About the Cover
For Mark van der Laan, the beauty of mathematical problems and the journeys he takes on the
road to solving them are the driving force behind his work in the field of biostatistics. In particular,
he is intrigued by the variety of approaches there may be to solve a problem and the fact that it
requires a large diversity of scientists and people working in the field to find the most elegant and
satisfying solutions. As he moved from an M.A. in mathematics to a Ph.D. in mathematical statistics
from the University of Utrecht to his current position in the Department of Biostatistics at the
University of California, Berkeley, Mark has found that the most interesting and creative mathematical problems are present in real-life applications. He says, “I have always realized, and have been
told by experienced researchers, that solving these applied problems requires a thorough education
in mathematics and that probability theory is fundamental. However, as in real life, the approach
taken toward the solution is often by far the most important step and requires philosophical and
abstract thinking.”
MARK VAN DER LAAN
Every day Mark is engaged in creatively solving mathematical problems that have implications
Biostatistician
in the fields of medical research, biology, and public health. For example, in collaboration with
medical researchers at the University of California, San Francisco, Mark is investigating the effects
of antiretroviral treatment (ART) on HIV/AIDS progression. As represented by the images on the cover, he is also involved in establishing the causal effect of air pollution on asthma in children, the causal effect of leisure-time activity and lean-to-fat ratio on health
outcomes in the elderly, as well as the identification of regulatory networks in basic biology.
Recognized for his progressive work in these fields, Mark van der Laan has received numerous awards. In April 2005, he was
awarded the van Dantzig Award for his theoretical and practical contributions made to the fields of operation research and statistics.
In August 2005, he received the COPSS (Committee of Presidents of Statistical Societies) Award, which is presented annually to a
young researcher in recognition of outstanding contributions to the statistics profession. Mark currently holds the UC Berkeley
Chancellor Endowed Chair 2005–2008, as well as the long-term Jiann-Ping Hsu/Karl E. Peace Endowed Chair in Biostatistics at
University of California, Berkeley.
Look for other featured applied researchers in forthcoming titles in the Tan applied mathematics series:
PETER BLAIR HENRY
International Economist
Stanford University
CHRIS SHANNON
Mathematical Economist
University of California,
Berkeley
JONATHAN D. FARLEY
Applied Mathematician
California Institute of
Technology
NAVIN KHANEJA
Applied Scientist
Harvard University
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EDITION
5
APPLIED
MATHEMATICS
FOR THE MANAGERIAL, LIFE,
AND SOCIAL SCIENCES
SOO T. TAN
STONEHILL COLLEGE
Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States
©2010, 2007 Brooks/Cole, Cengage Learning
Applied Mathematics: For the Managerial,
Life, and Social Sciences, Fifth Edition
Soo T. Tan
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1 2 3 4 5 6
7
12
11
10
09
08
TO PAT, BILL, AND MICHAEL
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CONTENTS
Preface xi
CHAPTER 1
Fundamentals of Algebra
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
CHAPTER 2
1
Real Numbers 2
Polynomials 7
Factoring Polynomials 14
Rational Expressions 20
Integral Exponents 26
Solving Equations 30
Rational Exponents and Radicals 36
Quadratic Equations 44
Inequalities and Absolute Value 53
Chapter 1 Summary of Principal Formulas and Terms 62
Chapter 1 Concept Review Questions 63
Chapter 1 Review Exercises 63
Chapter 1 Before Moving On 65
Functions and Their Graphs
2.1
2.2
2.3
2.4
2.5
The Cartesian Coordinate System and Straight Lines 68
Equations of Lines 74
Using Technology: Graphing a Straight Line 84
Functions and Their Graphs 87
Using Technology: Graphing a Function 100
The Algebra of Functions 103
Linear Functions 111
PORTFOLIO: Esteban Silva
2.6
67
115
Using Technology: Linear Functions 120
Quadratic Functions 123
PORTFOLIO: Deb Farace 128
2.7
