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1

PrecalculusA Prelude to Calculus

withStudent Solutions Manual

Sheldon AxlerSan Francisco State University

JOHN WILEY & SONS, INC.

Publisher Laurie RosatoneAcquisitions Editor Jessica JacobsSenior Editorial Assistant Jeff BensonDevelopment Editor Anne Scanlan-RohrerSenior Production Editor Ken SantorPhoto Editor Elle WagnerMarketing Manager Jaclyn ElkinsSenior Marketing Assistant Chelsee PengalDesign Director Madelyn LesureMedia Editor Melissa Edwards

This book was typeset in LATEX by the author. Printing and binding by RR Donnelley,Jefferson City. Cover printed by Phoenix Color Corporation.

The diagram on the cover contains the crucial definitions of trigonometry.About the Cover

The 1 shows that the trigonometric functions are defined in the context ofthe unit circle. The arrow shows that angles are measured counterclockwisefrom the positive horizontal axis. The point labeled (cosθ, sinθ) shows thatcosθ is the first coordinate of the endpoint of the radius corresponding tothe angle θ, and sinθ is the second coordinate of this endpoint. Because thisendpoint is on the unit circle, the identity cos2θ + sin2θ = 1 immediatelyfollows. The equation slope = tanθ shows that tanθ is the slope of theradius corresponding to the angle θ; thus tanθ = sinθ

cosθ .

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10 9 8 7 6 5 4 3 2 1

About the Author

Sheldon Axler is Dean ofthe College of Science & En-gineering at San FranciscoState University, where he

joined the faculty as Chair of the Mathematics Department in 1997.Axler was valedictorian of his high school in Miami, Florida. He received

his AB from Princeton University with highest honors, followed by a PhD inMathematics from the University of California at Berkeley.

As a postdoctoral Moore Instructor at MIT, Axler received a university-wideteaching award. Axler was then an assistant professor, associate professor,and professor at Michigan State University, where he received the first J.Sutherland Frame Teaching Award and the Distinguished Faculty Award.

Axler received the Lester R. Ford Award for expository writing from theMathematical Association of America in 1996. In addition to publishing nu-merous research papers, Axler is the author of Linear Algebra Done Right(which has been adopted as a textbook at over 225 universities and col-leges) and co-author of Harmonic Function Theory (a graduate/research-levelbook).

Axler has served as Editor-in-Chief of the Mathematical Intelligencer andas Associate Editor of the American Mathematical Monthly. He has been amember of the Council of the American Mathematical Society and a memberof the Board of Trustees of the Mathematical Sciences Research Institute.Axler currently serves on the editorial board of Springer’s series Undergrad-uate Texts in Mathematics, Graduate Texts in Mathematics, and Universitext.

v

Contents

About the Author v

Preface to the Instructor xv

Acknowledgments xx

Preface to the Student xxiii

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


Chapter Summary and Chapter Review Questions 32

vi

Contents vii

1 Functions and Their Graphs 33

1.1 Functions 34

Definition and Examples 34

Equality of Functions 36

The Domain of a Function 37

Functions via Tables 38

The Range of a Function 38


1.2 The Coordinate Plane and Graphs 47

The Coordinate Plane 47

The Graph of a Function 49

Determining a Function from Its Graph 51

Which Sets Are Graphs of Functions? 53

Determining the Range of a Function from Its Graph 53


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 in the Horizontal or Vertical Axis 67

