ANNOTATED TEACHER’S EDITION Calculus - …p1cdn4static.sharpschool.com/UserFiles/Servers/Server_20226865/... · CHAPTER 1 Prerequisites for Calculus 2 1.1 Lines 3 • Increments

CalculusGraphical, Numerical, Algebraic

Ross L. FinneyFranklin D. Demana The Ohio State University

Bert K. Waits The Ohio State University

Daniel Kennedy Baylor School

*AP is a registered trademark of the College Board, which was not involved in the production of, and doesnot endorse, this product.

T H I R D E D I T I O N

A N N O TAT E D T E A C H E R ’ S E D I T I O N

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Library of Congress Cataloging-in-Publication Data

Calculus : graphical, numerical, algebraic / authors, Ross L. Finney … [et al.].--3rd ed.p. cm.

Includes index.ISBN 0-13-201409-2 (annotated teacher’s edition)1. Calculus--Textbooks. I. Finney, Ross L.

QA303.C1755 2006515--dc22 2005052702

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Foreword

This text, as the edition before it, was especially designed and written for teachers and stu-dents of Advanced Placement Calculus. Combining the scholarship of Ross Finney andFrank Demana, the technological expertise of Bert Waits, and the intimate knowledge ofand experience with the Advanced Placement Program of Dan Kennedy, this text is trulyunique among calculus texts. It may be used, in perfect order and without supplementa-tion, from the first day of the course until the day of the AP* exam. Teachers who are newto teaching calculus, as well as those who are very experienced, will be amazed at theinsightful and unique treatment of many topics.

The text is a perfect balance of exploration and theory. Students are asked to explore manytopics before theoretical proof. The topic of slope fields, studied at the beginning of Chapter6 when differential equations are first introduced, has been considerably expanded. Locallinearity, stressed throughout the text, permits the early introduction of l’Hôpital’s Rule.When the definite integral is introduced, students are first asked to find total change givenover a specific period of time given a rate of change before they consider geometric appli-cations. The section on logistic growth––so important in real-life situations––has beenexpanded. Functions are defined graphically, with tables, and with words as well as alge-braically throughout the text. Problems and exercises throughout are based on real-life sit-uations, and many are similar to questions appearing on the AP* exams. The series chapteruses technology to enhance understanding. This is a brilliant approach, and is the way thatseries should be presented. Students studying series from this chapter will gain a unique andthorough understanding of the topic. This textbook is one of a very few that teaches whatconditional convergence means. Chapter 10, Parametric, Vector, and Polar Functions, cov-ers vectors of two dimensions, and is perfect for students of Calculus BC. This chapterteaches exactly what the AP* student is expected to know about vector functions.

Ross Finney has passed away since this new edition was started, but his influence andscholarship are still keenly felt in the text. Throughout his life, Ross was always a masterteacher, but even he was amazed at the insight and brilliance of the team of Dan, Frank,and Bert. This new edition is well prepared to take student and teacher on their journeythrough AP* Calculus, and I recommend it with the highest enthusiasm. There is no morecomfortable, complete conveyance available anywhere.

—Judith Broadwin

Judy Broadwin taught AP* Calculus at Jericho High School for many years. In addition, she was a reader, tableleader, and eventually BC Exam leader of the AP* exam. She was a member to the Development Committee for AP*Calculus during the years that the AP* course descriptions were undergoing significant change. Judy now teachescalculus at Baruch College of the City of New York.


Foreword iii

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CHAPTER 1 Prerequisites for Calculus 2

1.1 Lines 3

• Increments • Slope of a Line • Parallel and Perpendicular Lines • Equations ofLines • Applications

1.2 Functions and Graphs 12

• Functions • Domains and Ranges • Viewing and Interpreting Graphs • EvenFunctions and Odd Functions—Symmetry • Functions Defined in Pieces •Absolute Value Function • Composite Functions

1.3 Exponential Functions 22

• Exponential Growth • Exponential Decay • Applications • The Number e

1.4 Parametric Equations 30

• Relations • Circles • Ellipses • Lines and Other Curves

1.5 Functions and Logarithms 37

• One-to-One Functions • Inverses • Finding Inverses • Logarithmic Functions • Properties of Logarithms • Applications

1.6 Trigonometric Functions 46

• Radian Measure • Graphs of Trigonometric Functions • Periodicity • Even andOdd Trigonometric Functions • Transformations of Trigonometric Graphs •Inverse Trigonometric Functions

Key Terms 55

Review Exercises 56

CHAPTER 2 Limits and Continuity 58

2.1 Rates of Change and Limits 59

• Average and Instantaneous Speed • Definition of Limit • Properties of Limits • One-sided and Two-sided Limits • Sandwich Theorem

2.2 Limits Involving Infinity 70

• Finite Limits as x → � � • Sandwich Theorem Revisited • Infinite Limits as x → a • End Behavior Models • “Seeing” Limits as x → � �

2.3 Continuity 78

• Continuity at a Point • Continuous Functions • Algebraic Combinations •Composites • Intermediate Value Theorem for Continuous Functions

2.4 Rates of Change and Tangent Lines 87

• Average Rates of Change • Tangent to a Curve • Slope of a Curve • Normal toa Curve • Speed Revisited

iv

Contents

Every section throughout the bookalso includes “Exploration” and“Extending the Ideas” features whichfollow the exercises.

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Key Terms 95

Review Exercises 95

CHAPTER 3 Derivatives 98

3.1 Derivative of a Function 99

• Definition of a Derivative • Notation • Relationship Between the Graphs of ƒ and ƒ� • Graphing the Derivative from Data • One-sided Derivatives

3.2 Differentiability 109

• How ƒ�(a) Might Fail to Exist • Differentiability Implies Local Linearity •Derivatives on a Calculator • Differentiability Implies Continuity • IntermediateValue Theorem for Derivatives

3.3 Rules for Differentiation 116

• Positive Integer Powers, Multiples, Sums, and Differences • Products andQuotients • Negative Integer Powers of x • Second and Higher OrderDerivatives

3.4 Velocity and Other Rates of Change 127

• Instantaneous Rates of Change • Motion along a Line • Sensitivity to Change • Derivatives in Economics

3.5 Derivatives of Trigonometric Functions 141

• Derivative of the Sine Function • Derivative of the Cosine Function • SimpleHarmonic Motion • Jerk • Derivatives of Other Basic Trigonometric Functions

3.6 Chain Rule 148

• Derivative of a Composite Function • “Outside-Inside” Rule • Repeated Useof the Chain Rule • Slopes of Parametrized Curves • Power Chain Rule

3.7 Implicit Differentiation 157

• Implicitly Defined Functions • Lenses, Tangents, and Normal Lines •Derivatives of Higher Order • Rational Powers of Differentiable Functions

3.8 Derivatives of Inverse Trigonometric Functions 165

• Derivatives of Inverse Functions • Derivative of the Arcsine • Derivative of theArctangent • Derivative of the Arcsecant • Derivatives of the Other Three

3.9 Derivatives of Exponential and Logarithmic Functions 172

• Derivative of ex • Derivative of ax • Derivative of ln x • Derivative of logax •Power Rule for Arbitrary Real Powers

Calculus at Work 181

Key Terms 181

Review Exercises 181

Contents v

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CHAPTER 4 Applications of Derivatives 186

4.1 Extreme Values of Functions 187

• Absolute (Global) Extreme Values • Local (Relative) Extreme Values • FindingExtreme Values

4.2 Mean Value Theorem 196

• Mean Value Theorem • Physical Interpretation • Increasing and DecreasingFunctions • Other Consequences

4.3 Connecting ƒ� and ƒ� with the Graph of ƒ 205

• First Derivative Test for Local Extrema • Concavity • Points of Inflection •Second Derivative Test for Local Extrema • Learning about Functions fromDerivatives

4.4 Modeling and Optimization 219

• Examples from Mathematics • Examples from Business and Industry •Examples from Economics • Modeling Discrete Phenomena with DifferentiableFunctions

4.5 Linearization and Newton’s Method 233

• Linear Approximation • Newton’s Method • Differentials • Estimating Changewith Differentials • Absolute, Relative, and Percentage Change • Sensitivity toChange

