Calculus With Concepts in Calculus

1211 Pages·2006·140.493 MB·English

by  Denny Gulick, Robert Ellis

CALCULUS CCAALLCCUULLUUSS CALCULUS
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PREFACE
Like its predecessors, Calculus contains all the topics that normally constitute a
course in calculus of one and several variables. It is suitable for sequences taught
in three semesters or in four or five quarters. In the three-semester case, the first
semester would usually include the introductory chapter (Chapter 1), the three
chapters on limits and derivatives (Chapters 2 to 4), and the initial chapter on
integrals (Chapter 5). The second semester would then include the rest of the
discussion of integration (Chapters 6 to 8) and some combination of the chapters
on sequences and series (Chapter 9), polar coordinates and conic sections (Chapter
10), and the introduction to vectors and vector-valued functions (Chapters 11 and
12). The third semester would include the remainder of those chapters, along with
the material on calculus of several variables (Chapters 13 and 14) and Chapter 15,
which contains the theorems of Green and Stokes as well as the Divergence
Theorem.
New to this Edition
Noteworthy features that are new to this edition include:
Many passages, especially the introductory passages, have been rewritten to
be more accessible to the reader and to motivate the main ideas under
discussion.
At the end of nearly every section’s exercise set throughout Chapters 2-12 and
in many sections in the remainder of the text there are mini-projects that are
designed to expand on the concepts put forth in the section.
Quadratic approximation of functions has been added to Section 3.8 to
complement linear approximation and better prepare the reader for Taylor
polynomials. The chapter on applications of the integral (Chapter 6) now
vii
viii Preface
appears before the chapter on techniques of integration (Chapter 8). This
change reflects the general availability of software packages such as
Mathematica, MATLAB, Maple, and Derive that perform symbolic
integration. One technique—substitution in integrals—is still presented in
Section 5.6, before the applications of the integral; the other standard
techniques have been retained and appear in Chapter 8. Nevertheless, with a
little care in selecting exercises, Chapter 8 could easily be presented before
Chapter 6 if desirable.
An initial discussion of parametrized curves has been placed in Chapter 6 so
that the applications of length and area can involve such curves.
Anew section (Section 14.9) on parametrized surfaces expands the class of
surfaces whose areas can be computed and to which one can apply Stokes’s
and the Divergence Theorems in Chapter 15. New examples and exercises
have been added in both Chapter 14 and Chapter 15 to reflect this change.
New exercises, both applied and mathematical, appear through the text.
Organization
Although we have been careful in selecting the order in which the topics appear
in this edition, there is flexibility in the choice of topics and the order in which
they are introduced. Chapter 1 (which includes an introduction to the
trigonometric, logarithmic, and exponential functions) is preliminary and can be
covered quickly if the reader’s preparation is sufficient. With a little care,
techniques of integration (Chapter 8) can be discussed before applications of the
integral (Chapter 6). In addition, sequences and series (Chapter 9) can be studied
any time after Chapter 8, and conic sections (Sections 10.3 to 10.5) can be
considered any time after Chapter 4.
Pedagogical Features
The pedagogical features that have made the earlier editions so helpful to both
students and instructors have been retained or improved for this edition.
Whenever possible, we use geometric and intuitive motivation to introduce
concepts and results so that readers may readily absorb the definitions and
theorems that follow.
The concepts are supported by numerous examples, with graphical and
numerical emphasis where appropriate.
The topical development, in which we employ numerous worked examples
and approximately 1000 illustrations, aims for clarity and precision without
overburdening the reader with formalism. In keeping with this goal, we have
placed the more difficult proofs in the Appendix and have retained in the body
of the text the more illuminating and illustrative proofs from first-year
calculus. In the chapters on calculus of several variables we have proved
selected theorems that aid comprehension of the material.
Preface ix
Exercises appear both at the ends of the sections and, for review, at the end of
each chapter. Each set begins with a full complement of routine exercises
(graphical, numerical, and analytical) to provide practice in using the ideas and
methods presented in the text. These are followed by applied problems and by
other exercises of a more challenging nature. The especially difficult exercises
are identified with an asterisk.
