About the Author v

Preface to the Instructor xv

Acknowledgments xx

Preface to the Student xxii

**0 The Real Numbers** 1

0.1 The Real Line 2

Construction of the Real Line 2

Is Every Real Number Rational? 3

Problems 6

0.2 Algebra of the Real Numbers 7

Commutativity and Associativity 7

The Order of Algebraic Operations 8

The Distributive Property 10

Additive Inverses and Subtraction 11

Multiplicative Inverses and Division 12

Exercises, Problems, and Worked-out Solutions 14

0.3 Inequalities 18

Positive and Negative Numbers 18

Lesser and Greater 19

Intervals 21

Absolute Value 24

Exercises, Problems, and Worked-out Solutions 26

Chapter Summary and Chapter Review Questions 32

**1 Functions and Their Graphs** 33

1.1 Functions 34

Examples of Functions 34

Equality of Functions 35

The Domain of a Function 37

Functions via Tables 38

The Range of a Function 38

Exercises, Problems, and Worked-out Solutions 40

1.2 The Coordinate Plane and Graphs 47

The Coordinate Plane 47

The Graph of a Function 49

Determining a Function from Its Graph 50

Which Sets Are Graphs? 52

Determining the Range of a Function from Its Graph 53

Exercises, Problems, and Worked-out Solutions 54

1.3 Function Transformations and Graphs 62

Shifting a Graph Up or Down 62

Shifting a Graph Right or Left 63

Stretching a Graph Vertically or Horizontally 65

Reflecting a Graph Vertically or Horizontally 67

Even and Odd Functions 68

Exercises, Problems, and Worked-out Solutions 70

1.4 Composition of Functions 80

Definition of Composition 80

Order Matters in Composition 81

The Identity Function 82

Decomposing Functions 82

Exercises, Problems, and Worked-out Solutions 83

1.5 Inverse Functions 88

Examples of Inverse Functions 88

One-to-one Functions 89

The Definition of an Inverse Function 90

Finding a Formula for an Inverse Function 92

The Domain and Range of an Inverse Function 92

The Composition of a Function and Its Inverse 93

Comments about Notation 95

Exercises, Problems, and Worked-out Solutions 96

1.6 A Graphical Approach to Inverse Functions 102

The Graph of an Inverse Function 102

Inverse Functions via Tables 104

Graphical Interpretation of One-to-One 104

Increasing and Decreasing Functions 105

Exercises, Problems, and Worked-out Solutions 108

Chapter Summary and Chapter Review Questions 113

**2 Linear, Quadratic, Polynomial, and Rational Functions** 115

2.1 Linear Functions and Lines 116

Slope 116

The Equation of a Line 117

Parallel Lines 120

Perpendicular Lines 122

Exercises, Problems, and Worked-out Solutions 125

2.2 Quadratic Functions and Parabolas 133

The Vertex of a Parabola 133

Completing the Square 135

The Quadratic Formula 138

Exercises, Problems, and Worked-out Solutions 140

2.3 Integer Exponents 146

Exponentiation by Positive Integers 146

Properties of Exponentiation 147

Defining x0 148

Exponentiation by Negative Integers 149

Manipulations with Powers 150

Exercises, Problems, and Worked-out Solutions 152

2.4 Polynomials 158

The Degree of a Polynomial 158

The Algebra of Polynomials 160

Zeros and Factorization of Polynomials 161

The Behavior of a Polynomial Near ±∞ 163

Graphs of Polynomials 166

Exercises, Problems, and Worked-out Solutions 168

2.5 Rational Functions 173

Ratios of Polynomials 173

The Algebra of Rational Functions 174

Division of Polynomials 175

The Behavior of a Rational Function Near ±∞ 177

Graphs of Rational Functions 180

Exercises, Problems, and Worked-out Solutions 181

2.6 Complex Numbers 188

The Complex Number System 188

