Multivariable Calculus Concepts and Contexts | 4e
Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
Courtesy of Frank O. Gehry
Courtesy of Frank O. Gehry
The cover photograph shows the DZ Bank in Berlin, designed and built 1995–2001 by Frank Gehry and Associates. The interior atrium is dominated by a curvaceous fourstory stainless steel sculptural shell that suggests a prehistoric creature and houses a central conference space. The highly complex structures that Frank Gehry designs would be impossible to build without the computer. The CATIA software that his architects and engineers use to produce the computer models is based on principles of calculus—fitting curves by matching tangent lines, making sure the curvature isn’t too large, and controlling parametric surfaces. “Consequently,” says Gehry, “we have a lot of freedom. I can play with shapes.” The process starts with Gehry’s initial sketches, which are translated into a succession of physical models. (Hundreds of different physical models were constructed during the design of the building, first with basic wooden blocks and then evolving into more sculptural forms.) Then an engineer uses a digitizer to record the coordinates of a series of points on a physical model. The digitized points are fed into a computer and the CATIA software is used to link these points with smooth curves. (It joins curves so that their tangent lines coincide; you can use the same idea to design the shapes of letters in the Laboratory Project on page 208 of this book.) The architect has considerable freedom in creating these curves, guided by displays of the curve, its derivative, and its curvature. Then the curves are
Courtesy of Frank O. Gehry
Calculus and the Architecture of Curves
Courtesy of Frank O. Gehry thomasmayerarchive.com
Courtesy of Frank O. Gehry
connected to each other by a parametric surface, and again the architect can do so in many possible ways with the guidance of displays of the geometric characteristics of the surface. The CATIA model is then used to produce another physical model, which, in turn, suggests modifications and leads to additional computer and physical models.
The CATIA program was developed in France by Dassault Systèmes, originally for designing airplanes, and was subsequently employed in the automotive industry. Frank Gehry, because of his complex sculptural shapes, is the first to use it in architecture. It helps him answer his question, “How wiggly can you get and still make a building?”
Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
Multivariable Calculus Concepts and Contexts | 4e
James Stewart McMaster University and University of Toronto
Australia . Brazil . Japan . Korea . Mexico . Singapore . Spain . United Kingdom . United States
Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
Multivariable Calculus: Concepts and Contexts, Fourth Edition James Stewart Publisher: Richard Stratton Senior Developmental Editor: Jay Campbell Associate Developmental Editor: Jeannine Lawless Editorial Assistant: Elizabeth Neustaetter Media Editor: Peter Galuardi Senior Marketing Manager: Jennifer Jones Marketing Assistant: Angela Kim Marketing Communications Manager: Mary Anne Payumo Senior Project Manager, Editorial Production: Cheryll Linthicum Creative Director: Rob Hugel Senior Art Director: Vernon Boes Senior Print Buyer: Becky Cross Permissions Editor: Bob Kauser
© 2010, 2005 Brooks/Cole, Cengage Learning ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher. For product information and technology assistance, contact us at Cengage Learning Customer & Sales Support, 1-800-354-9706 For permission to use material from this text or product, submit all requests online at www.cengage.com/permissions. Further permissions questions can be e-mailed to
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Library of Congress Control Number: 2008940620 ISBN-13: 978-0-495-56054-8 ISBN-10: 0-495-56054-5
Production Service: TECHarts Text Designer: Jeanne Calabrese Photo Researcher: Nina Smith Copy Editor: Kathi Townes Illustrator: Brian Betsill
Brooks/Cole 10 Davis Drive Belmont, CA 94002-3098 USA
Cover Designer: Irene Morris Cover Image and page iv: thomasmayerarchive.com Compositor: Stephanie Kuhns, TECHarts
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Printed in Canada 1 2 3 4 5 6 7 1 3 12 11 10 09
Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
K03T09
Trademarks Derive is a registered trademark of Soft Warehouse, Inc. Maple is a registered trademark of Waterloo Maple, Inc. Mathematica is a registered trademark of Wolfram Research, Inc. Tools for Enriching is a trademark used herein under license.
Contents Preface
xi
To the Student
8
xx
Infinite Sequences and Series 8.1
Sequences
554
Laboratory Project 8.2 8.3 8.4 8.5 8.6 8.7
thomasmayerarchive.com
Writing Project
Logistic Sequences
■
■
An Elusive Limit
Applied Project
■
618
9.3 9.4
Radiation from the Stars
9.7
■
■
Laboratory Project
Review
634
662
663
Putting 3D in Perspective
Functions and Surfaces 673 Cylindrical and Spherical Coordinates ■
633
The Geometry of a Tetrahedron
Equations of Lines and Planes Laboratory Project
9.6
627
631
Three-Dimensional Coordinate Systems Vectors 639 The Dot Product 648 The Cross Product 654 Discovery Project
9.5
618
619
Vectors and the Geometry of Space
9.2
575
628
Focus on Problem Solving
9.1
564
How Newton Discovered the Binomial Series
Applications of Taylor Polynomials Review
9
thomasmayerarchive.com
■
Series 565 The Integral and Comparison Tests; Estimating Sums Other Convergence Tests 585 Power Series 592 Representations of Functions as Power Series 598 Taylor and Maclaurin Series 604 Laboratory Project
8.8
553
Families of Surfaces
672
682 687
688
Focus on Problem Solving
691
vii
Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
viii
CONTENTS
10 Vector Functions 10.1 10.2
Courtesy of Frank O. Gehry
10.3 10.4
Vector Functions and Space Curves 694 Derivatives and Integrals of Vector Functions 701 Arc Length and Curvature 707 Motion in Space: Velocity and Acceleration 716 Applied Project
10.5
693
■
Kepler’s Laws
Parametric Surfaces Review 733
727
Focus on Problem Solving
11 Partial Derivatives 11.1 11.2 11.3 11.4 11.5 11.6 11.7
737
■
Courtesy of Frank O. Gehry
Discovery Project
Designing a Dumpster ■
Lagrange Multipliers
12.3 12.4 12.5 12.6 12.7
Rocket Science
Applied Project
■
820
Hydro-Turbine Optimization
821
822 827
829
Double Integrals over Rectangles 830 Iterated Integrals 838 Double Integrals over General Regions 844 Double Integrals in Polar Coordinates 853 Applications of Double Integrals 858 Surface Area 868 Triple Integrals 873 Discovery Project
12.8
812
813
Applied Project
12 Multiple Integrals
thomasmayerarchive.com
Quadratic Approximations and Critical Points
Focus on Problem Solving
12.2
811
■
Review
12.1
735
