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Licensed to: iChapters User
Essential Calculus
Early Transcendental Functions
Ron Larson
The Pennsylvania State University
The Behrend College
Robert Hostetler
The Pennsylvania State University
The Behrend College
Bruce H. Edwards
University of Florida
Houghton Mifflin Company
Copyright 2008 Cengage Learning. All Rights Reserved.
May not be copied, scanned, or duplicated, in whole or in part.
Boston New York
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Publisher: Richard Stratton
Sponsoring Editor: Cathy Cantin
Senior Marketing Manager: Jennifer Jones
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Editorial Assistant: Joanna Carter
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Cover Design Manager: Anne Katzeff
Photo Editor: Jennifer Meyer Dare
We have included examples and exercises that use real-life data as well as technology output from a variety of software. This would not have been possible without the help of many
people and organizations. Our wholehearted thanks goes to all for their time and effort.
Cover photograph: “Music of the Spheres” by English sculptor John Robinson is a
three-foot-tall sculpture in bronze that has one continuous edge. You can trace its edge
three times around before returning to the starting point. To learn more about this and other
works by John Robinson, see the Centre for the Popularisation of Mathematics, University
of Wales, at http://www.popmath.org.uk/sculpture/gallery2.html.
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Copyright © 2008 by Houghton Mifflin Company. All rights reserved.
No part of this work may be reproduced or transmitted in any form or by any means,
electronic or mechanical, including photocopying and recording, or by any information
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Company unless such copying is expressly permitted by federal copyright law. Address
inquiries to College Permissions, Houghton Mifflin Company, 222 Berkeley Street, Boston,
MA 02116-3764.
Printed in the U.S.A.
Library of Congress Control Number: 2006934184
ISBN 13: 978-0-618-87918-2
ISBN 10: 0-618-87918-8
1 2 3 4 5 6 7 8 9-DOW-11 10 09 08 07
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Contents
A Word from the Authors
viii
Integrated Learning System for Calculus
Features
xv
Chapter 1
Limits and Their Properties
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Chapter 2
x
I
Linear Models and Rates of Change
1
Functions and Their Graphs
9
Inverse Functions
20
Exponential and Logarithmic Functions
31
Finding Limits Graphically and Numerically
Evaluating Limits Analytically
48
Continuity and One-Sided Limits
58
Infinite Limits
70
Review Exercises
77
Differentiation
38
80
2.1 The Derivative and the Tangent Line Problem
80
2.2 Basic Differentiation Rules and Rates of Change
90
2.3 Product and Quotient Rules and Higher-Order
Derivatives
102
2.4 The Chain Rule
112
2.5 Implicit Differentiation
126
2.6 Derivatives of Inverse Functions
134
2.7 Related Rates
140
2.8 Newton’s Method
148
Review Exercises
153
iii
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iv
CONTENTS
Chapter 3
Applications of Differentiation
157
3.1 Extrema on an Interval
157
3.2 Rolle’s Theorem and the Mean Value Theorem
164
3.3 Increasing and Decreasing Functions and the
First Derivative Test
170
3.4 Concavity and the Second Derivative Test
180
3.5 Limits at Infinity
187
3.6 Optimization Problems
197
3.7 Differentials
208
Review Exercises
214
Chapter 4
Integration
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
Chapter 5
217
Antiderivatives and Indefinite Integration
217
Area
227
Riemann Sums and Definite Integrals
238
The Fundamental Theorem of Calculus
248
Integration by Substitution
260
Numerical Integration
273
The Natural Logarithmic Function: Integration
279
Inverse Trigonometric Functions: Integration
287
Hyperbolic Functions
294
Review Exercises
304
Applications of Integration
5.1
5.2
5.3
5.4
5.5
5.6
306
Area of a Region Between Two Curves
306
Volume: The Disk Method
315
Volume: The Shell Method
325
Arc Length and Surfaces of Revolution
333
Applications in Physics and Engineering
343
Differential Equations: Growth and Decay
359
Review Exercises
366
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CONTENTS
Chapter 6
Integration Techniques, L’Hôpital’s Rule,
and Improper Integrals
368
6.1
6.2
6.3
6.4
6.5
6.6
6.7
Chapter 7
Integration by Parts
368
Trigonometric Integrals
376
Trigonometric Substitution
384
Partial Fractions
392
Integration by Tables and Other Integration Techniques
Indeterminate Forms and L’Hôpital’s Rule
405
Improper Integrals
415
Review Exercises
425
Infinite Series
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
Chapter 8
v
427
Sequences
427
Series and Convergence
438
The Integral and Comparison Tests
448
Other Convergence Tests
456
Taylor Polynomials and Approximations
466
Power Series
476
Representation of Functions by Power Series
485
Taylor and Maclaurin Series
491
Review Exercises
502
Conics, Parametric Equations, and
Polar Coordinates
504
8.1
8.2
8.3
8.4
8.5
Plane Curves and Parametric Equations
504
Parametric Equations and Calculus
513
Polar Coordinates and Polar Graphs
522
Area and Arc Length in Polar Coordinates
531
Polar Equations and Conics and Kepler’s Laws
539
Review Exercises
546
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400
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vi
CONTENTS
Chapter 9
Vectors and the Geometry of Space
9.1
9.2
9.3
9.4
9.5
9.6
9.7
Chapter 10
549
Vectors in the Plane
549
Space Coordinates and Vectors in Space
559
The Dot Product of Two Vectors
566
The Cross Product of Two Vectors in Space
574
Lines and Planes in Space
581
Surfaces in Space
592
Cylindrical and Spherical Coordinates
601
Review Exercises
607
Vector-Valued Functions
609
10.1 Vector-Valued Functions
609
10.2 Differentiation and Integration of Vector-Valued
Functions
616
10.3 Velocity and Acceleration
623
10.4 Tangent Vectors and Normal Vectors
631
10.5 Arc Length and Curvature
640
Review Exercises
651
Chapter 11
Functions of Several Variables
11.1
11.2
11.3
11.4
11.5
11.6
11.7
11.8
653
Introduction to Functions of Several Variables
Limits and Continuity
664
Partial Derivatives
673
Differentials and the Chain Rule
682
Directional Derivatives and Gradients
694
Tangent Planes and Normal Lines
704
Extrema of Functions of Two Variables
712
Lagrange Multipliers
720
Review Exercises
726
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653
Licensed to: iChapters User
vii
CONTENTS
Chapter 12
Multiple Integration
728
12.1
12.2
12.3
12.4
12.5
12.6
12.7
Iterated Integrals and Area in the Plane
728
Double Integrals and Volume
735
Change of Variables: Polar Coordinates
745
Center of Mass and Moments of Inertia
752
Surface Area
759
Triple Integrals and Applications
765
Triple Integrals in Cylindrical and Spherical
Coordinates
775
12.8 Change of Variables: Jacobians
781
Review Exercises
787
Chapter 13
Vector Analysis
13.1
13.2
13.3
13.4
13.5
13.6
13.7
13.8
790
Vector Fields
790
Line Integrals
800
Conservative Vector Fields and Independence of Path
Green’s Theorem
822
Parametric Surfaces
830
Surface Integrals
839
Divergence Theorem
850
Stokes’s Theorem
858
Review Exercises
864
Appendix A Proofs of Selected Theorems
A1
Appendix B Integration Tables
A17
Appendix C Business and Economic Applications
Answers to Odd-Numbered Exercises
Index
A109
Additional Appendices
A22
A29
The following appendices are available at the textbook website at
college.hmco.com/pic/larsonEC.
Appendix D Precalculus Review
D.1 Real Numbers and the Real Number Line
D.2 The Cartesian Plane
D.3 Review of Trigonometric Functions
Appendix E Rotation and General Second-Degree Equation
Appendix F Complex Numbers
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813
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A Word from the Authors
A Streamlined Text
In recent years, we have heard from some users, reviewers, and colleagues that calculus
books are too long and too expensive. To address these concerns, we developed
a streamlined version of our calculus text. In doing so, it was important not to
compromise our core philosophies: (1) to write a precise, readable book for students
with the basic rules and concepts clearly defined and demonstrated; and (2) to design
a comprehensive teaching instrument that employs proven pedagogical techniques,
freeing the instructor to make the most efficient use of classroom time.
To write Essential Calculus: Early Transcendental Functions, we asked our readers:
Exactly what are the essential topics for a three-semester calculus sequence?
Essential Calculus
The resulting textbook is approximately 23 the size of our mainstream text. The
structure and coverage of the topics enable a faster-paced course to cover the material
in a mathematically sound, thorough, and rigorous manner.
While developing the streamlined text, we recognized that some instructors and their
students may need additional practice problems or reference materials, so we moved
the following material from the text to the website college.hmco.com/pic/larson/EC:
•
•
•
•
•
•
Fourth Edition
Material on differential equations
Material on conics
Section projects
PS Problem Solving Exercises
Chapter overview application and graphics
Index of Applications
A Text Formed by Its Users
Third Edition
Much has changed since we began writing calculus textbooks in 1973—over 30 years
ago. With each edition we have listened to our users, colleagues, and students and
incorporated many of your suggestions for improvement.
Through your support and suggestions, versions of this text have evolved over the
years to include these extensive enhancements:
Second Edition
First Edition
• Comprehensive exercise sets with a wide variety of problems such as skill-building
exercises, applications, explorations, writing exercises, critical thinking exercises,
theoretical problems, and problems from the Putnam Exam
• Comprehensive and mathematically rigorous text
• Abundant, real-life applications (many with real data) that accurately represet the
diverse uses of calculus
• Open-ended activities and investigations that help students develop an understanding
of mathematical concepts intuitively
• Clear, uncluttered text presentation with full annotations and labels and a carefully
planned page layout that is designed for maximum readability
• Comprehensive, four-color art program
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A WORD FROM THE AUTHORS
ix
• Technology used throughout as both a problem-solving and an investigative tool
• Comprehensive program of additional resources available in print and online
• With 9 different volumes of the text available, you can choose the sequence, amount
of content, and teaching approach that is best for you and your students.
• References to the history of calculus and to the mathematicians who developed it
• References to over 50 articles from mathematical journals available at
www.MathArticles.com
• Worked out solutions to the odd-numbered text exercises provided in the printed
Student Study and Solutions Guide, in Eduspace, and at www.CalcChat.com
Although we streamlined the text to create Essential Calculus: Early Transcendental
Functions, we did not change many of the things our colleagues and the over two
million students who have used other versions of the texts have told us work for them.
We hope you will enjoy this text. We welcome any comments, as well as suggestions
for continued improvements.
Ron Larson
Robert Hostetler
Copyright 2008 Cengage Learning. All Rights Reserved.
May not be copied, scanned, or duplicated, in whole or in part.
Bruce H. Edwards
Licensed to: iChapters User
Integrated Learning System for Calculus
Over 25 Years of Success, Leadership, and Innovation
The best-selling authors Larson, Hostetler, and Edwards continue to offer instructors
and students more flexible teaching and learning options for the calculus course.
Calculus Textbook Options
The early transcendental functions calculus course is available in a variety of textbook
configurations to address the different ways instructors teach—and students take—their classes.
CALCULUS: Early Transcendental Functions
Designed for the
three-semester sequence
Designed for the
two-semester sequence
Designed for the
single-semester courses
CALCULUS with Late Trigonometry
Designed for the
three-semester sequence
Designed for third
semester of Calculus
CALCULUS, Eighth Edition
Designed for the
three-semester sequence
Designed for the
two-semester sequence
Designed for the
single-semester courses
Designed for third
semester of Calculus
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Designed for the foursemester precalculus
with calculus sequence
Licensed to: iChapters User
An Integrated Learning System
that makes a difference to you and your students
Many more resources are available with the Larson/Hostetler/Edwards Calculus
series to address your needs and the needs of your students.
Integrate Graphing Calculators
• Access to a COLOR graphing Calculator for all students
• Graphing calculators Explorations in the textbook and
additional activities in Eduspace®
• Calculator programs that do routine calculus computations
so students can focus on more complex problems
• Graphing Calculator Guides for various calculator models
Use Computer Algebra Systems:
Maple™, Mathematica®, Mathcad®,
and Derive®
• Open Explorations in Eduspace® expand
textbook examples using a Computer Algebra
System (CAS)
• Calculus Labs guide students to explore more
complex problems using a Computer Algebra
System
• Download data and other materials to make it
easier to integrate a CAS
Teach Online
• Eduspace® with online homework,
tutoring, and testing resources,
including SMARTHINKING™ tutoring
• Instructional DVDs
• CalcChat student discussion forums
• Eduspace® online course management
tools
Your
Calculus
Course
Present Calculus Visually
• Animations and Rotatable 3D Graphs visualize
concepts and enhance student understanding
• Digital (PowerPoint) Figures enable you to
create presentation materials using textbook
artwork
• Math Graphs allow students to print enlarged
textbook graphs for homework
Eduspace® online tutorials
• Live, Smarthinking™ tutoring
• Algorithmically-generated practice problems
• Point-of-use video clips
• Additional examples and practice problems
Make Calculus Relevant
• Math Articles enrich students’
understanding of calculus
• Math Trends and Biographies help
students see how Calculus developed
over time
• Simulations engage students and enable
them to explore calculus concepts
• Real-World Videos and Connections
show how calculus concepts connect to
students’ lives and fields of study
The Integrated Learning System for Essential Calculus: Early Transcendental Functions, offers dynamic
teaching teaching tools for instructors and interactive learning resources for students in the following flexible
course delivery formats.
• Eduspace® online learning system
• HM Testing CD-ROM
• Study and Solutions Guide in two volumes
available in print and electronically
• Instructional DVDs and videos
• Companion Textbook Websites for students and instructors
• Complete Solutions Guide (for instructors only) available
only electronically
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Licensed to: iChapters User
Integrated Learning System for Calculus
Eduspace® Online Calculus
Eduspace®, powered by Blackboard®, is ready to use and easy to integrate into the calculus course. It provides comprehensive
homework exercises, tutorials, and testing keyed to the textbook by section.
Features
• Online Multimedia eBook interactive textbook content
organized at the section level
• Algorithmic Homework (Practice and Graded) based on
•
•
•
•
•
•
•
•
•
•
•
•
examples from the textbook, organized at the chapter and
section level; includes full Equation Editor to support free
response question types
Explore It First (with editable graphs) organized at the
section level
Video Explanations organized at the section level
eSolutions (worked-out solutions to the odd-numbered
textbook exercises) organized at the chapter and section level
Online Color Graphing Calculator
Full Test Bank content
One-click access to SMARTHINKING® live, online
tutoring for students
Comprehensive problem sets for graded homework
Ample prerequisite skills review with customized student
self-study plan
Chapter Tests
Link to CalcChat
Electronic version of all textbook exercises
Tech Support available when you need it most
Instructor Resources
Student Resources
• Algorithmically generated tutorial exercises for
• Algorithmically generated tutorial questions for
•
•
•
•
•
•
•
unlimited practice
Comprehensive problem sets for graded homework
Electronic version of all textbook exercises
Interactive (multimedia) textbook pages with video
lectures, animations, and much more
Symbol palette for free response questions
Ample prerequisite skills review with customized student
self-study plan
Electronic gradebook
HM Testing
unlimited practice of prerequisite skills
• Point-of-use links to additional tools, animations,
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and simulations
Symbol palette for writing math notation
SMARTHINKING® live, online tutoring
Graphing calculator
Customized self-study plan
Chapter tests
Interactive ebook
3-D rotatable graphs
Prerequisite skills review exercises
Video instruction which corresponds to sections of text
Worked-out solutions to odd-numbered exercises
For additional information about the Larson, Hostetler, and Edwards
Calculus program, go to college.hmco.com/info/larsoncalculus.
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Licensed to: iChapters User
This website contains an array of useful instructor resources keyed to the textbook.
Features
54
Chapter 2
Differentiation
Test Form A
Name
__________________________________________
Date
Chapter 2
Class
__________________________________________
Section _______________________
• Complete Solutions Guide by Bruce Edwards
____________________________
1. If f x ⫽ 2x2 ⫹ 4, which of the following will calculate the derivative of f ?
(a)
2x ⫹ ⌬x2 ⫹ 4 ⫺ 2x2 ⫹ 4
⌬x
(b) lim
2x2 ⫹ 4 ⫹ ⌬x ⫺ 2x2 ⫹ 4
⌬x
(c) lim
2x ⫹ ⌬x2 ⫹ 4 ⫺ 2x2 ⫹ 4
⌬x
⌬x→0
⌬x→0
(d)
2x2
⫹ 4 ⫹ ⌬x ⫺ 2x2 ⫹ 4
⌬x
(e) None of these
2. Differentiate: y ⫽
1 ⫹ cos x
.
1 ⫺ cos x
(a) ⫺1
(d)
(b) ⫺2 csc x
⫺2 sin x
1 ⫺ cos x2
(c) 2 csc x
(e) None of these
3. Find dydx for y ⫽ x3x ⫹ 1.
3x2
2x ⫹ 1
(b)
(d)
7x ⫹
2x ⫹ 1
(e) None of these
3
x2
(c) 3x2x ⫹ 1
4. Find f⬘x for f x ⫽ 4 ⫹ e2x.
(a)
e2x
4 ⫹ e2x
(d) e x
(b)
1
22e2x
(c)
xe2x⫺1
4 ⫹ e2x
(e) None of these
5. The position equation for the movement of a particle is given by s ⫽ t 2 ⫺ 13 when s is
measured in feet and t is measured in seconds. Find the acceleration at two seconds.
(a) 342 unitssec2
(b) 18 unitssec2
(d) 90 unitssec2
(e) None of these
(c) 288 unitssec2
© Houghton Mifflin Company. All rights reserved.
x27x ⫹ 6
2x ⫹ 1
(a)
This resource contains worked-out solutions to all
textbook exercises in electronic format.
• Instructor’s Resource Guide by Ann Rutledge Kraus
This resource contains an abundance of resources keyed
to the textbook by chapter and section, including chapter
summaries, teaching strategies, multiple versions of
chapter tests, final exams, and gateway tests, and
suggested solutions to the Chapter Openers, Explorations,
Section Projects, and Technology features in the text in
electronic format.
• Test Item File The Test Item File contains a sample
question for every algorithm in HM Testing in electronic
format.
• Digital textbook art including 3-D rotatable graphs.
HM Testing (powered by Diploma™)
For the instructor, HM Testing is a robust test-generating system.
Features
• Comprehensive set of algorithmic test items
• Can produce chapter tests, cumulative tests, and final
exams
• Online testing
• Gradebook function
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Licensed to: iChapters User
Integrated Learning System for Calculus
Instructional DVDs and Videos
These comprehensive DVD and video presentations complement the textbook topic
coverage and have a variety of uses, including supplementing an online or hybrid
course, giving students the opportunity to catch up if they miss a class, and providing
substantial course material for self-study and review.
Features
• Comprehensive topic coverage from Calculus correlated to
section topics
• Additional explanations of calculus concepts,
sample problems, and applications
Companion Textbook Website
The free Houghton Mifflin website at college.hmco.com/pic/larsonEC contains an
abundance of instructor and student resources.
Features
• Downloadable graphing calculator programs
• Textbook Appendices D – F, containing additional presentations with exercises
covering precalculus review, rotation and the general second-degree equation,
and complex numbers
• Algebra Review Summary
• Calculus Labs
• 3-D rotatable graphs
Printed Resources
For the convenience of students, the Study and Solutions Guides are available
as printed supplements, but are also available in electronic format.
Study and Solutions Guide by Bruce Edwards
This student resource contains detailed, worked-out solutions to all
odd-numbered textbook exercises. It is available in two volumes: Volume I
covers Chapters 1–8 and Volume II covers Chapters 9–13.
For additional information about the Larson, Hostetler, and Edwards
Calculus program, go to college.hmco.com/info/larsoncalculus.