Using Technology: Finding the Points of Intersection of Two Graphs 132
Functions and Mathematical Models 134
Using Technology: Constructing Mathematical Models from Raw Data 144
Chapter 2 Summary of Principal Formulas and Terms 148
Chapter 2 Concept Review Questions 149
Chapter 2 Review Exercises 149
Chapter 2 Before Moving On 151
vi
CONTENTS
CHAPTER 3
Exponential and Logarithmic Functions
3.1
3.2
3.3
CHAPTER 4
4.2
4.3
4.4
CHAPTER 5
Exponential Functions 154
Using Technology 160
Logarithmic Functions 162
Exponential Functions as Mathematical Models 171
Using Technology: Analyzing Mathematical Models 180
Chapter 3 Summary of Principal Formulas and Terms 181
Chapter 3 Concept Review Questions 182
Chapter 3 Review Exercises 182
Chapter 3 Before Moving On 183
Mathematics of Finance
4.1
153
185
Compound Interest 186
Using Technology: Finding the Accumulated Amount of an Investment, the Effective Rate
of Interest, and the Present Value of an Investment 201
Annuities 204
Using Technology: Finding the Amount of an Annuity 212
Amortization and Sinking Funds 215
Using Technology: Amortizing a Loan 225
Arithmetic and Geometric Progressions 228
Chapter 4 Summary of Principal Formulas and Terms 236
Chapter 4 Concept Review Questions 237
Chapter 4 Review Exercises 238
Chapter 4 Before Moving On 239
Systems of Linear Equations and Matrices
5.1
5.2
5.3
5.4
5.5
5.6
241
Systems of Linear Equations: An Introduction 242
Systems of Linear Equations: Unique Solutions 249
Using Technology: Systems of Linear Equations: Unique Solutions 263
Systems of Linear Equations: Underdetermined and Overdetermined Systems 265
Using Technology: Systems of Linear Equations: Underdetermined and Overdetermined
Systems 274
Matrices 275
Using Technology: Matrix Operations 284
Multiplication of Matrices 287
Using Technology: Matrix Multiplication 299
The Inverse of a Square Matrix 301
Using Technology: Finding the Inverse of a Square Matrix 313
Chapter 5 Summary of Principal Formulas and Terms 316
Chapter 5 Concept Review Questions 316
Chapter 5 Review Exercises 317
Chapter 5 Before Moving On 319
CONTENTS
CHAPTER 6
Linear Programming
6.1
6.2
6.3
6.4
321
Graphing Systems of Linear Inequalities in Two Variables 322
Linear Programming Problems 330
Graphical Solution of Linear Programming Problems 338
The Simplex Method: Standard Maximization Problems 351
PORTFOLIO: Morgan Wilson 352
6.5
CHAPTER 7
Using Technology: The Simplex Method: Solving Maximization Problems 372
The Simplex Method: Standard Minimization Problems 376
Using Technology: The Simplex Method: Solving Minimization Problems 387
Chapter 6 Summary of Principal Terms 390
Chapter 6 Concept Review Questions 390
Chapter 6 Review Exercises 391
Chapter 6 Before Moving On 393
Sets and Probability
395
7.1
7.2
7.3
Sets and Set Operations 396
The Number of Elements in a Finite Set 405
The Multiplication Principle 411
7.4
Permutations and Combinations 417
Using Technology: Evaluating n!, P(n, r), and C(n, r)
Experiments, Sample Spaces, and Events 431
Definition of Probability 439
Rules of Probability 449
PORTFOLIO: Stephanie Molina 413
7.5
7.6
7.7
430
PORTFOLIO: Todd Good 451
Chapter 7 Summary of Principal Formulas and Terms 459
Chapter 7 Concept Review Questions 460
Chapter 7 Review Exercises 460
Chapter 7 Before Moving On 462
CHAPTER 8
Additional Topics in Probability
8.1
8.2
8.3
8.4
8.5
463
Use of Counting Techniques in Probability 464
Conditional Probability and Independent Events 471
Bayes’ Theorem 485
Distributions of Random Variables 494
Using Technology: Graphing a Histogram 502
Expected Value 504
PORTFOLIO: Ann-Marie Martz 510
8.6
Variance and Standard Deviation 516
Using Technology: Finding the Mean and Standard Deviation 527
Chapter 8 Summary of Principal Formulas and Terms 528
Chapter 8 Concept Review Questions 529
Chapter 8 Review Exercises 530
Chapter 8 Before Moving On 531
vii
viii
CONTENTS
CHAPTER 9
The Derivative
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
533
Limits 534
Using Technology: Finding the Limit of a Function 552
One-Sided Limits and Continuity 554
Using Technology: Finding the Points of Discontinuity of a Function 568
The Derivative 570
Using Technology: Graphing a Function and Its Tangent Line 586
Basic Rules of Differentiation 589
Using Technology: Finding the Rate of Change of a Function 600