Functions of Several Variables 738 Limits and Continuity 749 Partial Derivatives 756 Tangent Planes and Linear Approximations 770 The Chain Rule 780 Directional Derivatives and the Gradient Vector 789 Maximum and Minimum Values 802 Applied Project
11.8
726
■
Volumes of Hyperspheres
883
Triple Integrals in Cylindrical and Spherical Coordinates Applied Project
■
Discovery Project
Roller Derby ■
883
889
The Intersection of Three Cylinders
890
Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
CONTENTS
12.9
Change of Variables in Multiple Integrals Review 899 Focus on Problem Solving
13 Vector Calculus 13.1 13.2 13.3 13.4 13.5 13.6
thomasmayerarchive.com
13.7
13.9
903
905
Vector Fields 906 Line Integrals 913 The Fundamental Theorem for Line Integrals Green’s Theorem 934 Curl and Divergence 941 Surface Integrals 949 Stokes’ Theorem 960 Writing Project
13.8
891
■
Three Men and Two Theorems
The Divergence Theorem Summary 973 Review 974 Focus on Problem Solving
Appendixes
925
966
967
977
A1
D
Precise Definitions of Limits
A2
E
A Few Proofs
H
Polar Coordinates
I
Complex Numbers
J
Answers to Odd-Numbered Exercises
A3 A6 A22 A31
Index A51
Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
ix
Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
Preface When the first edition of this book appeared twelve years ago, a heated debate about calculus reform was taking place. Such issues as the use of technology, the relevance of rigor, and the role of discovery versus that of drill were causing deep splits in mathematics departments. Since then the rhetoric has calmed down somewhat as reformers and traditionalists have realized that they have a common goal: to enable students to understand and appreciate calculus. The first three editions were intended to be a synthesis of reform and traditional approaches to calculus instruction. In this fourth edition I continue to follow that path by emphasizing conceptual understanding through visual, verbal, numerical, and algebraic approaches. I aim to convey to the student both the practical power of calculus and the intrinsic beauty of the subject.
What’s New In the Fourth Edition? The changes have resulted from talking with my colleagues and students at the University of Toronto and from reading journals, as well as suggestions from users and reviewers. Here are some of the many improvements that I’ve incorporated into this edition: ■
The majority of examples now have titles.
■
Some material has been rewritten for greater clarity or for better motivation. See, for instance, the introduction to series on page 565.
■
New examples have been added and the solutions to some of the existing examples have been amplified.
■
A number of pieces of art have been redrawn.
■
The data in examples and exercises have been updated to be more timely.
■
Sections 8.7 and 8.8 have been merged into a single section. I had previously featured the binomial series in its own section to emphasize its importance. But I learned that some instructors were omitting that section, so I decided to incorporate binomial series into 8.7.
■
More than 25% of the exercises in each chapter are new. Here are a few of my favorites: 8.2.35, 9.1.42, 11.1.10–11, 11.6.37–38, 11.8.20–21, and 13.3.21–22.
■
There are also some good new problems in the Focus on Problem Solving sections. See, for instance, Problem 13 on page 632, Problem 8 on page 692, Problem 9 on page 736, and Problem 11 on page 904.
Features Conceptual Exercises
The most important way to foster conceptual understanding is through the problems that we assign. To that end I have devised various types of problems. Some exercise sets begin with requests to explain the meanings of the basic concepts of the section. (See, for instance, the first couple of exercises in Sections 8.2, 11.2, and 11.3. I often use them as xi
Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
xii
PREFACE
a basis for classroom discussions.) Similarly, review sections begin with a Concept Check and a True-False Quiz. Other exercises test conceptual understanding through graphs or tables (see Exercises 8.7.2, 10.2.1–2, 10.3.33–37, 11.1.1–2, 11.1.9–18, 11.3.3–10, 11.6.1–2, 11.7.3–4, 12.1.5–10, 13.1.11–18, 13.2.15–16, and 13.3.1–2). Graded Exercise Sets
Each exercise set is carefully graded, progressing from basic conceptual exercises and skill-development problems to more challenging problems involving applications and proofs.
Real-World Data
My assistants and I have spent a great deal of time looking in libraries, contacting companies and government agencies, and searching the Internet for interesting real-world data to introduce, motivate, and illustrate the concepts of calculus. As a result, many of the examples and exercises deal with functions defined by such numerical data or graphs. See, for instance, Example 3 in Section 9.6 (wave heights). Functions of two variables are illustrated by a table of values of the wind-chill index as a function of air temperature and wind speed (Example 1 in Section 11.1). Partial derivatives are introduced in Section 11.3 by examining a column in a table of values of the heat index (perceived air temperature) as a function of the actual temperature and the relative humidity. This example is pursued further in connection with linear approximations (Example 3 in Section 11.4). Directional derivatives are introduced in Section 11.6 by using a temperature contour map to estimate the rate of change of temperature at Reno in the direction of Las Vegas. Double integrals are used to estimate the average snowfall in Colorado on December 20–21, 2006 (Example 4 in Section 12.1). Vector fields are introduced in Section 13.1 by depictions of actual velocity vector fields showing San Francisco Bay wind patterns.
Projects
One way of involving students and making them active learners is to have them work (perhaps in groups) on extended projects that give a feeling of substantial accomplishment when completed. Applied Projects involve applications that are designed to appeal to the imagination of students. The project after Section 11.8 uses Lagrange multipliers to determine the masses of the three stages of a rocket so as to minimize the total mass while enabling the rocket to reach a desired velocity. Discovery Projects explore aspects of geometry: tetrahedra (after Section 9.4), hyperspheres (after Section 12.7), and intersections of three cylinders (after Section 12.8). The Laboratory Project on page 687 uses technology to discover how interesting the shapes of surfaces can be and how these shapes evolve as the parameters change in a family. The Writing Project on page 966 explores the historical and physical origins of Green’s Theorem and Stokes’ Theorem and the interactions of the three men involved. Many additional projects are provided in the Instructor’s Guide.