Even and Odd Functions 69


1.4 Composition of Functions 81

Definition of Composition 81

Order Matters in Composition 82

Decomposing Functions 83

Composing More than Two Functions 84


1.5 Inverse Functions 90

The Inverse Problem 90

One-to-one Functions 91

The Definition of an Inverse Function 92

The Domain and Range of an Inverse Function 94

The Composition of a Function and Its Inverse 95

Comments about Notation 97


viii Contents

1.6 A Graphical Approach to Inverse Functions 104

The Graph of an Inverse Function 104

Inverse Functions via Tables 106

Graphical Interpretation of One-to-One 106

Increasing and Decreasing Functions 108



2 Linear, Quadratic, Polynomial, and Rational Functions 117

2.1 Linear Functions and Lines 118

Slope 118

The Equation of a Line 119

Parallel Lines 122

Perpendicular Lines 124


2.2 Quadratic Functions and Parabolas 134

The Vertex of a Parabola 134

Completing the Square 136

The Quadratic Formula 139


2.3 Integer Exponents 147

Exponentiation by Positive Integers 147

Properties of Exponentiation 148

Defining x0 149

Exponentiation by Negative Integers 150

Manipulations with Powers 151


2.4 Polynomials 159

The Degree of a Polynomial 159

The Algebra of Polynomials 160

Zeros and Factorization of Polynomials 162

The Behavior of a Polynomial Near ±∞ 165

Graphs of Polynomials 168


Contents ix

2.5 Rational Functions 175

Ratios of Polynomials 175

The Algebra of Rational Functions 176

Division of Polynomials 177

The Behavior of a Rational Function Near ±∞ 180

Graphs of Rational Functions 183


2.6 Complex Numbers∗ 190

The Complex Number System 190

Arithmetic with Complex Numbers 191

Complex Conjugates and Division of Complex Numbers 192

Zeros and Factorization of Polynomials, Revisited 195


2.7 Systems of Equations and Matrices∗ 203

Solving a System of Equations 203

Systems of Linear Equations 205

Matrices and Linear Equations 209



3 Exponents and Logarithms 224

3.1 Rational and Real Exponents 225

Roots 225

Rational Exponents 228

Real Exponents 230


3.2 Logarithms as Inverses of Exponentiation 238

Logarithms Base 2 238

Logarithms with Arbitrary Base 239

Change of Base 241


3.3 Algebraic Properties of Logarithms 248

Logarithm of a Product 248

Logarithm of a Quotient 249

*Section can be skipped if focusing only on material needed for first-semester calculus.

