About the Author v
Preface to the Instructor xiii
Acknowledgments xviii
Preface to the Student xx
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 . . . . . . . . 13
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 . . . . . . . . . . . 31
1 Functions and Their Graphs 32
1.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
• Examples of Functions . . . . . . . . . . . . . . . . . . . . . 33
• Equality of Functions . . . . . . . . . . . . . . . . . . . . . . 34
• The Domain of a Function . . . . . . . . . . . . . . . . . . . 35
• Functions via Tables . . . . . . . . . . . . . . . . . . . . . . 36
• The Range of a Function . . . . . . . . . . . . . . . . . . . . 37
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 39
1.2 The Coordinate Plane and Graphs . . . . . . . . . . . . . . . . . . 44
• The Coordinate Plane. . . . . . . . . . . . . . . . . . . . . . 44
• The Graph of a Function . . . . . . . . . . . . . . . . . . . . 46
vi
Contents vii
• Determining a Function from Its Graph . . . . . . . . . . . . 47
• Which Sets Are Graphs? . . . . . . . . . . . . . . . . . . . . 49
• Determining the Range of a Function from Its Graph. . . . . . 50
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 51
1.3 Function Transformations and Graphs . . . . . . . . . . . . . . . 59
• Shifting a Graph Up or Down . . . . . . . . . . . . . . . . . . 59
• Shifting a Graph Right or Left . . . . . . . . . . . . . . . . . 61
• Stretching a Graph Vertically or Horizontally. . . . . . . . . . 62
• Reflecting a Graph Vertically or Horizontally. . . . . . . . . . 64
• Even and Odd Functions . . . . . . . . . . . . . . . . . . . . 65
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 67
1.4 Composition of Functions . . . . . . . . . . . . . . . . . . . . . . . 77
• Definition of Composition . . . . . . . . . . . . . . . . . . . 77
• Order Matters in Composition . . . . . . . . . . . . . . . . . 78
• The Identity Function. . . . . . . . . . . . . . . . . . . . . . 79
• Decomposing Functions . . . . . . . . . . . . . . . . . . . . 79
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 80
1.5 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
• Examples of Inverse Functions . . . . . . . . . . . . . . . . . 85
• One-to-one Functions. . . . . . . . . . . . . . . . . . . . . . 86
• The Definition of an Inverse Function . . . . . . . . . . . . . 87
• Finding a Formula for an Inverse Function . . . . . . . . . . . 89
• The Domain and Range of an Inverse Function. . . . . . . . . 89
• The Composition of a Function and Its Inverse. . . . . . . . . 90
• Comments about Notation . . . . . . . . . . . . . . . . . . . 92
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 93
1.6 A Graphical Approach to Inverse Functions . . . . . . . . . . . . 99
• The Graph of an Inverse Function . . . . . . . . . . . . . . . 99
• Inverse Functions via Tables . . . . . . . . . . . . . . . . . . 101
• Graphical Interpretation of One-to-One. . . . . . . . . . . . . 101
• Increasing and Decreasing Functions. . . . . . . . . . . . . . 102
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 105
Chapter Summary and Chapter Review Questions . . . . . . . . . . . 109
2 Linear, Quadratic, Polynomial, and Rational Functions 111
2.1 Linear Functions and Lines . . . . . . . . . . . . . . . . . . . . . . 112
• Slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
• The Equation of a Line . . . . . . . . . . . . . . . . . . . . . 113
• Parallel Lines . . . . . . . . . . . . . . . . . . . . . . . . . . 116
• Perpendicular Lines. . . . . . . . . . . . . . . . . . . . . . . 119
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 121
2.2 Quadratic Functions and Parabolas . . . . . . . . . . . . . . . . . 129
• The Vertex of a Parabola . . . . . . . . . . . . . . . . . . . . 129
viii Contents
• Completing the Square . . . . . . . . . . . . . . . . . . . . . 131
• The Quadratic Formula . . . . . . . . . . . . . . . . . . . . . 133
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 135
2.3 Integer Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
• Exponentiation by Positive Integers . . . . . . . . . . . . . . 141
• Properties of Exponentiation . . . . . . . . . . . . . . . . . . 142
• Defining x0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
• Exponentiation by Negative Integers . . . . . . . . . . . . . . 144
