Geoffrey c berresford, andrew m rockett brief applied calculus, 5th edition brooks cole (2010)(1)

1.3 Functions: Linear and Quadratic 331.4 Functions: Polynomial, Rational, and Exponential 50 Chapter Summar y with Hints and Suggestions 70 Review Exercises for Chapter 1 71 DERIVATIVES

Altitude and Olympic games, 272

Athletic field design, 219

Pole vaulting improvement, 295

Pythagorean baseball standings,

507, 520

World record 100-meter run, 303

World record mile run, 3, 17

and Reynolds number, 304

Blood vessel volume, 570

Body surface area, 506

Drug sensitivity, 162

Efficiency of animal motion, 220, 252

Epidemics, 121, 338, 345, 363, 480,

484, 492Fever, 150Fever thermometers, 491, 492Fick’s law, 471

Future life expectancy, 19

Gene frequency, 424, 430Glucose levels, 469, 485

Gompertz growth curve, 289, 304,

443, 486

Half-life of a drug, 289

Heart function, 471Heart medication, 191

Heart rate, 31 Height of a child, 363 Heterozygosity, 289 Leukemic cell growth, 68 Life expectancy, 19 Life expectancy and education, 124 Longevity and exercise, 218 Lung cancer and asbestos, 123

Medication ingestion, 254Mosquitoes, 271

Murrell’s rest allowance, 139Nutrition, 586

Oxygen consumption, 507

Penicillin dosage, 281

Poiseuille’s law and blood flow,

249, 365Pollen count, 217Population and individualbirthrate, 486

Reed-Frost epidemic model, 272 Ricker recruitment, 289, 304

Smoking and longevity, 543

Tainted meat, 50

Tumor growth, 248

Weight of a teenager, 401, 405

Environmental Sciences

Air temperature and altitude, 73 Animal size, 72

Average air pollution, 582Beverton-Holt recruitment curve,

20, 139

Biodiversity, 31 Carbon dioxide pollution, 73

Carbon monoxide pollution, 162

Consumption of natural resources,

341, 346, 347, 363, 404, 490Cost of cleaner water, 128Deer population, 484

Flexfast Rubber Company, 469 Global temperatures, 150, 335 Greenhouse gases and global warming, 162, 405

Growth of an oil slick, 156Harvest yield, 223, 228Light penetrating seawater, 271Maximizing farm revenue, 253Maximum sustainable yield, 235,

237, 237, 254 Nuclear waste, 288

Pollution, 218, 335, 375, 396,

401, 420Pollution and absenteeism, 544Predicting animal population, 479

Radioactive medical tracers, 287

Radioactive waste, 489

Rain forest depletion, 287 Sea level, 150

Sulfur oxide pollution, 245

Tag and recapture estimates, 506

Water quality, 122, 137, 491Water reservoir average depth, 397Water usage, 59

Wind power, 49, 69, 218 World solar cell production, 302

Management Science, Business, and

Titles or page numbers in italics indicate applications of greater generality or significance, most including source

citations that allow those interested to pursue these topics in more detail

Apple stock price, 315

Asset appreciation, 270

AT&T net income, 197, 205, 335

AT&T stock price, 347

Capital value of an asset, 365, 442

Car phone sales, 484

Complementary commodities, 521

Compound interest, 376

Compound interest growth

times, 289Computer expenditures, 492

Energy usage, 17

Estimating additional profit, 563

EZCie LED flashlight, 115

Gini index, 406

Gross domestic product, 307, 331

Gross world product, 454 Handheld computers, 68 Honeywell International, 108

Income tax, 67Insurance reserves, 68Interest compoundedcontinuously, 93Investment growth, 453Learning curve in airplane

production, 26, 31, 172

Least cost rule, 559Lot size, 236–238, 254

Macintosh computers, 43, 101, 376

Marginal and average cost, 191

Marginal average cost, 130,

Marketing to young adults, 121

Maximizing present value, 321Maximizing production, 552, 558Maximum production, 587Maximum profit, 41, 48, 74, 211,

218, 222, 227, 229, 253, 526,

531, 532, 585, 586Maximum revenue, 219, 227, 303,

321, 532

MBA salaries, 17 Microsoft net income, 205, 335, 376

Mineral deposit value, 582Minimizing inventory costs,

231, 233Minimizing package materials, 224Minimum cost, 253

Mobile phones, 27, 32 MP3 players, 32, 197, 484

National debt, 150, 316Net savings, 377Oil demand, 320Oil prices, 227Oil well output, 442

Optical computer mice, 173 Pareto’s law of income distribution, 365

Pasteurization temperature, 107

Per capita cigarette production, 218 Per capita national debt, 138 Per capita personal income, 17 POD (printing on demand), 130 PowerZip, 121

Predicting sales, 534, 542Present value of a continuousstream of income, 415, 419,

420, 489Present value of preferred stock, 443

Price and quantity, 221Price discrimination, 531, 532, 585Producers’ surplus, 383

Product recognition, 420, 483Product reliability, 442Production possibilities, 558Production runs, 237, 254Profit, 150, 151, 195, 244, 248, 254,

377, 515, 569Pulpwood forest value, 220Quality control, 272

Quest Communication, 49

Research expenditures, 68

Research In Motion stock price,

315, 347Returns to scale, 506

Revenue, 74, 108, 204, 248, 254,

334, 401, 419Rule of 6, 30Rule of 72, 289Salary, 47Sales, 137, 173, 204, 247, 248, 250,

302, 320, 345, 363, 364, 375,

443, 470, 481, 493, 520, 587Sales from celebrity

endorsement, 369

Satellite radio, 19, 73

Simple interest, 73

Slot machines, 18 Southwest Airlines, 49

Stock “limiting” market value, 484Straight-line depreciation, 18, 72

Super Bowl ticket costs, 545

Supply, 248, 250Tax revenue, 226, 228Temperature, 204Timber forest value, 209, 218Total productivity, 357

Total profit, 404

Total sales, 340, 400, 423, 430,

483, 492Total savings, 346

U.S oil production, 69

Automobile depreciation, 302

Automobile driving costs, 542

Central Bank of Brazil bonds, 288

Cézanne painting appreciation, 271

College trust fund, 270

Comparing interest rates, 266,

271, 319Compound interest, 162, 174, 261,

302, 319, 406Continuous compounding, 265

Cost of maintaining a home, 346

Depreciating a car, 262, 286

Earnings and calculus, 273, 302

European Bank bonds, 288

Federal income tax, 56

Parking space in Manhattan, 72

Stock price, 406

Stock yield, 505

Toyota Corolla depreciation, 271

Value of an investment, 346

Zero coupon bond, 270, 271

Social and Behavioral Sciences

Absenteeism, 532Advertising effectiveness, 288

Age at first marriage, 18

Campaign expenses, 229Cell phone usage, 288Cephalic index, 506

Cigarette tax revenue, 253

Cobb-Douglas productionfunction, 498

Cost of congressional victory, 545

Cost of labor contract, 378Crime, 543

Dating older women, 287Demand for oil, 316Diffusion of information, 284, 287,

302, 303, 320, 475

Divorces, 346

Early human ancestors, 287

Ebbinghaus model of memory, 302 Education and income, 287 Election costs, 272

Employment seekers, 431

Equal pay for equal work, 18

Forgetting, 272, 287Fund raising, 420GDP relative growth rate, 321

Gender pay gap, 495 Gini index of income distribution,

Procrastination, 521

Repetitive tasks, 364, 400, 405Response rate, 431

Smoking and education, 122 Smoking and income, 18

Spread of rumors, 480, 484, 492

Status, income, and education, 161,

162, 205, 520 Stevens’ Law of Psychophysics, 177 Stimulus and response, 205

Traffic accidents, 249

Violent crime, 586

Voting, 484Welfare, 249

World energy output, 286 World population, 67, 146, 271

Topics of General Interest

Accidents and driving speed, 124 Aging of America, 586

Aging world population, 545

Airplane flight path, 190, 205Approximation of , 454Area between curves, 364, 400,