The Product and Quotient Rules; Higher-Order Derivatives 602
Using Technology: The Product and Quotient Rules 615
The Chain Rule 618
Using Technology: Finding the Derivative of a Composite Function 629
Differentiation of Exponential and Logarithmic Functions 630
Using Technology 640
Marginal Functions in Economics 642
PORTFOLIO: Richard Mizak 647
Chapter 9 Summary of Principal Formulas and Terms 651
Chapter 9 Concept Review Questions 653
Chapter 9 Review Exercises 653
Chapter 9 Before Moving On 656
CHAPTER 10
Applications of the Derivative
10.1
10.2
10.3
10.4
10.5
657
Applications of the First Derivative 658
Using Technology: Using the First Derivative to Analyze a Function 675
Applications of the Second Derivative 678
Using Technology: Finding the Inflection Points of a Function 695
Curve Sketching 697
Using Technology: Analyzing the Properties of a Function 709
Optimization I 711
Using Technology: Finding the Absolute Extrema of a Function 724
Optimization II 726
PORTFOLIO: Gary Li 731
Chapter 10 Summary of Principal Terms 738
Chapter 10 Concept Review Questions 738
Chapter 10 Review Exercises 738
Chapter 10 Before Moving On 740
CHAPTER 11
Integration
11.1
11.2
11.3
11.4
741
Antiderivatives and the Rules of Integration 742
Integration by Substitution 755
Area and the Definite Integral 765
The Fundamental Theorem of Calculus 774
Using Technology: Evaluating Definite Integrals 784
CONTENTS
11.5
11.6
11.7
CHAPTER 12
Evaluating Definite Integrals 785
Using Technology: Evaluating Definite Integrals for Piecewise-Defined Functions 795
Area between Two Curves 797
Using Technology: Finding the Area between Two Curves 807
Applications of the Definite Integral to Business and Economics 808
Using Technology: Business and Economic Applications/Technology Exercises 820
Chapter 11 Summary of Principal Formulas and Terms 821
Chapter 11 Concept Review Questions 822
Chapter 11 Review Exercises 823
Chapter 11 Before Moving On 826
Calculus of Several Variables
12.1
12.2
12.3
827
Functions of Several Variables 828
Partial Derivatives 837
Using Technology: Finding Partial Derivatives at a Given Point 849
Maxima and Minima of Functions of Several Variables 850
PORTFOLIO: Kirk Hoiberg 852
Chapter 12 Summary of Principal Terms 860
Chapter 12 Concept Review Questions 860
Chapter 12 Review Exercises 861
Chapter 12 Before Moving On 862
Answers to Odd-Numbered Exercises 863
Index 915
How-To Technology Index 920
ix
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PREFACE
M
ath is an integral part of our increasingly complex daily life. Applied Mathematics
for the Managerial, Life, and Social Sciences, Fifth Edition, attempts to illustrate this point
with its applied approach to mathematics. Our objective for this Fifth Edition is threefold:
(1) to write an applied text that motivates students while providing the background in the
quantitative techniques necessary to better understand and appreciate the courses normally
taken in undergraduate training, (2) to lay the foundation for more advanced courses, such
as statistics and operations research, and (3) to make the text a useful tool for instructors.
Since the book contains more than enough material for the usual two-semester or
three-semester course, the instructor may be flexible in choosing the topics most suitable
for his or her course. The following chart on chapter dependency is provided to help the
instructor design a course that is most suitable for the intended audience.
1
2
3
7
Fundamentals
of Algebra
Functions and
Their Graphs
Exponential and
Logarithmic
Functions
Sets and
Probability
4
5
9
8
Mathematics
of Finance
Systems of
Linear Equations
and Matrices
The Derivative
Additional Topics
in Probability
6
10
Linear Programming
Applications
of the
Derivative
11
12
Integration
Calculus of
Several Variables
xii
PREFACE
THE APPROACH
Level of Presentation
My approach is intuitive, and I state the results informally. However, I have taken special care
to ensure that this approach does not compromise the mathematical content and accuracy.
Problem-Solving Approach
A problem-solving approach is stressed throughout the book. Numerous examples and
applications illustrate each new concept and result. Special emphasis is placed on helping
students formulate, solve, and interpret the results of the problems involving applications.
