Thư viện tri thức trực tuyến
Kho tài liệu với 50,000+ tài liệu học thuật
© 2023 Siêu thị PDF - Kho tài liệu học thuật hàng đầu Việt Nam
Calculus For Business, Economics, and the Social and Life Sciences
Nội dung xem thử
Mô tả chi tiết
ISBN 978-0-07-353231-8
MHID 0-07-353231-2
Part of
ISBN 978-0-07-729273-7
MHID 0-07-729273-1
www.mhhe.com
CALCULUS
For Business,
Economics,
and the Social
and Life Sciences
BRIEF EDITION
CALCULUS
LAURENCE D. HOFFMANN * GERALD L. BRADLEY
Tenth Edition
Tenth
Edition
BRIEF
EDITION
HOFFMANN
BRADLEY
Tools for Success in Calculus
Calculus for Business, Economics, and the Social and Life Sciences, Brief Edition provides a sound, intuitive
understanding of the basic concepts students need as they pursue careers in business, economics, and the life
and social sciences. Students achieve success using this text as a result of the authors’ applied and real-world
orientation to concepts, problem-solving approach, straightforward and concise writing style, and comprehensive
exercise sets.
In addition to the textbook, McGraw-Hill offers the following tools to help you succeed in calculus.
ALEKS®
(Assessment and LEarning in Knowledge Spaces)
www.aleks.com
What is ALEKS?
ALEKS is an intelligent, tutorial-based learning system for mathematics and statistics courses proven to help
students succeed.
ALEKS offers:
completion.
What can ALEKS do for you?
ALEKS Prep:
material.
ALEKS Placement:
preparedness.
Other Tools for Success for Instructors and Students
Resources available on the textbook’s website at www.mhhe.com/hoffmann
to allow for unlimited practice.
MD DALIM #997580 12/02/08 CYAN MAG YEL BLK
Calculus
For Business, Economics, and the Social and Life Sciences
hof32312_fm_i-xvi.qxd 12/4/08 08:21am Page i ntt 201:MHDQ089:mhhof10%0:hof10fm:
hof32312_fm_i-xvi.qxd 12/4/08 08:21am Page ii ntt 201:MHDQ089:mhhof10%0:hof10fm:
Calculus
For Business, Economics, and the Social and Life Sciences
Laurence D. Hoffmann
Smith Barney
Gerald L. Bradley
Claremont McKenna College
BRIEF
Tenth Edition
hof32312_fm_i-xvi.qxd 12/4/08 08:21am Page iii ntt 201:MHDQ089:mhhof10%0:hof10fm:
CALCULUS FOR BUSINESS, ECONOMICS, AND THE SOCIAL AND LIFE SCIENCES, BRIEF EDITION,
TENTH EDITION
Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas,
New York, NY 10020. Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. Previous editions © 2007, 2004, and 2000. No part of this publication may be reproduced or distributed in any form or by any
means, or stored in a database or retrieval system, without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for
distance learning.
Some ancillaries, including electronic and print components, may not be available to customers outside the United
States.
This book is printed on acid-free paper.
1 2 3 4 5 6 7 8 9 0 VNH/VNH 0 9
ISBN 978–0–07–353231–8
MHID 0–07–353231–2
Editorial Director: Stewart K. Mattson
Senior Sponsoring Editor: Elizabeth Covello
Director of Development: Kristine Tibbetts
Developmental Editor: Michelle Driscoll
Marketing Director: Ryan Blankenship
Senior Project Manager: Vicki Krug
Senior Production Supervisor: Kara Kudronowicz
Senior Media Project Manager: Sandra M. Schnee
Designer: Laurie B. Janssen
Cover/Interior Designer: Studio Montage, St. Louis, Missouri
(USE) Cover Image: ©Spike Mafford/Gettyimages
Senior Photo Research Coordinator: Lori Hancock
Supplement Producer: Mary Jane Lampe
Compositor: Aptara®, Inc.
Typeface: 10/12 Times
Printer: R. R. Donnelley, Jefferson City, MO
Chapter Opener One, Two: © Corbis Royalty Free; p. 188(left): © Nigel Cattlin/Photo Researchers, Inc.;
p. 188(right): © Runk/Schoenberger/Grant Heilman; Chapter Opener Three: © Getty Royalty Free; Chapter Opener
Four: © The McGraw-Hill Companies, Inc./Jill Braaten, photographer; p. 368: © Getty Royalty Free; Chapter
Opener Five: © Richard Klune/Corbis; p. 472: © Gage/Custom Medical Stock Photos; Chapter Opener Six:
© AFP/Getty Images; p. 518: © Alamy RF; Chapter Opener Seven(right): US Geological Survey; (left): Maps a la
carte, Inc.; Chapter Opener Eight: © Mug Shots/Corbis; p. 702: © Corbis Royalty Free; Chapter Opener Nine,
p. 755: © Getty Royalty Free; Chapter Opener Ten: © Corbis Royalty Free; p. 829: Courtesy of Zimmer Inc.;
Chapter Opener Eleven, p. 890, Appendix Opener: Getty Royalty Free.