• Manipulations with Powers . . . . . . . . . . . . . . . . . . . 145
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 147
2.4 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
• The Degree of a Polynomial. . . . . . . . . . . . . . . . . . . 153
• The Algebra of Polynomials . . . . . . . . . . . . . . . . . . 155
• Zeros and Factorization of Polynomials . . . . . . . . . . . . 156
• The Behavior of a Polynomial Near ±∞. . . . . . . . . . . . . 158
• Graphs of Polynomials . . . . . . . . . . . . . . . . . . . . . 161
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 163
2.5 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
• Ratios of Polynomials. . . . . . . . . . . . . . . . . . . . . . 168
• The Algebra of Rational Functions . . . . . . . . . . . . . . . 169
• Division of Polynomials. . . . . . . . . . . . . . . . . . . . . 170
• The Behavior of a Rational Function Near ±∞ . . . . . . . . . 173
• Graphs of Rational Functions. . . . . . . . . . . . . . . . . . 175
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 176
Chapter Summary and Chapter Review Questions . . . . . . . . . . . 183
3 Exponents and Logarithms 185
3.1 Rational and Real Exponents . . . . . . . . . . . . . . . . . . . . . 186
• Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
• Rational Exponents . . . . . . . . . . . . . . . . . . . . . . . 189
• Real Exponents . . . . . . . . . . . . . . . . . . . . . . . . . 191
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 193
3.2 Logarithms as Inverses of Exponentiation . . . . . . . . . . . . . 199
• Logarithms Base 2 . . . . . . . . . . . . . . . . . . . . . . . 199
• Logarithms with Arbitrary Base. . . . . . . . . . . . . . . . . 200
• Change of Base . . . . . . . . . . . . . . . . . . . . . . . . . 202
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 204
3.3 Algebraic Properties of Logarithms . . . . . . . . . . . . . . . . . 209
• Logarithm of a Product . . . . . . . . . . . . . . . . . . . . . 209
• Logarithm of a Quotient . . . . . . . . . . . . . . . . . . . . 210
• Common Logarithms and the Number of Digits . . . . . . . . 211
• Logarithm of a Power. . . . . . . . . . . . . . . . . . . . . . 212
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 213
Contents ix
3.4 Exponential Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
• Functions with Exponential Growth . . . . . . . . . . . . . . 220
• Population Growth . . . . . . . . . . . . . . . . . . . . . . . 222
• Compound Interest. . . . . . . . . . . . . . . . . . . . . . . 224
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 228
3.5 Additional Applications of Exponents and Logarithms . . . . . 234
• Radioactive Decay and Half-Life . . . . . . . . . . . . . . . . 234
• Earthquakes and the Richter Scale . . . . . . . . . . . . . . . 236
• Sound Intensity and Decibels. . . . . . . . . . . . . . . . . . 238
• Star Brightness and Apparent Magnitude. . . . . . . . . . . . 239
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 241
Chapter Summary and Chapter Review Questions . . . . . . . . . . . 247
4 Area, e, and the Natural Logarithm 249
4.1 Distance, Length, and Circles . . . . . . . . . . . . . . . . . . . . . 250
• Distance between Two Points. . . . . . . . . . . . . . . . . . 250
• Midpoints. . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
• Distance between a Point and a Line . . . . . . . . . . . . . . 253
• Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
• Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 257
4.2 Areas of Simple Regions . . . . . . . . . . . . . . . . . . . . . . . . 263
• Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
• Rectangles . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
• Parallelograms . . . . . . . . . . . . . . . . . . . . . . . . . 264
• Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
• Trapezoids . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
• Stretching. . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
• Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
• Ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 272
4.3 e and the Natural Logarithm . . . . . . . . . . . . . . . . . . . . . . 280
• Estimating Area Using Rectangles . . . . . . . . . . . . . . . 280
• Defining e. . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
• Defining the Natural Logarithm. . . . . . . . . . . . . . . . . 284
