About the Cover
Peter Blair Henry received his first lesson in international economics at the age of eight, when
his family moved from the Caribbean island of Jamaica to affluent Wilmette, Illinois. Upon
arrival in the United States, he wondered why people in his new home seemed to have so
much more than people in Jamaica. The elusive answer to the question of why the average
standard of living can be so different from one country to another still drives him today as a
Professor of Economics in the Graduate School of Business at Stanford University.
Peter began his academic career on the campus of the University of North Carolina at Chapel
Hill, where he was a wide receiver on the varsity football team and a Phi Beta Kappa graduate
in economics. With an intrinsic love of learning and a desire to make the world a better place,
he knew that he wanted a career as an economist. He also knew that a firm foundation in
mathematics would help him to answer the real-life questions that fueled his passion for economics—a passion that earned him a Rhodes Scholarship to Oxford University, where he
received a B.A. in mathematics.
PETER BLAIR HENRY
International Economist
This foundation in mathematics prepared Peter for graduate study at the Massachusetts Institute
of Technology (MIT), where he received his Ph.D. in economics. While in graduate school, he
served as a consultant to the Governors of the Bank of Jamaica and the Eastern Caribbean Central Bank (ECCB). His research at the
ECCB helped provide the intellectual foundation for establishing the first stock market in the Eastern Caribbean Currency Area. His
research and teaching at Stanford has been funded by the National Science Foundation’s Early Career Development Program
(CAREER), which recognizes and supports the early career-development activities of those teacher-scholars who are most likely to
become the academic leaders of the 21st century. Peter is also a member of the National Bureau of Economic Research (NBER), a
nonpartisan economics think tank based in Cambridge, Massachusetts.
Peter Blair Henry’s love of learning and his questioning nature have led him to his desired career as an international economist
whose research positively impacts and addresses the tough decisions that face the world’s economies. It is his foundation in mathematics that enables him to grapple objectively with complex and emotionally charged issues of international economic policy reform,
such as debt relief for developing countries and its effect on international stock markets. The images accompanying Peter Blair
Henry’s portrait on the cover represent these vital issues faced by developing countries.
Look for other featured applied researchers in the following forthcoming titles in the Tan applied mathematics series:
CHRIS SHANNON
Economics and Finance
University of California,
Berkeley
MARK VAN DER LAAN
Biostatistician
University of California,
Berkeley
JONATHAN D. FARLEY
Applied Mathematician
California Institute
of Technology
NAVIN KHANEJA
Applied Scientist
Harvard University
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EDITION
8
APPLIED CALCULUS
FOR THE MANAGERIAL, LIFE,
AND SOCIAL SCIENCES
A BRIEF APPROACH
S. T. TAN
STONEHILL COLLEGE
Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States
Applied Calculus for the Managerial, Life,
and Social Sciences: A Brief Approach,
Eighth Edition
S. T. Tan
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1 2 3 4 5 6 7 12 11 10 09 08
CONTENTS
Preface vi
CHAPTER 1
Preliminaries
1.1
1.2
1.3
1.4
CHAPTER 2
1
Precalculus Review I 3
Precalculus Review II 15
The Cartesian Coordinate System 25
Straight Lines 33
Chapter 1 Summary of Principal Formulas and Terms 46
Chapter 1 Concept Review Questions 46
Chapter 1 Review Exercises 47
Chapter 1 Before Moving On 48
Functions, Limits, and the Derivative
2.1
2.2
2.3
49
Functions and Their Graphs 50
Using Technology: Graphing a Function 63
The Algebra of Functions 67
Functions and Mathematical Models 75
PORTFOLIO: Deb Farace 82
2.4
2.5
2.6
CHAPTER 3
Using Technology: Finding the Points of Intersection of Two Graphs and Modeling 92
Limits 97
Using Technology: Finding the Limit of a Function 115
One-Sided Limits and Continuity 117