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Licensed to: iChapters User
Features
Essential Calculus offers a number of proven
pedagogical features developed by the Larson team
to promote student mastery of Calculus. In order to
streamline this version for faster paced courses, we
have moved some content to online resources, leaving
the essential course content presented in a variety of
ways to appeal to different learning styles, instructional
approaches, and course configurations.
2
Differentiation
Section 2.1
The Derivative and the Tangent Line Problem
• Find the slope of the tangent line to a curve at a point.
• Use the limit definition to find the derivative of a function.
• Understand the relationship between differentiability and continuity.
The Tangent Line Problem
Calculus grew out of four major problems that European mathematicians were working on during the seventeenth century.
1. The tangent line problem (this section)
y
2. The velocity and acceleration problem (Sections 2.2 and 2.3)
3. The minimum and maximum problem (Section 3.1)
4. The area problem (Section 4.2)
P
x
Tangent line to a circle
Figure 2.1
E X P L O R AT I O N
Integrating a Radical Function
Up to this point in the text, you have
not evaluated the following integral.
⫺1
1 ⫺
y
y
y
y = f(x)
FOR FURTHER INFORMATION For
more information on the crediting of
mathematical discoveries to the first
“discoverer,” see the article
“Mathematical Firsts—Who Done It?”
by Richard H. Williams and Roy D.
Mazzagatti in Mathematics Teacher.
To view this article, go to the website
www.matharticles.com.
1
x2
Each problem involves the notion of a limit, and calculus can be introduced with any
of the four problems.
Although partial solutions to the tangent line problem were given by Pierre de
Fermat (1601–1665), René Descartes (1596–1650), Christian Huygens (1629–1695),
and Isaac Barrow (1630 –1677), credit for the first general solution is usually given to
Isaac Newton (1642–1727) and Gottfried Leibniz (1646–1716). Newton’s work on
this problem stemmed from his interest in optics and light refraction.
What does it mean to say that a line is tangent to a curve at a point? For a circle,
the tangent line at a point P is the line that is perpendicular to the radial line at point
P, as shown in Figure 2.1.
For a general curve, however, the problem is more difficult. For example, how
would you define the tangent lines shown in Figure 2.2? You might say that a line is
tangent to a curve at a point P if it touches, but does not cross, the curve at point P.
This definition would work for the first curve shown in Figure 2.2, but not for the
second. Or you might say that a line is tangent to a curve if the line touches or
intersects the curve at exactly one point. This definition would work for a circle but
not for more general curves, as the third curve in Figure 2.2 shows.
dx
From geometry, you should be able to
find the exact value of this integral—
what is it? Using numerical integration with Simpson’s Rule or the
Trapezoidal Rule, you can’t be sure
of the accuracy of the approximation.
Why?
P
P
P
x
y = f (x)
y = f (x)
x
x
Tangent line to a curve at a point
Figure 2.2
80
Section Openers
Try finding the exact value using the
substitution
Every section begins with an outline of the key
concepts covered in the section. This serves as a
class planning resource for the instructor and
a study and review guide for the student.
x ⫽ sin ␪ and dx ⫽ cos␪ d␪.
Does your answer agree with the
value you obtained using geometry?
Explorations
Mary Evans Picture Library
For selected topics, Explorations offer the opportunity
to discover calculus concepts before they are formally
introduced in the text, thus enhancing student understanding. This optional feature can be omitted at the
discretion of the instructor with no loss of continuity
in the coverage of the material.
ARCHIMEDES (287–212 B.C.)
Archimedes used the method of exhaustion to
derive formulas for the areas of ellipses,
parabolic segments, and sectors of a spiral.
He is considered to have been the greatest
applied mathematician of antiquity.
Historical Notes
Integrated throughout the text, Historical Notes help
students grasp the basic mathematical foundations
of calculus.
xv
Copyright 2008 Cengage Learning. All Rights Reserved.
May not be copied, scanned, or duplicated, in whole or in part.
Licensed to: iChapters User
xvi
FEATURES
96
CHAPTER 2
All Theorems and Definitions are highlighted for
emphasis and easy reference. Proofs are shown for
selected theorems to enhance student understanding.
Derivatives of Exponential Functions
E X P L O R AT I O N
One of the most intriguing (and useful) characteristics of the natural exponential function is that it is its own derivative. Consider the following.
Use a graphing utility to graph the
function
f x ⫽
Theorems
Differentiation
Let f x ⫽ e x.
e x⫹⌬x ⫺ e x
⌬x
f⬘ x ⫽ lim
for ⌬x ⫽ 0.01. What does this function represent? Compare this graph
with that of the exponential function.
What do you think the derivative of
the exponential function equals?
⌬x→0
f x ⫹ ⌬ x ⫺ f x
⌬x
e x⫹⌬x ⫺ e x
⫽ lim
⌬x→0
⌬x
e xe ⌬x ⫺ 1
⫽ lim
⌬x→0
⌬x
Study Tips
The definition of e
Located at point of use throughout the text, Study
Tips advise students on how to avoid common errors,
address special cases, and expand upon theoretical
concepts.
lim 1 ⫹ ⌬ x1⌬x ⫽ e
⌬x→0
tells you that for small values of ⌬ x, you have e 1 ⫹ ⌬ x1⌬x, which implies that
e ⌬x 1 ⫹ ⌬ x. Replacing e ⌬x by this approximation produces the following.
e x e ⌬x ⫺ 1
⌬x
e x 1 ⫹ ⌬ x ⫺ 1
⌬x→0
⌬x
e x⌬ x
⫽ lim
⌬x→0 ⌬ x
⫽ ex
f⬘ x ⫽ lim
STUDY TIP The key to the formula for
the derivative of f x ⫽ e x is the limit
lim 1 ⫹ x
1x
x→0
⌬x→0
⫽ lim
⫽ e.
This important limit was introduced on
page 33 and formalized later on page 54.
It is used to conclude that for ⌬x 0,
1 ⫹ ⌬x1⌬ x e.
Graphics
This result is stated in the next theorem.
THEOREM 2.7
Numerous graphics throughout the text enhance
student understanding of complex calculus concepts
(especially in three-dimensional representations), as
well as real-life applications.
Derivative of the Natural Exponential Function
d x
e ⫽ e x
dx
y
At the point (1, e),
the slope is e ≈ 2.72.
4
You can interpret Theorem 2.7 graphically by saying that the slope of the graph
of f x ⫽ e x at any point x, e x is equal to the y-coordinate of the point, as shown in
Figure 2.20.
Derivatives of Exponential Functions
3
EXAMPLE 9
2
Find the derivative of each function.
a. f x ⫽ 3e
f (x) = e x
At the point (0, 1),
the slope is 1.
1
2
b. f x ⫽ x 2 ⫹ e x
c. f x ⫽ sin x ⫺ e x
Solution
x
−2
Figure 2.20
x
a. f⬘ x ⫽ 3
d x
e ⫽ 3e x
dx
b. f⬘ x ⫽
d 2
d
x ⫹ e x ⫽ 2x ⫹ e x
dx
dx
c. f⬘ x ⫽
d
d
sin x ⫺ e x ⫽ cos x ⫺ e x
dx
dx
128
CHAPTER 2
Differentiation
y
Examples
It is meaningless to solve for dydx in an equation that has no solution points.
(For example, x 2 ⫹ y 2 ⫽ ⫺4 has no solution points.) If, however, a segment of a
graph can be represented by a differentiable function, dydx will have meaning as the
slope at each point on the segment. Recall that a function is not differentiable at (1)
points with vertical tangents and (2) points at which the function is not continuous.
1
x2 + y2 = 0
Numerous examples enhance the usefulness of the
text as a study and learning tool. The detailed,
worked-out Solutions (many with side comments
to clarify the steps or the method) are presented
graphically, analytically, and/or numerically to
provide students with opportunities for practice and
further insight into calculus concepts. Many Examples
incorporate real-data analysis.
(0, 0)
x
−1
1
EXAMPLE 3
−1
a. x 2 ⫹ y 2 ⫽ 0
y
1 − x2
y=
1
(−1, 0)
(1, 0)
−1
−1
a. The graph of this equation is a single point. So, the equation does not define y as
a differentiable function of x.
b. The graph of this equation is the unit circle, centered at 0, 0. The upper semicircle
is given by the differentiable function
x
1 − x2
y=−
y ⫽ 1 ⫺ x 2,
(b)
y=
y ⫽ ⫺ 1 ⫺ x 2, ⫺1 < x < 1.
1−x
At the points ⫺1, 0 and 1, 0, the slope of the graph is undefined.
c. The upper half of this parabola is given by the differentiable function
1
x
y ⫽ 1 ⫺ x,
1
−1
y=−
y ⫽ ⫺ 1 ⫺ x,
Some graph segments can be represented by
differentiable functions.
Figure 2.28
x < 1.
At the point 1, 0, the slope of the graph is undefined.
EXAMPLE 4
Finding the Slope of a Graph Implicitly
Determine the slope of the tangent line to the graph of
x 2 ⫹ 4y 2 ⫽ 4
at the point 2, ⫺12 . See Figure 2.29.
y
Solution
2
x 2 + 4y 2 = 4
x
−1
1
−2
Figure 2.29
Instructional Notes accompany many of the
Theorems, Definitions, and Examples to offer
additional insights or describe generalizations.
x < 1
and the lower half of this parabola is given by the differentiable function
1−x
(c)
Notes
⫺1 < x < 1
and the lower semicircle is given by the differentiable function
y
−1
Eduspace® contains Open Explorations, which
investigate selected Examples using computer algebra
systems (Maple, Mathematica, Derive, and Mathcad).
The icon
identifies these Examples.
c. x ⫹ y 2 ⫽ 1
b. x 2 ⫹ y 2 ⫽ 1
Solution
1
(1, 0)
Open Exploration
Representing a Graph by Differentiable Functions
If possible, represent y as a differentiable function of x (see Figure 2.28).
(a)
(
2, − 1
2
)
x 2 ⫹ 4y 2 ⫽ 4
dy
⫽0
dx
dy ⫺2x ⫺x
⫽
⫽
dx
8y
4y
2x ⫹ 8y
Write original equation.
Differentiate with respect to x.
Solve for
dy
.
dx
So, at 2, ⫺12 , the slope is
⫺ 2
dy
1
⫽
⫽ .
dx ⫺42 2
Evaluate
dy
1
when x ⫽ 2 and y ⫽ ⫺
.
dx
2
NOTE To see the benefit of implicit differentiation, try doing Example 4 using the explicit
function y ⫽ ⫺ 124 ⫺ x 2.
Copyright 2008 Cengage Learning. All Rights Reserved.
May not be copied, scanned, or duplicated, in whole or in part.
Licensed to: iChapters User
xvii
FEATURES
Exercises
In Exercises 25 and 26, find an equation of the line that is
tangent to the graph of f and parallel to the given line.
The core of every calculus text, Exercises provide
opportunities for exploration, practice, and comprehension. Essential Calculus contains over 7500
Section and Chapter Review Exercises, carefully
graded in each set from skill-building to challenging.
The extensive range of problem types includes
true/false, writing, conceptual, real-data modeling,
and graphical analysis.
Function
Line
25. f x ⫽ x 3
Putnam Exam Challenge
3x ⫺ y ⫹ 1 ⫽ 0
1
26. f x ⫽
x ⫺ 1
61. Find the maximum value of f x ⫽ x3 ⫺ 3x on the set of all real
numbers x satisfying x 4 ⫹ 36 ≤ 13x2. Explain your reasoning.
x ⫹ 2y ⫹ 7 ⫽ 0
62. Find the minimum value of
27. The tangent line to the graph of y ⫽ gx at the point 5, 2
In Exercises 59– 62, describe the x-values at which f is
x ⫹ 1x6 ⫺ x 6 ⫹ 1x 6 ⫺ 2
9,
0
.
g
5
g
⬘
5
.
passes
through
the
point
Find
and
differentiable.
x ⫹ 1x3 ⫹ x3 ⫹ 1x3
59. f x ⫽
1
x⫹1
for x > 0.
60. f x ⫽ x 2 ⫺ 9
These problems were composed by the Committee on the Putnam Prize Competition.
© The Mathematical Association of America. All rights reserved.
y
y
12
10
1
6
4
2
x
−2
−1
1
−4
−2
Putnam Exam Challenge
61. f x ⫽ x ⫺ 3 23
Problems from the William Lowell Putnam
Mathematical Competitions, administered by the
Mathematical Association of America, are included
at the end of certain exercise sets to provide students
with additional challenging exercises. These can be
assigned as a group project or individually for more
advanced students. Many professors enjoy pointing
them out as an additional challenge exercise of the
math concept just covered.
−2
62. f x ⫽
2
4
−4
65. The graph of every cubic polynomial has precisely one point of
inflection.
x ≤ 0
x >66.0 The graph of f x ⫽ 1x is concave downward for x < 0 and
concave upward for x > 0, and thus it has a point of inflection
at x ⫽ 0.
x 2 ⫺ 4,
4 ⫺ x 2,
y
y
5
4
3
True or False? In Exercises 65–70, determine whether the
statement
is true or false. If it is false, explain why or give an
x
example that shows it is false.
67. The maximum value of y ⫽ 3 sin x ⫹ 2 cos x is 5.
4
g
68. The maximum slope of the graph of y ⫽ sinbx is b.
2
x
−4
1
4
x
1 2 3 4 5 6
−4
122. Modeling Data The normal daily maximum temperatures
T (in degrees Fahrenheit) for Denver, Colorado, are shown in
the table. (Source: National Oceanic and Atmospheric
Administration)
Month
Jan
Feb
Mar
Apr
May
Jun
Temperature
43.2
47.2
53.7
60.9
70.5
82.1
Month
Jul
Aug
Sep
Oct
Nov
Dec
Temperature
88.0
86.0
77.4
66.0
51.5
44.1
(a) Use a graphing utility to plot the data and find a model for
the data of the form
Tt ⫽ a ⫹ b sin ␲ t6 ⫺ c
where T is the temperature and t is the time in months,
with t ⫽ 1 corresponding to January.
(b) Use a graphing utility to graph the model. How well does
the model fit the data?
(c) Find T⬘ and use a graphing utility to graph the derivative.
TECHNOLOGY PITFALL When using a graphing utility to graph a function
involving radicals or rational exponents, be sure you understand the way the utility
evaluates radical expressions. For instance, even though
f x ⫽ x 2 ⫺ 423
and
gx ⫽ x 2 ⫺ 4 2 13
are the same algebraically, some graphing utilities distinguish between these two
functions. Which of the graphs shown in Figure 3.21 is incorrect? Why did the
graphing utility produce an incorrect graph?
f(x) = (x 2 − 4) 2/3
g(x) = [(x 2 − 4)2 ]1/3
5
4
−4
−1
5
4
−4
Technology
Throughout the text, the use of a graphing utility or
computer algebra system is suggested as appropriate
for problem-solving as well as exploration and
discovery. For example, students may choose to
use a graphing utility to execute complicated
computations, to visualize theoretical concepts, to
discover alternative approaches, or to verify the
results of other solution methods. However, students
are not required to have access to a graphing utility
to use this text effectively. In addition to describing
the benefits of using technology to learn calculus,
the text also addresses its possible misuse or
misinterpretation.
−1
Which graph is incorrect?
Figure 3.21
Additional Features
Additional teaching and learning resources are
integrated throughout the textbook, including journal
references, and Writing About Concepts exercises.
Visit college.hmco.com/pic/larsonEC for even more
teaching and learning resources.
Copyright 2008 Cengage Learning. All Rights Reserved.
May not be copied, scanned, or duplicated, in whole or in part.
Licensed to: iChapters User
Acknowledgments
Essential Calculus: Early Transcendental Functions
For their invaluable advice on Essential Calculus, we would like to thank:
John Annulis, University of Arkansas at Monticello; Karline Feller, Georgia Perimeter
College; Irvin R. Hentzel, Iowa State University; Matt Hudelson, Washington State
University; Laura Jacyna, Northern Virginia Community College—Loudoun; Charles
Lam, California State University, Bakersfield; Barbara Tozzi, Brookdale Community
College; Dennis Reissig, Suffolk County Community College.
We would also like to thank the many people who have helped us at various stages of
this project over the years. Their encouragement, criticisms, and suggestions have been
invaluable to us.
Calculus Early Transcendental Functions, Fourth Edition
Andre Adler, Illinois Institute of Technology; Evelyn Bailey, Oxford College of Emory
University; Katherine Barringer, Central Virginia Community College; Robert Bass,
Gardner-Webb University; Joy Becker, University of Wisconsin Stout; Michael Bezusko,
Pima Community College; Bob Bradshaw, Ohlone College; Robert Brown, The
Community College of Baltimore County (Essex Campus); Joanne Brunner, DePaul
University; Minh Bui, Fullerton College; Fang Chen, Oxford College of Emory
University; Alex Clark, University of North Texas; Jeff Dodd, Jacksonville State
University; Daniel Drucker, Wayne State University; Pablo Echeverria, Camden County
College; Angela Hare, Messiah College; Karl Havlak, Angelo State University; James
Herman, Cecil Community College; Xuezhang Hou, Towson University; Gene Majors,
Fullerton College; Suzanne Molnar, College of St. Catherine; Karen Murany, Oakland
Community College; Keith Nabb, Moraine Valley Community College; Stephen
Nicoloff, Paradise Valley Community College; James Pommersheim, Reed College;
James Ralston, Hawkeye Community College; Chip Rupnow, Martin Luther College;
Mark Snavely, Carthage College; Ben Zandy, Fullerton College
For the Fourth Edition Technology Program
Jim Ball, Indiana State University; Marcelle Bessman, Jacksonville University; Tim
Chappell, Penn Valley Community College; Oiyin Pauline Chow, Harrisburg Area
Community College; Julie M. Clark, Hollins University; Jim Dotzler, Nassau
Community College; Murray Eisenberg, University of Massachusetts at Amherst; Arek
Goetz, San Francisco State University; John Gosselin, University of Georgia; Shahryar
Heydari, Piedmont College; Douglas B. Meade, University of South Carolina; Teri
Murphy, University of Oklahoma; Howard Speier, Chandler-Gilbert Community
College
xviii
Copyright 2008 Cengage Learning. All Rights Reserved.
May not be copied, scanned, or duplicated, in whole or in part.
Licensed to: iChapters User
ACKNOWLEDGMENTS
xix
Reviewers of Previous Editions
Raymond Badalian, Los Angeles City College; Norman A. Beirnes, University of
Regina; Christopher Butler, Case Western Reserve University; Dane R. Camp, New
Trier High School, IL; Jon Chollet, Towson State University; Barbara Cortzen, DePaul
University; Patricia Dalton, Montgomery College; Luz M. DeAlba, Drake University;
Dewey Furness, Ricks College; Javier Garza, Tarleton State University; Claire Gates,
Vanier College; Lionel Geller, Dawson College; Carollyne Guidera, University College
of Fraser Valley; Irvin Roy Hentzel, Iowa State University; Kathy Hoke, University of
Richmond; Howard E. Holcomb, Monroe Community College; Gus Huige, University of
New Brunswick; E. Sharon Jones, Towson State University; Robert Kowalczyk,
University of Massachusetts–Dartmouth; Anne F. Landry, Dutchess Community
College; Robert F. Lax, Louisiana State University; Beth Long, Pellissippi State
Technical College; Gordon Melrose, Old Dominion University; Bryan Moran, Radford
University; David C. Morency, University of Vermont; Guntram Mueller, University of
Massachusetts–Lowell; Donna E. Nordstrom, Pasadena City College; Larry Norris,
North Carolina State University; Mikhail Ostrovskii, Catholic University of America;
Jim Paige, Wayne State College; Eleanor Palais, Belmont High School, MA; James V.
