Precalculus Preliminary Edition
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More About This Title Precalculus Preliminary Edition
English
For anyone who needs to learn calculus, the best place to start is by gaining a solid foundation in precalculus concepts. This new book provides that foundation. It includes only the topics that they’ll need to succeed in calculus. Axler explores the necessary topics in greater detail. Readers will benefit from the straightforward definitions and examples of complex concepts. Step-by-step solutions for odd-numbered exercises are also included so they can model their own applications of what they’ve learned. In addition, chapter openers and end-of-chapter summaries highlight the material to be learned. Any reader who needs to learn precalculus will benefit from this book.
English
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
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
