Evaluate the double integral. 7xy2 da, d is enclosed by x = 0 and x = 4 − y2 d

TO JULIE –Jon TO ALEXA AND COLTON –Colin

Publisher: Terri Ward Developmental Editors: Tony Palermino, Katrina Wilhelm Marketing Manager: Cara LeClair Market Development Manager: Shannon Howard Executive Media Editor: Laura Judge Associate Editor: Marie Dripchak Editorial Assistant: Victoria Garvey Director of Editing, Design, and Media Production: Tracey Kuehn Managing Editor: Lisa Kinne Project Editor: Kerry O’Shaughnessy Production Manager: Paul Rohloff Cover and Text Designer: Blake Logan Illustration Coordinator: Janice Donnola Illustrations: Network Graphics and Techsetters, Inc. Photo Editors: Eileen Liang, Christine Buese Photo Researcher: Eileen Liang Composition: John Rogosich/Techsetters, Inc. Printing and Binding: RR Donnelley Cover photo: ayzek/Shutterstock

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ABOUT THE AUTHORS

COLIN ADAMS

C olin Adams is the Thomas T. Read professor of Mathematics at Williams College,where he has taught since 1985. Colin received his undergraduate degree from MIT and his PhD from the University of Wisconsin. His research is in the area of knot theory and low-dimensional topology. He has held various grants to support his research, and written numerous research articles.

Colin is the author or co-author of The Knot Book, How toAce Calculus: The Streetwise Guide, How to Ace the Rest of Calculus: The Streetwise Guide, Riot at the Calc Exam and Other Mathematically Bent Stories, Why Knot?, Introduction to Topology: Pure and Applied, and Zombies & Calculus. He co-wrote and appears in the videos “The Great Pi vs. E Debate” and “Derivative vs. Integral: the Final Smackdown.”

He is a recipient of the Haimo National Distinguished Teaching Award from the Mathematical Association of America (MAA) in 1998, an MAA Polya Lecturer for 1998- 2000, a Sigma Xi Distinguished Lecturer for 2000-2002, and the recipient of the Robert Foster Cherry Teaching Award in 2003.

Colin has two children and one slightly crazy dog, who is great at providing the entertainment.

JON ROGAWSKI

A s a successful teacher for more than 30 years, Jon Rogawski listened and learnedmuch from his own students. These valuable lessons made an impact on his thinking, his writing, and his shaping of a calculus text.

Jon Rogawski received his undergraduate and master’s degrees in mathematics si- multaneously from Yale University, and he earned his PhD in mathematics from Princeton University, where he studied under Robert Langlands. Before joining the Department of Mathematics at UCLAin 1986, where he was a full professor, he held teaching and visiting positions at the Institute for Advanced Study, the University of Bonn, and the University of Paris at Jussieu and Orsay.

Jon’s areas of interest were number theory, automorphic forms, and harmonic analy- sis on semisimple groups. He published numerous research articles in leading mathemat- ics journals, including the research monograph Automorphic Representations of Unitary Groups in Three Variables (Princeton University Press). He was the recipient of a Sloan Fellowship and an editor of the Pacific Journal of Mathematics and the Transactions of the AMS.

Sadly, Jon Rogawski passed away in September 2011. Jon’s commitment to present- ing the beauty of calculus and the important role it plays in students’ understanding of the wider world is the legacy that lives on in each new edition of Calculus.

CONTENTS CALCULUS Early Transcendentals

Chapter 1 PRECALCULUS REVIEW 1

1.1 Real Numbers, Functions, and Graphs 1 1.2 Linear and Quadratic Functions 12 1.3 The Basic Classes of Functions 19 1.4 Trigonometric Functions 23 1.5 Inverse Functions 32 1.6 Exponential and Logarithmic Functions 40 1.7 Technology: Calculators and Computers 48

Chapter Review Exercises 53

Chapter 2 LIMITS 55

2.1 Limits, Rates of Change, and Tangent Lines 55 2.2 Limits: A Numerical and Graphical Approach 63 2.3 Basic Limit Laws 72 2.4 Limits and Continuity 75 2.5 Evaluating Limits Algebraically 84 2.6 Trigonometric Limits 89 2.7 Limits at Infinity 94 2.8 Intermediate Value Theorem 100 2.9 The Formal Definition of a Limit 103

Chapter Review Exercises 110

Chapter 3 DIFFERENTIATION 113

3.1 Definition of the Derivative 113 3.2 The Derivative as a Function 121 3.3 Product and Quotient Rules 135 3.4 Rates of Change 142 3.5 Higher Derivatives 151 3.6 Trigonometric Functions 156 3.7 The Chain Rule 159 3.8 Implicit Differentiation 167 3.9 Derivatives of General Exponential and Logarithmic

Functions 175 3.10 Related Rates 182

Chapter Review Exercises 189

Chapter 4 APPLICATIONS OF THE DERIVATIVE 193

4.1 Linear Approximation and Applications 193 4.2 Extreme Values 200 4.3 The Mean Value Theorem and Monotonicity 210 4.4 The Shape of a Graph 217 4.5 L’Hôpital’s Rule 224 4.6 Graph Sketching and Asymptotes 231 4.7 Applied Optimization 239 4.8 Newton’s Method 251

Chapter Review Exercises 256

Chapter 5 THE INTEGRAL 259

5.1 Approximating and Computing Area 259 5.2 The Definite Integral 272 5.3 The Indefinite Integral 281 5.4 The Fundamental Theorem of Calculus, Part I 288 5.5 The Fundamental Theorem of Calculus, Part II 294 5.6 Net Change as the Integral of a Rate of Change 300 5.7 Substitution Method 306 5.8 Further Transcendental Functions 313 5.9 Exponential Growth and Decay 318

Chapter Review Exercises 328

Chapter 6 APPLICATIONS OF THE INTEGRAL 333

6.1 Area Between Two Curves 333 6.2 Setting Up Integrals: Volume, Density, Average Value 341 6.3 Volumes of Revolution 351 6.4 The Method of Cylindrical Shells 359 6.5 Work and Energy 365