Rauff, Millikin University; Lila Roberts, Georgia Southern University; David Salusbury,
John Abbott College; John Santomas, Villanova University; Lynn Smith, Gloucester
County College; Linda Sundbye, Metropolitan State College of Denver; Anthony
Thomas, University of Wisconsin–Platteville; Robert J. Vojack, Ridgewood High School,
NJ; Michael B. Ward, Bucknell University; Charles Wheeler, Montgomery College
We would like to thank the staff at Larson Texts, Inc., who assisted with
proofreading the manuscript, preparing and proofreading the art package, and checking
and typesetting the supplements.
On a personal level, we are grateful to our wives, Deanna Gilbert Larson, Eloise
Hostetler, and Consuelo Edwards, for their love, patience, and support. Also, a special
note of thanks goes to R. Scott O’Neil.
If you have suggestions for improving this text, please feel free to write to us. Over
the years we have received many useful comments from both instructors and students,
and we value these very much.
Ron Larson
Robert Hostetler
Copyright 2008 Cengage Learning. All Rights Reserved.
May not be copied, scanned, or duplicated, in whole or in part.
Bruce H. Edwards
Licensed to: iChapters User
Copyright 2008 Cengage Learning. All Rights Reserved.
May not be copied, scanned, or duplicated, in whole or in part.
Licensed to: iChapters User
A
Proofs of Selected Theorems
THEOREM 1.2
Properties of Limits (Properties 2, 3, 4, and 5)
(page 48)
Let b and c be real numbers, let n be a positive integer, and let f and g be
functions with the following limits.
lim f x L
lim g x K
and
x→c
2. Sum or difference:
x→c
lim f x ± gx L ± K
x→c
lim f xgx LK
3. Product:
x→c
f x
L
, provided K 0
gx K
n
lim f x Ln
4. Quotient:
lim
x→c
5. Power:
x→c
Proof To prove Property 2, choose > 0. Because 2 > 0, you know that there
exists 1 > 0 such that 0 < x c < 1 implies f x L < 2. You also know
that there exists 2 > 0 such that 0 < x c < 2 implies gx K < 2. Let be the smaller of 1 and 2; then 0 < x c < implies that
f x L <
and gx K < .
2
2
So, you can apply the triangle inequality to conclude that
f x gx L K ≤ f x L gx K < 2 2 which implies that
lim f x gx L K lim f x lim gx.
x→c
x→c
x→c
The proof that
lim f x gx L K
x→c
is similar.
To prove Property 3, given that
lim f x L
x→c
and
lim gx K
x→c
you can write
f xgx f x L gx Lgx f x LK.
Because the limit of f x is L, and the limit of gx is K, you have
lim f x L 0 and
x→c
lim gx K 0.
x→c
A1
Copyright 2008 Cengage Learning. All Rights Reserved.
May not be copied, scanned, or duplicated, in whole or in part.
Licensed to: iChapters User
A2
APPENDIX A
Proofs of Selected Theorems
Let 0 < < 1. Then there exists > 0 such that if 0 < x c < , then
f x L 0 < and
gx K 0 < which implies that
f x L gx K 0 f x L gx K < < .
So,
lim [ f x L gx K 0.
x→c
Furthermore, by Property 1, you have
lim Lgx LK
lim Kf x KL.
and
x→c
x→c
Finally, by Property 2, you obtain
lim f xgx lim f x L gx K lim Lgx lim Kf x lim LK
x→c
x→c
x→c
x→c
x→c
0 LK KL LK
LK.
To prove Property 4, note that it is sufficient to prove that
lim
x→c
1
1
.
gx K
Then you can use Property 3 to write
lim
x→c
f x
1
1
L
lim f x
lim f x lim
.
x→c gx
gx x→c
gx x→c
K
Let > 0. Because lim gx K, there exists 1 > 0 such that if
x→c
K
2
0 < x c < 1, then gx K <
which implies that
K gx K gx ≤ gx K gx < gx 2 .
K
That is, for 0 < x c < 1,
K < gx
2
1
2 .
<
gx
K
or
Similarly, there exists a 2 > 0 such that if 0 < x c < 2, then
K
gx K < 2
2
.
Let be the smaller of 1 and 2. For 0 < x c < , you have
1
K gx
1
1
gx K
gxK
K
So, lim
x→c
1
1
.
gx K
1
K gx
gx <
1
K
2 K2
.
K 2
Finally, the proof of Property 5 can be obtained by a straightforward application of
mathematical induction coupled with Property 3.
Copyright 2008 Cengage Learning. All Rights Reserved.
May not be copied, scanned, or duplicated, in whole or in part.
Licensed to: iChapters User
APPENDIX A
THEOREM 1.4
Proofs of Selected Theorems
A3
The Limit of a Function Involving a Radical
(page 49)
Let n be a positive integer. The following limit is valid for all c if n is odd, and
is valid for c > 0 if n is even.
n x n c.
lim x→c
Proof Consider the case for which c > 0 and n is any positive integer. For a given
> 0, you need to find > 0 such that
n x n c < whenever
0 < xc < which is the same as saying
n
n
< x
c < < x c < .
whenever
n
n
Assume < c, which implies that 0 < c < c. Now, let be the smaller
of the two numbers
n
n
c c n
n
c n
and
c.
Then you have
n
n c c n
n c c
n
n
c n c THEOREM 1.5
< xc
< xc
< xc
< n c < c
n
n c < c
n
< x
n c < n x
<
n c < n
n x n c < .
< The Limit of a Composite Function (page 50)
If f and g are functions such that lim gx L and lim f x f L, then
x→c
x→L
lim f g x f lim gx f L.
x→c
Proof
x→c
For a given > 0, you must find > 0 such that
f gx f L < whenever 0 < x c < .
Because the limit of f x as x → L is f L, you know there exists 1 > 0 such that
f u f L < whenever
u L < 1.
Moreover, because the limit of gx as x → c is L, you know there exists > 0 such
that
gx L < 1
whenever 0 < x c < .
Finally, letting u gx, you have
f gx f L < whenever 0 < x c < .
Copyright 2008 Cengage Learning. All Rights Reserved.
May not be copied, scanned, or duplicated, in whole or in part.
Licensed to: iChapters User
A4
APPENDIX A
Proofs of Selected Theorems
THEOREM 1.7
Functions That Agree at All But One Point
(page 51)
Let c be a real number and let f x gx for all x c in an open interval
containing c. If the limit of gx as x approaches c exists, then the limit of f x
also exists and
lim f x lim gx.
x→c
x→c
Proof Let L be the limit of gx as x → c. Then, for each > 0 there exists a > 0
such that f x gx in the open intervals c , c and c, c , and
gx L < whenever 0 < x c < .
f x L < whenever 0 < x c < .
Because f x gx for all x in the open interval other than x c, it follows that
So, the limit of f x as x → c is also L.
THEOREM 1.8
The Squeeze Theorem (page 54)
If hx ≤ f x ≤ gx for all x in an open interval containing c, except possibly
at c itself, and if
lim hx L lim gx
x→c
x→c
then lim f x exists and is equal to L.
x→c
Proof
For > 0 there exist 1 > 0 and 2 > 0 such that
hx L < whenever 0 < x c < 1
gx L < whenever 0 < x c < 2.
and
Because hx ≤ f x ≤ gx for all x in an open interval containing c, except possibly
at c itself, there exists 3 > 0 such that hx ≤ f x ≤ gx for 0 < x c < 3. Let
be the smallest of 1, 2, and 3. Then, if 0 < x c < , it follows that
hx L < and gx L < , which implies that
< hx L < and
L < hx and
< gx L < gx < L .
Now, because hx ≤ f x ≤ gx, it follows that L < f x < L , which
implies that f x L < . Therefore,
lim f x L.
x→c
Copyright 2008 Cengage Learning. All Rights Reserved.
May not be copied, scanned, or duplicated, in whole or in part.
Licensed to: iChapters User
APPENDIX A
THEOREM 1.14
Proofs of Selected Theorems
A5
Vertical Asymptotes (page 72)
Let f and g be continuous on an open interval containing c. If f c 0, gc 0,
and there exists an open interval containing c such that gx 0 for all x c in
the interval, then the graph of the function given by
h x f x
gx
has a vertical asymptote at x c.
Proof Consider the case for which f c > 0 and there exists b > c such that
c < x < b implies gx > 0. Then, for M > 0, choose 1 such that
0 < x c < 1
implies
f c
3f c
< f x <
2
2
and 2 such that
0 < x c < 2
implies 0 < gx <
f c
.
2M
Now let be the smaller of 1 and 2. Then it follows that
0 < xc < implies
f x
f c 2M
>
M.
gx
2
f c
So, it follows that
lim
x→c
f x
gx and the line x c is a vertical asymptote of the graph of h.
Alternative Form of the Derivative (page 85)
The derivative of f at c is given by
fc lim
x→c
f x f c
xc
provided this limit exists.
Proof
The derivative of f at c is given by
fc lim
x→0
f c x f c
.
x
Let x c x. Then x → c as x → 0. So, replacing c x by x, you have
fc lim
x→0
f c x f c
f x f c
lim
.
x→c
x
xc
Copyright 2008 Cengage Learning. All Rights Reserved.
May not be copied, scanned, or duplicated, in whole or in part.
Licensed to: iChapters User
A6
APPENDIX A
Proofs of Selected Theorems
THEOREM 2.11
The Chain Rule (page 113)
If y f u is a differentiable function of u, and u gx is a differentiable
function of x, then y f gx is a differentiable function of x and
dy
dy
dx du
du
dx
or, equivalently,
d
f gx fgxgx.
dx
Proof In Section 2.4, you let hx f gx and used the alternative form of the
derivative to show that hc fgcgc, provided gx gc for values of x
other than c. Now consider a more general proof. Begin by considering the derivative
of f.
fx lim
x→0
f x x f x
y
lim
x→0 x
x
For a fixed value of x, define a function such that
x 0
.
x 0
0,
x y
fx,
x
Because the limit of x as x → 0 doesn’t depend on the value of 0, you have
lim x lim
x→0
x→0
fx 0
y
x
and you can conclude that is continuous at 0. Moreover, because y 0 when
x 0, the equation
y x
x xfx
is valid whether x is zero or not. Now, by letting u gx x gx, you can
use the continuity of g to conclude that
lim u lim gx x gx 0
x→0
x→0
which implies that
lim u 0.
x→0
Finally,
y u
u ufu →
y u
u
u fu,
x
x
x
and taking the limit as x → 0, you have
dy du
dx
dx
lim
x→0
u du
dy
du
fu 0 fu
dx
dx
dx
du
fu
dx
du dy
.
dx du
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May not be copied, scanned, or duplicated, in whole or in part.
x 0
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APPENDIX A
THEOREM 2.16
Proofs of Selected Theorems
A7
Continuity and Differentiability of Inverse
Functions (page 134)
Let f be a function whose domain is an interval I. If f has an inverse, then the
following statements are true.
1. If f is continuous on its domain, then f 1 is continuous on its domain.
2. If f is differentiable at c and fc 0, then f 1 is differentiable at f c.
Proof To prove Property 1, you first need to define what is meant by a strictly
increasing function or a strictly decreasing function. A function f is strictly increasing on an entire interval I if for any two numbers x1 and x2 in the interval, x1 < x2
implies f x1 < f x2 . The function f is strictly decreasing on the entire interval I if
x1 < x2 implies f x1 > f x2 . The function f is strictly monotonic on the interval I
if it is either strictly increasing or strictly decreasing. Now show that if f is continuous on I, and has an inverse, then f is strictly monotonic on I. Suppose that f were
not strictly monotonic. Then there would exist numbers x1, x2, x3 in I such that
x1 < x 2 < x3, but f x2 is not between f x1 and f x3. Without loss of generality,
assume f x1 < f x3 < f x2. By the Intermediate Value Theorem, there exists a
number x0 between x1 and x2 such that f x0 f x3. So, f is not one-to-one and
cannot have an inverse. So, f must be strictly monotonic.
Because f is continuous, the Intermediate Value Theorem implies that the set of values
of f, f x: x , forms an interval J. Assume that a is an interior point of J. From
the previous argument, f 1a is an interior point of I. Let > 0. There exists
0 < 1 < such that
I1 f 1a 1, f 1a 1 I.
Because f is strictly monotonic on I1, the set of values f x: x I1 forms an interval
J1 J. Let > 0 such that a , a J1. Finally, if y a < , then
f 1 y f 1a < 1 < . So, f 1 is continuous at a. A similar proof can be given
if a is an endpoint.
To prove Property 2, consider the limit
f 1y f 1a
ya
f 1a lim
y→a
where a is in the domain of f 1 and f 1a c. Because f is differentiable at c, f is
continuous at c, and so is f 1 at a. So, y → a implies that x → c, and you have
f 1a lim
x→c
xc
1
1
1
lim
.
f x f c x→c f x f c
f x f c
f c
lim
x→c
xc
xc
So, f 1a exists, and f 1 is differentiable at f c.
THEOREM 2.17
The Derivative of an Inverse Function (page 134)
Let f be a function that is differentiable on an interval I. If f has an inverse
function g, then g is differentiable at any x for which f gx 0. Moreover,
gx 1
,
f gx
fgx 0.
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Licensed to: iChapters User
A8
APPENDIX A
Proofs of Selected Theorems
Proof From the proof of Theorem 2.16, letting a x, you know that g is differentiable. Using the Chain Rule, differentiate both sides of the equation x f gx
to obtain
1 fgx
d
gx.
dx
Because fgx 0, you can divide by this quantity to obtain
d
1
gx .
dx
fgx
Concavity Interpretation (page 180)
1. Let f be differentiable on an open interval I. If the graph of f is concave
upward on I, then the graph of f lies above all of its tangent lines on I.
2. Let f be differentiable on an open interval I. If the graph of f is concave
downward on I, then the graph of f lies below all of its tangent lines on I.
Proof Assume that f is concave upward on I a, b. Then, f is increasing on
a, b. Let c be a point in the interval I a, b. The equation of the tangent line to
the graph of f at c is given by
gx f c fcx c.
If x is in the open interval c, b, then the directed distance from the point x, f x (on
the graph of f ) to the point x, gx (on the tangent line) is given by
d f x f c fcx c
f x f c fcx c.
Moreover, by the Mean Value Theorem, there exists a number z in c, x such that
fz f x f c
.
xc
So, you have
d f x f c fcx c
fzx c fcx c
fz fcx c.
The second factor x c is positive because c < x. Moreover, because f is increasing,
it follows that the first factor fz fc is also positive. Therefore, d > 0 and you
can conclude that the graph of f lies above the tangent line at x. If x is in the open interval
a, c, a similar argument can be given. This proves the first statement. The
proof of the second statement is similar.
THEOREM 3.10
Limits at Infinity (page 188)
If r is a positive rational number and c is any real number, then
lim
x→ c
0.
xr
Furthermore, if x r is defined when x < 0, then lim c 0.
x→ x r
Copyright 2008 Cengage Learning. All Rights Reserved.
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Licensed to: iChapters User
APPENDIX A
Proof
Proofs of Selected Theorems
A9
Begin by proving that
lim
x→ 1
0.
x
For > 0, let M 1. Then, for x > M, you have
x > M
1
< x
1
1
0 < .
x
So, by the definition of a limit at infinity, you can conclude that the limit of 1x as
x → is 0. Now, using this result and letting r mn, you can write the following.
lim
x→ c
c
lim mn
r
x→
x
x
c lim
x→ 1
n
x
m
m
1
c lim x
c lim
n
x→ 1
x
m
n
x→ n
0
c
0
m
The proof of the second part of the theorem is similar.
THEOREM 4.2
nn 1
2
i1
n
2n 12
n
i3 4.
4
i1
n
c cn
1.
3.
Summation Formulas (page 228)
i1
n
i
2
i1
2.
nn 12n 1
6
n
i Proof The proof of Property 1 is straightforward. By adding c to itself n times, you
obtain a sum of cn.
To prove Property 2, write the sum in increasing and decreasing order and add
corresponding terms, as follows.
n
2
3
. . . n 1 n 1 n 2 . . . 1
→
→
→
→
n
2
2
n
→
→
n
→
i1
→
n
i 1
→
i
i1
i n 1 n 1 n 1 . . . n 1 n 1
i1
n terms
So,
n
i
i1
nn 1
.
2
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Licensed to: iChapters User
A10
APPENDIX A
Proofs of Selected Theorems
To prove Property 3, use mathematical induction. First, if n 1, the result is true
because
1
i
i1
2
11 12 1
.
6
12 1 Now, assuming the result is true for n k, you can show that it is true for n k 1,
as follows.
k1
k
i
i2 i1
2
i1
k 12
kk 12k 1
k 12
6
k1
2k2 k 6k 6
6
k1
2k 3k 2
6
k 1k 22k 1 1
6
Property 4 can be proved using a similar argument with mathematical induction.
THEOREM 4.8
Preservation of Inequality (page 244)
1. If f is integrable and nonnegative on the closed interval a, b, then
b
0 ≤
a
f x dx.
2. If f and g are integrable on the closed interval a, b and f x ≤ gx for
every x in a, b, then
b
a
Proof
b
f x dx ≤
a
gx dx.
To prove Property 1, suppose, on the contrary, that
b
a
f x dx I < 0.
Then, let a x0 < x1 < x2 < . . . < xn b be a partition of a, b, and let
R
n
f c x
i
i1
i
be a Riemann sum. Because f x ≥ 0, it follows that R ≥ 0. Now, for sufficiently
small, you have R I < I2, which implies that
n
I
f c x R < I 2 < 0
i1
i
i
which is not possible. From this contradiction, you can conclude that
b
0 ≤
a
f x dx.
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Licensed to: iChapters User
APPENDIX A
Proofs of Selected Theorems
A11
To prove Property 2 of the theorem, note that f x ≤ gx implies that
gx f x ≥ 0. So, you can apply the result of Property 1 to conclude that
b
0 ≤
gx f x dx
a
b
0 ≤
b
a
a
b
gx dx a
f x dx
b
f x dx ≤
a
THEOREM 6.3
gx dx.
The Extended Mean Value Theorem (page 406)
If f and g are differentiable on an open interval a, b and continuous on a, b
such that gx 0 for any x in a, b, then there exists a point c in a, b such
that
fc
f b f a
.
gc gb ga
Proof You can assume that ga gb, because otherwise, by Rolle’s Theorem, it
would follow that gx 0 for some x in a, b. Now, define hx to be
hx f x gf bb fgaa gx.
Then
ha f a gf bb fgaa ga f aggbb gf baga
hb f b gf bb fgaa gb f aggbb gf baga
and
and, by Rolle’s Theorem, there exists a point c in a, b such that
hc fc f b f a
gc 0
gb ga
which implies that
f b f a
fc
.
gc gb ga
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Licensed to: iChapters User
A12
APPENDIX A
Proofs of Selected Theorems
THEOREM 6.4
L’Hôpital’s Rule (page 406)
Let f and g be functions that are differentiable on an open interval a, b containing c, except possibly at c itself. Assume that gx 0 for all x in a, b,
except possibly at c itself. If the limit of f xgx as x approaches c produces
the indeterminate form 00, then
lim
x→c
fx
f x
lim
x→c gx
gx
provided the limit on the right exists (or is infinite). This result also applies if
the limit of f xgx as x approaches c produces any one of the indeterminate
forms , , , or .
You can use the Extended Mean Value Theorem to prove L’Hôpital’s Rule. Of the
several different cases of this rule, the proof of only one case is illustrated. The
remaining cases, where x → c and x → c, are left for you to prove.
Proof
Consider the case for which
lim f x 0 and
x→c
lim gx 0.
x→c
Define the following new functions:
Fx f0,x,
xc
gx,
and Gx xc
0,
xc
.
xc
For any x, c < x < b, F and G are differentiable on c, x and continuous on c, x.