420, 442Automobile age, 490

Automobile fatalities, 485

Average population, 400, 588Average temperature, 375, 577, 582

Birthrate in Africa, 377

Boiling point and altitude, 47Box design, 253, 254, 506Building design, 558Bus shelter design, 229Carbon 14 dating, 257, 282,

287, 288

Cave paintings, 287

Cell phones, 68 Cigarette smoking, 323, 364 College tuition, 123

Dead Sea Scrolls, 282

Designing a tin can, 549

Emergency stopping distance, 569

Estimating error in calculating

volume, 564, 566Eternal recognition, 409

Ice cream cone price increases, 363

Impact time of a projectile, 48

Impact velocity, 48, 150

Internet access, 13, 122

Internet host computers, 420

Largest enclosed area, 213, 219,

228, 252, 547

Largest postal package, 228, 557, 558

Largest product with fixed sum, 219

Lives saved by seat belts, 378

Manhattan Island purchase, 270Maximum height of a bullet, 150Measurement errors, 569Melting ice, 254

Mercedes-Benz Brabus Rocket speed, 335

Millwright’s water wheel rule, 218

Minimizing cost of materials, 228Minimum perimeter rectangle, 229Moore’s law of computer

253, 558Page layout, 229Parking lot design, 219Permanent endowments, 437, 441,

444, 490Population, 107, 162, 174, 316, 364,

404, 420, 430, 486, 489

Porsche Cabriolet speed, 334 Postage stamps, 492, 545 Potassium 40 dating, 287, 288 Raindrops, 485

Rate of growth of a circle, 172Rate of growth of a sphere, 173Relative error in calculations,

569, 587Relativity, 93Repetitive tasks, 363Richter scale, 31Rocket tracking, 249

Scuba dive duration, 506, 569 Seat belt use, 19

Shroud of Turin, 257, 287

Smoking, 543

Smoking and education, 49

Smoking mortality rates, 536Snowballs, 248

Soda can design, 253

Speed and skid marks, 32

Speeding, 249

St Louis Gateway Arch, 273 Stopping distance, 47 Superconductivity, 77, 93

Survival rate, 175Suspension bridge, 454Telephone calls, 569Temperature conversion, 17Thermos bottle temperature, 320

Time of a murder, 469

Time saved by speeding, 131Total population of a region, 582Total real estate value, 588

Traffic safety, 122 Tsunamis, 48

Volume of a building, 582Volume under a tent, 576Warming beer, 303Water pressure, 47Waterfalls, 31Wheat yield, 543Wind speed, 71

Windchill index, 150, 498, 507,

520, 569Window design, 219

Wine appreciation, 228 World oil consumption, 453 World population, 302, 539

World’s largest city: now and later, 319

Young-adult population, 335

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Brief Applied Calculus

F I F T H E D I T I O N

Geoffrey C Berresford

Long Island University

Andrew M Rockett

Long Island University

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Brief Applied Calculus, Fifth Edition

Geoffrey C Berresford

Andrew M Rockett

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1 2 3 4 5 6 7 12 11 10 09 08

1.3 Functions: Linear and Quadratic 33

1.4 Functions: Polynomial, Rational, and Exponential 50 Chapter Summar y with Hints and Suggestions 70 Review Exercises for Chapter 1 71

DERIVATIVES AND THEIR USES

2.2 Rates of Change, Slopes, and Derivatives 94

2.6 The Chain Rule and the Generalized Power Rule 152

2.7 Nondifferentiable Functions 164 Chapter Summar y with Hints and Suggestions 169 Review Exercises for Chapter 2 171

FURTHER APPLICATIONS OF DERIVATIVES

3.1 Graphing Using the First Derivative 178

3.2 Graphing Using the First and Second Derivatives 193

3

2 1 Contents

v

3.4 Further Applications of Optimization 221

3.5 Optimizing Lot Size and Harvest Size 230

3.6 Implicit Differentiation and Related Rates 238 Chapter Summar y with Hints and Suggestions 250 Review Exercises for Chapter 3 252

Cumulative Review for Chapters 1–3 255

EXPONENTIAL AND LOGARITHMIC FUNCTIONS

4.3 Differentiation of Logarithmic and Exponential Functions 290

4.4 Two Applications to Economics: Relative Rates and Elasticity of Demand 306

Chapter Summar y with Hints and Suggestions 318 Review Exercises for Chapter 4 319

INTEGRATION AND ITS APPLICATIONS

5.1 Antiderivatives and Indefinite Integrals 324

5.2 Integration Using Logarithmic and Exponential Functions 336

5.3 Definite Integrals and Areas 348

5.4 Further Applications of Definite Integrals: Average Value

5.5 Two Applications to Economics: Consumers’ Surplus

5.6 Integration by Substitution 388 Chapter Summar y with Hints and Suggestions 402 Review Exercises for Chapter 5 403

INTEGRATION TECHNIQUES AND DIFFERENTIAL EQUATIONS

6.6 Further Applications of Differential Equations:

Chapter Summar y with Hints and Suggestions 487 Review Exercises for Chapter 6 489

CALCULUS OF SEVERAL VARIABLES

7.1 Functions of Several Variables 496

7.2 Partial Derivatives 508

7.3 Optimizing Functions of Several Variables 522

7.5 Lagrange Multipliers and Constrained Optimization 546

7.6 Total Differentials and Approximate Changes 559

7.7 Multiple Integrals 570 Chapter Summar y with Hints and Suggestions 583 Review Exercises for Chapter 7 585

Cumulative Review for Chapters 1–7 588

Answers to Selected Exercises A1

7

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Preface

A scientific study of yawning found that more yawns occurred in calculus class than anywhere else.* This book hopes to remedy that situation Rather than being another dry recitation of standard results, our presentation ex- hibits some of the many fascinating and useful applications of mathematics

in business, the sciences, and everyday life Even beyond its utility, ever, there is a beauty to calculus, and we hope to convey some of its ele- gance and simplicity.

how-This book is an introduction to calculus and its applications to the agement, social, behavioral, and biomedical sciences, and other fields The

man-seven-chapter Brief Applied Calculus contains more than enough material for a one-semester course, and the eleven-chapter Applied Calculus contains

additional chapters on trignometry, differential equations, sequences and series, and probability for a two-semester course The only prerequisites are some knowledge of algebra, functions, and graphing, which are reviewed

in Chapter 1.