x Contents

Common Logarithms and the Number of Digits 250

Logarithm of a Power 251


3.4 Exponential Growth 259

Functions with Exponential Growth 260

Population Growth 262

Compound Interest 264


3.5 Additional Applications of Exponents and Logarithms 275

Radioactive Decay and Half-Life 275

Earthquakes and the Richter Scale 277

Sound Intensity and Decibels 279

Star Brightness and Apparent Magnitude 280



4 Area, e, and the Natural Logarithm 290

4.1 Distance, Length, and Circles 291

Distance between Two Points 291

Midpoints 292

Distance between a Point and a Line 293

Circles 294

Length 296


4.2 Areas of Simple Regions 304

Squares 304

Rectangles 305

Parallelograms 305

Triangles 305

Trapezoids 306

Stretching 307

Circles 308

Ellipses 310


4.3 e and the Natural Logarithm 321

Estimating Area Using Rectangles 321

Contents xi

Defining e 323

Defining the Natural Logarithm 326

Properties of the Exponential Function and ln 327


4.4 Approximations with e and ln 336

Approximation of the Natural Logarithm 336

Inequalities with the Natural Logarithm 337

Approximations with the Exponential Function 338

An Area Formula 339


4.5 Exponential Growth Revisited 346

Continuously Compounded Interest 346

Continuous Growth Rates 347

Doubling Your Money 348



5 Trigonometric Functions 357

5.1 The Unit Circle 358

The Equation of the Unit Circle 358

Angles in the Unit Circle 359

Negative Angles 361

Angles Greater Than 360◦ 362

Length of a Circular Arc 363

Special Points on the Unit Circle 364


5.2 Radians 371

A Natural Unit of Measurement for Angles 371

Negative Angles 374

Angles Greater Than 2π 375

Length of a Circular Arc 376

Area of a Slice 376

Special Points on the Unit Circle 377


5.3 Cosine and Sine 383

Definition of Cosine and Sine 383

xii Contents

Cosine and Sine of Special Angles 385

The Signs of Cosine and Sine 386

The Key Equation Connecting Cosine and Sine 388

The Graphs of Cosine and Sine 389


5.4 More Trigonometric Functions 396

Definition of Tangent 396

Tangent of Special Angles 397

The Sign of Tangent 398

Connections between Cosine, Sine, and Tangent 399

The Graph of Tangent 399

Three More Trigonometric Functions 401


5.5 Trigonometry in Right Triangles 408

Trigonometric Functions via Right Triangles 408

Two Sides of a Right Triangle 410

One Side and One Angle of a Right Triangle 411


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


5.7 Inverse Trigonometric Functions 431

The Arccosine Function 431

The Arcsine Function 434

The Arctangent Function 436


5.8 Inverse Trigonometric Identities 442

The Arccosine, Arcsine, and Arctangent of −t:Graphical Approach 442

The Arccosine, Arcsine, and Arctangent of −t:Algebraic Approach 444

Arccosine Plus Arcsine 445

The Arctangent of 1t 445

Composition of Trigonometric Functions and Their Inverses 446

Contents xiii

More Compositions with Inverse Trigonometric Functions 447



6 Applications of Trigonometry 456

6.1 Using Trigonometry to Compute Area 457

The Area of a Triangle via Trigonometry 457

Ambiguous Angles 458

The Area of a Parallelogram via Trigonometry 460

The Area of a Polygon 461


6.2 The Law of Sines and the Law of Cosines 468

The Law of Sines 468

Using the Law of Sines 469

The Law of Cosines 471

Using the Law of Cosines 472

When to Use Which Law 474


6.3 Double-Angle and Half-Angle Formulas 482

The Cosine of 2θ 482

The Sine of 2θ 483

The Tangent of 2θ 484

The Cosine and Sine of θ2 485

The Tangent of θ2 487


6.4 Addition and Subtraction Formulas 496

The Cosine of a Sum and Difference 496

The Sine of a Sum and Difference 498

The Tangent of a Sum and Difference 499


6.5 Transformations of Trigonometric Functions 506

Amplitude 506

Period 508

Phase Shift 511


xiv Contents

6.6 Polar Coordinates∗ 522

Defining Polar Coordinates 522

Converting from Polar to Rectangular Coordinates 523

Converting from Rectangular to Polar Coordinates 524

Graphs of Polar Equations 528


6.7 Vectors and the Complex Plane∗ 533

An Algebraic and Geometric Introduction to Vectors 533

The Dot Product 539

The Complex Plane 541

De Moivre’s Theorem 545



7 Sequences, Series, and Limits 552

7.1 Sequences 553

Introduction to Sequences 553

Arithmetic Sequences 555

Geometric Sequences 556

Recursive Sequences 558


7.2 Series 567

Sums of Sequences 567

Arithmetic Series 567

Geometric Series 569

Summation Notation 571


7.3 Limits 577

Introduction to Limits 577

Infinite Series 581

Decimals as Infinite Series 583

Special Infinite Series 585



Index 593

*Section can be skipped if focusing only on material needed for first-semester calculus.

Preface to the Instructor

Goals and Prerequisites

This book seeks to prepare students to succeed in calculus. Thus this book focuseson topics that students need for calculus, especially first-semester calculus. Manyimportant subjects that should be known by all educated citizens but that are irrel-evant to calculus have been excluded.

Precalculus is a one-semester course at most universities. Nevertheless, typicalprecalculus textbooks contain about a thousand pages (not counting a student solu-tions manual), far more than can be covered in one semester. By emphasizing topicscrucial to success in calculus, this book has a more manageable size even though itincludes a student solutions manual. A thinner textbook should indicate to studentsthat they are truly expected to master most of the content of the book.