Technology
The availability of technology makes it not less important but more important to understand clearly the concepts that underlie the images on the screen. But, when properly used, graphing calculators and computers are powerful tools for discovering and understanding those concepts. I assume that the student has access to either a graphing calculator or a computer algebra system. The icon ; indicates an exercise that definitely requires the use of such technology, but that is not to say that a graphing device can’t be used on the other exercises as well. The symbol CAS is reserved for problems in which the full resources of a computer algebra system (like Derive, Maple, Mathematica, or the TI-89/92) are required. But technology doesn’t make pencil and paper obsolete. Hand calculation and sketches are often preferable to technology for illustrating and reinforcing some concepts. Both instructors and students need to develop the ability to decide where the hand or the machine is appropriate.
Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
PREFACE
xiii
Tools for Enriching™ Calculus
TEC is a companion to the text and is intended to enrich and complement its contents. (It is now accessible from the Internet at www.stewartcalculus.com.) Developed by Harvey Keynes, Dan Clegg, Hubert Hohn, and myself, TEC uses a discovery and exploratory approach. In sections of the book where technology is particularly appropriate, marginal icons direct students to TEC modules that provide a laboratory environment in which they can explore the topic in different ways and at different levels. Visuals are animations of figures in the text; Modules are more elaborate activities and include exercises. Instructors can choose to become involved at several different levels, ranging from simply encouraging students to use the Visuals and Modules for independent exploration, to assigning specific exercises from those included with each Module, or to creating additional exercises, labs, and projects that make use of the Visuals and Modules. TEC also includes Homework Hints for representative exercises (usually odd-numbered) in every section of the text, indicated by printing the exercise number in red. These hints are usually presented in the form of questions and try to imitate an effective teaching assistant by functioning as a silent tutor. They are constructed so as not to reveal any more of the actual solution than is minimally necessary to make further progress.
Enhanced WebAssign
Technology is having an impact on the way homework is assigned to students, particularly in large classes. The use of online homework is growing and its appeal depends on ease of use, grading precision, and reliability. With the fourth edition we have been working with the calculus community and WebAssign to develop an online homework system. Many of the exercises in each section are assignable as online homework, including free response, multiple choice, and multi-part formats. The system also includes Active Examples, in which students are guided in step-by-step tutorials through text examples, with links to the textbook and to video solutions.
Website: www.stewartcalculus.com
This website includes the following. ■
Algebra Review
■
Lies My Calculator and Computer Told Me
■
History of Mathematics, with links to the better historical websites
■
Additional Topics (complete with exercise sets): Trigonometric Integrals, Trigonometric Substitution, Strategy for Integration, Strategy for Testing Series, Fourier Series, Formulas for the Remainder Term in Taylor Series, Linear Differential Equations, Second-Order Linear Differential Equations, Nonhomogeneous Linear Equations, Applications of Second-Order Differential Equations, Using Series to Solve Differential Equations, Rotation of Axes, and (for instructors only) Hyperbolic Functions
■
Links, for each chapter, to outside Web resources
■
Archived Problems (drill exercises that appeared in previous editions, together with their solutions)
■
Challenge Problems (some from the Focus on Problem Solving sections of prior editions)
Content 8
■
Infinite Sequences and Series
Tests for the convergence of series are considered briefly, with intuitive rather than formal justifications. Numerical estimates of sums of series are based on which test was used to prove convergence. The emphasis is on Taylor series and polynomials and their applications to physics. Error estimates include those from graphing devices.
Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
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PREFACE
9 ■ Vectors and The Geometry of Space
The dot product and cross product of vectors are given geometric definitions, motivated by work and torque, before the algebraic expressions are deduced. To facilitate the discussion of surfaces, functions of two variables and their graphs are introduced here. 10 ■ Vector Functions
The calculus of vector functions is used to prove Kepler’s First Law of planetary motion, with the proofs of the other laws left as a project. In keeping with the introduction of parametric curves in Chapter 1, parametric surfaces are introduced as soon as possible, namely, in this chapter. I think an early familiarity with such surfaces is desirable, especially with the capability of computers to produce their graphs. Then tangent planes and areas of parametric surfaces can be discussed in Sections 11.4 and 12.6. 11
■
Partial Derivatives
Functions of two or more variables are studied from verbal, numerical, visual, and algebraic points of view. In particular, I introduce partial derivatives by looking at a specific column in a table of values of the heat index (perceived air temperature) as a function of the actual temperature and the relative humidity. Directional derivatives are estimated from contour maps of temperature, pressure, and snowfall. 12
■
Multiple Integrals
Contour maps and the Midpoint Rule are used to estimate the average snowfall and average temperature in given regions. Double and triple integrals are used to compute probabilities, areas of parametric surfaces, volumes of hyperspheres, and the volume of intersection of three cylinders. 13 ■ Vector Fields
Vector fields are introduced through pictures of velocity fields showing San Francisco Bay wind patterns. The similarities among the Fundamental Theorem for line integrals, Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem are emphasized.
Ancillaries Multivariable Calculus: Concepts and Contexts, Fourth Edition, is supported by a complete set of ancillaries developed under my direction. Each piece has been designed to enhance student understanding and to facilitate creative instruction. The table on pages xviii and xix lists ancillaries available for instructors and students.
