## TABLE OF CONTENTS

About the Author v

Preface to the Instructor xv

Acknowledgments xxi

Preface to the Student xxiii

**Chapter 0 The Real Numbers 1**

0.1 The Real Line 2

Construction of the Real Line 2

Is Every Real Number Rational? 3

Problems 5

0.2 Algebra of the Real Numbers 6

Commutativity and Associativity 6

The Order of Algebraic Operations 7

The Distributive Property 8

Additive Inverses and Subtraction 9

Multiplicative Inverses and the Algebra of Fractions 10

Symbolic Calculators 13

Exercises, Problems, and Worked-out Solutions 15

0.3 Inequalities, Intervals, and Absolute Value 20

Positive and Negative Numbers 20

Inequalities 21

Intervals 23

Absolute Value 25

Exercises, Problems, and Worked-out Solutions 29

Chapter Summary and Chapter Review Questions 35

**Chapter 1 Functions and Their Graphs 37**

1.1 Functions 38

Definition and Examples 38

The Domain of a Function 41

The Range of a Function 42

Functions via Tables 44

Exercises, Problems, and Worked-out Solutions 45

1.2 The Coordinate Plane and Graphs 50

The Coordinate Plane 50

The Graph of a Function 52

Determining the Domain and Range from a Graph 54

Which Sets are Graphs of Functions? 56

Exercises, Problems, and Worked-out Solutions 56

1.3 Function Transformations and Graphs 63

Vertical Transformations: Shifting, Stretching, and Flipping 63

Horizontal Transformations: Shifting, Stretching, Flipping 66

Combinations of Vertical Function Transformations 68

Even Functions 71

Odd Functions 72

Exercises, Problems, and Worked-out Solutions 73

1.4 Composition of Functions 81

Combining Two Functions 81

Definition of Composition 82

Decomposing Functions 85

Composing More than Two Functions 85

Function Transformations as Compositions 86

Exercises, Problems, and Worked-out Solutions 88

1.5 Inverse Functions 93

The Inverse Problem 93

One-to-one Functions 94

The Definition of an Inverse Function 95

The Domain and Range of an Inverse Function 97

The Composition of a Function and Its Inverse 98

Comments About Notation 99

Exercises, Problems, and Worked-out Solutions 101

1.6 A Graphical Approach to Inverse Functions 106

The Graph of an Inverse Function 106

Graphical Interpretation of One-to-One 107

Increasing and Decreasing Functions 108

Inverse Functions via Tables 110

Exercises, Problems, and Worked-out Solutions 111

Chapter Summary and Chapter Review Questions 115

**Chapter 2 Linear, Quadratic, Polynomial, and Rational Functions 119**

2.1 Lines and Linear Functions 120

Slope 120

The Equation of a Line 121

Parallel Lines 125

Perpendicular Lines 126

Exercises, Problems, and Worked-out Solutions 128

2.2 Quadratic Functions and Conics 135

Completing the Square and the Quadratic Formula 135

Parabolas and Quadratic Functions 138

Circles 140

Ellipses 142

Hyperbolas 144

Exercises, Problems, and Worked-out Solutions 146

2.3 Exponents 157

Positive Integer Exponents 157

Defining *x*^{0} 159

Negative Integer Exponents 160

Roots 161

Rational Exponents 164

Properties of Exponents 165

Exercises, Problems, and Worked-out Solutions 166∂

2.4 Polynomials 174

The Degree of a Polynomial 174

The Algebra of Polynomials 175

Zeros and Factorization of Polynomials 177

The Behavior of a Polynomial Near ±∞ 179

Graphs of Polynomials 181

Exercises, Problems, and Worked-out Solutions 182

2.5 Rational Functions 187

The Algebra of Rational Functions 187

Division of Polynomials 188

The Behavior of a Rational Function Near ±∞ 191

Graphs of Rational Functions 194

Exercises, Problems, and Worked-out Solutions 195

Chapter Summary and Chapter Review Questions 201

**Chapter 3 Exponential Functions, Logarithms, and e 203**

3.1 Logarithms as Inverses of Exponential Functions 204

Exponential Functions 204

Logarithms Base 2 206

Logarithms with Any Base 207

Common Logarithms and the Number of Digits 208

Exercises, Problems, and Worked-out Solutions 209

3.2 The Power Rule for Logarithms 214

Logarithm of a Power 214

Radioactive Decay and Half-Life 215

Change of Base 217

Exercises, Problems, and Worked-out Solutions 219

3.3 The Product and Quotient Rules for Logarithms 223

Logarithm of a Product 223

Logarithm of a Quotient 224

Earthquakes and the Richter Scale 225

Sound Intensity and Decibels 226

Star Brightness and Apparent Magnitude 227

Exercises, Problems, and Worked-out Solutions 228

3.4 Exponential Growth 235

Functions with Exponential Growth 236

Population Growth 239

Compound Interest 241

Exercises, Problems, and Worked-out Solutions 245

3.5 *e *and the Natural Logarithm 250

Estimating Area Using Rectangles 250

Defining *e *252

Defining the Natural Logarithm 254

Properties of the Exponential Function and Natural Logarithm 255

Exercises, Problems, and Worked-out Solutions 256

3.6 Approximations and Area with *e *and ln 262

Approximation of the Natural Logarithm 262

Approximations with the Exponential Function 263

An Area Formula 265

Exercises, Problems, and Worked-out Solutions 267

3.7 Exponential Growth Revisited 270

Continuously Compounded Interest 270

Continuous Growth Rates 271

Doubling Your Money 272

Exercises, Problems, and Worked-out Solutions 274

Chapter Summary and Chapter Review Questions 278

**Chapter 4 Trigonometric Functions 281**

4.1 The Unit Circle 282

The Equation of the Unit Circle 282

Angles in the Unit Circle 283

Negative Angles 284

Angles Greater than 360^{◦ }286

Length of a Circular Arc 287

Special Points on the Unit Circle 287

Exercises, Problems, and Worked-out Solutions 289

4.2 Radians 295

A Natural Unit of Measurement for Angles 295

The Radius Corresponding to an Angle 298

Length of a Circular Arc 300

Area of a Slice 301

Special Points on the Unit Circle 301

Exercises, Problems, and Worked-out Solutions 302

4.3 Cosine and Sine 307

Definition of Cosine and Sine 307

The Signs of Cosine and Sine 309

The Key Equation Connecting Cosine and Sine 310

The Graphs of Cosine and Sine 311

Exercises, Problems, and Worked-out Solutions 313

4.4 More Trigonometric Functions 317

Definition of Tangent 317

The Sign of Tangent 318

Connections Among Cosine, Sine, and Tangent 319

The Graph of Tangent 320

Three More Trigonometric Functions 321

Exercises, Problems, and Worked-out Solutions 322

4.5 Trigonometry in Right Triangles 327

Trigonometric Functions via Right Triangles 327

Two Sides of a Right Triangle 328

One Side and One Angle of a Right Triangle 329

Exercises, Problems, and Worked-out Solutions 331

4.6 Trigonometric Identities 336

The Relationship Among Cosine, Sine, and Tangent 336

Trigonometric Identities for the Negative of an Angle 338

Trigonometric Identities with *π/*2 339

Trigonometric Identities Involving a Multiple of *π *341

Exercises, Problems, and Worked-out Solutions 343

Chapter Summary and Chapter Review Questions 348

**Chapter 5 Trigonometric Algebra and Geometry 351**

5.1 Inverse Trigonometric Functions 352

The Arccosine Function 352

The Arcsine Function 354

The Arctangent Function 357

Exercises, Problems, and Worked-out Solutions 359

5.2 Inverse Trigonometric Identities 365

Composition of Trigonometric Functions and Their Inverses 365

More Inverse Functions 366

More Compositions with Inverse Trigonometric Functions 367

The Arccosine, Arcsine, and Arctangent of *−t *369

Arccosine Plus Arcsine 370

Exercises, Problems, and Worked-out Solutions 371

5.3 Using Trigonometry to Compute Area 375

The Area of a Triangle via Trigonometry 375

Ambiguous Angles 376

The Area of a Parallelogram via Trigonometry 377

The Area of a Polygon 378

Trigonometric Approximations 380

Exercises, Problems, and Worked-out Solutions 383

5.4 The Law of Sines and the Law of Cosines 388

The Law of Sines 388

The Law of Cosines 390

When to Use Which Law 393

Exercises, Problems, and Worked-out Solutions 395

5.5 Double-Angle and Half-Angle Formulas 402

The Cosine of 2*θ *402

The Sine of 2*θ *403

The Tangent of 2*θ *404

The Cosine and Sine of *θ/*2 404

The Tangent of *θ/*2 406

Exercises, Problems, and Worked-out Solutions 407

5.6 Addition and Subtraction Formulas 414

The Cosine of a Sum and Difference 414

The Sine of a Sum and Difference 416

The Tangent of a Sum and Difference 417

Products of Trigonometric Functions 418

Exercises, Problems, and Worked-out Solutions 418

5.7 Transformations of Trigonometric Functions 423

Amplitude 423

Period 425

Phase Shift 426

Fitting Transformations of Trigonometric Functions to Data 429

Exercises, Problems, and Worked-out Solutions 430

Chapter Summary and Chapter Review Questions 437

**Chapter 6 Sequences, Series, and Limits 439**

6.1 Sequences 440

Introduction to Sequences 440

Arithmetic Sequences 442

Geometric Sequences 443

Recursively Defined Sequences 445

Exercises, Problems, and Worked-out Solutions 448

6.2 Series 453

Sums of Sequences 453

Arithmetic Series 453

Geometric Series 455

Summation Notation 457

Pascal’s Triangle 459

The Binomial Theorem 462

Exercises, Problems, and Worked-out Solutions 465

6.3 Limits 470

Introduction to Limits 470

Infinite Series 473

Decimals as Infinite Series 476

Special Infinite Series 477

Exercises, Problems, and Worked-out Solutions 479

Chapter Summary and Chapter Review Questions 482

**Chapter 7 Polar Coordinates, Vectors, and Complex Numbers 483**

7.1 Polar Coordinates 484

Defining Polar Coordinates 484

Converting from Polar to Rectangular Coordinates 485

Converting from Rectangular to Polar Coordinates 485

Graphs of Polar Equations 488

Exercises, Problems, and Worked-out Solutions 491

7.2 Vectors 494

An Algebraic and Geometric Introduction to Vectors 494

Vector Addition 496

Vector Subtraction 498

Scalar Multiplication 500

The Dot Product 500

Exercises, Problems, and Worked-out Solutions 503

7.3 Complex Numbers 506

The Complex Number System 506

Arithmetic with Complex Numbers 507

Complex Conjugates and Division of Complex Numbers 508

Zeros and Factorization of Polynomials, Revisited 511

Exercises, Problems, and Worked-out Solutions 514

7.4 The Complex Plane 518

Complex Numbers as Points in the Plane 518

Geometric Interpretation of Complex Multiplication and Division 519

De Moivre’s Theorem 522

Finding Complex Roots 523

Exercises, Problems, and Worked-out Solutions 524

Chapter Summary and Chapter Review Questions 526

Appendix: Area 527

Circumference 527

Squares, Rectangles, and Parallelograms 528

Triangles and Trapezoids 529

Stretching 531

Circles and Ellipses 531

Exercises, Problems, and Worked-out Solutions 534

Photo Credits 543

Index 545

Colophon: Notes on Typesetting 551