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CONFIRMING PAGES CalculusFourth Edition ROBERT T. SMITH
Millersville University of Pennsylvania ROLAND B. MINTON Roanoke
College i
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CONFIRMING PAGES CALCULUS, FOURTH EDITION Published by McGraw-Hill,
a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of
the Americas, New York, NY 10020. Copyright c 2012 by The
McGraw-Hill Companies, Inc. All rights reserved. Previous editions
c 2008, 2002, and 2000. No part of this publication may be
reproduced or distributed in any form or by any means, or stored in
a database or retrieval system, without the prior written consent
of The McGraw-Hill Companies, Inc., including, but not limited to,
in any network or other electronic storage or transmission, or
broadcast for distance learning. Some ancillaries, including
electronic and print components, may not be available to customers
outside the United States. This book is printed on acid-free paper.
1 2 3 4 5 6 7 8 9 0 QVR/QVR 1 0 9 8 7 6 5 4 3 2 1 ISBN
9780073383118 MHID 0073383112 Vice President, Editor-in-Chief:
Marty Lange Vice President, EDP: Kimberly Meriwether David Senior
Director of Development: Kristine Tibbetts Editorial Director:
Stewart K. Mattson Sponsoring Editor: John R. Osgood Developmental
Editor: Eve L. Lipton Marketing Manager: Kevin M. Ernzen Lead
Project Manager: Peggy J. Selle Senior Buyer: Sandy Ludovissy Lead
Media Project Manager: Judi David Senior Designer: Laurie B.
Janssen Cover Designer: Ron Bissell Cover Image: c
Gettyimages/George Diebold Photography Senior Photo Research
Coordinator: John C. Leland Compositor: Aptara, Inc. Typeface:
10/12 Times Roman Printer: Quad/Graphics All credits appearing on
page or at the end of the book are considered to be an extension of
the copyright page. Library of Congress Cataloging-in-Publication
Data Smith, Robert T. (Robert Thomas), 1955- Calculus / Robert T.
Smith, Roland B. Minton. 4th ed. p. cm. Includes index. ISBN
9780073383118ISBN 0073383112 (hard copy : alk. paper) 1.
Transcendental functionsTextbooks. 2. CalculusTextbooks. I. Minton,
Roland B., 1956 II. Title. QA353.S649 2012 515 .22dc22 2010030314
www.mhhe.com ii
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CONFIRMING PAGES DEDICATION To Pam, Katie and Michael To Jan, Kelly
and Greg And in memory of our parents: George and Anne Smith and
Paul and Mary Frances Minton iii
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CONFIRMING PAGES About the Authors Robert T. Smith is Professor of
Mathematics and Dean of the School of Science and
MathematicsatMillersvilleUniversityofPennsylvania,wherehehasbeenafacultymember
since 1987. Prior to that, he was on the faculty at Virginia Tech.
He earned his Ph.D. in mathematics from the University of Delaware
in 1982. Professor Smiths mathematical interests are in the
application of mathematics to prob-
lemsinengineeringandthephysicalsciences.Hehaspublishedanumberofresearcharticles
on the applications of partial differential equations as well as on
computational problems in x-ray tomography. He is a member of the
American Mathematical Society, the Mathematical Association of
America, and the Society for Industrial and Applied Mathematics.
Professor Smith lives in Lancaster, Pennsylvania, with his wife
Pam, his daughter Katie and his son Michael. His ongoing
extracurricular goal is to learn to play golf well enough to not
come in last in his annual mathematicians/statisticians tournament.
Roland B. Minton is Professor of Mathematics and Chair of the
Department of Mathemat- ics, Computer Science and Physics at
Roanoke College, where he has taught since 1986. Prior to that, he
was on the faculty at Virginia Tech. He earned his Ph.D. from
Clemson University in 1982. He is the recipient of Roanoke College
awards for teaching excellence and professional achievement, as
well as the 2005 Virginia Outstanding Faculty Award and the 2008
George Polya Award for mathematics exposition. Professor Mintons
current research program is in the mathematics of golf, especially
the analysis of ShotLink statistics. He has published articles on
various aspects of sports science, and co-authored with Tim
Pennings an article on Pennings dog Elvis and his ability to solve
calculus problems. He is co-author of a technical monograph on
control theory. He has supervised numerous independent studies and
held workshops for local high school teachers. He is an active
member of the Mathematical Association of America. Professor Minton
lives in Salem, Virginia, with his wife Jan and occasionally with
his daughter Kelly and son Greg when they visit. He enjoys playing
golf when time permits and watching sports events even when time
doesnt permit. Jan also teaches at Roanoke College and is very
active in mathematics education. In addition to Calculus: Early
Transcendental Functions, Professors Smith and Minton are also
coauthors of Calculus: Concepts and Connections c 2006, and three
earlier books for McGraw-Hill Higher Education. Earlier editions of
Calculus have been translated into Spanish, Chinese and Korean and
are in use around the world. iv
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CONFIRMING PAGES Brief Table of Contents CHAPTER 0 .. Preliminaries
1 CHAPTER 1 .. Limits and Continuity 47 CHAPTER 2 ..
Differentiation 107 CHAPTER 3 .. Applications of Differentiation
173 CHAPTER 4 .. Integration 251 CHAPTER 5 .. Applications of the
Definite Integral 315 CHAPTER 6 .. Exponentials, Logarithms and
Other Transcendental Functions 375 CHAPTER 7 .. Integration
Techniques 421 CHAPTER 8 .. First-Order Differential Equations 491
CHAPTER 9 .. Infinite Series 531 CHAPTER 10 .. Parametric Equations
and Polar Coordinates 625 CHAPTER 11 .. Vectors and the Geometry of
Space 687 CHAPTER 12 .. Vector-Valued Functions 749 CHAPTER 13 ..
Functions of Several Variables and Partial Differentiation 809
CHAPTER 14 .. Multiple Integrals 901 CHAPTER 15 .. Vector Calculus
977 CHAPTER 16 .. Second-Order Differential Equations 1073 APPENDIX
A .. Proofs of Selected Theorems A-1 APPENDIX B .. Answers to
Odd-Numbered Exercises A-13 v
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CONFIRMING PAGES Table of Contents Seeing the Beauty and Power of
Mathematics xiii Applications Index xxiv CHAPTER 0 .. Preliminaries
1 0.1 The Real Numbers and the Cartesian Plane 2 . The Real Number
System and Inequalities . The Cartesian Plane 0.2 Lines and
Functions 9 . Equations of Lines . Functions 0.3 Graphing
Calculators and Computer Algebra Systems 21 0.4 Trigonometric
Functions 27 0.5 Transformations of Functions 36 CHAPTER 1 ..
Limits and Continuity 47 1.1 A Brief Preview of Calculus: Tangent
Lines and the Length of a Curve 47 1.2 The Concept of Limit 52 1.3
Computation of Limits 59 1.4 Continuity and Its Consequences 68 .
The Method of Bisections 1.5 Limits Involving Infinity; Asymptotes
78 . Limits at Infinity 1.6 Formal Definition of the Limit 87 .
Exploring the Definition of Limit Graphically . Limits Involving
Infinity 1.7 Limits and Loss-of-Significance Errors 98 . Computer
Representation of Real Numbers CHAPTER 2 .. Differentiation 107 2.1
Tangent Lines and Velocity 107 . The General Case . Velocity 2.2
The Derivative 118 . Alternative Derivative Notations . Numerical
Differentiation vi
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CONFIRMING PAGES Table of Contents vii 2.3 Computation of
Derivatives: The Power Rule 127 . The Power Rule . General
Derivative Rules . Higher Order Derivatives . Acceleration 2.4 The
Product and Quotient Rules 135 . Product Rule . Quotient Rule .
Applications 2.5 The Chain Rule 142 2.6 Derivatives of
Trigonometric Functions 147 . Applications 2.7 Implicit
Differentiation 155 2.8 The Mean Value Theorem 162 CHAPTER 3 ..
Applications of Differentiation 173 3.1 Linear Approximations and
Newtons Method 174 . Linear Approximations . Newtons Method 3.2
Maximum and Minimum Values 185 3.3 Increasing and Decreasing
Functions 195 . What You See May Not Be What You Get 3.4 Concavity
and the Second Derivative Test 203 3.5 Overview of Curve Sketching
212 3.6 Optimization 223 3.7 Related Rates 234 3.8 Rates of Change
in Economics and the Sciences 239 CHAPTER 4 .. Integration 251 4.1
Antiderivatives 252 4.2 Sums and Sigma Notation 259 . Principle of
Mathematical Induction 4.3 Area 266 4.4 The Definite Integral 273 .
Average Value of a Function 4.5 The Fundamental Theorem of Calculus
284 4.6 Integration by Substitution 292 . Substitution in Definite
Integrals 4.7 Numerical Integration 298 . Simpsons Rule . Error
Bounds for Numerical Integration CHAPTER 5 .. Applications of the
Definite Integral 315 5.1 Area Between Curves 315 5.2 Volume:
Slicing, Disks and Washers 324 . Volumes by Slicing . The Method of
Disks . The Method of Washers
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CONFIRMING PAGES viii Table of Contents 5.3 Volumes by Cylindrical
Shells 338 5.4 Arc Length and Surface Area 345 . Arc Length .
Surface Area 5.5 Projectile Motion 352 5.6 Applications of
Integration to Physics and Engineering 361 CHAPTER 6 ..
Exponentials, Logarithms and Other Transcendental Functions 375 6.1
The Natural Logarithm 375 . Logarithmic Differentiation 6.2 Inverse
Functions 384 6.3 The Exponential Function 391 . Derivative of the
Exponential Function 6.4 The Inverse Trigonometric Functions 399
6.5 The Calculus of the Inverse Trigonometric Functions 405 .
Integrals Involving the Inverse Trigonometric Functions 6.6 The
Hyperbolic Functions 411 . The Inverse Hyperbolic Functions .
