Concepts and Applications - Prek...

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P A U L A . F O E R S T E R

C o n c e p t s a n d A p p l i c a t i o n sSecond Edition



Project Editor: Christopher DavidConsulting Editor: Josephine NoahEditorial Assistants: Lori Dixon, Shannon MillerReviewer: Judith BroadwinAccuracy Checkers: Jenn Berg, Dudley BrooksProduction Director: Diana Jean RayProduction Editor: Kristin FerraioliCopyeditors: Tara Joffe, Luana Richards, Mary Roybal, Joan SaundersProduction Coordinator: Michael HurtikText Designers: Adriane Bosworth, Jenny SomervilleArt Editors: Jason Luz, Laura MurrayPhoto Researcher: Margee RobinsonArt and Design Coordinator: Kavitha BeckerIllustrator: Jason LuzTechnical Art: Matthew PerryCover Designer: Jenny SomervilleCover Photo Credit: Alec Pytlowany/MasterfileComposition and Prepress: The GTS Companies/York, PAPrinter: Von Hoffmann Printers

Executive Editor: Casey FitzSimonsPublisher: Steven Rasmussen

C© 2005 by Key Curriculum Press. All rights reserved.

No part of this publication may be reproduced, stored in a retrieval system, ortransmitted, in any form or by any means, electronic, photocopying, recording, orotherwise, without the prior written permission of the publisher.

®Key Curriculum Press and ®The Geometer’s Sketchpad are registered trademarks of KeyCurriculum Press. ™Sketchpad is a trademark of Key Curriculum Press. All otherregistered trademarks and trademarks in this book are the property of their respectiveholders.

Key Curriculum Press1150 65th StreetEmeryville, CA [email protected]

Printed in the United States of America10 9 8 7 6 5 4 3 2 1 08 07 06 05 04ISBN 1-55953-654-3

Photograph credits appear on the last two pages of the book.

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To people from the past, including James H. Marable of Oak RidgeNational Laboratory, from whom I first understood the conceptsof calculus; Edmund Eickenroht, my former student, whose desireit was to write his own calculus text; and my late wife, Jo Ann.

To my wife, Peggy, who shares my zest for life and accomplishment.

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Consultants to the First Edition

Donald J. Albers, Mathematical Association of America, Washington D.C.

Judith Broadwin, Jericho High School, Jericho, New York

Joan Ferrini-Mundy, University of New Hampshire, Durham, New Hampshire

Gregory D. Foley, Sam Houston State University, Huntsville, Texas

John Kenelly, Clemson University, Clemson, South Carolina

Dan Kennedy, Baylor School, Chattanooga, Tennessee

Deborah B. Preston, Keystone School, San Antonio, Texas

Field Testers of the First Edition

Betty Baker, Bogan High School, Chicago, Illinois

Glenn C. Ballard, William Henry Harrison High School, Evansville, Indiana

Bruce Cohen, Lick-Wilmerding High School, San Francisco, California

Christine J. Comins, Pueblo County High School, Pueblo, Colorado

Deborah Davies, University School of Nashville, Nashville, Tennessee

Linda E. de Sola, Plano Senior High School, Plano, Texas

Paul A. Foerster, Alamo Heights High School, San Antonio, Texas

Joan M. Gell, Palos Verdes Peninsula High School, Rolling Hills Estates, California