Using Technology: Finding the Points of Discontinuity of a Function 131
The Derivative 133
Using Technology: Graphing a Function and Its Tangent Line 150
Chapter 2 Summary of Principal Formulas and Terms 152
Chapter 2 Concept Review Questions 152
Chapter 2 Review Exercises 153
Chapter 2 Before Moving On 156
Differentiation
3.1
3.2
3.3
3.4
3.5
157
Basic Rules of Differentiation 158
Using Technology: Finding the Rate of Change of a Function 169
The Product and Quotient Rules 171
Using Technology: The Product and Quotient Rules 180
The Chain Rule 182
Using Technology: Finding the Derivative of a Composite Function 193
Marginal Functions in Economics 194
Higher-Order Derivatives 208
Using Technology: Finding the Second Derivative of a Function at a Given Point 214
iv
CONTENTS
3.6
3.7
CHAPTER 4
Applications of the Derivative
4.1
4.2
4.3
4.4
4.5
CHAPTER 5
Implicit Differentiation and Related Rates 215
Differentials 227
Using Technology: Finding the Differential of a Function 236
Chapter 3 Summary of Principal Formulas and Terms 237
Chapter 3 Concept Review Questions 238
Chapter 3 Review Exercises 239
Chapter 3 Before Moving On 242
243
Applications of the First Derivative 244
Using Technology: Using the First Derivative to Analyze a Function 261
Applications of the Second Derivative 264
Using Technology: Finding the Inflection Points of a Function 282
Curve Sketching 283
Using Technology: Analyzing the Properties of a Function 295
Optimization I 298
Using Technology: Finding the Absolute Extrema of a Function 311
Optimization II 312
Chapter 4 Summary of Principal Terms 323
Chapter 4 Concept Review Questions 324
Chapter 4 Review Exercises 324
Chapter 4 Before Moving On 327
Exponential and Logarithmic Functions
5.1
5.2
5.3
329
Exponential Functions 330
Using Technology 336
Logarithmic Functions 338
Compound Interest 346
PORTFOLIO: Richard Mizak 349
5.4
5.5
5.6
CHAPTER 6
Using Technology: Finding the Accumulated Amount of an Investment, the Effective Rate
of Interest, and the Present Value of an Investment 359
Differentiation of Exponential Functions 360
Using Technology 370
Differentiation of Logarithmic Functions 372
Exponential Functions as Mathematical Models 380
Using Technology: Analyzing Mathematical Models 390
Chapter 5 Summary of Principal Formulas and Terms 393
Chapter 5 Concept Review Questions 393
Chapter 5 Review Exercises 394
Chapter 5 Before Moving On 396
Integration
6.1
6.2
6.3
6.4
397
Antiderivatives and the Rules of Integration 398
Integration by Substitution 411
Area and the Definite Integral 421
The Fundamental Theorem of Calculus 430
CONTENTS
6.5
6.6
6.7
CHAPTER 7
Evaluating Definite Integrals 441
Using Technology: Evaluating Definite Integrals for Piecewise-Defined Functions 451
Area between Two Curves 453
Using Technology: Finding the Area between Two Curves 463
Applications of the Definite Integral to Business and Economics 464
Using Technology: Business and Economic Applications 476
Chapter 6 Summary of Principal Formulas and Terms 477
Chapter 6 Concept Review Questions 478
Chapter 6 Review Exercises 479
Chapter 6 Before Moving On 482
Additional Topics in Integration
7.1
7.2
7.3
7.4
7.5
483
Integration by Parts 484
Integration Using Tables of Integrals 491
Numerical Integration 497
Improper Integrals 511
Applications of Calculus to Probability 520
PORTFOLIO: Gary Li 526
Chapter 7 Summary of Principal Formulas and Terms 531
Chapter 7 Concept Review Questions 532
Chapter 7 Review Exercises 532
Chapter 7 Before Moving On 534
CHAPTER 8
Calculus of Several Variables
8.1
8.2
8.3
535
Functions of Several Variables 536
Partial Derivatives 545
Using Technology: Finding Partial Derivatives at a Given Point 557
Maxima and Minima of Functions of Several Variables 558
PORTFOLIO: Kirk Hoiberg 560
8.4
8.5
8.6
APPENDIX
The Method of Least Squares 568
Using Technology: Finding an Equation of a Least-Squares Line 577
Constrained Maxima and Minima and the Method of Lagrange Multipliers 579
Double Integrals 589
Chapter 8 Summary of Principal Terms 603
Chapter 8 Concept Review Questions 603
Chapter 8 Review Exercises 604
Chapter 8 Before Moving On 606
Inverse Functions
607
Answers to Odd-Numbered Exercises
Index
651
613
v
PREFACE