Derivation of the Catenary CHAPTER 7 .. Integration Techniques 421
7.1 Review of Formulas and Techniques 422 7.2 Integration by Parts
426 7.3 Trigonometric Techniques of Integration 433 . Integrals
Involving Powers of Trigonometric Functions . Trigonometric
Substitution 7.4 Integration of Rational Functions Using Partial
Fractions 442 . Brief Summary of Integration Techniques 7.5
Integration Tables and Computer Algebra Systems 450 . Using Tables
of Integrals . Integration Using a Computer Algebra System 7.6
Indeterminate Forms and LHopitals Rule 457 . Other Indeterminate
Forms 7.7 Improper Integrals 467 . Improper Integrals with a
Discontinuous Integrand . Improper Integrals with an Infinite Limit
of Integration . A Comparison Test 7.8 Probability 479 CHAPTER 8 ..
First-Order Differential Equations 491 8.1 Modeling with
Differential Equations 491 . Growth and Decay Problems . Compound
Interest 8.2 Separable Differential Equations 501 . Logistic
Growth
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CONFIRMING PAGES Table of Contents ix 8.3 Direction Fields and
Eulers Method 510 8.4 Systems of First-Order Differential Equations
521 . Predator-Prey Systems CHAPTER 9 .. Infinite Series 531 9.1
Sequences of Real Numbers 532 9.2 Infinite Series 544 9.3 The
Integral Test and Comparison Tests 554 . Comparison Tests 9.4
Alternating Series 565 . Estimating the Sum of an Alternating
Series 9.5 Absolute Convergence and the Ratio Test 571 . The Ratio
Test . The Root Test . Summary of Convergence Tests 9.6 Power
Series 579 9.7 Taylor Series 587 . Representation of Functions as
Power Series . Proof of Taylors Theorem 9.8 Applications of Taylor
Series 599 . The Binomial Series 9.9 Fourier Series 607 . Functions
of Period Other Than 2 . Fourier Series and Music Synthesizers
CHAPTER 10 .. Parametric Equations and Polar Coordinates 625 10.1
Plane Curves and Parametric Equations 625 10.2 Calculus and
Parametric Equations 634 10.3 Arc Length and Surface Area in
Parametric Equations 641 10.4 Polar Coordinates 649 10.5 Calculus
and Polar Coordinates 660 10.6 Conic Sections 668 . Parabolas .
Ellipses . Hyperbolas 10.7 Conic Sections in Polar Coordinates 677
CHAPTER 11 .. Vectors and the Geometry of Space 687 11.1 Vectors in
the Plane 688 11.2 Vectors in Space 697 . Vectors in R3 11.3 The
Dot Product 704 . Components and Projections
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CONFIRMING PAGES x Table of Contents 11.4 The Cross Product 714
11.5 Lines and Planes in Space 726 . Planes in R3 11.6 Surfaces in
Space 734 . Cylindrical Surfaces . Quadric Surfaces . An
Application CHAPTER 12 .. Vector-Valued Functions 749 12.1
Vector-Valued Functions 750 . Arc Length in R3 12.2 The Calculus of
Vector-Valued Functions 758 12.3 Motion in Space 769 . Equations of
Motion 12.4 Curvature 779 12.5 Tangent and Normal Vectors 786 .
Tangential and Normal Components of Acceleration . Keplers Laws
12.6 Parametric Surfaces 799 CHAPTER 13 .. Functions of Several
Variables and Partial Differentiation 809 13.1 Functions of Several
Variables 809 13.2 Limits and Continuity 822 13.3 Partial
Derivatives 833 13.4 Tangent Planes and Linear Approximations 844 .
Increments and Differentials 13.5 The Chain Rule 854 . Implicit
Differentiation 13.6 The Gradient and Directional Derivatives 864
13.7 Extrema of Functions of Several Variables 874 . Proof of the
Second Derivatives Test 13.8 Constrained Optimization and Lagrange
Multipliers 887 A M.F. Mood Standard Racket Greater Than 3THROAT
THROAT Prince Racket FIRST STRING FIRST STRING Greater Than 4
Greater Than 5 Greater Than 6 COEFFICENT OF RESTITUTION RACKETS
HELD BY VISE BALL VELOCITY OF 385 M PA FRAME BALL HITS FRAME IN
THIS AREA 25 25 17 2627 32 45 40 44 32 74 77 43 45 50 42 42 47 44
55 55 55 52 52 52 53 53 52 41 34 2742 30 20 2634 25 35 35 35 32 45
27 28 30 40 45 45 47 66 65 22 45 24 27 CHAPTER 14 .. Multiple
Integrals 901 14.1 Double Integrals 901 . Double Integrals over a
Rectangle . Double Integrals over General Regions 14.2 Area, Volume
and Center of Mass 916 . Moments and Center of Mass 14.3 Double
Integrals in Polar Coordinates 926 14.4 Surface Area 933 14.5
Triple Integrals 938 . Mass and Center of Mass
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CONFIRMING PAGES Table of Contents xi 14.6 Cylindrical Coordinates
948 14.7 Spherical Coordinates 956 . Triple Integrals in Spherical
Coordinates 14.8 Change of Variables in Multiple Integrals 962
CHAPTER 15 .. Vector Calculus 977 15.1 Vector Fields 977 15.2 Line
Integrals 990 15.3 Independence of Path and Conservative Vector
Fields 1003 15.4 Greens Theorem 1014 15.5 Curl and Divergence 1022
15.6 Surface Integrals 1032 . Parametric Representation of Surfaces
15.7 The Divergence Theorem 1044 15.8 Stokes Theorem 1053 15.9
Applications of Vector Calculus 1061 CHAPTER 16 .. Second-Order
Differential Equations 1073 16.1 Second-Order Equations with
Constant Coefficients 1074 16.2 Nonhomogeneous Equations:
Undetermined Coefficients 1082 16.3 Applications of Second-Order
Equations 1090 16.4 Power Series Solutions of Differential
Equations 1098 Appendix A: Proofs of Selected Theorems A-1 Appendix
B: Answers to Odd-Numbered Exercises A-13 Credits C-1 Index I-1
Bibliography See www.mhhe.com/Smithminton
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CONFIRMING PAGES McGraw-Hill Higher Education and Blackboard have
teamed up. Blackboard, the Web-based course-management system, has
partnered with McGraw- Hill to better allow students and faculty to
use online materials and activities to complement face-to-face
teaching. Blackboard features exciting social learning and teaching
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Whether youre choosing a book for your course or building ConnectTM
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you easy access to industry leading tech- nology and content,
whether your campus hosts it, or we do. Be sure to ask your local
McGraw-Hill representative for details. xii
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CONFIRMING PAGES Seeing the Beauty and Power of Mathematics The
calculus course is a critical course for science, technology,
engineering, and math majors. This course sets the stage for many
majors and is where students see the beauty of mathematics,
encouraging them to take upper-level math courses. In a calculus
market-research study conducted in 2008, calculus faculty pointed
out three critical components to student success in the calculus.
The most critical is mastery of the prerequisite algebra and
trigonometry skills. Our market research study showed that 58
percent of faculty mentioned that students struggled
withcalculusbecauseofpooralgebraskillsand72percentsaidbecauseofpoortrigonometry
skills. This is the number one learning challenge preventing
students from being successful in the rst calculus course. The
second critical component for student success is a text that
presents calculus concepts, especially the most challenging
concepts, in a clear and elegant manner. This helps students see
and appreciate the beauty and power of mathematics. Lastly,
calculus faculty told us that it is critical for a calculus text to
include all the classic calculus problems. Other calculus textbooks
may reect one or two of these critical components. However, there
is only ONE calculus textbook that includes all three:
Smith/Minton, 4e. Read on to understand how Smith/Minton handles
all three issues, helping your students to see the beauty and power
of mathematics. Mastery of Prerequisite Algebra and Trigonometry
Skills ALEKS Prep for Calculus is a Web-based program that focuses
on prerequisite and introductory material for Calculus, and can be
used during the rst six weeks of the term to prepare students for
success in the course. ALEKS uses articial intelligence and
adaptive questioning to assess precisely a students preparedness
and provide personalized instruction on the exact topics the
student is most ready to learn. By providing comprehensive
explanations, practice and feedback, ALEKS allows students to
quickly ll in gaps in prerequisite knowledge on their own time,
while also allowing instructors to focus on core course concepts.
Use ALEKS Prep for Calculus during the rst six weeks of the term to
see improved student condence and performance, as well as fewer
drops. xiii
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CONFIRMING PAGES xiv Seeing the Beauty and Power of Mathematics
ALEKS PREP FOR CALCULUS FEATURES: r Articial Intelligence: Targets
Gaps in Individual Student Knowledge r Adaptive, Open-Response
Environment: Avoids Multiple-Choice Questions and Ensures Student
Mastery r Automated Reports: Monitor Student and Class Progress For
more information about ALEKS, please visit:
www.aleks.com/highered/math ALEKS is a registered trademark of
ALEKS Corporation. Elegant Presentation of Calculus Concepts
Calculus reviewers and focus groups worked with the authors to
provide a more concise, streamlined presentation that maintains the
clarity of past editions. New examples and exercises illustrate the
physical meaning of the derivative and give real counterexamples.