Acknowledgments I am grateful to the following reviewers for sharing their knowledge and judgment with me. I have learned something from each of them. Fourth Edition Reviewers
Jennifer Bailey, Colorado School of Mines Lewis Blake, Duke University James Cook, North Carolina State University Costel Ionita, Dixie State College Lawrence Levine, Stevens Institute of Technology Scott Mortensen, Dixie State College
Drew Pasteur, North Carolina State University Jeffrey Powell, Samford University Barbara Tozzi, Brookdale Community College Kathryn Turner, Utah State University Cathy Zucco-Tevelof, Arcadia University
Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
PREFACE Previous Edition Reviewers
Irfan Altas, Charles Sturt University William Ardis, Collin County Community College Barbara Bath, Colorado School of Mines Neil Berger, University of Illinois at Chicago Jean H. Bevis, Georgia State University Martina Bode, Northwestern University Jay Bourland, Colorado State University Paul Wayne Britt, Louisiana State University Judith Broadwin, Jericho High School (retired) Charles Bu, Wellesley University Meghan Anne Burke, Kennesaw State University Robert Burton, Oregon State University Roxanne M. Byrne, University of Colorado at Denver Maria E. Calzada, Loyola University–New Orleans Larry Cannon, Utah State University Deborah Troutman Cantrell, Chattanooga State Technical Community College Bem Cayco, San Jose State University John Chadam, University of Pittsburgh Robert A. Chaffer, Central Michigan University Dan Clegg, Palomar College Camille P. Cochrane, Shelton State Community College James Daly, University of Colorado Richard Davis, Edmonds Community College Susan Dean, DeAnza College Richard DiDio, LaSalle University Robert Dieffenbach, Miami University–Middletown Fred Dodd, University of South Alabama Helmut Doll, Bloomsburg University William Dunham, Muhlenberg College David A. Edwards, The University of Georgia John Ellison, Grove City College Joseph R. Fiedler, California State University–Bakersfield Barbara R. Fink, DeAnza College James P. Fink, Gettysburg College Joe W. Fisher, University of Cincinnati Robert Fontenot, Whitman College Richard L. Ford, California State University Chico Laurette Foster, Prairie View A & M University Ronald C. Freiwald, Washington University in St. Louis Frederick Gass, Miami University Gregory Goodhart, Columbus State Community College John Gosselin, University of Georgia
Daniel Grayson, University of Illinois at Urbana–Champaign Raymond Greenwell, Hofstra University Gerrald Gustave Greivel, Colorado School of Mines John R. Griggs, North Carolina State University Barbara Bell Grover, Salt Lake Community College Murli Gupta, The George Washington University John William Hagood, Northern Arizona University Kathy Hann, California State University at Hayward Richard Hitt, University of South Alabama Judy Holdener, United States Air Force Academy Randall R. Holmes, Auburn University Barry D. Hughes, University of Melbourne Mike Hurley, Case Western Reserve University Gary Steven Itzkowitz, Rowan University Helmer Junghans, Montgomery College Victor Kaftal, University of Cincinnati Steve Kahn, Anne Arundel Community College Mohammad A. Kazemi, University of North Carolina, Charlotte Harvey Keynes, University of Minnesota Kandace Alyson Kling, Portland Community College Ronald Knill, Tulane University Stephen Kokoska, Bloomsburg University Kevin Kreider, University of Akron Doug Kuhlmann, Phillips Academy David E. Kullman, Miami University Carrie L. Kyser, Clackamas Community College Prem K. Kythe, University of New Orleans James Lang, Valencia Community College–East Campus Carl Leinbach, Gettysburg College William L. Lepowsky, Laney College Kathryn Lesh, University of Toledo Estela Llinas, University of Pittsburgh at Greensburg Beth Turner Long, Pellissippi State Technical Community College Miroslav Lovri´c, McMaster University Lou Ann Mahaney, Tarrant County Junior College–Northeast John R. Martin, Tarrant County Junior College Andre Mathurin, Bellarmine College Prep R. J. McKellar, University of New Brunswick Jim McKinney, California State Polytechnic University–Pomona
Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
xv
xvi
PREFACE
Richard Eugene Mercer, Wright State University David Minda, University of Cincinnati Rennie Mirollo, Boston College Laura J. Moore-Mueller, Green River Community College Scott L. Mortensen, Dixie State College Brian Mortimer, Carleton University Bill Moss, Clemson University Tejinder Singh Neelon, California State University San Marcos Phil Novinger, Florida State University Richard Nowakowski, Dalhousie University Stephen Ott, Lexington Community College Grace Orzech, Queen’s University Jeanette R. Palmiter, Portland State University Bill Paschke, University of Kansas David Patocka, Tulsa Community College–Southeast Campus Paul Patten, North Georgia College Leslie Peek, Mercer University Mike Pepe, Seattle Central Community College Dan Pritikin, Miami University Fred Prydz, Shoreline Community College Denise Taunton Reid, Valdosta State University James Reynolds, Clarion University Hernan Rivera, Texas Lutheran University Richard Rochberg, Washington University Gil Rodriguez, Los Medanos College David C. Royster, University of North Carolina–Charlotte Daniel Russow, Arizona Western College Dusty Edward Sabo, Southern Oregon University
Daniel S. Sage, Louisiana State University N. Paul Schembari, East Stroudsburg University Dr. John Schmeelk, Virginia Commonwealth University Bettina Schmidt, Auburn University at Montgomery Bernd S.W. Schroeder, Louisiana Tech University Jeffrey Scott Scroggs, North Carolina State University James F. Selgrade, North Carolina State University Brad Shelton, University of Oregon Don Small, United States Military Academy–West Point Linda E. Sundbye, The Metropolitan State College of Denver Richard B. Thompson,The University of Arizona William K. Tomhave, Concordia College Lorenzo Traldi, Lafayette College Alan Tucker, State University of New York at Stony Brook Tom Tucker, Colgate University George Van Zwalenberg, Calvin College Dennis Watson, Clark College Paul R. Wenston, The University of Georgia Ruth Williams, University of California–San Diego Clifton Wingard, Delta State University Jianzhong Wang, Sam Houston State University JingLing Wang, Lansing Community College Michael B. Ward, Western Oregon University Stanley Wayment, Southwest Texas State University Barak Weiss, Ben Gurion University–Be’er Sheva, Israel Teri E. Woodington, Colorado School of Mines James Wright, Keuka College
In addition, I would like to thank Ari Brodsky, David Cusick, Alfonso Gracia-Saz, Emile LeBlanc, Tanya Leise, Joe May, Romaric Pujol, Norton Starr, Lou Talman, and Gail Wolkowicz for their advice and suggestions; Al Shenk and Dennis Zill for permission to use exercises from their calculus texts; COMAP for permission to use project material; Alfonso Gracia-Saz, B. Hovinen, Y. Kim, Anthony Lam, Romaric Pujol, Felix Recio, and Paul Sally for ideas for exercises; Dan Drucker for the roller derby project; and Tom Farmer, Fred Gass, John Ramsay, Larry Riddle, V. K. Srinivasan, and Philip Straffin for ideas for projects. I’m grateful to Dan Clegg, Jeff Cole, and Tim Flaherty for preparing the answer manuscript and suggesting ways to improve the exercises. As well, I thank those who have contributed to past editions: Ed Barbeau, George Bergman, David Bleecker, Fred Brauer, Andy Bulman-Fleming, Tom DiCiccio, Martin Erickson, Garret Etgen, Chris Fisher, Stuart Goldenberg, Arnold Good, John Hagood, Gene Hecht, Victor Kaftal, Harvey Keynes, E. L. Koh, Zdislav Kovarik, Kevin Kreider, Jamie Lawson, David Leep, Gerald Leibowitz, Larry Peterson, Lothar Redlin, Peter Rosenthal, Carl Riehm, Ira Rosenholtz, Doug Shaw, Dan Silver, Lowell Smylie, Larry Wallen, Saleem Watson, and Alan Weinstein.
Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
PREFACE
xvii
I also thank Stephanie Kuhns, Rebekah Million, Brian Betsill, and Kathi Townes of TECH-arts for their production services; Marv Riedesel and Mary Johnson for their careful proofing of the pages; Thomas Mayer for the cover image; and the following Brooks/ Cole staff: Cheryll Linthicum, editorial production project manager; Jennifer Jones, Angela Kim, and Mary Anne Payumo, marketing team; Peter Galuardi, media editor; Jay Campbell, senior developmental editor; Jeannine Lawless, associate editor; Elizabeth Neustaetter, editorial assistant; Bob Kauser, permissions editor; Becky Cross, print/media buyer; Vernon Boes, art director; Rob Hugel, creative director; and Irene Morris, cover designer. They have all done an outstanding job. I have been very fortunate to have worked with some of the best mathematics editors in the business over the past three decades: Ron Munro, Harry Campbell, Craig Barth, Jeremy Hayhurst, Gary Ostedt, Bob Pirtle, and now Richard Stratton. Special thanks go to all of them. JAMES STEWART
Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
Ancillaries for Instructors PowerLecture CD-ROM with JoinIn and ExamView ISBN 0-495-56049-9
Contains all art from the text in both jpeg and PowerPoint formats, key equations and tables from the text, complete pre-built PowerPoint lectures, and an electronic version of the Instructor’s Guide. Also contains JoinIn on TurningPoint personal response system questions and ExamView algorithmic test generation. See below for complete descriptions. TEC Tools for Enriching™ Calculus by James Stewart, Harvey Keynes, Dan Clegg, and developer Hu Hohn TEC provides a laboratory environment in which students can explore selected topics. TEC also includes homework hints for representative exercises. Available online at www.stewartcalculus.com.
ExamView Create, deliver, and customize tests and study guides (both print and online) in minutes with this easy-to-use assessment and tutorial software on CD. Includes full algorithmic generation of problems and complete questions from the Printed Test Bank. JoinIn on TurningPoint Enhance how your students interact with you, your lecture, and each other. Brooks/Cole, Cengage Learning is now pleased to offer you book-specific content for Response Systems tailored to Stewart’s Calculus, allowing you to transform your classroom and assess your students’ progress with instant in-class quizzes and polls. Contact your local Cengage representative to learn more about JoinIn on TurningPoint and our exclusive infrared and radio-frequency hardware solutions. Text-Specific DVDs ISBN 0-495-56050-2
Instructor’s Guide by Douglas Shaw and James Stewart ISBN 0-495-56047-2
Each section of the main text is discussed from several viewpoints and contains suggested time to allot, points to stress, text discussion topics, core materials for lecture, workshop/ discussion suggestions, group work exercises in a form suitable for handout, and suggested homework problems. An electronic version is available on the PowerLecture CD-ROM. Instructor’s Guide for AP ® Calculus by Douglas Shaw
Text-specific DVD set, available at no charge to adopters. Each disk features a 10- to 20-minute problem-solving lesson for each section of the chapter. Covers both single- and multivariable calculus. Solution Builder www.cengage.com/solutionbuilder The online Solution Builder lets instructors easily build and save personal solution sets either for printing or posting on password-protected class websites. Contact your local sales representative for more information on obtaining an account for this instructor-only resource.
ISBN 0-495-56059-6
Taking the perspective of optimizing preparation for the AP exam, each section of the main text is discussed from several viewpoints and contains suggested time to allot, points to stress, daily quizzes, core materials for lecture, workshop/ discussion suggestions, group work exercises in a form suitable for handout, tips for the AP exam, and suggested homework problems. Complete Solutions Manual, Multivariable by Dan Clegg ISBN 0-495-56056-1
Ancillaries for Instructors and Students eBook Option ISBN 0-495-56121-5
Whether you prefer a basic downloadable eBook or a premium multimedia eBook with search, highlighting, and note taking capabilities as well as links to videos and simulations, this new edition offers a range of eBook options to fit how you want to read and interact with the content.
Includes worked-out solutions to all exercises in the text. Printed Test Bank by William Tomhave and Xuequi Zeng ISBN 0-495-56123-1
Contains multiple-choice and short-answer test items that key directly to the text.
xviii
Stewart Specialty Website www.stewartcalculus.com Contents: Algebra Review Additional Topics Drill Web Links History of exercises Challenge Problems Mathematics Tools for Enriching Calculus (TEC) N
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Enhanced WebAssign Instant feedback, grading precision, and ease of use are just three reasons why WebAssign is the most widely used homework system in higher education. WebAssign’s homework delivery system lets instructors deliver, collect, grade and record assignments via the web. And now, this proven system has been enhanced to include end-of-section problems from Stewart’s Calculus: Concepts and Contexts—incorporating exercises, examples, video skillbuilders and quizzes to promote active learning and provide the immediate, relevant feedback students want.
Student Solutions Manual, Multivariable by Dan Clegg ISBN 0-495-56055-3
Provides completely worked-out solutions to all odd-numbered exercises within the text, giving students a way to check their answers and ensure that they took the correct steps to arrive at an answer. CalcLabs with Maple, Multivariable by Philip B. Yasskin and Art Belmonte ISBN 0-495-56058-8
The Brooks/Cole Mathematics Resource Center Website www.cengage.com/math When you adopt a Brooks/Cole, Cengage Learning mathematics text, you and your students will have access to a variety of teaching and learning resources. This website features everything from book-specific resources to newsgroups. It’s a great way to make teaching and learning an interactive and intriguing experience. Maple CD-ROM ISBN 0-495-01492-3 (Maple 10) ISBN 0-495-39052-6 (Maple 11)
Maple provides an advanced, high performance mathematical computation engine with fully integrated numerics & symbolics, all accessible from a WYSIWYG technical document environment. Available for bundling with your Stewart Calculus text at a special discount.
