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Precalculus: A Prelude to Calculus

Sheldon Axler
Publisher: 
John Wiley
Publication Date: 
2008
Number of Pages: 
599
Format: 
Hardcover
Price: 
96.95
ISBN: 
9780470416747
Category: 
Textbook
[Reviewed by
Allen Stenger
, on
04/22/2009
]

This is a competent but not very innovative precalculus text. It has a fairly conventional coverage of high school algebra, functions, and trigonometry, and an unusual amount on area.

It does have some unusual approaches to particular topics. These are often improvements over the usual approach, but they don't reach very far. The most interesting innovation is to express exponential decay in terms of powers of 2 instead of powers of e, because that makes the half-life very obvious. One reason for the emphasis on area is to define the constant e in terms of an area under the curve y = 1/x; I wasn't convinced that this was an improvement over more traditional approaches, although it does give a glimpse into what's coming in calculus.

The book includes complete solutions for all the odd-numbered exercises, and each even-numbered exercise is constructed to use the same techniques as the immediately-preceding odd-numbered exercise. The solutions are well-written and easy to follow.

The existence of precalculus texts and courses raises the question: Is precalculus a real subject? The present book answers in the negative, saying in the Preface, "This book seeks to prepare students to succeed in calculus". The book takes this to its logical conclusion by omitting those portions of algebra and trigonometry that are not useful in calculus.

I think on the whole this streamlined approach is unsuccessful. It does not really cut out that much, and by cutting it gives the impression that no part of precalculus is interesting in itself: it's just something you have to suffer through so you can enjoy the good stuff later.

In particular motivations are weak. For example, on p. 147 we launch into an investigation of how to define exponentiation by positive integers, which is explained clearly but is explained without any hints about whether this operation, if we could figure out how to do it, would be interesting or useful. This approach is used throughout the book and could be thought of as an axiomatic approach to precalculus: define everything first, ensure that everything is consistent, but don't worry about where the subject came from or where it's going.


See also our review of the preliminary edition of this book.


Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.

 

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 2π 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 π2  420
            Trigonometric Identities Involving a Multiple of π 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 2θ 483
            The Sine of 2θ 484
            The Tangent of 2θ 485
            The Cosine and Sine of θ2  485
            The Tangent of θ2  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 Moivre’s 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

 

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