CHANGES IN THE FIFTH EDITION

First, what has not changed is the essential character of the book: simple,

clear, and mathematically correct explanations of calculus, alternating with relevant and engaging examples.

Exercises We have added many new exercises, including new Applied

Exer-cises and Conceptual ExerExer-cises, and have updated others with new data Many

exercises now have sources (book or journal names or website addresses) to

establish their factual basis and enable further research In Chapter 1 we have

added regression (modeling) exercises, in which students use calculators to fit

equations to actual data (see, for example, pages 19 and 32) Throughout the

book we have added what may be termed Wall Street exercises (pages 205 and

315), applications based on financial data from sources that are provided The regression exercises in Chapter 1 illustrate the methods used to develop the models in the Applied Exercises throughout the book.

New or Modified Topics We have expanded our treatment of the following

topics: limits involving infinity (pages 83–85), graphing rational functions (pages 184–187), and elasticity of demand (pages 309–315) To show how to

solve the regression (modeling) exercises in Chapter 1 we have added

(optional) examples on regression (linear on page 13, power on page 27,

quadratic on page 43, and exponential on page 62) In addition to these

expanded applications, we have included some more difficult exercises (see,

*Ronald Baenninger, “Some Comparative Aspects of Yawning in Betta splendens, Homo

sapiens, Panthera leo, and Papoi spinx,” Journal of Comparative Psychology 101 (4).

for example, pages 136 and 161), and provided a complete proof of the Chain Rule based on Carathédory’s definition of the derivative (page 163) To ac- commodate these additions without substantially lengthening the book we have tightened the exposition in every chapter.

Pedagogy We have redrawn many graphs for improved accuracy and ity We have relocated some examples immediately to the right of the boxes

clar-that summarize results, calling them Brief Examples, thereby providing

im-mediate reinforcement of the concepts (see, for example, pages 21 and 23).

FEATURES

Realistic Applications The basic nature of courses using this book is very

“applied” and therefore this book contains an unusually large number of applications, many appearing in no other textbook We explore learning curves in airplane production (pages 26–27 and 31), corporate operating revenues (page 49), the age of the Dead Sea Scrolls (pages 282–283), the distance traveled by sports cars (pages 334–335), lives saved by seat belts (page 378), as well as the cost of a congressional victory (page 545) These and many other applications convincingly show that mathematics is more than just the manipulation of abstract symbols and is deeply connected to everyday life.

Graphing Calculators (Optional) Using this book does not require a graphing

calculator, but having one will enable you to do many problems more easily and at the same time deepen your understanding by allowing you to concen-

trate on concepts Throughout the book are Graphing Calculator Explorations and Graphing Calculator Exercises (marked by the symbol ), which

explore interesting applications, such as when men and women will achieve

equal pay (page 18),

carry out otherwise “messy” calculations, such as the population growth

comparisons on pages 268 and 272, and

show the advantages and limitations of technology, such as the differences

between ln x2 and 2 ln x on page 279.

While any graphing calculator (or a computer) may be used, the displays

shown in the text are from the Texas Instruments TI-84, except for a few from the TI-89 A discussion of the essentials of graphing calculators follows this

preface For those not using a graphing calculator, the Graphing Calculator

Explorations have been carefully planned so that most can also be read

simply for enrichment (as with the concavity and maximization problems

on pages 195 and 216) Students, however, will need a calculator with keys

like and for powers and natural logarithms.

Graphing Calculator Programs (Optional) Some topics require extensive calculation, and for them we have created (optional) graphing calculator programs for use with this book We provide these programs for free to all

In

yx

students and faculty (see “How to Obtain Graphing Calculator Programs” later in this preface) The topics covered are: Riemann sums (page 350), trapezoidal approximation (page 447), Simpson’s rule (page 451), and slope fields (page 461) These programs allow the student to concentrate on the results rather than the computation.

Spreadsheets (Optional) While access to a computer is not necessary for

this book, the Spreadsheet Explorations allow deeper exploration of some

topics We have included spreadsheet explorations of: nondifferentiable functions (pages 167–168), maximizing an enclosed area (pages 213–214), elasticity of demand (page 313), consumption of natural resources (page 343), improper integrals (page 436), and graphing a function of two variables (page 502) Ancillary materials for Microsoft Excel are also available (see

“Resources for the Student” later in this preface).

Enhanced Readability We have added space around all in-line mathematics

to make them stand out from the narrative An elegant four-color design increases the visual appeal and readability For the sake of continuity, refer- ences to earlier material are minimized by restating results whenever they are used Where references are necessary, explicit page numbers are given.

Application Previews Each chapter begins with an Application Preview

that presents an interesting application of the mathematics developed in that chapter Each is self-contained (although some exercises may later refer

to it) and serves to motivate interest in the coming material Topics include: world records in the mile run (pages 3–4), Stevens’ law of psychophysics (page 177), and cigarette smoking (pages 323–324).

Practice Problems Learning mathematics requires your active tion—“mathematics is not a spectator sport.” Throughout the readings are

participa-short pencil-and-paper Practice Problems designed to consolidate your

understanding of one topic before moving ahead to another, such as using negative exponents (page 22) or finding and checking an indefinite integral (page 325).

Annotations Notes to the right of many mathematical formulas and

manipulations state the results in words, assisting the important skill of

reading mathematics, as well as providing explanations and justifications for

the steps in calculations (see page 100) and interpretations of the results (see page 198).

Extensive Exercises Anyone who ever learned any mathematics did so

by solving many many problems, and the exercises are the most essential part of the learning process The exercises (see, for instance, pages 286–289) are graded from routine drills to significant applications, and some conclude

with Explorations and Excursions that extend and augment the material

presented in the text The Conceptual Exercises were described earlier in this preface Exercises marked with the symbol require a graphing calculator

Answers to odd-numbered exercises and answers to all Chapter Review

ex-ercises are given at the end of the book (full solutions are given in the Student

Solutions Manual).

Explorations and Excursions At the end of some exercise sets are optional problems of a more advanced nature that carry the development of certain top- ics beyond the level of the text, such as: the Beverton-Holt recruitment curve (page 20), average and marginal cost (page 192), elasticity of supply (page 317), and competitive and complementary commodities (page 521).

Conceptual Exercises These short problems are true/false, yes/no, or in-the-blank quick-answer questions to reinforce understanding of a subject without calculations (see, for example, page 93) We have found that students actually enjoy these simple and intuitive questions at the end of a long chal- lenging assignment.

fill-This “Be Careful” icon warns students of possible misunderstandings (see page 52) or particular difficulties (see page 127).

Just-in-Time Review We understand that many students have weak algebra skills Therefore, rather than just “reviewing” material that they never mas- tered in the first place, we keep the review chapter brief and then reinforce algebraic skills throughout the exposition with blue annotations immedi- ately to the right of the mathematics in every example We also review expo- nential and logarithmic functions again just before they are differentiated in Section 4.3 This puts the material where it is relevant and more likely to be remembered.