You can apply the Extended Mean Value Theorem to conclude that there exists a
number z in c, x such that
Fz
Fx Fc
Gz Gx Gc
Fx
Gx
fz
gz
f x
.
gx
Finally, by letting x approach c from the right, x → c, you have z → c because
c < z < x, and
lim
x→c
f x
fz
lim
gx x→c gz
lim
fz
gz
lim
fx
.
gx
z→c
x→c
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Licensed to: iChapters User
APPENDIX A
THEOREM 7.19
Proofs of Selected Theorems
A13
Taylor’s Theorem (page 472)
If a function f is differentiable through order n 1 in an interval I containing
c, then, for each x in I, there exists z between x and c such that
f x f c fcx c f c
f nc
2
n
x c . . . x c Rnx
2!
n!
where
Rnx Proof
f n1z
x c n1.
n 1!
To find Rnx, fix x in I x c and write
Rnx f x Pnx
where Pnx is the nth Taylor polynomial for f x. Then let g be a function of t defined
by
x tn1
f nt
gt f x f t ftx t . . . x tn Rnx
.
n!
x cn1
The reason for defining g in this way is that differentiation with respect to t has a
telescoping effect. For example, you have
d
f t ftx t ft ft f tx t
dt
f tx t.
The result is that the derivative gt simplifies to
gt x tn
f n1t
x tn n 1Rnx
n!
x cn1
for all t between c and x. Moreover, for a fixed x,
gc f x Pnx Rnx f x f x 0
and
gx f x f x 0 . . . 0 f x f x 0.
Therefore, g satisfies the conditions of Rolle’s Theorem, and it follows that there is a
number z between c and x such that gz 0. Substituting z for t in the equation for
gt and then solving for Rnx, you obtain
x zn
f n1z
x zn n 1Rnx
0
n!
x cn1
f n1z
Rnx x cn1.
n 1!
gz Finally, because gc 0, you have
f nc
0 f x f c fcx c . . . x cn Rnx
n!
f nc
f x f c fcx c . . . x cn Rnx.
n!
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Licensed to: iChapters User
A14
APPENDIX A
Proofs of Selected Theorems
THEOREM 7.20
Convergence of a Power Series (page 477)
For a power series centered at c, precisely one of the following is true.
1. The series converges only at c.
2. There exists a real number R > 0 such that the series converges absolutely for
x c < R, and diverges for x c > R.
3. The series converges absolutely for all x.
The number R is the radius of convergence of the power series. If the series
converges only at c, the radius of convergence is R 0, and if the series
converges for all x, the radius of convergence is R . The set of all values
of x for which the power series converges is the interval of convergence of the
power series.
Proof In order to simplify the notation, the theorem for the power series an x n
centered at x 0 will be proved. The proof for a power series centered at x c
follows easily. A key step in this proof uses the completeness property of the set of
real numbers: If a nonempty set S of real numbers has an upper bound, then it must
have a least upper bound (see page 434).
It must be shown that if a power series an x n converges at x d, d 0, then it
converges for all b satisfying b < d . Because an x n converges, lim an d n 0.
x→ So, there exists N > 0 such that an d n < 1 for all n ≥ N. Then, for n ≥ N,
an b n an b n
So, for b < d ,
bn
dn
n
bn
dn
n b
a
d
<
.
n
dn
dn
dn
b
< 1, which implies that
d
is a convergent geometric series. By the Comparison Test, the series an b n converges.
Similarly, if the power series an x n diverges at x b, where b 0, then it diverges
for all d satisfying d > b . If an d n converged, then the argument above would
imply that an b n converged as well.
Finally, to prove the theorem, suppose that neither case 1 nor case 3 is true. Then there
exist points b and d such that an x n converges at b and diverges at d. Let
S x: an x n converges. S is nonempty because b S. If x S, then x ≤ d ,
which shows that d is an upper bound for the nonempty set S. By the completeness
property, S has a least upper bound, R.
Now, if x > R, then xS, so an x n diverges. And if x < R, then x is not an upper
bound for S, so there exists b in S satisfying b > x . Since b S, an b n
converges, which implies that an x n converges.
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Licensed to: iChapters User
APPENDIX A
THEOREM 8.10
Proofs of Selected Theorems
A15
Classification of Conics by Eccentricity
(page 539)
Let F be a fixed point ( focus) and let D be a fixed line (directrix) in the plane.
Let P be another point in the plane and let e (eccentricity) be the ratio of the
distance between P and F to the distance between P and D. The collection of
all points P with a given eccentricity is a conic.
1. The conic is an ellipse if 0 < e < 1.
2. The conic is a parabola if e 1.
3. The conic is a hyperbola if e > 1.
y
P
Proof If e 1, then, by definition, the conic must be a parabola. If e 1, then you
can consider the focus F to lie at the origin and the directrix x d to lie to the right
of the origin, as shown in Figure A.1. For the point P r, x, y, you have
PF r and PQ d r cos . Given that e PF PQ , it follows that
Q
PF PQe
r
r ed r cos .
By converting to rectangular coordinates and squaring each side, you obtain
x 2 y 2 e2d x2 e2d 2 2dx x2.
θ
x
F
Completing the square produces
x=d
2
x 1 ede 2
Figure A.1
2
y2
e 2d 2
.
2
1e
1 e 22
If e < 1, this equation represents an ellipse. If e > 1, then 1 e 2 < 0, and the
equation represents a hyperbola.
THEOREM 11.4
Sufficient Condition for Differentiability
(page 683)
If f is a function of x and y, where fx and fy are continuous in an open region
R, then f is differentiable on R.
Proof Let S be the surface defined by z f x, y, where f, fx , and fy are continuous
at x, y. Let A, B, and C be points on surface S, as shown in Figure A.2. From this
figure, you can see that the change in f from point A to point C is given by
z
C
B
∆z2
∆z1
z f x x, y y f x, y
∆z
f x x, y f x, y f x x, y y f x x, y
z1 z 2.
A
Between A and B, y is fixed and x changes. So, by the Mean Value Theorem, there is
a value x1 between x and x x such that
y
x
(x + ∆x, y + ∆y)
(x, y)
(x + ∆x, y)
z f x x, y y f x, y
Figure A.2
z1 f x x, y f x, y fxx1, y x.
Similarly, between B and C, x is fixed and y changes, and there is a value y1 between
y and y y such that
z2 f x x, y y f x x, y fy x x, y1 y.
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Licensed to: iChapters User
A16
APPENDIX A
Proofs of Selected Theorems
By combining these two results, you can write
z z1 z 2 fx x1, y
x fy x x, y1 y.
If you define 1 and 2 as
1 fxx1, y fxx, y and 2 fy x x, y1 fyx, y
it follows that
z z1 z 2 1 fxx, y x 2 fy x, y y
fx x, y x fy x, y y 1
x 2
y.
By the continuity of fx and fy and the fact that x ≤ x1 ≤ x x and y ≤ y1 ≤ y y,
it follows that 1 → 0 and 2 → 0 as x → 0 and y → 0. Therefore, by definition, f is
differentiable.
THEOREM 11.6
Chain Rule: One Independent Variable (page 686)
Let w f x, y, where f is a differentiable function of x and y. If x gt and
y h t, where g and h are differentiable functions of t, then w is a differentiable function of t, and
d w w dx w dy
.
dt
x dt
y dt
Proof Because g and h are differentiable functions of t, you know that both x and
y approach zero as t approaches zero. Moreover, because f is a differentiable
function of x and y, you know that
w w
w
x y 1 x 2 y
x
y
where both 1 and 2 → 0 as x, y → 0, 0. So, for t 0,
x
y
w w x w y
1
2
t
x t
y t
t
t
from which it follows that
w w dx w dy
dx
dy
dw
0
lim
0
t→0 t
dt
x dt
y dt
dt
dt
w dx w dy
.
x dt
y dt
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May not be copied, scanned, or duplicated, in whole or in part.
Licensed to: iChapters User
B
Integration Tables
Forms Involving un
1.
2.
un du un1
C, n 1
n1
1
du ln u C
u
Forms Involving a bu
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
u
1
du 2 bu a ln a bu C
a bu
b
C
a
u
1
du 2
ln a bu
2
a bu
b a bu
1
u
1
a
du 2
C, n 1, 2
a bun
b n 2a bun2 n 1a bun1
u2
bu
1
du 3 2a bu a2 ln a bu
a bu
b
2
C
u2
1
a2
du
bu
2a ln a bu
a bu2
b3
a bu
C
2a
u2
1
a2
du
ln a bu
a bu3
b 3 a bu 2a bu2
C
1
1
2a
a2
u2
du
C,
a bun
b 3 n 3a bun3 n 2a bun2 n 1a bun1
n 1, 2, 3
u
1
1
du ln
C
ua bu
a a bu
1
u
1
1
1
du ln
ua bu2
a a bu a a bu
u
1
1 1 b
du ln
u 2a bu
a u a a bu
C
C
u
1
1 a 2bu
2b
du 2
ln
u 2a bu2
a ua bu
a
a bu
C
A17
Copyright 2008 Cengage Learning. All Rights Reserved.
May not be copied, scanned, or duplicated, in whole or in part.
Licensed to: iChapters User
A18
APPENDIX B
Integration Tables
Forms Involving a bu cu2, b2 4ac
2
arctan
2cu b
14.
1
du a bu cu2
15.
u
1
du ln a bu cu 2 b
2
a bu cu
2c
4ac b2
4ac b2
17.
18.
19.
20.
21.
22.
una bu du 1
du ua bu
u
2
una bu32 na
b2n 3
1
a
ln
2
a
a bu a
arctan
1
du
a bu cu 2
a > 0
bu
C,
a a
a < 0
a bu
u
n
1
du
ua bu
du 2a bu a
du a bu32 2n 5b
1
an 1
un1
2
u
22a bu
a bu C
du a bu
3b 2
un
2
du una bu na
a bu
2n 1b
un1a bu du
1
a bu
2n 3b
1
du a bu
an 1
un1
2
u
C,
a bu a
n
a bu
b2 < 4ac
2cu b b2 4ac
1
ln
C, b2 > 4ac
b2 4ac
2cu b b2 4ac
Forms Involving a bu
16.
C,
1
du , n 1
a bu
n1
u
a bu
un1
un1
du
a bu
du , n 1
Forms Involving a2 ± u2, a > 0
23.
24.
25.
1
1
u
du arctan C
a2 u2
a
a
1
du u2 a2
ua
1
1
du C
ln
a2 u2
2a u a
u
1
1
du 2
2n 3
a2 ± u2n
2a n 1 a2 ± u2n1
1
du , n 1
a2 ± u2n1
Forms Involving u2 ± a2, a > 0
26.
27.
28.
u2 ± a2 du 1
uu2 ± a2 ± a2 ln u u2 ± a2 C
2
u2u2 ± a2 du u2 a2
u
1
u2u2 ± a2u2 ± a2 a4 ln u u2 ± a2 C
8
du u2 a2 a ln
a u2 a2
C
u
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May not be copied, scanned, or duplicated, in whole or in part.
Licensed to: iChapters User
APPENDIX B
29.
30.
31.
32.
33.
34.
35.
36.
u2 a2
du u2 a2 a arcsec
u
u2 ± a2
2
u
du 1
u2 ± a2
ln u u2 ± a2 C
u
1
uu a
2
du 1 a u2 a2
C
ln
a
u
1
1
u
du arcsec
C
uu2 a2
a
a
u2
2
u ± a2
a
du ln u u2 ± a2 C
u2 ± a2
2
u C
du 1
uu2 ± a2 a2 ln u u2 ± a2 C
2
u2 ± a2
1
du C
2
a2u
u ±a
2 2
u
±u
1
du 2 2
C
u2 ± a232
a u ± a2
Forms Involving a2 u2, a > 0
37.
38.
39.
40.
41.
42.
43.
44.
45.
a2 u2 du 1
u
u2u2 a2a2 u2 a4 arcsin
C
8
a
u2a2 u2 du a2 u2
u
a2 u2
u2
1
a2 u2
du a2 u2 a ln
du u2
du a a2 u2
C
u
a2 u2
u
arcsin C
u
a
du arcsin
1
ua2
1
u
ua2 u2 a2 arcsin
C
2
a
u
C
a
1 a a2 u2
C
ln
a
u
u2
1
u
du ua2 u2 a2 arcsin
C
u2
2
a
a2
1
u2a2
u2
du a2 u2
C
a2u
u
1
du 2 2
C
a2 u232
a a u2
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Integration Tables
A19
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A20
APPENDIX B
Integration Tables
Forms Involving sin u or cos u
46.
48.
50.
52.
54.
56.
58.
sin u du cos u C
47.
1
sin2 u du u sin u cos u C
2
49.
sinn u du sinn1 u cos u n 1
n
n
51.
sinn2 u du
u sin u du sin u u cos u C
un sin u du un cos u n
53.
55.
un1 cos u du
1
du tan u sec u C
1 ± sin u
57.
61.
62.
63.
65.
67.
68.
69.
70.
71.
73.
cos u du sin u C
1
cos2 u du u sin u cos u C
2
cosn u du cosn1 u sin u n 1
n
n
cosn2 u du
u cos u du cos u u sin u C
un cos u du un sin u n
un1 sin u du
1
du cot u ± csc u C
1 ± cos u
1
du ln tan u C
sin u cos u
Forms Involving tan u, cot u, sec u, csc u
59.
tan u du ln cos u C
60.
cot u du ln sin u C
sec u du ln sec u tan u C
csc u du ln csc u cot u C
or
tan2 u du u tan u C
64.
sec2 u du tan u C
66.
tann u du tann1 u
n1
cot n u du secn u du cot n1u
n1
cot2 u du u cot u C
csc2 u du cot u C
cot n2 u du, n 1
cscn2 u cot u n 2
n1
n1
secn2 u du, n 1
cscn2u du, n 1
1
1
du u ± ln cos u ± sin u C
1 ± tan u
2
72.
1
du u cot u csc u C
1 ± sec u
74.
tann2 u du, n 1
secn2 u tan u n 2
n1
n1
cscn u du csc u du ln csc u cot u C
1
1
du u ln sin u ± cos u C
1 ± cot u
2
1
du u tan u ± sec u C
1 ± csc u
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Licensed to: iChapters User
APPENDIX B
Forms Involving Inverse Trigonometric Functions
75.
77.
79.
80.
arcsin u du u arcsin u 1 u2 C
76.
arctan u du u arctan u ln1 u2 C
78.
arccsc u du u arccsc u ln u 83.
85.
86.
2
eu du eu C
82.
uneu du uneu n
89.
90.
eau sin bu du eau cos bu du un1eu du
a2
eau
a sin bu b cos bu C
b2
a2
eau
a cos bu b sin bu C
b2
84.
ln u du u1 ln u C
un ln u du 94.
96.
91.
ln un du uln un n
cosh u du sinh u C
93.
sech2 u du tanh u C
95.
sech u tanh u du sech u C
97.
100.
du
u2 ± a2
ln u u2 ± a2 C
1
du u ln1 eu C
1 eu
u ln u du Forms Involving Inverse Hyperbolic Functions (in logarithmic form)
98.
ueu du u 1eu C
un1
1 n 1 ln u C, n 1
n 12
ln u2 du u 2 2 ln u ln u2 C
88.
Forms Involving Hyperbolic Functions
92.
arccot u du u arccot u ln1 u2 C
u 1 C
Forms Involving ln u
87.
arccos u du u arccos u 1 u2 C
arcsec u du u arcsec u ln u u2 1 C
Forms Involving eu
81.
Integration Tables
99.
u2
1 2 ln u C
4
ln un1 du
sinh u du cosh u C
csch2 u du coth u C
csch u coth u du csch u C
du
au
1
C
ln
a2 u2 2a a u
du
1 a a2 ± u2
C
ln
a
u
ua2 ± u2
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A21
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A22
CHAPTER 6
C
Differential Equations
Business and Economic Applications
Previously, you learned that one of the most common ways to measure change is with
respect to time. In this section, you will study some important rates of change in
economics that are not measured with respect to time. For example, economists refer
to marginal profit, marginal revenue, and marginal cost as the rates of change of
the profit, revenue, and cost with respect to the number of units produced or sold.
Summary of Business Terms and Formulas
Basic Terms
x is the number of units produced (or sold).
p is the price per unit.
R is the total revenue from selling x units.
C is the total cost of producing x units.
Basic Formulas
R xp
C
x
PRC
P is the total profit from selling x units.
The break-even point is the number of units for which R C.
C is the average cost per unit.
C
Marginals
dR
Marginal revenue extra revenue from selling one additional unit
dx
Marginal
revenue
1 unit
dC
Marginal cost extra cost of producing one additional unit
dx
dP
Marginal profit extra profit from selling one additional unit
dx
Extra revenue
for one unit
A revenue function
Figure C.1
In this summary, note that marginals can be used to approximate the extra
revenue, cost, or profit associated with selling or producing one additional unit. This
is illustrated graphically for marginal revenue in Figure C.1.
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Licensed to: iChapters User
APPENDIX C
EXAMPLE 1
600
Marginal
profit
(50, 525)
Using Marginals as Approximations
P 0.0002x 3 10x.
a. Find the marginal profit for a production level of 50 units.
b. Compare this with the actual gain in profit obtained by increasing production from
50 to 51 units. (See Figure C.2.)
500
Profit (in dollars)
A23
A manufacturer determines that the profit P (in dollars) derived from selling x units of
an item is given by
(51, 536.53)
P
Business and Economic Applications
400
300
Solution
a. Because the profit is P 0.0002x3 10x, the marginal profit is given by the
derivative
200
100
P = 0.0002 x 3 + 10x
x
10
20
30
40
50
Number of units
Marginal profit is the extra profit from
selling one additional unit.
Figure C.2
dP
0.0006x 2 10.
dx
When x 50, the marginal profit is
dP
0.0006502 10
dx
Marginal profit for x 50
$11.50.
b. For x 50 and 51, the actual profits are
P 0.000250 3 1050
25 50
$525.00
P 0.000251 3 1051
26.53 510
$536.53.
So, the additional profit obtained by increasing the production level from 50 to 51
units is
$536.53 $525.00 $11.53.
Extra profit for one unit
The profit function in Example 1 is unusual in that the profit continues to increase
as long as the number of units sold increases. In practice, it is more common to
encounter situations in which sales can be increased only by lowering the price per
item. Such reductions in price ultimately cause the profit to decline.
The number of units x that consumers are willing to purchase at a given price p
per unit is defined as the demand function
p f x.
Demand function
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A24
APPENDIX C
Business and Economic Applications
EXAMPLE 2
A business sells 2000 items per month at a price of $10 each. It is estimated that
monthly sales will increase by 250 items for each $0.25 reduction in price. Find the
demand function corresponding to this estimate.
p
Solution From the given estimate, x increases 250 units each time p drops $0.25
from the original cost of $10. This is described by the equation
Price (in dollars)
20
x
p = 12 −
1000
15
x 2000 250
10
p
100.25
12,000 1000p
5
or
x
1000 2000 3000 4000 5000
Number of units
p 12 A demand function p
EXAMPLE 3
p
3.00
x
,
1000
Demand function
Finding the Marginal Revenue
A fast-food restaurant has determined that the monthly demand for its hamburgers is
p=
60,000 − x
20,000
p
2.50
60,000 x
.
20,000
Find the increase in revenue per hamburger (marginal revenue) for monthly sales of
20,000 hamburgers. (See Figure C.4.)
2.00
1.50
1.00
Solution Because the total revenue is given by R xp, you have
0.50
x
20,000
40,000
R xp x
60,000
Number of units
As the price decreases, more hamburgers are
sold.
Figure C.4
x ≥ 2000.
The graph of the demand function is shown in Figure C.3.
