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F O E R S T E R
Concepts and Applications Second Edition
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Project Editor: Christopher David Consulting Editor: Josephine Noah Editorial Assistants: Lori Dixon, Shannon Miller Reviewer: Judith Broadwin Accuracy Checkers: Jenn Berg, Dudley Brooks Production Director: Diana Jean Ray Production Editor: Kristin Ferraioli Copyeditors: Tara Joffe, Luana Richards, Mary Roybal, Joan Saunders Production Coordinator: Michael Hurtik Text Designers: Adriane Bosworth, Jenny Somerville Art Editors: Jason Luz, Laura Murray Photo Researcher: Margee Robinson Art and Design Coordinator: Kavitha Becker Illustrator: Jason Luz Technical Art: Matthew Perry Cover Designer: Jenny Somerville Cover Photo Credit: Alec Pytlowany/Masterfile Composition and Prepress: The GTS Companies/York, PA Printer: Von Hoffmann Printers Executive Editor: Casey FitzSimons Publisher: Steven Rasmussen
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2005 by Key Curriculum Press. All rights reserved.
No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, photocopying, recording, or otherwise, without the prior written permission of the publisher.
®Key Curriculum Press and ®The Geometer’s Sketchpad are registered trademarks of Key Curriculum Press. ™Sketchpad is a trademark of Key Curriculum Press. All other registered trademarks and trademarks in this book are the property of their respective holders. Key Curriculum Press 1150 65th Street Emeryville, CA 94608
[email protected] www.keypress.com Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 08 07 06 05 04 ISBN 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 Ridge National Laboratory, from whom I first understood the concepts of calculus; Edmund Eickenroht, my former student, whose desire it 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 technology was making radical changes in the teaching and learning of calculus. The fundamental differences embodied in the text have arisen from teaching my own students using this technology. In addition, the text has been thoroughly revised to incorporate comments and suggestions from the many consultants and field testers 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 to teaching calculus and to put up with the difficulties of receiving materials at the last minute. Special thanks to Bill Medigovich for editing the first edition, coordinating the field test program, and organizing the first two summer institutes for instructors. Special thanks also to Debbie Preston for drafting the major part of the Instructor’s Guide and parts of the Solutions Manual, and for working with the summer institutes for instructors. By serving as both instructors and consultants, these two have given this text an added dimension of clarity and teachability. Thanks also to my students for enduring all those handouts, and for finding things to be changed! Special thanks to my students Craig Browning, Meredith Fast, William Fisher, Brad Wier, and Matthew Willis for taking good class notes so that the text materials could include classroom-tested examples. Thanks to the late Richard V. Andree and his wife, Josephine, for allowing their children, Phoebe Small and Calvin Butterball, to make occasional appearances in my texts. Finally, thanks to Chris Sollars, Debbie Davies, and Debbie Preston for their ideas and 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 in San Antonio, Texas, which he has done since 1961. After earning a bachelor’s degree in chemical engineering, he served four years in the U.S. Navy. Following his first five years at Alamo Heights, he earned a master’s degree in mathematics. He has published five textbooks, based on problems he wrote for his own students to let them see more realistically how mathematics is applied in the real world. In 1983 he received the Presidential Award for Excellence in Mathematics Teaching, the first year of the award. He raised three children with the late Jo Ann Foerster, and he also has two grown stepchildren through his wife Peggy Foerster, as well as three grandchildren. Paul plans to continue teaching for the foreseeable future, relishing the excitement of the ever-changing content of the evolving mathematics curriculum.