Chapter Review Exercises 371

Chapter 7 TECHNIQUES OF INTEGRATION 373

7.1 Integration by Parts 373 7.2 Trigonometric Integrals 379 7.3 Trigonometric Substitution 386 7.4 Integrals Involving Hyperbolic and Inverse Hyperbolic

Functions 392 7.5 The Method of Partial Fractions 398 7.6 Strategies for Integration 407 7.7 Improper Integrals 414 7.8 Probability and Integration 425 7.9 Numerical Integration 431

Chapter Review Exercises 440

Chapter 8 FURTHER APPLICATIONS OF THE INTEGRAL AND TAYLOR POLYNOMIALS 443

8.1 Arc Length and Surface Area 443 8.2 Fluid Pressure and Force 450 8.3 Center of Mass 456 8.4 Taylor Polynomials 465

Chapter Review Exercises 476

Chapter 9 INTRODUCTION TO DIFFERENTIAL EQUATIONS 479

9.1 Solving Differential Equations 479 9.2 Models Involving y′ = k(y − b) 487

iv

CONTENTS v

9.3 Graphical and Numerical Methods 492 9.4 The Logistic Equation 500 9.5 First-Order Linear Equations 504

Chapter Review Exercises 510

Chapter 10 INFINITE SERIES 513

10.1 Sequences 513 10.2 Summing an Infinite Series 523 10.3 Convergence of Series with Positive Terms 534 10.4 Absolute and Conditional Convergence 543 10.5 The Ratio and Root Tests and Strategies for

Choosing Tests 548 10.6 Power Series 553 10.7 Taylor Series 563

Chapter Review Exercises 575

Chapter 11 PARAMETRIC EQUATIONS, POLAR COORDINATES, AND CONIC SECTIONS 579

11.1 Parametric Equations 579 11.2 Arc Length and Speed 590 11.3 Polar Coordinates 596 11.4 Area and Arc Length in Polar Coordinates 604 11.5 Conic Sections 609

Chapter Review Exercises 622

Chapter 12 VECTOR GEOMETRY 625

12.1 Vectors in the Plane 625 12.2 Vectors in Three Dimensions 635 12.3 Dot Product and the Angle Between Two Vectors 645 12.4 The Cross Product 653 12.5 Planes in 3-Space 664 12.6 A Survey of Quadric Surfaces 670 12.7 Cylindrical and Spherical Coordinates 678

Chapter Review Exercises 685

Chapter 13 CALCULUS OF VECTOR-VALUED FUNCTIONS 689

13.1 Vector-Valued Functions 689 13.2 Calculus of Vector-Valued Functions 697 13.3 Arc Length and Speed 706 13.4 Curvature 711 13.5 Motion in 3-Space 722 13.6 Planetary Motion According to Kepler and Newton 731

Chapter Review Exercises 737

Chapter 14 DIFFERENTIATION IN SEVERAL VARIABLES 739

14.1 Functions of Two or More Variables 739 14.2 Limits and Continuity in Several Variables 750 14.3 Partial Derivatives 757 14.4 Differentiability and Tangent Planes 767

14.5 The Gradient and Directional Derivatives 774 14.6 The Chain Rule 787 14.7 Optimization in Several Variables 795 14.8 Lagrange Multipliers: Optimizing with a Constraint 809

Chapter Review Exercises 818

Chapter 15 MULTIPLE INTEGRATION 821

15.1 Integration in Two Variables 821 15.2 Double Integrals over More General Regions 832 15.3 Triple Integrals 845 15.4 Integration in Polar, Cylindrical, and Spherical

Coordinates 856 15.5 Applications of Multiple Integrals 866 15.6 Change of Variables 878

Chapter Review Exercises 891

Chapter 16 LINE AND SURFACE INTEGRALS 895

16.1 Vector Fields 895 16.2 Line Integrals 905 16.3 Conservative Vector Fields 919 16.4 Parametrized Surfaces and Surface Integrals 930 16.5 Surface Integrals of Vector Fields 944

Chapter Review Exercises 954

Chapter 17 FUNDAMENTAL THEOREMS OF VECTOR ANALYSIS 957

17.1 Green’s Theorem 957 17.2 Stokes’ Theorem 971 17.3 Divergence Theorem 981

Chapter Review Exercises 993

APPENDICES A1 A. The Language of Mathematics A1 B. Properties of Real Numbers A7 C. Induction and the Binomial Theorem A12 D. Additional Proofs A16

ANSWERS TO ODD-NUMBERED EXERCISES ANS1

REFERENCES R1

INDEX I1

Additional content can be accessed online via LaunchPad:

ADDITIONAL PROOFS

• L’Hôpital’s Rule • Error Bounds for Numerical Integration • Comparison Test for Improper Integrals

ADDITIONAL CONTENT

• Second Order Differential Equations • Complex Numbers

PREFACE

ABOUT CALCULUS

On Teaching Mathematics I consider myself very lucky to have a career as a teacher and practitioner of mathematics. When I was young, I decided I wanted to be a writer. I loved telling stories. But I was also good at math, and, once in college, it didn’t take me long to become enamored with it. I loved the fact that success in mathematics does not depend on your presentation skills or your interpersonal relationships. You are either right or you are wrong and there is little subjective evaluation involved. And I loved the satisfaction of coming up with a solution. That intensified when I started solving problems that were open research questions that had previously remained unsolved.

So, I became a professor of mathematics. And I soon realized that teaching mathe- matics is about telling a story. The goal is to explain to students in an intriguing manner, at the right pace, and in as clear a way as possible, how mathematics works and what it can do for you. I find mathematics immensely beautiful. I want students to feel that way, too.

On Writing a Calculus Text I had always thought I might write a calculus text. But that is a daunting task. These days, calculus books average over a thousand pages. And I would need to convince myself that I had something to offer that was different enough from what already appears in the existing books. Then, I was approached about writing the third edition of Jon Rogawski’s calculus book. Here was a book for which I already had great respect. Jon’s vision of what a calculus book should be fit very closely with my own. Jon believed that as math teachers, how we say it is as important as what we say. Although he insisted on rigor at all times, he also wanted a book that was written in plain English, a book that could be read and that would entice students to read further and learn more. Moreover, Jon strived to create a text in which exposition, graphics, and layout would work together to enhance all facets of a student’s calculus experience.