Figure C.3
Price (in dollars)
Finding a Demand Function
x
1
60,000x x ).
60,000
20,000 20,000
2
By differentiating, you can find the marginal revenue to be
dR
1
60,000 2x.
dx
20,000
When x 20,000, the marginal revenue is
dR
1
60,000 220,000
dx
20,000
20,000
20,000
$1 per unit.
NOTE The demand function in Example 3 is typical in that a high demand corresponds to a
low price, as shown in Figure C.4.
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Licensed to: iChapters User
APPENDIX C
P
Profit (in dollars)
Suppose that in Example 3 the cost C (in dollars) of producing x hamburgers is
x 2 − 5,000
20,000
C 5000 0.56x,
(24,400, 24,768)
25,000
0 ≤ x ≤ 50,000.
Find the total profit and the marginal profit for 20,000, 24,400, and 30,000 units.
20,000
Solution Because P R C, you can use the revenue function in Example 3 to
obtain
15,000
10,000
5,000
x
20,000
−5,000
40,000
Number of units
The maximum profit corresponds to the
point where the marginal profit is 0. When
more than 24,400 hamburgers are sold, the
marginal profit is negative—increasing
production beyond this point will reduce
rather than increase profit.
Figure C.5
1
60,000x x 2 5000 0.56x
20,000
x2
2.44x 5000.
20,000
P
So, the marginal profit is
dP
x .
2.44 dx
10,000
The table shows the total profit and the marginal profit for each of the three indicated
demands. Figure C.5 shows the graph of the profit function.
Demand
20,000
24,400
30,000
Profit
$23,800
$24,768
$23,200
$0.44
$0.00
$0.56
Marginal profit
Finding the Maximum Profit
EXAMPLE 5
3500
Profit (in dollars)
A25
Finding the Marginal Profit
EXAMPLE 4
P = 2.44x −
Business and Economic Applications
R = 50
3000
In marketing an item, a business has discovered that the demand for the item is represented by
x
p
2500
Maximum
profit:
dR = dC
dx dx
2000
1500
1000
500
x
2
3
4
5
Number of units (in thousands)
dR
dC
Maximum profit occurs when .
dx
dx
.
Demand function
The cost C (in dollars) of producing x items is given by C 0.5x 500. Find the
price per unit that yields a maximum profit (see Figure C.6).
C = 0.5x + 500
1
50
x
Solution From the given cost function, you obtain
P R C xp 0.5x 500.
Primary equation
Substituting for p (from the demand function) produces
Px
Figure C.6
50x 0.5x 500 50x 0.5x 500.
Setting the marginal profit equal to 0,
dP
25
0.5 0
dx
x
yields x 2500. From this, you can conclude that the maximum profit occurs when
the price is
p
50
2500
50
$1.00.
50
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A26
APPENDIX C
Business and Economic Applications
NOTE To find the maximum profit in Example 5, the profit function, P R C,
was differentiated and set equal to 0. From the equation
dP dR dC
0
dx
dx
dx
it follows that the maximum profit occurs when the marginal revenue is equal to the
marginal cost, as shown in Figure C.6.
Minimizing the Average Cost
EXAMPLE 6
Cost per unit (in dollars)
C
C=
800
x + 0.04 + 0.0002 x
A company estimates that the cost C (in dollars) of producing x units of a product is
given by C 800 0.04x 0.0002x 2. Find the production level that minimizes the
average cost per unit.
2.00
1.50
Solution Substituting the given equation for C produces
1.00
C
0.50
x
1000
2000
3000
C 800 0.04x 0.0002x 2 800
0.04 0.0002x.
x
x
x
Setting the derivative d Cdx equal to 0 yields
4000
dC
800
2 0.0002 0
dx
x
Number of units
Minimum average cost occurs when
dC
0.
dx
x2 800
4,000,000 ⇒ x 2000 units.
0.0002
See Figure C.7.
Figure C.7
Exercises for Appendix C
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
1. Think About It The figure shows the cost C of producing x
units of a product.
(a) What is C0 called?
3. R 900x 0.1x 2
(b) Sketch a graph of the marginal cost function.
(c) Does the marginal cost function have an extremum? If so,
describe what it means in economic terms.
C
C
4. R 600x 2 0.02x 3
5. R 1,000,000x
0.02x 2 1800
6. R 30x 23 2x
C
C
In Exercises 3–6, find the number of units x that produces a
maximum revenue R.
R
In Exercises 7–10, find the number of units x that produces the
minimum average cost per unit C.
7. C 0.125x 2 20x 5000
8. C 0.001x 3 5x 250
C(0)
x
Figure for 1
x
Figure for 2
2. Think About It The figure shows the cost C and revenue R for
producing and selling x units of a product.
9. C 3000x x 2300 x
10. C 2x 3 x 2 5000x
x 2 2500
(a) Sketch a graph of the marginal revenue function.
(b) Sketch a graph of the profit function. Approximate the position of the value of x for which profit is maximum.
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Licensed to: iChapters User
APPENDIX C
In Exercises 11–14, find the price per unit p (in dollars) that
produces the maximum profit P.
Demand Function
Cost Function
11. C 100 30x
p 90 x
12. C 2400x 5200
p 6000 0.4x 2
13. C 4000 40x 0.02x 2
p 50 14. C 35x 2x 1
p 40 x 1
x
100
Average Cost In Exercises 15 and 16, use the cost function to
find the value of x at which the average cost is a minimum. For
that value of x, show that the marginal cost and average cost
are equal.
15. C 2x 2 5x 18
16. C x 3 6x 2 13x
17. Prove that the average cost is a minimum at the value of x
where the average cost equals the marginal cost.
18. Maximum Profit The profit P for a company is
where s is the amount (in hundreds of dollars) spent on advertising. What amount of advertising produces a maximum profit?
19. Numerical, Graphical, and Analytic Analysis The cost per
unit for the production of a radio is $60. The manufacturer
charges $90 per unit for orders of 100 or less. To encourage
large orders, the manufacturer reduces the charge by $0.15 per
radio for each unit ordered in excess of 100 (for example, there
would be a charge of $87 per radio for an order size of 120).
(a) Analytically complete six rows of a table such as the one
below. (The first two rows are shown.)
Price
A27
21. Minimum Cost A power station is on one side of a river that
1
is 2 mile wide, and a factory is 6 miles downstream on the other
side. It costs $12 per foot to run power lines over land and $16
per foot to run them underwater. Find the most economical path
for the transmission line from the power station to the
factory.
22. Maximum Revenue When a wholesaler sold a product at $25
per unit, sales were 800 units per week. After a price increase
of $5, the average number of units sold dropped to 775 per
week. Assume that the demand function is linear, and find the
price that will maximize the total revenue.
23. Minimum Cost The ordering and transportation cost C (in
thousands of dollars) of the components used in manufacturing
a product is
C 100
x
,
200
x
x 30 2
1 ≤ x
where x is the order size (in hundreds). Find the order size that
minimizes the cost. (Hint: Use Newton’s Method or the zero
feature of a graphing utility.)
24. Average Cost A company estimates that the cost C (in
dollars) of producing x units of a product is
P 230 20s 12s 2
x
Business and Economic Applications
Profit
102 90 20.15 10290 20.15 10260 3029.40
104 90 40.15 10490 40.15 10460 3057.60
(b) Use a graphing utility to generate additional rows of the
table. Use the table to estimate the maximum profit. (Hint:
Use the table feature of the graphing utility.)
C 800 0.4x 0.02x 2 0.0001x 3.
Find the production level that minimizes the average cost per
unit. (Hint: Use Newton’s Method or the zero feature of a
graphing utility.)
25. Revenue The revenue R for a company selling x units is
R 900x 0.1x 2.
Use differentials to approximate the change in revenue if sales
increase from x 3000 to x 3100 units.
26. Analytic and Graphical Analysis A manufacturer of fertilizer finds that the national sales of fertilizer roughly follow the
seasonal pattern
F 100,000 1 sin
2 t 60
365
(c) Write the profit P as a function of x.
where F is measured in pounds. Time t is measured in days,
with t 1 corresponding to January 1.
(d) Use calculus to find the critical number of the function in
part (c), and find the required order size.
(a) Use calculus to determine the day of the year when the
maximum amount of fertilizer is sold.
(e) Use a graphing utility to graph the function in part (c) and
verify the maximum profit from the graph.
(b) Use a graphing utility to graph the function and approximate the day of the year when sales are minimum.
20. Maximum Profit A real estate office handles 50 apartment
units. When the rent is $720 per month, all units are occupied.
However, on the average, for each $40 increase in rent, one unit
becomes vacant. Each occupied unit requires an average of $48
per month for service and repairs. What rent should be charged
to obtain a maximum profit?
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A28
CHAPTER 6
Differential Equations
27. Modeling Data The table shows the monthly sales G (in
thousands of gallons) of gasoline at a gas station in 2004. The
time in months is represented by t, with t 1 corresponding to
January.
(a) Use a graphing utility to plot the data.
(b) Find a model of the form S a bt c sin t for the
data. (Hint: Start by finding . Next, use a graphing utility
to find a bt. Finally, approximate c.)
(c) Use a graphing utility to graph the model with the data and
make any adjustments necessary to obtain a better fit.
t
1
2
3
4
5
6
G
8.91
9.18
9.79
9.83
10.37
10.16
(d) Use the model to predict the maximum quarterly sales in
the year 2006.
t
7
8
9
10
11
12
G
10.37
10.81
10.03
9.97
9.85
9.51
30. Think About It Match each graph with the function it best
represents—a demand function, a revenue function, a cost
function, or a profit function. Explain your reasoning. [The
graphs are labeled (a), (b), (c), and (d).]
A model for these data is
(a)
t
0.62 .
G 9.90 0.64 cos
6
40,000
30,000
(a) Use a graphing utility to plot the data and graph the model.
10,000
(b)
20,000
20,000
(b) Use the model to approximate the month when gasoline
sales were greatest.
(c) What factor in the model causes the seasonal variation in
sales of gasoline? What part of the model gives the average
monthly sales of gasoline?
10,000
x
x
2,000
(c)
8,000
2,000
8,000
2,000
8,000
(d)
40,000
30,000
40,000
30,000
20,000
20,000
10,000
(d) Suppose the gas station added the term 0.02t to the model.
What does the inclusion of this term mean? Use this model
to estimate the maximum monthly sales in the year 2008.
28. Airline Revenues The annual revenue R (in millions of
dollars) for an airline for the years 1995–2004 can be
modeled by
40,000
30,000
10,000
x
2,000
x
8,000
R 4.7t 4 193.5t 3 2941.7t 2 19,294.7t 52,012
Elasticity The relative responsiveness of consumers to a change
in the price of an item is called the price elasticity of demand. If
p f x is a differentiable demand function, the price elasticity of demand is
where t 5 corresponds to 1995.
(a) During which year (between 1995 and 2004) was the airline’s revenue the least?
(b) During which year was the revenue the greatest?
(c) Find the revenues for the years in which the revenue was
least and greatest.
(d) Use a graphing utility to confirm your results in parts (a)
and (b).
29. Modeling Data The manager of a department store recorded
the quarterly sales S (in thousands of dollars) of a new seasonal product over a period of 2 years, as shown in the table, where
t is the time in quarters, with t 1 corresponding to the winter
quarter of 2002.
t
1
2
3
4
5
6
7
8
S
7.5
6.2
5.3
7.0
9.1
7.8
6.9
8.6
p/x
.
dp / dx
For a given price, if < 1, the demand is inelastic, and if
> 1, the demand is elastic. In Exercises 31–34, find for
the demand function at the indicated x-value. Is the demand
elastic, inelastic, or neither at the indicated x-value?
31. p 400 3x
x 20
33. p 400 0.5x 2
x 20
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May not be copied, scanned, or duplicated, in whole or in part.
32. p 5 0.03x
x 100
34. p 500
x2
x 23
Licensed to: iChapters User
Index
A
B
Barrow, Isaac (1630–1677), 130
Base(s)
of an exponential function, 120
of a logarithmic function, 120
other than e, derivatives for, 120
Basic differentiation rules for elementary
functions, 138
Basic equation, 394
guidelines for solving, 398
Basic integration rules, 219, 290
Basic limits, 48
Basic types of transformations, 13
Bearing, 556
Bernoulli, James (1654–1705), 510
Bernoulli, John (1667–1748), 392
Bifolium, 132
Binomial series, 496
Bisection method, 66
Boundary point of a region R, 664
Bounded
above, 434
below, 434
monotonic sequence, 434
region R, 712
sequence, 434
Brachistochrone problem, 510
Breteuil, Emilie de (1706–1749), 344
Bullet-nose curve, 124
C
Cardioid, 527, 528
Catenary, 297
Cauchy, Augustin-Louis (1789–1857), 63
Cauchy-Schwarz Inequality, 573
Center
of curvature, 645
of gravity, 348
of a one-dimensional system, 347
of a two-dimensional system, 348
of mass, 347, 348, 349
of a one-dimensional system, 347
of a planar lamina, 349
of a planar lamina of variable
density, 754, 770
of a two-dimensional system, 348
of a power series, 476
Centered at c, 466
Central force field, 791
Centripetal component of acceleration, 635
Centroid, 350
of a simple region, 754
A109
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May not be copied, scanned, or duplicated, in whole or in part.
INDEX
Abel, Niels Henrik (1802–1829), 151
Absolute convergence, 459
Absolute maximum of a function, 157
of two variables, 712
Absolute minimum of a function, 157
of two variables, 712
Absolute value
derivative involving, 119
function, 12
Absolute Value Theorem for sequences, 431
Absolute zero, 62
Absolutely convergent, 459
Acceleration, 108, 624, 646
centripetal component of, 635
tangential and normal components of,
634, 635, 648
vector, 634, 648
Accumulation function, 254
Addition
of ordinates, 295
of vectors
in the plane, 551
in space, 561
Additive Identity Property of Vectors, 552
Additive Interval Property, 243
Additive Inverse Property of Vectors, 552
Agnesi, Maria Gaetana (1718–1799), 190
d'Alembert, Jean Le Rond (1717–1783),
673
Algebraic function(s), 14, 15, 137
derivatives of, 121
Algebraic properties of the cross product,
575
Alternating series, 456
geometric series, 456
harmonic series, 457, 459
remainder, 458
Alternating Series Test, 456
Alternative form
of the derivative, A5
of the directional derivative, 697
of Green's Theorem, 827, 828
Angle
between two nonzero vectors, 567
between two planes, 583
of inclination of a plane, 708
Angular speed, 757
Antiderivative, 217
of f with respect to x, 218
general, 218
representation of, 217
of a vector-valued function, 620
Antidifferentiation, 218
of a composite function, 260
Aphelion, 545
Approximating zeros
bisection method, 66
Intermediate Value Theorem, 65
with Newton's Method, 148
Approximation
linear, 208, 684
tangent line, 208
Arc length, 333, 334
function, 641
parameter, 641, 642
in parametric form, 516
of a polar curve, 535
of a space curve, 640
in the xy-plane, 760
Arccosecant function, 24, 25
Arccosine function, 24, 25
Arccotangent function, 24, 25
Archimedes (287–212 B.C.), 229
Arcsecant function, 24, 25
Arcsine function, 24, 25
series for, 497
Arctangent function, 24, 25
series for, 497
Area
line integral for, 825
of a parametric surface, 834
in polar coordinates, 531
of a rectangle, 229
of a region between two curves, 307
of a region in the plane, 233, 730
of a surface of revolution, 338
in parametric form, 517
in polar coordinates, 536
of the surface S, 760
in the xy-plane, 760
Associative Property of Vector Addition,
552
Astroid, 132
Asymptote(s)
horizontal, 188
slant, 193
vertical, 71, 72, A5
Average rate of change, 3
Average value
of a continuous function over a solid
region Q, 774
of a function on an interval, 252
of a function over a region R, 744
Average velocity, 97
Axis of revolution, 315
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A110
INDEX
Chain Rule, 112, 113, 121, A6
implicit differentiation, 690
one independent variable, 686, A16
and trigonometric functions, 117
two independent variables, 688
Change in x, 81
Change in y, 81
Change of variables, 263
for definite integrals, 266
for double integrals, 783
guidelines for making, 264
to polar form, 747
using a Jacobian, 781
Charles’s Law, 62
Circle, 132, 528
Circle of curvature, 645
Circulation of F around C, 861
Circumscribed rectangle, 231
Cissoid, 132
Classification of conics by eccentricity,
539, A15
Closed
curve, 818
disk, 664
region R, 664
surface, 850
Cobb-Douglas production function, 658
Coefficient, 14
leading, 14
Collinear, 8
Common types of behavior associated with
nonexistence of a limit, 41
Commutative Property
of the dot product, 566
of vector addition, 552
Comparison Test
Direct, 451
Limit, 452
Complete, 434
Completeness, 65
Completing the square, 288
Component of acceleration
centripetal, 635
normal, 635, 648
tangential, 635, 648
Component form of a vector in the plane,
550
Component functions, 609
Components of a vector, 570
along v, 570
in the direction of v, 571
orthogonal to v, 570
in the plane, 550
Composite function, 15
antidifferentiation of, 260
continuity of, 63
limit of, 50, A3
of two variables, 654
continuity of, 669
Composition of functions, 15
Concave downward, 180
Concave upward, 180
Concavity, 180
interpretation, A8
test for, 181
Conditional convergence, 459
Conditionally convergent, 459
Conic(s)
classification by eccentricity, 539, A15
directrix of, 539
eccentricity of, 539
focus of, 539
polar equations of, 540
Connected region, 816
Conservative vector field, 793, 813
independence of path, 816
test for, 794, 796
Constant
force, 343
function, 14
of integration, 218
Multiple Rule, 93, 121
differential form, 211
of proportionality, 360
Rule, 90, 121
term of a polynomial function, 14
Constraint, 720
Continuity
of a composite function, 63
of a composite function of two
variables, 669
differentiability implies, 87, 685
and differentiability of inverse
functions, 134, A7
implies integrability, 240
properties of, 63
of a vector-valued function, 613
Continuous, 58
at c, 48, 58
on the closed interval a, b, 61
everywhere, 58
function of two variables, 668
on an interval, 613
from the left and from the right, 61
on an open interval a, b, 58
in the open region R, 668, 670
at a point, 613
x0 , y0, 668
x0 , y0, z 0, 670
vector field, 790
Continuously differentiable, 333
Contour lines, 656
Converge, 150, 428, 438
Convergence
absolute, 459
conditional, 459
of a geometric series, 440
of improper integral with infinite
discontinuities, 418
of improper integral with infinite
integration limits, 415
interval of, 477, 481
of p-series, 449
of a power series, 477, A14
radius of, 477, 481
of a sequence, 428
of a series, 438
of Taylor series, 493
tests for series
Alternating Series Test, 456
Direct Comparison Test, 451
geometric series, 440
guidelines, 463
Integral Test, 448
Limit Comparison Test, 452
p-series, 449
Ratio Test, 461
Root Test, 462
Convergent series, nth term of, 442
Convex limaçon, 528
Coordinate conversion, 523
cylindrical to rectangular, 601
cylindrical to spherical, 604
rectangular to cylindrical, 601
rectangular to spherical, 604
spherical to cylindrical, 604
spherical to rectangular, 604
Coordinate planes, 559
xy-plane, 559
xz-plane, 559
yz-plane, 559
Coordinate system
cylindrical, 601
polar, 522
spherical, 604
three-dimensional, 559
Coordinates, polar, 522
Cosecant function
derivative of, 106, 121
integral of, 284
inverse of, 24, 25
Cosine function, 12
derivative of, 95, 121
integral of, 284
inverse of, 24, 25
series for, 497
Cotangent function
derivative of, 106, 121
integral of, 284
inverse of, 24, 25
Copyright 2008 Cengage Learning. All Rights Reserved.