Because students often have difficulty setting up and solving word problems, extra care
has been taken to help students master these skills:
■
■
■
■
Very early on in the text students are given practice in solving word problems (see
Example 7, Section 1.8).
Guidelines are given to help students formulate and solve word problems (see Section
2.7).
One entire section is devoted to modeling and setting up linear programming problems
(see Section 6.2).
Optimization problems are covered in two sections. In Section 10.4 students are asked
to solve problems in which the objective function to be optimized is given, and in Section 10.5 students are asked to solve problems in which the optimization problems must
first be formulated.
Intuitive Introduction to Concepts
Mathematical concepts are introduced with concrete, real-life examples wherever appropriate. An illustrative list of some of the topics introduced in this manner follows:
■
■
■
■
■
■
■
■
■
■
The algebra of functions: The U.S. Budget Deficit
Mathematical modeling: Social Security Trust Fund Assets
Limits: The Motion of a Maglev
The chain rule: The Population of Americans Age 55 and Older
Increasing and decreasing functions: The Fuel Economy of a Car
Concavity: U.S. and World Population Growth
Inflection points: The Point of Diminishing Returns
Curve sketching: The Dow Jones Industrial Average on “Black Monday”
Exponential functions: Income Distribution of American Families
Area between two curves: Petroleum Saved with Conservation Measures
Connections
One example (the maglev) is used as a common thread throughout the development of calculus—from limits through integration. The goal here is to show students the connections
between the concepts presented—limits, continuity, rates of change, the derivative, the
definite integral, and so on.
Motivation
Illustrating the practical value of mathematics in applied areas is an important objective of
my approach. Many of the applications are based on mathematical models (functions) that
I have constructed using data drawn from various sources, including current newspapers,
magazines, and the Internet. Sources are given in the text for these applied problems.
xiii
PREFACE
Modeling
I believe that one of the important skills that a student should acquire is the ability to translate a real problem into a mathematical model that can provide insight into the problem.
In Section 2.7, the modeling process is discussed, and students are asked to use models
(functions) constructed from real-life data to answer questions. Students get hands-on
experience constructing these models in the Using Technology sections.
NEW TO THIS EDITION
Algebra Review Where
Students Need It Most
Well-placed algebra review notes,
keyed to the early algebra review
chapters, appear where students often
need help in the calculus portion of the
text. These are indicated by the (x )
icon. See this feature in action on
pages 542 and 577.
2
EXAMPLE 6 Evaluate:
lim
hS0
11 ⫹ h ⫺ 1
h
Letting h approach zero, we obtain the indeterminate form 0/0. Next,
we rationalize the numerator of the quotient by multiplying both the numerator
and the denominator by the expression 1 11 ⫹ h ⫹ 12 , obtaining
Solution
1 11 ⫹ h ⫺ 12 1 11 ⫹ h ⫹ 12
11 ⫹ h ⫺ 1
⫽
* (x ) See page 41.
h
h1 11 ⫹ h ⫹ 12
1⫹h⫺1
⫽
1 1a ⫺ 1b 2 1 1a ⫹ 1b 2 ⫽ a ⫺ b
h1 11 ⫹ h ⫹ 12
2
h
h1 11 ⫹ h ⫹ 12
1
⫽
11 ⫹ h ⫹ 1
⫽
Motivating Real-World
Applications
More than 220 new applications have
been added to the Applied Examples
and Exercises. Among these
applications are global warming,
depletion of Social Security trust fund
assets, driving costs for a 2008
medium-sized sedan, hedge fund
investments, mobile instant messaging
accounts, hiring lobbyists, Web
conferencing, the autistic brain, the
revenue of Polo Ralph Lauren, U.S.
health-care IT spending, and
consumption of bottled water.
APPLIED EXAMPLE 1 Global Warming The increase in carbon dioxide (CO2) in the atmosphere is a major cause of global warming. The
Keeling curve, named after Charles David Keeling, a professor at Scripps Institution of Oceanography, gives the average amount of CO2, measured in parts per
million volume (ppmv), in the atmosphere from the beginning of 1958 through
2007. Even though data were available for every year in this time interval, we’ll
construct the curve based only on the following randomly selected data points.