The proofs of basic differentiation rules in sections 2.42.6 have
been revised, making them more elegant and easy for students to
follow, as were the theorems and proofs of basic integration and
integration rules. Further, an extensive revision of multivariable
calculus chapters includes a revision of the denition and proof of
the derivative of a vector-valued function, the normal vector, the
gradiant and both path and surface integration. More than any other
text, I believe Smith/Minton approaches deep concepts from a
thoughtful perspective in a very friendly style.Louis Rossi,
University of Delaware [Smith-Minton is] sufciently rigorous
without being considered to too mathy. It is a very readable book
with excellent graphics and outstanding applications sections.Todd
King, Michigan Technological University Rigorous, more interesting
to read than Stewart. Full of great application examples.Fred
Bourgoin, Laney College The material is very well presented in a
rigorous manner . . . very readable, numerous examples for students
with a wide variety of interests.John Heublein, Kansas State
UniversitySalina
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CONFIRMING PAGES Seeing the Beauty and Power of Mathematics xv
Classic Calculus Problems Many new classic calculus exercises have
been added. From basic derivative problems to related rates
applications involving ow, and multivariable applications to
electricity and magnetism, these exercises give students the
opportunity to challenge themselves and allow instructors exibility
when choosing assignments. The authors have also reorganized the
exercises to move consistently from the simplest to most difcult
problems, making it easier for instructors to choose exercises of
the appropriate level for their students. They moved the
applications to a separate section within the exercise sets and
were careful to include many examples from the common calculus
majors such as engineering, physical sciences, computer science and
biology. Thought provoking, clearly organized, challenging,
excellent problem sets, guarantee that students will actually read
the book and ask questions about concepts and topics. Donna Latham,
Sierra College [Smith-Minton is] a traditional calculus book that
is easy to read and has excellent applications.Hong Liu,
Embry-Riddle Aeronautical UniversityDaytona Beach The rigorous
treatment of calculus with an easy conversational style that has a
wealth of examples and problems. The topics are nicely arranged so
that theoretical topics are segregated from applications.Jayakumar
Ramanatan, Eastern Michigan University
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CONFIRMING PAGESxvi
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CONFIRMING PAGESxvii
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CONFIRMING PAGES the future of custom publishing is here.
Introducing McGraw-Hill Createa new, self- service website that
allows you to create custom course materials by drawing upon
McGraw-Hills comprehensive, cross-disciplinary content and other
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in minutes begin creating now: www.mcgrawhillcreate.com Start by
registering for a free Create account. If you already have a
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need to register on Create. Simply Sign In and enter your username
and password to begin using the site. ws you to create custom ing
upon McGraw-Hills ciplinary content and other ect, then arrange the
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dierent sources ur own content w copy in minutes If you already
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not need to register on Create. Simply Sign In and enter your
username and password to begin using the site. xviii
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CONFIRMING PAGES edit, share and approve like never before. After
you've completed your project, simply go to the My Projects tab at
the top right corner of the page to access and manage all of your
McGraw-Hill Create projects. Here you will be able to edit your
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once your review copy has been ordered. Click Approve to make your
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or bookstore modify your projects projects as needed. xix
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CONFIRMING PAGES xx Seeing the Beauty and Power of Mathematics
SUPPLEMENTS ONLINE INSTRUCTORS SOLUTIONS MANUAL An invaluable,
timesaving resource, the Instructors Solutions Manual contains
comprehen- sive, worked-out solutions to the odd- and even-numbered
exercises in the text. STUDENT SOLUTIONS MANUAL (ISBN
978-0-07-7256968) The Student Solutions Manual is a helpful
reference that contains comprehensive, worked- out solutions to the
odd-numbered exercises in the text. ONLINE TESTBANK AND
PREFORMATTED TESTS Brownstone Diploma testing software offers
instructors a quick and easy way to cre- ate customized exams and
view student results. Instructors may use the software to sort
questions by section, difculty level, and type; add questions and
edit existing questions; create multiple versions of questions
using algorithmically-randomized variables; prepare multiple-choice
quizzes; and construct a grade book. ONLINE CALCULUS CONCEPTS
VIDEOS Students will see essential concepts explained and brought
to life through dynamic ani- mations in this new video series
available on DVD and on the Smith/Minton website. The twenty-ve key
concepts, chosen after consultation with calculus instructors
across the country, are the most commonly taught topics that
students need help with and that also lend themselves most readily
to on-camera demonstration. CALCULUS AND TECHNOLOGY It is our
conviction that graphing calculators and computer algebra systems
must not be used indiscriminately. The focus must always remain on
the calculus. We have en- sured that each of our exercise sets
offers an extensive array of problems that should be worked by
hand. We also believe, however, that calculus study supplemented
with an intelligent use of technology gives students an extremely
powerful arsenal of problem- solving skills. Many passages in the
text provide guidance on how to judiciously use and not
abusegraphing calculators and computers. We also provide ample
opportunity for students to practice using these tools. Exercises
that are most easily solved with the aid of a graphing calculator
or a computer algebra system are easily identied with a icon.
IMPROVEMENTS IN THE FOURTH EDITION Building upon the success of the
Third Edition of Calculus, we have made the following revisions to
produce an even better Fourth Edition: Presentation r A key goal of
the Fourth Edition revision was to offer a clearer presentation of
calculus. With this goal in mind, the authors were able to reduce
the amount of material by nearly 150 pages. r The level of rigor
has been carefully balanced to ensure that concepts are presented
in a rigorously correct manner without allowing technical details
to overwhelm beginning calculus students. For example, the sections
on continuity, sum rule, chain rule, the denite integral and
Riemann sums, introductory vectors, and advanced multivariable
calculus (including the Greens Theorem section) have been revised
to improve the theorems, denitions, and/or proofs. r The exercise
sets were redesigned in an effort to aid instructors by allowing
them to more easily identify and assign problems of a certain type.
r The derivatives of hyperbolic functions are developed in Section
6.6, giving this important class of functions a full development.
Separating these functions from the
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CONFIRMING PAGES Seeing the Beauty and Power of Mathematics xxi
exponential and trigonometric functions allows for early and
comprehensive exploration of the relationship between these
functions, exponential functions, trigonometric functions, and
their derivatives and integrals. Exercises r More than 1,000 new
classic calculus problems were added, covering topics from
polynomials to multivariable calculus, including optimization,
related rates, integration techniques and applications, parametric
and polar equations, vectors, vector calculus, and differential
equations. r A reorganization of the exercise sets makes the range
of available exercises more transparent. Earlier exercises focus on
fundamentals, as developed in examples in the text. Later exercises
explore interesting extensions of the material presented in the
text. r Multi-step exercises help students make connections among
concepts and require students to become more critical readers.
Closely related exercises are different parts of the same numbered
exercise, with follow-up questions to solidify lessons learned. r
Application exercises have been separated out in all appropriate
sections. A new header identies the location of applied exercises
which are designed to show students the connection between what
they learn in class, other areas of study, and outside life. This
differentiates the applications from exploratory exercises that
allow students to discover connections and extensions for
themselves. ACKNOWLEDGMENTS A project of this magnitude requires
the collaboration of an incredible number of talented and dedicated
individuals. Our editorial staff worked tirelessly to provide us
with countless surveys, focus group reports, and reviews, giving us
the best possible read on the current state of calculus
instruction. First and foremost, we want to express our
appreciation to our sponsoring editor John Osgood and our
developmental editor Eve Lipton for their encouragement and support
to keep us on track throughout this project. They challenged us to
make this a better book. We also wish to thank our editorial
director Stewart Mattson, and director of development Kris Tibbets
for their ongoing strong support. We are indebted to the
McGraw-Hill production team, especially project manager Peggy Selle
and design coordinator Laurie Janssen, for (among other things)
producing a
beautifullydesignedtext.TheteamatMRCChasprovideduswithnumeroussuggestionsfor
clarifying and improving the exercise sets and ensuring the texts
accuracy. Our marketing manager Kevin Ernzen has been instrumental
in helping to convey the story of this book to a wider audience,
and media project manager Sandy Schnee created an innovative suite
of media supplements. Our work on this project beneted tremendously
from the insightful comments we received from many reviewers,
survey respondents and symposium attendees. We wish to thank the
following individuals whose contributions helped to shape this
book: REVIEWERS OF THE FOURTH EDITION Andre Adler, Illinois
Institute of Technology Daniel Balaguy, Sierra College Frank
Bauerle, University of California Santa Cruz Fred Bourgoin, Laney
College Kris Chatas, Washtenaw Community College Raymond Clapsadle,
University of Memphis Dan Edidin, University of Missouri Columbia
Timothy Flaherty, Carnegie Mellon University Gerald Greivel,
Colorado School of Mines Jerrold Grossman, Oakland University Murli
Gupta, George Washington University Ali Hajjafar, University of
Akron
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CONFIRMING PAGES xxii Seeing the Beauty and Power of Mathematics
Donald Hartig, California Polytechnic State UniversitySan Luis
Obispo John Heublein, Kansas State UniversitySalina Joseph Kazimir,
East Los Angeles College Harihar Khanal, Embry-Riddle Aeronautical
UniversityDaytona Beach Todd King, Michigan Technological
University Donna Latham, Sierra College Rick Leborne, Tennessee
Technological University Hong Liu, Embry-Riddle Aeronautical
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Patrick Ward, Illinois Central College Jay Zimmerman, Towson
University WITH MANY THANKS TO OUR PREVIOUS REVIEWER PANEL: Kent
Aeschliman, Oakland Community College Stephen Agard, University of
Minnesota Charles Akemann, University of California, Santa Barbara
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CONFIRMING PAGES Seeing the Beauty and Power of Mathematics xxiii
Tan Ban Pin, National University of Singapore Linda Powers,
Virginia Polytechnic Institute and State University Mohammad A.