Levels of Reinforcement Because there are many new ideas and niques in this book, learning checks are provided at several different lev-

tech-els As noted above, Practice Problems encourage mastery of new skills directly after they are introduced Section Summaries briefly state both essential formulas and key concepts (see page 202) Chapter Summaries

review the major developments of the chapter and are keyed to particular

chapter review exercises (see pages 250–251) Hints and Suggestions at

the end of each chapter summary unify the chapter, give specific minders of essential facts or “tricks” that might be otherwise overlooked

re-or fre-orgotten, and list a selection of the review exercises fre-or a Practice Test

of the chapter material (see page 251) Cumulative Reviews at the end of

groups of chapters unify the materials developed up to that point (see page 255).

Accuracy and Proofs All of the answers and other mathematics have been carefully checked by several mathematicians The statements of defi- nitions and theorems are mathematically accurate Because the treatment

is applied rather than theoretical, intuitive and geometric justifications have often been preferred to formal proofs Such a justification or proof ac- companies every important mathematical idea; we never resort to phrases

like “it can be shown that ” When proofs are given, they are correct and honest.

Philosophy We wrote this book with several principles in mind One is that

to learn something, it is best to begin doing it as soon as possible Therefore, the preliminary material is brief, so that students begin calculus without delay An early start allows more time during the course for interesting applications and necessary review Another principle is that the mathematics should be done with the applications Consequently, every section contains applications (there are no “pure math” sections).

Prerequisites The only prerequisite for most of this book is some edge of algebra, graphing, and functions, and these are reviewed in Chapter 1 Other review material has been placed in relevant locations in later chapters.

knowl-Resources on the Web Additional materials available on the Internet at www.cengage.com/math/berresford include:

Suggestions for Projects and Essays, open-ended topics that ask

stu-dents (individually or in groups) to research a relevant person or idea, to compare several different mathematical ideas, or to relate a concept to their lives (such as marginal and average cost, why two successive 10% increases don’t add up to a 20% increase, elasticity of supply of drugs and alcohol, and arithmetic versus geometric means).

An expanded collection of Application Previews, short essays that were

used in an earlier edition to introduce each section Topics include

Exponential Functions and the World’s Worst Currency; Size, Shape, and Exponents; and The Confused Creation of Calculus.

HOW TO OBTAIN GRAPHING CALCULATOR PROGRAMS AND EXCEL SPREADSHEETS

The optional graphing calculator programs used in the text have been ten for a variety of Texas Instruments Graphing Calculators (including the

writ-TI-83, TI-84, TI-85, TI-86, TI-89, and TI-92), and may be obtained for free, in

any of the following ways:

■ If you know someone who already has the programs on a Texas ments graphing calculator like yours, you can easily transfer the pro- grams from their calculator to yours using the black cable that came with the calculator and the LINK button.

Instru-■ You may download the programs and instructions from the Cengage website at www.cengage.com/math/berresford onto a computer and then to your calculator using a USB cable.

The Microsoft Excel spreadsheets used in the Spreadsheet Explorations may be obtained for free by downloading the spreadsheet files from the Cengage website at www.cengage.com/math/berresford.

RESOURCES FOR THE INSTRUCTOR

Instructor’s Solutions Manual The Instructor’s Solutions Manual contains worked-out solutions for all exercises in the text It is available on the Instruc- tor’s book companion website.

Computerized Test Bank Create, deliver and customize tests and study guides in minutes with this easy-to-use assessment software on CD The thousands of algorithmic questions in the test bank are derived from the text- book exercises, ensuring consistency between exams and the book.

WebAssign Instant feedback, grading precision, and ease of use are just three reasons why WebAssign is the most widely used homework system in higher education WebAssign’s homework delivery system lets instructors deliver, collect, grade and record assignments via the web And now, this proven system has been enhanced to include additional resources for instructors and students.

RESOURCES FOR THE STUDENT

Student Solutions Manual Need help with your homework or to prepare for an exam? The Student Solutions Manual contains worked-out solutions for all odd-numbered exercises in the text It is a great resource to help you work through those tough problems.

DVD Lecture Series These comprehensive, instructional lecture tions serve a number of uses They are great if you need to catch up after missing a class, need to supplement online or hybrid instruction, or need material for self-study or review.

presenta-Microsoft Excel Guide by Revathi Narasimhan This guide provides list of exercises from the text that can be completed after each step-by-step Excel example No prior knowledge of Excel is necessary.

WebAssign WebAssign, the most widely used homework system in higher education, offers instant feedback and repeatable problems—everything you could ask for in an online homework system WebAssign’s homework sys- tem lets you practice and submit homework via the web It is easy to use and loaded with extra resources.

We are indebted to many people for their useful suggestions, conversations, and correspondence during the writing and revising of this book We thank Chris and Lee Berresford, Anne Burns, Richard Cavaliere, Ruth Enoch, Theodore Faticoni, Jeff Goodman, Susan Halter, Brita and Ed Immergut, Ethel Matin, Gary Patric, Shelly Rothman, Charlene Russert, Stuart Saal, Bob Sickles, Michael Simon, John Stevenson, and all of our “Math 6” students at C.W Post for serving as proofreaders and critics over the past years.

We had the good fortune to have had supportive and expert editors at Cengage Learning: Molly Taylor (senior sponsoring editor), Maria Morelli (development editor), Kerry Falvey (production editor), Roger Lipsett (accuracy reviewer), and Holly McLean-Aldis (proofreader) They made the difficult tasks seem easy, and helped beyond words We also express our gratitude to the many others at Cengage Learning who made important contributions too numerous to mention.

The following reviewers have contributed greatly to the development of the fifth edition of this text:

Frederick Adkins Indiana University of Pennsylvania

David Allen Iona College, NY

Joel M Berman Valencia Community College, FL

Julane Crabtree Johnson Community College, KS

Biswa Datta Northern Illinois University

Allan Donsig University of Nebraska—Lincoln

Sally Edwards Johnson Community College, KS

Frank Farris Santa Clara University, CA

Brad Feldser Kennesaw State University, GA

Abhay Gaur Duquesne University, PA

Jerome Goldstein University of Memphis, TN

John B Hawkins Georgia Southern University

John Karloff University of North Carolina

Todd King Michigan Technical University

Richard Leedy Polk Community College, FL

Sanjay Mundkur Kennesaw State University, GA

David Parker Salisbury University, MD

Shahla Peterman University of Missouri—Rolla

Susan Pfiefer Butler Community College, KS

Daniel Plante Stetson University, FL

Xingping Sun Missouri State University

Jill Van Valkenburg Bowling Green State University

Erica Voges New Mexico State University

We would also like to thank the reviewers of the previous edition:

John A Blake, Oakwood College; Dave Bregenzer, Utah State University; Kelly Brooks, Pierce College; Donald O Clayton, Madisonville Community

College; Charles C Clever, South Dakota State University; Dale L Craft, South Florida Community College; Kent Craghead, Colby Community College;

Lloyd David, Montreat College; John Haverhals, Bradley University; Randall Helmstutler, University of Virginia; Heather Hulett, University of

Wisconsin—La Crosse; David Hutchison, Indiana State University; Dan

Jelsovsky, Florida Southern College; Alan S Jian, Solano Community College;

Dr Hilbert Johs, Wayne State College; Hideaki Kaneko, Old Dominion

University; Michael Longfritz, Rensselear Polytechnic Institute; Dr Hank

Martel, Broward Community College; Kimberly McGinley Vincent,

Washington State University; Donna Mills, Frederick Community College; Pat

Moreland, Cowley College; Sue Neal, Wichita State University; Cornelius Nelan, Quinnipiac University; Catherine A Roberts, University of Rhode

Island; George W Schultz, St Petersburg College; Paul H Stanford, University

of Texas—Dallas; Jaak Vilms, Colorado State University; Jane West, Trident Technical College; Elizabeth White, Trident Technical College; Kenneth J.