4.6 Related Rates 246

• Related Rate Equations • Solution Strategy • Simulating Related Motion

Key Terms 255


CHAPTER 5 The Definite Integral 262

5.1 Estimating with Finite Sums 263

• Distance Traveled • Rectangular Approximation Method (RAM) • Volume of aSphere • Cardiac Output

5.2 Definite Integrals 274

• Riemann Sums • Terminology and Notation of Integration • Definite Integraland Area • Constant Functions • Integrals on a Calculator • DiscontinuousIntegrable Functions

5.3 Definite Integrals and Antiderivatives 285

• Properties of Definite Integrals • Average Value of a Function • Mean ValueTheorem for Definite Integrals • Connecting Differential and Integral Calculus

5.4 Fundamental Theorem of Calculus 294

• Fundamental Theorem, Part 1 • Graphing the Function ∫xa ƒ(t)dt • FundamentalTheorem, Part 2 • Area Connection • Analyzing Antiderivatives Graphically

vi Contents

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5.5 Trapezoidal Rule 306

• Trapezoidal Approximations • Other Algorithms • Error Analysis

Key Terms 315



CHAPTER 6 Differential Equations and Mathematical Modeling 320

6.1 Slope Fields and Euler’s Method 321

• Differential Equations • Slope Fields • Euler’s Method

6.2 Antidifferentiation by Substitution 331

• Indefinite Integrals • Leibniz Notation and Antiderivatives • Substitution inIndefinite Integrals • Substitution in Definite Integrals

6.3 Antidifferentiation by Parts 341

• Product Rule in Integral Form • Solving for the Unknown Integral • TabularIntegration • Inverse Trigonometric and Logarithmic Functions

6.4 Exponential Growth and Decay 350

• Separable Differential Equations • Law of Exponential Change • ContinuouslyCompounded Interest • Radioactivity • Modeling Growth with Other Bases •Newton’s Law of Cooling

6.5 Logistic Growth 362

• How Populations Grow • Partial Fractions • The Logistic Differential Equation • Logistic Growth Models

Key Terms 372



CHAPTER 7 Applications of Definite Integrals 378

7.1 Integral As Net Change 379

• Linear Motion Revisited • General Strategy • Consumption Over Time • NetChange from Data • Work

7.2 Areas in the Plane 390

• Area Between Curves • Area Enclosed by Intersecting Curves • Boundarieswith Changing Functions • Integrating with Respect to y • Saving Time withGeometry Formulas

7.3 Volumes 399

• Volume As an Integral • Square Cross Sections • Circular Cross Sections •Cylindrical Shells • Other Cross Sections

Contents vii

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7.4 Lengths of Curves 412

• A Sine Wave • Length of Smooth Curve • Vertical Tangents, Corners, andCusps

7.5 Applications from Science and Statistics 419

• Work Revisited • Fluid Force and Fluid Pressure • Normal Probabilities


Key Terms 430


CHAPTER 8 Sequences, L’Hôpital’s Rule, and Improper Integrals 434

8.1 Sequences 435

• Defining a Sequence • Arithmetic and Geometric Sequences • Graphing aSequence • Limit of a Sequence

8.2 L’Hôpital’s Rule 444

• Indeterminate Form 0/0 • Indeterminate Forms �/�, � � 0, and � � � •Indeterminate Forms 1�, 00, �0

8.3 Relative Rates of Growth 453

• Comparing Rates of Growth • Using L’Hôpital’s Rule to Compare GrowthRates • Sequential versus Binary Search

8.4 Improper Integrals 459

• Infinite Limits of Integration • Integrands with Infinite Discontinuities • Testfor Convergence and Divergence • Applications

Key Terms 470


CHAPTER 9 Infinite Series 472

9.1 Power Series 473

• Geometric Series • Representing Functions by Series • Differentiation andIntegration • Identifying a Series

9.2 Taylor Series 484

• Constructing a Series • Series for sin x and cos x • Beauty Bare • Maclaurinand Taylor Series • Combining Taylor Series • Table of Maclaurin Series

9.3 Taylor’s Theorem 495

• Taylor Polynomials • The Remainder • Remainder Estimation Theorem •Euler’s Formula

9.4 Radius of Convergence 503

• Convergence • nth-Term Test • Comparing Nonnegative Series • Ratio Test •Endpoint Convergence

viii Contents

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9.5 Testing Convergence at Endpoints 513

• Integral Test • Harmonic Series and p-series • Comparison Tests • AlternatingSeries • Absolute and Conditional Convergence • Intervals of Convergence •A Word of Caution

Key Terms 526



CHAPTER 10 Parametric, Vector, and Polar Functions 530

10.1 Parametric Functions 531

• Parametric Curves in the Plane • Slope and Concavity • Arc Length • Cycloids

10.2 Vectors in the Plane 538

• Two-Dimensional Vectors • Vector Operations • Modeling Planar Motion •Velocity, Acceleration, and Speed • Displacement and Distance Traveled

10.3 Polar Functions 548

• Polar Coordinates • Polar Curves • Slopes of Polar Curves • Areas Enclosedby Polar Curves • A Small Polar Gallery

Key Terms 559


APPENDIX

A1 Formulas from Precalculus Mathematics 562

A2 Mathematical Induction 566

A3 Using the Limit Definition 569

A4 Proof of the Chain Rule 577

A5 Conic Sections 578

A6 Hyperbolic Functions 603

A7 A Brief Table of Integrals 612

Glossary 618

Additional Answers 629

Applications Index 691

Index 695

Contents ix

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About the Authors

Ross L. FinneyRoss Finney received his undergraduate degree and Ph.D. from the University of Michigan at Ann Arbor. He taught at the Universityof Illinois at Urbana–Champaign from 1966 to 1980 and at the Massachusetts Institute of Technology (MIT) from 1980 to 1990. Dr.Finney worked as a consultant for the Educational Development Center in Newton, Massachusetts. He directed the UndergraduateMathematics and its Applications Project (UMAP) from 1977 to 1984 and was founding editor of the UMAP Journal. In 1984, hetraveled with a Mathematical Association of America (MAA) delegation to China on a teacher education project through People toPeople International.

Dr. Finney coauthored a number of Addison-Wesley textbooks, including Calculus; Calculus and Analytic Geometry; ElementaryDifferential Equations with Linear Algebra; and Calculus for Engineers and Scientists. Dr. Finney’s coauthors were deeply saddenedby the death of their colleague and friend Ross Finney on August 4, 2000.

Franklin D. DemanaFrank Demana received his master’s degree in mathematics and his Ph.D. from Michigan State University. Currently, he is ProfessorEmeritus of Mathematics at The Ohio State University. As an active supporter of the use of technology to teach and learn mathemat-ics, he is cofounder of the national Teachers Teaching with Technology (T3) professional development program. He has been the direc-tor and codirector of more than $10 million of National Science Foundation (NSF) and foundational grant activities. He is currentlya co-principal investigator on a $3 million grant from the U.S. Department of Education Mathematics and Science EducationalResearch program awarded to The Ohio State University. Along with frequent presentations at professional meetings, he has publisheda variety of articles in the areas of computer- and calculator-enhanced mathematics instruction. Dr. Demana is also cofounder (withBert Waits) of the annual International Conference on Technology in Collegiate Mathematics (ICTCM). He is co-recipient of the 1997Glenn Gilbert National Leadership Award presented by the National Council of Supervisors of Mathematics, and co-recipient of the1998 Christofferson-Fawcett Mathematics Education Award presented by the Ohio Council of Teachers of Mathematics.

Dr. Demana coauthored Precalculus: Graphical, Numerical, Algebraic; Essential Algebra: A Calculator Approach; Transition toCollege Mathematics; College Algebra and Trigonometry: A Graphing Approach; College Algebra: A Graphing Approach;Precalculus: Functions and Graphs; and Intermediate Algebra: A Graphing Approach.

Bert K. WaitsBert Waits received his Ph.D. from The Ohio State University and is currently Professor Emeritus of Mathematics there. Dr. Waits iscofounder of the national Teachers Teaching with Technology (T3) professional development program, and has been codirector or prin-cipal investigator on several large National Science Foundation projects. Dr. Waits has published articles in more than 50 nationally rec-ognized professional journals. He frequently gives invited lectures, workshops, and minicourses at national meetings of the MAA and theNational Council of Teachers of Mathematics (NCTM) on how to use computer technology to enhance the teaching and learning of math-ematics. He has given invited presentations at the International Congress on Mathematical Education (ICME-6, -7, and -8) in Budapest(1988), Quebec (1992), and Seville (1996). Dr. Waits is co-recipient of the 1997 Glenn Gilbert National Leadership Award presented bythe National Council of Supervisors of Mathematics, and is the cofounder (with Frank Demana) of the ICTCM. He is also co-recipientof the 1998 Christofferson-Fawcett Mathematics Education Award presented by the Ohio Council of Teachers of Mathematics.