The prerequisite for this course is the usual course in intermediate algebra. Manystudents in precalculus classes have had a trigonometry course previously, but thisbook does not assume that the students remember any trigonometry. In fact thebook is fairly self-contained, starting with a review of the real numbers in Chapter 0, Chapter 0 could have

been titledA Prelude to APrelude to Calculus.

whose numbering is intended to indicate that many instructors will prefer to skipthis beginning material or cover it quickly.

Inverse Functions

The unifying concept of inverse functions are introduced early in the book in Sec-tion 1.5. This crucial idea has its first major use in this book in the definition ofy1/m as the number that when raised to the mth power gives y (in other words, thefunction y �→ y1/m is the inverse of the function x �→ xm; see Section 3.1). The sec-ond major use of inverse functions occurs in the definition of logb y as the numbersuch that b raised to this number gives y (in other words, the function y �→ logb yis the inverse of the function x �→ bx ; see Section 3.2).

Thus students should be comfortable with using inverse functions by the timethey reach the inverse trigonometric functions (arccosine, arcsine, and arctangent)in Section 5.7. This familiarity with inverse functions should help students deal withinverse operations (such as antidifferentiation) when they reach calculus.

Algebraic Properties of Logarithms

Logarithms play a key role in calculus, but many calculus instructors complain thattoo many students lack appropriate algebraic manipulation skills with logarithms.In Chapter 3 logarithms are defined as the inverse functions of exponentiation. Thebase for logarithms here is arbitrary, although most of the examples and motiva-tion in Chapter 3 use logarithms base 2 or logarithms base 10. In Chapter 3, the

xv

xvi Preface to the Instructor

algebraic properties of logarithms are easily derived from the algebraic propertiesof exponentiation.

The crucial concepts of e and natural logarithms are saved for a later chapter.The initial separationof logarithms and e

should help studentsmaster both concepts.

Thus students can concentrate in Chapter 3 on understanding logarithms (arbitrarybase) and their properties without at the same time worrying about grasping con-cepts related to e. Similarly, when natural logarithms arise naturally in Chapter 4,students should be able to concentrate on issues surrounding e without at the sametime learning properties of logarithms.

Half-life and Exponential Growth

All precalculus textbooks present radioactive decay as an example of exponentialdecay. Amazingly, the typical precalculus textbook states that if a radioactive iso-tope has a half-life of h, then the amount left at time t will equal e−kt times theamount present at time 0, where k = ln 2

h .A much clearer formulation would state, as this textbook does, that the amount

left at time t will equal 2−t/h times the amount present at time 0. The unnecessaryuse of e and ln 2 in this context may suggest to students that e and natural logarithmshave only contrived and artificial uses, which is not the message that students shouldreceive from their textbook. Using 2−t/h helps students understand the concept ofhalf-life, with a formula connected to the meaning of the concept.

Similarly, many precalculus textbooks consider, for example, a colony of bacteriadoubling in size every 3 hours, with the textbook then producing the formula e(t ln 2)/3

for the growth factor after t hours. The simpler and natural formula 2t/3 seems notto be mentioned in such books. This book presents the more natural approach tosuch issues of exponential growth and decay.

Area

About half of calculus (namely, integration) deals with area, but most precalculustextbooks barely mention the subject.

Chapter 4 in this book builds the intuitive notion of area starting with squares,and then quickly derives formulas for the area of rectangles, triangles, parallelo-grams, and trapezoids. A discussion of the effects of stretching either horizontallyor vertically easily leads to the familiar formula for the area enclosed by a circle.Chapter 4 uses the same ideas to derive the formula for the area inside an ellipse.

Chapter 4 then turns to the question of estimating the area under parts of thecurve y = 1

x by using rectangles. This easy nontechnical introduction, with itsemphasis on ideas without the clutter of the notation of Riemann sums, will servestudents well when they reach integral calculus in a later course.

e, The Exponential Function, and the Natural Logarithm

Most precalculus textbooks either present no motivation for e or motivate e via con-tinuously compounding interest or through the limit of an indeterminate expressionof the form 1∞; these concepts are difficult for students at this level to understand.