Year
Amount
1958
1970
1974
1978
1985
1991
1998
2003
2007
315
325
330
335
345
355
365
375
380
The scatter plot associated with these data is shown in Figure 54a. A mathematical model giving the approximate amount of CO2 in the atmosphere during this
period is given by
A(t) ⫽ 0.010716t 2 ⫹ 0.8212t ⫹ 313.4
(1 ⱕ t ⱕ 50)
xiv
PREFACE
Modeling with Data
Modeling with Data exercises are
now found in many of the Using
Technology sections throughout
the text. Students can actually see
how some of the functions found
in the exercises are constructed.
(See Internet Users in China,
Exercise 40, page 159, and the
corresponding exercise where the
model is derived in Exercise 14,
page 161.)
40. INTERNET USERS IN CHINA The number of Internet users in
China is projected to be
N(t) ⫽ 94.5e0.2t
(1 ⱕ t ⱕ 6)
where N(t) is measured
in millions
andDtATA
is measured
in of Internet users in
14. MODELING
WITH
The number
years, with t ⫽ 1 corresponding
the beginning
of 2005.
China (in to
millions)
from the
beginning of 2005 through
a. How many Internet 2010
usersare
were
thereinat the
the following
beginningtable:
of
shown
2005? At the beginning of 2006?
b. How many Internet users are there expected to be at the
2005
2006
2007
2008
2009
2010
beginning of 2010? Year
Number
116.1
141.9
169.0
209.0
258.1
314.8
c. Sketch the graph of N.
Source: C. E. Unterberg
a. Use ExpReg to find an exponential regression model
for the data. Let t ⫽ 1 correspond to the beginning of
2005.
Hint: a x ⫽ e x ln a.
b. Plot the scatter diagram and the graph of the function f
found in part (a).
Making Connections
with Technology
Many Using Technology sections
have been updated. A new example—Market for CholesterolReducing Drugs—has been added
to Using Technology 2.5. Using
Technology 3.3 includes a new
example in which an exponential
model is constructed—Internet
Gaming Sales—using the logistic
function of a graphing utility.
Another new example—TV
Mobile Phones—has been added
to Using Technology 10.3. Additional graphing calculator screens
in some sections and Exploring
with Technology examples illustrating the use of the graphing
calculator to solve inequalities
and to generate random numbers
have been added throughout.
APPLIED EXAMPLE 2 Market for Cholesterol-Reducing Drugs In
a study conducted in early 2000, experts projected a rise in the market for
cholesterol-reducing drugs. The U.S. market (in billions of dollars) for such drugs
from 1999 through 2004 is approximated by
M(t) ⫽ 1.95t ⫹ 12.19
where t is measured in years, with t ⫽ 0 corresponding to 1999.
a. Plot the graph of the function M in the viewing window [0, 5] ⫻ [0, 25].
b. Assuming that the projection held and the trend continued, what was the market for cholesterol-reducing drugs in 2005 (t ⫽ 6)?
APPLIED EXAMPLE 1 Internet Gaming Sales The estimated growth
c. What was the rate of increase of the market for cholesterol-reducing drugs
in global Internet-gaming revenue (in billions of dollars), as predicted by
over the period in question?
industry analysts, is given in the following table:
Source: S. G. Cowen
Year
Revenue
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
3.1
3.9
5.6
8.0
11.8
15.2
18.2
20.4
22.7
24.5
a. Use Logistic to find a regression model for the data. Let t ⫽ 0 correspond to
2001.
b. Plot the scatter diagram and the graph of the function f found in part (a) using
the viewing window [0, 9] ⫻ [0, 30].
Source: Christiansen Capital/Advisors
PREFACE
Variety of Problem
Types
Additional rote questions, true
or false questions, and concept
questions have been added
throughout the text to enhance
the exercise sets. (See, for
example, the graphical questions
added to Concept Questions 2.3,
page 95.)
2.3 Concept Questions
1. a. What is a function?
b. What is the domain of a function? The range of a function?
c. What is an independent variable? A dependent variable?
c.
d.
y
2. a. What is the graph of a function? Use a drawing to illustrate the graph, the domain, and the range of a function.
b. If you are given a curve in the xy-plane, how can you tell
if the graph is that of a function f defined by y ⫽ f(x)?
y
y
x
0
x
0
3. Are the following graphs of functions? Explain.
a.
b.
y
4. What are the domain and range of the function f with the
following graph?
y
4
0
x
0
x
3
2
1
(2, 12 )
1
Action-Oriented
Study Tabs
Convenient color-coded study
tabs, similar to Post-it® flags,
make it easy for students to tab
pages that they want to return to
later, whether it be for
additional review, exam
preparation, online exploration,
or identifying a topic to be
discussed with the instructor.
xv
2
3
x
4
5
xvi
PREFACE
Specific Content Changes
More applications have been added to the algebra review chapters.
