Applied Calculus for the Managerial, Life

APPLIED CALCULUSFOR THE MANAGERIAL, LIFE, AND SOCIAL SCIENCESA BRIEF APPROACH

S. T. TANSTONEHILL COLLEGE

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Applied Calculus for the Managerial, Life,

and Social Sciences: A Brief Approach,

Eighth Edition

S. T. Tan

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ISBN-13: 978-0-495-38754-1

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CONTENTS

Preface vi

1.1 Precalculus Review I 31.2 Precalculus Review II 151.3 The Cartesian Coordinate System 251.4 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

2.1 Functions and Their Graphs 50Using Technology: Graphing a Function 63

2.2 The Algebra of Functions 672.3 Functions and Mathematical Models 75

PORTFOLIO: Deb Farace 82

Using Technology: Finding the Points of Intersection of Two Graphs and Modeling 92

2.4 Limits 97Using Technology: Finding the Limit of a Function 115

2.5 One-Sided Limits and Continuity 117Using Technology: Finding the Points of Discontinuity of a Function 131

2.6 The Derivative 133Using Technology: Graphing a Function and Its Tangent Line 150





3.1 Basic Rules of Differentiation 158Using Technology: Finding the Rate of Change of a Function 169

3.2 The Product and Quotient Rules 171Using Technology: The Product and Quotient Rules 180

3.3 The Chain Rule 182Using Technology: Finding the Derivative of a Composite Function 193

3.4 Marginal Functions in Economics 1943.5 Higher-Order Derivatives 208

Using Technology: Finding the Second Derivative of a Function at a Given Point 214

Preliminaries 1CHAPTER 1

Functions, Limits, and the Derivative 49CHAPTER 2

Differentiation 157CHAPTER 3

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3.6 Implicit Differentiation and Related Rates 2153.7 Differentials 227

Using Technology: Finding the Differential of a Function 236





4.1 Applications of the First Derivative 244Using Technology: Using the First Derivative to Analyze a Function 261

4.2 Applications of the Second Derivative 264Using Technology: Finding the Inflection Points of a Function 282

4.3 Curve Sketching 283Using Technology: Analyzing the Properties of a Function 295

4.4 Optimization I 298Using Technology: Finding the Absolute Extrema of a Function 311

4.5 Optimization II 312Chapter 4 Summary of Principal Terms 323




5.1 Exponential Functions 330Using Technology 336

5.2 Logarithmic Functions 3385.3 Compound Interest 346

PORTFOLIO: Richard Mizak 349

Using Technology: Finding the Accumulated Amount of an Investment, the Effective Rateof Interest, and the Present Value of an Investment 359

5.4 Differentiation of Exponential Functions 360Using Technology 370

5.5 Differentiation of Logarithmic Functions 3725.6 Exponential Functions as Mathematical Models 380

Using Technology: Analyzing Mathematical Models 390





6.1 Antiderivatives and the Rules of Integration 3986.2 Integration by Substitution 4116.3 Area and the Definite Integral 4216.4 The Fundamental Theorem of Calculus 430

iv CONTENTS

Applications of the Derivative 243CHAPTER 4

Exponential and Logarithmic Functions 329CHAPTER 5

Integration 397CHAPTER 6

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CONTENTS v

6.5 Evaluating Definite Integrals 441Using Technology: Evaluating Definite Integrals for Piecewise-Defined Functions 451

6.6 Area between Two Curves 453Using Technology: Finding the Area between Two Curves 463

6.7 Applications of the Definite Integral to Business and Economics 464Using Technology: Business and Economic Applications 476





7.1 Integration by Parts 4847.2 Integration Using Tables of Integrals 4917.3 Numerical Integration 4977.4 Improper Integrals 5117.5 Applications of Calculus to Probability 520

PORTFOLIO: Gary Li 526





8.1 Functions of Several Variables 5368.2 Partial Derivatives 545

Using Technology: Finding Partial Derivatives at a Given Point 557

8.3 Maxima and Minima of Functions of Several Variables 558PORTFOLIO: Kirk Hoiberg 560

8.4 The Method of Least Squares 568Using Technology: Finding an Equation of a Least-Squares Line 577

8.5 Constrained Maxima and Minima and the Method of Lagrange Multipliers 5798.6 Double Integrals 589

Chapter 8 Summary of Principal Terms 603




Answers to Odd-Numbered Exercises 613

Index 651

Additional Topics in Integration 483CHAPTER 7

Calculus of Several Variables 535CHAPTER 8

Inverse Functions 607APPENDIX

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PREFACE

Math 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 withits applied approach to mathematics. This text is appropriate for use in a one-semester 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 appliedtext 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.

Level of PresentationMy approach is intuitive, and I state the results informally. However, I have taken special careto ensure that this approach does not compromise the mathematical content and accuracy.

Problem-Solving ApproachA problem-solving approach is stressed throughout the book. Numerous examples andapplications illustrate each new concept and result. Special emphasis is placed on helpingstudents formulate, solve, and interpret the results of the problems involving applications.Because students often have difficulty setting up and solving word problems, extra carehas been taken to help students master these skills:

■ Very early on in the text, students are given practice in setting up word problems (seeSection 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 additionalstep of formulating the problem are solved.

Intuitive Introduction to ConceptsMathematical concepts are introduced with concrete, real-life examples wherever appro-priate. 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

THE APPROACH

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ConnectionsOne example (the maglev) is used as a common thread throughout the development of cal-culus—from limits through integration. The goal here is to show students the connectionsbetween the concepts presented—limits, continuity, rates of change, the derivative, thedefinite integral, and so on.

MotivationIllustrating the practical value of mathematics in applied areas is an important objective ofmy approach. Many of the applications are based on mathematical models (functions) thatI 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.

ModelingI believe that one of the important skills that a student should acquire is the ability to trans-late 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-onexperience constructing these models in the Using Technology sections.

PREFACE vii

Algebra Review Gives Students a Plan of ActionA Diagnostic Test now precedes theprecalculus review. Each question isreferenced by the section and example inthe text where the relevant topic can bereviewed. Students can now use this testto diagnose their weaknesses and reviewthe material on an as needed basis.

NEW TO THIS EDITION

Algebra Review WhereStudents Need It MostWell-placed algebra review notes, keyedto the review chapter, appear wherestudents need them most throughout thetext. These are indicated by the icon.See this feature in action on pages 105and 537.

(x2)

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viii PREFACE

Modeling with DataModeling with Data exercises arenow found in many of the UsingTechnology sections throughoutthe text. Students can actually seehow some of the functions found inthe exercises are constructed. (SeeInternet users in China, Exercise44, page 335, and thecorresponding exercise where themodel is derived in Exercise 14,page 337.)

Making Connectionswith TechnologyMany Using Technologysections have been updated. Anew example—TV mobilephones—has been added toUsing Technology 4.3. A newUsing Technology section hasbeen added to Section 5.3(“Compound Interest”). UsingTechnology 5.6 includes a newexample in which anexponential model isconstructed—Internet gamingsales—using the logisticfunction of a graphing utility.Additional graphing calculatorscreens have been added insome sections.

Motivating Real-WorldApplicationsMore than 140 new applications havebeen added to the Applied Examplesand Exercises. Among theseapplications are global warming,depletion of Social Security trust fundassets, driving costs for a 2007medium-sized sedan, hedge fundinvestments, mobile instant messagingaccounts, hiring lobbyists, Webconferencing, the autistic brain, therevenue of Polo Ralph Lauren, U.S.health-care IT spending, andconsumption of bottled water.

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PREFACE ix

Variety of ProblemTypesAdditional rote questions, trueor false questions, and conceptquestions have been addedthroughout the text to enhancethe exercise sets. (See, forexample, the graphical questionsadded to Concept Questions 2.1,page 57.)

Action-OrientedStudy TabsConvenient color-coded studytabs, similar to Post it® flags,make it easy for students to tabpages that they want to return tolater, whether it be foradditional review, exampreparation, online exploration,or identifying a topic to bediscussed with the instructor.

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In addition to the new features, we have retained many of the following hallmarks thathave 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

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 andabsolute value are covered in Section 1.2. Two examples illustrating how nonlinearinequalities are solved have been added.

■ Section 2.3 on functions and mathematical models has been reorganized, and new mod-els have been introduced. Here, students are now asked to use a model describing globalwarming to predict the amount of carbon dioxide (CO2) that will be present in theatmosphere in 2010 and a model describing the assets of the Social Security trust fundto determine when those assets are expected to be depleted.

■ The chain rule in Section 3.3 is now introduced with an application—the population ofAmericans aged 55 years and older.

■ A How-To Technology Index has been added for easy reference.

Self-Check ExercisesOffering students immediate feedbackon key concepts, these exercises begineach end of section exercise set. Fullyworked-out solutions can be found atthe end of each exercise section.

Concept QuestionsDesigned to test students’understanding of the basic conceptsdiscussed in the section, these questionsencourage students to explain learnedconcepts in their own words.

ExercisesEach exercise section contains an ampleset of problems of a routinecomputational nature followed by anextensive set of application-orientedproblems.

TRUSTED FEATURES

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PREFACE xi

Using TechnologyThese optional features appear after thesection exercises. They can be used inthe classroom if desired or as materialfor self-study by the student. Here, thegraphing calculator is used as a tool tosolve problems. These sections arewritten in the traditional example–exercise format, with answers given atthe back of the book. Illustrationsshowing graphing calculator screens areextensively used. In keeping with thetheme of motivation through real-lifeexamples, many sourced applications areagain included. Students can constructtheir own models using real-life data inmany of the Using Technology sections.These include models for the growth ofthe Indian gaming industry, health-carespending, TIVO owners, nicotinecontent of cigarettes, computer security,and online gaming, among others.

Exploring with TechnologyDesigned to explore mathematical concepts andto shed further light on examples in the text,these optional questions appear throughout themain body of the text and serve to enhance thestudent’s understanding of the concepts andtheory presented. Often the solution of anexample in the text is augmented with agraphical or numerical solution. Completesolutions to these exercises are given in theInstructor’s Solutions Manual.

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xii PREFACE

Before Moving On . . .Found at the end of each chapter review, theseexercises give students a chance to see if theyhave mastered the basic computational skillsdeveloped in each chapter. If they solve aproblem incorrectly, they can go to theCompanion Website and try again. In fact, theycan keep on trying until they get it right. Ifstudents need step-by-step help, they can use theCengageNOW Tutorials that are keyed to the textand work out similar problems at their own pace.

Review ExercisesOffering a solid review of the chaptermaterial, the Review Exercises containroutine computational exercises followedby applied problems.

Summary of PrincipalFormulas and TermsEach review section begins with the Sum-mary highlighting important equations andterms with page numbers given for quickreview.

Concept Review QuestionsThese questions give students a chance tocheck their knowledge of the basic defini-tions and concepts given in each chapter.

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PREFACE xiii

Explore & DiscussThese optional questions can bediscussed in class or assigned ashomework. These questions generallyrequire more thought and effort than theusual exercises. They may also be usedto add a writing component to the classor as team projects. Complete solutionsto these exercises are given in theInstructor’s Solutions Manual.

PortfoliosThe real-life experiences of a varietyof professionals who usemathematics in the workplace arerelated in these interviews. Amongthose interviewed are a senioraccount manager at PepsiCo and anassociate on Wall Street who usesstatistics and calculus in writingoptions.

Example VideosAvailable through the Online Resource Center and Enhanced WebAssign, these video examples offer hours of instruction from award-winning teacher Deborah Upton of Stonehill College. Watch as she walks students through key examples from the text, step by step—giving them a foundation in the skills that they need to know. Eachexample available online is identified by the video icon located in the margin.

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xiv PREFACE

INSTRUCTOR’S SOLUTIONS MANUAL (ISBN 0-495-38897-1) by Soo T. TanThe complete solutions manual provides worked out solutions to all problems in the text,as well as “Exploring with Technology” and “Explore & Discuss” questions.

POWERLECTURE (ISBN 0-495-38899-8)This comprehensive CD-ROM includes the Instructor’s Solutions Manual, PowerPointSlides, and ExamView® Computerized Testing featuring algorithmically generated ques-tions to create, deliver, and customize tests.

ENHANCED WEBASSIGNInstant feedback and ease of use are just two reasons why WebAssign is the most widelyused homework system in higher education. WebAssign allows you to assign, collect,grade, and record homework assignments via the Web. Now this proven homework sys-tem has been enhanced to include links to textbook sections, video examples, and prob-lem-specific tutorials. Enhanced WebAssign is more than a homework system—it is acomplete learning system for math students.

STUDENT SOLUTIONS MANUAL (ISBN 0-495-38898-X) by Soo T. TanGiving you more in-depth explanations, this insightful resource includes fully worked-outsolutions for the answers to select exercises included at the back of the textbook, as wellas problem-solving strategies, additional algebra steps, and review for selected problems.

ONLINE RESOURCE CENTER (ISBN 0-495-56369-2)Sign in, save time, and get the grade you want! One code will give you access to great toolsfor Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach,Eighth Edition. It includes Personal Tutor (online tutoring with an expert that offers helpright now), CengageNOW (an online diagnostic, homework, and tutorial system), andaccess to new Solution Videos on the password-protected Premium Website.

I wish to express my personal appreciation to each of the following reviewers of the EighthEdition, whose many suggestions have helped make a much improved book.

Paul Abraham Kent State University—Stark

James Adair Missouri Valley College

Debra D. Bryant Tennessee Technological University

Christine DevenaMiles Community College

Kevin Ferland Bloomsburg University

Murray Lieb New Jersey Institute of Technology

Rebecca Lynn Colorado State University

Carol Overdeep Saint Martin's University

Mohammed Rajah Miracosta College

Brian Rodas Santa Monica College

Mohammad Siddique Virginia Union University

Jennifer Strehler Oakton Community College

Ray Toland Clarkson University

Justin Wyss-Gallifent University of Maryland at College Park

TEACHING AIDS

LEARNING AIDS

ACKNOWLEDGMENTS

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PREFACE xv

I also thank the following reviewers whose comments and suggestions for previouseditions have greatly shaped the current form of this edition.

Faiz Al-Rubaee University of North Florida

James V. Balch Middle Tennessee State University

Albert Bronstein Purdue University

Kimberly Jordan Burch Montclair State University

Michael Button San Diego City College

Peter Casazza University of Missouri–Columbia

Matthew P. Coleman Fairfield University

William Coppage Wright State University

Lisa Cox Texas A&M University

Frank Deutsch Penn State University

Carl Droms James Madison University

Bruce Edwards University of Florida at Gainesville

Janice Epstein Texas A&M University

Gary J. Etgen University of Houston

Charles S. Frady Georgia State University

Howard Frisinger Colorado State University

Larry Gerstein University of California at Santa Barbara

Matthew Gould Vanderbilt University

Harvey Greenwald California Polytechnic State University—SanLuis Obispo

John Haverhals Bradley University

Yvette Hester Texas A&M University

Frank Jenkins John Carroll University

David E. Joyce Clark University

Mohammed Kazemi University of North Carolina–Charlotte

James H. Liu James Madison University

Norman R. Martin Northern Arizona University

Sandra Wray McAfee University of Michigan

Maurice Monahan South Dakota State University

Dean Moore Florida Community College at Jacksonville

Ralph J. Neuhaus University of Idaho

Gertrude Okhuysen Mississippi State University

James Olsen North Dakota State University

Lloyd Olson North Dakota State University

Wesley Orser Clark College

Richard Porter Northeastern University

Virginia Puckett Miami Dade College

Richard Quindley Bridgewater State College

Mary E. Rerick University of North Dakota

Thomas N. Roe South Dakota State University

Donald R. Sherbert University of Illinois

Anne Siswanto East Los Angeles College

Jane Smith University of Florida

Devki Talwar Indiana University of Pennsylvania

Larry TaylorNorth Dakota State University

Giovanni Viglino Ramapo College of New Jersey

Hiroko K. Warshauer Texas State University–San Marcos

Lawrence V. Welch Western Illinois University

Jennifer Whitfield Texas A&M University

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xvi PREFACE

I also wish to thank Kevin Charlwood and Tao Guo for their many helpful sugges-tions for improving the text. I am also grateful to Kevin for helping with the accuracychecking of this text. I also thank the editorial, production, and marketing staffs ofBrooks/Cole: Carolyn Crockett, Danielle Derbenti, Catie Ronquillo, Cheryll Linthicum,Mandy Jellerichs, Sam Subity, Jennifer Liang, and Rebecca Dashiell for all of their helpand support during the development and production of this edition. I also thank MarthaEmry and Betty Duncan who both did an excellent job of ensuring the accuracy and read-ability of this edition. Simply stated, the team I have been working with is outstanding, andI truly appreciate all their hard work and efforts. Finally, a special thanks to the mathe-maticians—Chris Shannon and Mark van der Lann at Berkeley, Peter Blair Henry at Stan-ford, Jonathan D. Farley at Cal Tech, and Navin Khaneja at Harvard for taking time offfrom their busy schedules to describe how mathematics is used in their research. Their pic-tures and applications of their research appear on the covers of my applied mathematicsseries.

S. T. Tan

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ABOUT THE AUTHOR

SOO T. TAN received his S.B. degree from Massachusetts Institute

of Technology, his M.S. degree from the University of

Wisconsin–Madison, and his Ph.D. from the University of California

at Los Angeles. He has published numerous papers in Optimal

Control Theory, Numerical Analysis, and Mathematics of Finance. He

is currently a Professor of Mathematics at Stonehill College.

By the time I started writing the first of what turned out to be a series of textbooks in

mathematics for students in the managerial, life, and social sciences, I had quite a few

years of experience teaching mathematics to non-mathematics majors. One of the most

important lessons I learned from my early experience teaching these courses is that many

of the students come into these courses with some degree of apprehension. This aware-

ness led to the intuitive approach I have adopted in all of my texts. As you will see, I try to

introduce each abstract mathematical concept through an example drawn from a com-

mon, real-life experience. Once the idea has been conveyed, I then proceed to make it

precise, thereby assuring that no mathematical rigor is lost in this intuitive treatment of

the subject. Another lesson I learned from my students is that they have a much greater

appreciation of the material if the applications are drawn from their fields of interest and

from situations that occur in the real world. This is one reason you will see so many exer-

cises in my texts that are modeled on data gathered from newspapers, magazines, jour-

nals, and other media. Whether it be the market for cholesterol-reducing drugs, financing

a home, bidding for cable rights, broadband Internet households, or Starbucks’ annual

sales, I weave topics of current interest into my examples and exercises to keep the book

relevant to all of my readers.

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TO PAT, BILL, AND MICHAEL

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Applied Calculus for the Managerial, Life

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