Precalculus. A prelude to calculus

Cover......Page 1
Title Page......Page 5
Copyright......Page 6
About the Author......Page 7
Contents......Page 9
Preface to the Instructor......Page 17
Acknowledgments......Page 23
Preface to the Student......Page 25
Chapter 0 The Real Numbers......Page 27
  Construction of the Real Line......Page 28
  Is Every Real Number Rational?......Page 29
  Problems......Page 31
  Commutativity and Associativity......Page 32
  The Order of Algebraic Operations......Page 33
  The Distributive Property......Page 34
  Additive Inverses and Subtraction......Page 35
  Multiplicative Inverses and the Algebra of Fractions......Page 36
  Symbolic Calculators......Page 39
  Exercises, Problems, and Worked-out Solutions......Page 41
  Positive and Negative Numbers......Page 46
  Inequalities......Page 47
  Intervals......Page 49
  Absolute Value......Page 51
  Exercises, Problems, and Worked-out Solutions......Page 55
 Chapter Summary and Chapter Review Questions......Page 61
Chapter 1 Functions and Their Graphs......Page 63
  Definition and Examples......Page 64
  The Domain of a Function......Page 67
  The Range of a Function......Page 68
  Functions via Tables......Page 70
  Exercises, Problems, and Worked-out Solutions......Page 71
  The Coordinate Plane......Page 76
  The Graph of a Function......Page 78
  Determining the Domain and Range from a Graph......Page 80
  Exercises, Problems, and Worked-out Solutions......Page 82
  Vertical Transformations: Shifting, Stretching, and Flipping......Page 89
  Horizontal Transformations: Shifting, Stretching, Flipping......Page 92
  Combinations of Vertical Function Transformations......Page 94
  Even Functions......Page 97
  Odd Functions......Page 98
  Exercises, Problems, and Worked-out Solutions......Page 99
  Combining Two Functions......Page 107
  Definition of Composition......Page 108
  Composing More than Two Functions......Page 111
  Function Transformations as Compositions......Page 112
  Exercises, Problems, and Worked-out Solutions......Page 114
  The Inverse Problem......Page 119
  One-to-one Functions......Page 120
  The Definition of an Inverse Function......Page 121
  The Domain and Range of an Inverse Function......Page 123
  The Composition of a Function and Its Inverse......Page 124
  Comments About Notation......Page 125
  Exercises, Problems, and Worked-out Solutions......Page 127
  The Graph of an Inverse Function......Page 132
  Graphical Interpretation of One-to-One......Page 133
  Increasing and Decreasing Functions......Page 134
  Inverse Functions via Tables......Page 136
  Exercises, Problems, and Worked-out Solutions......Page 137
 Chapter Summary and Chapter Review Questions......Page 141
Chapter 2 Linear, Quadratic, Polynomial, and Rational Functions......Page 145
  Slope......Page 146
  The Equation of a Line......Page 147
  Parallel Lines......Page 151
  Perpendicular Lines......Page 152
  Exercises, Problems, and Worked-out Solutions......Page 154
  Completing the Square and the Quadratic Formula......Page 161
  Parabolas and Quadratic Functions......Page 164
  Circles......Page 166
  Ellipses......Page 168
  Hyperbolas......Page 170
  Exercises, Problems, and Worked-out Solutions......Page 172
  Positive Integer Exponents......Page 183
  Defining x°......Page 185
  Negative Integer Exponents......Page 186
  Roots......Page 187
  Rational Exponents......Page 190
  Properties of Exponents......Page 191
  Exercises, Problems, and Worked-out Solutions......Page 192
  The Degree of a Polynomial......Page 200
  The Algebra of Polynomials......Page 201
  Zeros and Factorization of Polynomials......Page 203
  The Behavior of a Polynomial Near ±∞......Page 205
  Graphs of Polynomials......Page 207
  Exercises, Problems, and Worked-out Solutions......Page 208
  The Algebra of Rational Functions......Page 213
  Division of Polynomials......Page 214
  The Behavior of a Rational Function Near ±∞......Page 217
  Graphs of Rational Functions......Page 220
  Exercises, Problems, and Worked-out Solutions......Page 221
 Chapter Summary and Chapter Review Questions......Page 227
Chapter 3 Exponential Functions, Logarithms, and e......Page 229
  Exponential Functions......Page 230
  Logarithms Base 2......Page 232
  Logarithms with Any Base......Page 233
  Common Logarithms and the Number of Digits......Page 234
  Exercises, Problems, and Worked-out Solutions......Page 235
  Logarithm of a Power......Page 240
  Radioactive Decay and Half-Life......Page 241
  Change of Base......Page 243
  Exercises, Problems, and Worked-out Solutions......Page 245
  Logarithm of a Product......Page 249
  Logarithm of a Quotient......Page 250
  Earthquakes and the Richter Scale......Page 251
  Sound Intensity and Decibels......Page 252
  Star Brightness and Apparent Magnitude......Page 253
  Exercises, Problems, and Worked-out Solutions......Page 254
 3.4 Exponential Growth......Page 261
  Functions with Exponential Growth......Page 262
  Population Growth......Page 265
  Compound Interest......Page 267
  Exercises, Problems, and Worked-out Solutions......Page 271
  Estimating Area Using Rectangles......Page 276
  Defining e......Page 278
  Defining the Natural Logarithm......Page 280
  Properties of the Exponential Function and Natural Logarithm......Page 281
  Exercises, Problems, and Worked-out Solutions......Page 282
  Approximation of the Natural Logarithm......Page 288
  Approximations with the Exponential Function......Page 289
  An Area Formula......Page 291
  Exercises, Problems, and Worked-out Solutions......Page 293
  Continuously Compounded Interest......Page 296
  Continuous Growth Rates......Page 297
  Doubling Your Money......Page 298
  Exercises, Problems, and Worked-out Solutions......Page 300
 Chapter Summary and Chapter Review Questions......Page 304
Chapter 4 Trigonometric Functions......Page 307
  The Equation of the Unit Circle......Page 308
  Angles in the Unit Circle......Page 309
  Negative Angles......Page 310
  Angles Greater than 360°......Page 312
  Special Points on the Unit Circle......Page 313
  Exercises, Problems, and Worked-out Solutions......Page 315
  A Natural Unit of Measurement for Angles......Page 321
  The Radius Corresponding to an Angle......Page 324
  Length of a Circular Arc......Page 326
  Special Points on the Unit Circle......Page 327
  Exercises, Problems, and Worked-out Solutions......Page 328
  Definition of Cosine and Sine......Page 333
  The Signs of Cosine and Sine......Page 335
  The Key Equation Connecting Cosine and Sine......Page 336
  The Graphs of Cosine and Sine......Page 337
  Exercises, Problems, and Worked-out Solutions......Page 339
  Definition of Tangent......Page 343
  The Sign of Tangent......Page 344
  Connections Among Cosine, Sine, and Tangent......Page 345
  The Graph of Tangent......Page 346
  Three More Trigonometric Functions......Page 347
  Exercises, Problems, and Worked-out Solutions......Page 348
  Trigonometric Functions via Right Triangles......Page 353
  Two Sides of a Right Triangle......Page 354
  One Side and One Angle of a Right Triangle......Page 355
  Exercises, Problems, and Worked-out Solutions......Page 357
  The Relationship Among Cosine, Sine, and Tangent......Page 362
  Trigonometric Identities for the Negative of an Angle......Page 364
  Trigonometric Identities with π/2......Page 365
  Trigonometric Identities Involving a Multiple of π......Page 367
  Exercises, Problems, and Worked-out Solutions......Page 369
 Chapter Summary and Chapter Review Questions......Page 374
Chapter 5 Trigonometric Algebra and Geometry......Page 377
  The Arccosine Function......Page 378
  The Arcsine Function......Page 380