Library of Congress Cataloging-in-Publication Data
Hoffmann, Laurence D., 1943-
Calculus for business, economics, and the social and life sciences — Brief 10th ed. / Laurence D. Hoffmann,
Gerald L. Bradley.
p. cm.
Includes index.
ISBN 978–0–07–353231–8 — ISBN 0–07–353231–2 (hard copy : alk. paper)
1. Calculus—Textbooks. I. Bradley, Gerald L., 1940- II. Title.
QA303.2.H64 2010
515—dc22 2008039622
www.mhhe.com
hof32312_fm_i-xvi.qxd 12/4/08 08:21am Page iv ntt 201:MHDQ089:mhhof10%0:hof10fm:
CONTENTS
Preface vii
CHAPTER 1 Functions, Graphs, and Limits
1.1 Functions 2
1.2 The Graph of a Function 15
1.3 Linear Functions 29
1.4 Functional Models 45
1.5 Limits 63
1.6 One-Sided Limits and Continuity 78
Chapter Summary 90
Important Terms, Symbols, and Formulas 90
Checkup for Chapter 1 90
Review Exercises 91
Explore! Update 96
Think About It 98
CHAPTER 2 Differentiation: Basic Concepts 101
2.1 The Derivative 102
2.2 Techniques of Differentiation 117
2.3 Product and Quotient Rules; Higher-Order Derivatives 129
2.4 The Chain Rule 142
2.5 Marginal Analysis and Approximations Using Increments 156
2.6 Implicit Differentiation and Related Rates 167
Chapter Summary 179
Important Terms, Symbols, and Formulas 179
Checkup for Chapter 2 180
Review Exercises 181
Explore! Update 187
Think About It 189
v
hof32312_fm_i-xvi.qxd 12/4/08 08:21am Page v ntt 201:MHDQ089:mhhof10%0:hof10fm:
CHAPTER 3 Additional Applications of the Derivative
3.1 Increasing and Decreasing Functions; Relative Extrema 192
3.2 Concavity and Points of Inflection 208
3.3 Curve Sketching 225
3.4 Optimization; Elasticity of Demand 240
3.5 Additional Applied Optimization 259
Chapter Summary 277
Important Terms, Symbols, and Formulas 277
Checkup for Chapter 3 278
Review Exercises 279
Explore! Update 285
Think About It 287
CHAPTER 4 Exponential and Logarithmic Functions
4.1 Exponential Functions; Continuous Compounding 292
4.2 Logarithmic Functions 308
4.3 Differentiation of Exponential and Logarithmic Functions 325
4.4 Applications; Exponential Models 340
Chapter Summary 357
Important Terms, Symbols, and Formulas 357
Checkup for Chapter 4 358
Review Exercises 359
Explore! Update 365
Think About It 367
CHAPTER 5 Integration 371
5.1 Antidifferentiation: The Indefinite Integral 372
5.2 Integration by Substitution 385
5.3 The Definite Integral and the Fundamental
Theorem of Calculus 397
5.4 Applying Definite Integration: Area Between
Curves and Average Value 414
5.5 Additional Applications to Business and Economics 432
5.6 Additional Applications to the Life and Social Sciences 445
Chapter Summary 462
Important Terms, Symbols, and Formulas 462
Checkup for Chapter 5 463
Review Exercises 464
Explore! Update 469
Think About It 472
vi CONTENTS
hof32312_fm_i-xvi.qxd 12/4/08 08:21am Page vi ntt 201:MHDQ089:mhhof10%0:hof10fm:
CONTENTS vii
CHAPTER 6 Additional Topics in Integration
6.1 Integration by Parts; Integral Tables 476
6.2 Introduction to Differential Equations 490
6.3 Improper Integrals; Continuous Probability 509
6.4 Numerical Integration 526
Chapter Summary 540
Important Terms, Symbols, and Formulas 540
Checkup for Chapter 6 541
Review Exercises 542
Explore! Update 548
Think About It 551
CHAPTER 7 Calculus of Several Variables
7.1 Functions of Several Variables 558
7.2 Partial Derivatives 573
7.3 Optimizing Functions of Two Variables 588
7.4 The Method of Least-Squares 601
7.5 Constrained Optimization: The Method of Lagrange Multipliers 613
7.6 Double Integrals 624
Chapter Summary 644
Important Terms, Symbols, and Formulas 644
Checkup for Chapter 7 645
Review Exercises 646
Explore! Update 651
Think About It 653
APPENDIX A Algebra Review
A.1 A Brief Review of Algebra 658
A.2 Factoring Polynomials and Solving Systems of Equations 669
A.3 Evaluating Limits with L'Hôpital's Rule 682
A.4 The Summation Notation 687
Appendix Summary 668
Important Terms, Symbols, and Formulas 668
Review Exercises 689
Think About It 692
TABLES I Powers of e 693
II The Natural Logarithm (Base e) 694
TEXT SOLUTIONS Answers to Odd-Numbered Excercises, Chapter Checkup
Exercises, and Odd-Numbered Chapter Review Exercises 695
Index 779
hof32312_fm_i-xvi.qxd 12/4/08 08:21am Page vii ntt 201:MHDQ089:mhhof10%0:hof10fm:
Calculus for Business, Economics, and the Social and Life Sciences, Brief Edition,
provides a sound, intuitive understanding of the basic concepts students need as they
pursue careers in business, economics, and the life and social sciences. Students
achieve success using this text as a result of the author’s applied and real-world orientation to concepts, problem-solving approach, straightforward and concise writing
style, and comprehensive exercise sets. More than 100,000 students worldwide have
studied from this text!
