before or

after triangle trigonometry or both approaches can be combined (as explained

in the chart on page xiv);

Graphing Explorations, in which students discover and develop useful mathe-

matics on their own;

Cautions that alert students to common misconceptions and mistakes;

Technology Tips that provide assistance in carrying out various procedures on

specific calculators;

Exercises that proceed from routine drill to those requiring some thought,

including graph interpretation and applied problems, as well as Thinkers that

challenge students to “think outside the box” (most of these are not difficult—

just different);

Chapter Reviews that include a list of important concepts (referenced by section

and page number), a summary of important facts and formulas, and a set of

review questions;

Algebra Review Appendix for students who need a review of basic algebra;

Geometry Review Appendix that summarizes frequently used facts from plane

geometry, with examples and exercises;

Program Appendix that provides a small number of programs that are useful for

updating older calculator models or for more easily carrying out procedures

discussed in the text.

Acknowledgments

First of all, the senior author is happy to welcome Doug

Shaw as a coauthor. His insights and enthusiasm have had

a very positive impact on this revision and will, I trust,

continue to do so in the future.

We are particularly grateful to three people who have

been associated with several editions of this book:

Leslie Lahr, our Developmental Editor, who has been

involved in almost all aspects of the book, and

from whose sage advice we have greatly benefited;

Phil Embree of William Woods University, one of our

accuracy checkers and the author of several Discovery

Projects, whose helpful suggestions have definitely

improved the final product;

Laura Horowitz of Hearthside Publication Services,

whose calm coordination of all aspects of production

has kept us sane and happy.

Thanks are also due to Heidi A. Howard, who pitched in

on a variety of tasks whenever needed.

It is a pleasure to acknowledge the invaluable assis-

tance of the Brooks/Cole staff:

Gary Whalen, Acquisitions Editor

Joe Rogove, Marketing Manager

Natasha Coats, Assistant Editor

Lynh Pham, Editorial Assistant

Hal Humphrey, Senior Content Project Manager

Vernon Boes, Senior Art Director

We also want to thank the outside production staff:

Grace Lefrancois, Copy Editor

Jade Myers, Art Rendering

Terri Wright of Terri Wright Design

Preface xiii

Special thanks go to

John Samons, Florida Community College,

who did most of the accuracy checking of the manuscript,

and to

Jerret Dumouchel, Florida Community College

Arkady Hanjiev, North Hills College,

who prepared the Solution Manuals.

The students who have assisted in manuscript preparation

also have our thanks: Varun Khanna (St. Louis Univer-

sity), and Jordan Meyer, Ren Waddell and Kevan Irvine

(University of Northern Iowa).

Finally, we want to thank the reviewers whose con-

structive comments have played a crucial role in revising

this text.

Jared Abwawo, Tacoma Community College

Tony Akhlaghi, Bellevue Community College

Donna Bernardy, Lane Community College

Patrick DeFazio, Onondaga Community College

Linda Horner, Columbia State Community College

Douglas Nelson, Central Oregon Community College

Stephen Nicoloff, Paradise Valley Community College

Bogdan Nita, Montclair State University

Dennis Reissig, Suffolk County Community College

Arnavaz Taraporevala, New York City College of

Technology

Thanks also go the reviewers of previous editions:

Deborah Adams, Jacksonville University

Kelly Bach, University of Kansas

David Blankenbaker, University of New Mexico

Kathleen A. Cantone, Onondaga Community College

Bettyann Daley, University of Delaware

Margaret Donlan, University of Delaware

Patricia Dueck, Arizona State University

Betsy Farber, Bucks County Community College

Alex Feldman, Boise State University

Robert Fliess, West Liberty State College

Betty Givan, Eastern Kentucky University

William Grimes, Central Missouri State University

Frances Gulick, University of Maryland

John Hamm, University of New Mexico

Lonnie Hass, North Dakota State University

Larry Howe, Rowan University

Conrad D. Krueger, San Antonio College

Ann Lawrance, Wake Technical Community College

Charles Laws, Cleveland State Community College

Anatoly S. Libgober, University of Illinois at Chicago

Martha Lisle, Prince George’s Community College

Matthew Liu, University of Wisconsin-Stevens Point

Sergey Lvin, University of Maine

George Matthews, Onondaga Community College

Nancy Matthews, University of Oklahoma

Ruth Meyering, Grand Valley State University

William Miller, Central Michigan University

Philip Montgomery, University of Kansas

Roger Nelsen, Lewis and Clark College

Jack Porter, University of Kansas

Robert Rogers, University of New Mexico

Barbara Sausen, Fresno City College

Hugo Sun, California State University at Fresno

Stuart Thomas, University of Oregon

Trung G. Tran, Tacoma Community College

Bettie Truitt, Black Hawk College

Jan Vandever, South Dakota State University

Judith Wolbert, Kettering University

Cathleen M. Zucco-Teveloff, Trinity College

The last word, as always, goes to our wives, Mary

Alice Hungerford and Laurel Shaw, who have, as always,

provided understanding and support when it was most

needed.

Thomas W. Hungerford

Douglas J. Shaw

This book contains more than enough material for two semesters. By using the

chart on the facing page (and the similar ones at the beginning of each chapter that

show the interdependence of sections within the chapter), you can easily design a

course to fit the needs of your students and the constraints of time. When planning

your syllabus, two things are worth noting.

Special Topics Sections with this label are usually related to the immediately

preceding section and are not prerequisites for other sections of the text. Judg-

ments vary as to which of these topics are essential and which can be omitted.

So feel free to include as many or as few Special Topics sections as you wish.

Trigonometry We believe that a precalculus course should emphasize trigono-

metric functions of a real variable from the beginning. We also think that the

advent of technology has made the cotangent, secant, and cosecant functions

less necessary than was the case when calculations were done by hand. So

they are not introduced until the end of Chapter 6. This is the approach given

in the first column of the chart below.

However, we are well aware that some instructors feel strongly that triangle

trigonometry should be introduced first, or that both approaches should be

presented together, or that all six trig functions should be introduced at

the same time, etc. The following chart (and its footnotes) show how to

arrange things to suit your personal preferences.

To the Instructor

OPTIONS FOR COVERING TRIGONOMETRY

Mixed

[Trig functions of a real variable and basic triangle trig

Real Variable First Triangles First introduced as soon as possible]

Real Variable First Triangles First

Chapter 6* Chapter 8 (using Alternate Sections 6.1–6.4

Section 6.1

8.1 in place of 8.1) 8.1–8.2 Alternate 8.1

6.5–6.6 8.2

6.2–6.6*

Chapter 6 (using Alternate

6.2 in place of 6.2)

Chapters 7, 8, 9 in Chapters 7 & 9 in either Chapter 7, Sections 8.3–8.4 Chapter 7, Sections 8.3–8.4

any order

†

order

†

and Chapter 9 in any order

†

and Chapter 9 in any order

†

*For early introduction of cotangent, secant, and cosecant, incorporate Section 6.6 into Sections 6.2–6.4 by covering

Part I of 6.6 at the end of 6.2;

Part II of 6.6 at the end of 6.3;

Part III of 6.6 at the end of 6.4.

†

Within each of these chapters, sections can be covered in several different orders. See the interdependence-of-sections chart at the beginning of the chapter.

xiv

Interdependence of Chapters

Basics

Graphs and

Technology

Systems of

Equations*

Functions

and Graphs

Polynomial

and Rational

Functions

Exponential and

Logarithmic

Functions

Trigonometric

Functions

Limits and

Continuity

Trigonometric

Identities and

Equations

Applications of

Trigonometry

Analytic

Geometry

(Parametric &

Polar Graphing)

†

Triangle

Trigonometry

Discrete

Algebra

*Section 2.1 (graphs) is a prerequisite for Special Topics 11.1.A (Systems of Nonlinear Equations).

†

Sections 10.1–10.4 (Conic Sections) depend only on Chapter 2. Trigonometry is an additional prerequisite for Special Topics 10.3.A and 10.4.A

and Sections 10.5–10.7.

Analytic

Geometry

(Conic Sections)

†