Word, Central Texas College.

Finally, and most importantly, we thank our wives, Barbara and Kathryn, for their encouragement and support.

COMMENTS WELCOMED

With the knowledge that any book can always be improved, we welcome rections, constructive criticisms, and suggestions from every reader.

cor-geoffrey.berresford@liu.edu andrew.rockett@liu.edu

A User’s Guide to Features

Application PreviewFound on every chapter opener page, Application Previews motivate thechapter They offer a unique “mathematics in your world” application or aninteresting historical note A page with further information on the topic,and often a related exercise number, is referenced

Moroccan runner Hicham

El Guerrouj, current world record holder for the mile run, bested the record set

6 years earlier

by 1.26 seconds.

World Record Mile Runs

The dots on the graph below show the world record times for the mile run Moroccan runner Hicham El Guerrouj These points fall roughly along a line, calculator to find a regression line (see Example 8 and Exercises 69–74),

based on a method called least squares, whose mathematical basis will be

explained in Chapter 7.

History of the Record for the Mile Run

Notice that the times do not level off as you might expect, but continue to decrease.

1860 1880 1900 1920 1940 1960 1980 2000

3:50 3:40 4:00 4:10 4:20 4:30 4:40

= record regression line

World record mile runs 1865–1999

The equation of the regression line is y 0.356x  257.44, where x represents years after 1900 and y is the time in seconds The regression line

most recent world record would have been predicted quite accurately by this line, since the rightmost dot falls almost exactly on the line.

4:36.5 1865 Richard Webster 4:29.0 1868 William Chinnery 4:28.8 1868 Walter Gibbs 4:26.0 1874 Walter Slade 4:24.5 1875 Walter Slade 4:23.2 1880 Walter George 4:21.4 1882 Walter George 4:18.4 1884 Walter George 4:18.2 1894 Fred Bacon 4:17.0 1895 Fred Bacon 4:15.6 1895 Thomas Conneff 4:15.4 1911 John Paul Jones 4:14.4 1913 John Paul Jones 4:12.6 1915 Norman Taber 4:10.4 1923 Paavo Nurmi

4:09.2 1931 Jules Ladoumegue 4:07.6 1933 Jack Lovelock 4:06.8 1934 Glenn Cunningham 4:06.4 1937 Sydney Wooderson 4:06.2 1942 Gunder Hägg 4:06.2 1942 Arne Andersson 4:04.6 1942 Gunder Hägg 4:02.6 1943 Arne Andersson 4:01.6 1944 Arne Andersson 4:01.4 1945 Gunder Hägg 3:59.4 1954 Roger Bannister 3:58.0 1954 John Landy 3:57.2 1957 Derek Ibbotson 3:54.5 1958 Herb Elliott 3:54.4 1962 Peter Snell

3:54.1 1964 Peter Snell 3:53.6 1965 Michel Jazy 3:51.3 1966 Jim Ryun 3:51.1 1967 Jim Ryun 3:51.0 1975 Filbert Bayi 3:49.4 1975 John Walker 3:49.0 1979 Sebastian Coe 3:48.8 1980 Steve Ovett 3:48.53 1981 Sebastian Coe 3:48.40 1981 Steve Ovett 3:47.33 1981 Sebastian Coe 3:46.31 1985 Steve Cram 3:44.39 1993 Noureddine Morceli 3:43.13 1999 Hicham El Guerrouj Source: USA Track & Field

Real World IconThis globe icon marks examples in which calculus

is connected to every-day life

xvii

Graphing Calculator Explorations

To allow for optional use of the graphingcalculator, the Explorations are boxed

Most can also be read simply for richment Exercises and examples thatare designed to be done with a graphingcalculator are marked with an icon

en-C(x)

Total cost

9x

Unit cost Number

of units Fixed cost

EXAMPLE 4 FINDING A COMPANY’S COST FUNCTION

An electronics company manufactures pocket calculators at a cost of $9 each, and the company’s fixed costs (such as rent) amount to $400 per day Find a

function C(x) that gives the total cost of producing x pocket calculators in a day.

Solution

Each calculator costs $9 to produce, so x calculators will cost 9x dollars, to

which we must add the fixed costs of $400.

Graphing Calculator Exploration

win-dow [5, 5] by [10, 10] How does the shape of the parabola change

when the coefficient of x2 increases?

b.Graph What did the negative sign do to the parabola?

c.Predict the shape of the parabolas and Then check your predictions by graphing the functions.

2.7 NONDIFFERENTIABLE FUNCTIONS 167

Practice Problem For the function graphed below, find the x-values at which the derivative is

undefined.

➤ Solution on next page

B E C A R E F U L : All differentiable functions are continuous (see page 134),

but not all continuous functions are differentiable—for example,

These facts are shown in the following diagram.

Differentiable functions

Continuous functions

For example, cell B5 evaluates h 1/3at h  obtaining 1/3 

1000 1/3   10 Column B evaluates this different quotient for the

positive values of h in column A, while column E evaluates it for the

corre-sponding negative values of h in column D.

*To obtain this and other Spreadsheet Explorations, go to http://college.hmco.com/PIC/

sheet Explorations.

√ 3 1000

 1

1000  1 1000

168 CHAPTER 2 DERIVATIVES AND THEIR USES

h

A B C D E (f(0+h)-f(0))/h (f(0+h)-f(0))/h B5

1.0000000 -1.0000000 -1.0000000

-4.6415888 -10.0000000 -46.4158883 -100.0000000

h

=A5^(-1/3)

-0.1000000 -0.0010000 -0.0000100 -0.0000001

1.0000000 4.6415888 0.1000000 0.0010000 10.0000000 46.4158883 100.0000000 0.0001000 0.0000010

1 3

5

6 8

Notice that the values in column B are becoming arbitrarily large, while the values in column E are becoming arbitrarily small, so the difference quotient does not approach a limit as This shows that the deriv-

ative of ƒ(x)  x2/3at 0 does not exist, so the function ƒ(x)  x2/3is not

differentiable at x 0.

h S 0.