Enhanced Topic Coverage
Every section in the text underwent careful analysis and extensive review to ensure
the most beneficial and clear presentation. Additional steps and definition boxes were
added when necessary for greater clarity and precision, and discussions and introductions were added or rewritten as needed to improve presentation.
Improved Exercise Sets
Almost 300 new routine and application exercises have been added to the already extensive problem sets. A wealth of new applied problems has been added to help demonstrate the practicality of the material. These new problems come from many fields of
study, but in particular more applications focused on economics have been added. Exercise sets have been rearranged so that odd and even routine exercises are paired and the
applied portion of each set begins with business and economics questions.
Just-in-Time Reviews
More Just-in-Time Reviews have been added in the margins to provide students with
brief reminders of important concepts and procedures from college algebra and precalculus without distracting from the material under discussion.
Graphing Calculator Introduction
The Graphing Calculator Introduction can now be found on the book’s website at
www.mhhe.com/hoffmann. This introduction includes instructions regarding common
calculator keystrokes, terminology, and introductions to more advanced calculator
applications that are developed in more detail at appropriate locations in the text.
Appendix A: Algebra Review
The Algebra Review has been heavily revised to include many new examples and figures, as well as over 75 new exercises. The discussions of inequalities and absolute
value now include property lists, and there is new material on factoring and rationalizing expressions, completing the square, and solving systems of equations.
New Design
The Tenth Edition design has been improved with a rich, new color palette; updated
writing and calculator exercises; and Explore! box icons, and all figures have been
revised for a more contemporary and visual aesthetic. The goal of this new design is
to provide a more approachable and student-friendly text.
Chapter-by-Chapter Changes
Chapter-by-chapter changes are available on the book’s website,
www.mhhe.com/hoffmann.
Overview of the
Tenth Edition
PREFACE
viii
Improvements to
This Edition
hof32312_fm_i-xvi.qxd 12/4/08 08:21am Page viii ntt 201:MHDQ089:mhhof10%0:hof10fm:
ix
KEY FEATURES OF THIS TEXT
Applications
Throughout the text great effort is made to
ensure that topics are applied to practical
problems soon after their introduction, providing
methods for dealing with both routine
computations and applied problems. These
problem-solving methods and strategies are
introduced in applied examples and practiced
throughout in the exercise sets.
ix
EXAMPLE 5.1.3
Find the following integrals:
a. (2x5 8x3 3x2 5) dx
b.
c.
Solution
a. By using the power rule in conjunction with the sum and difference rules and the
multiple rule, you get
b. There is no “quotient rule” for integration, but at least in this case, you can still divide
the denominator into the numerator and then integrate using the method in part (a):
c.
3
1
5 e
5t
1
3/2 t
3/2 C 3
5 e
5t
2
3 t
3/2 C
(3e5t
t) dt (3e5t
t
1/2) dt
1
3 x3 2x 7 ln |x| C
x3 2x 7
x dx x2 2 7
x dx
1
3 x6 2x4 x3 5x C
2
x6
6 8
x4
4 3
x3
3 5x C
(2x5 8x3 3x2 5) dx 2x5 dx 8x3 dx 3x2 dx 5 dx
(3e5t
t) dt
x3 2x 7
x dx
y
EXPLORE!
Refer to Example 5.1.4. Store
the function f(x ) 3x2 1 into
Y1. Graph using a bold
graphing style and the window
[0, 2.35]0.5 by [2, 12]1.
Place into Y2 the family of
antiderivatives
F(x ) x3 x L1
where L1 is the list of integer
values 5 to 5. Which of
these antiderivatives passes
through the point (2, 6)?
Repeat this exercise for
f(x ) 3x2 2.
Integration Rules This list of rules can be used to simplify the computation of definite integrals.
Rules for Definite Integrals
Let f and g be any functions continuous on a x b. Then,
1. Constant multiple rule: k f(x) dx k f(x) dx for constant k
2. Sum rule: [ f(x) g(x)] dx f(x) dx g(x) dx
3. Difference rule: [ f(x) g(x)] dx f(x) dx g(x) dx
4. f(x) dx 0
5. f(x) dx f(x) dx
6. Subdivision rule: f(x) dx f(x) dx f(x) dx
b
c
c
a
b
a
b
a
a
b
a
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b. We want to find a time with such that . Solving
this equation, we find that
Since t 0.39 is outside the time interval (8 A.M. to 5 P.M.), it follows that the temperature in the city is the same as the average temperature only
when t 7.61, that is, at approximately 1:37 P.M.
2 ta 11
0.39 or 7.61
ta 4 13
take square roots on both sides ta 4 13
multiply both sides by 3 (ta 4)2 (3)13
3 13
subtract 3 from both sides 1
3
(ta 4)2 4
3 3 13
3
3 1
3
(ta 4)2 4
3
T(ta) 4
3 2 t t ta a 11
Just-In-Time REVIEW
Since there are 60 minutes in
an hour, 0.61 hour is the same
as 0.61(60) minutes.