CalcLabs with Mathematica, Multivariable by Selwyn Hollis ISBN 0-495-82722-3
Each of these comprehensive lab manuals will help students learn to effectively use the technology tools available to them. Each lab contains clearly explained exercises and a variety of labs and projects to accompany the text. A Companion to Calculus, Second Edition by Dennis Ebersole, Doris Schattschneider, Alicia Sevilla, and Kay Somers ISBN 0-495-01124-X
Written to improve algebra and problem-solving skills of students taking a calculus course, every chapter in this companion is keyed to a calculus topic, providing conceptual background and specific algebra techniques needed to understand and solve calculus problems related to that topic. It is designed for calculus courses that integrate the review of precalculus concepts or for individual use.
Student Resources TEC Tools for Enriching™ Calculus by James Stewart, Harvey Keynes, Dan Clegg, and developer Hu Hohn TEC provides a laboratory environment in which students can explore selected topics. TEC also includes homework hints for representative exercises. Available online at www.stewartcalculus.com.
Linear Algebra for Calculus by Konrad J. Heuvers, William P. Francis, John H. Kuisti, Deborah F. Lockhart, Daniel S. Moak, and Gene M. Ortner ISBN 0-534-25248-6
This comprehensive book, designed to supplement the calculus course, provides an introduction to and review of the basic ideas of linear algebra.
Study Guide, Multivariable by Robert Burton and Dennis Garity ISBN 0-495-56057-X
Contains key concepts, skills to master, a brief discussion of the ideas of the section, and worked-out examples with tips on how to find the solution.
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xix
To the Student
Reading a calculus textbook is different from reading a newspaper or a novel, or even a physics book. Don’t be discouraged if you have to read a passage more than once in order to understand it. You should have pencil and paper and calculator at hand to sketch a diagram or make a calculation. Some students start by trying their homework problems and read the text only if they get stuck on an exercise. I suggest that a far better plan is to read and understand a section of the text before attempting the exercises. In particular, you should look at the definitions to see the exact meanings of the terms. And before you read each example, I suggest that you cover up the solution and try solving the problem yourself. You’ll get a lot more from looking at the solution if you do so. Part of the aim of this course is to train you to think logically. Learn to write the solutions of the exercises in a connected, step-by-step fashion with explanatory sentences—not just a string of disconnected equations or formulas. The answers to the odd-numbered exercises appear at the back of the book, in Appendix J. Some exercises ask for a verbal explanation or interpretation or description. In such cases there is no single correct way of expressing the answer, so don’t worry that you haven’t found the definitive answer. In addition, there are often several different forms in which to express a numerical or algebraic answer, so if your answer differs from mine, don’t immediately assume you’re wrong. For example, if the answer given in the back of the book is s2 ⫺ 1 and you obtain 1兾(1 ⫹ s2 ), then you’re right and rationalizing the denominator will show that the answers are equivalent. The icon ; indicates an exercise that definitely requires the use of either a graphing calculator or a computer with graphing software. (Section 1.4 discusses the use of these graphing devices and some of the pitfalls that you may
encounter.) But that doesn’t mean that graphing devices can’t be used to check your work on the other exercises as well. The symbol CAS is reserved for problems in which the full resources of a computer algebra system (like Derive, Maple, Mathematica, or the TI-89/92) are required. You will also encounter the symbol |, which warns you against committing an error. I have placed this symbol in the margin in situations where I have observed that a large proportion of my students tend to make the same mistake. Tools for Enriching Calculus, which is a companion to this text, is referred to by means of the symbol TEC and can be accessed from www.stewartcalculus.com. It directs you to modules in which you can explore aspects of calculus for which the computer is particularly useful. TEC also provides Homework Hints for representative exercises that are indicated by printing the exercise number in red: 15. These homework hints ask you questions that allow you to make progress toward a solution without actually giving you the answer. You need to pursue each hint in an active manner with pencil and paper to work out the details. If a particular hint doesn’t enable you to solve the problem, you can click to reveal the next hint. I recommend that you keep this book for reference purposes after you finish the course. Because you will likely forget some of the specific details of calculus, the book will serve as a useful reminder when you need to use calculus in subsequent courses. And, because this book contains more material than can be covered in any one course, it can also serve as a valuable resource for a working scientist or engineer. Calculus is an exciting subject, justly considered to be one of the greatest achievements of the human intellect. I hope you will discover that it is not only useful but also intrinsically beautiful. JAMES STEWART
xx
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Infinite Sequences and Series
8
thomasmayerarchive.com
Infinite sequences and series were introduced briefly in A Preview of Calculus in connection with Zeno’s paradoxes and the decimal representation of numbers. Their importance in calculus stems from Newton’s idea of representing functions as sums of infinite series. For instance, in finding areas he often integrated a function by first expressing it as a series and then integrating each term of the series. We will pursue his idea in Section 8.7 in order to integrate such functions 2 as e⫺x . (Recall that we have previously been unable to do this.) Many of the functions that arise in mathematical physics and chemistry, such as Bessel functions, are defined as sums of series, so it is important to be familiar with the basic concepts of convergence of infinite sequences and series. Physicists also use series in another way, as we will see in Section 8.8. In studying fields as diverse as optics, special relativity, and electromagnetism, they analyze phenomena by replacing a function with the first few terms in the series that represents it. 553
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554
CHAPTER 8
INFINITE SEQUENCES AND SERIES
8.1 Sequences A sequence can be thought of as a list of numbers written in a definite order: a1 , a2 , a3 , a4 , . . . , an , . . . The number a 1 is called the first term, a 2 is the second term, and in general a n is the nth term. We will deal exclusively with infinite sequences and so each term a n will have a successor a n⫹1 . Notice that for every positive integer n there is a corresponding number a n and so a sequence can be defined as a function whose domain is the set of positive integers. But we usually write a n instead of the function notation f n for the value of the function at the number n. Notation: The sequence {a 1 , a 2 , a 3 , . . .} is also denoted by
a n
⬁
a n n苷1
or
EXAMPLE 1 Describing sequences Some sequences can be defined by giving a formula for the nth term. In the following examples we give three descriptions of the sequence: one by using the preceding notation, another by using the defining formula, and a third by writing out the terms of the sequence. Notice that n doesn’t have to start at 1.