• Properties of the Exponential Function and ln . . . . . . . . . 285
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 287
4.4 Approximations with e and ln. . . . . . . . . . . . . . . . . . . . . 294
• Approximations of the Natural Logarithm . . . . . . . . . . . 294
• Inequalities with the Natural Logarithm . . . . . . . . . . . . 295
• Approximations with the Exponential Function . . . . . . . . 296
• An Area Formula . . . . . . . . . . . . . . . . . . . . . . . . 297
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 300
x Contents
4.5 Exponential Growth Revisited . . . . . . . . . . . . . . . . . . . . . 304
• Continuously Compounded Interest . . . . . . . . . . . . . . 304
• Continuous Growth Rates . . . . . . . . . . . . . . . . . . . 305
• Doubling Your Money. . . . . . . . . . . . . . . . . . . . . . 306
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 308
Chapter Summary and Chapter Review Questions . . . . . . . . . . . 313
5 Trigonometric Functions 315
5.1 The Unit Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
• The Equation of the Unit Circle. . . . . . . . . . . . . . . . . 316
• Angles in the Unit Circle . . . . . . . . . . . . . . . . . . . . 317
• Negative Angles. . . . . . . . . . . . . . . . . . . . . . . . . 319
• Angles Greater Than 360◦ . . . . . . . . . . . . . . . . . . . 320
• Length of a Circular Arc . . . . . . . . . . . . . . . . . . . . 321
• Special Points on the Unit Circle . . . . . . . . . . . . . . . . 322
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 323
5.2 Radians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
• A Natural Unit of Measurement for Angles . . . . . . . . . . . 329
• Negative Angles. . . . . . . . . . . . . . . . . . . . . . . . . 332
• Angles Greater Than 2π . . . . . . . . . . . . . . . . . . . . 333
• Length of a Circular Arc . . . . . . . . . . . . . . . . . . . . 334
• Area of a Slice . . . . . . . . . . . . . . . . . . . . . . . . . 334
• Special Points on the Unit Circle . . . . . . . . . . . . . . . . 335
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 336
5.3 Cosine and Sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
• Definition of Cosine and Sine. . . . . . . . . . . . . . . . . . 341
• Cosine and Sine of Special Angles . . . . . . . . . . . . . . . 343
• The Signs of Cosine and Sine . . . . . . . . . . . . . . . . . . 344
• The Key Equation Connecting Cosine and Sine . . . . . . . . . 346
• The Graphs of Cosine and Sine . . . . . . . . . . . . . . . . . 347
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 350
5.4 More Trigonometric Functions . . . . . . . . . . . . . . . . . . . . 355
• Definition of Tangent. . . . . . . . . . . . . . . . . . . . . . 355
• Tangent of Special Angles . . . . . . . . . . . . . . . . . . . 356
• The Sign of Tangent . . . . . . . . . . . . . . . . . . . . . . 357
• Connections between Cosine, Sine, and Tangent . . . . . . . . 358
• The Graph of Tangent . . . . . . . . . . . . . . . . . . . . . 358
• Three More Trigonometric Functions. . . . . . . . . . . . . . 360
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 361
5.5 Trigonometry in Right Triangles . . . . . . . . . . . . . . . . . . . 367
• Trigonometric Functions via Right Triangles . . . . . . . . . . 367
• Two Sides of a Right Triangle. . . . . . . . . . . . . . . . . . 369
• One Side and One Angle of a Right Triangle . . . . . . . . . . 370
Contents xi
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 371
5.6 Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . . 377
• The Relationship Between Cosine and Sine . . . . . . . . . . . 377
• Trigonometric Identities for the Negative of an Angle . . . . . 379
• Trigonometric Identities with π2
. . . . . . . . . . . . . . . . 380
• Trigonometric Identities Involving a Multiple of π . . . . . . . 382
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 386
5.7 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . 392
• The Arccosine Function. . . . . . . . . . . . . . . . . . . . . 392
• The Arcsine Function. . . . . . . . . . . . . . . . . . . . . . 395
• The Arctangent Function . . . . . . . . . . . . . . . . . . . . 397
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 400
5.8 Inverse Trigonometric Identities . . . . . . . . . . . . . . . . . . . 403
• The Arccosine, Arcsine, and Arctangent of −t:
Graphical Approach . . . . . . . . . . . . . . . . . . . . . 403
