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 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 420
Trigonometric Identities Involving a Multiple of 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 483
The Sine of 484
The Tangent of 485
The Cosine and Sine of 485
The Tangent of 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 Moivre갽s 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