Thus, 7.61 hours after 6 A.M.
is 37 minutes past 1 P.M. or
1.37 P.M.
Just-In-Time Reviews 37
These references, located in the margins, are
used to quickly remind students of important
concepts from college algebra or precalculus as
they are being used in examples and review.
Definitions
Definitions and key concepts are set off in shaded
boxes to provide easy referencing for the student.
5.1.5 through 5.1.8). However, since Q(x) is an antiderivative of Q(x), the fundamental theorem of calculus allows us to compute net change by the following definite integration formula.
Net Change ■ If Q(x) is continuous on the interval a x b, then the net
change in Q(x) as x varies from x a to x b is given by
Q(b) Q(a)
b
a
Q(x) dx
Here are two examples involving net change.
EXAMPLE 5.3.9
At a certain factory, the marginal cost is 3(q 4)2 dollars per unit when the level of
production is q units. By how much will the total manufacturing cost increase if the
level of production is raised from 6 units to 10 units?
Procedural Examples and Boxes
Each new topic is approached with careful clarity by
providing step-by-step problem-solving techniques
through frequent procedural examples and summary
boxes.
hof32312_fm_i-xvi.qxd 12/4/08 6:21 PM Page ix User-S198 201:MHDQ089:mhhof10%0:hof10fm:
Writing Exercises
These problems, designated by writing icons, challenge a
student’s critical thinking skills and invite students to research
topics on their own.
Calculator Exercises
Calculator icons designate problems within each section that
can only be completed with a graphing calculator.
CHAPTER SUMMARY 364 CHAPTER 4 Exponential and Logarithmic Functions 4-74
where t is the number of years after a fixed base
year and D0 is the mortality rate when t 0.
a. Suppose the initial mortality rate of a particular
group is 0.008 (8 deaths per 1,000 women).
What is the mortality rate of this group 10 years
later? What is the rate 25 years later?
b. Sketch the graph of the mortality function D(t)
for the group in part (a) for 0 t 25.
82. GROSS DOMESTIC PRODUCT The gross
domestic product (GDP) of a certain country was
100 billion dollars in 1990 and 165 billion dollars
in 2000. Assuming that the GDP is growing
exponentially, what will it be in the year 2010?
83. ARCHAEOLOGY “Lucy,” the famous prehuman
whose skeleton was discovered in Africa, has been
found to be approximately 3.8 million years old.
a. Approximately what percentage of original 14C
would you expect to find if you tried to apply carbon dating to Lucy? Why would this be a problem if you were actually trying to “date” Lucy?
b. In practice, carbon dating works well only for
relatively “recent” samples—those that are no
more than approximately 50,000 years old. For
older samples, such as Lucy, variations on
carbon dating have been developed, such as
potassium-argon and rubidium-strontium dating.
Read an article on alternative dating methods
and write a paragraph on how they are used.*
84. RADIOLOGY The radioactive isotope
gallium-67 (67Ga), used in the diagnosis of
malignant tumors, has a half-life of 46.5 hours. If
we start with 100 milligrams of the isotope, how
many milligrams will be left after 24 hours? When
will there be only 25 milligrams left? Answer
these questions by first using a graphing utility to
graph an appropriate exponential function and then
using the TRACE and ZOOM features.
85. A population model developed by the U.S. Census
Bureau uses the formula
to estimate the population of the United States (in
millions) for every tenth year from the base year
P(t) 202.31
1 e3.9380.314t
1790. Thus, for instance, t 0 corresponds to
1790, t 1 to 1800, t 10 to 1890, and so on.
The model excludes Alaska and Hawaii.
a. Use this formula to compute the population of
the United States for the years 1790, 1800,
1830, 1860, 1880, 1900, 1920, 1940, 1960,
1980, 1990, and 2000.
b. Sketch the graph of P(t). When does this model
predict that the population of the United States
will be increasing most rapidly?
c. Use an almanac or some other source to find the
actual population figures for the years listed in
part (a). Does the given population model seem
to be accurate? Write a paragraph describing
some possible reasons for any major differences
between the predicted population figures and the
actual census figures.
86. Use a graphing utility to graph y 2x
, y 3x
,
y 5x
, and y (0.5)x on the same set of axes.
How does a change in base affect the graph of the
exponential function? (Suggestion: Use the
graphing window [3, 3]1 by [3, 3]1.)
87. Use a graphing utility to draw the graphs of
y , y , and y 3x on the same set
of axes. How do these graphs differ? (Suggestion:
Use the graphing window [3, 3]1 by [3, 3]1.)
88. Use a graphing utility to draw the graphs of y 3x
and y 4 ln on the same axes. Then use
TRACE and ZOOM to find all points of
intersection of the two graphs.
89. Solve this equation with three decimal place
accuracy:
log5 (x 5) log2 x 2 log10 (x
2 2x)
90. Use a graphing utility to draw the graphs of
y ln (1 x
2
) and y
on the same axes. Do these graphs intersect?
