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Tables of Contents for Precalculus

Chapter/Section Title

Page #

Page Count

PREFACE

XV

CHAPTER 1 MODELING AND PROBLEM SOLVING

1

34

1.1 A Case for Algebra

1

6

Mathematics Is an Analytical Tool

4

1

Mathematics Is Rich in Concepts and Contexts

4

1

Mathematics Is Part of Our Intellectual Heritage

5

1

You May Actually Use Mathematics!

5

1

Some Suggestions for Feeling at Home with This Book

5

2

1.2 Models of Quantitative Relationships

7

11

Three Types of Models

7

1

Information Provided by the Three Types of Models

8

1

Creating One Model from Another

9

9

1.3 Problem-Solving Strategies

18

9

First Step: Understand the Problem

19

1

Second Step: Devise a Plan

20

1

Third Step: Carry Out the Plan

21

2

Fourth Step: Look Back

23

4

1.4 Solving Equations and Inequalities Graphically

27

7

Solving Equations Graphically

27

4

Solving Inequalities Graphically

31

3

Chapter Review

34

1

CHAPTER 2 FUNCTIONS

35

66

2.1 An Overview of Functions

35

14

The Definition of Function

35

1

Functional Notation

36

2

Domain and Range

38

1

Three Views of Functions

38

6

Reasons to Study Functions and Their Models

44

4

Supplementary Topic: Nonnumerical Functions

48

1

2.2 Combinations of Functions

49

7

Adding, Subtracting, Multiplying, and Dividing Functions

49

1

Composing Functions

50

6

2.3 Linear Functions and Average Rates of Change

56

16

Rate of Change for a Linear Function

56

4

Linearization

60

2

Average Rate of Change

62

4

Dynamic Behavior: Increasing and Decreasing Functions

66

5

Supplementary Topic: Linearizing a Situation

71

1

Supplementary Topic: Local Linearization: A "Propinquity Principle"

72

1

2.4 Sequences

72

8

Arithmetic Sequences

75

2

Arithmetic Series

77

3

2.5 Piecewise Defined Functions

80

11

Three Views of Piecewise Defined Functions

81

2

Three Views of Linear Absolute Value Functions

83

5

Solving Linear Absolute Value Inequalities

88

3

2.6 Parametric Equations

91

7

Eliminating the Parameter

93

1

Parametrizing a Path

94

2

Domain and Ranges of Linear Parametric Functions

96

2

Chapter Review

98

3

CHAPTER 3 LINEAR SYSTEMS

101

46

3.1 Systems of Linear Equations

101

10

Methods of Solving 2 X 2 Systems

102

2

Methods of Solving 3 X 3 Systems

104

1

Numbers of Solutions to 2 X 2 and 3 X 3 Systems

105

5

Supplementary Topic: Number of Solutions to m X n Linear Systems

110

1

3.2 Matrix Solutions of Systems of Linear Equations

111

13

Augmented Matrix of a System

111

1

Matrix Row Operations

112

2

Gauss-Jordan Elimination

114

2

Reduced Row-Echelon Matrices

116

1

Matrices and Numbers of Solutions

117

5

Summary: Matrix Versus Nonmatrix Methods

122

2

3-3 Determinants and Cramer's Rule

124

12

Second-Order Determinants and Cramer's Rule for 2 X 2 Systems

126

2

Third-Order Determinants and Cramer's Rule for 3 X 3 Systems

128

3

Alternate Methods of Evaluating a 3 X 3 Determinant

131

2

Higher-Order Determinants and Cramer's Rule for Larger Systems

133

3

3-4 Systems of Linear Inequalities and Linear Programming

136

8

Graphs of Linear Inequalities in Two Variables

136

2

Graphs of Systems of Inequalities in Two Variables

138

2

Linear Programming

140

4

Chapter Review

144

3

CHAPTER 4 QUADRATIC FUNCTIONS AND RELATIONS

147

94

4-1 Quadratic Functions

147

17

A Graphical View of Quadratic Functions

148

4

An Analytical View of Quadratic Functions

152

6

A Numerical View of Quadratic Functions

158

5

Supplementary Topic: Effect of the Viewing Window on the Apparent Steepness of Graphs