Exercises

2.7

1–4 For each function graphed below, find the

x-values at which the derivative does not exist.

The domain of a rational function is the set of numbers for which the

denominator is not zero For example, the domain of the function f(x) on the left above is {x x 2} (since x  2 makes the denominator zero), and the domain of g(x) on the right is the set of all real numbers  (since x2  1

is never zero) The graphs of these functions are shown below Notice that ally reach.





A rational function

is a polynomial over a polynomial

x  2

4 2

horizontal

y  0 (x-axis)

Practice Problem 2 What is the domain of ? ➤ Solution on page 64

B E C A R E F U L : Simplifying a rational function by canceling a common factor from the numerator and the denominator can change the domain of the func- they have different domains) For example, the rational function on the left below is not defined at while the simplified version on the right is

defined at so that the two functions are technically not equal.

understanding of the material using Excel, an alternative for those whoprefer spreadsheet technology See “Integrating Excel” on page xx for a list

of exercises that can be done with Excel

Practice ProblemsStudents can check their understanding of atopic as they read the text or do homework by

working out a Practice Problem Complete solutions are found at the end of each

section, just before the Section Summary

Be Careful The “Be Careful” icon marks places where the authors help students avoid common errors

➤

➤

Section SummaryFound at the end of every section, these summaries briefly state the main ideas ofthe section, providing a study tool or reminder for students.

Exercises

The Applied Exercises are labeled with general

and specific titles so instructors can assign

problems appropriate for the class Conceptual Exercises encourage students to “think outside the box,” and Explorations and Excursions push

students further

End of Chapter Material ➤

To help students study, each chapter ends

with a Chapter Summary with Hints and

Suggestions and Review Exercises The last

bullet of the Hints and Suggestions lists the

Review Exercises that a student could use to

self-test Both even and odd answers are

supplied in the back of the book

Cumulative Review

There is a Cumulative Review after every

3–4 chapters Even and odd answers are

supplied in the back of the book

4.

Using the point-slope form with

5.

6.

Slope is m  3 and y-intercept is (0, 6).

Multiplying each side by 3

Section Summary

An interval is a set of real numbers corresponding to a section of the real line.

none of its endpoints.

The nonvertical line through two points (x1, y1) and (x2, y2) has slope

The slope of a vertical line is undefined or, equivalently, does not exist.

There are five equations or forms for lines:

General linear equation

A graphing calculator can find the regression line for a set of points, which can then be used to predict future trends

1.1 Real Numbers, Inequalities, and Lines

Translate an interval into set notation and graph

it on the real line.

(Review Exercises 1 – 4.)

[a, b] (a, b) [a, b) (a, b]

(, b] (, b) [a, ) (a, ) (, ) Express given information in interval form.

(Review Exercises 5 – 6.)

Find an equation for a line that satisfies certain

conditions (Review Exercises 7 – 12.)

Find an equation of a line from its graph.

(Review Exercises 13 – 14.)

Use straight-line depreciation to find the value of

an asset (Review Exercises 15 – 16.)

Use real-world data to find a regression line and

make a prediction (Review Exercise 17.)

1.2 Exponents

Evaluate negative and fractional exponents

without a calculator (Review Exercises 18– 25.)

Evaluate an exponential expression using a

calculator (Review Exercises 26–29.)

Use real-world data to find a power regression curve and make a prediction.

(Review Exercise 30.)

1.3 Functions: Linear and Quadratic

Evaluate and find the domain and range of a

function (Review Exercises 31–34.)

A function f is a rule that assigns to each number x in a set (the domain) a (single) number f(x) The range is the set of all resulting values f(x).

Use the vertical line test to see if a graph defines a

function (Review Exercises 35–36.)

Graph a linear function:

Graph a quadratic function:

Use a graphing calculator to graph a quadratic

function (Review Exercises 45–46.)

xb √b2 4ac 2a

xb 2a

f(x)  ax2 bx  c

f(x)  mx  b

More About Compositions

105 a Find the composition f(g(x)) of the two

linear functions and

g(x)  cx  d (for constants a, b, c, and d).

b.Is the composition of two linear functions always a linear function?

f (x)  ax  b

106 a Is the composition of two quadratic functions

always a quadratic function? [Hint: Find the

81 Use the rule of 6 to find how costs change if a

com-pany wants to quadruple its capacity.

82 Use the rule of 6 to find how costs change if a

company wants to triple its capacity.

83– 84.ALLOMETRY :Heart RateIt is well known that the hearts of smaller animals beat faster than the hearts of larger animals The actual relationship is approximately

where the heart rate is in beats per minute and the weight is in pounds Use this relationship to estimate the heart rate of:

83 A 16-pound dog.

84 A 625-pound grizzly bear.

Source: Biology Review 41

85– 86.BUSINESS:Learning Curves in Airplane ProductionRecall (pages 26–27) that the learning curve for the production of Boeing 707 airplanes is

150n0.322 (thousand work-hours) Find how many work-hours it took to build:

85 The 50th Boeing 707.

86 The 250th Boeing 707.

87.GENERAL:Richter ScaleThe Richter scale (developed by Charles Richter in 1935) is widely used to measure the strength of earthquakes Every increase of 1 on the Richter scale corresponds to a 10-fold increase in ground motion Therefore, an

increase on the Richter scale from A to B means

that ground motion increases by a factor of (for Find the increase in ground motion between the following earthquakes:

a.The 1994 Northridge, California, earthquake, measuring 6.8 on the Richter scale, and the

1906 San Francisco earthquake, measuring 8.3.

(The San Francisco earthquake resulted in

4 square miles of San Francisco.)

88.GENERAL:Richter Scale (continuation) Every

increase of 1 on the Richter scale corresponds

released Therefore, an increase on the Richter

scale from A to B means that the energy

released increases by a factor of (for

a.Find the increase in energy released between the

earthquakes in Exercise 87a.

b.Find the increase in energy released between the

earthquakes in Exercise 87b.

89– 90.GENERAL:WaterfallsWater falling from

a waterfall that is x feet high will hit the ground

with speed miles per hour (neglecting air resistance).

89 Find the speed of the water at the bottom of the

highest waterfall in the world, Angel Falls in Venezuela (3281 feet high).

90 Find the speed of the water at the bottom of the

highest waterfall in the United States, Ribbon Falls

in Yosemite, California (1650 feet high).

91– 92.ENVIRONMENTAL SCIENCE:Biodiversity

It is well known that larger land areas can support

multiplying the land area by a factor of x multiplies the number of species by a factor of x0.239 Use a graphing calculator to graph Use the window [0, 100] by [0, 4].

Source: Robert H MacArthur and Edward O Wilson, The Theory of Island Biogeography

91 Find the multiple x for the land area that leads

to double the number of species That is, find the value of x such that [Hint: Either use TRACE or find where

INTERSECTs

92 Find the multiple x for the land area that leads

to triple the number of species That is, find

the value of x such that [Hint: Either use TRACE or find where INTERSECTs y2  3.]