91. Make a table for the quantities and
, with n 8, 9, 12, 20, 25, 31, 37,
38, 43, 50, 100, and 1,000. Which of the two
quantities seems to be larger? Do you think this
inequality holds for all n 8?
(n 1)
n
(n)
n1
1
x
x
3 x 3x
*A good place to start your research is the article by Paul J. Campbell,
“How Old Is the Earth?”, UMAP Modules 1992: Tools for Teaching,
Arlington, MA: Consortium for Mathematics and Its Applications,
1993.
Chapter Review
Chapter Review material aids the student in
synthesizing the important concepts discussed within
the chapter, including a master list of key technical
terms and formulas introduced in the chapter.
Antiderivative; indefinite integral: (372, 374)
Power rule: (375)
(375)
(375)
(375)
Sum rule: (376)
Initial value problem (378)
Integration by substitution: (386)
where u u(x)
du u(x) dx
g(u(x))u(x) dx g(u) du
[ f(x) g(x)] dx f(x) dx g(x) dx
Constant rule: kdx kx C
Exponential rule: ekx dx 1
k
ekx C
Logarithmic rule: 1
x dx ln |x| C
xn dx xn1
n 1 C for n
1
f(x)dx F(x) C if and only if F(x) f(x)
Important Terms, Symbols, and Formulas
Definite integral: (401)
Area under a curve: (399, 401)
Special rules for definite integrals: (404)
a
b
f(x) dx
b
a
f(x) dx
a
a
f(x) dx 0
a b
x
y
R
y = f (x)
b
a
f(x) dx lim
n→ [ f(x1) f(x n)] x
Area of R
b
a
f(x) dx
CHAPTER SUMMARY
1. Evaluate each of these expressions:
a.
b.
c. log2 4 log4161
d.
2. Simplify each of these expressions:
a. (9x4
y
2
)
3/2
b. (3x2
y4/3)
1/2
c.
d.
x0.2y1.2
x1.5y0.4
5
y
x
3/2
x2/3
y1/6
2
8
27
2/3
16
81
3/2
3
(25)1.5
8
27
(32
(92
)
(27)2/3
3. Find all real numbers x that satisfy each of these
equations.
a.
b. e
1/x 4
c. log4 x
2 2
d.
4. In each case, find the derivative . (In some
cases, it may help to use logarithmic
differentiation.)
a.
b. y ln (x3 2x
2 3x)
c. y x3 ln x
d. y e2x
(2x 1
3
1 x2
y ex
x2 3x
dy
dx
25
1 2e0.5t 3
42xx
2
1
64
Checkup for Chapter 4
Chapter Checkup
Chapter Checkups provide a quick quiz for students
to test their understanding of the concepts introduced
in the chapter.
x KEY FEATURES OF THIS TEXT
CONSUMERS’ WILLINGNESS TO SPEND For
the consumers’ demand functions D(q) in Exercises 1
through 6:
(a) Find the total amount of money consumers are
willing to spend to get q0 units of the
commodity.
(b) Sketch the demand curve and interpret the
consumers’ willingness to spend in part (a) as
an area.
1. D(q) 2(64 q2
) dollars per unit; q0 6 units
2. D(q) dollars per unit; q0 5 units
3. D(q) dollars per unit; q0 12 units
4. D(q) dollars per unit; q0 10 units
5. D(q) 40e0.05q dollars per unit; q0 10 units
6. D(q) 50e0.04q dollars per unit; q0 15 units
CONSUMERS’ SURPLUS In Exercises 7 through
10, p D(q) is the price (dollars per unit) at which q
units of a particular commodity will be demanded by
the market (that is, all q units will be sold at this
price), and q0 is a specified level of production. In
each case, find the price p0 D(q0) at which q0 units
will be demanded and compute the corresponding consumers’ surplus CS. Sketch the demand curve y D(q)
and shade the region whose area represents the
consumers’ surplus.
7. D(q) 2(64 q2
); q0 3 units
8. D(q) 150 2q 3q2
; q0 6 units
9. D(q) 40e0.05q
; q0 5 units
10. D(q) 75e0.04q
; q0 3 units
PRODUCERS’ SURPLUS In Exercises 11 through
14, p S(q) is the price (dollars per unit) at which q
units of a particular commodity will be supplied to the
market by producers, and q0 is a specified level of
production. In each case, find the price p0 S(q0) at
which q0 units will be supplied and compute the
corresponding producers’ surplus PS. Sketch the supply
curve y S(q) and shade the region whose area
represents the producers’ surplus.
300
4q 3
400
0.5q 2
300
(0.1q 1)2
11. S(q) 0.3q2 30; q0 4 units
12. S(q) 0.5q 15; q0 5 units
13. S(q) 10 15e0.03q
; q0 3 units
14. S(q) 17 11e
0.01q
; q0 7 units
CONSUMERS’ AND PRODUCERS’ SURPLUS AT
EQUILIBRIUM In Exercises 15 through 19,
the demand and supply functions, D(q) and S(q), for a
particular commodity are given. Specifically, q
thousand units of the commodity will be demanded
(sold) at a price of p D(q) dollars per unit, while q
thousand units will be supplied by producers when the
price is p S(q) dollars per unit. In each case:
(a) Find the equilibrium price pe (where supply
equals demand).