May not be copied, scanned, or duplicated, in whole or in part.
Licensed to: iChapters User
INDEX
D
Decomposition of NxDx into partial
fractions, 393
Decreasing function, 170
test for, 170
Definite integral(s), 240
as the area of a region, 241
change of variables, 266
evaluation of a line integral as a, 802
properties of, 244
two special, 243
of a vector-valued function, 620
Degree of a polynomial function, 14
Delta, , 664
-neighborhood, 664
Demand, 8
Density, 349
Density function , 752, 770
Dependent variable, 9
of a function of two variables, 653
Derivative(s)
of algebraic functions, 121
alternative form, A5
for bases other than e, 120, 121
Chain Rule, 112, 113, 121
implicit differentiation, 690
one independent variable, 686, A16
and trigonometric functions, 117
two independent variables, 688
Constant Multiple Rule, 93, 121
Constant Rule, 90, 121
of cosecant function, 106, 121
of cosine function, 95, 121
of cotangent function, 106, 121
Difference Rule, 94, 121
directional, 694, 695, 701
of an exponential function, base a, 120,
121
of a function, 83
General Power Rule, 114, 121
higher-order, 108
of hyperbolic functions, 296
implicit, 127
of an inverse function, 134, A7
of inverse trigonometric functions, 136
involving absolute value, 119
from the left and from the right, 85
of a logarithmic function, base a, 120,
121
of the natural exponential function, 96,
121
of the natural logarithmic function, 118,
121
notation, 83
parametric form, 513
partial, 673
first, 673
Power Rule, 91, 121
of power series, 481
Product Rule, 102, 121
Quotient Rule, 104, 121
of secant function, 106, 121
second, 108
Simple Power Rule, 91, 121
of sine function, 95, 121
Sum Rule, 94, 121
of tangent function, 106, 121
third, 108
of trigonometric functions, 106, 121
of a vector-valued function, 616
properties of, 618
Determinant form of cross product, 574
Difference quotient, 10, 81
Difference Rule, 94, 121
differential form, 211
Difference of two vectors, 551
Differentiability implies continuity, 87, 685
of a function of two variables, 685
Differentiable at x, 83
Differentiable, continuously, 333
Differentiable function
on the closed interval a, b, 85
on an open interval a, b, 83
in a region R, 683
of three variables, 684
of two variables, 683
vector-valued, 616
Differential, 209
function of three variables, 684
function of two variables, 682
of x, 209
of y, 209
Differential equation, 218
general solution of, 218
initial condition, 222
particular solution of, 222
solutions of
general, 218
particular, 222
Differential form, 211
of a line integral, 808
Differential formulas, 211
constant multiple, 211
product, 211
quotient, 211
sum or difference, 211
Differential operator, 796, 798
Laplacian, 799
Differentiation, 83
implicit, 126
Chain Rule, 690
guidelines for, 127
Copyright 2008 Cengage Learning. All Rights Reserved.
May not be copied, scanned, or duplicated, in whole or in part.
INDEX
Coulomb's Law, 344, 791
Critical number(s)
of a function, 159
relative extrema occur only at, 159
Critical point
of a function of two variables, 713
relative extrema occur at, 713
Cross product of two vectors in space, 574
algebraic properties of, 575
determinant form, 574
geometric properties of, 576
torque, 578
Cruciform, 132
Cubic function, 14
Cubing function, 12
Curl of a vector field, 795
and divergence, 798
Curtate cycloid, 512
Curvature, 643
center of, 645
circle of, 645
formulas for, 644, 648
radius of, 645
in rectangular coordinates, 645, 648
related to acceleration and speed, 646
Curve
closed, 818
lateral surface area over, 811
level, 656
orientation of, 800
piecewise smooth, 509, 800
plane, 504, 609
pursuit, 299, 301
rectifiable, 333
rose, 525, 528
simple, 822
smooth, 333, 509, 618, 631
piecewise, 509, 800
space, 609
tangent line to, 632
Cusps, 618
Cycloid, 509
curtate, 512
prolate, 515
Cylinder, 592
directrix of, 592
equations of, 592
generating curve of, 592
right, 592
rulings of, 592
Cylindrical coordinate system, 601
pole of, 601
Cylindrical coordinates
converting to rectangular coordinates,
601
converting to spherical coordinates, 604
Cylindrical surface, 592
A111
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A112
INDEX
involving inverse hyperbolic functions,
300
logarithmic, 131
numerical, 87
partial, 673
of a vector-valued function, 617
Differentiation rules
Chain, 112, 113, 117, 118
Constant, 90, 121
Constant Multiple, 93, 121
cosecant function, 106, 121
cosine function, 95, 121
cotangent function, 106, 121
Difference, 94, 121
for elementary functions, 138
general, 121
General Power, 114, 121
Power, 91, 121
Product, 102, 121
Quotient, 104, 121
secant function, 106, 121
Simple Power, 91, 121
sine function, 95, 121
Sum, 94, 121
summary of, 121
tangent function, 106, 121
Dimpled limaçon, 528
Direct Comparison Test, 451
Direct substitution, 48, 49
Directed line segment, 549
equivalent, 549
initial point of, 549
length of, 549
magnitude of, 549
terminal point of, 549
Direction angles of a vector, 569
Direction cosines of a vector, 569
Direction field, 225
Direction of motion, 623
Direction numbers, 581
Direction vector, 581
Directional derivative, 694, 695
alternative form, 697
of f in the direction of u, 695, 701
of a function of three variables, 701
Directrix
of a conic, 539
of a cylinder, 592
Dirichlet, Peter Gustav (1805–1859), 41
Dirichlet function, 41
Discontinuity, 59
infinite, 415
nonremovable, 59
removable, 59
Disk, 315, 664
closed, 664
method, 316
open, 664
Distance
between a point and a line in space, 587
between a point and a plane, 586
Distance Formula in space, 560
Distributive Property
for the dot product, 566
for vectors, 552
Diverge, 428, 438
Divergence
of improper integral with infinite
discontinuities, 418
of improper integral with infinite
integration limits, 415
of a sequence, 428
of a series, 438
tests for series
Direct Comparison Test, 451
geometric series, 440
guidelines, 463
Integral Test, 448
Limit Comparison Test, 452
nth-Term Test, 442
p-series, 449
Ratio Test, 461
Root Test, 462
of a vector field, 797
and curl, 798
Divergence-free vector field, 797
Divergence Theorem, 828, 850
Divide out common factors, 52
Domain
of a function, 9
of two variables, 653
of a vector-valued function, 610
Dot product
Commutative Property of, 566
Distributive Property for, 566
form of work, 572
projection using the, 571
properties of, 566
of vectors, 566
Double integral, 735, 736, 737
change of variables for, 783
of f over R, 737
properties of, 737
Doyle Log Rule, 663
Dummy variable, 242
Dyne, 343
E
e, the number, 33
Eccentricity, 539
classification of conics by, 539, A15
Electric force field, 791
Elementary function(s), 14, 137
basic differentiation rules for, 138
power series for, 497
Eliminating the parameter, 506
Ellipse, 539
rotated, 132
Ellipsoid, 593, 594
Elliptic cone, 593, 595
Elliptic paraboloid, 593, 595
Endpoint extrema, 157
Energy
kinetic, 819
potential, 819
Epicycloid, 512, 516
Epsilon-delta, -, 42
definition of limit, 42
Equal vectors
in the plane, 550
in space, 561
Equality of mixed partial derivatives, 678
Equation of a plane in space
general form, 582
standard form, 582
Equation(s)
basic, 394
of cylinders, 592
guidelines for solving, 398
harmonic, 799
Laplace’s, 799
of a line
general form, 5
horizontal, 5
point-slope form, 2, 5
slope-intercept form, 4, 5
in space, parametric, 581
in space, symmetric, 581
summary, 5
vertical, 5
parametric, 504, 830
primary, 197, 198
related-rate, 140
secondary, 198
of tangent plane, 705
Equilibrium, 347
Equipotential
lines, 656
Equivalent
conditions, 818
directed line segments, 549
Error
in approximating a Taylor polynomial,
472
in measurement, 210
percent error, 210
propagated error, 210
relative error, 210
in Simpson’s Rule, 277
in Trapezoidal Rule, 277
Euler, Leonhard (1707–1783), 14
Evaluate a function, 9
Evaluating a flux integral, 845
Evaluating a surface integral, 839
Copyright 2008 Cengage Learning. All Rights Reserved.
May not be copied, scanned, or duplicated, in whole or in part.
Licensed to: iChapters User
INDEX
F
Factorial, 430
Family of functions, 240
Famous curves
astroid, 132
bifolium, 132
bullet-nose curve, 124
circle, 132, 528
cissoid, 132
cruciform, 132
folium of Descartes, 132, 538
kappa curve, 130, 132
lemniscate, 129, 132, 528
parabola, 132
rotated ellipse, 132
rotated hyperbola, 132
serpentine, 110
top half of circle, 124
witch of Agnesi, 110, 132, 190
Faraday, Michael (1791–1867), 819
Fermat, Pierre de (1601–1665), 159
Field
central force, 791
electric force, 791
force, 790
gravitational, 791
inverse square, 791
vector, 790
over Q, 790
over R, 790
velocity, 790, 791
Finite Fourier series, 383
First Derivative Test, 172
First moments, 756, 770
First partial derivatives, 673
notation for, 674
Fixed plane, 650
Fluid force, 353, 354
Fluid pressure, 353
Flux integral
evaluating, 845
of F across S, 845
Focus
of a conic, 539
Folium of Descartes, 132, 538
Force, 343
constant, 343
exerted by a fluid, 353, 354
of friction, 647
resultant, 555
variable, 343, 344
Force field, 790
central, 791
electric, 791
work, 805
Form of a convergent power series, 491
Formulas
for curvature, 644, 648
summation, 228, A9
Fourier, Joseph (1768–1830), 485
Fourier Sine Series, 375
Friction, 647
Fubini’s Theorem, 739
for a triple integral, 766
Function, 9
absolute maximum of, 157
absolute minimum of, 157
absolute value, 12
accumulation, 254
addition of, 15
algebraic, 137
antiderivative of, 217
arc length, 333, 334, 641
average value of, 252
Cobb-Douglas production, 658
component, 609
composite, 15
of two variables, 654
composition of, 15
concave downward, 180
concave upward, 180
constant, 14
Copyright 2008 Cengage Learning. All Rights Reserved.
May not be copied, scanned, or duplicated, in whole or in part.
continuous, 58
continuously differentiable, 333
cosine, 12
critical number of, 159
cubic, 14
cubing, 12
decreasing, 170
test for, 170
defined by power series, properties of,
481
density, 752, 770
derivative of, 83
difference of, 15
Dirichlet, 41
domain of, 9
elementary, 14, 137
algebraic, 14, 15
exponential, 14
logarithmic, 14
trigonometric, 14
evaluate, 9
even, 16
explicit form, 9, 126
exponential to base a, 31, 120
extrema of, 157
extreme values of, 157
family of, 240
greatest integer, 60
harmonic, 821
hyperbolic, 294
cosecant, 294
cosine, 294
cotangent, 294
secant, 294
sine, 294
tangent, 294
identity, 12
implicit form, 9
implicitly defined, 126
increasing, 170
test for, 170
inner product of, 383
integrable, 240
inverse, 20
inverse hyperbolic, 298
cosecant, 298
cosine, 298
cotangent, 298
secant, 298
sine, 298
tangent, 298
inverse trigonometric, 24
cosecant, 24
cosine, 24
cotangent, 24
secant, 24
sine, 24
tangent, 24
INDEX
Evaluation by iterated integrals, 766
Evaluation of a line integral as a definite
integral, 802
Even function, 16
integration of, 268
test for, 16
Everywhere continuous, 58
Existence of an inverse function, 22
Existence of a limit, 61
Existence theorem, 65, 157
Expanded about c, 466
Explicit form of a function, 9, 126
Exponential decay, 360
Exponential function, 14, 31
to base a, 31, 120
derivative of, 120, 121
derivative of, 96, 121
properties of, 31
series for, 497
Exponential growth, 360
Exponential growth and decay model, 360
initial value, 360
proportionality constant, 360
Exponents, properties of, 31
Extended Mean Value Theorem, 406, A11
Extrema
endpoint, 157
of a function, 157
guidelines for finding, 160
relative, 158
Extreme Value Theorem, 157, 712
Extreme values of a function, 157
A113
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A114
INDEX
limit of, 38
linear, 14
logarithmic to base a, 120
logistic, 191
natural exponential, 33
natural logarithmic, 33, 34
notation, 9
odd, 16
one-to-one, 11
onto, 11
orthogonal, 383
point of inflection, 182, 183
polynomial, 14, 49
of two variables, 654
position, 97, 628
potential, 793
product of, 15
quadratic, 14
quotient of, 15
radius, 597
range of, 9
rational, 12, 15
of two variables, 654
real-valued, 9
relative extrema of, 159
relative maximum of, 158
relative minimum of, 158
Riemann zeta, 455
signum, 69
sine, 12
square root, 12
squaring, 12
step, 60
strictly monotonic, 171
of three variables
continuity of, 670
directional derivative, 701
gradient of, 701
transcendental, 15, 137
transformation of a graph of, 13
horizontal shift, 13
reflection about origin, 13
reflection about x-axis, 13
reflection about y-axis, 13
reflection in the line y x, 21
vertical shift, 13
of two variables, 653
absolute maximum of, 712
absolute minimum of, 712
continuity of, 668
critical point of, 713
dependent variable, 653
differentiability implies continuity,
685
differentiable, 683
differential of, 682
domain of, 653
gradient of, 696
graph of, 655
independent variables, 653
limit of, 665
maximum of, 712
minimum of, 712
nonremovable discontinuity of, 668
partial derivative of, 673
range of, 653
relative extrema of, 712
relative maximum of, 712, 715
relative minimum of, 712, 715
removable discontinuity of, 668
total differential of, 682
vector-valued, 609
Vertical Line Test, 12
of x and y, 653
zero of, 16
Functions that agree at all but one point,
51, A4
Fundamental Theorem
of Algebra, 850
of Calculus, 248
guidelines for using, 249
Second, 255
of Line Integrals, 813, 814
G
Gabriel’s Horn, 421, 832
Galilei, Galileo (1564–1642), 137
Galois, Evariste (1811–1832), 151
Gauss, Carl Friedrich (1777–1855), 850
Gauss’s Law, 847
Gauss’s Theorem, 850
General antiderivative, 218
General differentiation rules, 121
General form
of the equation of a line, 5
of the equation of a plane in space, 582
General harmonic series, 449
General Power Rule
for differentiation, 114, 121
for integration, 265
General solution of a differential equation,
218
Generating curve of a cylinder, 592
Geometric properties of the cross product,
576
Geometric property of triple scalar
product, 579
Geometric series, 440
alternating, 456
convergence of, 440
Gibbs, Josiah Willard (1839–1903), 800
Golden ratio, 437
Grad, 696
Gradient, 790, 793
of a function of three variables, 701
of a function of two variables, 696
normal to level curves, 700
normal to level surfaces, 709
properties of, 698
recovering a function from, 797
Graph(s)
of absolute value function, 12
of basic functions, 12
of cosine function, 12
of cubing function, 12
of a function
transformation of, 13
of two variables, 655
of identity function, 12
intercept of, 4
of parametric equations, 504
of rational function, 12
of sine function, 12
of square root function, 12
of squaring function, 12
Gravitational field, 791
Greatest integer function, 60
Green, George (1793–1841), 823
Green’s Theorem, 822
alternative forms, 827, 828
Gregory, James (1638–1675), 481
Guidelines
for evaluating integrals involving secant
and tangent, 379
for evaluating integrals involving sine
and cosine, 376
for finding extrema on a closed interval,
160
for finding intervals on which a function
is increasing or decreasing, 171
for finding an inverse function, 22
for finding limits at infinity of rational
functions, 190
for finding a Taylor series, 495
for implicit differentiation, 127
for integration, 282
for integration by parts, 368
for making a change of variables, 264
for solving applied minimum and
maximum problems, 198
for solving the basic equation, 398
for solving related-rate problems, 141
for testing a series for convergence or
divergence, 463
for using the Fundamental Theorem of
Calculus, 249
Gyration, radius of, 757
H
Half-life, 361
Hamilton, Isaac William Rowan
(1805–1865), 551
Harmonic equation, 799
Harmonic function, 821
Copyright 2008 Cengage Learning. All Rights Reserved.
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Licensed to: iChapters User
INDEX
I
Identities, hyperbolic, 295, 296
Identity function, 12
Image of x under f, 9
Implicit derivative, 127
Implicit differentiation, 126, 690
Chain Rule, 690
guidelines for, 127
Implicit form of a function, 9
Implicitly defined function, 126
Improper integral, 415
convergence of, 418
divergence of, 418
with infinite discontinuities, 418
with infinite integration limits, 415
special type, 421
Inclination, angle of, 708
Incompressible, 797, 855
Increasing function, 170
test for, 170
Increment of z, 682
Increments of x and y, 682
Indefinite integral, 218
of a vector-valued function, 620
Indefinite integration, 218
Independence of path and conservative
vector fields, 816
Independent of path, 816
Independent variable, 9
of a function of two variables, 653
Indeterminate form, 52, 72, 189, 405
Index of summation, 227
Inductive reasoning, 432
Inequality
Cauchy-Schwarz, 573
preservation of, 244, A10
triangle, 554
Inertia
moment of, 756, 770
polar, 756
Infinite discontinuity, 415
Infinite interval, 187
Infinite limit(s), 70
at infinity, 193
from the left and from the right, 70
properties of, 74
Infinite series (or series), 438
alternating, 456
convergence of, 438
divergence of, 438
geometric, 440
harmonic, alternating, 457, 459
nth partial sum, 438
properties of, 442
p-series, 449
sum of, 438
telescoping, 439
terms of, 438
Infinity, limit at, 187, 188, A8
Inflection point, 182, 183
Initial condition, 222
Initial point of a directed line segment, 549
Initial value, 360
Inner partition, 735, 765
polar, 746
Inner product
of two functions, 383
of two vectors, 566
Inner radius of a solid of revolution, 318
Inscribed rectangle, 231
Inside limits of integration, 729
Instantaneous rate of change, 3
Integrability and continuity, 240
Integrable function, 240, 737
Integral(s)
definite, 240
properties of, 244
two special, 243
double, 735, 736, 737
flux, 845
of hyperbolic functions, 296
improper, 415
indefinite, 218
involving inverse trigonometric
functions, 287
involving secant and tangent, guidelines
for evaluating, 379
involving sine and cosine, guidelines
for evaluating, 376
iterated, 729
line, 801
Mean Value Theorem, 251
of px Ax 2 Bx C, 275
single, 737
of the six basic trigonometric functions,
284
surface, 839
triple, 765
Integral Test, 448
Integration
additive interval property, 243
basic rules of, 219, 290
change of variables, 263
constant of, 218
of even and odd functions, 268
guidelines for, 282
indefinite, 218
involving inverse hyperbolic functions,
300
Log Rule, 279
lower limit of, 240
of power series, 481
preservation of inequality, 244, A10
region R of, 729
upper limit of, 240
of a vector-valued function, 620
Integration by parts, 368
guidelines for, 368
summary of common integrals using,
373
tabular method, 373
Integration by tables, 400
Integration formulas
reduction formulas, 402
special, 388
summary of, 862
Integration rules
basic, 290
General Power Rule, 265
Integration techniques
integration by parts, 368
method of partial fractions, 392
substitution for rational functions of
sine and cosine, 403
tables, 400
trigonometric substitution, 384
Intercept(s), 4
x-intercept, 16
y-intercept, 4
Interior point of a region R, 664, 670
Intermediate Value Theorem, 65
Interpretation of concavity, A8
Copyright 2008 Cengage Learning. All Rights Reserved.