Chapter 4 presents a clean and well-motivated approach to e and the naturallogarithm. We do this by looking at the area (intuitively defined) under the curvey = 1

x , above the x-axis, and between the lines x = 1 and x = c.

Preface to the Instructor xvii

A similar approach to e and the natural logarithm is common in calculus courses.However, this approach is not usually adopted in precalculus textbooks. Using ob-vious properties of area, the simple presentation given here shows how these ideascan come through clearly without the technicalities of calculus or Riemann sums.Indeed, this precalculus approach to the exponential function and the natural log-arithm shows that a good understanding of these subjects need not wait until thecalculus course. Students who have seen the approach given here should be wellprepared to deal with these concepts in their calculus courses.

The approach taken here also has the advantage that it easily leads, as we will seein Chapter 4, to the approximation ln(1+h) ≈ h for small values of h. Furthermore,the same methods show that if r is any number, then

(1+ r

x)x ≈ er for large values

of x. A final bonus of this approach is that the connection between continuouslycompounding interest and e becomes a nice corollary of natural considerations con-cerning area.

Trigonometry

Should the trigonometric functions be introduced via the unit circle or via right tri-angles? Calculus requires the unit-circle approach (because, for example, discussingthe Taylor series for cosx requires us to consider negative values of x and values ofx that are more than π

2 radians). Thus this textbook uses the unit-circle approach,but quickly gives applications to right triangles. The unit-circle approach also allowsfor a well-motivated introduction to radian measure.

The trigonometry section of this book concentrates almost exclusively on thefunctions cosine, sine, and tangent and their inverse functions, with only cursorymention of secant, cosecant, and cotangent. These latter three functions, which aresimply the multiplicative inverses of the three key trigonometric functions, add littlecontent or understanding.

Exercises and Problems

Students learn mathematics by actively working on a wide range of exercises and Each exercise has aunique correct an-swer, usually a num-ber or a function; eachproblem has multiplecorrect answers, usu-ally explanations orexamples.

problems. Ideally, a student who reads and understands the material in a section ofthis book should be able to do the exercises and problems in that section withoutfurther help. However, some of the exercises require application of the ideas in acontext that students may not have seen before, and many students will need helpwith these exercises. This help is available from the complete worked-out solutionsto all the odd-numbered exercises that appear at the end of each section.

Because the worked-out solutions were written solely by the author of the text-book, students can expect a consistent approach to the material. Furthermore,students will save money by not having to purchase a separate student solutionsmanual.

The exercises (but not the problems) occur in pairs, so that an odd-numbered This book containswhat is usually a sepa-rate book called thestudent solutionsmanual.

exercise is followed by an even-numbered exercise whose solution uses the sameideas and techniques. A student stumped by an even-numbered exercise should beable to tackle it after reading the worked-out solution to the corresponding odd-numbered exercise. This arrangement allows the text to focus more centrally onexplanations of the material and examples of the concepts.

My experience with teaching precalculus is that most students read the studentsolutions manual when they are assigned homework, even though they are reluctant

xviii Preface to the Instructor

to read the main text. The integration of the student solutions manual within thisbook might encourage students who would otherwise read only the student solutionsmanual to drift over and also read the main text. To reinforce this tendency, theworked-out solutions to the odd-numbered exercises at the end of each section aretypeset with a slightly less appealing style (smaller type, two-column format, andnot right justified) than the main text. The reader-friendly appearance of the maintext might nudge students to spend some time there.

Exercises and problems in this book vary greatly in difficulty and purpose. Someexercises and problems are designed to hone algebraic manipulation skills; otherexercises and problems are designed to push students to genuine understandingbeyond rote algorithmic calculation.