In writing his book, Jon paid special attention to certain aspects of the text:

1. Clear, accessible exposition that anticipates and addresses student difficulties. 2. Layout and figures that communicate the flow of ideas. 3. Highlighted features that emphasize concepts and mathematical reasoning: Conceptual Insight, Graphical Insight, Assumptions Matter, Reminder, and Historical Perspective.

4. A rich collection of examples and exercises of graduated difficulty that teach basic skills, problem-solving techniques, reinforce conceptual understanding, and motivate cal- culus through interesting applications. Each section also contains exercises that develop additional insights and challenge students to further develop their skills.

Coming into the project of creating the third edition, I was somewhat apprehensive. Here was an already excellent book that had attained the goals set for it by its author. First and foremost, I wanted to be sure that I did it no harm. On the other hand, I have been teaching calculus now for 30 years, and in that time, I have come to some conclusions about what does and does not work well for students.

As a mathematician, I want to make sure that the theorems, proofs, arguments and development are correct. There is no place in mathematics for sloppiness of any kind. As a teacher, I want the material to be accessible. The book should not be written at the mathematical level of the instructor. Students should be able to use the book to learn the material, with the help of their instructor. Working from the high standard that Jon set, I have tried hard to maintain the level of quality of the previous edition while making the changes that I believe will bring the book to the next level.

vi

PREFACE vii

Placement of Taylor Polynomials Taylor polynomials appear in Chapter 8, before infinite series in Chapter 10. The goal here is to present Taylor polynomials as a natural extension of linear approximation. When teaching infinite series, the primary focus is on convergence, a topic that many students find challenging. By the time we have covered the basic convergence tests and studied the convergence of power series, students are ready to tackle the issues involved in representing a function by its Taylor series. They can then rely on their previous work with Taylor polynomials and the error bound from Chapter 8. However, the section on Taylor polynomials is written so that you can cover this topic together with the materials on infinite series if this order is preferred.

Careful, Precise Development W. H. Freeman is committed to high quality and precise textbooks and supplements. From this project’s inception and throughout its development and production, quality and precision have been given significant priority. We have in place unparalleled procedures to ensure the accuracy of the text:

• Exercises and Examples • Exposition • Figures • Editing • Composition

Together, these procedures far exceed prior industry standards to safeguard the quality and precision of a calculus textbook.

New to the Third Edition There are a variety of changes that have been implemented in this edition. Following are some of the most important.

MORE FOCUS ON CONCEPTS The emphasis has been shifted to focus less on the memo- rization of specific formulas, and more on understanding the underlying concepts. Memo- rization can never be completely avoided, but it is in no way the crux of calculus. Students will remember how to apply a procedure or technique if they see the logical progression that generates it. And they then understand the underlying concepts rather than seeing the topic as a black box in which you insert numbers. Specific examples include:

• (Section 1.2) Removed the general formula for the completion of a square and instead, emphasized the method so students need not memorize the formula.

• (Section 7.2) Changed the methods for evaluating trigonometric integrals to focus on techniques to apply rather than formulas to memorize.

• (Chapter 9) Discouraged the memorization of solutions of specific types of differ- ential equations and instead, encouraged the use of methods of solution.

• (Section 12.2) Decreased number of formulas for parametrizing a line from two to one, as the second can easily be derived from the first.

• (Section 12.6) De-emphasized the memorization of the various formulas for quadric surfaces. Instead, moved the focus to slicing with planes to find curves and using those to determine the shape of the surface. These methods will be useful regardless of the type of surface it is.

• (Section 14.4) Decreased the number of essential formulas for linear approximation of functions of two variables from four to two, providing the background to derive the others from these.

CHANGES IN NOTATION There are numerous notational changes. Some were made to bring the notation more into line with standard usage in mathematics and other fields in which mathematics is applied. Some were implemented to make it easier for students to remember the meaning of the notation. Some were made to help make the corresponding concepts that are represented more transparent. Specific examples include:

viii PREFACE

• (Section 4.6) Presented a new notation for graphing that gives the signs of the first and second derivative and then simple symbols (slanted up and down arrows and up and down u’s) to help the student keep track of when the graph is increasing or decreasing and concave up or concave down over the given interval.

• (Section 7.1) Simplified the notation for integration by parts and provided a visual method for remembering it.

• (Chapter 10) Changed names of the various tests for convergence/divergence of infinite series to evoke the usage of the test and thereby make it easier for students to remember them.

• (Chapters 13–17) Rather than using c(t) for a path, we consistently switched to the vector-valued function r(t). This also allowed us to replace ds with dr as a differential, which means there is less likely to be confusion with ds, dS and dS.

MORE EXPLANATIONS OF DERIVATIONS Occasionally, in the previous edition, a result was given and verified, without motivating where the derivation came from. I believe it is important for students to understand how someone might come up with a particular result, thereby helping them to picture how they might themselves one day be able to derive results.

• (Section 14.4) Developed the equation of the tangent plane in a manner that makes geometric sense.

• (Section 14.5) Included a proof of the fact the gradient of a function f of three variables is orthogonal to the surfaces that are the level sets of f .

• (Section 14.8) Gave an intuitive explanation for why the Method of Lagrange Multipliers works.

• (Section 15.5) Developed the center of mass formulas by first discussing the one- dimensional case of a seesaw.

REORDERING AND ADDING TOPICS There were some specific rearrangements among the sections and additions. These include:

• A subsection on piecewise-defined functions has been added to Section 1.3. • The section on implicit differentiation in Chapter 3 (previously Section 3.10) has

been moved up to become Section 3.8 and has absorbed the previous Section 3.8 (in- verse functions) so that implicit differentiation can be applied to derive the various derivatives as necessary.

• The section on indefinite integrals (previously Section 4.9) has been moved from Chapter 4 (Applications of the Derivative) to Chapter 5 (The Integral). This is a more natural placement for it.