(a)
(b) (c) (d)
⬁
n n⫹1
an 苷
n n⫹1
an 苷
⫺1nn ⫹ 1 3n
n苷1
⫺1nn ⫹ 1 3n
{sn ⫺ 3 } ⬁n苷3
a n 苷 sn ⫺ 3 , n 艌 3
a n 苷 cos
n cos 6
v
⬁
n苷0
n , n艌0 6
1 2 3 4 n , , , ,..., ,... 2 3 4 5 n⫹1
2 3 4 5 ⫺1nn ⫹ 1 ⫺ , ,⫺ , ,..., ,... 3 9 27 81 3n
{0, 1, s2 , s3 , . . . , sn ⫺ 3 , . . .}
1,
n s3 1 , , 0, . . . , cos ,... 2 2 6
EXAMPLE 2 Find a formula for the general term a n of the sequence
3 4 5 6 7 ,⫺ , ,⫺ , ,... 5 25 125 625 3125
assuming that the pattern of the first few terms continues. SOLUTION We are given that
a1 苷
3 5
a2 苷 ⫺
4 25
a3 苷
5 125
a4 苷 ⫺
6 625
a5 苷
7 3125
Notice that the numerators of these fractions start with 3 and increase by 1 whenever we go to the next term. The second term has numerator 4, the third term has numerator 5; in general, the nth term will have numerator n ⫹ 2. The denominators are the powers of 5,
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SECTION 8.1
SEQUENCES
555
so a n has denominator 5 n. The signs of the terms are alternately positive and negative, so we need to multiply by a power of ⫺1. In Example 1(b) the factor ⫺1 n meant we started with a negative term. Here we want to start with a positive term and so we use ⫺1 n⫺1 or ⫺1 n⫹1. Therefore a n 苷 ⫺1 n⫺1
n⫹2 5n
EXAMPLE 3 Here are some sequences that don’t have simple defining equations.
(a) The sequence pn , where pn is the population of the world as of January 1 in the year n. (b) If we let a n be the digit in the nth decimal place of the number e, then a n is a welldefined sequence whose first few terms are 7, 1, 8, 2, 8, 1, 8, 2, 8, 4, 5, . . . (c) The Fibonacci sequence fn is defined recursively by the conditions f1 苷 1
f2 苷 1
fn 苷 fn⫺1 ⫹ fn⫺2
n艌3
Each term is the sum of the two preceding terms. The first few terms are 1, 1, 2, 3, 5, 8, 13, 21, . . . This sequence arose when the 13th-century Italian mathematician known as Fibonacci solved a problem concerning the breeding of rabbits (see Exercise 47). a¡
a™ a£
1 2
0
A sequence such as the one in Example 1(a), a n 苷 nn ⫹ 1, can be pictured either by plotting its terms on a number line, as in Figure 1, or by plotting its graph, as in Figure 2. Note that, since a sequence is a function whose domain is the set of positive integers, its graph consists of isolated points with coordinates
a¢ 1
FIGURE 1
1, a1
an
2, a2
3, a3
...
n, a n
...
From Figure 1 or Figure 2 it appears that the terms of the sequence a n 苷 nn ⫹ 1 are approaching 1 as n becomes large. In fact, the difference
1
1⫺
7
a¶= 8 0
1 2 3 4 5 6 7
n
n 1 苷 n⫹1 n⫹1
can be made as small as we like by taking n sufficiently large. We indicate this by writing lim a n 苷 lim
FIGURE 2
nl⬁
nl⬁
n 苷1 n⫹1
In general, the notation lim a n 苷 L
nl⬁
means that the terms of the sequence a n approach L as n becomes large. Notice that the following definition of the limit of a sequence is very similar to the definition of a limit of a function at infinity given in Section 2.5.
Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
556
CHAPTER 8
INFINITE SEQUENCES AND SERIES
1
Definition A sequence a n has the limit L and we write
lim a n 苷 L
nl⬁
A more precise definition of the limit of a sequence is given in Appendix D.
or
a n l L as n l ⬁
if we can make the terms a n as close to L as we like by taking n sufficiently large. If lim n l ⬁ a n exists, we say the sequence converges (or is convergent). Otherwise, we say the sequence diverges (or is divergent). Figure 3 illustrates Definition 1 by showing the graphs of two sequences that have the limit L. an
an
L
L
FIGURE 3
Graphs of two sequences with lim an= L
0
0
n
n
n `
If you compare Definition 1 with Definition 2.5.4 you will see that the only difference between lim n l ⬁ a n 苷 L and lim x l ⬁ f x 苷 L is that n is required to be an integer. Thus we have the following theorem, which is illustrated by Figure 4. 2 Theorem If lim x l ⬁ f x 苷 L and f n 苷 a n when n is an integer, then lim n l ⬁ a n 苷 L.
y
y=ƒ
L
0
FIGURE 4
x
1 2 3 4
In particular, since we know from Section 2.5 that lim x l ⬁ 1x r 苷 0 when r ⬎ 0, we have 3
lim
nl⬁
1 苷0 nr
if r ⬎ 0
If an becomes large as n becomes large, we use the notation lim a n 苷 ⬁
nl⬁
In this case the sequence a n is divergent, but in a special way. We say that a n diverges to ⬁. The Limit Laws given in Section 2.3 also hold for the limits of sequences and their proofs are similar.
Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
SECTION 8.1
SEQUENCES
557
If a n and bn are convergent sequences and c is a constant, then
Limit Laws for Sequences
lim a n ⫹ bn 苷 lim a n ⫹ lim bn
nl⬁
nl⬁
nl⬁
lim a n ⫺ bn 苷 lim a n ⫺ lim bn
nl⬁
nl⬁
nl⬁
lim ca n 苷 c lim a n
nl⬁
lim c 苷 c
nl⬁
nl⬁
lim a n bn 苷 lim a n ⴢ lim bn
nl⬁
nl⬁
lim
lim a n an nl⬁ 苷 bn lim bn
nl⬁
if lim bn 苷 0 nl⬁
nl⬁
lim a np 苷 lim a n
nl⬁
nl⬁
nl⬁
p
if p ⬎ 0 and a n ⬎ 0
The Squeeze Theorem can also be adapted for sequences as follows (see Figure 5). If a n 艋 bn 艋 cn for n 艌 n 0 and lim a n 苷 lim cn 苷 L, then lim bn 苷 L.