Some exercises and problems intentionally reinforce material from earlier in thebook. For example, Exercise 27 in Section 5.3 asks students to find the smallestnumber x such that sin(ex) = 0; students will need to understand that they want tochoose x so that ex = π and thus x = lnπ . Although such exercises require morethought than most exercises in the book, they allow students to see crucial conceptsmore than once, sometimes in unexpected contexts.

A Book Designed to be Read

Mathematics faculty frequently complain, with justification, that most students inlower-division mathematics courses do not read the textbook. When doing home-work, a typical precalculus student looks only at the relevant section of the textbookor the student solutions manual for an example similar to the homework problemat hand. The student reads enough of that example to imitate the procedure, doesthe homework problem, and then follows the same process with the next homeworkproblem. Little understanding may take place.

In contrast, this book is designed to be read by students. The writing style andlayout are meant to induce students to read and understand the material. Explana-tions are more plentiful than typically found in precalculus books, with examples ofthe concepts making the ideas concrete whenever possible.

The Calculator Issue

The issue of whether and how calculators should be used by students has generatedTo aid instructorsin presenting the

kind of course theywant, the symbol

appears with exer-cises and problems

that require studentsto use a calculator.

immense controversy.Some sections of this book have many exercises and problems designed for calcu-

lators (for example Section 3.4 on exponential growth and Section 6.2 on the law ofsines and the law of cosines), but some sections deal with material not as amenable tocalculator use. The emphasis throughout the text has been on giving students boththe understanding and the skills they need to succeed in calculus. Thus the bookdoes not aim for an artificially predetermined percentage of exercises and problemsin each section requiring calculator use.

Some exercises and problems that require a calculator are intentionally designedto make students realize that by understanding the material, they can overcome thelimitations of calculators. As one example among many, Exercise 41 in Section 3.3asks students to find the number of digits in the decimal expansion of 74000. Bruteforce with a calculator will not work with this problem because the number involvedhas too many digits. However, a few moments’ thought should show students thatthey can solve this problem by using logarithms (and their calculators!).

Preface to the Instructor xix

The calculator icon can be interpreted for some exercises, depending on the Regardless of whatlevel of calculator usean instructor expects,students should notturn to a calculator tocompute somethinglike cos 0, becausethen cos has becomejust a button on thecalculator.

instructor’s preference, to mean that the solution should be a decimal approxima-tion rather than the exact answer. For example, Exercise 3 in Section 4.5 asks howmuch would need to be deposited in a bank account paying 4% interest compoundedcontinuously so that at the end of 10 years the account would contain $10,000. Theexact answer to this exercise is 10000/e0.4 dollars, but it may be more satisfying tothe student (after obtaining the exact answer) to use a calculator to see that approx-imately $6,703 needs to be deposited. For such exercises, instructors can decidewhether to ask for exact answers or decimal approximations (the worked-out solu-tions for the odd-numbered exercises will usually contain both).

What to Cover

Different instructors will want to cover different sections of this book. I usuallycover Chapter 0 (The Real Numbers), even though it should be review, because itdeals with familiar topics in a deeper fashion than students may have previouslyseen. I frequently cover Section 2.5 (Rational Functions) only lightly because graph-ing rational functions, and in particular finding local minima and maxima, is betterdone with calculus. Many instructors will prefer to skip Chapter 7 (Sequences, Series,and Limits), leaving that material to a calculus course. A one-semester precalculuscourse will probably not have time to cover the sections denoted with an asterisk (∗);those sections can safely be skipped by courses focusing only on material neededfor first-semester calculus.

What’s New as Compared to the Preliminary Edition

Numerous improvements have been made throughout the text based upon sugges-tions from faculty and students who used the Preliminary Edition. For example, theintroduction to e now gives instructors a gentler path to help lead students to dis-cover this remarkable number. More exercises and problems have been added tomany sections.

Some faculty requested coverage of additional topics because their precalculuscourses serve other purposes beyond preparing students for first-semester calculus.Thus three new optional sections have been added, dealing with complex numbers,systems of equations and matrices, and vectors.