• A new section on choosing from amongst the various methods of integration has been added to Chapter 7.

• A subsection on choosing the appropriate convergence/divergence test has been added to Section 10.5.

• An explanation of how to find indefinite limits using power series has been added to Section 10.6.

• The definitions of divergence and curl have been moved from Chapter 17 to Section 16.1. This allows us to utilize them at an appropriate earlier point in the text.

• A list all of the different types of integrals that have been introduced in Chapter 16 has been added to Section 16.5.

• A subsection on the Vector Form of Green’s Theorem has been added to Section 17.1.

NEW EXAMPLES, FIGURES, AND EXERCISES Numerous examples and accompanying figures have been added to clarify concepts. A variety of exercises have also been added throughout the text, particularly where new applications are available or further conceptual development is advantageous. Figures marked with a icon have been made dynamic and can be accessed via LaunchPad. A selection of these figures also includes brief tutorial videos explaining the concepts at work.

ONLINE HOMEWORK OPTIONS ix

SUPPLEMENTS

For Instructors Instructor’s Solutions Manual Contains worked-out solutions to all exercises in the text.

Test Bank Computerized (CD-ROM), ISBN:1-3190-0939-5 Includes a comprehensive set of multiple-choice test items.

Instructor’s Resource Manual Provides sample course outlines, suggested class time, key points, lecture material, discussion topics, class activities, work- sheets, projects, and questions to accompany the Dynamic Fig- ures.

For Students Student Solutions Manual Single Variable ISBN: 1-4641-7188-2 Multivariable ISBN: 1-4641-7189-0 Contains worked-out solutions to all odd-numbered exercises in the text.

Software Manuals Maple™ and Mathematica® software manuals serve as basic introductions to popular mathematical software options.

ONLINE HOMEWORK OPTIONS

Our new course space, LaunchPad, combines an interactive e-Book with high-quality multimedia content and ready-made assessment options, including LearningCurve adap- tive quizzing. Pre-built, curated units are easy to assign or adapt with your own material, such as readings, videos, quizzes, discussion groups, and more. LaunchPad includes a gradebook that provides a clear window on performance for your whole class, for individ- ual students, and for individual assignments. While a streamlined interface helps students focus on what’s due next, social commenting tools let them engage, make connections, and learn from each other. Use LaunchPad on its own or integrate it with your school’s learning management system so your class is always on the same page. Contact your rep to make sure you have access.

Assets integrated into LaunchPad include:

Interactive e-Book: Every LaunchPad e-Book comes with powerful study tools for stu- dents, video and multimedia content, and easy customization for instructors. Students can search, highlight, and bookmark, making it easier to study and access key content. And instructors can make sure their class gets just the book they want to deliver: customize and rearrange chapters, add and share notes and discussions, and link to quizzes, activities, and other resources.

LearningCurve provides students and instructors with powerful adaptive quizzing, a game-like format, direct links to the e-Book, and instant feedback. The quizzing system features questions tailored specifically to the text and adapts to students’ responses, pro- viding material at different difficulty levels and topics based on student performance.

Dynamic Figures: Over 250 figures from the text have been recreated in a new interactive format for students and instructors to manipulate and explore, making the visual aspects and dimensions of calculus concepts easier to grasp. Brief tutorial videos accompany selected figures and explain the concepts at work.

CalcClips: These whiteboard tutorials provide animated and narrated step-by-step solu- tions to exercises that are based on key problems in the text.

SolutionMaster offers an easy-to-use Web-based version of the instructor’s solutions, allowing instructors to generate a solution file for any set of homework exercises.

x FEATURES

www.webassign.net/freeman.com WebAssign Premium integrates the book’s exercises into the world’s most popular and trusted online homework system, making it easy to assign algorithmically generated homework and quizzes. Algorithmic exercises offer the instructor optional algorith- mic solutions. WebAssign Premium also offers access to resources, including Dynamic Figures, CalcClips whiteboard tutorials, and a “Show My Work” feature. In addition, WebAssign Premium is available with a fully customizable e-Book option.

webwork.maa.org W. H. Freeman offers thousands of algorithmically generated questions (with full solu- tions) through this free, open-source online homework system created at the University of Rochester. Adopters also have access to a shared national library test bank with thou- sands of additional questions, including 2,500 problem sets matched to the book’s table of contents.

FEATURES

CONCEPTUAL INSIGHT Leibniz notation is widely used for several reasons. First, it re- minds us that the derivative df/dx, although not itself a ratio, is in fact a limit of ratios

. Second, the notation specifies the independent variable. This is useful when variables other than x are used. For example, if the independent variable is t , we write df/dt . Third, we often think of d/dx as an “operator” that performs differentiation on functions. In other words, we apply the operator d/dx to f to obtain the derivative df/dx. We will see other advantages of Leibniz notation when we discuss the Chain Rule in Section 3.7.

Ch. 3, p. 123

Conceptual Insights encourage students to develop a conceptual understanding of calculus by explaining important ideas clearly but informally.

GRAPHICAL INSIGHT Can we visualize the rate represented by f (x)? The second derivative is the rate at which f (x) is changing, so f (x) is large if the slopes of the tangent lines change rapidly, as in Figure 3(A). Similarly, f (x) is small if the slopes of the tangent lines change slowly—in this case, the curve is relatively flat, as in Figure 3(B). If f is a linear function [Figure 3(C)], then the tangent line does not change at all and f (x) = 0. Thus, f (x) measures the “bending” or concavity of the graph.

(A) Large second derivative: Tangent lines turn rapidly.

(B) Smaller second derivative: Tangent lines turn slowly.

(C) Second derivative is zero: Tangent line does not change.

FIGURE 3 Ch. 3, p. 153

Graphical Insights enhance students’ visual understanding by making the crucial connections between graphical properties and the underlying concepts.

FEATURES xi

EXAMPLE 3 Evaluate sin2 x dx.