Valmore E. Guernon, Lincoln Junior/Senior High School, Lincoln, Rhode Island

David S. Heckman, Monmouth Academy, Monmouth, Maine

Don W. Hight, Pittsburg State University, Pittsburg, Kansas

Edgar Hood, Dawson High School, Dawson, Texas

Ann Joyce, Issaquah High School, Issaquah, Washington

John G. Kelly, Arroyo High School, San Lorenzo, California

Linda Klett, San Domenico School, San Anselmo, California

George Lai, George Washington High School, San Francisco, California

Katherine P. Layton, Beverly Hills High School, Beverly Hills, California

Debbie Lindow, Reynolds High School, Troutdale, Oregon

Robert Maass, International Studies Academy, San Francisco, California

Guy R. Mauldin, Science Hill High School, Johnson City, Tennessee

Windle McKenzie, Brookstone School, Columbus, Georgia

Bill Medigovich, Redwood High School, Larkspur, California

Sandy Minkler, Redlands High School, Redlands, California

Deborah B. Preston, Keystone School, San Antonio, Texas

Sanford Siegel, School of the Arts, San Francisco, California

Susan M. Smith, Ysleta Independent School District, El Paso, Texas

Gary D. Starr, Girard High School, Girard, Kansas

Tom Swartz, George Washington High School, San Francisco, California

Tim Trapp, Mountain View High School, Mesa, Arizona

Dixie Trollinger, Mainland High School, Daytona Beach, Florida

David Weinreich, Queen Anne School, Upper Marlboro, Maryland

John P. Wojtowicz, Saint Joseph’s High School, South Bend, Indiana

Tim Yee, Malibu High School, Malibu, California

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Author’s Acknowledgments

This text was written during the period when graphing calculator technologywas making radical changes in the teaching and learning of calculus. Thefundamental differences embodied in the text have arisen from teaching my ownstudents using this technology. In addition, the text has been thoroughly revisedto incorporate comments and suggestions from the many consultants and fieldtesters listed on the previous page.

Thanks in particular to the original field test people—Betty Baker, Chris Comins,Debbie Davies, Val Guernon, David Heckman, Don Hight, Kathy Layton,Guy Mauldin, Windle McKenzie, Debbie Preston, Gary Starr, and John Wojtowicz.These instructors were enterprising enough to venture into a new approach toteaching calculus and to put up with the difficulties of receiving materials at thelast minute.

Special thanks to Bill Medigovich for editing the first edition, coordinatingthe field test program, and organizing the first two summer institutes forinstructors. Special thanks also to Debbie Preston for drafting the major part ofthe Instructor’s Guide and parts of the Solutions Manual, and for working withthe summer institutes for instructors. By serving as both instructors andconsultants, these two have given this text an added dimension of clarity andteachability.

Thanks also to my students for enduring all those handouts, and for findingthings to be changed! Special thanks to my students Craig Browning,Meredith Fast, William Fisher, Brad Wier, and Matthew Willis for taking goodclass notes so that the text materials could include classroom-tested examples.

Thanks to the late Richard V. Andree and his wife, Josephine, for allowing theirchildren, Phoebe Small and Calvin Butterball, to make occasional appearances inmy texts.

Finally, thanks to Chris Sollars, Debbie Davies, and Debbie Preston for their ideasand encouragement as I worked on the second edition of Calculus.

Paul A. Foerster

About the Author

Paul Foerster enjoys teaching mathematics at Alamo Heights High School inSan Antonio, Texas, which he has done since 1961. After earning a bachelor’sdegree in chemical engineering, he served four years in the U.S. Navy. Followinghis first five years at Alamo Heights, he earned a master’s degree inmathematics. He has published five textbooks, based on problems he wrote forhis own students to let them see more realistically how mathematics is appliedin the real world. In 1983 he received the Presidential Award for Excellence inMathematics Teaching, the first year of the award. He raised three children withthe late Jo Ann Foerster, and he also has two grown stepchildren through his wifePeggy Foerster, as well as three grandchildren. Paul plans to continue teachingfor the foreseeable future, relishing the excitement of the ever-changing contentof the evolving mathematics curriculum.

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Forewordby John Kenelly, Clemson University

In the explosion of the information age and the resulting instructional reforms,we have all had to deal repeatedly with the question: “When machines domathematics, what do mathematicians do?” Many feel that our historical rolehas not changed, but that the emphasis is now clearly on selection andinterpretation rather than manipulation and methods. As teachers, we continueto sense the need for a major shift in the instructional means we employ toimpart mathematical understanding to our students. At the same time, werecognize that behind any technology there must be human insight.