A major redesign using full color has led to considerable improvements in theappearance of the book.

Finally, a comprehensive index now allows users to locate topics within the bookquickly.

Comments Welcome

I seek your help in making this a better book. Please send me your comments andyour suggestions for improvements. Thanks!


e-mail: [email protected] page: www.axler.net

Acknowledgments

As usual in a textbook, as opposed to a research article, little attempt hasMost of the results inthis book belong to

the common heritageof mathematics, cre-ated over thousands

of years by cleverand curious people.

been made to provide proper credit to the original creators of the ideas pre-sented in this book. Where possible, I have tried to improve on standardapproaches to this material. However, the absence of a reference does notimply originality on my part. I thank the many mathematicians who havecreated and refined our beautiful subject.

I chose Wiley as the publisher of this book because of the company’s com-mitment to excellence. The people at Wiley have made outstanding contri-butions to this project, providing wise editorial advice, superb design exper-tise, high-level production skill, and insightful marketing savvy. I am trulygrateful to the following Wiley folks, all of whom helped make this a bet-ter and more successful book than it would have been otherwise: AngelaBattle, Jeff Benson, Melissa Edwards, Jaclyn Elkins, Jessica Jacobs, MadelynLesure, Chelsee Pengal, Laurie Rosatone, Christopher Ruel, Ken Santor, AnneScanlan-Rohrer, Elle Wagner.

The accuracy checkers (Victoria Green, Celeste Hernandez, Nancy Mat-thews, Yan Tian, and Charles Waiveris) and copy editors (Katrina Avery andPatricia Brecht) excelled at catching my mathematical and linguistic mistakes.

The instructors and students who used the Preliminary Edition of thisbook provided wonderfully useful feedback. Numerous reviewers gave meterrific suggestions as the book progressed through various stages of devel-opment. I am grateful to all the class testers and reviewers whose names arelisted on the following pages.

Like most mathematicians, I owe thanks to Donald Knuth, who inventedTEX, and to Leslie Lamport, who invented LATEX, which I used to typeset thisbook.

Thanks also to Wolfram Research for producing Mathematica, which isthe software I used to create the graphics in this book.

My awesome partner Carrie Heeter deserves considerable credit for her as-tute advice and continual encouragement throughout the long book-writingprocess.

Many thanks to all of you!