Rammaha, University of NebraskaLincoln Richard Rebarber, University
of Nebraska Lincoln Kim Rescorla, Eastern Michigan University Edgar
Reyes, Southeastern Louisiana University Mark Smith, University of
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of Malaya Teri Woodington, Colorado School of Mines Gordon
Woodward, University of NebraskaLincoln Haidong Wu, University of
Mississippi Adil Yaqub, University of California, Santa Barbara
Hong-Ming Yin, Washington State University, Pullman Paul Yun, El
Camino College Jennifer Zhao, University of Michigan at Dearborn In
addition, a number of our colleagues graciously gave their time and
energy to help create or improve portions of the manuscript. We
would especially like to thank Richard Grant, Bill Ergle, Jack
Steehler, Ben Huddle, Chris Lee, Dave Taylor, Dan Larsen and Jan
Minton of Roanoke College for sharing their expertise in calculus
and related applications; student assistants Danielle Shiley and
Hannah Green for their insight and hard work; Tim Pennings and Art
Benjamin for inspirations both mathematical and personal; Tom Burns
for help with an industrial application; Gregory Minton and James
Albrecht for suggesting several brilliant problems; Dorothee Blum
of Millersville University for helping to class-test an early
version of the manuscript; Bruce Ikenaga of Millersville University
for generously sharing his expertise in TeX and Corel Draw and Pam
Vercellone-Smith, for lending us her expertise in many of the
biological applications. We also wish to thank Dorothee Blum, Bob
Buchanan, Antonia Cardwell, Roxana Costinescu, Chuck Denlinger,
Bruce Ikenaga, Zhoude Shao, Ron Umble and Zenaida Uy of
Millersville University for offering numerous helpful suggestions
for improvement. In addition, we would like to thank all of our
students throughout the years, who have (sometimes unknowingly)
eld-tested innumerable ideas, some of which worked and the rest of
which will not be found in this book. Ultimately, this book is for
our families. We simply could not have written a book of this
magnitude without their strong support. We thank them for their
love and inspiration throughout our growth as textbook authors.
Their understanding, in both the technical and the personal sense,
was essential. They provide us with the reason why we do all of the
things we do. So, it is tting that we especially thank our wives,
Pam Vercellone-Smith and Jan Minton and our children, Katie and
Michael Smith and Kelly and Greg Minton; and our parents, George
and Anne Smith and Paul and Mary Frances Minton. Robert T. Smith
Lancaster, Pennsylvania Roland B. Minton Salem, Virginia
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CONFIRMING PAGES Applications Index Biology Alligator sex, 77
Anthills, 344 Bacterial growth, 491492, 499 Biological
oscillations, 668 Bird ight, 246 Birth rates, 283 Cell age, 484
Cell division, 398 Circulatory system, 232 Competitive exclusion
principle, 526 Coyote, 358 Cricket chirping, 20 Decay, 491496
Dinosaurs, 491 Dog swimming, 886 E. coli growth, 499 Firey ashes,
203 Fish population, 509 Fish speed, 246 Fossils, 491 Genetics, 886
Growth, 491496 Height of person, 485 Human speed, 117 Kangaroo
legs, 315 Leg width and body weight, 134 Paleontology, 491
Pollutant density, 947 Pollution, 169 Predator-prey system, 293,
527 Pupil dilation, 47, 246 Pupil size, 84, 466 Reproductive rate,
184 Shark prey tracking, 873 Spruce budworm, 184, 222 Tree
infestation, 520 Zebra stripes, 520 Chemistry Acid-base titration,
383 Autocatalytic reaction, 242243 Bimolecular reactions, 509
Concentration analysis, 396, 397 Methane, 713 Product, 246 Rate of
reaction, 242243 Reactants, 246 Second-order chemical reaction, 246
Temperature, entropy, and Gibbs free energy, 843 Construction
Building height, 237 Sand pile height, 238 Sliding ladder, 234
Demographics Birthrate, 323 Critical threshold, 512513 Maximum
sustainable population, 506 Population and time, 20 Population
carrying capacity, 505 Population estimation, 920 Population growth
maximum, 244 Population prediction, 12 Rumor spread, 398 Urban
population growth, 126 Economics Advertising costs, 211, 238
Airline ticket sales, 35 Annual percentage yield, 496 Asset
depreciation, 497498 Bank account balance, 210 Barge costs, 246
Capital expenditure, 863 Complementary commodities, 843 Compound
interest, 496498, 499 Consumer surplus, 291 Cost minimization,
229230, 231 Coupon collectors problem, 564 Demand, 244 Diminishing
returns, 233 Economic Order Quantity, 283, 291 Elasticity of
demand, 241242, 244 Endowment, 508 Future value, 501 Gini index,
272 Gross domestic product, 265, 272 Ice cream sales, 713 Income
calculation, 862 Income stream, 500 Income tax, 68, 127 Initial
investment, 508 Investing, 210 Investment strategies, 506
Investment value, 843 Just-in-time inventory, 251, 291
Manufacturing costs, 211 Marginal cost, 239 Marginal prot, 239
Mortgages, 508 Multiplier effect, 553 National debt, 135 Oil
consumption, 323 Oil prices, 925 Packing, 544 Parking fees, 57
Present value, 500, 553 Product sales, 202 Production costs, 238,
240241, 324 Production optimization, 891892 Prot maximization, 324
Rate of change, 236, 239244 Relative change in demand, 240241
Relative change in price, 240 Resale value, 509 Retirement fund,
508 Revenue, 139 Revenue maximization, 233 Rule of 72, 501 Salary
increase, 77, 391 Stock investing, 873 Substitute commodities, 843
Supply and demand, 323 Tax rates, 500 Tax tables, 73 xxiv
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CONFIRMING PAGES Applications Index xxv Unit price, 141 Worker
productivity, 211 Engineering Airplane design, 227 Beam sag, 851
Capacitors, 1090 Car design, 977 Catenary, 411 Ceiling shape, 676
Circuit charge, 1091 Circuitry, 35, 1096 Cooling towers, 745
Electric dipole, 586 Falling ladder, 646 Flashlight design, 670
Highway construction costs, 229230, 231 Hydrostatic force on dam,
368 International Space Station, 375 Light pole height, 35 Metal
sheet gauge, 851 Norman window, 232 Oil pipeline, 231 Oil rig beam,
713 Oil tank capacity, 404 Parabolic dish, 741, 742 Pyramids, 335
Reliability testing, 421 RoboCup, 749 Robot vision, 239 Rocket
fuel, 369 Rocket height, 86 Rocket launch, 35 Rocket thrust, 890
Saddle Dome, 744 Screen door closing, 1081 Series circuit, 1096
Soda can design, 227228 Spacecraft launch, 238 Spiral staircase,
758 St. Louis Gateway Arch, 411, 418 Telegraph cable, 383
Thrust-time curve, 370 Tower height, 3334 Useful life phase, 421
Voltage, 10911092 Water ow, 323 Water pumping, 239, 363 Water
reservoir, 211 Water system, 117 Water tank, 288, 290 Water tower,
369 Environment Oil spill, 234, 237 Hobbies Jewelry making, 344
Photography, 20 Home Garden construction, 223224 Medicine Achilles
tendon, 315 Allosteric enzyme, 142 Antibiotics, 500
Antidepressants, 500 Brain neurons, 398 Computed tomography, 327
Drug concentration, 398 Drug dosage, 553 Drug half-life, 500 Drug
injection, 86 Drug sensitivity, 246 Enzymatic reaction, 194 Foot
arch, 322 Glucose concentration, 1097 HIV, 283 Infection, 237
Pneumotachograph, 310 Tendon force, 322 Music Digital, 531 Guitar
string, 238, 843 Octaves, 8 Piano tuning, 35, 619 Synthesizers,
617618 Timbre, 617 Tuning, 8 Physics AC circuit, 232, 293
Acceleration, 135 Air resistance, 356 Atmospheric pressure, 398
Ball motion, 353 Black body radiation, 606 Boiling point of water
at elevation, 20 Change in position, 277 Distance fallen, 288
Electric potential, 607 Electromagnetic eld, 511 Electromagnetic
radiation, 606 Falling object position, 257 Falling object
velocity, 418 Freezing point, 117 Frequency modulation, 222
Friction, 77 Gas laws, 238 Gravitation, 184 Half-life, 499
Hydrostatic forces, 371 Hyperbolic mirrors, 674 Impulse-momentum
equation, 283 Light path, 211, 231 Light reection, 232 Newtons Law
of Cooling, 494, 499 Object distance fallen, 6 Object launch, 358
Object velocity, 310 Pendulum, 1097 Plancks law, 398, 606 Planetary
orbits, 640 Projectiles, 265 Radio waves, 2728 Radioactive decay,
494 Raindrop evaporation, 237 Relativity, 86, 184 Rod density,
243244 Solar and Heliospheric Observatory, 173 Sound waves, 632
Spring motion, 154, 1081 Spring-mass system, 142 Terminal velocity,
509 Thrown ball, 231 Velocity, 57 Velocity required to reach
height, 354 Voltage, 293 Volume and pressure, 157 Weightlessness,
360 Sports/ Entertainment Auto racing, 687 Badminton, 283 Ball
height, 233 Baseball bat corking, 371 Baseball bat hit, 238, 321
Baseball bat mass, 366 Baseball bat sweet spot, 367 Baseball
batting average, 863 Baseball impulse, 365 Baseball knuckleball, 57
Baseball outelding, 404 Baseball pitching, 57, 359 Baseball player
gaze, 406407 Baseball spin, 724 Baseball statistics analysis, 8
Baseball velocity, 383 Basketball free throws, 359, 486 Basketball
perfect swish, 86 Bicycling, 20, 553
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Transcendental) CONFIRMING PAGES C H A P T E R 0 Preliminaries In
this chapter, we present a collection of familiar topics, primarily
those that we consider essential for the study of calculus. While
we do not intend this chapter to be a comprehensive review of
precalculus math- ematics, we have tried to hit the highlights and
provide you with some standard notation and language that we will
use throughout the text. As it grows, a chambered nautilus creates
a spiral shell. Behind this beautiful geometry is a surprising
amount of mathematics. The nautilus grows in such a way that the
overall proportions of its shell remain constant. That is, if you
draw a rectangle to circumscribe the shell, the ratio of height to
width of the rectangle remains nearly constant. There are several
ways to represent this property mathematically. In polar
coordinates (which we present in Chapter 10), we study logarith-
mic spirals that have the property that the angle of growth is
constant, corresponding to the constant proportions of a nautilus
shell. Using basic geometry, you can divide the circumscribing
rectangle into a sequence of squares as in the gure. The relative
sizes of the squares form the famous Fibonacci sequence 1, 1, 2, 3,
5, 8, . . . , where each number in the sequence is the sum of the
preceding two numbers. 21 13 8 5 32 A nautilus shell The Fibonacci
sequence has an amazing list of interesting properties. (Search on
the Internet to see what we mean!) Numbers in the sequence have a
surprising habit of showing up in nature, such as the number of
petals on a lily (3), buttercup (5), marigold (13), black-eyed
Susan (21) and pyrethrum (34). Although we have a very simple
description of how to generate the Fibonacci sequence, think about
howyoumightdescribeitasafunction.Aplot of the rst several numbers
in the sequence (shown in Figure 0.1) should give you the
impression of a graph curving up, perhaps a parabola or an
exponential curve. Two aspects of this problem are impor- tant
themes throughout the calculus. One of these is the importance of
looking for pat- terns to help us better describe the world. A
second theme is the interplay between graphs and functions. By
connecting the techniques of algebra with the visual images
provided by graphs, you will signicantly improve your ability to
solve real-world problems mathematically. 1
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2 3 4 5 6 7 8 x y 5 10 15 20 25 30 35 0 FIGURE 0.1 The Fibonacci
sequence 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE The Real
Number System and Inequalities Our journey into calculus begins
with the real number system, focusing on those properties that are
of particular interest for calculus. The set of integers consists
of the whole numbers and their additive inverses: 0, 1, 2, 3, . . .