Dr. Waits coauthored Precalculus: Graphical, Numerical, Algebraic; College Algebra and Trigonometry: A Graphing Approach;College Algebra: A Graphing Approach; Precalculus: Functions and Graphs; and Intermediate Algebra: A Graphing Approach.

Daniel KennedyDan Kennedy received his undergraduate degree from the College of the Holy Cross and his master’s degree and Ph.D. in mathemat-ics from the University of North Carolina at Chapel Hill. Since 1973 he has taught mathematics at the Baylor School in Chattanooga,Tennessee, where he holds the Cartter Lupton Distinguished Professorship. Dr. Kennedy became an Advanced Placement Calculusreader in 1978, which led to an increasing level of involvement with the program as workshop consultant, table leader, and examleader. He joined the Advanced Placement Calculus Test Development Committee in 1986, then in 1990 became the first high schoolteacher in 35 years to chair that committee. It was during his tenure as chair that the program moved to require graphing calculatorsand laid the early groundwork for the 1998 reform of the Advanced Placement Calculus curriculum. The author of the 1997 Teacher’sGuide––AP® Calculus, Dr. Kennedy has conducted more than 50 workshops and institutes for high school calculus teachers. His arti-cles on mathematics teaching have appeared in the Mathematics Teacher and the American Mathematical Monthly, and he is a fre-quent speaker on education reform at professional and civic meetings. Dr. Kennedy was named a Tandy Technology Scholar in 1992and a Presidential Award winner in 1995.

Dr. Kennedy coauthored Precalculus: Graphical, Numerical, Algebraic; Prentice Hall Algebra I; Prentice Hall Geometry; andPrentice Hall Algebra 2.

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xi

To the Teacher

The main goal of this third edition is to realign the content with the changes in theAdvanced Placement (AP*) calculus syllabus and the new type of AP* exam questions. Wehave also more carefully connected examples and exercises and updated the data used inexamples and exercises. Cumulative Quick Quizzes are now provided two or three timesin each chapter.

The course outlines for AP* Calculus reflect changes in the goals and philosophy of cal-culus courses now being taught in colleges and universities. The following objectivesreflect the goals of the curriculum.

• Students should understand the meaning of the derivative in terms of rate of changeand local linear approximations.

• Students should be able to work with functions represented graphically, numerically,analytically, or verbally, and should understand the connections among these repre-sentations.

• Students should understand the meaning of the definite integral both as a limit ofRiemann sums and as a net accumulation of a rate of change, and understand the rela-tionship between the derivative and integral.

• Students should be able to model problem situations with functions, differential equa-tions, or integrals, and communicate both orally and in written form.

• Students should be able to represent differential equations with slope fields, solveseparable differential equations analytically, and solve differential equations usingnumerical techniques such as Euler’s method.

• Students should be able to interpret convergence and divergence of series using tech-nology, and to use technology to help solve problems. They should be able to repre-sent functions with series and find the Lagrange error bound for Taylor polynomials.

This revision of Finney/Thomas/Demana/Waits Calculus completely supports the con-tent, goals, and philosophy of the new advanced placement calculus course description.

Calculus is explored through the interpretation of graphs and tables as well as analyticmethods (multiple representation of functions). Derivatives are interpreted as rates ofchange and local linear approximation. Local linearity is used throughout the book. Thedefinite integral is interpreted as total change over a specific interval and as a limit ofRiemann sums. Problem situations are modeled with integrals. Chapter 6 focuses on theuse of differential equations to model problems. We interpret differential equations usingslope fields and then solve them analytically or numerically. Convergence and divergenceof series are interpreted graphically and the Lagrange error bound is used to measure theaccuracy of approximating functions with Taylor polynomials.

The use of technology is integrated throughout the book to provide a balanced approachto the teaching and learning of calculus that involves algebraic, numerical, graphical, andverbal methods (the rule of four). Students are expected to use a multirepresentationalapproach to investigate and solve problems, to write about their conclusions, and often towork in groups to communicate mathematics orally. This book reflects what we have learnedabout the appropriate use of technology in the classroom during the last decade.

The visualizations and technological explorations pioneered by Demana and Waits areincorporated throughout the book. A steady focus on the goals of the advanced placementcalculus curriculum has been skillfully woven into the material by Kennedy, a master highschool calculus teacher. Suggestions from numerous teachers have helped us shape thismodern, balanced, technological approach to the teaching and learning of calculus.


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CHANGES FOR THIS EDITIONThe course descriptions for the two Advanced Placement courses (Calculus AB andCalculus BC) have changed over the years to respond to new technology and to new pointsof emphasis in college and university courses. The updated editions of this textbook haveconsistently responded to those changes to make it easier for students and teachers to adjust.This latest edition contains significantly enhanced coverage of the following topics:

• Slope fields, now a topic for both AB and BC students, are studied in greater depthand are used to visualize differential equations from the beginning.

• Euler’s method, currently a BC topic, is used as a numerical technique (with multipleexamples) for solving differential equations using the insights gained from slope fields.

• Local linearity, a point of emphasis in previous editions but now more important thanever for understanding various applications of the derivative, is now a thread runningthroughout the book.

• More examples and exercises have been added to illustrate the connections betweenthe graph of a function and the graph of its derivative (or the graph of ƒ and a func-tion defined as an integral of ƒ).

• The logistic differential equation, a BC topic that is covered weakly in most textbooksdespite its many applications, now has its own section.

Similarly, the coverage of some other topics has been trimmed to reflect the intent of theirinclusion in the AP* courses:

• The use of partial fractions for finding antiderivatives has been narrowed to distinctlinear factors in the denominator and has been more directly linked to the logisticdifferential equation;

• The treatment of vector calculus has been revised to focus on planar motion prob-lems, which are easily solved using earlier results componentwise;

• The treatment of polar functions has been narrowed to the polar topics in the BC coursedescription and has been linked more directly to the treatment of parametric functions.

Moreover, this latest edition continues to explore the ways teachers and students can usegraphing calculator technology to enhance their understanding of calculus topics.

This edition of the text also includes new features to further assist students in their studyof calculus:

• What You’ll Learn About… and Why introduces the big ideas in each section andexplains their purpose.

• At the end of each example students are encouraged to Now Try a related exercise atthe end of the section to check their comprehension.

• A Quick Quiz for AP* Preparation appears every few sections, requiring students toanswer questions about topics covered in multiple sections, to assist them in obtaininga conceptual understanding of the material.

• Each exercise set includes a group of Standardized Test Questions. Additionally, anAP* Examination Preparation appears at the end of each set of chapter reviewexercises.

For further information about new and continuing features, please consult the To theStudent material.

xii To the Teacher

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To the Teacher xiii

CONTINUING FEATURESBalanced ApproachA principal feature of this edition is the balance attained among the rule of four:analytic/algebraic, numerical, graphical, and verbal methods of representing problems. Webelieve that students must value all of these methods of representation, understand howthey are connected in a given problem, and learn how to choose the one(s) most appropri-ate for solving a particular problem.

The Rule of FourIn support of the rule of four, we use a variety of techniques to solve problems. Forinstance, we obtain solutions algebraically or analytically, support our results graphicallyor numerically with technology, and then interpret the result in the original problem con-text. We have written exercises where students are asked to solve problems by onemethod and then support or confirm their solutions by using another method. We wantstudents to understand that technology can be used to support (but not prove) results, andthat algebraic or analytic techniques are needed to prove results. We want students tounderstand that mathematics provides the foundation that allows us to use technology tosolve problems.

ApplicationsThe text includes a rich array of interesting applications from biology, business, chem-istry, economics, engineering, finance, physics, the social sciences, and statistics. Someapplications are based on real data from cited sources. Students are exposed to func-tions as mechanisms for modeling data and learn about how various functions canmodel real-life problems. They learn to analyze and model data, represent data graphi-cally, interpret from graphs, and fit curves. Additionally, the tabular representations ofdata presented in the text highlight the concept that a function is a correspondencebetween numerical variables, helping students to build the connection between thenumbers and the graphs.