(b) Find the consumers’ surplus and the
producers’ surplus at equilibrium.
15. D(q) 131 q2
; S(q) 50 q2
16. D(q) 65 q2
; S(q) q2 2q 5
17. D(q) 0.3q2 70; S(q) 0.1q2 q 20
18. D(q) ; S(q) 5 q
19. D(q) 3; S(q) (q 1)
20. PROFIT OVER THE USEFUL LIFE OF A
MACHINE Suppose that when it is t years old,
a particular industrial machine generates revenue
at the rate R(t) 6,025 8t
2 dollars per year
and that operating and servicing costs accumulate
at the rate C(t) 4,681 13t
2 dollars per year.
a. How many years pass before the profitability
of the machine begins to decline?
b. Compute the net profit generated by the
machine over its useful lifetime.
c. Sketch the revenue rate curve y R(t) and
the cost rate curve y C(t) and shade the
region whose area represents the net profit
computed in part (b).
1
3
16
q 2
245 2q
1
3
2
3
1
3
EXERCISES ■ 5.5 Exercise Sets
Almost 300 new problems have been added to increase the
effectiveness of the highly praised exercise sets! Routine
problems have been added where needed to ensure students
have enough practice to master basic skills, and a variety of
applied problems have been added to help demonstrate the
practicality of the material.
hof32312_fm_i-xvi.qxd 12/4/08 08:21am Page x ntt 201:MHDQ089:mhhof10%0:hof10fm:
KEY FEATURES OF THIS TEXT xi
Think About It Essays
The modeling-based Think About It essays show students
how material introduced in the chapter can be used to
construct useful mathematical models while explaining the
modeling process, and providing an excellent starting
point for projects or group discussions.
Explore! Technology
Utilizing the graphing, Explore Boxes
challenge a student’s understanding of the
topics presented with explorations tied to
specific examples. Explore! Updates provide
solutions and hints to selected boxes
throughout the chapter.
6. If you invest $2,000 at 5% compounded
continuously, how much will your account be
worth in 3 years? How long does it take before
your account is worth $3,000?
7. PRESENT VALUE Find the present value of
$8,000 payable 10 years from now if the annual
interest rate is 6.25% and interest is compounded:
a. Semiannually
b. Continuously
8. PRICE ANALYSIS A product is introduced and
t months later, its unit price is p(t) hundred
dollars, where
p ln(t 1)
t 1 5
1 e What is the maximum revenue?
10. CARBON DATING An archaeological artifact is
found to have 45% of its original 14C. How old is
the artifact? (Use 5,730 years as the half-life of
14C.)
11. BACTERIAL GROWTH A toxin is introduced
into a bacterial colony, and t hours later, the
population is given by
N(t) 10,000(8 t)e0.1t
a. What was the population when the toxin was
introduced?
b. When is the population maximized? What is the
maximum population?
c. What happens to the population in the long run
(as )? t→
In Exercises 1 through 4, sketch the graph of the given
exponential or logarithmic function without using
calculus.
1. f(x) 5x
2. f(x) 2ex
3. f(x) ln x2
4. f(x) log3 x
5. a. Find f(4) if f(x) Aekx and f(0) 10,
f(1) 25.
b. Find f(3) if f(x) Aekx and f(1) 3,
f(2) 10.
c. Find f(9) if f(x) 30 Aekx and f(0) 50,
f(3) 40.
d. Find f(10) if and f(0) 3,
f(5) 2.
f(t) 6
1 Aekt
Review Exercises
6. Evaluate the following expressions without using
tables or a calculator.
a. ln e
5
b. e
ln 2
c. e
3 ln 4ln 2
d. ln 9e
2 ln 3e
2
In Exercises 7 through 13, find all real numbers x that
satisfy the given equation.
7. 8 2e0.04x
8. 5 1 4e
6x
9. 4 ln x 8
10. 5x e3
11. log9 (4x 1) 2
12. ln (x 2) 3 ln (x 1)
Review Problems
A wealth of additional routine and applied problems
is provided within the end-of-chapter exercise sets,
offering further opportunities for practice.
THINK ABOUT ITTHINK ABOUT IT
JUST NOTICEABLE DIFFERENCES
IN PERCEPTION
Calculus can help us answer questions about human perception, including questions
relating to the number of different frequencies of sound or the number of different
hues of light people can distinguish (see the accompanying figure). Our present goal
is to show how integral calculus can be used to estimate the number of steps a person can distinguish as the frequency of sound increases from the lowest audible frequency of 15 hertz (Hz) to the highest audible frequency of 18,000 Hz. (Here hertz,
abbreviated Hz, equals cycles per second.)
A mathematical model* for human auditory perception uses the formula
y 0.767x
0.439, where y Hz is the smallest change in frequency that is detectable at
frequency x Hz. Thus, at the low end of the range of human hearing, 15 Hz, the smallest change of frequency a person can detect is y 0.767 150.439 2.5 Hz, while
at the upper end of human hearing, near 18,000 Hz, the least noticeable difference is
approximately y 0.767 18,0000.439 57 Hz. If the smallest noticeable change of
frequency were the same for all frequencies that people can hear, we could find the
number of noticeable steps in human hearing by simply dividing the total frequency
range by the size of this smallest noticeable change. Unfortunately, we have just seen
that the smallest noticeable change of frequency increases as frequency increases, so
the simple approach will not work. However, we can estimate the number of distinguishable steps using integration.