. A rational number is any number of the form p q , where p and q
are integers and q = 0. For example, 2 3 , 7 3 and 27 125 are all
rational numbers. Notice that every integer n is also a rational
number, since we can write it as the quotient of two integers: n =
n 1 . The irrational numbers are all those real numbers that cannot
be written in the form p q , where p and q are integers. Recall
that rational numbers have decimal expansions that either terminate
or repeat. For instance, 1 2 = 0.5, 1 3 = 0.33333, 1 8 = 0.125 and
1 6 = 0.166666 are all rational numbers. By contrast, irrational
numbers have decimal expansions that do not repeat or terminate.
For instance, three familiar irrational numbers and their decimal
expansions are 2 = 1.41421 35623 . . . , = 3.14159 26535 . . . and
e = 2.71828 18284 . . . . We picture the real numbers arranged
along the number line displayed in Figure 0.2 (the real line). The
set of real numbers is denoted by the symbol R. 0 5432112345 2 3 p
e FIGURE 0.2 The real line For real numbers a and b, where a <
b, we dene the closed interval [a, b] to be the set of numbers
between a and b, including a and b (the endpoints). That is, [a, b]
= {x R | a x b}, as illustrated in Figure 0.3, where the solid
circles indicate that a and b are included in [a, b]. a b FIGURE
0.3 A closed interval
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Transcendental) CONFIRMING PAGES 0-3 SECTION 0.1 .. The Real
Numbers and the Cartesian Plane 3 Similarly, the open interval (a,
b) is the set of numbers between a and b, but not including the
endpoints a and b, that is, (a, b) = {x R | a < x < b}, as
illustrated in Figure 0.4, where the open circles indicate that a
and b are not included in (a, b). Similarly, we denote the set {x R
| x > a} by the interval notation (a, ) and {x R | x < a} by
(, a). In both of these cases, it is important to recognize that
and are not real numbers and we are using this notation as a
convenience. a b FIGURE 0.4 An open interval You should already be
very familiar with the following properties of real numbers.
THEOREM 1.1 If a and b are real numbers and a < b, then (i) For
any real number c, a + c < b + c. (ii) For real numbers c and d,
if c < d, then a + c < b + d. (iii) For any real number c
> 0, a c < b c. (iv) For any real number c < 0, a c > b
c. REMARK 1.1 We need the properties given in Theorem 1.1 to solve
inequalities. Notice that (i) says that you can add the same
quantity to both sides of an inequality. Part (iii) says that you
can multiply both sides of an inequality by a positive number.
Finally, (iv) says that if you multiply both sides of an inequality
by a negative number, the inequality is reversed. We illustrate the
use of Theorem 1.1 by solving a simple inequality. EXAMPLE 1.1
Solving a Linear Inequality Solve the linear inequality 2x + 5 <
13. Solution We can use the properties in Theorem 1.1 to solve for
x. Subtracting 5 from both sides, we obtain (2x + 5) 5 < 13 5 or
2x < 8. Dividing both sides by 2, we obtain x < 4. We often
write the solution of an inequality in interval notation. In this
case, we get the interval (, 4). You can deal with more complicated
inequalities in the same way. EXAMPLE 1.2 Solving a Two-Sided
Inequality Solve the two-sided inequality 6 < 1 3x 10. Solution
First, recognize that this problem requires that we nd values of x
such that 6 < 1 3x and 1 3x 10. It is most efcient to work with
both inequalities simultaneously. First, subtract 1 from each term,
to get 6 1 < (1 3x) 1 10 1 or 5 < 3x 9.
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Now, divide by 3, but be careful. Since 3 < 0, the inequalities
are reversed. We have 5 3 > 3x 3 9 3 or 5 3 > x 3. We usually
write this as 3 x < 5 3 , or in interval notation as [3, 5 3 ).
y 4224 8 4 4 8 x 1 FIGURE 0.5 y = x 1 x + 2 You will often need to
solve inequalities involving fractions. We present a typical
example in the following. EXAMPLE 1.3 Solving an Inequality
Involving a Fraction Solve the inequality x 1 x + 2 0. Solution In
Figure 0.5, we show a graph of the function, which appears to
indicate that the solution includes all x < 2 and x 1. Carefully
read the inequality and observe that there are only three ways to
satisfy this: either both numerator and denominator are positive,
both are negative or the numerator is zero. To visualize this, we
draw number lines for each of the individual terms, indicating
where each is positive, negative or zero and use these to draw a
third number line indicating the value of the quotient, as shown in
the margin. In the third number line, we have placed an above the 2
to indicate that the quotient is undened at x = 2. From this last
number line, you can see that the quotient is nonnegative whenever
x < 2 or x 1. We write the solution in interval notation as (,
2) [1, ). Note that this solution is consistent with what we see in
Figure 0.5. For inequalities involving a polynomial of degree 2 or
higher, factoring the polynomial and determining where the
individual factors are positive and negative, as in example 1.4,
will lead to a solution. EXAMPLE 1.4 Solving a Quadratic Inequality
Solve the quadratic inequality x2 + x 6 > 0. (1.1) Solution In
Figure 0.6, we show a graph of the polynomial on the left side of
the inequality. Since this polynomial factors, (1.1) is equivalent
to (x + 3)(x 2) > 0. (1.2) This can happen in only two ways:
when both factors are positive or when both factors are negative.
As in example 1.3, we draw number lines for both of the individual
factors, indicating where each is positive, negative or zero and
use these to draw a number line representing the product. We show
these in the margin. Notice that the third number line indicates
that the product is positive whenever x < 3 or x > 2. We
write this in interval notation as (, 3) (2, ). x 1 x 21 0 2 x 1 1
0 x 2 2 0 y x 10 10 20 62 4246 FIGURE 0.6 y = x2 + x 6 (x 3)(x 2) 2
00 3 x 2 2 0 x 3 3 0 No doubt, you will recall the following
standard denition. DEFINITION 1.1 The absolute value of a real
number x is |x| = x, if x 0. x, if x < 0
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Transcendental) CONFIRMING PAGES 0-5 SECTION 0.1 .. The Real
Numbers and the Cartesian Plane 5 Make certain that you read
Denition 1.1 correctly. If x is negative, then x is positive. This
says that |x| 0 for all real numbers x. For instance, using the
denition, |4| = (4) = 4. Notice that for any real numbers a and b,
|a b| = |a| |b|, although |a + b| = |a| + |b|, in general. (To
verify this, simply take a = 5 and b = 2 and compute both
quantities.) However, it is always true that |a + b| |a| + |b|.