• The Arccosine, Arcsine, and Arctangent of −t:
Algebraic Approach . . . . . . . . . . . . . . . . . . . . . 405
• Arccosine Plus Arcsine . . . . . . . . . . . . . . . . . . . . . 406
• The Arctangent of 1t
. . . . . . . . . . . . . . . . . . . . . . 406
• Composition of Trigonometric Functions and Their Inverses. . 407
• More Compositions with Inverse Trigonometric Functions . . . 408
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 411
Chapter Summary and Chapter Review Questions . . . . . . . . . . . 415
6 Applications of Trigonometry 417
6.1 Using Trigonometry to Compute Area . . . . . . . . . . . . . . . . 418
• The Area of a Triangle via Trigonometry . . . . . . . . . . . . 418
• Ambiguous Angles . . . . . . . . . . . . . . . . . . . . . . . 419
• The Area of a Parallelogram via Trigonometry . . . . . . . . . 421
• The Area of a Polygon . . . . . . . . . . . . . . . . . . . . . 422
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 423
6.2 The Law of Sines and the Law of Cosines . . . . . . . . . . . . . . 429
• The Law of Sines . . . . . . . . . . . . . . . . . . . . . . . . 429
• Using the Law of Sines . . . . . . . . . . . . . . . . . . . . . 430
• The Law of Cosines . . . . . . . . . . . . . . . . . . . . . . . 432
• Using the Law of Cosines . . . . . . . . . . . . . . . . . . . . 433
• When to Use Which Law . . . . . . . . . . . . . . . . . . . . 435
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 436
6.3 Double-Angle and Half-Angle Formulas . . . . . . . . . . . . . . . 444
• The Cosine of 2θ . . . . . . . . . . . . . . . . . . . . . . . . 444
• The Sine of 2θ . . . . . . . . . . . . . . . . . . . . . . . . . 445
• The Tangent of 2θ . . . . . . . . . . . . . . . . . . . . . . . 446
• The Cosine and Sine of θ2
. . . . . . . . . . . . . . . . . . . . 447
xii Contents
• The Tangent of θ2
. . . . . . . . . . . . . . . . . . . . . . . . 449
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 450
6.4 Addition and Subtraction Formulas . . . . . . . . . . . . . . . . . 458
• The Cosine of a Sum and Difference . . . . . . . . . . . . . . 458
• The Sine of a Sum and Difference. . . . . . . . . . . . . . . . 460
• The Tangent of a Sum and Difference . . . . . . . . . . . . . 461
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 462
6.5 Transformations of Trigonometric Functions . . . . . . . . . . . 468
• Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
• Period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
• Phase Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 475
6.6 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484
• Defining Polar Coordinates . . . . . . . . . . . . . . . . . . . 484
• Converting from Polar to Rectangular Coordinates. . . . . . . 485
• Converting from Rectangular to Polar Coordinates. . . . . . . 486
• Graphs of Polar Equations . . . . . . . . . . . . . . . . . . . 490
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 492
Chapter Summary and Chapter Review Questions . . . . . . . . . . . 495
7 Sequences, Series, and Limits 497
7.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
• Introduction to Sequences . . . . . . . . . . . . . . . . . . . 498
• Arithmetic Sequences. . . . . . . . . . . . . . . . . . . . . . 500
• Geometric Sequences . . . . . . . . . . . . . . . . . . . . . . 501
• Recursive Sequences . . . . . . . . . . . . . . . . . . . . . . 503
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 505
7.2 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512
• Sums of Sequences . . . . . . . . . . . . . . . . . . . . . . . 512
• Arithmetic Series . . . . . . . . . . . . . . . . . . . . . . . . 512
• Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . 514
• Summation Notation . . . . . . . . . . . . . . . . . . . . . . 516
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 517
7.3 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522
• Introduction to Limits . . . . . . . . . . . . . . . . . . . . . 522
• Infinite Series. . . . . . . . . . . . . . . . . . . . . . . . . . 526
• Decimals as Infinite Series . . . . . . . . . . . . . . . . . . . 528
• Special Infinite Series . . . . . . . . . . . . . . . . . . . . . . 530
• Exercises, Problems, and Worked-out Solutions . . . . . . . . 531
Chapter Summary and Chapter Review Questions . . . . . . . . . . . 535
Index of Definitions 536
Index of Boxed Items 538