Toward this end, let y f(x) represent the just noticeable difference of frequency
people can distinguish at frequency x. Next, choose numbers x0, x1, . . . , xn beginning
at x0 15 Hz and working up through higher frequencies to xn 18,000 Hz in such
a way that for j 0, 2, . . . , n 1,
xj f(xj) xj1
*Part of this essay is based on Applications of Calculus: Selected Topics from the Environmental and
Life Sciences, by Anthony Barcellos, New York: McGraw-Hill, 1994, pp. 21–24.
EXPLORE! UPDATE
Complete solutions for all EXPLORE! boxes throughout the text can be accessed at
the book-specific website, www.mhhe.com/hoffmann.
Store the constants {4, 2, 2, 4} into L1 and write Y1 X^3 and Y2 Y1 L1.
Graph Y1 in bold, using the modified decimal window [4.7, 4.7]1 by [6, 6]1. At
x 1 (where we have drawn a vertical line), the slopes for each curve appear equal.
Solution for Explore!
on Page 373
Using the tangent line feature of your graphing calculator, draw tangent lines at
x 1 for several of these curves. Every tangent line at x 1 has a slope of 3,
although each line has a different y intercept.
The numerical integral, fnInt(expression, variable, lower limit, upper limit) can be
found via the MATH key, 9:fnInt(, which we use to write Y1 below. We obtain a family of graphs that appear to be parabolas with vertices on the y axis at y 0, 1,
and 4. The antiderivative of f(x) 2x is F(x) x2 C, where C 0, 1, and 4,
in our case.
Solution for Explore!
on Page 374
EXPLORE! UPDATE
hof32312_fm_i-xvi.qxd 12/4/08 7:50 PM Page xi User-S198 201:MHDQ089:mhhof10%0:hof10fm:
xii SUPPLEMENTS
Applied Calculus for Business, Economics, and the Social and Life
Sciences, Expanded Tenth Edition
ISBN – 13: 9780073532332 (ISBN-10: 0073532339)
Expanded Tenth Edition contains all of the material present in the Brief Tenth Edition of Calculus for Business, Economics, and the Social and Life Sciences, plus four
additional chapters covering Differential Equations, Infinite Series and Taylor Approximations, Probability and Calculus, and Trigonometric Functions.
Supplements
Also available . . .
Student's Solution Manual
The Student’s Solutions Manual contains comprehensive, worked-out solutions for
all odd-numbered problems in the text with the exception of the Checkup section for
which solutions to all problems are provided. Detailed calculator instructions and
keystrokes are also included for problems marked by the calculator icon. ISBN–13:
9780073349022 (ISBN–10: 0-07-33490-X)
Instructor's Solutions Manual
The Instructor’s Solutions Manual contains comprehensive, worked-out solutions for
all even-numbered problems in the text and is available on the book’s website,
www.mhhe.com/hoffmann.
Computerized Test Bank
Brownstone Diploma testing software, available on the book’s website, offers instructors a quick and easy way to create customized exams and view student results. The
software utilizes an electronic test bank of short answer, multiple choice, and true/false
questions tied directly to the text, with many new questions added for the Tenth Edition. Sample chapter tests and final exams in Microsoft Word and PDF formats are also
provided.
MathZone—www.mathzone.com
McGraw-Hill’s MathZone is a complete online homework system for mathematics
and statistics. Instructors can assign textbook-specific content from over 40 McGrawHill titles as well as customize the level of feedback students receive, including the
ability to have students show their work for any given exercise.
Within MathZone, a diagnostic assessment tool powered by ALEKS is available
to measure student preparedness and provide detailed reporting and personalized
remediation.
For more information, visit the book’s website (www.mhhe.com/hoffmann) or
contact your local McGraw-Hill sales representative (www.mhhe.com/rep).
hof32312_fm_i-xvi.qxd 12/4/08 08:21am Page xii ntt 201:MHDQ089:mhhof10%0:hof10fm:
SUPPLEMENTS xiii
ALEKS—www.aleks.com/highered
ALEKS (Assessment and LEarning in Knowledge Spaces) is a dynamic online learning system for mathematics education, available over the Web 24/7. ALEKS assesses
students, accurately determines their knowledge, and then guides them to the material that they are most ready to learn. With a variety of reports, Textbook Integration
Plus, quizzes, and homework assignment capabilities, ALEKS offers flexibility and
ease of use for instructors.
• ALEKS uses artificial intelligence to determine exactly what each student
knows and is ready to learn. ALEKS remediates student gaps and provides
highly efficient learning and improved learning outcomes.
• ALEKS is a comprehensive curriculum that aligns with syllabi or specified
textbooks. Used in conjunction with McGraw-Hill texts, students also receive
links to text-specific videos, multimedia tutorials, and textbook pages.