May not be copied, scanned, or duplicated, in whole or in part.
INDEX
Harmonic series, 449
alternating, 457, 459
general, 449
Helix, 610
Higher-order derivative, 108
Hooke’s Law, 344
Horizontal asymptote, 188
Horizontal component of a vector, 554
Horizontal line, 14
Horizontal Line Test, 22
Horizontal shift of a graph of a function, 13
to the left, 13
to the right, 13
Horizontally simple region of integration,
730
Huygens, Christian (1629–1695), 333
Hyperbola, 539
rotated, 132
Hyperbolic functions, 294
derivatives of, 296
graph of, addition of ordinates, 295
identities, 295, 296
integrals of, 296
inverse, 298
differentiation involving, 300
integration involving, 300
Hyperbolic identities, 295, 296
Hyperbolic paraboloid, 593, 595
Hyperboloid
of one sheet, 593, 594
of two sheets, 593, 594
A115
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A116
INDEX
Interval
of convergence, 477, 481
infinite, 187
Inverse function, 20
continuity and differentiability of, 134,
A7
derivative of, 134, A7
existence of, 22
guidelines for finding, 22
Horizontal Line Test, 22
reflective property of, 21
Inverse hyperbolic functions, 298
differentiation involving, 300
graphs of, 299
integration involving, 300
Inverse square field, 791
Inverse trigonometric functions, 24
derivatives of, 136
graphs of, 25
integrals involving, 287
properties of, 26
Irrotational vector field, 795
Isobars, 656
Isothermal surface, 659
Isotherms, 656
Iterated integral, 729
evaluation by, 766
inside limits of integration, 729
outside limits of integration, 729
Iteration, 148
ith term of a sum, 227
J
Jacobi, Carl Gustav (1804–1851), 781
Jacobian, 781
K
Kappa curve, 130, 132
Kepler, Johannes (1571–1630), 542
Kepler’s Laws, 542
Kinetic energy, 819
Kovalevsky, Sonya (1850–1891), 664
L
Lagrange, Joseph-Louis (1736–1813), 166,
721
Lagrange form of the remainder, 472
Lagrange multiplier, 720, 721
Lagrange’s Theorem, 721
Lambert, Johann Heinrich (1728–1777),
294
Lamina, planar, 349
Laplace, Pierre Simon de (1749–1827), 775
Laplace’s equation, 799
Laplacian, 799
Lateral surface area over a curve, 811
Law of Conservation of Energy, 819
Leading coefficient
of a polynomial function, 14
test, 14
Least upper bound, 434
Left-handed orientation, 559
Leibniz, Gottfried Wilhelm (1646–1716),
211
Leibniz notation, 211
Lemniscate, 129, 132, 528
Length
of an arc, 333, 334
of a directed line segment, 549
of the moment arm, 346
of a scalar multiple, 553
of a vector in the plane, 550
of a vector in space, 561
on x-axis, 760
Level curve, 656
gradient is normal to, 700
Level surface, 658
gradient is normal to, 709
L’Hôpital, Guillaume (1661–1704), 406
L’Hôpital’s Rule, 406, A12
Limaçon, 528
convex, 528
dimpled, 528
with inner loop, 528
Limit(s), 38, 39
basic, 48
of a composite function, 50, A3
definition of, 42
- definition of, 42
evaluating
direct substitution, 48, 49
divide out common factors, 52
rationalize the numerator, 52
existence of, 61
of a function involving a radical, 49, A3
of a function of two variables, 665
indeterminate form, 52, 72
infinite, 70
from the left and from the right, 70
properties of, 74
at infinity, 187, 188, A8
infinite, 193
of a rational function, guidelines for
finding, 190
of integration
inside, 729
lower, 240
outside, 729
upper, 240
involving e, 33, 54
from the left, 60
of the lower and upper sums, 233
nonexistence of, common types of
behavior, 41
of nth term of a convergent series, 442
one-sided, 60
from the left, 60
from the right, 60
of polynomial and rational functions, 49
properties of, 48, A1
from the right, 60
of a sequence, 428
properties of, 429
strategy for finding, 51
three special, 54
of transcendental functions, 50
of trigonometric functions, 50
of a vector-valued function, 612
Limit Comparison Test, 452
Line(s)
contour, 656
equation of
general form, 5
horizontal, 5
point-slope form, 2, 5
slope-intercept form, 4, 5
summary, 5
vertical, 5
equipotential, 656
moment about, 346
normal, 704, 705
at a point, 133
parallel, 5
perpendicular, 5
secant, 81
slope of, 1
in space
direction number of, 581
direction vector of, 581
parametric equations of, 581
symmetric equations of, 581
tangent, 81
with slope m, 81
vertical, 83
Line of impact, 704
Line integral, 801
for area, 825
differential form of, 808
evaluation of as a definite integral, 802
of f along C, 801
independent of path, 816
summary of, 848
of a vector field, 805
Line segment, directed, 549
Linear approximation, 208, 684
Linear combination, 554
Linear function, 14
Log Rule for Integration, 279
Logarithmic differentiation, 131
Logarithmic function, 14
to base a, 120
natural, 33, 34
derivative of, 118
properties of, 34
Copyright 2008 Cengage Learning. All Rights Reserved.
May not be copied, scanned, or duplicated, in whole or in part.
Licensed to: iChapters User
INDEX
of inertia, 756, 770
polar, 756
for a space curve, 812
of mass, 754
of a one-dimensional system, 347
of a planar lamina, 349
second, 756, 770
of a two-dimensional system, 348
Monotonic sequence, 433
bounded, 434
M
Macintyre, Sheila Scott (1910–1960), 376
Maclaurin, Colin (1698–1746), 491
Maclaurin polynomial, 468
Maclaurin series, 492
Magnitude
of a directed line segment, 549
of a vector in the plane, 550
Mass, 346, 845
center of, 347, 348, 349
of a planar lamina of variable
density, 754, 770
two-dimensional system, 348
moments of, 754
of a planar lamina of variable density,
752
Maximum
absolute, 157
of f on I, 157
of a function of two variables, 712
relative, 158
Mean Value Theorem, 166
Extended, 406, A11
for Integrals, 251
Method of Lagrange Multipliers, 720, 721
Method of partial fractions, 392
basic equation, 394
guidelines for solving, 398
Midpoint Rule, 237
in space, 560
Minimum
absolute, 157
of f on I, 157
of a function of two variables, 712
relative, 158
Mixed partial derivatives, 677
equality of, 678
Möbius Strip, 838
Moment(s)
about a line, 346
about the origin, 346, 347
about a point, 346
about the x-axis, 348
about the x- and y-axes, 349
about the y-axis, 348
arm, length of, 346
first, 756, 770
of a force about a point, 578
N
n factorial, 430
Napier, John (1550–1617), 119
Natural exponential function, 33
derivative of, 96, 121
series for, 497
Natural logarithmic function, 33, 34
derivative of, 118
graph of, 34
properties of, 34
series for, 497
Negative of a vector, 551
Newton, Isaac (1642–1727), 81
Newton’s Law of Cooling, 363
Newton’s Law of Gravitation, 791
Newton’s Law of Universal Gravitation,
344
Newton’s Method, 148
for approximating the zeros of a
function, 148
convergence of, 150
iteration, 148
Newton’s Second Law of Motion, 627
Nodes, 618
Noether, Emmy (1882–1935), 553
Nonexistence of a limit, common types of
behavior, 41
Nonremovable discontinuity, 59
of a function of two variables, 668
Norm
of a partition, 239, 735, 746, 765
polar, 746
of a vector in the plane, 550
Normal component
of acceleration, 634, 635, 648
of a vector field, 845
Normal line, 704, 705
at a point, 133
to S at P, 705
Normal vectors, 568
principal unit, 632, 648
to a smooth parametric surface, 833
Normalization of v, 553
Notation
derivative, 83
for first partial derivatives, 674
function, 9
Leibniz, 211
sigma, 227
nth Maclaurin polynomial for f at c, 468
nth partial sum, 438
nth Taylor polynomial for f at c, 468
nth term
of a convergent series, 442
of a sequence, 427
nth-Term Test for Divergence, 442
Number, critical, 159
Number e, 33
limit involving, 33, 54
Numerical differentiation, 87
O
Octants, 559
Odd function, 16
integration of, 268
test for, 16
Ohm’s Law, 213
One-dimensional system
center of mass of, 347
moment of, 347
One-sided limit, 60
One-to-one function, 11
Onto function, 11
Open disk, 664
Open interval
continuous on, 58
differentiable on, 83
Open region R, 664, 670
continuous in, 668, 670
Open sphere, 670
Operations with power series, 487
Orientable surface, 844
Orientation
of a curve, 800
of a plane curve, 505
of a space curve, 609
Oriented surface, 844
Origin
moment about, 346, 347
of a polar coordinate system, 522
reflection about, 13
Orthogonal, 383
graphs, 133
vectors, 568
Outer radius of a solid of revolution, 318
Outside limits of integration, 729
P
Pappus
Second Theorem of, 357
Theorem of, 352
Parabola, 132, 539
Parabolic spandrel, 357
Copyright 2008 Cengage Learning. All Rights Reserved.
May not be copied, scanned, or duplicated, in whole or in part.
INDEX
Logarithmic properties, 35
Logarithmic spiral, 538
Logistic function, 191
Lower bound of a sequence, 434
Lower bound of summation, 227
Lower limit of integration, 240
Lower sum, 231
limit of, 233
A117
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A118
INDEX
Parallel
lines, 5
planes, 583
vectors, 562
Parameter, 504
arc length, 641, 642
eliminating, 506
Parametric equations, 504
graph of, 504
of a line in space, 581
for a surface, 830
Parametric form
of arc length, 516
of area of a surface of revolution, 517
of the derivative, 513
Parametric surface, 830
area of, 834
equations for, 830
partial derivatives of, 833
smooth, 833
normal vector to, 833
surface area of, 834
Partial derivative(s), 673
equality of mixed, 678
first, 673
of a function of two variables, 673
mixed, 677
notation for, 674
of a parametric surface, 833
of r, 833
Partial differentiation, 673
Partial fractions, 392
decomposition of NxDx into, 393
method of, 392
Partial sums, sequence of, 438
Particular solution of a differential
equation, 222
Partition
inner, 735, 765
polar, 746
norm of, 239, 735, 765
polar, 746
regular, 239
Pascal, Blaise (1623–1662), 353
Pascal’s Principle, 353
Path, 665, 800
Percent error, 210
Perihelion, 545
Perpendicular
lines, 5
planes, 583
vectors, 568
Piecewise smooth curve, 509, 800
Planar lamina, 349
center of mass of, 349
moment of, 349
Plane
angle of inclination of, 708
distance between a point and, 586
region, simply connected, 822
tangent, 705
equation of, 705
vector in, 549
Plane curve, 504, 609
orientation of, 505
smooth, 800
Plane in space
angle between two, 583
equation of
general form, 582
standard form, 582
parallel, 583
to the axis, 585
to the coordinate plane, 585
perpendicular, 583
trace of, 585
Point
of diminishing returns, 207
of inflection, 182, 183
moment about, 346
in a vector field
incompressible, 855
sink, 855
source, 855
Point-slope equation of a line, 2, 5
Polar axis, 522
Polar coordinate system, 522
origin of, 522
polar axis of, 522
pole of, 522
Polar coordinates, 522
area in, 531
area of a surface of revolution in, 536
converting to rectangular coordinates,
523
Polar curve, arc length of, 535
Polar equations of conics, 540
Polar form of slope, 526
Polar moment of inertia, 756
Polar sectors, 745
Pole, 522
of cylindrical coordinate system, 601
tangent lines at, 527
Polynomial
Maclaurin, 468
Taylor, 468
Polynomial approximation, 466
centered at c, 466
expanded about c, 466
Polynomial function, 14, 49
constant term of, 14
degree, 14
leading coefficient of, 14
limit of, 49
of two variables, 654
Position function, 97
for a projectile, 628
Potential energy, 819
Potential function for a vector field, 793
Power Rule
for differentiation, 91, 121
for integration, 265
Power series, 476
centered at c, 476
convergence of, 477, A14
convergent form, 491
derivative of, 481
for elementary functions, 497
integration of, 481
interval of convergence of, 477
operations with, 487
properties of functions defined by, 481
interval of convergence of, 481
radius of convergence of, 481
radius of convergence of, 477
Preservation of inequality, 244, A10
Pressure, 353
fluid, 353
Primary equation, 197, 198
Principal unit normal vector, 632, 648
Probability density function, 423
Product Rule, 102, 121
differential form, 211
Projectile, position function for, 628
Projection form of work, 572
Projection of u onto v, 570
using the dot product, 571
Prolate cycloid, 515
Propagated error, 210
Properties
of continuity, 63
of the cross product
algebraic, 575
geometric, 576
of definite integrals, 244
of the derivative of a vector-valued
function, 618
of the dot product, 566
of double integrals, 737
of exponential functions, 32
of exponents, 31
of functions defined by power series,
481
of the gradient, 698
of infinite limits, 74
of infinite series, 442
of inverse trigonometric functions, 26
of limits, 48, A1
of limits of sequences, 429
logarithmic, 35
of the natural logarithmic function, 34
of vector operations, 552
Proportionality constant, 360
p-series, 449
convergence of, 449
harmonic, 449
Pursuit curve, 299, 301
Copyright 2008 Cengage Learning. All Rights Reserved.
May not be copied, scanned, or duplicated, in whole or in part.
Licensed to: iChapters User
INDEX
Q
R
Radial lines, 522
Radical, limit of a function involving a, 49,
A3
Radius
of convergence, 477, 481
of curvature, 645
function, 597
of gyration, 757
Ramanujan, Srinivasa (1887–1920), 488
Range of a function, 9
of two variables, 653
Raphson, Joseph (1648–1715), 148
Rate of change, 3, 676
average, 3
instantaneous, 3
Ratio, 3
Ratio Test, 461
Rational function, 12, 15
guidelines for finding limits at infinity
of, 190
limit of, 49
of two variables, 654
Rationalize the numerator, 52
Real-valued function f of a real variable x,
9
Recovering a function from its gradient,
797
Rectangle
area of, 229
circumscribed, 231
inscribed, 231
representative, 306
Rectangular coordinates
converting to cylindrical coordinates,
601
converting to polar coordinates, 523
converting to spherical coordinates, 604
curvature in, 645, 648
Rectifiable curve, 333
Recursively defined sequence, 427
Remainder
alternating series, 458
of a Taylor polynomial, 472
Lagrange form, 472
Removable discontinuity, 59
of a function of two variables, 668
Representation of antiderivatives, 217
Representative element, 311
disk, 315
rectangle, 306
washer, 318
Resultant force, 555
Resultant vector, 551
Revolution
axis of, 315
solid of, 315
surface of, 337
area of, 338
Riemann, Georg Friedrich Bernhard
(1826–1866), 239
Riemann sum, 239
Riemann zeta function, 455
Right cylinder, 592
Right-handed orientation, 559
Rolle, Michel (1652–1719), 164
Rolle’s Theorem, 164
Root Test, 462
Rose curve, 525, 528
Rotated ellipse, 132
Rotated hyperbola, 132
Rotation of F about N, 861
r-simple region of integration, 747
Rule(s)
basic integration, 219
Midpoint, 237
Simpson’s, 276
Trapezoidal, 274
Rulings of a cylinder, 592
S
Saddle point, 715
Scalar, 549
field, 656
multiple, 551
multiplication, 551, 561
product of two vectors, 566
quantity, 549
Secant function
derivative of, 106, 121
integral of, 284
inverse of, 24
Secant line, 81
Second derivative, 108
Second Derivative Test, 184
Second Fundamental Theorem of Calculus,
255
Second moment, 756, 770
Second Partials Test, 715
Copyright 2008 Cengage Learning. All Rights Reserved.
May not be copied, scanned, or duplicated, in whole or in part.
INDEX
Quadratic function, 14
Quadric surface, 593
ellipsoid, 593, 594
elliptic cone, 593, 595
elliptic paraboloid, 593, 595
hyperbolic paraboloid, 593, 595
hyperboloid of one sheet, 593, 594
hyperboloid of two sheets, 593, 594
standard form of the equations of, 593,
594, 595
Quaternions, 551
Quotient, difference, 10, 81
Quotient Rule, 104, 121
differential form, 211
Reduction formulas, 402
Reflection
about the origin, 13
about the x-axis, 13
about the y-axis, 13
in the line y x, 21
Reflective property of inverse functions, 21
Refraction, 206, 725
Region of integration R, 729
horizontally simple, 730
r-simple, 747
-simple, 747
vertically simple, 730
Region in the plane
area of, 233, 730
between two curves, 307
centroid of, 350
connected, 816
Region R
boundary point of, 664
bounded, 712
closed, 664
differentiable in, 683
interior point of, 664, 670
open, 664, 670
continuous in, 668, 670
simply connected, 822
Regular partition, 239
Related-rate equation, 140
Related-rate problems, guidelines for
solving, 141
Relation, 9
Relationship between divergence and curl,
798
Relative error, 210
Relative extrema
First Derivative Test for, 172
of a function, 158
of two variables, 712
occur only at critical numbers, 159
occur only at critical points, 713
Second Derivative Test for, 184
Second Partials Test for, 715
Relative maximum
at c, f c, 158
First Derivative Test for, 172
of a function, 158
of two variables, 712, 715
Second Derivative Test for, 184
Second Partials Test for, 715
Relative minimum
at c, f c, 158
First Derivative Test for, 172
of a function, 158
of two variables, 712, 715
Second Derivative Test for, 184
Second Partials Test for, 715
A119
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A120
INDEX
Second Theorem of Pappus, 357
Secondary equation, 198
Separation of variables, 359
Sequence, 427
Absolute Value Theorem, 431
bounded, 434
bounded above, 434
bounded below, 434
convergence of, 428
divergence of, 428
least upper bound of, 434
limit of, 428
properties of, 429
lower bound of, 434
monotonic, 433
nth term of, 427
of partial sums, 438
recursively defined, 427
Squeeze Theorem, 430
terms of, 427
upper bound of, 434
Series, 438
absolutely convergent, 459
alternating, 456
binomial, 496
conditionally convergent, 459
convergence of, 438
divergence of, 438
nth-term test for, 442
geometric, 440
alternating, 456
convergence of, 440
guidelines for testing for convergence
or divergence, 463
harmonic, alternating, 457, 459
infinite, 438
properties of, 442
Maclaurin, 492
nth partial sum, 438
nth term of convergent, 442
power, 476
p-series, 449
sum of, 438
Taylor, 491, 492
telescoping, 439
terms of, 438
Serpentine, 110
Shell method, 325, 326
Shift of a graph
horizontal, 13
to the left, 13
to the right, 13
vertical, 13
downward, 13
upward, 13
Sigma notation, 227
index of summation, 227
ith term, 227
lower bound of summation, 227
upper bound of summation, 227
Signum function, 69
Simple curve, 822
Simple Power Rule, 91, 121
Simple solid region, 851
Simply connected plane region, 822
Simpson’s Rule, 276
error in, 277
Sine function, 12
derivative of, 95, 121
integral of, 284
inverse of, 24
series for, 497
Single integral, 737
Sink, 855
Slant asymptote, 193
Slope(s)
field, 225, 269
of the graph of f at x c, 81
of a line, 1
in polar form, 526
of the surface in the x- and y-directions,
674
of a tangent line, 81
Slope-intercept equation of a line, 4, 5
Smooth
curve, 333, 509, 618, 631
on an open interval, 618
piecewise, 509, 800
parametric surface, 833
plane curve, 800
space curve, 800
Snell’s Law of Refraction, 206, 725
Solenoidal, 797
Solid of revolution, 315
inner radius of, 318
outer radius, 318
Solid region, simple, 851
Solution
general of a differential equation, 218
of an equation, by radicals, 151
by radicals, 151
Some basic limits, 48
Somerville, Mary Fairfax (1780–1872),
653
Source, 855
Space curve, 609
arc length of, 640
moment of inertia for, 812
smooth, 800
Special integration formulas, 388
Special polar graphs, 528
Special type of improper integral, 421
Speed, 98, 623, 624, 646, 648
angular, 757
Sphere, 560
open, 670
standard equation of, 560
Spherical coordinate system, 604
converting to cylindrical coordinates,
604
converting to rectangular coordinates,
604
Spiral
of Archimedes, 517, 524, 538
logarithmic, 538
Square root function, 12
Squaring function, 12
Squeeze Theorem, 54, A4
for Sequences, 430
Standard equation of a sphere, 560
Standard form
of the equation of a plane in space, 582
of the equations of quadric surfaces,
593, 594, 595
Standard position of a vector, 550
Standard unit vector, 554
notation, 561
Step function, 60
Stokes, George Gabriel (1819–1903), 858
Stokes’s Theorem, 827, 858
Strategy for finding limits, 51
Strictly monotonic function, 171
Substitution for rational functions of sine
and cosine, 403
Sufficient condition for differentiability,
683, A15
Sum
ith term of, 227
lower, 231
limit of, 233
Riemann, 239
Rule, 94, 121
differential form, 211
of a series, 438
of two vectors, 551
upper, 231
limit of, 233
Summary
of common integrals using integration
by parts, 373
of differentiation rules, 121
of equations of lines, 5
of integration formulas, 862
of line and surface integrals, 848
of velocity, acceleration, and curvature,
648
Summation
formulas, 228, A9
index of, 227
lower bound of, 227
upper bound of, 227
Surface
closed, 850
cylindrical, 592
isothermal, 659
level, 658
Copyright 2008 Cengage Learning. All Rights Reserved.