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Foreword by 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 do mathematics, what do mathematicians do?” Many feel that our historical role has not changed, but that the emphasis is now clearly on selection and interpretation rather than manipulation and methods. As teachers, we continue to sense the need for a major shift in the instructional means we employ to impart mathematical understanding to our students. At the same time, we recognize 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 education remains 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 his famous best. The material is presented in an easily understood fashion with ample technology-based examples and exercises. The applications are intimately connected with the topic and amplify the key elements in the section. The material is a wealth of both fresh items and ancient insights that have stood the test of time. For example, alongside Escalante’s “cross hatch” method of repeated integration by parts, you’ll find Heaviside’s thumb trick for solving partial fractions! The students are repeatedly sent to their “graphers.” Early on, when differentiation is introduced, Paul discusses local linearity, and later he utilizes the zoom features of calculators in the coverage of l’Hospital’s rule—that’s fresh. Later still, he presents the logistic curve and slope fields in differential equations. All of these are beautiful examples of how computing technology has changed the calculus course. The changes and additions found in this second edition exhibit the timeliness of the text. Exponentials and logarithms have been given an even more prominent role that reflects their greater emphasis in today’s calculus instruction. The narrative, 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 or the other” but correctly both. Rich, substantive, in-depth questions bring to mind superb Advanced Placement free response questions, or it might be that many AP questions remind you of Foerster’s style! Throughout, you see how comprehensive Paul is in his study of the historical role of calculus and the currency of his understanding of the AP community and collegiate “calculus reform.” Brilliant, timely, solid, and loaded with tons of novel applications—your typical Foerster! John Kenelly has been involved with the Advanced Placement Calculus program for over 30 years. He was Chief Reader and later Chair of the AP Calculus Committee when Paul Foerster was grading the AP exams in the 1970s. He is a leader in development of the graphing calculator and in pioneering its use in college and school classrooms. He served as president of the IMO 2001 USA, the organization 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 CHAPTER
1
Limits, Derivatives, Integrals, and Integrals 1-1 1-2 1-3 1-4 1-5 1-6
CHAPTER
CHAPTER
2
3
The Concept of Instantaneous Rate Rate of Change by Equation, Graph, or Table One Type of Integral of a Function Definite Integrals by Trapezoids, from Equations and Data Calculus Journal Chapter Review and Test
xiii 1 3 6 14 18 24 25
Properties of Limits
31
2-1 2-2 2-3 2-4 2-5 2-6 2-7
33 34 40 45 52 60 64
Numerical Approach to the Definition of Limit Graphical and Algebraic Approaches to the Definition of Limit The Limit Theorems Continuity and Discontinuity Limits Involving Infinity The Intermediate Value Theorem and Its Consequences Chapter Review and Test
Derivatives, Antiderivatives, and Indefinite Integrals
71
3-1 3-2 3-3 3-4
73 74 78
3-5 3-6 3-7 3-8 3-9 3-10
Graphical Interpretation of Derivative Difference Quotients and One Definition of Derivative Derivative Functions, Numerically and Graphically Derivative of the Power Function and Another Definition of Derivative Displacement, Velocity, and Acceleration Introduction to Sine, Cosine, and Composite Functions Derivatives of Composite Functions—The Chain Rule Proof and Application of Sine and Cosine Derivatives Exponential and Logarithmic Functions Chapter Review and Test
85 92 100 102 107 115 122
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CHAPTER
CHAPTER
4
5
6
129
4-1 4-2 4-3 4-4 4-5 4-6 4-7 4-8 4-9 4-10
131 132 137 142 146 153 160 169 174 180 187
5-1 5-2 5-3 5-4 5-5 5-6 5-7 5-8 5-9 5-10
189 190 197 204 211 221 227 233 242 252 259 267
6-1 6-2
269
Integral of the Reciprocal Function: A Population Growth Problem Antiderivative of the Reciprocal Function and Another Form of the Fundamental Theorem The Uniqueness Theorem and Properties of Logarithmic Functions The Number e, Exponential Functions, and Logarithmic Differentiation Limits of Indeterminate Forms: l’Hospital’s Rule Derivative and Integral Practice for Transcendental Functions Chapter Review and Test Cumulative Review: Chapters 1–6
270 280 288 295 301 306 311
The Calculus of Growth and Decay
315
7-1 7-2 7-3 7-4 7-5
317 318 324 333
7-6
x
A Definite Integral Problem Linear Approximations and Differentials Formal Definition of Antiderivative and Indefinite Integral Riemann Sums and the Definition of Definite Integral The Mean Value Theorem and Rolle’s Theorem The Fundamental Theorem of Calculus Definite Integral Properties and Practice Definite Integrals Applied to Area and Other Problems Volume of a Solid by Plane Slicing Definite Integrals Numerically by Grapher and by Simpson’s Rule Chapter Review and Test
The Calculus of Exponential and Logarithmic Functions
6-5 6-6 6-7 6-8
7
Combinations of Two Functions Derivative of a Product of Two Functions Derivative of a Quotient of Two Functions Derivatives of the Other Trigonometric Functions Derivatives of Inverse Trigonometric Functions Differentiability and Continuity Derivatives of a Parametric Function Graphs and Derivatives of Implicit Relations Related Rates Chapter Review and Test
Definite and Indefinite Integrals
6-3 6-4
CHAPTER
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Products, Quotients, and Parametric Functions
5-11
CHAPTER
13:29
Direct Proportion Property of Exponential Functions Exponential Growth and Decay Other Differential Equations for Real-World Applications Graphical Solution of Differential Equations by Using Slope Fields Numerical Solution of Differential Equations by Using Euler’s Method The Logistic Function, and Predator-Prey Population Problems
341 348
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CHAPTER L? A? V?
8
p.i.?