(continues)

➤

Integrating Excel

If you would like to use Excel or another spreadsheet software when working the exercises in this text, refer to the chart below It lists exercises from many sections that you might find instructive to do with spreadsheet technology Please note that none of these exercises are dependent on Excel If you would

like help using Excel, please consider the Excel Guide for Finite Mathematics

and Applied Calculus, which is available from Cengage Additionally, the Getting Started with Excel chapter of the guide is available on the website.

Graphing Calculator Basics

While the (optional) Graphing Calculator Explorations may be carried out on

most graphing calculators, the screens shown in this book are from the

Texas Instruments TI-83, TI-84, and TI-84 Plus calculators Any specific

in-structions are also for these calculators (We occasionally show a screen

from a TI-89 calculator, but for illustration purposes only.) To carry out the

Graphing Calculator Explorations, you should be familiar with the terms

described in Graphing Calculator Terminology below To do the regression (or

modeling) examples in Chapter 1 (again optional), you should be familiar

with the techniques in the following section headed Entering Data.

GRAPHING CALCULATOR TERMINOLOGY

The viewing or graphing WINDOW is the part of the Cartesian plane shown in the display screen of your graphing calculator XMIN and XMAX

are the smallest and largest x-values shown, and YMIN and YMAX are the

smallest and largest y-values shown These values can be set by using the

WINDOW or RANGE command and are changed automatically by using any of the ZOOM operations XSCALE and YSCALE define the distance

between tick marks on the x- and y-axes.

xxi

XSCALE and YSCALE are each set at

1, so the tick marks are 1 unit apart

The unit distances in the x- and

y-directions on the screen may differ.YMAX

marks (generally 2 to 20) on each axis The x- and y-axes will not be visible

if the viewing window does not include the origin.

Pixel, an abbreviation for picture element, refers to a tiny rectangle on the

screen that can be darkened to represent a dot on a graph Pixels are arranged in a rectangular array on the screen In the above window, the axes and tick marks are formed by darkened pixels The size of the screen and number of pixels varies with different calculators.

TRACE allows you to move a flashing pixel, or cursor, along a curve in the viewing window with the x- and y-coordinates shown at the bottom of the

screen.

Useful Hint: To make the x-values in TRACE take simple values like

.1, 2, and 3, choose XMIN and XMAX to be multiples of one less than

the number of pixels across the screen For example, on the TI-84, which has 95 pixels across the screen, using an x-window like [–9.4, 9.4] or [–4.7, 4.7] or [940, –940] will TRACE with simpler x-values than the

standard windows stated in this book.

ZOOM IN allows you to magnify any part of the viewing window to see

finer detail around a chosen point ZOOM OUT does the opposite, like

stepping back to see a larger portion of the plane but with less detail These

and other ZOOM commands change the viewing window.

VALUE or EVALUATE finds the value of a previously entered expression at a

specified x-value.

SOLVE or ROOT finds the x-value that solves ƒ(x)  0, equivalently, the

x-intercepts of a curve When applied to a difference ƒ(x) – g(x), it finds

the x-value where the two curves meet (also done by the INTERSECT

command).

entered curve between specified x-values.

NDERIV or DERIV or dy/dx approximates the derivative of a function at

a point FnInt or ƒ(x)dx approximates the definite integral of a function on

an interval.

In CONNECTED MODE your calculator will darken pixels to connect lated points on a graph to show it as a continuous or “unbroken” curve However, this may lead to “false lines” in a graph that should have breaks

calcu-or “jumps.” False lines can be eliminated by using DOT MODE.

The TABLE command lists in table form the values of a function, just as you

have probably done when graphing a curve The x-values may be chosen by

you or by the calculator.

The Order of Operations used by most calculators evaluates operations in

the following order: first powers and roots, then operations like LN and

LOG, then multiplication and division, then addition and subtraction—left to

right within each level For example, 5 ^ 2x means (5 ^ 2)x, not 5 ^ (2x) Also, 1/x + 1 means (1/x) + 1, not 1/(x + 1) See your calculator’s in-

struction manual for further information.

B E C A R E F U L : Some calculators evaluate 1/2x as (1/2)x and some as 1/(2x) When in doubt, use parentheses to clarify the expression.

Much more information can be found in the manual for your graphing culator Other features will be discussed later as needed.

cal-ENTERING DATA

To ensure that your calculator is properly prepared to accept data, turn off all statistical plots by pressing 2ND Y  4 ENTER

clear all data lists by pressing 2ND  4 ENTER

set up the statistical data editor by pressing STAT 5 ENTER

and finally clear all functions by pressing and repeatedly using and the down-arrow key until all functions are cleared You will need to carry out the preceding steps only once if you are using your calculator only with this book.

To enter the data shown in the following table (which is taken from Example 8

on pages 13–14), first put the x-values into list L1 by pressing

and entering each value followed by an Then put the y-values into

list L2 by pressing the right-arrow key and entering each y-value followed

by an ENTER

ENTER STAT

y 47 50.7 54 58.5 61.5 64.9

Turn on Plot1 by pressing and then use the

for list L2) to set Plot1 as shown below Press (which gives ZoomStat) to plot this data on an appropriate viewing window You may then use TRACE and the arrow keys to check the values of your data points.

9 ZOOM 2

2ND

1 2ND ENTER

ENTER ENTER

Y

2ND

When you are finished studying the data you have entered, press

to turn Plot1 off and then 2ND  4 ENTER to clear all lists ENTER

Y

Brief Applied Calculus

Moroccan runner Hicham

El Guerrouj,current worldrecord holder for the mile run,bested therecord set

6 years earlier

by 1.26seconds

World Record Mile Runs

The dots on the graph below show the world record times for the mile run from 1865 to the 1999 world record of 3 minutes 43.13 seconds, set by the Moroccan runner Hicham El Guerrouj These points fall roughly along a line,

called the regression line In this section we will see how to use a graphing

calculator to find a regression line (see Example 8 and Exercises 69–74),

based on a method called least squares, whose mathematical basis will be

explained in Chapter 7.

History of the Record for the Mile Run

Notice that the times do not level off as you might expect, but continue to decrease.

1860 1880 1900 1920 1940 1960 1980 2000

3:503:40

4:004:104:204:304:40

= recordregression line

World record mile runs 1865–1999

The equation of the regression line is y  0.356x  257.44, where x represents years after 1900 and y is the time in seconds The regression line

can be used to predict the world mile record in future years Notice that the most recent world record would have been predicted quite accurately by this line, since the rightmost dot falls almost exactly on the line.