The discussion of mortgages has been enhanced with a new example on adjustable-rate
mortgages and the addition of many new applied exercises.
Section 2.7, on functions and mathematical models, has been reorganized and new models have been introduced. Here, students are now asked to use a model describing global
warming to predict the amount of carbon dioxide (CO2) that will be present in the
atmosphere in 2010, and a model describing the assets of the Social Security trust fund
to determine when those assets are expected to be depleted.
The chain rule in Section 9.6 is now introduced with an application: the population of
Americans aged 55 years and older.
A How-To Technology Index has been added for easy reference.
New Using Technology Excel sections for Microsoft Office 2007 are now available on
the Web.
■
■
■
■
■
■
TRUSTED FEATURES
In addition to the new features, we have retained many of the following hallmarks that
have made this series so usable and well-received in past editions:
■
■
■
■
Self-Check Exercises
Offering students immediate
feedback on key concepts, these
exercises begin each end-ofsection exercise set. Fully workedout solutions can be found at the
end of each exercise section.
Section exercises to help students understand and apply concepts
Optional technology sections to explore mathematical ideas and solve problems
End-of-chapter review sections to assess understanding and problem-solving skills
Features to motivate further exploration
2.5 Self-Check Exercises
A manufacturer has a monthly fixed cost of $60,000 and a production cost of $10 for each unit produced. The product sells
for $15/unit.
1. What is the cost function?
2. What is the revenue function?
Concept Questions
2.5 Concept Questions
Designed to test students’
understanding of the basic
concepts discussed in the section,
these questions encourage
students to explain learned
concepts in their own words.
1. a. What is a linear function? Give an example.
b. What is the domain of a linear function? The range?
c. What is the graph of a linear function?
Exercises
Each exercise section contains an
ample set of problems of a
routine computational nature
followed by an extensive set of
application-oriented problems.
2. What is the general form of a linear cost function? A linear
revenue function? A linear profit function?
3. What is the profit function?
4. Compute the profit (loss) corresponding to production levels of 10,000 and 14,000 units/month.
Solutions to Self-Check Exercises 2.5 can be found on
page 120.
3. Explain the meaning of each term:
a. Break-even point
b. Break-even quantity
c. Break-even revenue
2.5 Exercises
In Exercises 1–10, determine whether the equation
defines y as a linear function of x. If so, write it in the
form y ⴝ mx ⴙ b.
1. 2x ⫹ 3y ⫽ 6
2. ⫺2x ⫹ 4y ⫽ 7
3. x ⫽ 2y ⫺ 4
4. 2x ⫽ 3y ⫹ 8
5. 2x ⫺ 4y ⫹ 9 ⫽ 0
6. 3x ⫺ 6y ⫹ 7 ⫽ 0
7. 2 x 2 ⫺ 8y ⫹ 4 ⫽ 0
9. 2x ⫺ 3y2 ⫹ 8 ⫽ 0
8. 3 1x ⫹ 4y ⫽ 0
10. 2x ⫹ 1y ⫺ 4 ⫽ 0
11. A manufacturer has a monthly fixed cost of $40,000 and a
production cost of $8 for each unit produced. The product
sells for $12/unit.
In Exercises 15–20, find the point of intersection of each
pair of straight lines.
15. y ⫽ 3 x ⫹ 4
y ⫽ ⫺2 x ⫹ 14
16.
y ⫽ ⫺4x ⫺ 7
⫺y ⫽ 5x ⫹ 10
17. 2x ⫺ 3y ⫽ 6
3x ⫹ 6y ⫽ 16
18.
2 x ⫹ 4y ⫽ 11
⫺5x ⫹ 3y ⫽ 5
19.
1
x⫺5
4
3
2x ⫺ y ⫽ 1
2
y⫽
20. y ⫽
2
x⫺4
3
x ⫹ 3y ⫹ 3 ⫽ 0
PREFACE
Using Technology
These optional features appear after the
section exercises. They can be used in
the classroom if desired or as material
for self-study by the student. Here, the
graphing calculator and Microsoft Excel
2003 are used as a tool to solve
problems. (Instructions for Microsoft
Excel 2007 are given at the Companion
Website.) These sections are written in
the traditional example–exercise format,
with answers given at the back of the
book. Illustrations showing graphing
calculator screens are extensively used.