  The Arctangent Function......Page 383
  Exercises, Problems, and Worked-out Solutions......Page 385
  Composition of Trigonometric Functions and Their Inverses......Page 391
  More Inverse Functions......Page 392
  More Compositions with Inverse Trigonometric Functions......Page 393
  The Arccosine, Arcsine, and Arctangent of -t......Page 395
  Arccosine Plus Arcsine......Page 396
  Exercises, Problems, and Worked-out Solutions......Page 397
  The Area of a Triangle via Trigonometry......Page 401
  Ambiguous Angles......Page 402
  The Area of a Parallelogram via Trigonometry......Page 403
  The Area of a Polygon......Page 404
  Trigonometric Approximations......Page 406
  Exercises, Problems, and Worked-out Solutions......Page 409
  The Law of Sines......Page 414
  The Law of Cosines......Page 416
  When to Use Which Law......Page 419
  Exercises, Problems, and Worked-out Solutions......Page 421
  The Cosine of 2θ......Page 428
  The Sine of 2θ......Page 429
  The Cosine and Sine of θ/2......Page 430
  The Tangent of θ/2......Page 432
  Exercises, Problems, and Worked-out Solutions......Page 433
  The Cosine of a Sum and Difference......Page 440
  The Sine of a Sum and Difference......Page 442
  The Tangent of a Sum and Difference......Page 443
  Exercises, Problems, and Worked-out Solutions......Page 444
  Amplitude......Page 449
  Period......Page 451
  Phase Shift......Page 452
  Fitting Transformations of Trigonometric Functions to Data......Page 455
  Exercises, Problems, and Worked-out Solutions......Page 456
 Chapter Summary and Chapter Review Questions......Page 463
Chapter 6 Sequences, Series, and Limits......Page 465
  Introduction to Sequences......Page 466
  Arithmetic Sequences......Page 468
  Geometric Sequences......Page 469
  Recursively Defined Sequences......Page 471
  Exercises, Problems, and Worked-out Solutions......Page 474
  Arithmetic Series......Page 479
  Geometric Series......Page 481
  Summation Notation......Page 483
  Pascal’s Triangle......Page 485
  The Binomial Theorem......Page 488
  Exercises, Problems, and Worked-out Solutions......Page 491
  Introduction to Limits......Page 496
  Infinite Series......Page 499
  Decimals as Infinite Series......Page 502
  Special Infinite Series......Page 503
  Exercises, Problems, and Worked-out Solutions......Page 505
 Chapter Summary and Chapter Review Questions......Page 508
Chapter 7 Polar Coordinates, Vectors, and Complex Numbers......Page 509
  Defining Polar Coordinates......Page 510
  Converting from Rectangular to Polar Coordinates......Page 511
  Graphs of Polar Equations......Page 514
  Exercises, Problems, and Worked-out Solutions......Page 517
  An Algebraic and Geometric Introduction to Vectors......Page 520
  Vector Addition......Page 522
  Vector Subtraction......Page 524
  The Dot Product......Page 526
  Exercises, Problems, and Worked-out Solutions......Page 529
  The Complex Number System......Page 532
  Arithmetic with Complex Numbers......Page 533
  Complex Conjugates and Division of Complex Numbers......Page 534
  Zeros and Factorization of Polynomials, Revisited......Page 537
  Exercises, Problems, and Worked-out Solutions......Page 540
  Complex Numbers as Points in the Plane......Page 544
  Geometric Interpretation of Complex Multiplication and Division......Page 545
  De Moivre’s Theorem......Page 548
  Finding Complex Roots......Page 549
  Exercises, Problems, and Worked-out Solutions......Page 550
 Chapter Summary and Chapter Review Questions......Page 552
 Circumference......Page 553
 Squares, Rectangles, and Parallelograms......Page 554
 Triangles and Trapezoids......Page 555
 Circles and Ellipses......Page 557
 Exercises, Problems, and Worked-out Solutions......Page 560
Photo Credits......Page 569
Index......Page 571
Colophon: Notes on Typesetting......Page 577
EULA......Page 578