• Textbook Integration Plus allows ALEKS to be automatically aligned with
syllabi or specified McGraw-Hill textbooks with instructor-chosen dates,
chapter goals, homework, and quizzes.
• ALEKS with AI-2 gives instructors increased control over the scope and
sequence of student learning. Students using ALEKS demonstrate a steadily
increasing mastery of the content of the course.
• ALEKS offers a dynamic classroom management system that enables instructors
to monitor and direct student progress toward mastery of course objectives.
ALEKS Prep for Calculus
ALEKS Prep delivers students the individualized instruction needed in the first weeks
of class to help them master core concepts they should have learned prior to entering their present course, freeing up lecture time for instructors and helping more students succeed.
ALEKS Prep course products feature:
• Artificial intelligence. Targets gaps in individual student knowledge.
• Assessment and learning. Directed toward individual student needs.
• Automated reports. Monitor student and course progress.
• Open response environment. Includes realistic input tools.
• Unlimited Online Access. PC and Mac compatible.
CourseSmart Electronic Textbook
CourseSmart is a new way for faculty to find and review e-textbooks. It’s also a
great option for students who are interested in accessing their course materials digitally and saving money. At CourseSmart, students can save up to 50% off the cost of
a print book, reduce their impact on the environment, and gain access to powerful
Web tools for learning including full text search, notes and highlighting, and e-mail
tools for sharing notes between classmates. www.CourseSmart.com
hof32312_fm_i-xvi.qxd 12/4/08 08:21am Page xiii ntt 201:MHDQ089:mhhof10%0:hof10fm:
xiv ACKNOWLEDGEMENTS
Acknowledgements
James N. Adair, Missouri Valley College
Faiz Al-Rubaee, University of North Florida
George Anastassiou, University of Memphis
Dan Anderson, University of Iowa
Randy Anderson, Craig School of Business
Ratan Barua, Miami Dade College
John Beachy, Northern Illinois University
Don Bensy, Suffolk County Community College
Neal Brand, University of North Texas
Lori Braselton, Georgia Southern University
Randall Brian, Vincennes University
Paul W. Britt, Louisiana State University—Baton Rouge
Albert Bronstein, Purdue University
James F. Brooks, Eastern Kentucky University
Beverly Broomell, SUNY—Suffolk
Roxanne Byrne, University of Colorado at Denver
Laura Cameron, University of New Mexico
Rick Carey, University of Kentucky
Steven Castillo, Los Angeles Valley College
Rose Marie Castner, Canisius College
Deanna Caveny, College of Charleston
Gerald R. Chachere, Howard University
Terry Cheng, Irvine Valley College
William Chin, DePaul University
Lynn Cleaveland, University of Arkansas
Dominic Clemence, North Carolina A&T State
University
Charles C. Clever, South Dakota State University
Allan Cochran, University of Arkansas
Peter Colwell, Iowa State University
Cecil Coone, Southwest Tennessee Community College
Charles Brian Crane, Emory University
Daniel Curtin, Northern Kentucky University
Raul Curto, University of Iowa
Jean F. Davis, Texas State University—San Marcos
John Davis, Baylor University
Karahi Dints, Northern Illinois University
Ken Dodaro, Florida State University
Eugene Don, Queens College
Dora Douglas, Wright State University
Peter Dragnev, Indiana University–Purdue University,
Fort Wayne
Bruce Edwards, University of Florida
Margaret Ehrlich, Georgia State University
Maurice Ekwo, Texas Southern University
George Evanovich, St. Peters’ College
Haitao Fan, Georgetown University
Brad Feldser, Kennesaw State University
Klaus Fischer, George Mason University
Michael Freeze, University of North Carolina—
Wilmington
Constantine Georgakis, DePaul University
Sudhir Goel, Valdosta State University
Hurlee Gonchigdanzan, University of Wisconsin—
Stevens Point
Ronnie Goolsby, Winthrop College
Lauren Gordon, Bucknell University
Angela Grant, University of Memphis
John Gresser, Bowling Green State University
Murli Gupta, George Washington University
Doug Hardin, Vanderbilt University
Marc Harper, University of Illinois at Urbana—
Champaign
Jonathan Hatch, University of Delaware
John B. Hawkins, Georgia Southern University
Celeste Hernandez, Richland College
William Hintzman, San Diego State University
Matthew Hudock, St. Philips College
Joel W. Irish, University of Southern Maine
Zonair Issac, Vanderbilt University
Erica Jen, University of Southern California
Jun Ji, Kennesaw State University
Shafiu Jibrin, Northern Arizona University
Victor Kaftal, University of Cincinnati
Sheldon Kamienny, University of Southern California
Georgia Katsis, DePaul University
Fritz Keinert, Iowa State University
Melvin Kiernan, St. Peter’s College
As in past editions, we have enlisted the feedback of professors teaching from our
text as well as those using other texts to point out possible areas for improvement.
Our reviewers provided a wealth of detailed information on both our content and
the changing needs of their course, and many changes we have made were a direct
result of consensus among these review panels. This text owes its considerable success to their valuable contributions, and we thank every individual involved in this
process.
hof32312_fm_i-xvi.qxd 12/4/08 08:21am Page xiv ntt 201:MHDQ089:mhhof10%0:hof10fm: