Precalculus: A Prelude to Calculus

✍ Scribed by Axler, Sheldon

• Publisher: Wiley
• Year: 2012
• Tongue: English
• Leaves: 673
• Edition: 2
• Category: Library

✦ Synopsis

Sheldon Axler'sPrecalculusfocuses only on topics that students actually need to succeed in calculus. Because of this, Precalculus is a very manageable size even though it includes a student solutions manual. The book is geared towards courses with intermediate algebra prerequisites and it does not assume that students remember any trigonometry. It covers topics such as inverse functions, logarithms, half-life and exponential growth, area, e, the exponential function, the natural logarithm and trigonometry. TheStudent Solutions Manualis integrated at the end of every section. The proximity of the solutions encourages students to go back and read the main text as they are working through the problems and exercises. The inclusion of the manual also saves students money. Axlers Precalculus is available with WileyPLUS, a research-based, online environment for effective teaching and learning.

WileyPLUS sold separately from text.

✦ Table of Contents

Cover......Page 1
Title......Page 5
Copyright......Page 6
About the Author......Page 7
Contents......Page 8
A Book Designed to be Read......Page 17
Exercises and Problems......Page 18
Inverse Functions......Page 19
Logarithms, e, and Exponential Growth......Page 20
What’s New in this Second Edition......Page 21
Comments Welcome......Page 22
WileyPLUS......Page 23
Acknowledgments......Page 24
Reviewers......Page 25
Preface to the Student......Page 26
0: The Real Numbers......Page 27
Construction of the Real Line......Page 28
Is Every Real Number Rational?......Page 29
Problems......Page 32
Commutativity and Associativity......Page 33
The Order of Algebraic Operations......Page 34
The Distributive Property......Page 36
Additive Inverses and Subtraction......Page 37
Multiplicative Inverses and the Algebra of Fractions......Page 38
Symbolic Calculators......Page 41
Exercises......Page 44
Problems......Page 45
Worked-out Solutions To Odd-Numbered Exercises......Page 46
Positive and Negative Numbers......Page 49
Inequalities......Page 50
Intervals......Page 52
Absolute Value......Page 55
Exercises......Page 59
Problems......Page 60
Worked-out Solutions To Odd-Numbered Exercises......Page 61
Summary and Review Questions......Page 66
1: Functions and Their Graphs......Page 67
Definition and Examples......Page 68
The Domain of a Function......Page 72
The Range of a Function......Page 73
Functions via Tables......Page 74
Exercises......Page 75
Problems......Page 76
Worked-out Solutions To Odd-Numbered Exercises......Page 77
The Coordinate Plane......Page 81
The Graph of a Function......Page 83
Determining the Domain and Range from a Graph......Page 85
Which Sets Are Graphs of Functions?......Page 87
Exercises......Page 88
Worked-out Solutions To Odd-Numbered Exercises......Page 89
Vertical Transformations: Shifting, Stretching, and Flipping......Page 95
Horizontal Transformations: Shifting, Stretching, Flipping......Page 98
Combinations of Vertical Function Transformations......Page 102
Even Functions......Page 105
Odd Functions......Page 106
Exercises......Page 107
Problems......Page 108
Worked-out Solutions To Odd-Numbered Exercises......Page 109
Combining Two Functions......Page 116
Definition of Composition......Page 117
Order Matters in Composition......Page 120
Decomposing Functions......Page 121
Composing More than Two Functions......Page 122
Function Transformations as Compositions......Page 123
Exercises......Page 125
Worked-out Solutions To Odd-Numbered Exercises......Page 126
The Inverse Problem......Page 130
One-to-one Functions......Page 131
The Definition of an Inverse Function......Page 132
The Domain and Range of an Inverse Function......Page 134
The Composition of a Function and Its Inverse......Page 135
Comments About Notation......Page 137
Exercises......Page 139
Problems......Page 140
Worked-out Solutions To Odd-Numbered Exercises......Page 141
The Graph of an Inverse Function......Page 145
Graphical Interpretation of One-to-One......Page 147
Increasing and Decreasing Functions......Page 148
Inverse Functions via Tables......Page 150
Exercises......Page 151
Worked-out Solutions To Odd-Numbered Exercises......Page 152
Summary and Review Questions......Page 155
2: Linear, Quadratic, Polynomial, and Rational Functions......Page 159
Slope......Page 160
The Equation of a Line......Page 161
Parallel Lines......Page 165
Perpendicular Lines......Page 166
Exercises......Page 168
Problems......Page 169
Worked-out Solutions To Odd-Numbered Exercises......Page 171
Completing the Square and the Quadratic Formula......Page 176
Parabolas and Quadratic Functions......Page 179
Circles......Page 181
Ellipses......Page 183
Hyperbolas......Page 186
Exercises......Page 188
Problems......Page 191
Worked-out Solutions To Odd-Numbered Exercises......Page 192
Positive Integer Exponents......Page 200
Defining x0......Page 202
Negative Integer Exponents......Page 203
Roots......Page 204
Rational Exponents......Page 208
Properties of Exponents......Page 209
Exercises......Page 210
Problems......Page 211
Worked-out Solutions To Odd-Numbered Exercises......Page 212
The Degree of a Polynomial......Page 219
The Algebra of Polynomials......Page 220
Zeros and Factorization of Polynomials......Page 222
The Behavior of a Polynomial Near ± infinity......Page 225
Exercises......Page 228
Problems......Page 229
Worked-out Solutions To Odd-Numbered Exercises......Page 231
The Algebra of Rational Functions......Page 234
Division of Polynomials......Page 235
The Behavior of a Rational Function Near ± infinity......Page 238
Graphs of Rational Functions......Page 241
Exercises......Page 242
Problems......Page 243
Worked-out Solutions To Odd-Numbered Exercises......Page 244
Summary and Review Questions......Page 249
3: Exponential Functions, Logarithms, and e......Page 251
Exponential Functions......Page 252
Logarithms Base 2......Page 254
Logarithms with Any Base......Page 255
Common Logarithms and the Number of Digits......Page 256
Exercises......Page 257
Problems......Page 258
Worked-out Solutions To Odd-Numbered Exercises......Page 259
Logarithm of a Power......Page 263
Radioactive Decay and Half-Life......Page 264
Change of Base......Page 266
Exercises......Page 268
Worked-out Solutions To Odd-Numbered Exercises......Page 269
Logarithm of a Product......Page 273
Logarithm of a Quotient......Page 274
Earthquakes and the Richter Scale......Page 275
Sound Intensity and Decibels......Page 276
Star Brightness and Apparent Magnitude......Page 277
Exercises......Page 279
Problems......Page 280
Worked-out Solutions To Odd-Numbered Exercises......Page 281
3.4 Exponential Growth......Page 286
Functions with Exponential Growth......Page 287
Population Growth......Page 291
Compound Interest......Page 293
Exercises......Page 297
Problems......Page 299
Worked-out Solutions To Odd-Numbered Exercises......Page 300
Estimating Area Using Rectangles......Page 304
Defining e......Page 306
Defining the Natural Logarithm......Page 309
Properties of the Exponential Function and ln......Page 310
Exercises......Page 312
Problems......Page 313
Worked-out Solutions To Odd-Numbered Exercises......Page 314
Approximation of the Natural Logarithm......Page 318
Approximations with the Exponential Function......Page 320
An Area Formula......Page 322
Exercises......Page 324
Problems......Page 325
Worked-out Solutions To Odd-Numbered Exercises......Page 326
Continuously Compounded Interest......Page 328
Continuous Growth Rates......Page 329
Doubling Your Money......Page 330
Exercises......Page 332
Problems......Page 333
Worked-out Solutions To Odd-Numbered Exercises......Page 334
Summary and Review Questions......Page 337
4: Trigonometric Functions......Page 339
The Equation of the Unit Circle......Page 340
Angles in the Unit Circle......Page 341
Negative Angles......Page 343
Angles Greater than 360°......Page 344
Length of a Circular Arc......Page 345
Special Points on the Unit Circle......Page 346
Exercises......Page 348
Worked-out Solutions To Odd-Numbered Exercises......Page 349
A Natural Unit of Measurement for Angles......Page 355
The Radius Corresponding to an Angle......Page 358
Length of a Circular Arc......Page 361
Area of a Slice......Page 362
Special Points on the Unit Circle......Page 363
Problems......Page 364
Worked-out Solutions To Odd-Numbered Exercises......Page 365
Definition of Cosine and Sine......Page 369
The Signs of Cosine and Sine......Page 372
The Key Equation Connecting Cosine and Sine......Page 374
The Graphs of Cosine and Sine......Page 375
Problems......Page 377
Worked-out Solutions To Odd-Numbered Exercises......Page 378
Definition of Tangent......Page 381
The Sign of Tangent......Page 383
Connections Among Cosine, Sine, and Tangent......Page 384
The Graph of Tangent......Page 385
Three More Trigonometric Functions......Page 386
Problems......Page 388
Worked-out Solutions To Odd-Numbered Exercises......Page 389
Trigonometric Functions via Right Triangles......Page 393
Two Sides of a Right Triangle......Page 395
One Side and One Angle of a Right Triangle......Page 396
Exercises......Page 397
Worked-out Solutions To Odd-Numbered Exercises......Page 399
The Relationship Among Cosine, Sine, and Tangent......Page 403
Trigonometric Identities for the Negative of an Angle......Page 405
Trigonometric Identities with π/2......Page 407
Trigonometric Identities Involving a Multiple of π......Page 408
Exercises......Page 411
Problems......Page 412
Worked-out Solutions To Odd-Numbered Exercises......Page 413
Summary and Review Questions......Page 416
5: Trigonometric Algebra and Geometry......Page 419
The Arccosine Function......Page 420
The Arcsine Function......Page 423
The Arctangent Function......Page 426
Exercises......Page 429
Problems......Page 430
Worked-out Solutions To Odd-Numbered Exercises......Page 431
Composition of Trigonometric Functions and Their Inverses......Page 434
More Compositions with Inverse Trigonometric Functions......Page 435
The Arccosine, Arcsine, and Arctangent of –t......Page 437
Arccosine Plus Arcsine......Page 438
Problems......Page 439
Worked-out Solutions To Odd-Numbered Exercises......Page 440
The Area of a Triangle via Trigonometry......Page 443
Ambiguous Angles......Page 444
The Area of a Parallelogram via Trigonometry......Page 445
The Area of a Polygon......Page 446
Trigonometric Approximations......Page 449
Exercises......Page 452
Problems......Page 453
Worked-out Solutions To Odd-Numbered Exercises......Page 454
The Law of Sines......Page 457
Using the Law of Sines......Page 458
The Law of Cosines......Page 460
Using the Law of Cosines......Page 461
When to Use Which Law......Page 463
Exercises......Page 465
Problems......Page 466
Worked-out Solutions To Odd-Numbered Exercises......Page 468
The Cosine of 2θ......Page 474
The Sine of 2θ......Page 475
The Tangent of 2θ......Page 476
The Cosine and Sine of θ/2......Page 477
The Tangent of θ/2......Page 479
Exercises......Page 480
Problems......Page 481
Worked-out Solutions To Odd-Numbered Exercises......Page 483
The Cosine of a Sum and Difference......Page 488
The Sine of a Sum and Difference......Page 490
The Tangent of a Sum and Difference......Page 491
Products of Trigonometric Functions......Page 492
Problems......Page 493
Worked-out Solutions To Odd-Numbered Exercises......Page 494
Summary and Review Questions......Page 498
6: Applications of Trigonometry......Page 501
Amplitude......Page 502
Period......Page 504
Phase Shift......Page 507
Fitting Transformations of Trigonometric Functions to Data......Page 509
Exercises......Page 511
Problems......Page 512
Worked-out Solutions To Odd-Numbered Exercises......Page 514
Defining Polar Coordinates......Page 519
Converting from Rectangular to Polar Coordinates......Page 520
Graphs of Polar Equations......Page 523
Exercises......Page 527
Worked-out Solutions To Odd-Numbered Exercises......Page 528
An Algebraic and Geometric Introduction to Vectors......Page 530
Vector Addition......Page 532
Vector Subtraction......Page 535
Scalar Multiplication......Page 537
The Dot Product......Page 538
Exercises......Page 540
Problems......Page 541
Worked-out Solutions To Odd-Numbered Exercises......Page 542
The Complex Number System......Page 544
Arithmetic with Complex Numbers......Page 545
Complex Conjugates and Division of Complex Numbers......Page 546
Zeros and Factorization of Polynomials, Revisited......Page 549
Exercises......Page 552
Problems......Page 553
Worked-out Solutions To Odd-Numbered Exercises......Page 554
Complex Numbers as Points in the Plane......Page 557
Geometric Interpretation of Complex Multiplication and Division......Page 559
De Moivre’s Theorem......Page 562
Finding Complex Roots......Page 563
Problems......Page 564
Worked-out Solutions To Odd-Numbered Exercises......Page 565
Summary and Review Questions......Page 566
7: Sequences, Series, and Limits......Page 567
Introduction to Sequences......Page 568
Arithmetic Sequences......Page 570
Geometric Sequences......Page 571
Recursively Defined Sequences......Page 574
Exercises......Page 577
Problems......Page 578
Worked-out Solutions To Odd-Numbered Exercises......Page 579
Arithmetic Series......Page 583
Geometric Series......Page 585
Summation Notation......Page 587
Pascal’s Triangle......Page 590
The Binomial Theorem......Page 593
Exercises......Page 596
Problems......Page 597
Worked-out Solutions To Odd-Numbered Exercises......Page 598
Introduction to Limits......Page 602
Infinite Series......Page 605
Decimals as Infinite Series......Page 608
Special Infinite Series......Page 610
Exercises......Page 612
Worked-out Solutions To Odd-Numbered Exercises......Page 613
Summary and Review Questions......Page 615
8: Systems of Linear Equations......Page 617
How Many Solutions?......Page 618
Linear Equations......Page 620
Gaussian Elimination......Page 622
Exercises......Page 624
Worked-out Solutions To Odd-Numbered Exercises......Page 626
Representing Systems of Linear Equations by Matrices......Page 630
Gaussian Elimination with Matrices......Page 632
Systems of Linear Equations with No Solutions......Page 634
Systems of Linear Equations with Infinitely Many Solutions......Page 636
How Many Solutions?, Revisited......Page 637
Exercises......Page 638
Worked-out Solutions To Odd-Numbered Exercises......Page 639
Summary and Review Questions......Page 642
Circumference......Page 643
Squares, Rectangles, and Parallelograms......Page 644
Triangles and Trapezoids......Page 646
Stretching......Page 647
Circles and Ellipses......Page 648
Exercises......Page 651
Problems......Page 653
Curves in the Coordinate Plane......Page 655
Graphing Inverse Functions as Parametric Curves......Page 660
Shifting, Stretching, or Flipping a Parametric Curve......Page 661
Exercises......Page 664
Problems......Page 665
Photo Credits......Page 666
C......Page 667
D......Page 668
G......Page 669
L......Page 670
Q......Page 671
S......Page 672
Z......Page 673