NOTES For any two real numbers a and b, |a b| gives the distance
between a and b. (See Figure 0.7.) This is referred to as the
triangle inequality. The interpretation of |a b| as the distance
between a and b (see the note in the margin) is particularly useful
for solving inequalities involving absolute values. Wherever
possible, we suggest that you use this interpretation to read what
the inequality means, rather than merely following a procedure to
produce a solution. a b a b FIGURE 0.7 The distance between a and b
EXAMPLE 1.5 Solving an Inequality Containing an Absolute Value
Solve the inequality |x 2| < 5. (1.3) Solution First, take a few
moments to read what this inequality says. Since |x 2| gives the
distance from x to 2, (1.3) says that the distance from x to 2 must
be less than 5. So, nd all numbers x whose distance from 2 is less
than 5. We indicate the set of all numbers within a distance 5 of 2
in Figure 0.8. You can now read the solution directly from the
gure: 3 < x < 7 or in interval notation: (3, 7). 2 5 3 2 5 72
5 5 FIGURE 0.8 |x 2| < 5 Many inequalities involving absolute
values can be solved simply by reading the in- equality correctly,
as in example 1.6. EXAMPLE 1.6 Solving an Inequality with a Sum
Inside an Absolute Value Solve the inequality |x + 4| 7. (1.4)
Solution To use our distance interpretation, we must rst rewrite
(1.4) as |x (4)| 7. This now says that the distance from x to 4 is
less than or equal to 7. We illustrate the solution in Figure 0.9,
from which it follows that the solution is 11 x 3 or [11, 3]. 4 7
11 4 7 34 7 7 FIGURE 0.9 |x + 4| 7 Recall that for any real number
r > 0, |x| < r is equivalent to the following inequality not
involving absolute values: r < x < r. In example 1.7, we use
this to revisit the inequality from example 1.5. EXAMPLE 1.7 An
Alternative Method for Solving Inequalities Solve the inequality |x
2| < 5. Solution This is equivalent to the two-sided inequality
5 < x 2 < 5. Adding 2 to each term, we get the solution 3
< x < 7, or in interval notation (3, 7), as before.
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The Cartesian Plane For any two real numbers x and y we visualize
the ordered pair (x, y) as a point in two dimensions. The Cartesian
plane is a plane with two real number lines drawn at right angles.
The horizontal line is called the x-axis and the vertical line is
called the y-axis. The point where the axes cross is called the
origin, which represents the ordered pair (0, 0). To represent the
ordered pair (1, 2), start at the origin, move 1 unit to the right
and 2 units up and mark the point (1, 2), as in Figure 0.10. In
example 1.8, we analyze a small set of experimental data by
plotting some points in the Cartesian plane. This simple type of
graph is sometimes called a scatter plot. EXAMPLE 1.8 Using a Graph
Obtained from a Table of Data Suppose that you drop an object from
the top of a building and record how far the object has fallen at
different times, as shown in the following table. Time (sec) 0 0.5
1.0 1.5 2.0 Distance (ft) 0 4 16 36 64 Plot the points in the
Cartesian plane and discuss any patterns you notice. In particular,
use the graph to predict how far the object will have fallen in 2.5
seconds. Solution Taking the rst coordinate (x) to represent time
and the second coordinate (y) to represent distance, we plot the
points (0, 0), (0.5, 4), (1, 16), (1.5, 36) and (2, 64), as seen in
Figure 0.11. (1, 2) y x 2112 2 1 1 2 FIGURE 0.10 The Cartesian
plane y x 1.0 2.00.5 1.5 40 60 20 Time Distance FIGURE 0.11 Scatter
plot of data Notice that the points appear to be curving upward
(like a parabola). To predict the y-value corresponding to x = 2.5
(i.e., the distance fallen at time 2.5 seconds), we assume that
this pattern continues, so that the y-value would be much higher
than 64. But, how much higher is reasonable? It helps now to refer
back to the data. Notice that the change in height from x = 1.5 to
x = 2 is 64 36 = 28 feet. Since Figure 0.11 suggests that the curve
is bending upward, the change in height between successive points
should be getting larger and larger. You might reasonably predict
that the height will change by more than 28. If you look carefully
at the data, you might notice a pattern. Observe that the distances
given at 0.5-second intervals are 02 , 22 , 42 , 62 and 82 . A
reasonable guess for the distance at time 2.5 seconds might then be
102 = 100. Further, notice that this corresponds to a change of 36
from the distance at x = 2.0 seconds. At this stage, this is only
an educated guess and other guesses (98 or 102, for example) might
be equally reasonable. We urge that you think carefully about
example 1.8. You should be comfortable with the interplay between
the graph and the numerical data. This interplay will be a
recurring theme in our study of calculus. The distance between two
points in the Cartesian plane is a simple consequence of the
Pythagorean Theorem, as follows. THEOREM 1.2 The distance between
the points (x1, y1) and (x2, y2) in the Cartesian plane is given by
d{(x1, y1), (x2, y2)} = (x2 x1)2 + (y2 y1)2 (1.5) PROOF We have
oriented the points in Figure 0.12 so that (x2, y2) is above and to
the right of (x1, y1). y x x2x1 y2 y1 y2 y1 x2 x1 Distance (x1, y1)
(x2, y2) FIGURE 0.12 Distance Referring to the right triangle shown
in Figure 0.12, notice that regardless of the orientation of the
two points, the length of the horizontal side of the triangle is
|x2 x1|
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Transcendental) CONFIRMING PAGES 0-7 SECTION 0.1 .. The Real
Numbers and the Cartesian Plane 7 and the length of the vertical
side of the triangle is |y2 y1|. The distance between the two
points is the length of the hypotenuse of the triangle, given by
the Pythagorean Theorem as (x2 x1)2 + (y2 y1)2. We illustrate the
use of the distance formula in example 1.9. EXAMPLE 1.9 Using the
Distance Formula Find the distances between each pair of points (1,
2), (3, 4) and (2, 6). Use the distances to determine if the points
form the vertices of a right triangle. y x 42 6 4 6 2 FIGURE 0.13 A
right triangle? Solution The distance between (1, 2) and (3, 4) is
d{(1, 2), (3, 4)} = (3 1)2 + (4 2)2 = 4 + 4 = 8. The distance
between (1, 2) and (2, 6) is d{(1, 2), (2, 6)} = (2 1)2 + (6 2)2 =
1 + 16 = 17. Finally, the distance between (3, 4) and (2, 6) is
d{(3, 4), (2, 6)} = (2 3)2 + (6 4)2 = 1 + 4 = 5. From a plot of the
points (see Figure 0.13), it is unclear whether a right angle is
formed at (3, 4). However, the sides of a right triangle must
satisfy the Pythagorean Theorem. This would require that 8 2 + 5 2
= 17 2 . This is incorrect! Since this statement is not true, the
triangle is not a right triangle. Again, we want to draw your
attention to the interplay between graphical and algebraic
techniques. If you master the relationship between graphs and
equations, your study of calculus will be a more rewarding and
enjoyable learning experience. EXERCISES 0.1 WRITING EXERCISES 1.
To understand Denition 1.1, you must believe that |x| = x for
negative xs. Using x = 3 as an example, explain in words why
multiplying x by 1 produces the same result as taking the absolute
value of x. 2. A common shortcut used to write inequalities is 4
< x < 4 in place of 4 < x and x < 4. Unfortunately,
many people mistakenly write 4 < x < 4 in place of 4 < x
or x < 4. Explain why the string 4 < x < 4 could never be
true. (Hint: Whatdoesthisinequalitystringimplyaboutthenumbersonthe
far left and far right? Here, you must write 4 < x or x < 4.)
3. Explain the result of Theorem 1.1 (ii) in your own words, as-
suming that all constants involved are positive. 4. Suppose a
friend has dug holes for the corner posts of a rect- angular deck.
Explain how to use the Pythagorean Theorem to determine whether or
not the holes truly form a rectangle (90 angles). In exercises 128,
solve the inequality. 1. 3x + 2 < 11 2. 4x + 1 < 5 3. 2x 3
< 7 4. 3x 1 < 9 5. 4 3x < 6 6. 5 2x < 9 7. 4 x + 1 <
7 8. 1 < 2 x < 3 9. 2 < 2 2x < 3 10. 0 < 3 x < 1
11. x2 + 3x 4 > 0 12. x2 + 4x + 3 < 0 13. x2 x 6 < 0 14.
x2 + 1 > 0 15. 3x2 + 4 > 0 16. x2 + 3x + 10 > 0 17. |x 3|
< 4 18. |2x + 1| < 1 19. |3 x| < 1 20. |3 + x| > 1 21.
|2x + 1| > 2 22. |3x 1| < 4 23. x + 2 x 2 > 0 24. x 4 x +
1 < 1 25. x2 x 2 (x + 4)2 > 0 26. 3 2x (x + 1)2 < 0 27. 8x
(x + 1)3 < 0 28. x 2 (x + 2)3 > 0
............................................................
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In exercises 2934, nd the distance between the pair of points. 29.
(2, 1), (4, 4) 30. (2, 1), (1, 4) 31. (1, 2), (3, 2) 32. (1, 2),
(3, 6) 33. (0, 2), (2, 6) 34. (4, 1), (2, 1)
............................................................ In
exercises 3538, determine if the set of points forms the ver- tices
of a right triangle. 35. (1, 1), (3, 4), (0, 6) 36. (0, 2), (4, 8),
(2, 12) 37. (2, 3), (2, 9), (4, 13) 38. (2, 3), (0, 6), (3, 8)
............................................................ In
exercises 3942, the data represent populations at various times.
Plot the points, discuss any patterns that are evident and predict
the population at the next step. 39. (0, 1250), (1, 1800), (2,
2450), (3, 3200) 40. (0, 3160), (1, 3250), (2, 3360), (3, 3490) 41.
(0, 4000), (1, 3990), (2, 3960), (3, 3910) 42. (0, 2100), (1,
2200), (2, 2100), (3, 1700)
............................................................ 43. As
discussed in the text, a number is rational if and only if its
decimal representation terminates or repeats. Calculators and
computers perform their calculations using a nite number of digits.
Explain why such calculations can only produce rational numbers.