ExplorationsStudents are expected to be actively involved in understanding calculus concepts and solv-ing problems. Often the explorations provide a guided investigation of a concept. Theexplorations help build problem-solving ability by guiding students to develop a mathe-matical model of a problem, solve the mathematical model, support or confirm the solu-tion, and interpret the solution. The ability to communicate their understanding is just asimportant to the learning process as reading or studying, not only in mathematics but inevery academic pursuit. Students can gain an entirely new perspective on their knowledgewhen they explain what they know in writing.

Graphing UtilitiesThe book assumes familiarity with a graphing utility that will produce the graph of afunction within an arbitrary viewing window, find the zeros of a function, compute thederivative of a function numerically, and compute definite integrals numerically.Students are expected to recognize that a given graph is reasonable, identify all theimportant characteristics of a graph, interpret those characteristics, and confirm themusing analytic methods. Toward that end, most graphs appearing in this book resemblestudents’ actual grapher output or suggest hand-drawn sketches. This is one of the firstcalculus textbooks to take full advantage of graphing calculators, philosophicallyrestructuring the course to teach new things in new ways to achieve new understanding,while (courageously) abandoning some old things and old ways that are no longer serv-ing a purpose.

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Exercise SetsThe exercise sets were revised extensively for this edition, including many new ones.There are nearly 4,000 exercises, with more than 80 Quick Quiz exercises and 560 QuickReview exercises. The different types of exercises included are:

Algebraic and analytic manipulation

Interpretation of graphs

Graphical representations

Numerical representations

Explorations

Writing to learn

Group activities

Data analyses

Descriptively titled applications

Extending the ideas

Each exercise set begins with the Quick Review feature, which can be used to introducelessons, support Examples, and review prerequisite skills. The exercises that follow aregraded from routine to challenging. An additional block of exercises, Extending the Ideas,may be used in a variety of ways, including group work. We also provide Review Exercisesand AP* Examination Preparation at the end of each chapter.

SUPPLEMENTS AND RESOURCESFor the StudentStudent Edition, ISBN 0-13-201408-4

Preparing for the Calculus AP* Exam, ISBN 0-321-33574-0• Introduction to the AP* AB and BC Calculus Exams

• Precalculus Review of Calculus Prerequisites

• Review of AP* Calculus AB and Calculus BC Topics

• Practice Exams

• Answers and Solutions

Student Practice Workbook, ISBN 0-13-201411-4• New examples that parallel key examples from each section in the book are provided

along with a detailed solution

• Related practice problems follow each example

Texas Instruments Graphing Calculator Manual, ISBN 0-13-201415-7• An introduction to Texas Instruments’ graphing calculators, as they are used for

calculus

• Features the TI-84 Plus Silver Edition, the TI-86, and the TI-89 Titanium. The key-strokes, menus and screens for the TI-83 Plus, TI-83 Plus Silver Edition, and the TI-84 Plus are similar to the TI-84 Plus Silver Edition and the TI-89, TI-92 Plus, andVoyage™ 200 are similar to the TI-89 Titanium.

For the TeacherAnnotated Teacher Edition, ISBN 0-13-201409-2

• Answers included on the same page as the problem appears, for most exercises

xiv To the Teacher

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• Solutions to Chapter Opening Problems, Teaching Notes, Common Errors, Notes onExamples and Exploration Extensions, and Assignment Guide included at the begin-ning of the book.

Teacher’s AP* Correlations and Preparation Guide, 0-13-201413-0

• Calculus AB/BC topic correlations, Pacing Guides for AB/BC, Assignment Guides,Concepts Worksheets, Group Activity Explorations, Sample Tests, and Answers

Assessment Resources, 0-13-201412-2

• Chapter quizzes, chapter tests, semester tests, final tests, and alternate assessments,along with all answers

Solutions Manual, ISBN 0-13-201414-9

• Complete solutions for Quick Reviews, Exercises, Explorations, and Chapter Reviews

Transparencies, ISBN 0-13-201410-6

• Full color transparencies for key figures from the text

Technology Resources

MathXL® www.mathxl.comMathXL® is a powerful online homework, tutorial, and assessment system that accompa-nies our textbooks in mathematics or statistics. With MathXL, instructors can create, edit,and assign online homework and tests using algorithmically generated exercises correlat-ed at the objective level to the textbook. They can also create and assign their own onlineexercises and import TestGen tests for added flexibility. All student work is tracked inMathXL’s online gradebook. Students can take chapter tests in MathXL and receive per-sonalized study plans based on their test results. The study plan diagnoses weaknesses andlinks students directly to tutorial exercises for the objectives they need to study and retest.Students can also access supplemental animations and video clips directly from selectedexercises. For more information, visit our Web site at www.mathxl.com, or contact yourlocal sales representative.

InterAct Math Tutorial Web site, www.interactmath.comGet practice and tutorial help online! This interactive tutorial Web site provides algorith-mically generated practice exercises that correlate directly to the exercises in the textbook.Students can retry an exercise as many times as they like with new values each time forunlimited practice and mastery. Every exercise is accompanied by an interactive guidedsolution that provides helpful feedback for incorrect answers, and students can also viewa worked-out sample problem that steps them through an exercise similar to the one they'reworking on.

Video Lectures on CD, ISBN 0-13-2030709-5The video lectures feature engaging mathematics instructors who present comprehensivecoverage of the core topics of the text. The presentations include examples and exercisesfrom the text and support an approach that emphasizes visualization and problem-solving.

TestGen®, ISBN 0-13-201419-XTestGen® enables instructors to build, edit, print, and administer tests using a computer-ized bank of questions developed to cover all the objectives of the text. TestGen is algo-rithmically based, allowing instructors to create multiple but equivalent versions of thesame question or test with the click of a button. Instructors can also modify test questionsor add new questions by using the built-in question editor, which allows users to create

To the Teacher xv

5128_FM_TE_ppi-xxiv 1/18/06 1:08 PM Page xv

graphs, import graphics, and insert math notation, variable numbers, or text. Tests can beprinted or administered online via the Internet or another network. TestGen comes pack-aged with QuizMaster, which allows students to take tests on a local area network. Thesoftware is available on a dual-platform Windows/Macintosh CD-ROM.

Presentation Express CD-ROM, ISBN 0-13-201420-3This time saving component includes all the transparencies in PowerPoint format aswell as section-by-section lecture notes for the entire book, making it easier for you toteach and to customize based on your teaching preferences. All slides can be cus-tomized and edited.

Teacher Express CD-ROM (with LessonView), ISBN 0-13-201422-XPlan – Teach – Assess. TeacherEXPRESS is a new suite of instructional tools on CD-ROMto help teachers plan, teach, and assess at the click of a mouse. Powerful lesson planning,resource management, testing, and an interactive teacher’s edition all in one place makeclass preparation quick and easy! Contents: Planning Express, Teacher’s Edition, ProgramTeaching Resources, Correlations, and Links to Other Resources.

Student Express CD-ROM, ISBN 0-13-201421-1An interactive textbook on CD-ROM makes this the perfect student tool for studying ortest review.

Technology Resource Manual: Casio and HP CalculatorsAvailable for download from the PHSchool.com Web site (http://www.phschool.com/).Enter the code aze-0002 in the Web Codes box in the upper-left corner of the home page.Please note the Web Code is case sensitive.

xvi To the AP* Student

To the AP* Student

We know that as you study for your AP* course, you’re preparing along the way for theAP* exam. By tying the material in this book directly to AP* course goals and exam top-ics, we help you to focus your time most efficiently. And that’s a good thing!

The AP* exam is an important milestone in your education. A high score will positionyou optimally for college acceptance—and possibly will give you college credits that putyou a step ahead. Our primary commitment is to provide you with the tools you need toexcel on the exam ... the rest is up to you!

Test-Taking Strategies for an Advanced Placement*Calculus Examination You should approach the AP* Calculus Examination the same way you would any majortest in your academic career. Just remember that it is a one-shot deal—you should be atyour peak performance level on the day of the test. For that reason you should do every-thing that your “coach” tells you to do. In most cases your coach is your classroom teacher.It is very likely that your teacher has some experience, based on workshop information orprevious students’ performance, to share with you.

You should also analyze your own test-taking abilities. At this stage in your education,you probably know your strengths and weaknesses in test-taking situations. You may bevery good at multiple choice questions but weaker in essays, or perhaps it is the other wayaround. Whatever your particular abilities are, evaluate them and respond accordingly.Spend more time on your weaker points. In other words, rather than spending time in yourcomfort zone where you need less work, try to improve your soft spots. In all cases, con-centrate on clear communication of your strategies, techniques, and conclusions.

5128_FM_TE_ppi-xxiv 1/18/06 1:08 PM Page xvi

To the AP* Student xvii

The following table presents some ideas in a quick and easy form.

General Strategies for AP* Examination Preparation

Time DosThrough the Year • Register with your teacher/coordinator

• Pay your fee (if applicable) on time• Take good notes• Work with others in study groups• Review on a regular basis• Evaluate your test-taking strengths and weaknesses—

keep track of how successful you are when guessingThe Week Before • Combine independent and group review

• Get tips from your teacher• Do lots of mixed review problems• Check your exam date, time, and location• Review the appropriate AP* Calculus syllabus (AB or BC)

The Night Before • Put new batteries in your calculator• Make sure your calculator is on the approved list• Lay out your clothes and supplies so that you are ready to

go out the door• Do a short review• Go to bed at a reasonable hour

Exam Day • Get up a little earlier than usual• Eat a good breakfast/lunch• Put some hard candy in your pocket in case you need an

energy boost during the test• Get to your exam location 15 minutes early

Exam Night • Relax—you earned it

Topics from the Advanced Placement* Curriculum for Calculus AB, Calculus BCAs an AP* Student, you are probably well aware of the good study habits that are neededto be a successful student in high school and college:

• attend all the classes

• ask questions (either during class or after)

• take clear and understandable notes

• make sure you understand the concepts rather than memorizing formulas

• do your homework; extend your test-prep time over several days or weeks, instead ofcramming

• use all the resources—text and people—that are available to you.

No doubt this list of “good study habits” is one that you have seen or heard before. Youshould know that there is powerful research that suggests a few habits or routines willenable you to go beyond “knowing about” calculus, to more deeply “understanding” cal-culus. Here are three concrete actions for you to consider:

• Review your notes at least once a week and rewrite them in summary form.

• Verbally explain concepts (theorems, etc.) to a classmate.

5128_FM_TE_ppi-xxiv 1/18/06 1:08 PM Page xvii

xviii To the AP* Student

Topic Outline for AP* Calculus AB and AP* Calculus BC

(excerpted from the College Board's Course Description - Calculus: Calculus AB, Calculus BC, May 2007)

I. Calculus Exam Functions, Graphs, and Limits Calculus

A AB BC Analysis of graphs 1.2–1.6B AB BC Limits of functions (including one-sided limits)

B1 AB BC An intuitive understanding of the limiting process 2.1, 2.2B2 AB BC Calculating limits using algebra 2.1, 2.2B3 AB BC Estimating limits from graphs or tables of data 2.1, 2.2

C AB BC Asymptotic and unbounded behaviorC1 AB BC Understanding asymptotes in terms of graphical behavior 2.2C2 AB BC Describing asymptotic behavior in terms of limits involving infinity 2.2C3 AB BC Comparing relative magnitudes of functions and their rates of change 2.2, 2.4, 8.3

D AB BC Continuity as a property of functionsD1 AB BC An intuitive understanding of continuity 2.3D2 AB BC Understanding continuity in terms of limits 2.3D3 AB BC Geometric understanding of graphs of continuous functions 2.3, 4.1–4.3

E BC Parametric, polar, and vector functions 10.1 10.3

II. Calculus Exam Derivatives Calculus

A AB BC Concept of the derivativeA1 AB BC Derivative presented graphically, numerically, and analytically 2.4–4.5A2 AB BC Derivative interpreted as an instantaneous rate of change 2.4A3 AB BC Derivative defined as the limit of the difference quotient 2.4–3.1A4 AB BC Relationship between differentiability and continuity 3.2

B AB BC Derivative at a pointB1 AB BC Slope of a curve at a point 2.4B2 AB BC Tangent line to a curve at a point and local linear approximation 2.4, 4.5B3 AB BC Instantaneous rate of change as the limit of average rate of change 2.4, 3.4B4 AB BC Approximate rate of change from graphs and tables of values 2.4, 3.4

C AB BC Derivative as a functionC1 AB BC Corresponding characteristics of graphs of ƒ and ƒ� 3.1, 4.3

• Form a study group that meets regularly to do homework and discuss reading and lec-ture notes.

Most of these tips boil down to one mantra, which all mathematicians believe in:

Math is not a spectator sport.

The AP* Calculus Examination is based on the following Topic Outline. For your con-venience, we have noted all Calculus AB and Calculus BC objectives with clear indica-tions of topics required only by the Calculus BC Exam. The outline cross-references eachAP* Calculus objective with the appropriate section(s) of this textbook: Calculus:Graphical, Numerical, Algebraic, Third Edition, by Finney, Demana, Waits, and Kennedy.

Use this outline to track your progress through the AP* exam topics. Be sure to coverevery topic associated with the exam you are taking. Check it off when you have studiedand/or reviewed the topic.

Even as you prepare for your exam, I hope this book helps you map—and enjoy—yourcalculus journey!

—John BrunstingHinsdale Central High School

5128_FM_TE_ppi-xxiv 1/18/06 1:08 PM Page xviii

To the AP* Student xix

C2 AB BC Relationship between the increasing and decreasing behavior of ƒ and the sign of ƒ� 4.1, 4.3

C3 AB BC The Mean Value Theorem and its geometric consequences. 4.2C4 AB BC Equations involving derivatives. Verbal descriptions are translated 3.4, 3.5,

into equations involving derivatives and vice versa 4.6, 6.4, 6.5D AB BC Second Derivatives

D1 AB BC Corresponding characteristics of graphs of ƒ, ƒ� and ƒ� 4.3D2 AB BC Relationship between the concavity of ƒ and the sign of ƒ� 4.3D3 AB BC Points of inflection as places where concavity changes 4.3

E AB BC Applications of derivativesE1 AB BC Analysis of curves, including the notions of monotonicity and concavity 4.1–4.3E2 BC Analysis of planar curves given in parametric form, polar form, and

vector form, including velocity and acceleration vectors 10.1–10.3E3 AB BC Optimization, both absolute (global) and relative (local) extrema 4.3, 4.4E4 AB BC Modeling rates of change, including related rates problems 4.6E5 AB BC Use of implicit differentiation to find the derivative of an inverse function 3.7E6 AB BC Interpretation of the derivative as a rate of change in varied applied

contexts, including velocity, speed, and acceleration 3.4E7 AB BC Geometric interpretation of differential equations via slope fields and

the relationship between slope fields and solution curves for differential equations 6.1

E8 BC Numerical solution of differential equations using Euler’s method 6.1E9 BC L’Hopital’s Rule, including its use in determining limits and convergence

of improper integrals and series 8.1, 9.5F AB BC Computation of derivatives

F1 AB BC Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric 3.3, 3.5, functions 3.8, 3.9

F2 AB BC Basic rules for the derivative of sums, products, and quotients of functions 3.3F3 AB BC Chain rule and implicit differentiation 3.6, 3.7F4 BC Derivatives of parametric, polar, and vector functions 10.1–10.3

III. Calculus Exam Integrals Calculus

A Interpretations and properties of definite integralsA1 AB BC Definite integral as a limit of Riemann sums 5.1, 5.2A2 AB BC Definite integral of the rate of change of a quantity over an interval

interpreted as the change of the quantity over the closed interval [a,b] of ∫ƒ�(x)dx = ƒ(b) – ƒ(a) 5.1, 5.4

A3 AB BC Basic properties of definite integrals (Examples include additivity and linearity.) 5.2 - 5.3

B Applications of integralsB1a AB BC Appropriate integrals are used in a variety of applications to model

physical, biological, or economic situations. ... students should be able to adapt their knowledge and techniques. Emphasis is on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. … specific applications should include using the integral of a rate of change to give accumulated change, finding the area of a region, the volume of a solid with known cross 5.4, 5.5,sections, the average value of a function, and the distance traveled by 6.4, 6.5,a particle along a line 7.1–7.5

B1b BC Appropriate integrals are used … specific applications should include … finding the area of a region bounded by polar curves ... and the length 7.4, of a curve (including a curve given in parametric form) 10.1, 10.3

C Fundamental Theorem of CalculusC1 AB BC Use of the Fundamental Theorem to evaluate definite integrals 5.4C2 AB BC Use of the Fundamental Theorem to represent a particular antiderivative,

and the analytical and graphical analysis of functions so derived 5.4, 6.1

5128_FM_TE_ppi-xxiv 1/18/06 1:08 PM Page xix

xx To the AP* Student

D Techniques of antidifferentiationD1 AB BC Antiderivatives following directly from derivatives of basic functions 4.2, 6.1, 6.2D2a AB BC Antiderivatives by substitution of variables (including change of limits

for definite integrals) 6.2D2b BC Antiderivatives by … parts, and simple partial fractions (nonrepeating

linear factors only) 6.3, 6.5D3 BC Improper integrals (as limits of definite integrals) 8.3

E Applications of antidifferrentiationE1 AB BC Finding specific antiderivatives using initial conditions, including

applications to motion along a line 6.1, 7.1E2 AB BC Solving separable differential equations and using them in modeling

In particular, studying the equations y � = ky and exponential growth 6.4E3 BC Solving logistic differential equations and using them in modeling 6.5

F Numerical approximations to definite integralsF1 AB BC Use of Riemann and trapezoidal sums to approximate definite integrals of

functions represented algebraically, graphically, and by tables of values 5.2, 5.5

IV. Calculus Exam Polynomial Approximations and Series Calculus

A Concept of seriesA1 BC A series is defined as a sequence of partial sums, and convergence is

defined in terms of the limit of the sequence of partial sums. Technology can be used to explore convergence or divergence 9.1

B Series of constantsB1 BC Motivating examples, including decimal expansion 9.1B2 BC Geometric series with applications 9.1B3 BC The harmonic series 9.5B4 BC Alternating series with error bound 9.5B5 BC Terms of series as areas of rectangles and their relationship to

improper integrals, including the integral test and its use in testing the convergence of p-series 9.5

B6 BC The ratio test for convergence or divergence 9.4B7 BC Comparing series to test for convergence and divergence 9.4

C Taylor seriesC1 BC Taylor polynomial approximation with graphical demonstration of

convergence (For example, viewing graphs of various Taylor polynomials of the sine function approximating the sine curve.) 9.2

C2 BC Maclaurin series and the general Taylor series centered at x = a 9.2C3 BC Maclaurin series for the functions ex, sin x, cos x, and 1/(1 – x) 9.2C4 BC Formal manipulation of Taylor series and shortcuts to computing

Taylor series, including substitution, differentiation, antidifferentiation, and the formation of new series from known series 9.1, 9.2

C5 BC Functions defined by power series 9.1, 9.2C6 BC Radius and interval of convergence of power series 9.1, 9.4, 9.5C7 BC Lagrange error bound for Taylor polynomials 9.3

5128_FM_TE_ppi-xxiv 1/18/06 1:08 PM Page xx

Using the Book for Maximum EffectivenessSo, how can this book help you to join in the game of mathematics for a winning future?Let us show you some unique tools that we have included in the text to help prepare younot only for the AP* Calculus exam, but also for success beyond this course.

Chapter Openers provide a motivating photograph and applicationto show you an example that illustrates the relevance of what you’llbe learning in the chapter.

A Chapter Overview then follows to give you a sense of what youare going to learn. This overview provides a roadmap of the chapteras well as tells how the different topics in the chapter are connectedunder one big idea. It is always helpful to remember that mathemat-ics isn’t modular, but interconnected, and that the different skillsyou are learning throughout the course build on one another to helpyou understand more complex concepts.

Similarly, the What you’ll learn about…and why feature gives you the big ideas ineach section and explains their purpose. You should read this as you begin the sectionand always review it after you have completed the section to make sure you under-stand all of the key topics that you have just studied.

Margin Notes appear throughout the book on various topics. Some notes providemore information on a key concept or an example. Other notes offer practical adviceon using your graphing calculator to obtain themost accurate results.

Brief Historical Notes present the stories ofpeople and the research that they have done toadvance the study of mathematics. Reading thesenotes will often provide you with additionalinsight for solving problems that you can uselater when doing the homework or completingthe AP* Exam.

Chapter6 Differential Equationsand MathematicalModeling

One way to measure how light in the ocean di-minishes as water depth increases involvesusing a Secchi disk. This white disk is 30

centimeters in diameter, and is lowered into theocean until it disappears from view. The depth of thispoint (in meters), divided into 1.7, yields the coeffi-cient k used in the equation lx � l0e�kx. This equationestimates the intensity lx of light at depth x using l0,the intensity of light at the surface.

In an ocean experiment, if the Secchi disk disap-pears at 55 meters, at what depth will only 1% ofsurface radiation remain? Section 6.4 will help youanswer this question.

320

Chapter 6 Overview

One of the early accomplishments of calculus was predicting the future position of aplanet from its present position and velocity. Today this is just one of a number of occa-sions on which we deduce everything we need to know about a function from one of itsknown values and its rate of change. From this kind of information, we can tell how long asample of radioactive polonium will last; whether, given current trends, a population willgrow or become extinct; and how large major league baseball salaries are likely to be inthe year 2010. In this chapter, we examine the analytic, graphical, and numerical tech-niques on which such predictions are based.

6.1What you’ll learn about

• Differential Equations

• Slope Fields

• Euler’s Method

. . . and why

Differential equations have alwaysbeen a prime motivation for thestudy of calculus and remain soto this day.

Differential Equation Mode

If your calculator has a differentialequation mode for graphing, it is intended for graphing slope fields. Theusual “Y�” turns into a “dy�dx �”screen, and you can enter a function ofx and/or y. The grapher draws a slopefield for the differential equation whenyou press the GRAPH button.

Charles Richard Drew(1904–1950)

Millions of people arealive today becauseof Charles Drew’s pioneering work onblood plasma and thepreservation of humanblood for transfusion.After directing the Red

Cross program that collected plasma for the Armed Forces in World War II, Dr. Drew went on to become Head ofSurgery at Howard University and Chiefof Staff at Freedmen’s Hospital inWashington, D.C.

To the AP* Student xxi

5128_FM_TE_ppi-xxiv 2/3/06 3:33 PM Page xxi

Many examples include solutions to Solve Algebraically, SolveGraphically, or Solve Numerically. You should be able to use differentapproaches for finding solutions to problems. For instance, you would obtaina solution algebraically when that is the most appropriate technique to use,and you would obtain solutions graphically or numerically when algebra isdifficult or impossible to use. We urge you to solve problems by onemethod, then support or confirm your solution by using another method, andfinally, interpret the results in the context of the problem. Doing so rein-forces the idea that to understand a problem fully, you need to understand italgebraically, graphically, and numerically whenever possible.

Each example ends with a suggestion to Now Try a related exercise. Workingthe suggested exercise is an easy way for you to check your comprehensionof the material while reading each section, instead of waiting until the end ofeach section or chapter to see if you “got it.” True comprehension of the text-book is essential for your success on the AP* Exam.

Explorations appear throughoutthe text and provide you with theperfect opportunity to become an

active learner and discover mathematics onyour own. Honing your critical thinking andproblem-solving skills will ultimately benefityou on all of your AP* Exams.

Each exercise set begins with a QuickReview to help you review skills needed inthe exercise set, reminding you again thatmathematics is not modular. Each QuickReview includes section references to showwhere these skills were covered earlier in the text. If you find these problems overlychallenging, you should go back through the book and your notes to review the materialcovered in previous chapters. Remember, you need to understand the material from theentire calculus course for the AP* Calculus Exam, not just memorize the concepts fromthe last part of the course.

Along with the standard types of exercises, including skill-based, application,writing, exploration, and extension questions, each exercise set includes a group ofStandardized Test Questions. Each group includes two true-false with justificationsand four multiple-choice questions, with instructions about the permitted use of yourgraphing calculator.

Section 4.4 Modeling and Optimization 221

Examples from Business and Industry

To optimize something means to maximize or minimize some aspect of it. What is the sizeof the most profitable production run? What is the least expensive shape for an oil can?What is the stiffest rectangular beam we can cut from a 12-inch log? We usually answersuch questions by finding the greatest or smallest value of some function that we haveused to model the situation.

EXAMPLE 3 Fabricating a Box

An open-top box is to be made by cutting congruent squares of side length x from the cor-ners of a 20- by 25-inch sheet of tin and bending up the sides (Figure 4.38). How largeshould the squares be to make the box hold as much as possible? What is the resultingmaximum volume?

SOLUTION

Model The height of the box is x, and the other two dimensions are �20 � 2x� and�25 � 2x�. Thus, the volume of the box is

V�x� � x �20 � 2x��25 � 2x�.

Solve Graphically Because 2x cannot exceed 20, we have 0 x 10. Figure 4.39suggests that the maximum value of V is about 820.53 and occurs at x � 3.68.

Confirm Analytically Expanding, we obtain V�x� � 4x3 � 90x2 � 500x. The firstderivative of V is

V�x� � 12x2 � 180x � 500.

The two solutions of the quadratic equation V�x� � 0 are

c1 � � 3.68 and

c2 � � 11.32.

Only c1 is in the domain �0, 10� of V. The values of V at this one critical point and thetwo endpoints are

Critical point value: V�c1� � 820.53

Endpoint values: V�0� � 0, V�10� � 0.

Interpret Cutout squares that are about 3.68 in. on a side give the maximum volume,about 820.53 in3. Now try Exercise 7.

180 � �1�8�0�2�� 4�8��5�0�0��

24

180 � �1�8�0�2�� 4�8��5�0�0��

24

Figure 4.38 An open box made by cut-ting the corners from a piece of tin. (Example 3)

20"

x x

x x

x

x

x

x

x

20 � 2x

25 � 2x

(a)

25"

(b)

Figure 4.39 We chose the �300 in�300 y 1000 so that the coordinates of the local maximum at the bottom of thescreen would not interfere with the graph.(Example 3)

[0, 10] by [–300, 1000]

y � x(20 � 2x)(25 � 2x)

MaximumX = 3.6811856 Y = 820.52819

Constructing Cones

A cone of height h and radius r is constructed from a flat, circular disk of radius 4 in. byremoving a sector AOC of arc length x in. and then connecting the edges OA and OC.What arc length x will produce the cone of maximum volume, and what is that volume?

1. Show that

r � �8p

2p� x� , h � �1�6� �� r�2�, and

V�x� � �p

3� (�8p2p� x

�)2

�16 � (�8p2p� x�)2

.

2. Show that the natural domain of V is 0 x 16p. Graph V over this domain.

3. Explain why the restriction 0 x 8p makes sense in the problem situation.Graph V over this domain.

4. Use graphical methods to find where the cone has its maximum volume, andwhat that volume is.

5. Confirm your findings in part 4 analytically. [Hint: Use V�x� � �1�3�pr2h,h2 � r2 � 16, and the Chain Rule.]

EXPLORATION 1

O

A

C

4"

4" O A

C

h

r

4"

x

NOT TO SCALE

Quick Review 6.3 (For help, go to Sections 3.8 and 3.9.)

In Exercises 1–4, find dy�dx.

1. y � x3 sin 2x 2. y � e2 x ln �3x � 1�

3. y � tan�1 2x 4. y � sin�1 �x � 3�

In Exercises 5 and 6, solve for x in terms of y.

5. y � tan�1 3x x � �13

� tan y 6. y � cos�1 �x � 1�

7. Find the area under the arch of the curve y � sin �x from x � 0to x � 1. �

p

2�

8. Solve the differential equation dy/dx � e2 x. y � �12

� e2x � C

9. Solve the initial value problem dy/dx � x � sin x, y�0� � 2.

10. Use differentiation to confirm the integration formula

ex sin x dx � �12

� ex �sin x � cos x�.

2x3 cos 2x � 3x2 sin 2x�3x

3e�

2x

1� � 2e2x ln (3x � 1)

�1 �

24x2�

��1 �

1

(x� � 3)2��

x � cos y � 1

9. y2

x cos x � 3

�ddx� �

12

�ex(sin x � cos x)� � ex sin x

Standardized Test QuestionsYou may use a graphing calculator to solve the following problems.

61. True or False There is exactly one point in the plane withpolar coordinates (2, 2). Justify your answer. True. Polar

62. True or False The total area enclosed by the 3-petaled rose

r � sin 3� is �2�

0�12

� sin2 3�d�. Justify your answer.

63. Multiple Choice The area of the region enclosed by the polar

graph of r � �3 � co�s �� is given by which integral? D

(A) �2�

0�3 � co�s �� d� (B) �

�

0�3 � co�s �� d�

(C) 2��2

0(3 � cos�) d� (D) �

�

0(3 � cos�) d�

(E) ��2

0�3 � co�s �� d�

64. Multiple Choice The area enclosed by one petal of the3-petaled rose r � 4 cos(3�) is given by which integral? E

(A) 16��3

��3cos(3�) d� (B) 8�

��6

��6cos(3�) d�

(C) 8��3

��3cos2(3�) d� (D) 16�

��6

��6cos2(3�) d�

(E) 8��6

��6cos2(3�) d�

16

�

Let � tan 1 (3/5). Then the area is 0 2��cos2 � � sin2 �

��d�.

coordinates determine a unique point.

xxii To the AP* Student

5128_FM_TE_ppi-xxiv 2/3/06 3:33 PM Page xxii

Each chapter concludes with a list of Key Terms, with references back towhere they are covered in the chapter, as well as Chapter ReviewExercises to check your comprehension of the chapter material.

The Quick Quiz for AP* Preparation provides another opportunity toreview your understanding as you progress through each chapter. A quizappears after every two or three sections and asks you to answer questionsabout topics covered in those sections. Each quiz contains three multiple-choice questions and one free-response question of the AP* type. This con-tinual reinforcement of ideas steers you away from rote memorization andtoward the conceptual understanding needed for the AP* Calculus Exam.

An AP* ExaminationPreparation section appearsat the end of each set ofchapter review exercises andincludes three free-responsequestions of the AP* type.This set of questions, which also may or may not permit the use of your graphing cal-culator, gives you additional opportunity to practice skills and problem-solving tech-niques needed for the AP* Calculus Exam.

Calculus at Work featuresindividuals who are usingcalculus in their jobs, provid-ing you with some insight asto when you will use calculusin your careers. Some of theapplications of calculus theyencounter are mentionedthroughout the text.

In addition to this text, Preparing for the AP* Calculus AB or BC Examinations, writtenby experienced AP* teachers, is also available to help you prepare for the AP* CalculusExam. What does it include?

• Text-specific correlations between key AP* test topics and Calculus: Graphical,Numerical, Algebraic

• Reinforcement of the important connections between what you’ll learn and whatyou’ll be tested on in May

• 2 full sample AB exams & 2 sample BC exams including answers and explanation

• Test Taking strategies

You can order Preparing for the AP* Calculus AB or BC Examinations by going online toPHSchool.com/catalog or calling 1-800-848-9500 and requesting ISBN 0-321-33574-0.

To the AP* Student xxiii

arc length (p. 413)

area between curves (p. 390)

Cavalieri’s theorems (p. 404)

center of mass (p. 389)

constant-force formula (p. 384)

cylindrical shells (p. 402)

displacement (p. 380)

fluid force (p. 421)

fluid pressure (p. 421)

foot-pound (p. 384)

force constant (p. 385)

Gaussian curve (p. 423)

Hooke’s Law (p. 385)

inflation rate (p. 388)

joule (p. 384)

length of a curve (p. 413)

mean (p. 423)

moment (p. 389)

net change (p. 379)

newton (p. 384)

normal curve (p. 423)

normal pdf (p. 423)

probability density function (pdf) (p. 422)

68-95-99.7 rule (p. 423)

smooth curve (p. 413)

smooth function (p. 413)

solid of revolution (p. 400)

standard deviation (p. 423)

surface area (p. 405)

total distance traveled (p. 381)

universal gravitational constant (p. 428)

volume by cylindrical shells (p. 402)

volume by slicing (p. 400)

volume of a solid (p. 399)

weight-density (p. 421)

work (p. 384)

Chapter 7 Key Terms

Chapter 7 Review Exercises

The collection of exercises marked in red could be used as a chaptertest.In Exercises 1–5, the application involves the accumulation of smallchanges over an interval to give the net change over that entire inter-val. Set up an integral to model the accumulation and evaluate it toanswer the question.

1. A toy car slides down a ramp and coasts to a stop after 5 sec. Its velocity from t � 0 to t � 5 is modeled by v�t� � t2 � 0.2t3 ft �sec. How far does it travel? �10.417 ft

2. The fuel consumption of a diesel motor between weeklymaintenance periods is modeled by the function c�t� �4 � 0.001t 4 gal �day, 0 � t � 7. How many gallons does it consume in a week? �31.361 gal

3. The number of billboards per mile along a 100-mile stretch of aninterstate highway approaching a certain city is modeled by thefunction B�x� � 21 � e0.03x, where x is the distance from the cityin miles. About how many billboards are along that stretch ofhighway? �1464

Calculus at WorkI am working toward becoming anarcheaoastronomer and ethnoastronomerof Africa. I have a Bachelor’s degree inPhysics, a Master’s degree in Astronomy, and a Ph.D. in Astronomy and Astro-physics. From 1988 to 1990 I was a mem-ber of the Peace Corps, and I taughtmathematics to high school students inthe Fiji Islands. Calculus is a requiredcourse in high schools there.

For my Ph.D. dissertation, I investigatedthe possibility of the birthrate of starsbeing related to the composition of starformation clouds. I collected data on theabsorption of electromagnetic emissionsemanating from these regions. The inten-sity of emissions graphed versus wave-

length produces a flat curve with down-ward spikes at the characteristic wave-lengths of the elements present. An esti-mate of the area between a spike and theflat curve results in a concentration inmolecules/cm3 of an element. This area isthe difference in the integrals of the flatand spike curves. In particular, I was look-ing for a large concentration of water-ice,which increases the probability of planetsforming in a region.

Currently, I am applying for two researchgrants. One will allow me to use the NASAinfrared telescope on Mauna Kea tosearch for C3S2 in comets. The other willhelp me study the history of astronomy inTunisia.

Jarita HolbrookLos Angeles, CA

You should solve these problems without using a graphing calculator.

1. Multiple Choice How many critical points does the functionf (x) � (x � 2)5 (x � 3)4 have?

(A) One (B) Two (C) Three (D) Five (E) Nine

2. Multiple Choice For what value of x does the function f (x) � (x � 2) (x � 3)2 have a relative maximum?

(A) �3 (B) �73

(C) �52

(D) 73

(E) 52

3. Multiple Choice If g is a differentiable function such thatg(x) � 0 for all real numbers x, and if f ′(x) � (x2 � 9)g(x), whichof the following is true?

(A) f has a relative maximum at x � �3 and a relative minimumat x � 3.

(B) f has a relative minimum at x � �3 and a relative maximumat x � 3.

(C) f has relative minima at x � �3 and at x � 3.

(D) f has relative maxima at x � �3 and at x � 3.

(E) It cannot be determined if f has any relative extrema.

4. Free Response Let f be the function given by f �x� � 3 ln (x2 � 2) � 2x with domain [–2, 4].

(a) Find the coordinate of each relative maximum point and eachrelative minimum point of f . Justify your answer.

(b) Find the x-coordinate of each point of inflection of the graphof f .

(c) Find the absolute maximum value of f �x�.

Quick Quiz for AP* Preparation: Sections 4.1–4.3; , f ( ) ( ) ( )( ) ( )( ) y , f

3. B; The sign graph of f � is determined. It changes from negative to positive at x � �3 and from positive to negative at x � 3. Choice (B) follows from the First Derivative Test

AP* Examination PreparationYou may use a graphing calculator to solve the following problems.

53. Let R be the region in the first quadrant enclosed by the y-axisand the graphs of y � 2 � sin x and y � sec x.

(a) Find the area of R.

(b) Find the volume of the solid generated when R is revolvedabout the x-axis.

(c) Find the volume of the solid whose base is R and whose crosssections cut by planes perpendicular to the x-axis are squares.

54. The temperature outside a house during a 24-hour period is given by

F(t) � 80 � 10 cos�1p

2t

�, 0 � t � 24,

where F(t) is measured in degrees Fahrenheit and t is measuredin hours.

(a) Find the average temperature, to the nearest degree Fahren-heit, between t � 6 and t � 14.

(b) An air conditioner cooled the house whenever the outsidetemperature was at or above 78 degrees Fahrenheit. For whatvalues of t was the air conditioner cooling the house?

(c) The cost of cooling the house accumulates at the rate of $0.05 per hour for each degree the outside temperature exceeds78 degrees Fahrenheit. What was the total cost, to the nearestcent, to cool the house for this 24-hour period?

55. The rate at which people enter an amusement park on a givenday is modeled by the function E defined by

E(t) �t2 �

12546t0�

0160

.

The rate at which people leave the same amusement park on thesame day is modeled by the function L defined by

L(t) �t2 � 3

988t9�

0370

.

Both E(t) and L(t) are measured in people per hour, and time t ismeasured in hours after midnight. These functions are valid for9 � t � 23, which are the hours that the park is open. At timet � 9, there are no people in the park.

(a) How many people have entered the park by 5:00 P.M.(t � 17)? Round your answer to the nearest whole number.

(b) The price of admission to the park is $15 until 5:00 P.M.(t � 17). After 5:00 P.M., the price of admission to the park is $11.How many dollars are collected from admissions to the park onthe given day? Round your answer to the nearest whole number.

(c) Let H(t) � �t

9(E(x) � L(x))dx for 9 � t � 23. The value of

H(17) to the nearest whole number is 3725. Find the value ofH�(17) and explain the meaning of H(17) and H�(17) in the con-text of the park.

(d) At what time t, for 9 � t � 23, does the model predict thatthe number of people in the park is a maximum?

5128_FM_TE_ppi-xxiv 2/3/06 6:05 PM Page xxiii

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Acknowledgments

Many individuals contributed to the development of this textbook that is written especiallyfor Advanced Placement Calculus teachers. To those of you who have labored on this andprevious editions of this text, we offer our deepest gratitude. We also extend our sincerethanks to the dedicated users and reviewers of the previous editions of this textbook whoseinvaluable insight forms the heart of each textbook revision. We apologize for any omissions:

ConsultantJudith BroadwinBaruch College of the City of New YorkNew York

Note to Instructors:The Annotated Teacher’s Edition contains answers to all exercises. Where space permits,answers appear on the same page as their corresponding exercises—either adjacent to theexercise or at the top or bottom of the exercise set—or answers appear in available spaceon the adjacent page. Where space does not permit answers to appear in the exercise set,those answers can be found in the back of the book.

Reviewers Linda AntinonePaschal High SchoolFort Worth, Texas

Pam ArthurBryan Adams High SchoolDallas, Texas

Ray BartonSalt Lake High SchoolSalt Lake City, Utah

Brenda BattenThomas Heyward AcademyRidgeland, South Carolina

Karen ClarkeRye Country Day SchoolRye, New York

Timothy M. DonougheMayfield High SchoolMayfield Village, Ohio

Helga EnkoA.R. Johnson High SchoolAugusta, Georgia

Robert FirmanBorah High SchoolBoise, Idaho

Nancy GauseCypress Springs HighSchoolCypress, Texas

Dan HallThe Bolles SchoolJacksonville, Florida

Heather LaJoieEast Mecklenburg HighSchoolCharlotte, North Carolina

Betty MayberryGallatin High SchoolGallatin, Tennessee

Mary Ann MolnarNorthern Valley RegionalHigh SchoolDemarest, New Jersey

Martha MontgomeryFremont City SchoolsFremont, Ohio

Steve Olson Hingham High SchoolHingham, Massachusetts

David H. Van LangeveldClearfield High SchoolClearfield, Utah

Virginia WilliamsBenjamin Mays HighSchoolAtlanta, Georgia

Gladys WoodMemorial High SchoolHouston, Texas

Jim YoungTerra Linda High SchoolSan Rafael, California

5128_FM_TE_ppi-xxiv 1/18/06 1:08 PM Page xxiv

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ANNOTATED TEACHER’S EDITION Calculus - …p1cdn4static.sharpschool.com/UserFiles/Servers/Server_20226865/... · CHAPTER 1 Prerequisites for Calculus 2 1.1 Lines 3 • Increments