Solution We could apply the reduction formula Eq. (5) from the last section. However, instead, we apply a method that does not rely on knowing that formula. We utilize the trigonometric identity called the double angle formula sin2 x = 12 (1 − cos 2x). Then

sin2 x dx = 1 2 (1 − cos 2x) dx = x

2 − sin 2x

4 + C

Using the trigonometric identities in the margin, we can also integrate cos2 x, obtain- ing the following:REMINDER Useful Identities:

sin2 x = 1 2 (1 − cos 2x)

cos2 x = 1 2 (1 + cos 2x)

sin 2x = 2 sin x cos x

cos 2x = cos2 x − sin2 x

sin2 x dx = x 2

− sin 2x 4

+ C = x 2

− 1 2

sin x cos x + C 1

cos2 x dx = x 2

+ sin 2x 4

+ C = x 2

+ 1 2

sin x cos x + C 2

Ch. 7, p. 380

Reminders are margin notes that link the current discussion to important concepts introduced earlier in the text to give students a quick review and make connections with related ideas.

EXAMPLE 1 Use L’Hôpital’s Rule to evaluate lim x→2

x3 − 8 x4 + 2x − 20 .

Solution Let f (x) = x3 − 8 and g(x) = x4 + 2x − 20. Both f and g are differentiable and f (x)/g(x) is indeterminate of type 0/0 at a = 2 because f (2) = g(2) = 0:

• Numerator: f (2) = 23 − 1 = 0 • Denominator: g(2) = 24 + 2(2) − 20 = 0

Furthermore, g (x) = 4x3 + 2 is nonzero near x = 2, so L’Hôpital’s Rule applies. We may replace the numerator and denominator by their derivatives to obtain

CAUTION When using L’Hôpital’s Rule, be sure to take the derivative of the numerator and denominator separately:

lim x→a

f (x)

g(x) = lim

x→a f (x)

g (x)

Do not differentiate the quotient function y = f (x)/g(x).

lim x→2

x3 − 8 x4 + 2x − 2 = limx→2

(x3 − 8) (x4 + 2x − 2)

L’Hôpital’s Rule

= lim x→2

3x2

4x3 + 2 = 3(22)

4(23) + 2 = 12 34

= 6 17

Ch. 4, p. 224

Caution Notes warn students of common pitfalls they may encounter in understanding the material.

Historical Perspectives are brief vignettes that place key discoveries and conceptual advances in their historical context. They give students a glimpse into some of the accomplishments of great mathematicians and an appreciation for their significance.

HISTORICAL

PERSPECTIVE

(Mechanics Magazine London, 1824)

Geometric series were used as early as the third century bce by Archimedes in a brilliant argu- ment for determining the area S of a “parabolic segment” (shaded region in Figure 3). Given two points A and C on a parabola, there is a point B between A and C where the tangent line is paral- lel to AC (apparently, Archimedes was aware of the Mean Value Theorem more than 2000 years before the invention of calculus). Let T be the area of triangle ABC. Archimedes proved that if D is chosen in a similar fashion relative to AB and E is chosen relative to BC, then

1 4 T = Area( ADB) + Area( BEC) 6

This construction of triangles can be continued. The next step would be to construct the four tri- angles on the segments AD, DB, BE, EC, of

total area 14 2 T . Then construct eight triangles

of total area 14 3 T , etc. In this way, we obtain in-

finitely many triangles that completely fill up the parabolic segment. By the formula for the sum of a geometric series, we get

S = T + 1 4 T + 1

16 T + · · · = T

∞

n=0

1 4n

= 4 3 T

For this and many other achievements, Archi-

medes is ranked together with Newton and Gauss as one of the greatest scientists of all time.

The modern study of infinite series began in the seventeenth century with Newton, Leib- niz, and their contemporaries. The divergence

of ∞

n=1 1/n (called the harmonic series) was

known to the medieval scholar Nicole d’Oresme (1323–1382), but his proof was lost for cen- turies, and the result was rediscovered on more than one occasion. It was also known that the

sum of the reciprocal squares ∞

n=1 1/n2 con-

verges, and in the 1640s, the Italian Pietro Men- goli put forward the challenge of finding its sum. Despite the efforts of the best mathematicians of the day, including Leibniz and the Bernoulli brothers Jakob and Johann, the problem resisted solution for nearly a century. In 1735 the great master Leonhard Euler (at the time, 28 years old) astonished his contemporaries by proving that

1

12 + 1

22 + 1

32 + 1

42 + 1

52 + 1

62 + · · · = π

2

6

This formula, surprising in itself, plays a role in a variety of mathematical fields. A theorem from number theory states that two whole num- bers, chosen randomly, have no common factor with probability 6/π2 ≈ 0.6 (the reciprocal of Euler’s result). On the other hand, Euler’s re- sult and its generalizations appear in the field of statistical mechanics.

Ch. 10, p. 530

xii ACKNOWLEDGMENTS

Assumptions Matter uses short explanations and well-chosen counterexamples to help students appreciate why hypotheses are needed in theorems.

EXAMPLE 3 Assumptions Matter Show that the Product Law cannot be applied to lim x→0

f (x)g(x) if f (x) = x and g(x) = x−1.

Solution For all x = 0, we have f (x)g(x) = x · x−1 = 1, so the limit of the product exists:

lim x→0

f (x)g(x) = lim x→0

1 = 1

However, lim x→0

x−1 does not exist because g(x) = x−1 approaches ∞ as x → 0+ and it approaches −∞ as x → 0−. Therefore, the Product Law cannot be applied and its conclusion does not hold:

lim x→0

f (x) lim x→0

g(x) = lim x→0

x lim x→0

x−1

Does not exist Ch. 2, p. 74

Section Summaries summarize a section’s key points in a concise and useful way and emphasize for students what is most important in each section.

Section Exercise Sets offer a comprehensive set of exercises closely coordinated with the text. These exercises vary in difficulty from routine, to moderate, to more challenging. Also included are icons indicating problems that require the student to give a written

response or require the use of technology .

Chapter Review Exercises offer a comprehensive set of exercises closely coordinated with the chapter material to provide additional problems for self-study or assignments.