In a world of change, we must build on the past and take advantage of the future.Applications and carefully chosen examples still guide us through what works.Challenges and orderly investigations still develop mature thinking and insights.As much as the instructional environment might change, quality educationremains our goal. What we need are authors and texts that bridge the transition.It is in this regard that Paul Foerster and his texts provide outstanding answers.

In Calculus: Concepts and Applications, Second Edition, Paul is again at hisfamous best. The material is presented in an easily understood fashion withample technology-based examples and exercises. The applications are intimatelyconnected with the topic and amplify the key elements in the section. Thematerial is a wealth of both fresh items and ancient insights that have stood thetest of time. For example, alongside Escalante’s “cross hatch” method ofrepeated integration by parts, you’ll find Heaviside’s thumb trick for solvingpartial fractions! The students are repeatedly sent to their “graphers.” Early on,when differentiation is introduced, Paul discusses local linearity, and later heutilizes the zoom features of calculators in the coverage of l’Hospital’srule—that’s fresh. Later still, he presents the logistic curve and slope fields indifferential equations. All of these are beautiful examples of how computingtechnology has changed the calculus course.

The changes and additions found in this second edition exhibit the timeliness ofthe text. Exponentials and logarithms have been given an even more prominentrole that reflects their greater emphasis in today’s calculus instruction. Thenarrative, problem sets, Explorations, and tests all support the position that the

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choice between technology and traditional methods is not exclusively “one orthe other” but correctly both. Rich, substantive, in-depth questions bring tomind superb Advanced Placement free response questions, or it might be thatmany AP questions remind you of Foerster’s style!

Throughout, you see how comprehensive Paul is in his study of the historicalrole of calculus and the currency of his understanding of the AP community andcollegiate “calculus reform.” Brilliant, timely, solid, and loaded with tons ofnovel applications—your typical Foerster!

John Kenelly has been involved with the Advanced Placement Calculus programfor over 30 years. He was Chief Reader and later Chair of the AP CalculusCommittee when Paul Foerster was grading the AP exams in the 1970s. He is aleader in development of the graphing calculator and in pioneering its use incollege and school classrooms. He served as president of the IMO 2001 USA, theorganization that acts as host when the International Mathematical Olympiad(IMO) comes to the United States.

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Contents

A Note to the Student from the Author xiii

C H A P T E R 1 Limits, Derivatives, Integrals, and Integrals 1

1-1 The Concept of Instantaneous Rate 31-2 Rate of Change by Equation, Graph, or Table 61-3 One Type of Integral of a Function 141-4 Definite Integrals by Trapezoids, from Equations and Data 181-5 Calculus Journal 241-6 Chapter Review and Test 25

C H A P T E R 2 Properties of Limits 31

2-1 Numerical Approach to the Definition of Limit 332-2 Graphical and Algebraic Approaches to the Definition of Limit 342-3 The Limit Theorems 402-4 Continuity and Discontinuity 452-5 Limits Involving Infinity 522-6 The Intermediate Value Theorem and Its Consequences 602-7 Chapter Review and Test 64

C H A P T E R 3 Derivatives, Antiderivatives, and Indefinite Integrals 71

3-1 Graphical Interpretation of Derivative 733-2 Difference Quotients and One Definition of Derivative 743-3 Derivative Functions, Numerically and Graphically 783-4 Derivative of the Power Function and Another

Definition of Derivative 853-5 Displacement, Velocity, and Acceleration 923-6 Introduction to Sine, Cosine, and Composite Functions 1003-7 Derivatives of Composite Functions—The Chain Rule 1023-8 Proof and Application of Sine and Cosine Derivatives 1073-9 Exponential and Logarithmic Functions 1153-10 Chapter Review and Test 122