In keeping with the theme of motivation
through real-life examples, many
sourced applications are again included.
Students can construct their own models
using real-life data in many of the Using
Technology sections. These include
models for the growth of the Indian
gaming industry, TIVO owners, nicotine
content of cigarettes, computer security,
and online gaming, among others.
USING
TECHNOLOGY
xvii
Finding the Accumulated Amount of an Investment,
the Effective Rate of Interest, and the Present Value
of an Investment
Graphing Utility
Some graphing utilities have built-in routines for solving problems involving the
mathematics of finance. For example, the TI-83/84 TVM SOLVER function incorporates
several functions that can be used to solve the problems that are encountered in Sections 4.1–4.3. To access the TVM SOLVER on the TI-83 press 2nd , press FINANCE ,
and then select 1: TVM Solver . To access the TVM Solver on the TI-83 plus and the
TI-84, press APPS , press 1: Finance , and then select 1: TVM Solver . Step-by-step
procedures for using these functions can be found on our Companion Web site.
EXAMPLE 1 Finding the Accumulated Amount of an Investment Find the
accumulated amount after 10 years if $5000 is invested at a rate of 10% per year
compounded monthly.
Solution
Using the TI-83/84 TVM SOLVER with the following inputs,
N ⫽ 120
N
= 120
I% = 10
PV = −5000
PMT = 0
FV = 13535.20745
P/Y = 12
C/Y = 12
PMT : END
BEGIN
I% ⫽ 10
PV ⫽ ⫺5000
PMT ⫽ 0
(10)(12)
Recall that an investment is an outflow.
FV ⫽ 0
P/Y ⫽ 12
The number of payments each year
C/Y ⫽ 12
The number of conversion periods each year
PMT:END BEGIN
FIGURE T1
The TI-83/84 screen showing the future
value (FV) of an investment
we obtain the display shown in Figure T1. We conclude that the required accumulated amount is $13,535.21.
EXAMPLE 2 Finding the Effective Rate of Interest Find the effective rate of
interest corresponding to a nominal rate of 10% per year compounded quarterly.
Eff (10, 4)
Excel
10.38128906
Excel has many built-in functions for solving problems involving the mathematics of
SolutiontheHere
functionvalue),
of the TI-83/84
calculator
to obtain
the
finance. Here we illustrate
usewe
of use
thethe
FVEff(future
EFFECT
(effective
rate),
result shown in Figure T2. The required effective rate is approximately 10.38% per
and the PV (presentyear.
value) functions to solve problems of the type that we have
encountered in Section 4.1.
FIGURE T2
EXAMPLE 3 Finding the Present Value of an Investment Find the present
value the
of $20,000
due in 5 years
if the interest
rateInvestment
is 7.5% per year
compounded
EXAMPLE 4 Finding
Accumulated
Amount
of an
Find
the
accumulated amountdaily.
after 10 years if $5000 is invested at a rate of 10% per year
compounded monthly.l i
Ui
h TI 83/84
i h h f ll i i
The TI-83/84 screen showing the effective rate of interest (Eff)
Exploring with Technology
Designed to explore mathematical concepts and
to shed further light on examples in the text,
these optional questions appear throughout the
main body of the text and serve to enhance the
student’s understanding of the concepts and
theory presented. Often the solution of an
example in the text is augmented with a
graphical or numerical solution. Complete
solutions to these exercises are given in the
Instructor’s Solutions Manual.
Exploring with
TECHNOLOGY
In the opening paragraph of Section 3.1, we pointed out that the accumulated
amount of an account earning interest compounded continuously will eventually
outgrow by far the accumulated amount of an account earning interest at the
same nominal rate but earning simple interest. Illustrate this fact using the following example.
Suppose you deposit $1000 in account I, earning interest at the rate of 10%
per year compounded continuously so that the accumulated amount at the end
of t years is A1(t) ⫽ 1000e0.1t. Suppose you also deposit $1000 in account II,
earning simple interest at the rate of 10% per year so that the accumulated
amount at the end of t years is A2(t) ⫽ 1000(1 ⫹ 0.1t). Use a graphing utility
to sketch the graphs of the functions A1 and A2 in the viewing window
[0, 20] ⫻ [0, 10,000] to see the accumulated amounts A1(t) and A2(t) over a
20-year period.