ACKNOWLEDGMENTS Colin Adams and W. H. Freeman and Company are grateful to the many instructors from across the United States and Canada who have offered comments that assisted in the development and refinement of this book. These contributions included class testing, manuscript reviewing, problems reviewing, and participating in surveys about the book and general course needs.

ALABAMA Tammy Potter, Gadsden State Community College; David Dempsey, Jacksonville State University; Edwin Smith, Jacksonville State University; Jeff Dodd, Jacksonville State University; Douglas Bailer, Northeast Alabama Community College; Michael Hicks, Shelton State Community College; Patricia C. Eiland, Troy University, Montgomery Campus; Chadia Affane Aji, Tuskegee University; James L. Wang, The University of Alabama; Stephen Brick, University of South Alabama; Jo- erg Feldvoss, University of South Alabama ALASKA Mark A. Fitch, University of Alaska Anchorage; Kamal Narang, University of Alaska An- chorage; Alexei Rybkin, University of Alaska Fairbanks; Martin Getz, University of Alaska Fairbanks ARIZONA Stefania Tracogna, Ari- zona State University; Bruno Welfert, Arizona State University; Light Bryant, Arizona Western College; Daniel Russow, Arizona Western Col- lege; Jennifer Jameson, Coconino College; George Cole, Mesa Com- munity College; David Schultz, Mesa Community College; Michael Bezusko, Pima Community College, Desert Vista Campus; Garry Car- penter, Pima Community College, Northwest Campus; Paul Flasch, Pima County Community College; Jessica Knapp, Pima Community College, Northwest Campus; Roger Werbylo, Pima County Community College; Katie Louchart, Northern Arizona University; Janet McShane, North- ern Arizona University; Donna M. Krawczyk, The University of Ari- zona ARKANSAS Deborah Parker, Arkansas Northeastern College;

J. Michael Hall, Arkansas State University; Kevin Cornelius, Ouachita Baptist University; Hyungkoo Mark Park, Southern Arkansas Univer- sity; Katherine Pinzon, University of Arkansas at Fort Smith; Denise LeGrand, University of Arkansas at Little Rock; John Annulis, University of Arkansas at Monticello; Erin Haller, University of Arkansas, Fayet- teville; Shannon Dingman, University of Arkansas, Fayetteville; Daniel J. Arrigo, University of Central Arkansas CALIFORNIA Michael S. Gagliardo, California Lutheran University; Harvey Greenwald, Califor- nia Polytechnic State University, San Luis Obispo; Charles Hale, Cali- fornia Polytechnic State University; John Hagen, California Polytechnic State University, San Luis Obispo; Donald Hartig, California Polytech- nic State University, San Luis Obispo; Colleen Margarita Kirk, California Polytechnic State University, San Luis Obispo; Lawrence Sze, California Polytechnic State University, San Luis Obispo; Raymond Terry, Califor- nia Polytechnic State University, San Luis Obispo; James R. McKinney, California State Polytechnic University, Pomona; Robin Wilson, Cali- fornia State Polytechnic University, Pomona; Charles Lam, California State University, Bakersfield ; David McKay, California State University, Long Beach; Melvin Lax, California State University, Long Beach; Wal- lace A. Etterbeek, California State University, Sacramento; Mohamed Al- lali, Chapman University; George Rhys, College of the Canyons; Janice Hector, DeAnza College; Isabelle Saber, Glendale Community College;

ACKNOWLEDGMENTS xiii

Peter Stathis, Glendale Community College; Douglas B. Lloyd, Golden West College; Thomas Scardina, Golden West College; Kristin Hartford, Long Beach City College; Eduardo Arismendi-Pardi, Orange Coast Col- lege; Mitchell Alves, Orange Coast College; Yenkanh Vu, Orange Coast College; Yan Tian, Palomar College; Donna E. Nordstrom, Pasadena City College; Don L. Hancock, Pepperdine University; Kevin Iga, Pep- perdine University; Adolfo J. Rumbos, Pomona College; Virginia May, Sacramento City College; Carlos de la Lama, San Diego City College; Matthias Beck, San Francisco State University; Arek Goetz, San Fran- cisco State University; Nick Bykov, San Joaquin Delta College; Eleanor Lang Kendrick, San Jose City College; Elizabeth Hodes, Santa Barbara City College; William Konya, Santa Monica College; John Kennedy, Santa Monica College; Peter Lee, Santa Monica College; Richard Salome, Scotts Valley High School; Norman Feldman, Sonoma State University; Elaine McDonald, Sonoma State University; John D. Eggers, University of California, San Diego; Adam Bowers, University of California, San Diego; Bruno Nachtergaele, University of California, Davis; Boumedi- ene Hamzi, University of California, Davis; Olga Radko, University of California, Los Angeles; Richard Leborne, University of California, San Diego; Peter Stevenhagen, University of California, San Diego; Jeffrey Stopple, University of California, Santa Barbara; Guofang Wei, Uni- versity of California, Santa Barbara; Rick A. Simon, University of La Verne; Alexander E. Koonce, University of Redlands; Mohamad A. Al- wash, West Los Angeles College; Calder Daenzer, University of California, Berkeley; Jude Thaddeus Socrates, Pasadena City College; Cheuk Ying Lam, California State University Bakersfield ; Borislava Gutarts, Califor- nia State University, Los Angeles; Daniel Rogalski, University of Cali- fornia, San Diego; Don Hartig, California Polytechnic State University; Anne Voth, Palomar College; Jay Wiestling, Palomar College; Lindsey Bramlett-Smith, Santa Barbara City College; Dennis Morrow, College of the Canyons; Sydney Shanks, College of the Canyons; Bob Tolar, College of the Canyons; Gene W. Majors, Fullerton College; Robert Diaz, Fuller- ton College; Gregory Nguyen, Fullerton College; Paul Sjoberg, Fullerton College; Deborah Ritchie, Moorpark College; Maya Rahnamaie, Moor- park College; Kathy Fink, Moorpark College; Christine Cole, Moor- park College; K. Di Passero, Moorpark College; Sid Kolpas, Glendale Community College; Miriam Castrconde, Irvine Valley College; Ilkner Erbas-White, Irvine Valley College; Corey Manchester, Grossmont Col- lege; Donald Murray, Santa Monica College; Barbara McGee, Cuesta College; Marie Larsen, Cuesta College; Joe Vasta, Cuesta College; Mike Kinter, Cuesta College; Mark Turner, Cuesta College; G. Lewis, Cuesta College; Daniel Kleinfelter, College of the Desert; Esmeralda Medrano, Citrus College; James Swatzel, Citrus College; Mark Littrell, Rio Hondo College; Rich Zucker, Irvine Valley College; Cindy Torigison, Palomar College; Craig Chamberline, Palomar College; Lindsey Lang, Diablo Valley College; Sam Needham, Diablo Valley College; Dan Bach, Dia- blo Valley College; Ted Nirgiotis, Diablo Valley College; Monte Collazo, Diablo Valley College; Tina Levy, Diablo Valley College; Mona Pan- chal, East Los Angeles College; Ron Sandvick, San Diego Mesa College; Larry Handa, West Valley College; Frederick Utter, Santa Rose Junior College; Farshod Mosh, DeAnza College; Doli Bambhania, DeAnza Col- lege; Charles Klein, DeAnza College; Tammi Marshall, Cauyamaca Col- lege; Inwon Leu, Cauyamaca College; Michael Moretti, Bakersfield Col- lege; Janet Tarjan, Bakersfield College; Hoat Le, San Diego City College; Richard Fielding, Southwestern College; Shannon Gracey, Southwestern College; Janet Mazzarella, Southwestern College; Christina Soderlund, California Lutheran University; Rudy Gonzalez, Citrus College; Robert Crise, Crafton Hills College; Joseph Kazimir, East Los Angeles College; Randall Rogers, Fullerton College; Peter Bouzar, Golden West College; Linda Ternes, Golden West College; Hsiao-Ling Liu, Los Angeles Trade Tech Community College; Yu-Chung Chang-Hou, Pasadena City College; Guillermo Alvarez, San Diego City College; Ken Kuniyuki, San Diego Mesa College; Laleh Howard, San Diego Mesa College; Sharareh Ma- sooman, Santa Barbara City College; Jared Hersh, Santa Barbara City College; Betty Wong, Santa Monica College; Brian Rodas, Santa Monica College; Veasna Chiek, Riverside City College COLORADO Tony

Weathers, Adams State College; Erica Johnson, Arapahoe Community College; Karen Walters, Arapahoe Community College; Joshua D. Lai- son, Colorado College; G. Gustave Greivel, Colorado School of Mines; Holly Eklund, Colorado School of the Mines; Mike Nicholas, Colorado School of the Mines; Jim Thomas, Colorado State University; Eleanor Storey, Front Range Community College; Larry Johnson, Metropolitan State College of Denver; Carol Kuper, Morgan Community College; Larry A. Pontaski, Pueblo Community College; Terry Chen Reeves, Red Rocks Community College; Debra S. Carney, Colorado School of the Mines; Louis A. Talman, Metropolitan State College of Denver; Mary A. Nel- son, University of Colorado at Boulder; J. Kyle Pula, University of Den- ver; Jon Von Stroh, University of Denver; Sharon Butz, University of Denver; Daniel Daly, University of Denver; Tracy Lawrence, Arapa- hoe Community College; Shawna Mahan, University of Colorado Den- ver; Adam Norris, University of Colorado at Boulder; Anca Radulescu, University of Colorado at Boulder; Mike Kawai, University of Colorado Denver; Janet Barnett, Colorado State University–Pueblo; Byron Hur- ley, Colorado State University–Pueblo; Jonathan Portiz, Colorado State University–Pueblo; Bill Emerson, Metropolitan State College of Denver; Suzanne Caulk, Regis University;Anton Dzhamay, University of Northern Colorado CONNECTICUT Jeffrey McGowan, Central Connecticut State University; Ivan Gotchev, Central Connecticut State University; CharlesWaiveris, Central Connecticut State University; Christopher Ham- mond, Connecticut College; Anthony Y. Aidoo, Eastern Connecticut State University; Kim Ward, Eastern Connecticut State University; Joan W. Weiss, Fairfield University; Theresa M. Sandifer, Southern Connecti- cut State University; Cristian Rios, Trinity College; Melanie Stein, Trinity College; Steven Orszag, Yale University DELAWARE Patrick F. Mw- erinde, University of Delaware DISTRICT OF COLUMBIA Jef- frey Hakim, American University; Joshua M. Lansky, American Univer- sity; James A. Nickerson, Gallaudet University FLORIDA Gregory Spradlin, Embry-Riddle University at Daytona Beach; Daniela Popova, Florida Atlantic University; Abbas Zadegan, Florida International Uni- versity; Gerardo Aladro, Florida International University; Gregory Hen- derson, Hillsborough Community College; Pam Crawford, Jacksonville University; Penny Morris, Polk Community College; George Schultz, St. Petersburg College; Jimmy Chang, St. Petersburg College; Carolyn Kistner, St. Petersburg College; Aida Kadic-Galeb, The University of Tampa; Constance Schober, University of Central Florida; S. Roy Choud- hury, University of Central Florida; Kurt Overhiser, Valencia Commu- nity College; Jiongmin Yong, University of Central Florida; Giray Okten, The Florida State University; Frederick Hoffman, Florida Atlantic Uni- versity; Thomas Beatty, Florida Gulf Coast University; Witny Librun, Palm Beach Community College North; Joe Castillo, Broward County College; Joann Lewin, Edison College; Donald Ransford, Edison Col- lege; Scott Berthiaume, Edison College; Alexander Ambrioso, Hillsbor- ough Community College; Jane Golden, Hillsborough Community Col- lege; Susan Hiatt, Polk Community College–Lakeland Campus; Li Zhou, Polk Community College–Winter Haven Campus; Heather Edwards, Semi- nole Community College; Benjamin Landon, Daytona State College; Tony Malaret, Seminole Community College; Lane Vosbury, Seminole Commu- nity College; William Rickman, Seminole Community College; Cheryl Cantwell, Seminole Community College; Michael Schramm, Indian River State College; Janette Campbell, Palm Beach Community College–Lake Worth; Kwai-Lee Chui, University of Florida; Shu-Jen Huang, Univer- sity of Florida GEORGIA Christian Barrientos, Clayton State Uni- versity; Thomas T. Morley, Georgia Institute of Technology; Doron Lu- binsky, Georgia Institute of Technology; Ralph Wildy, Georgia Military College; Shahram Nazari, Georgia Perimeter College; Alice Eiko Pierce, Georgia Perimeter College, Clarkson Campus; Susan Nelson, Georgia Perimeter College, Clarkson Campus; Laurene Fausett, Georgia South- ern University; Scott N. Kersey, Georgia Southern University; Jimmy L. Solomon, Georgia Southern University; Allen G. Fuller, Gordon Col- lege; Marwan Zabdawi, Gordon College; Carolyn A. Yackel, Mercer Uni- versity; Blane Hollingsworth, Middle Georgia State College; Shahryar Heydari, Piedmont College; Dan Kannan, The University of Georgia; June

xiv ACKNOWLEDGMENTS

Jones, Middle Georgia State College; Abdelkrim Brania, Morehouse Col- lege; Ying Wang, Augusta State University; James M. Benedict, Augusta State University; Kouong Law, Georgia Perimeter College; Rob Williams, Georgia Perimeter College; Alvina Atkinson, Georgia Gwinnett Col- lege; Amy Erickson, Georgia Gwinnett College HAWAII Shuguang Li, University of Hawaii at Hilo; Raina B. Ivanova, University of Hawaii at Hilo IDAHO Uwe Kaiser, Boise State University; Charles Kerr, Boise State University; Zach Teitler, Boise State University; Otis Kenny, Boise State University; Alex Feldman, Boise State University; Doug Bul- lock, Boise State University; Brian Dietel, Lewis-Clark State College; Ed Korntved, Northwest Nazarene University; Cynthia Piez, University of Idaho ILLINOIS Chris Morin, Blackburn College; Alberto L. Del- gado, Bradley University; John Haverhals, Bradley University; Herbert E. Kasube, Bradley University; Marvin Doubet, Lake Forest College; MarvinA. Gordon, Lake Forest Graduate School of Management; Richard J. Maher, Loyola University Chicago; Joseph H. Mayne, Loyola University Chicago; Marian Gidea, Northeastern Illinois University; John M.Alongi, Northwestern University; Miguel Angel Lerma, Northwestern Univer- sity; Mehmet Dik, Rockford College; Tammy Voepel, Southern Illinois University Edwardsville; Rahim G. Karimpour, Southern Illinois Univer- sity; Thomas Smith, University of Chicago; Laura DeMarco, University of Illinois; Evangelos Kobotis, University of Illinois at Chicago; Jennifer McNeilly, University of Illinois at Urbana-Champaign; Timur Oikhberg, University of Illinois at Urbana-Champaign; Manouchehr Azad, Harper College; Minhua Liu, Harper College; Mary Hill, College of DuPage; Arthur N. DiVito, Harold Washington College INDIANA Vania Mas- cioni, Ball State University; Julie A. Killingbeck, Ball State University; Kathie Freed, Butler University; Zhixin Wu, DePauw University; John P. Boardman, Franklin College; Robert N. Talbert, Franklin College; Robin Symonds, Indiana University Kokomo; Henry L. Wyzinski, In- diana University Northwest; Melvin Royer, Indiana Wesleyan Univer- sity; Gail P. Greene, Indiana Wesleyan University; David L. Finn, Rose- Hulman Institute of Technology; Chong Keat Arthur Lim, University of Notre Dame IOWA Nasser Dastrange, Buena Vista University; Mark A. Mills, Central College; Karen Ernst, Hawkeye Community College; Richard Mason, Indian Hills Community College; Robert S. Keller, Lo- ras College; Eric Robert Westlund, Luther College; Weimin Han, The University of Iowa KANSAS Timothy W. Flood, Pittsburg State Uni- versity; Sarah Cook, Washburn University; Kevin E. Charlwood, Wash- burn University; Conrad Uwe, Cowley County Community College; David N. Yetter, Kansas State University KENTUCKY Alex M. McAllister, Center College; Sandy Spears, Jefferson Community & Technical College; Leanne Faulkner, Kentucky Wesleyan College; Donald O. Clayton, Madis- onville Community College; Thomas Riedel, University of Louisville; Manabendra Das, University of Louisville; Lee Larson, University of Louisville; Jens E. Harlander, Western Kentucky University; Philip Mc- Cartney, Northern Kentucky University; Andy Long, Northern Kentucky University; Omer Yayenie, Murray State University; Donald Krug, North- ern Kentucky University LOUISIANA William Forrest, Baton Rouge Community College; Paul Wayne Britt, Louisiana State University; Galen Turner, Louisiana Tech University; Randall Wills, Southeastern Louisiana University; Kent Neuerburg, Southeastern Louisiana University; Guoli Ding, Louisiana State University; Julia Ledet, Louisiana State Univer- sity; Brent Strunk, University of Louisiana at Monroe MAINE An- drew Knightly, The University of Maine; Sergey Lvin, The University of Maine; Joel W. Irish, University of Southern Maine; Laurie Woodman, University of Southern Maine; David M. Bradley, The University of Maine; William O. Bray, The University of Maine MARYLAND Leonid Stern, Towson University; Jacob Kogan, University of Maryland Balti- more County; Mark E. Williams, University of Maryland Eastern Shore; Austin A. Lobo, Washington College; Supawan Lertskrai, Harford Com- munity College; Fary Sami, Harford Community College;Andrew Bulleri, Howard Community College MASSACHUSETTS Sean McGrath, Algonquin Regional High School; Norton Starr, Amherst College; Re- nato Mirollo, Boston College; Emma Previato, Boston University; Laura K Gross, Bridgewater State University; Richard H. Stout, Gordon Col-