xx

Acknowledgments xxi

Class Testers and Reviewers

• Alison Ahlgren, University of Illinois,Urbana-Champaign

• Margo Alexander, Georgia State University

• Ulrich Albrecht, Auburn University

• Caroline Autrey, University of West Georgia

• Robin Ayers, Western Kentucky University

• Robert Bass, Gardner Webb University

• Jo Battaglia, Penn State University

• Chris Bendixen, Lake Michigan College

• Kimberly Bennekin, Georgia Perimeter College

• Allan Berele, DePaul University

• Rebecca Berg, Bowie State University

• Andrew Beyer, San Francisco State University

• Michael Boardman, Pacific University

• Bob Bradshaw, Ohlone College

• Ellen Brook, Cuyahoga Community College

• David Buhl, Northern Michigan University

• William L. Burgin, Gaston College

• Brenda Burns-Williams, North Carolina StateUniversity

• Nick Bykov, San Joaquin Delta College

• Keith G. Calkins, Andrews University

• Tom Caplinger, University of Memphis

• Jamylle Carter, San Francisco State University

• Yu Chen, Idaho State University

• Charles Conley, University of North Texas

• Robert Crise Jr., Crafton Hills College

• Joanne Darken, Community College ofPhiladelphia

• Joyati Debnath, Winona State University

• Donna Densmore, Bossier Parish CommunityCollege

• Jeff Dodd, Jacksonville State University

• Benay Don, Suffolk County Community College

• Marcia Drost, Texas A & M University

• Douglas Dunbar, Okaloosa-Walton CommunityCollege

• Jason Edington, Mendocino College

• Thomas English, Penn State University, Erie

• Kevin Farrell, Lyndon State College

• Karline Feller, Georgia Perimeter College

• Terran Felter, California State University,Bakersfield

• Maggie Flint, Northeast State Community College

• Heng Fu, Thomas Nelson Community College

• Igor Fulman, Arizona State University

• Abel Gage, Skagit Valley College

• Gail Gonyo, Adirondack Community College

• Ivan Gotchev, Central Connecticut StateUniversity

• Peg Greene, Florida Community College

• Michael B. Gregory, University of North Dakota

• Julio Guillen, New Jersey City University

• Mako Haruta, University of Hartford

• Judy Hayes, Lake-Sumter Community College

• Richard Hill, Idaho State University

• Alan Hong, Santa Monica College

• Mizue Horiuchi, San Francisco State University

• Miles Hubbard, St. Cloud State University

• Stacey Hubbard, San Francisco State University

• Brian Jue, California State University, Stanislaus

• Dongrim Kim, Arizona State University

• Mohammed Kazemi, University of NorthCarolina, Charlotte

• Curtis Kifer, San Francisco State University

• Betty Larson, South Dakota State University

• Richard Leedy, Polk Community College

• Mary Legner, Riverside Community College

• Richard Low, San José State University

• Jane Mays, Grand Valley State University

• Eric Miranda, San Francisco State University

• Scot Morrison, Western Nevada College

• Scott Mortensen, Dixie State College

• Susan Nelson, Georgia Perimeter College

• Nicholas Passell, University of Wisconsin, EauClaire

• Vic Perera, Kent State University, Trumbull

xxii Acknowledgments

• David Ray, University of Tennessee, Martin

• Alexander Retakh, Stony Brook University

• Erika Rhett, Claflin University

• Randy Ross, Morehead State University

• Carol Rychly, Augusta State University

• David Santos, Community College ofPhiladelphia

• Virginia Sheridan, California State University,Bakersfield

• Barbara Shipman, University of Texas, Arlington

• Tatiana Shubin, San José State University

• Dave Sobecki, Miami University

• Jacqui Stone, University of Maryland

• Karel Stroethoff, University of Montana, Missoula

• Janet Tarjan, Bakersfield College

• Chia-chi Tung, Minnesota State University

• Hanson Umoh, Delaware State University

• Charles Waiveris, Central Connecticut StateUniversity

• Jeff Waller, Grossmont College

• Amy Wangsness, Fitchburg State College

• Rachel Winston, Cerro Coso Community College

• Elizabeth Wisniewski, Marymount College ofFordham University

• Tzu-Yi Alan Yang, Columbus State CommunityCollege

• David Zeigler, California State University,Sacramento

Preface to the Student

This book will help prepare you to succeed in calculus. If you master thematerial in this book, you will have the knowledge, the understanding, andthe skills needed to do well in a calculus course.

To learn this material well, you will need to spend serious time reading thisbook. You cannot expect to absorb mathematics the way you devour a novel.If you read through a section of this book in less than an hour, then you aregoing too fast. You should pause to ponder and internalize each definition,often by trying to invent some examples in addition to those given in thebook. For each result stated in the book, you should seek examples to showwhy each hypothesis is necessary. When steps in a calculation are left outin the book, you need to supply the missing pieces, which will require somewriting on your part. These activities can be difficult when attempted alone;try to work with a group of a few other students.

You will need to spend several hours per section doing the exercises and Complete worked-outsolutions to the odd-numbered exercisesare given at the end ofeach section.

problems. Make sure that you can do all the exercises and most of the prob-lems, not just the ones assigned for homework. By the way, the differencebetween an exercise and a problem in this book is that each exercise has aunique correct answer that is a mathematical object such as a number or afunction. In contrast, the solutions to problems consist of explanations orexamples; thus problems have multiple correct answers.

Have fun, and best wishes in your studies!


web page: www.axler.net

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