44. In example 1.8, we discussed how the tendency of the data
points to curve up corresponds to larger increases in consec- utive
y-values. Explain why this is true. APPLICATIONS 45. The ancient
Greeks analyzed music mathematically. They found that if pipes of
length L and L 2 are struck, they make tones that blend together
nicely. We say that these tones are one octave apart. In general,
nice harmonies are produced by pipes (or strings) with rational
ratios of lengths. For example, pipes of length L and 2 3 L form a
fth (i.e., middle C and the G above middle C). On a piano keyboard,
12 fths are equal to 7 octaves. A glitch in piano tuning, known as
the Pythagorean comma, results from the fact that 12 fths with
total length ratio 2 3 12 do not equal 7 octaves with length ratio
1 2 7 . Show that the difference is about 1.3%. 46. For the 12 keys
of a piano octave to have exactly the same length ratios (see
exercise 45), the ratio of consecutive lengths should be a number x
such that x12 = 2. Briey explain why. This means that x = 12 2.
There are two problems with this equal-tempered tuning. First, 12 2
is irrational. Explain why it would be difcult to get the pipe or
string exactly the right length. In any case, musicians say that
equal-tempered pianos sound dull. 47. The use of squares in the
Pythagorean Theorem has found a surprising use in the analysis of
baseball statistics. In Bill James Historical Abstract, a rule is
stated that a teams winning percentage P is approximately equal to
R2 R2 + G2 , where R is the number of runs scored by the team and G
is the number of runs scored against the team. For exam- ple, in
1996 the Texas Rangers scored 928 runs and gave up 799 runs. The
formula predicts a winning percentage of 9282 9282 + 7992 0.574. In
fact, Texas won 90 games and lost 72 for a winning percentage of 90
162 0.556. Fill out the follow- ing table (data from the 1996
season). What are possible ex- planations for teams that win more
(or fewer) games than the formula predicts? Team R G P wins losses
win % Yankees 871 787 92 70 Braves 773 648 96 66 Phillies 650 790
67 95 Dodgers 703 652 90 72 Indians 952 769 99 62 EXPLORATORY
EXERCISES 1. It can be very difcult to prove that a given number is
irrational. According to legend, the following proof that 2 is
irrational so upset the ancient Greek mathematicians that they
drowned a mathematician who revealed the result to the general
public. The proof is by contradiction; that is, we imagine that 2
is rational and then show that this cannot be true. If 2 were
rational, we would have that 2 = p q for some integers p and q.
Assume that p q is in simplied form (i.e., any common fac- tors
have been divided out). Square the equation 2 = p q to get 2 = p2
q2 . Explain why this can only be true if p is an even integer.
Write p = 2r and substitute to get 2 = 4r2 q2 . Then, rearrange
this expression to get q2 = 2r2 . Explain why this can only be true
if q is an even integer. Something has gone wrong: explain why p
and q cant both be even integers. Since this cant be true, we
conclude that 2 is irrational. 2. In the text, we stated that a
number is rational if and only if its decimal representation
repeats or terminates. In this exercise, we prove that the decimal
representation of any rational num- ber repeats or terminates. To
start with a concrete example, use long division to show that 1 7 =
0.142857142857. Note that when you get a remainder of 1, its all
over: you started with a 1 to divide into, so the sequence of
digits must repeat. For a general rational number p q , there are q
possible remainders (0, 1, 2, . . . , q 1). Explain why when doing
long division you must eventually get a remainder you have had
before. Explain why the digits will then either terminate or start
repeating.
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Transcendental) CONFIRMING PAGES 0-9 SECTION 0.2 .. Lines and
Functions 9 0.2 LINES AND FUNCTIONS Equations of Lines The federal
government conducts a nationwide census every 10 years to determine
the popu- lation. Population data for several recent decades are
shown in the accompanying table. Year U.S. Population 1960
179,323,175 1970 203,302,031 1980 226,542,203 1990 248,709,873 x y
0 179 10 203 20 227 30 249 Transformed data One difculty with
analyzing these data is that the numbers are so large. This problem
is remedied by transforming the data. We can simplify the year data
by dening x to be the number of years since 1960, so that 1960
corresponds to x = 0, 1970 corresponds to x = 10 and so on. The
population data can be simplied by rounding the numbers to the
nearest million. The transformed data are shown in the accompanying
table and a scatter plot of these data points is shown in Figure
0.14. The points in Figure 0.14 may appear to form a straight line.
(Use a ruler and see if you agree.) To determine whether the points
are, in fact, on the same line (such points are called colinear),
we might consider the population growth in each of the indicated
decades. From 1960 to 1970, the growth was 24 million. (That is, to
move from the rst point to the second, you increase x by 10 and
increase y by 24.) Likewise, from 1970 to 1980, the growth was 24
million. However, from 1980 to 1990, the growth was only 22
million. Since the rate of growth is not constant, the data points
do not fall on a line. This argument involves the familiar concept
of slope. y x 50 100 150 200 250 10 20 30 FIGURE 0.14 Population
data DEFINITION 2.1 For x1 = x2, the slope of the straight line
through the points (x1, y1) and (x2, y2) is the number m = y2 y1 x2
x1 . (2.1) When x1 = x2 and y1 = y2, the line through (x1, y1) and
(x2, y2) is vertical and the slope is undened. We often describe
slope as the change in y divided by the change in x, written y x ,
or more simply as Rise Run . (See Figure 0.15a.) Referring to
Figure 0.15b (where the line has positive slope), notice that for
any four points A, B, D and E on the line, the two right triangles
ABC and DEF are similar. Recall that for similar triangles, the
ratios of corresponding sides must be the same. x x x2 x1 Run y y2
y1 (x1, y1) (x2, y2) Rise y2 y1 x2x1 y y x x x y B E A D F C y
FIGURE 0.15a Slope FIGURE 0.15b Similar triangles and slope
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In this case, this says that y x = y x and so, the slope is the
same no matter which two points on the line are selected. Notice
that a line is horizontal if and only if its slope is zero. EXAMPLE
2.1 Finding the Slope of a Line Find the slope of the line through
the points (4, 3) and (2, 5). Solution From (2.1), we get m = y2 y1
x2 x1 = 5 3 2 4 = 2 2 = 1. EXAMPLE 2.2 Using Slope to Determine If
Points Are Colinear Use slope to determine whether the points (1,
2), (3, 10) and (4, 14) are colinear. Solution First, notice that
the slope of the line joining (1, 2) and (3, 10) is m1 = y2 y1 x2
x1 = 10 2 3 1 = 8 2 = 4. Similarly, the slope through the line
joining (3, 10) and (4, 14) is m2 = y2 y1 x2 x1 = 14 10 4 3 = 4.
Since the slopes are the same, the points must be colinear. Recall
that if you know the slope and a point through which the line must
pass, you have enough information to graph the line. The easiest
way to graph a line is to plot two points and then draw the line
through them. In this case, you need only to nd a second point.
EXAMPLE 2.3 Graphing a Line If a line passes through the point (2,
1) with slope 2 3 , nd a second point on the line and then graph
the line. Solution Since slope is given by m = y2 y1 x2 x1 , we
take m = 2 3 , y1 = 1 and x1 = 2, to obtain 2 3 = y2 1 x2 2 . You
are free to choose the x-coordinate of the second point. For
instance, to nd the point at x2 = 5, substitute this in and solve.
From 2 3 = y2 1 5 2 = y2 1 3 , we get 2 = y2 1 or y2 = 3. A second
point is then (5, 3). The graph of the line is shown in Figure
0.16a. An alternative method for nding a second point is to use the
slope m = 2 3 = y x . The slope of 2 3 says that if we move three
units to the right, we must move two units up to stay on the line,
as illustrated in Figure 0.16b. y x 1 1 2 3 4 1 5432 FIGURE 0.16a
Graph of straight line y x 1 1 2 3 4 1 5432 x 3 y2 FIGURE 0.16b
Using slope to nd a second point In example 2.3, the choice of x =
5 was entirely arbitrary; you can choose any x-value you want to nd
a second point. Further, since x can be any real number, you
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Transcendental) CONFIRMING PAGES 0-11 SECTION 0.2 .. Lines and
Functions 11 can leave x as a variable and write out an equation
satised by any point (x, y) on the line. In the general case of the
line through the point (x0, y0) with slope m, we have from (2.1)
that m = y y0 x x0 . (2.2) Multiplying both sides of (2.2) by (x
x0), we get y y0 = m(x x0) or POINT-SLOPE FORM OF A LINE y = m(x
x0) + y0. (2.3) Equation (2.3) is called the point-slope form of
the line. EXAMPLE 2.4 Finding the Equation of a Line Given Two
Points Find an equation of the line through the points (3, 1) and
(4, 1) and graph the line. Solution From (2.1), the slope is m = 1
1 4 3 = 2 1 = 2. Using (2.3) with slope m = 2, x-coordinate x0 = 3
and y-coordinate y0 = 1, we get the equation of the line: y = 2(x
3) + 1. (2.4) To graph the line, plot the points (3, 1) and (4, 1),
and you can easily draw the line seen in Figure 0.17. y x 1 1 2 3 4
5 6 7 1 2 3 4 FIGURE 0.17 y = 2(x 3) + 1 Although the point-slope
form of the equation is often the most convenient to work with, the
slope-intercept form is sometimes more convenient. This has the
form y = mx + b, where m is the slope and b is the y-intercept
(i.e., the place where the graph crosses the y-axis). In example
2.4, you simply multiply out (2.4) to get y = 2x + 6 + 1 or y = 2x
+ 7. As you can see from Figure 0.17, the graph crosses the y-axis
at y = 7. Theorem 2.1 presents a familiar result on parallel and
perpendicular lines. THEOREM 2.1 Two (nonvertical) lines are
parallel if they have the same slope. Further, any two vertical
lines are parallel. Two (nonvertical) lines of slope m1 and m2 are
perpendicular whenever the product of their slopes is 1 (i.e., m1
m2 = 1). Also, any vertical line and any horizontal line are
perpendicular. Since we can read the slope from the equation of a
line, its a simple matter to de- termine when two lines are
parallel or perpendicular. We illustrate this in examples 2.5 and
2.6.