163

1

Supplementary Topic: Complex Factors

163

1

4-2 Modeling and Problem Solving with Quadratic Functions

164

10

Optimization Problems

164

2

Solving Quadratic Inequalities

166

2

Fitting a Quadratic Function to a Table

168

2

Parametric Models for Quadratic Relationships

170

4

Supplementary Topic: Validity of a Quadratic Model

174

1

4-3 Relations

174

11

The Definition of Relation

176

1

Implicit Functions

177

3

Square Root Functions

180

5

4-4 A Graphical View of Conic Sections

185

15

A Graphical View of Parabolas

187

2

A Graphical View of Ellipses

189

3

A Graphical View of Circles

192

1

A Graphical View of Hyperbolas

193

5

Exceptional Graphs

198

2

4-5 Graphical Transformations

200

14

Stretches and Compressions

201

2

Shifts

203

1

Reflections

204

2

Applying a Sequence of Transformations

206

2

Summary

208

3

Supplementary Topic: Graphical Transformations in the Context of Functions

211

3

4-6 An Analytical View of Conic Sections

214

14

An Analytical View of Parabolas

214

4

An Analytical View of Ellipses

218

4

An Analytical View of Hyperbolas

222

6

4-7 Systems of Quadratic Equations and Inequalities

228

11

Solving Systems of Quadratic Equations Analytically

229

3

Solving Systems of Quadratic Equations Graphically

232

2

Solving Systems of Quadratic Inequalities

234

5

Chapter Review

239

2

CHAPTER 5 POLYNOMIAL FUNCTIONS

241

54

5-1 Power Functions

241

9

Basic Power Functions and Their Graphs

241

2

Graphical Transformations

243

1

Direct Variation

244

6

5-2 An Analytical View of Polynomial Functions

250

8

Factors and Zeros

251

2

The Fundamental Theorem of Algebra

253

4

Supplementary Topic: The Rational Root Theorem

257

1

5-3 A Graphical View of Polynomial Functions

258

14

Points of Interest

259

5

End Behavior

264

4

Symmetry

268

4

5-4 A Numerical View of Polynomial Functions

272

7

nth-Order Differences

272

2

Fitting a Polynomial Function to a Table

274

1

A Numerical Method for Finding Zeros

275

2

A Comparison of Methods for Finding Zeros

277

2

5-5 Solving Polynomial Inequalities

279

5

Solving Polynomial Inequalities Graphically

279

2

Solving Polynomial Inequalities Analytically

281

2

Supplementary Topic: Using the Scan Method to Graph Polynomial Functions

283

1

5-6 The Binomial Theorem

284

9

Pascal's Triangle

285

2

Binomial Coefficients

287

6

Chapter Review

293

2

CHAPTER 6 RATIONAL FUNCTIONS

295

28

6-1 Reciprocal Power Functions

295

9

Three Views of Reciprocal Power Functions

296

2

Inverse Variation

298

2

Graphical Transformations

300

4

6-2 Discontinuities and End Behavior of Rational Functions

304

10

Discontinuities and Vertical Asymptotes

305

3

End Behavior and Horizontal Asymptotes

308

6

Supplementary Topic: Slant Asymptotes

314

1

6-3 Solving Rational Inequalities

314

6

A Graphical Method

316

1

The Test-Value Method

316

1

The Scan Method

317

3

Chapter Review

320

3

CHAPTER 7 EXPONENTIAL AND LOGARITHMIC FUNCTIONS

323

68

7-1 Exponential Functions

323

15

A Numerical View of Exponential Functions

324

2

An Analytical View of Exponential Functions

326

3

A Graphical View of Exponential Functions

329

3

Geometric Sequences and Series

332

6

7-2 The Special Number XXX

338

7

The Definition of XXX

338

4

Using the Base XXX to Express Exponential Functions

342

3

7-3 Inverse Functions

345

14

One-to-One Functions

346

3

Finding Inverses of One-to-One Functions

349

9

Supplementary Topic: Parametric Representation of Inverse Functions

358

1

7-4 Logarithmic Functions

359

16

A Numerical View of Logarithmic Functions

360

3

A Graphical View of Logarithmic Functions

363

3

An Analytical View of Logarithmic Functions

366

9

Supplementary Topic: Slide Rules

375

1

7-5 Curve Fitting

375

14

Fitting Linear Functions to Data

376

3

Fitting Logarithmic Functions to Data

379

2

Fitting Exponential Functions to Data

381

1

Fitting Power Functions to Data

382

2

Curve Fitting on Your Calculator

384

5

Chapter Review

389

2

CHAPTER 8 TRIGONOMETRIC FUNCTIONS

391

64

8-1 Introduction to the Sine and Cosine Functions

391

15

Two Ways of Defining the Sine and Cosine Functions

392

4

Compatibility of the Two Definitions

396

4

Angular Velocity

400

1

Summary

401

5

8-2 Three Views of Sine and Cosine Functions

406

15

A Numerical View

406

2

An Analytical View

408

2

A Graphical View

410

5

Harmonic Motion

415

4

Supplementary Topic: Equivalent Expressions for Sine and Cosine Functions

419

2

8-3 The Six Trigonometric Functions

421

11

The Definitions of the Trigonometric Functions

421

3

An Analytical View of the Trigonometric Functions

424

2

A Numerical View of the Trigonometric Functions

426

1

A Graphical View of the Trigonometric Functions

427

2

Summary

429

3

8-4 The Trigonometric Functions in Right Triangles

432

8

Historical Origins of the Trigonometric Functions

432

2

Calculations Using Trigonometric Functions in Right Triangles

434

6

8-5 The Inverse Trigonometric Functions

440

12

The Inverse Sine Function

440

3

The Inverse Cosine Function

443

3

The Inverse Tangent Function

446

2

The Inverse Cotangent, Secant, and Cosecant Functions

448

1

Compositions of Trigonometric and Inverse Trigonometric Functions

449

3

Chapter Review

452

3

CHAPTER 9 TRIGONOMETRIC FUNCTIONS AS ANALYTICAL TOOLS

455

44

9-1 The Law of Cosines and the Law of Sines

455

10

The Law of Cosines

455

4

The Law of Sines

459

6

9-2 Trigonometric Identities

465

13

Fundamental Identities

465

1

Reflection and Rotation Identities

465

3

Sum and Difference Identities for Sine and Cosine

468

2

Double-Angle and Half-Angle Identities

470

2

Product-to-Sum Identities

472

1

Methods for Distinguishing Identities from Conditional Equations

473

5

9-3 Vectors

478

12

Geometric Representation of Vectors

479

4

Analytical Representation of Vectors

483

7

9-4 Trigonometric Equations

490

7

Three Special Types of Equations

492

1

Other Trigonometric Equations

493

4

Chapter Review

497

2

CHAPTER 10 TRIGONOMETRIC FUNCTIONS AS GRAPHING TOOLS

499

10-1 Graphs of General Conic Sections

499

10

Coordinate Systems Related by Rotation of Axes

499

5

Choosing an Appropriate Angle of Rotation for Graphing a Conic Section

504

3

Summary

507

1

Supplementary Topic: The Invariance of B(2) -- 4AC

508

1

10-2 Parametric Equations for Conic Sections

509

13

Circular Motion

509

6

Parametric Equations for Ellipses

515

3

Parametric Equations for Hyperbolas

518

4

10-3 Polar Coordinate System

522

7

Polar Coordinates of Points

523

3

Relationships Between Rectangular and Polar Coordinates

526

3

10-4 Graphing in Polar Coordinates

529

12

Polar and Rectangular Equations for the Same Curve

530

1

Graphs of Polar Equations

531

10

10-5 The Geometry of Complex Numbers

541

10

The Complex Plane

542

1

The Polar Form of a Complex Number

543

1

Adding and Subtracting Complex Numbers Geometrically

544

1

Multiplying and Dividing Complex Numbers Geometrically

545

2

De Moivre's Theorem

547

4

Chapter Review

551

APPENDIX A BASIC ALGEBRA REFERENCE

A1

A-1 Accuracy and Precision

A1

3

A-2 Linear Equations

A4

1

A-3 The Coordinate Plane

A5

2

A-4 The Pythagorean Theorem and the Distance Formula

A7

1

A-5 Basic Graphing Techniques

A8

3

A-6 Graphing Linear Equations

A11

4

A-7 Intervals

A15

2

A-8 Linear Inequalities

A17

2

A-9 Absolute Value Equations and Inequalities

A19

3

A-10 Systems of Linear Equations

A22

4

A-11 The Laws of Exponents

A26

2

A-12 Factoring

A28

4

A-13 Quadratic Equations

A32

4

A-14 Operations with Complex Numbers

A36

2

A-15 Division and Synthetic Division of Polynomials

A38

3

A-16 Algebraic Fractions

A41

7

A-17 Equations with Algebraic Fractions

A48

1

A-18 Radicals and Rational Exponents

A49

5

A-19 Equations with Radicals

A54

APPENDIX B TIPS FOR GRAPHING FUNCTIONS WITH A CALCULATOR

B1

B-1 The Viewing Window

B1

5

B-2 Graphing Linear Functions

B6

1

B-3 Graphing Quadratic Functions

B7

2

B-4 Graphing Quadratic Relations

B9

2

B-5 Graphing Polynomial Functions

B11

3

B-6 Graphing Rational Functions

B14

ANSWERS TO ODD-NUMBERED EXERCISES

ANS-1

INDEX

I-1