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C H A P T E R 4 Products, Quotients, and Parametric Functions 129

4-1 Combinations of Two Functions 1314-2 Derivative of a Product of Two Functions 1324-3 Derivative of a Quotient of Two Functions 1374-4 Derivatives of the Other Trigonometric Functions 1424-5 Derivatives of Inverse Trigonometric Functions 1464-6 Differentiability and Continuity 1534-7 Derivatives of a Parametric Function 1604-8 Graphs and Derivatives of Implicit Relations 1694-9 Related Rates 1744-10 Chapter Review and Test 180

C H A P T E R 5 Definite and Indefinite Integrals 187

5-1 A Definite Integral Problem 1895-2 Linear Approximations and Differentials 1905-3 Formal Definition of Antiderivative and Indefinite Integral 1975-4 Riemann Sums and the Definition of Definite Integral 2045-5 The Mean Value Theorem and Rolle’s Theorem 2115-6 The Fundamental Theorem of Calculus 2215-7 Definite Integral Properties and Practice 2275-8 Definite Integrals Applied to Area and Other Problems 2335-9 Volume of a Solid by Plane Slicing 2425-10 Definite Integrals Numerically by Grapher and

by Simpson’s Rule 2525-11 Chapter Review and Test 259

C H A P T E R 6 The Calculus of Exponential and Logarithmic Functions 267

6-1 Integral of the Reciprocal Function: A Population Growth Problem 2696-2 Antiderivative of the Reciprocal Function and Another

Form of the Fundamental Theorem 2706-3 The Uniqueness Theorem and Properties of Logarithmic Functions 2806-4 The Number e, Exponential Functions, and

Logarithmic Differentiation 2886-5 Limits of Indeterminate Forms: l’Hospital’s Rule 2956-6 Derivative and Integral Practice for Transcendental Functions 3016-7 Chapter Review and Test 3066-8 Cumulative Review: Chapters 1–6 311

C H A P T E R 7 The Calculus of Growth and Decay 315

7-1 Direct Proportion Property of Exponential Functions 3177-2 Exponential Growth and Decay 3187-3 Other Differential Equations for Real-World Applications 3247-4 Graphical Solution of Differential Equations by Using Slope Fields 3337-5 Numerical Solution of Differential Equations by

Using Euler’s Method 3417-6 The Logistic Function, and Predator-Prey Population Problems 348

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7-7 Chapter Review and Test 3597-8 Cumulative Review: Chapters 1–7 365

C H A P T E R 8p.i.?

A?L?

V ?

The Calculus of Plane and Solid Figures 369

8-1 Cubic Functions and Their Derivatives 3718-2 Critical Points and Points of Inflection 3728-3 Maxima and Minima in Plane and Solid Figures 3858-4 Volume of a Solid of Revolution by Cylindrical Shells 3958-5 Length of a Plane Curve—Arc Length 4018-6 Area of a Surface of Revolution 4078-7 Lengths and Areas for Polar Coordinates 4148-8 Chapter Review and Test 423

C H A P T E R 9 Algebraic Calculus Techniques forthe Elementary Functions 431

9-1 Introduction to the Integral of a Product of Two Functions 4339-2 Integration by Parts—A Way to Integrate Products 4349-3 Rapid Repeated Integration by Parts 4389-4 Reduction Formulas and Computer Algebra Systems 4449-5 Integrating Special Powers of Trigonometric Functions 4499-6 Integration by Trigonometric Substitution 4549-7 Integration of Rational Functions by Partial Fractions 4609-8 Integrals of the Inverse Trigonometric Functions 4669-9 Calculus of the Hyperbolic and Inverse Hyperbolic Functions 4699-10 Improper Integrals 4819-11 Miscellaneous Integrals and Derivatives 4889-12 Integrals in Journal 4939-13 Chapter Review and Test 494