Exercises requiring the use of a calculator or computer are indicated by the
symbol . One or two projects that require more thought appear at the end
of most sections from Chapter 2 on.
Topics for Discussion appear in the Review sections of Chapters 2 to 15. They
are ideal for discussion in class and for writing assignments.
Chapters 3 to 15 all end with a collection of Cumulative Review Exercises,
which are intended to reinforce the main ideas of the previous chapters.
In the interest of accuracy, every exercise has been completely worked by each
of the authors. Answers to odd-numbered exercises (except those requiring
longer explanations) appear at the back of the book.
Throughout the book, statements of definitions, theorems, lemmas, and
corollaries, as well as important formulas, are highlighted with tints for easy
identification. Numbering is consecutive throughout each chapter for
definitions and theorems, and consecutive within each section for examples
and formulas. We use the symbol (cid:1)to signal the end of a proof and (cid:2)for the
end of the solution to an example.
Lists of Key Terms and Expressions, Key Formulas, and Key Theorems appear
at the end of each chapter before the Review Exercises.
On the endpapers we have assembled important formulas and results to
facilitate reference.
Pronunciation of difficult terms and names is shown in footnotes on the pages
where they first appear.
Solutions Manuals
The Student Solutions Manual, by the authors, is available for purchase. It
contains worked-out solutions for all odd-numbered exercises in the text.
The Instructor’s Solutions Manual, by the authors, is available free of charge and
in two volumes to instructors who adopt this text. It contains worked-out solutions
for all exercises and a separate answer section for all exercises. (Answers to odd-
numbered exercises are given at the back of the book.)
x Preface
Acknowledgements
We are grateful to many people who have helped us in a variety of ways as we
prepared the various editions of this book. Our thanks go to these reviewers:
Linda J. S. Allen, Texas Tech University
Robert M. Anderson, Boise State University
James Angelos, Central Michigan University
Michael Bleicher, Clark Atlanta University
Gary A. Bogar, Montana State University
Jack Ceder, University of California–Santa Barbara
Richard M. Davitt, University of Louisville
Frank Glaser de Lugo, California State Polytechnic Institute–Pomona
Benny Evans, Oklahoma State University
W. E. Fitzgibbon, University of Houston
Martin Flashman, Humboldt State University
Herbert A. Gindler, San Diego State University
Stuart Goldenberg, California Polytechnic State University
Dorian Goldfeld, Columbia University
John Gosselin, University of Georgia
Kamel N. Haddad, California State University–Bakersfield
Judykay Hartzell, University of Nebraska–Omaha
Chung-Wu Ho, Southern Illinois University–Edwardsville
Arnold J. Insel, Illinois State University
David Johnson, Lehigh University
Ronald Knill, Tulane University
Cecilia Knoll, Florida Institute of Technology
Jack Lamoreaux, Brigham Young University
M. Paul Latiolais, Portland State University
Daniel McCallum, University of Arkansas–Little Rock
Giles Maloof, Boise State University
Maurice Monahan, South Dakota State University
Robert Myers, Oklahoma State University
Thomas Roe, South Dakota State University
David Ryeburn, Simon Fraser University
Laurence Small, Los Angeles Pierce College
Kirby C. Smith, Texas A&M University
Hugo S. H. Sun, California State University–Fresno
Stephen Willard, University of Alberta
Many thanks are also due these survey respondents:
John Akeroyd, University of Arkansas
Maria Calzada, Loyola University
Todd Cochrane, Kansas State University
Don Curlovic, University of South Carolina–Sumter
Tim Feeman, Villanova University
Norman J. Finizio, University of Rhode Island
Michael B. Gregory, University of North Dakota
Preface xi
Boris Hasselblatt, Tufts University
Lee Johnson, Virginia Polytechnic Institute and State University
Roger Johnson, Georgia Institute of Technology
Sidney Kolpas, Glendale College
Peter Kuhfittig, Milwaukee School of Engineering
David Minda, University of Cincinnati
S. James Taylor, University of Virginia
Sergey Yuzvinsky, University of Oregon
Many of our colleagues at the University of Maryland have made contributions
to the original writing of this book and to the revisions; we wish to express our
appreciation to
Jeffrey Adams Henry King
Stuart Antman Umberto Neri
Kenneth Berg John Osborn
Michael Boyle Jonathan Rosenberg
Jeff Cooper Jason Schultz
Jerome Dancis C. Robert Warner
Paul Green Peter Wolfe
Frances Gulick Scott Wolpert
John Horváth Mishael Zedek
We are also grateful for comments and suggestions from
Peter M. Gibson, University of Alabama–Huntsville
Gregory Grant, University of Maryland
Jonathan Sandow, Yeshiva College
Steven Schonefeld, Tri-State University
William V. Thayer, St. Louis Community College–Meramec
Finally we want to thank Jon Fuller and Andy Gates for their assistance in
bringing this edition to publication.