M
ath is an integral part of our daily life. Applied Calculus for the Managerial, Life,
and Social Sciences: A Brief Approach, Eighth Edition, attempts to illustrate this point with
its applied approach to mathematics. This text is appropriate for use in a onesemester or a two-quarter introductory calculus course for students in the managerial, life,
and social sciences. My objective for this Eighth Edition is twofold: (1) To write an applied
text that motivates students and (2) to make the book a useful teaching tool for instructors.
I hope that with the present edition I have come one step closer to realizing my goal.
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 setting up word problems (see
Section 2.3).
Guidelines are given to help formulate and solve related-rates problems in Section 3.6.
In Chapter 4, optimization problems are covered in two sections. First, the techniques
of calculus are used to solve problems in which the function to be optimized is given
(Section 4.4); second, in Section 4.5, optimization problems that require the additional
step of formulating the problem are solved.
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:
■
■
■
■
■
■
■
■
■
■
■
Limits: The Motion of a Maglev
The algebra of functions: The U.S. Budget Deficit
The chain rule: The Population of Americans Aged 55 Years and Older
Differentials: Calculating Mortgage Payments
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
Approximating definite integrals: The Cardiac Output of a Heart
PREFACE
vii
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.
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 2.3, 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 Gives
Students a Plan of Action
A Diagnostic Test now precedes the
precalculus review. Each question is
referenced by the section and example in
the text where the relevant topic can be
reviewed. Students can now use this test
to diagnose their weaknesses and review
the material on an as needed basis.
Algebra Review Where
Students Need It Most
Well-placed algebra review notes, keyed
to the review chapter, appear where
students need them most throughout the
text. These are indicated by the (x ) icon.
See this feature in action on pages 105
and 537.
2
viii
PREFACE
Motivating Real-World
Applications
More than 140 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 2007
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.
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
44, page 335, and the
corresponding exercise where the
model is derived in Exercise 14,
page 337.)
Making Connections
with Technology
Many Using Technology
sections have been updated. A
new example—TV mobile
phones—has been added to
Using Technology 4.3. A new
Using Technology section has
been added to Section 5.3
(“Compound Interest”). Using
Technology 5.6 includes a new
example in which an
exponential model is
constructed—Internet gaming
sales—using the logistic
function of a graphing utility.
Additional graphing calculator
screens have been added in
some sections.
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 2.1,
page 57.)
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.
ix
x
PREFACE
Specific Content Changes
■
■
■
■
The precalculus review in Sections 1.1 and 1.2 has been reorganized. Operations with
algebraic expressions and factoring are now covered in Section 1.1, and inequalities and
absolute value are covered in Section 1.2. Two examples illustrating how nonlinear
inequalities are solved have been added.
Section 2.3 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 3.3 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.
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:
■
■
■
■
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
Self-Check Exercises
Offering students immediate feedback
on key concepts, these exercises begin
each end of section exercise set. Fully
worked-out solutions can be found at
the end of each exercise section.
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.
Exercises
Each exercise section contains an ample
set of problems of a routine
computational nature followed by an
extensive set of ap ...

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