Arithmetic with Complex Numbers 189

Complex Conjugates and Division of Complex Numbers 190

Zeros and Factorization of Polynomials, Revisited 193

Exercises, Problems, and Worked-out Solutions 196

2.7 Systems of Equations and Matrices∗ 202

Solving a System of Equations 202

Systems of Linear Equations 204

Matrices and Linear Equations 208

Exercises, Problems, and Worked-out Solutions 215

Chapter Summary and Chapter Review Questions 221

**3 Exponents and Logarithms** 223

3.1 Rational and Real Exponents 224

Roots 224

Rational Exponents 227

Real Exponents 229

Exercises, Problems, and Worked-out Solutions 231

3.2 Logarithms as Inverses of Exponentiation 237

Logarithms Base 2 237

Logarithms with Arbitrary Base 238

Change of Base 240

Exercises, Problems, and Worked-out Solutions 242

3.3 Algebraic Properties of Logarithms 247

Logarithm of a Product 247

Logarithm of a Quotient 248

Common Logarithms and the Number of Digits 249

Logarithm of a Power 250

Exercises, Problems, and Worked-out Solutions 251

3.4 Exponential Growth 258

Functions with Exponential Growth 259

Population Growth 261

Compound Interest 263

Exercises, Problems, and Worked-out Solutions 268

3.5 Additional Applications of Exponents and Logarithms 274

Radioactive Decay and Half-Life 274

Earthquakes and the Richter Scale 276

Sound Intensity and Decibels 278

Star Brightness and Apparent Magnitude 279

Exercises, Problems, and Worked-out Solutions 281

Chapter Summary and Chapter Review Questions 287

4 **Area, e, and the Natural Logarithm** 289

4.1 Distance, Length, and Circles 290

Distance between Two Points 290

Midpoints 291

Distance between a Point and a Line 292

Circles 293

Length 295

Exercises, Problems, and Worked-out Solutions 297

4.2 Areas of Simple Regions 303

Squares 303

Rectangles 304

Parallelograms 304

Triangles 304

Trapezoids 305

Stretching 306

Circles 307

Ellipses 310

Exercises, Problems, and Worked-out Solutions 312

4.3 e and the Natural Logarithm 320

Estimating Area Using Rectangles 320

Defining e 322

Defining the Natural Logarithm 325

Properties of the Exponential Function and ln 326

Exercises, Problems, and Worked-out Solutions 328

4.4 Approximations with e and ln 335

Approximations of the Natural Logarithm 335

Inequalities with the Natural Logarithm 336

Approximations with the Exponential Function 337

An Area Formula 338

Exercises, Problems, and Worked-out Solutions 341

4.5 Exponential Growth Revisited 345

Continuously Compounded Interest 345

Continuous Growth Rates 346

Doubling Your Money 347

Exercises, Problems, and Worked-out Solutions 349

Chapter Summary and Chapter Review Questions 354

**5 Trigonometric Functions** 356

5.1 The Unit Circle 357

The Equation of the Unit Circle 357

Angles in the Unit Circle 358

Negative Angles 360

Angles Greater Than 360◦ 361

Length of a Circular Arc 362

Special Points on the Unit Circle 363

Exercises, Problems, and Worked-out Solutions 364

5.2 Radians 370

A Natural Unit of Measurement for Angles 370

Negative Angles 373

Angles Greater Than 2π 374

Length of a Circular Arc 375

Area of a Slice 375

Special Points on the Unit Circle 376

Exercises, Problems, and Worked-out Solutions 377

5.3 Cosine and Sine 382

Definition of Cosine and Sine 382

Cosine and Sine of Special Angles 384

The Signs of Cosine and Sine 385

The Key Equation Connecting Cosine and Sine 387

The Graphs of Cosine and Sine 388

Exercises, Problems, and Worked-out Solutions 390

5.4 More Trigonometric Functions 395

Definition of Tangent 395

Tangent of Special Angles 396

The Sign of Tangent 397

Connections between Cosine, Sine, and Tangent 398

The Graph of Tangent 398

Three More Trigonometric Functions 400

Exercises, Problems, and Worked-out Solutions 401

5.5 Trigonometry in Right Triangles 407

Trigonometric Functions via Right Triangles 407

Two Sides of a Right Triangle 409

One Side and One Angle of a Right Triangle 410

Exercises, Problems, and Worked-out Solutions 410

5.6 Trigonometric Identities 417

The Relationship Between Cosine and Sine 417

Trigonometric Identities for the Negative of an Angle 419

Trigonometric Identities with π2 420

Trigonometric Identities Involving a Multiple of π 422

Exercises, Problems, and Worked-out Solutions 426

5.7 Inverse Trigonometric Functions 432

The Arccosine Function 432

The Arcsine Function 435

The Arctangent Function 437

Exercises, Problems, and Worked-out Solutions 440

5.8 Inverse Trigonometric Identities 443

The Arccosine, Arcsine, and Arctangent of −t:

Graphical Approach 443

The Arccosine, Arcsine, and Arctangent of −t:

Algebraic Approach 445

Arccosine Plus Arcsine 446

The Arctangent of 1t 446

Composition of Trigonometric Functions and Their Inverses 447

More Compositions with Inverse Trigonometric Functions 448

Exercises, Problems, and Worked-out Solutions 451

Chapter Summary and Chapter Review Questions 455

**6 Applications of Trigonometry** 457

6.1 Using Trigonometry to Compute Area 458

The Area of a Triangle via Trigonometry 458

Ambiguous Angles 459

The Area of a Parallelogram via Trigonometry 461

The Area of a Polygon 462

Exercises, Problems, and Worked-out Solutions 463

6.2 The Law of Sines and the Law of Cosines 469

The Law of Sines 469

Using the Law of Sines 470

The Law of Cosines 472

Using the Law of Cosines 473

When to Use Which Law 475

Exercises, Problems, and Worked-out Solutions 476

6.3 Double-Angle and Half-Angle Formulas 483

The Cosine of 2θ 483

The Sine of 2θ 484

The Tangent of 2θ 485

The Cosine and Sine of θ2 485

The Tangent of θ2 488

Exercises, Problems, and Worked-out Solutions 489

6.4 Addition and Subtraction Formulas 497

The Cosine of a Sum and Difference 497

The Sine of a Sum and Difference 499

The Tangent of a Sum and Difference 500

Exercises, Problems, and Worked-out Solutions 501

6.5 Transformations of Trigonometric Functions 507

Amplitude 507

Period 509

Phase Shift 512

Exercises, Problems, and Worked-out Solutions 514

6.6 Polar Coordinates∗ 523

Defining Polar Coordinates 523

Converting from Polar to Rectangular Coordinates 524

Converting from Rectangular to Polar Coordinates 525

Graphs of Polar Equations 529

Exercises, Problems, and Worked-out Solutions 531

6.7 Vectors and the Complex Plane∗ 534

An Algebraic and Geometric Introduction to Vectors 534

The Dot Product 540

The Complex Plane 542

De Moivre’s Theorem 546

Exercises, Problems, and Worked-out Solutions 547

Chapter Summary and Chapter Review Questions 551

**7 Sequences, Series, and Limits** 553

7.1 Sequences 554

Introduction to Sequences 554

Arithmetic Sequences 556

Geometric Sequences 557

Recursive Sequences 559

Exercises, Problems, and Worked-out Solutions 562

7.2 Series 568

Sums of Sequences 568

Arithmetic Series 568

Geometric Series 570

Summation Notation 572

Exercises, Problems, and Worked-out Solutions 573

7.3 Limits 578

Introduction to Limits 578

Infinite Series 582

Decimals as Infinite Series 584

Special Infinite Series 586

Exercises, Problems, and Worked-out Solutions 588

Chapter Summary and Chapter Review Questions 591

Index 592