Robert Ellis
Denny Gulick
March 2003

TO THE READER
When you begin to study calculus, you will find that you have encountered many
of its concepts and techniques before. Calculus makes extensive use of plane
geometry and algebra, two branches of mathematics with which you are already
familiar. However, added to these is a third ingredient, which may be new to you:
the notion of limit and of limiting processes. From the idea of limit arise the two
principal concepts that form the nucleus of calculus; these are the derivative and
the integral.
The derivative can be thought of as a rate of change, and this interpretation has
many applications. For example, we may use the derivative to find the velocity of
an object, such as a rocket, or to determine the maximum and minimum values of
a function. In fact, the derivative provides so much information about the behavior
of functions that it greatly simplifies graphing them. Because of its broad
applicability, the derivative is as important in such disciplines as physics,
engineering, economics, and biology as it is in pure mathematics.
The definition of the integral is motivated by the familiar notion of area.
Although the methods of plane geometry enable us to calculate the areas of
polygons, they do not provide ways of finding the areas of plane regions whose
boundaries are curves other than circles. By means of the integral we can find the
areas of many such regions. We will also use it to calculate volumes, centers of
gravity, lengths of curves, work, and hydrostatic force.
The derivative and the integral have found many diverse uses. The following
list, taken from the examples and exercises in this book, illustrates the variety of
fields in which these powerful concepts are employed.
xiii
xiv To the Reader
Application Section
Marginal cost and marginal revenue 3.1, 3.3, 5.4
Weight of an astronaut 3.2
Angular velocity and acceleration of a pendulum 3.4, 3.5
Windpipe pressure during a cough 4.1
Dating of a bone or lunar rock sample 4.4
Atmospheric pressure 4.4
Population prediction 4.4, 9.2
Amount of an anesthetic needed during an operation 4.4
Cost of insulating an attic floor 4.6
Blood resistance in vascular branching 4.6
Surface area of a cell in a beehive 4.6
Probable distance of an electron from the center of an atom 4.6
Escape velocity from the earth’s gravitational field 4.8, 12.7
Terminal speed of a falling object 4.8, 7.4, 7.8
Blood alcohol levels Chapter 4 Review
Buffon’s needle problem 5.4
Rate of flow of blood in an artery 5.5
Probability of failure in certain systems 5.6
Equity of distribution of income 5.8
Volume of the great Pyramid of Cheops 6.1
Length of a (hanging) telephone line 6.4
Energy required to empty water from a swimming pool 6.4
Center of gravity 6.5, 14.7
Force of water on a dam 6.6
Work done on a piston Chapter 6 Review
Calibration of a water clock Chapter 6 Review
Contour of the Gateway Arch in St. Louis 7.4
Shape of a hanging cable 7.4, 7.8
Banking of curves 7.5
Intensity of thermal radiation 7.6, 8.7
Spread of a disease 8.4
Chemical reaction rate 8.4
Thermonuclear reactions 8.4
Present sale value 8.7
Mean life of an atom 8.7
Location of an electron in an atom 8.7
Pareto’s law of distribution of income 8.7
Harmonics of a stringed musical instrument 9.2
Compound and continuous interest 9.2
Multiplier effect in economics 9.4
Period of a pendulum 9.8
Shape of a suspended bridge cable 9.10, 10.3
Light-gathering power of a telescope Chapter 9 Review
Location of the source of a sound 10.3
Orbit of Mars 10.3
Corner mirror on the moon 11.2
Motion of a point on the rim of a tire 12.1