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Licensed to: iChapters User
INDEX
T
Tables, integration by, 400
Tabular method for integration by parts,
373
Tangent function
derivative of, 106, 121
integral of, 284
inverse of, 24
Tangent line(s), 81
approximation, 208
to a curve, 632
at the pole, 527
problem, 80
slope of, 81
with slope m, 81
vertical, 83
Tangent plane, 705
equation of, 705
to S at P, 705
Tangent vector, 623
Tangential component of acceleration, 634,
635, 648
Tautochrone problem, 510
Taylor, Brook (1685-1731), 468
Taylor polynomial, 468
error in approximating, 472
remainder, Lagrange form of, 472
Taylor series, 491, 492
convergence of, 493
guidelines for finding, 495
Taylor’s Theorem, 472, A13
Telescoping series, 439
Terminal point of a directed line segment,
549
Terms
of a sequence, 427
of a series, 438
Test(s)
for concavity, 181
conservative vector field in the plane,
794
conservative vector field in space, 796
for convergence
Alternating Series Test, 456
Direct Comparison Test, 451
geometric series, 440
guidelines, 463
Integral Test, 448
Limit Comparison Test, 452
p-series, 449
Ratio Test, 461
Root Test, 462
for even and odd functions, 16
for increasing and decreasing functions,
170
Theorem, existence, 157
Theorem of Pappus, 352
second, 357
Theta, simple region of integration, 747
Third derivative, 108
Three-dimensional coordinate system, 559
left-handed orientation, 559
right-handed orientation, 559
Top half of circle, 124
Topographic map, 656
Torque, 348, 578
Total differential, 682
Total mass
of a one-dimensional system, 347
of a two-dimensional system, 348
Trace
of a plane in space, 585
of a surface, 593
Tractrix, 299, 300
Transcendental function, 15, 137
limits of, 50
Transformation, 13, 782
Transformation of a graph of a function, 13
basic types, 13
horizontal shift, 13
reflection about origin, 13
reflection about x-axis, 13
reflection about y-axis, 13
reflection in the line y x, 21
vertical shift, 13
Trapezoidal Rule, 274
error in, 277
Triangle inequality, 554
Trigonometric function(s), 14
cosine, 12
derivative of, 106, 117, 121
integrals of the six basic, 284
inverse, 24
derivatives of, 136
integrals involving, 287
properties of, 26
limit of, 50
sine, 12
Trigonometric substitution, 384
Triple integral, 765
of f over Q, 765
Triple scalar product, 578
geometric property of, 579
Two-dimensional system
center of mass of, 348
moment of, 348
Two special definite integrals, 243
U
Unit tangent vector, 631, 648
Unit vector, 550
in the direction of v
in the plane, 553
in space, 561
standard, 554
Upper bound
least, 434
of a sequence, 434
of summation, 227
Upper limit of integration, 240
Upper sum, 231
limit of, 233
u-substitution, 260
V
Value of f at x, 9
Variable
dependent, 9
dummy, 242
force, 343, 344
independent, 9
Vector(s)
acceleration, 634, 648
addition
associative property of, 552
commutative property of, 552
in the plane, 549
in space, 561
Additive Identity Property, 552
Additive Inverse Property of, 552
angle between two, 567
component
of u along v, 570
of u orthogonal to v, 570
component form of, 550
components, 550, 570
cross product of, 574
difference of two, 551
direction, 581
direction angles of, 569
direction cosines of, 569
Copyright 2008 Cengage Learning. All Rights Reserved.
May not be copied, scanned, or duplicated, in whole or in part.
INDEX
orientable, 844
oriented, 844
parametric, 830
parametric equations for, 830
quadric, 593
trace of, 593
Surface area
of a parametric surface, 834
of a solid, 759, 760
Surface integral, 839
evaluating, 839
of f over S, 839
summary of, 848
Surface of revolution, 337, 597, 598
area of, 338
parametric form, 517
polar form, 536
Symmetric equations of a line in space,
581
A121
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A122
INDEX
Distributive Property, 552
dot product of, 566
equal, 550, 561
horizontal component of, 554
initial point, 549
inner product of, 566
length of, 550, 561
linear combination of, 554
magnitude of, 550
negative of, 551
norm of, 550
normal, 568
normalization of, 553
operations, properties of, 552
orthogonal, 568
parallel, 562
perpendicular, 568
in the plane, 549
principal unit normal, 632, 648
product of two vectors in space, 574
projection of, 570
resultant, 551
scalar multiplication, 551
scalar product of, 566
in space, 561
space, 553
axioms, 553
standard position, 550
standard unit notation, 561
sum, 551
tangent, 623
terminal point, 549
triple scalar product, 578
unit, 550
in the direction of v, 553, 561
standard, 554
unit tangent, 631, 648
velocity, 623, 648
vertical component of, 554
zero, 550, 561
Vector field, 790
circulation of, 861
conservative, 793, 813
continuous, 790
curl of, 795
divergence of, 797
divergence-free, 797
incompressible, 855
irrotational, 795
line integral of, 805
normal component of, 845
over Q, 790
over R, 790
potential function for, 793
rotation of, 861
sink, 855
source, 855
solenoidal, 797
test for, 794, 796
Vector-valued function(s), 609
antiderivative of, 620
continuity of, 613
continuous on an interval, 613
continuous at a point, 613
definite integral of, 620
derivative of, 616
properties of, 618
differentiation, 617
domain of, 610
indefinite integral of, 620
integration of, 620
limit of, 612
Velocity, 98, 624
average, 97
field, 790, 791
incompressible, 797
vector, 623, 648
Vertéré, 190
Vertical asymptote, 71, 72, A5
Vertical component of a vector, 554
Vertical line, 5
Vertical Line Test, 12
Vertical shift of a graph of a function, 13
downward, 13
upward, 13
Vertical tangent line, 83
Vertically simple region of integration, 730
Volume of a solid
disk method, 316
with known cross sections, 320
shell method, 325, 326
washer method, 318
Volume of a solid region, 737, 765
W
Wallis, John (1616–1703), 378
Wallis’s Formulas, 378
Washer, 318
Washer method, 318
Weierstrass, Karl (1815–1897), 713
Witch of Agnesi, 110, 132, 190
Work, 572
done by a constant force, 343
done by a variable force, 343, 344
dot product form, 572
force field, 805
projection form, 572
X
x-axis
moment about, 348
reflection about, 13
x-intercept, 16
xy-plane, 559
xz-plane, 559
Y
y-axis
moment about, 348
reflection about, 13
y-intercept, 4
yz-plane, 559
Z
Zero factorial, 430
Zero of a function, 16
approximating
bisection method, 66
Intermediate Value Theorem, 65
with Newton’s Method, 148
Zero vector
in the plane, 550
in space, 561
Copyright 2008 Cengage Learning. All Rights Reserved.
May not be copied, scanned, or duplicated, in whole or in part.
Licensed to: iChapters User
DERIVATIVES AND INTEGRALS
Basic Differentiation Rules
1.
4.
7.
10.
13.
16.
19.
22.
25.
28.
31.
34.
d
cu cu
dx
d u
vu uv
dx v
v2
d
x 1
dx
d u
e eu u
dx
d
sin u cos uu
dx
d
cot u csc2 uu
dx
u
d
arcsin u dx
1 u2
d
u
arccot u dx
1 u2
d
sinh u cosh uu
dx
d
coth u csch2 uu
dx
d
u
sinh1 u dx
u2 1
d
u
coth1 u dx
1 u2
2.
5.
8.
11.
14.
17.
20.
23.
26.
29.
32.
35.
d
u ± v u ± v
dx
d
c 0
dx
d
u
u
u , u 0
dx
u
d
u
loga u dx
ln au
d
cos u sin uu
dx
d
sec u sec u tan uu
dx
d
u
arccos u dx
1 u2
d
u
arcsec u dx
u u2 1
d
cosh u sinh uu
dx
d
sech u sech u tanh uu
dx
d
u
cosh1 u dx
u2 1
d
u
sech1 u dx
u1 u2
3.
5.
7.
9.
11.
13.
15.
17.
kf u du k f u du
2.
du u C
4.
eu du eu C
6.
cos u du sin u C
8.
cot u du ln sin u C
10.
6.
9.
Basic Integration Formulas
1.
3.
csc u du ln csc u cot u C
12.
csc2 u du cot u C
14.
csc u cot u du csc u C
16.
du
1
u
arctan C
a 2 u2 a
a
18.
12.
15.
18.
21.
24.
27.
30.
33.
36.
d
uv uv vu
dx
d n
u nu n1u
dx
d
u
ln u dx
u
d u
a ln aau u
dx
d
tan u sec2 uu
dx
d
csc u csc u cot uu
dx
u
d
arctan u dx
1 u2
d
u
arccsc u dx
u u2 1
d
tanh u sech2 uu
dx
d
csch u csch u coth uu
dx
d
u
tanh1 u dx
1 u2
d
u
csch1 u dx
u 1 u2
f u ± gu du au du ln1aa
u
f u du ±
C
sin u du cos u C
tan u du ln cos u C
sec u du ln sec u tan u C
sec2 u du tan u C
sec u tan u du sec u C
du
u
C
a
du
u
1
arcsec
C
a
uu2 a2 a
a2 u2
arcsin
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May not be copied, scanned, or duplicated, in whole or in part.
gu du
Licensed to: iChapters User
TRIGONOMETRY
Definition of the Six Trigonometric Functions
Opposite
Right triangle definitions, where 0 < < 2.
opp
hyp
sin csc e
s
u
hyp
opp
en
pot
Hy
adj
hyp
cos sec θ
hyp
adj
Adjacent
opp
adj
tan cot adj
opp
Circular function definitions, where is any angle.
y
r
y
sin csc r = x2 + y2
r
y
(x, y)
x
r
r
cos sec θ
y
r
x
x
y
x
x
cot tan x
y
Reciprocal Identities
1
sin x csc x
1
csc x sin x
sin x
cos x
cot x 1
tan x cot x
1
cot x tan x
cos x
sin x
(−1, 0) π 180°
210°
330°
sin 2u 2 sin u cos u
cos 2u cos2 u sin2 u 2 cos2 u 1 1 2 sin2 u
2 tan u
tan 2u 1 tan2 u
Power-Reducing Formulas
1 cos 2u
2
1 cos 2u
2
cos u 2
1
cos
2u
tan2 u 1 cos 2u
1 cot2 x csc2 x
Sum-to-Product Formulas
2 x cos x
csc
x sec x
2
sec
x csc x
2
sin u sin v 2 sin
2 x sin x
tan
x cot x
2
cot
x tan x
2
cos
Reduction Formulas
sinx sin x
cscx csc x
secx sec x
x
(− 23 , − 12) 76π 5π 225°240° 300°315°7π 116π ( 23 , − 21)
(− 22 , − 22 ) 4 43π 270° 32π 53π 4 ( 22 , − 22 )
1
3
(0, −1) ( 2 , − 2 )
(− 12 , − 23 )
Cofunction Identities
sin
0° 0
360° 2π (1, 0)
sin2 u Pythagorean Identities
sin2 x cos2 x 1
1 tan2 x sec2 x
(− 12 , 23 ) π (0, 1) ( 12 , 23 )
90°
(− 22 , 22 ) 3π 23π 2 π3 π ( 22 , 22 )
120°
60°
4 π
45°
( 23 , 21)
(− 23 , 12) 56π 4150°135°
6
30°
Double -Angle Formulas
1
sec x cos x
1
cos x sec x
Tangent and Cotangent Identities
tan x y
cosx cos x
tanx tan x
cotx cot x
Sum and Difference Formulas
sinu ± v sin u cos v ± cos u sin v
cosu ± v cos u cos v sin u sin v
tan u ± tan v
tanu ± v 1 tan u tan v
u 2 v cos
u 2 v
uv
uv
sin u sin v 2 cos
sin
2 2 uv
uv
cos u cos v 2 cos
cos
2 2 uv
uv
cos u cos v 2 sin
sin
2 2 Product-to-Sum Formulas
1
sin u sin v cosu v cosu v
2
1
cos u cos v cosu v cosu v
2
1
sin u cos v sinu v sinu v
2
1
cos u sin v sinu v sinu v
2
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May not be copied, scanned, or duplicated, in whole or in part.
Licensed to: iChapters User
ALGEBRA
Factors and Zeros of Polynomials
Let px an x n an1x n1 . . . a1x a0 be a polynomial. If pa 0, then a is a zero of the
polynomial and a solution of the equation px 0. Furthermore, x a is a factor of the polynomial.
Fundamental Theorem of Algebra
An nth degree polynomial has n (not necessarily distinct) zeros. Although all of these
zeros may be imaginary, a real polynomial of odd degree must have at least one real zero.
Quadratic Formula
If px ax 2 bx c, and 0 ≤ b2 4ac, then the real zeros of p are x b ± b2 4ac2a.
Special Factors
x 2 a 2 x ax a
x 3 a 3 x ax 2 ax a 2
x 3 a3 x ax 2 ax a 2
x 4 a 4 x 2 a 2x 2 a 2
Binomial Theorem
x y2 x 2 2xy y 2
x y2 x 2 2xy y 2
x y3 x 3 3x 2y 3xy 2 y 3
x y3 x 3 3x 2y 3xy 2 y 3
x y4 x 4 4x 3y 6x 2y 2 4xy3 y 4
x y4 x 4 4x 3y 6x 2y 2 4xy 3 y 4
nn 1 n2 2 . . .
y nxy n1 y n
x
2!
nn 1 n2 2 . . .
x yn x n nx n1y y ± nxy n1 y n
x
2!
x yn x n nx n1y Rational Zero Theorem
If px an x n a n1x n1 . . . a1x a0 has integer coefficients, then every
rational zero of p is of the form x rs, where r is a factor of a0 and s is a factor of an.
Factoring by Grouping
acx 3 adx 2 bcx bd ax 2cx d bcx d ax 2 bcx d
Arithmetic Operations
ab ac ab c
ab a d ad
c
d b c bc
b
ab
a c
c
a
c
ad bc
b d
bd
a
b
a
c
bc
ab a b
c
c
c
ab ba
cd
dc
ab ac
bc
a
a
ac
b
b
c
Exponents and Radicals
a0 1, a 0
ab x a xb x
a xa y a xy
n am amn
ax a
b
x
ax
bx
1
ax
a a12
ax
a xy
ay
n a a1n
n a
n b
n ab axy a xy
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May not be copied, scanned, or duplicated, in whole or in part.
n
n a
a n
b b
Licensed to: iChapters User
FORMULAS FROM GEOMETRY
Triangle
Sector of Circular Ring
h a sin 1
Area bh
2
(Law of Cosines)
c
p average radius,
w width of ring,
in radians
Area pw
a
θ
h
b
c2 a 2 b2 2ab cos c
(Pythagorean Theorem)
a2
b2
Area ab
a
Circumference 2
b
Equilateral Triangle
h
Area 3s2
s
h
A
Right Circular Cone
Trapezoid
a
h
Area a b
2
h
h
r 2h
3
Lateral Surface Area rr2 h2
h
Volume b
r
Frustum of Right Circular Cone
r
r 2 rR R 2h
3
Lateral Surface Area sR r
s
Volume b
a
h
b
h
Right Circular Cylinder
Circle
Area r 2
Circumference 2 r
Volume r 2h
Lateral Surface Area 2 rh
r
Sector of Circle
R
r
h
Sphere
4
Volume r 3
3
Surface Area 4 r 2
s
θ
r
Circular Ring
p average radius,
w width of ring
Area R 2 r 2
2 pw
a
s
Parallelogram
in radians
r2
Area 2
s r
b
a 2 b2
2
A area of base
Ah
Volume 3
s
h
4
Area bh
w
Cone
3s
2
θ
Ellipse
Right Triangle
c2
p
r
Wedge
r
p
R
w
A area of upper face,
B area of base
A B sec A
θ
B
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May not be copied, scanned, or duplicated, in whole or in part.
A108
Answers to Odd-Numbered Exercises
This page contains answers for this chapter only.
19. (a)
Order
size, x Price
Profit, P
90 20.15 10290 20.15 10260 3029.40
102
104
90 40.15 10490 40.15 10460 3057.60
106
90 60.15 10690 60.15 10660 3084.60
108
90 80.15 10890 80.15 10860 3110.40
110
90 100.15 11090 100.15 11060 3135.00
112
90 120.15 11290 120.15 11260 3158.40
(b)
Order
size, x Price
Profit, P
.
.
.
.
.
.
.
.
.
146
90 460.15 14690 460.15 14660 3372.60
148
90 480.15 14890 480.15 14860 3374.40
150
90 500.15 15090 500.15 15060 3375.00
152
90 520.15 15290 520.15 15260 3374.40
154
90 540.15 15490 540.15 15460 3372.60
.
.
.
.
.
.
.
.
.
Maximum profit: $3375.00
(c) P x90 x 1000.15 x60 45x 0.15x2,
x ≥ 100
(d) 150 units
(e) 4000
(150, 3375)
100
300
0
21. The line should run from the power station to a point across the
river 327 mile downstream.
23. x 40 units
25. $30,000
27. (a) 12
APPENDIX C (page A26)
1. (a) Fixed cost
(b)
0
8
13
(b) July (c) The cosine factor; 9.90
(d) The term 0.02t would mean a steady growth of sales over
time. In this case, the maximum sales in 2008 that is, on
49 ≤ t ≤ 60 would be about 11.6 thousand gallons.
29. (a) 10
(c) 10
(c) Yes; the extremum occurs when production costs are
increasing at their slowest rate.
3. 4500
5. 300
7. 200
9. 200
11. $60
13. $35
15. x 3 17. Proof
0
0
(b) S 6.2 0.25t 1.5 sin
31. 17
3 ; elastic
0
9
2 t
0
(d) $12,000
33. 12; inelastic
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May not be copied, scanned, or duplicated, in whole or in part.
9