C H A P T E R 10 The Calculus of Motion—Averages,Extremes, and Vectors 499

10-1 Introduction to Distance and Displacement forMotion Along a Line 501

10-2 Distance, Displacement, and Acceleration for Linear Motion 50210-3 Average Value Problems in Motion and Elsewhere 50810-4 Minimal Path Problems 51410-5 Maximum and Minimum Problems in Motion and Elsewhere 52010-6 Vector Functions for Motion in a Plane 52210-7 Chapter Review and Test 538

C H A P T E R 11 The Calculus of Variable-Factor Products 545

11-1 Review of Work—Force Times Displacement 54711-2 Work Done by a Variable Force 54811-3 Mass of a Variable-Density Object 55311-4 Moments, Centroids, Center of Mass, and the Theorem of Pappus 55811-5 Force Exerted by a Variable Pressure—Center of Pressure 567

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11-6 Other Variable-Factor Products 57311-7 Chapter Review and Test 580

C H A P T E R 12 The Calculus of Functions Defined by Power Series 587

12-1 Introduction to Power Series 58912-2 Geometric Sequences and Series as Mathematical Models 59012-3 Power Series for an Exponential Function 59712-4 Power Series for Other Elementary Functions 59812-5 Taylor and Maclaurin Series, and Operations on These Series 60512-6 Interval of Convergence for a Series—The Ratio Technique 61312-7 Convergence of Series at the Ends of the Convergence Interval 62112-8 Error Analysis for Series—The Lagrange Error Bound 63512-9 Chapter Review and Test 64312-10 Cumulative Reviews 648

Final Examination: A Guided Tour Through Calculus 655

Appendix: Summary of Properties ofTrigonometric Functions 659

Answers to Selected Problems 661

Glossary 755

Index of Problem Titles 761

General Index 767

Photograph Credits 777

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A Note to the Studentfrom the Author

In earlier courses you have learned about functions. Functions expressthe way one variable quantity, such as distance you travel, is related toanother quantity, such as time. Calculus was invented over 300 yearsago to deal with the rate at which a quantity varies, particularly if thatrate does not stay constant.

In your calculus course you will learn the algebraic formulas forvariable rates that will tie together the mathematics you have learnedin earlier courses. Fortunately, computers and graphing calculators(“graphers”) will give you graphical and numerical methods tounderstand the concepts even before you develop the formulas. In thisway you will be able to work calculus problems from the real worldstarting on day one. Later, once you understand the concepts, theformulas will give you time-efficient ways to work theseproblems.

The time you save by using technology for solving problems andlearning concepts can be used to develop your ability to write aboutmathematics. You will be asked to keep a written journal recording theconcepts and techniques you have been learning, and verbalizingthings you may not yet have mastered. Thus, you will learn calculus infour ways—algebraically, graphically, numerically, and verbally. Inwhichever of these areas your talents lie, you will have the opportunityto excel.

As in any mathematics course, you must learn calculus by doing it.Mathematics is not a “spectator sport.” As you work on theExplorations that introduce you to new concepts and techniques, youwill have a chance to participate in cooperative groups, learning fromyour classmates and improving your skills.

The Quick Review problems at the beginning of each problem set askyou to recall quickly things that you may have forgotten from earlier inthe text or from previous courses. Other problems, marked by ashaded star, will prepare you for a topic in a later section. Prior to theChapter Test at the end of each chapter, you will find review problemskeyed to each section. Additionally, the Concept Problems give you a

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chance to apply your knowledge to new and challenging situations. So,keeping up with your homework will help to ensure your success.

At times you may feel you are becoming submerged in details. Whenthat happens, just remember that calculus involves only four concepts:

• Limits• Derivatives• Integrals (one kind)• Integrals (another kind)

Ask yourself, “Which of these concepts does my present work applyto?” That way, you will better see the big picture. Best wishes as youventure into the world of higher mathematics!

Paul A. FoersterAlamo Heights High SchoolSan Antonio, Texas

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