Squeeze Theorem for Sequences
nl⬁
cn
nl⬁
nl⬁
Another useful fact about limits of sequences is given by the following theorem, which follows from the Squeeze Theorem because ⫺ a n 艋 a n 艋 a n .
bn an 0
4
If lim a n 苷 0, then lim a n 苷 0.
Theorem
nl⬁
nl⬁
n
FIGURE 5
The sequence bn is squeezed between the sequences a n and cn .
This shows that the guess we made earlier from Figures 1 and 2 was correct.
EXAMPLE 4 Find lim
nl⬁
n . n⫹1
SOLUTION The method is similar to the one we used in Section 2.5: Divide numerator
and denominator by the highest power of n that occurs in the denominator and then use the Limit Laws. lim 1 n 1 nl⬁ lim 苷 lim 苷 nl⬁ n ⫹ 1 nl⬁ 1 1 1⫹ lim 1 ⫹ lim nl⬁ nl⬁ n n 苷
1 苷1 1⫹0
Here we used Equation 3 with r 苷 1. EXAMPLE 5 Applying l’Hospital’s Rule to a related function
Calculate lim
nl⬁
ln n . n
SOLUTION Notice that both numerator and denominator approach infinity as n l ⬁. We
can’t apply l’Hospital’s Rule directly because it applies not to sequences but to functions
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558
CHAPTER 8
INFINITE SEQUENCES AND SERIES
of a real variable. However, we can apply l’Hospital’s Rule to the related function f x 苷 ln xx and obtain lim
xl⬁
ln x 1x 苷 lim 苷0 xl⬁ 1 x
Therefore, by Theorem 2, we have lim
nl⬁
an
ln n 苷0 n
EXAMPLE 6 Determine whether the sequence a n 苷 ⫺1 n is convergent or divergent.
1
SOLUTION If we write out the terms of the sequence, we obtain
0
1
2
3
4
⫺1, 1, ⫺1, 1, ⫺1, 1, ⫺1, . . .
n
_1
The graph of this sequence is shown in Figure 6. Since the terms oscillate between 1 and ⫺1 infinitely often, a n does not approach any number. Thus lim n l ⬁ ⫺1 n does not exist; that is, the sequence ⫺1 n is divergent.
FIGURE 6
EXAMPLE 7 Evaluate lim
The graph of the sequence in Example 7 is shown in Figure 7 and supports the answer.
nl⬁
⫺1 n if it exists. n
SOLUTION We first calculate the limit of the absolute value:
an 1
lim
nl⬁
⫺1 n n
苷 lim
nl⬁
1 苷0 n
Therefore, by Theorem 4, 0
1
n
lim
nl⬁
_1
⫺1 n 苷0 n
The following theorem says that if we apply a continuous function to the terms of a convergent sequence, the result is also convergent. The proof is given in Appendix E.
FIGURE 7
5
Theorem If lim a n 苷 L and the function f is continuous at L, then nl⬁
lim f a n 苷 f L
nl⬁
EXAMPLE 8 Find lim sinn. nl⬁
SOLUTION Because the sine function is continuous at 0, Theorem 5 enables us to write
lim sinn 苷 sin lim n 苷 sin 0 苷 0
nl⬁
v
nl⬁
EXAMPLE 9 Using the Squeeze Theorem
Discuss the convergence of the sequence
a n 苷 n!n n, where n! 苷 1 ⴢ 2 ⴢ 3 ⴢ ⭈ ⭈ ⭈ ⴢ n. SOLUTION Both numerator and denominator approach infinity as n l ⬁ but here we
have no corresponding function for use with l’Hospital’s Rule (x! is not defined when x is not an integer). Let’s write out a few terms to get a feeling for what happens to a n Copyright 2009 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
SECTION 8.1 Creating Graphs of Sequences Some computer algebra systems have special commands that enable us to create sequences and graph them directly. With most graphing calculators, however, sequences can be graphed by using parametric equations. For instance, the sequence in Example 9 can be graphed by entering the parametric equations x苷t
y 苷 t!t t
SEQUENCES
559
as n gets large: a1 苷 1
a2 苷 an 苷
6
1ⴢ2 2ⴢ2
a3 苷
1ⴢ2ⴢ3 3ⴢ3ⴢ3
1 ⴢ 2 ⴢ 3 ⴢ ⭈⭈⭈ ⴢ n n ⴢ n ⴢ n ⴢ ⭈⭈⭈ ⴢ n
It appears from these expressions and the graph in Figure 8 that the terms are decreasing and perhaps approach 0. To confirm this, observe from Equation 6 that
and graphing in dot mode, starting with t 苷 1 and setting the t-step equal to 1. The result is shown in Figure 8.
1 n
an 苷
1
2 ⴢ 3 ⴢ ⭈⭈⭈ ⴢ n n ⴢ n ⴢ ⭈⭈⭈ ⴢ n
Notice that the expression in parentheses is at most 1 because the numerator is less than (or equal to) the denominator. So 1 0 ⬍ an 艋 n We know that 1n l 0 as n l ⬁. Therefore a n l 0 as n l ⬁ by the Squeeze Theorem.
10
0
FIGURE 8
v
EXAMPLE 10 Limit of a geometric sequence For what values of r is the sequence r n
convergent? SOLUTION We know from Section 2.5 and the graphs of the exponential functions in
Section 1.5 that lim x l ⬁ a x 苷 ⬁ for a ⬎ 1 and lim x l ⬁ a x 苷 0 for 0 ⬍ a ⬍ 1. Therefore, putting a 苷 r and using Theorem 2, we have lim r n 苷
nl⬁
⬁ 0
if r ⬎ 1 if 0 ⬍ r ⬍ 1
For the cases r 苷 1 and r 苷 0 we have lim 1n 苷 lim 1 苷 1
nl⬁
lim 0 n 苷 lim 0 苷 0
and
nl⬁
nl⬁
nl⬁
If ⫺1 ⬍ r ⬍ 0, then 0 ⬍ r ⬍ 1, so
lim r n 苷 lim r
nl⬁
nl⬁
n
苷0
and therefore lim n l ⬁ r n 苷 0 by Theorem 4. If r 艋 ⫺1, then r n diverges as in Example 6. Figure 9 shows the graphs for various values of r. (The case r 苷 ⫺1 is shown in Figure 6.) an
an
r>1 1
1
_1