CHAPTER
CHAPTER
9
10
359 365
The Calculus of Plane and Solid Figures
369
8-1 8-2 8-3 8-4 8-5 8-6 8-7 8-8
371 372 385 395 401 407 414 423
Cubic Functions and Their Derivatives Critical Points and Points of Inflection Maxima and Minima in Plane and Solid Figures Volume of a Solid of Revolution by Cylindrical Shells Length of a Plane Curve—Arc Length Area of a Surface of Revolution Lengths and Areas for Polar Coordinates Chapter Review and Test
Algebraic Calculus Techniques for the Elementary Functions
431
9-1 9-2 9-3 9-4 9-5 9-6 9-7 9-8 9-9 9-10 9-11 9-12 9-13
433 434 438 444 449 454 460 466 469 481 488 493 494
Introduction to the Integral of a Product of Two Functions Integration by Parts—A Way to Integrate Products Rapid Repeated Integration by Parts Reduction Formulas and Computer Algebra Systems Integrating Special Powers of Trigonometric Functions Integration by Trigonometric Substitution Integration of Rational Functions by Partial Fractions Integrals of the Inverse Trigonometric Functions Calculus of the Hyperbolic and Inverse Hyperbolic Functions Improper Integrals Miscellaneous Integrals and Derivatives Integrals in Journal Chapter Review and Test
The Calculus of Motion—Averages, Extremes, and Vectors
10-2 10-3 10-4 10-5 10-6 10-7
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Chapter Review and Test Cumulative Review: Chapters 1–7
10-1
CHAPTER
13:29
Introduction to Distance and Displacement for Motion Along a Line Distance, Displacement, and Acceleration for Linear Motion Average Value Problems in Motion and Elsewhere Minimal Path Problems Maximum and Minimum Problems in Motion and Elsewhere Vector Functions for Motion in a Plane Chapter Review and Test
499 501 502 508 514 520 522 538
The Calculus of Variable-Factor Products
545
11-1 11-2 11-3 11-4 11-5
547 548 553 558 567
Review of Work—Force Times Displacement Work Done by a Variable Force Mass of a Variable-Density Object Moments, Centroids, Center of Mass, and the Theorem of Pappus Force Exerted by a Variable Pressure—Center of Pressure
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Other Variable-Factor Products Chapter Review and Test
573 580
The Calculus of Functions Defined by Power Series
587
12-1 12-2 12-3 12-4 12-5 12-6 12-7 12-8 12-9 12-10
589 590 597 598 605 613 621 635 643 648
Introduction to Power Series Geometric Sequences and Series as Mathematical Models Power Series for an Exponential Function Power Series for Other Elementary Functions Taylor and Maclaurin Series, and Operations on These Series Interval of Convergence for a Series—The Ratio Technique Convergence of Series at the Ends of the Convergence Interval Error Analysis for Series—The Lagrange Error Bound Chapter Review and Test Cumulative Reviews
Final Examination: A Guided Tour Through Calculus
655
Appendix: Summary of Properties of Trigonometric 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 Student from the Author In earlier courses you have learned about functions. Functions express the way one variable quantity, such as distance you travel, is related to another quantity, such as time. Calculus was invented over 300 years ago to deal with the rate at which a quantity varies, particularly if that rate does not stay constant. In your calculus course you will learn the algebraic formulas for variable rates that will tie together the mathematics you have learned in earlier courses. Fortunately, computers and graphing calculators (“graphers”) will give you graphical and numerical methods to understand the concepts even before you develop the formulas. In this way you will be able to work calculus problems from the real world starting on day one. Later, once you understand the concepts, the formulas will give you time-efficient ways to work these problems. The time you save by using technology for solving problems and learning concepts can be used to develop your ability to write about mathematics. You will be asked to keep a written journal recording the concepts and techniques you have been learning, and verbalizing things you may not yet have mastered. Thus, you will learn calculus in four ways—algebraically, graphically, numerically, and verbally. In whichever of these areas your talents lie, you will have the opportunity to excel. As in any mathematics course, you must learn calculus by doing it. Mathematics is not a “spectator sport.” As you work on the Explorations that introduce you to new concepts and techniques, you will have a chance to participate in cooperative groups, learning from your classmates and improving your skills. The Quick Review problems at the beginning of each problem set ask you to recall quickly things that you may have forgotten from earlier in the text or from previous courses. Other problems, marked by a shaded star, will prepare you for a topic in a later section. Prior to the Chapter Test at the end of each chapter, you will find review problems keyed 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. When that 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 apply to?” That way, you will better see the big picture. Best wishes as you venture into the world of higher mathematics! Paul A. Foerster Alamo Heights High School San Antonio, Texas
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