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EXAMPLE 2.5 Finding the Equation of a Parallel Line Find an
equation of the line parallel to y = 3x 2 and through the point (1,
3). y x 10 20 10 20 4224 FIGURE 0.18 Parallel lines Solution Its
easy to read the slope of the line from the equation: m = 3. The
equation of the parallel line is then y = 3[x (1)] + 3 or simply y
= 3x + 6. We show a graph of both lines in Figure 0.18. EXAMPLE 2.6
Finding the Equation of a Perpendicular Line Find an equation of
the line perpendicular to y = 2x + 4 and intersecting the line at
the point (1, 2). Solution The slope of y = 2x + 4 is 2. The slope
of the perpendicular line is then 1/(2) = 1 2 . Since the line must
pass through the point (1, 2), the equation of the perpendicular
line is y = 1 2 (x 1) + 2 or y = 1 2 x + 3 2 . We show a graph of
the two lines in Figure 0.19. y x 4 2 2 4 2 42 FIGURE 0.19
Perpendicular lines We now return to this sections introductory
example and use the equation of a line to estimate the population
in the year 2000. EXAMPLE 2.7 Using a Line to Predict Population
Given the population data for the census years 1960, 1970, 1980 and
1990, predict the population for the year 2000. Solution We began
this section by showing that the points in the corresponding table
are not colinear. Nonetheless, they are nearly colinear. So, why
not use the straight line connecting the last two points (20, 227)
and (30, 249) (corresponding to the populations in the years 1980
and 1990) to predict the population in 2000? (This is a simple
example of a more general procedure called extrapolation.) The
slope of the line joining the two data points is m = 249 227 30 20
= 22 10 = 11 5 . The equation of the line is then y = 11 5 (x 30) +
249. y x 5040302010 100 200 300 FIGURE 0.20 Population See Figure
0.20 for a graph of the line. If we follow this line to the point
corresponding to x = 40 (the year 2000), we have the predicted
population 11 5 (40 30) + 249 = 271. That is, the predicted
population is 271 million people. The actual census gure for 2000
was 281 million, which indicates that the U.S. population grew at a
faster rate between 1990 and 2000 than in the previous decade. f x
BA y Functions For any two subsets A and B of the real line, we
make the following familiar denition.
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Transcendental) CONFIRMING PAGES 0-13 SECTION 0.2 .. Lines and
Functions 13 DEFINITION 2.2 A function f is a rule that assigns
exactly one element y in a set B to each element x in a set A. In
this case, we write y = f (x). We call the set A the domain of f.
The set of all values f (x) in B is called the range of f , written
{y|y = f (x), for some x A}. Unless explicitly stated otherwise,
whenever a function f is given by a particular expression, the
domain of f is the largest set of real numbers for which the
expression is dened. We refer to x as the independent variable and
to y as the dependent variable. REMARK 2.1 Functions can be dened
by simple formulas, such as f (x) = 3x + 2, but in general, any
correspondence meeting the requirement of matching exactly one y to
each x denes a function. By the graph of a function f, we mean the
graph of the equation y = f (x). That is, the graph consists of all
points (x, y), where x is in the domain of f and where y = f (x).
Notice that not every curve is the graph of a function, since for a
function, only one y-value can correspond to a given value of x.
You can graphically determine whether a curve is the graph of a
function by using the vertical line test: if any vertical line
intersects the graph in more than one point, the curve is not the
graph of a function, since in this case, there are two y-values for
a given value of x. EXAMPLE 2.8 Using the Vertical Line Test
Determine which of the curves in Figures 0.21a and 0.21b correspond
to functions. y x 11 y x 0.5 21 1 1 FIGURE 0.21a FIGURE 0.21b
Solution Notice that the circle in Figure 0.21a is not the graph of
a function, since a vertical line at x = 0.5 intersects the circle
twice. (See Figure 0.22a.) The graph in Figure 0.21b is the graph
of a function, even though it swings up and down repeatedly.
Although horizontal lines intersect the graph repeatedly, vertical
lines, such as the one at x = 1.2, intersect only once. (See Figure
0.22b.) The functions with which you are probably most familiar are
polynomials. These are the simplest functions to work with because
they are dened entirely in terms of arithmetic. y x 10.51 FIGURE
0.22a Curve fails vertical line test y x 0.5 21 1 1 FIGURE 0.22b
Curve passes vertical line test DEFINITION 2.3 A polynomial is any
function that can be written in the form f (x) = an xn + an1xn1 + +
a1x + a0, where a0, a1, a2, . . . , an are real numbers (the
coefcients of the polynomial) with an = 0 and n 0 is an integer
(the degree of the polynomial).
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Note that every polynomial function can be dened for all xs on the
entire real line. Further, recognize that the graph of the linear
(degree 1) polynomial f (x) = ax + b is a straight line. EXAMPLE
2.9 Sample Polynomials The following are all examples of
polynomials: f (x) = 2 (polynomial of degree 0 or constant), f (x)
= 3x + 2 (polynomial of degree 1 or linear polynomial), f (x) = 5x2
2x + 2 3 (polynomial of degree 2 or quadratic polynomial), f (x) =
x3 2x + 1 (polynomial of degree 3 or cubic polynomial), f (x) = 6x4
+ 12x2 3x + 13 (polynomial of degree 4 or quartic polynomial) and f
(x) = 2x5 + 6x4 8x2 + x 3 (polynomial of degree 5 or quintic
polynomial). We show graphs of these six functions in Figures
0.23a0.23f. y x 1 3 y x 4224 15 10 5 5 10 15 y x 642246 40 80 120
FIGURE 0.23a f (x) = 2 FIGURE 0.23b f (x) = 3x + 2 FIGURE 0.23c f
(x) = 5x2 2x + 2 3 y x 321123 10 5 5 10 y x 2112 20 10 10 20 y x 13
2 1 10 10 20 FIGURE 0.23d f (x) = x3 2x + 1 FIGURE 0.23e f (x) =
6x4 + 12x2 3x + 13 FIGURE 0.23f f (x) = 2x5 + 6x4 8x2 + x 3
42. P1: OSO/OVY P2: OSO/OVY QC: OSO/OVY T1: OSO MHDQ256-Ch00
MHDQ256-Smith-v1.cls December 8, 2010 15:25 LT (Late
Transcendental) CONFIRMING PAGES 0-15 SECTION 0.2 .. Lines and
Functions 15 DEFINITION 2.4 Any function that can be written in the
form f (x) = p(x) q(x) , where p and q are polynomials, is called a
rational function. Notice that since p and q are polynomials, they
can both be dened for all x and so, the rational function f (x) =
p(x) q(x) can be dened for all x for which q(x) = 0. EXAMPLE 2.10 A
Sample Rational Function Find the domain of the function f (x) = x2
+ 7x 11 x2 4 . Solution Here, f is a rational function. We show a
graph in Figure 0.24. Its domain consists of those values of x for
which the denominator is nonzero. Notice that x2 4 = (x 2)(x + 2)
and so, the denominator is zero if and only if x = 2. This says
that the domain of f is {x R|x = 2} = (, 2) (2, 2) (2, ). y x 1 313
10 5 5 10 FIGURE 0.24 f (x) = x2 + 7x 11 x2 4 The square root
function is dened in the usual way. When we write y = x, we mean
that y is that number for which y2 = x and y 0. In particular, 4 =
2. Be careful not to write erroneous statements such as 4 = 2. In
particular, be careful to write x2 = |x|. Since x2 is asking for
the nonnegative number whose square is x2 , we are looking for |x|
and not x. We can say x2 = x, only for x 0. Similarly, for any
integer n 2, y = n x whenever yn = x, where for n even, x 0 and y
0. EXAMPLE 2.11 Finding the Domain of a Function Involving a Square
Root or a Cube Root Find the domains of f (x) = x2 4 and g(x) = 3
x2 4. Solution Since even roots are dened only for nonnegative
values, f (x) is dened only for x2 4 0. Notice that this is
equivalent to having x2 4, which occurs when x 2 or x 2. The domain
of f is then (, 2] [2, ). On the other hand, odd roots are dened
for both positive and negative values. Consequently, the domain of
g is the entire real line, (, ). We often nd it useful to label
intercepts and other signicant points on a graph. Finding these
points typically involves solving equations. A solution of the
equation f (x) = 0 is called a zero of the function f or a root of
the equation f (x) = 0. Notice that a zero of the function f
corresponds to an x-intercept of the graph of y = f (x).
43. P1: OSO/OVY P2: OSO/OVY QC: OSO/OVY T1: OSO MHDQ256-Ch00
MHDQ256-Smith-v1.cls December 8, 2010 15:25 LT (Late
Transcendental) CONFIRMING PAGES 16 CHAPTER 0 .. Preliminaries 0-16
EXAMPLE 2.12 Finding Zeros by Factoring Find all x- and
y-intercepts of f (x) = x2 4x + 3. Solution To nd the y-intercept,
set x = 0 to obtain y = 0 0 + 3 = 3. To nd the x-intercepts, solve
the equation f (x) = 0. In this case, we can factor to get f (x) =
x2 4x + 3 = (x 1)(x 3) = 0. You can now read off the zeros: x = 1
and x = 3, as indicated in Figure 0.25. y x 2 4 6 8 10 321 4 62
FIGURE 0.25 y = x2 4x + 3 Unfortunately, factoring is not always so
easy. Of course, for the quadratic equation ax2 + bx + c = 0 (for a
= 0), the solution(s) are given by the familiar quadratic formula:
x = b b2 4ac 2a . EXAMPLE 2.13 Finding Zeros Using the Quadratic
Formula Find the zeros of f (x) = x2 5
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