4:36.5 1865 Richard Webster4:29.0 1868 William Chinnery4:28.8 1868 Walter Gibbs4:26.0 1874 Walter Slade4:24.5 1875 Walter Slade4:23.2 1880 Walter George4:21.4 1882 Walter George4:18.4 1884 Walter George4:18.2 1894 Fred Bacon4:17.0 1895 Fred Bacon4:15.6 1895 Thomas Conneff4:15.4 1911 John Paul Jones4:14.4 1913 John Paul Jones4:12.6 1915 Norman Taber4:10.4 1923 Paavo Nurmi

4:09.2 1931 Jules Ladoumegue4:07.6 1933 Jack Lovelock4:06.8 1934 Glenn Cunningham4:06.4 1937 Sydney Wooderson4:06.2 1942 Gunder Hägg4:06.2 1942 Arne Andersson4:04.6 1942 Gunder Hägg4:02.6 1943 Arne Andersson4:01.6 1944 Arne Andersson4:01.4 1945 Gunder Hägg3:59.4 1954 Roger Bannister3:58.0 1954 John Landy3:57.2 1957 Derek Ibbotson3:54.5 1958 Herb Elliott3:54.4 1962 Peter Snell

3:54.1 1964 Peter Snell3:53.6 1965 Michel Jazy3:51.3 1966 Jim Ryun3:51.1 1967 Jim Ryun3:51.0 1975 Filbert Bayi3:49.4 1975 John Walker3:49.0 1979 Sebastian Coe3:48.8 1980 Steve Ovett3:48.53 1981 Sebastian Coe3:48.40 1981 Steve Ovett3:47.33 1981 Sebastian Coe3:46.31 1985 Steve Cram3:44.39 1993 Noureddine Morceli3:43.13 1999 Hicham El GuerroujSource: USA Track & Field

Linear trends, however, must not be extended too far The downward slope of this line means that it will eventually “predict” mile runs in a fraction

of a second, or even in negative time (see Exercises 57 and 58 on page 17).

Moral: In the real world, linear trends do not continue indefinitely This and

other topics in “linear” mathematics will be developed in Section 1.1.

REAL NUMBERS, INEQUALITIES, AND LINES

Introduction

Quite simply, calculus is the study of rates of change We will use calculus to

analyze rates of inflation, rates of learning, rates of population growth, and rates of natural resource consumption.

In this first section we will study linear relationships between two variable quantities—that is, relationships that can be represented by lines.

In later sections we will study nonlinear relationships, which can be sented by curves.

repre-Real Numbers and Inequalities

In this book the word “number” means real number, a number that can be represented by a point on the number line (also called the real line).

The order of the real numbers is expressed by inequalities For example,

a  b means “a is to the left of b” or, equivalently, “b is to the right of a.”

1.1

Inequalities

a  b a is less than (smaller than) b

a  b a is less than or equal to b

a  b a is greater than (larger than) b

a  b a is greater than or equal to b

The inequalities a  b and a  b are called strict inequalities, and a  b

and a  b are called nonstrict inequalities.

I M P O R T A N T N O T E : Throughout this book are many Practice Problems —

short questions designed to check your understanding of a topic before moving on to new material Full solutions are given at the end of the section Solve the following Practice Problem and then check your answer.

Practice Problem 1 Which number is smaller: or  1,000,000? ➤Solution on page 14

Multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality:

A double inequality, such as a  x  b, means that both the inequalities

a  x and x  b hold The inequality a  x  b can be interpreted graphically as “x is between a and b.”

a  x  b

Sets and Intervals

Braces { } are read “the set of all” and a vertical bar  is read “such that.”

a

1 100

a. means “the set of all x such that x is greater than 3.”

b. { x  2  x  5 } means “the set of all x such that x is between  2 and 5.”

the endpoints 2 and 5 in square brackets, [2, 5], to indicate that the

parenthe-ses, (2, 5), to indicate that the endpoints 2 and 5 are excluded An interval is

closed if it includes both endpoints, and open if it includes neither point The four types of intervals are shown below: a solid dot  on the

end-graph indicates that the point is included in the interval; a hollow dot °

indicates that the point is excluded.

{ x  2  x  5 } { x  2  x  5 }

{ x  x  1 }.

An interval may extend infinitely far to the right (indicated by the symbol

for infinity) or infinitely far to the left (indicated by  for negative

infin-ity ) Note that and  are not numbers, but are merely symbols to indicate that the interval extends endlessly in that direction The infinite intervals

in the next box are said to be closed or open depending on whether they

include or exclude their single endpoint.

Half-open or half-closed

We use parentheses rather than square brackets with and  since they are

not actual numbers.

The interval ( , ) extends infinitely far in both directions ing the entire real line) and is also denoted by  (the set of all real numbers).

(mean- (mean- (q, q)

[3, ) (3, ) (  , 5]

(  , 5)

*So named because it was originated by the French philosopher and mathematician RenéDescartes (1596 – 1650) Following the custom of the day, Descartes signed his scholarly paperswith his Latin name Cartesius, hence “Cartesian” plane.

Cartesian Plane

Two real lines or axes, one horizontal and one vertical, intersecting at their zero points, define the Cartesian plane.* The point where they meet is called the origin The axes divide the plane into four quadrants, I through IV, as

shown below.

Any point in the Cartesian plane can be specified uniquely by an

ordered pair of numbers (x, y); x, called the abscissa or x-coordinate, is the

number on the horizontal axis corresponding to the point; y, called the

ordinate or y-coordinate, is the number on the vertical axis corresponding

to the point.

Lines and Slopes

The symbol  (read “delta,” the Greek letter D) means “the change in.” For

any two points (x1, y1) and (x2, y2) we define

The change in x is the difference in the x-coordinates The change in y is the difference in the y-coordinates

Any two distinct points determine a line A nonvertical line has a slope that

measures the steepness of the line, and is defined as the change in y divided by

the change in x for any two points on the line.

2 y1

2 x1

1(2, 3)

(3, 3)

(3, 2)

(1, 2)(2, 1)

2 3

x y

1

2

3

123

1

2

3The Cartesian plane with several points.Order matters: (1, 2) is not the same as (2, 1)

(x, y) x y

Quadrant IQuadrant II

Quadrant IVQuadrant III

origin

abscissaordinate

The Cartesian plane

x y

Slope is the change in y over the change in x (x2 x1)

x2 x1

B E C A R E F U L : In slope, the x-values go in the denominator.

The changes and are often called, respectively, the “rise” and the

“run,” with the understanding that a negative “rise” means a “fall.” Slope is then “rise over run.”

Find the slope of the line through each pair of points, and graph the line.

a. (2, 1), (3, 4) b. (2, 4), (3, 1)

c. (  1, 3), (2, 3) d. (2,  1), (2, 3)

Solution

We use the slope formula for each pair (x1, y1), (x2, y2).

a. For ( 2 , 1 ) and ( 3 , 4 ) the slope is b. For ( 2 4 ) and ( 3 , 1 ) the slope

34

x

y

slope 0

(2, 3)(1, 3)

5

21

1 2 3 4 5

34

x y

Notice that when the x-coordinates are the same [as in part (d) above], the line is vertical, and when the y-coordinates are the same [as in part (c) above], the line is horizontal.

If giving an alternative definition for slope:

Practice Problem 3 A company president is considering four

dif-ferent business strategies, called S1, S2, S3, and

S4, each with different projected future profits.

The graph on the right shows the annual jected profit for the first few years for each of the strategies Which strategy yields:

pro-a. the highest projected profit in year 1?

b. the highest projected profit in the long run?

➤Solutions on page 14

Equations of Lines

The point where a nonvertical line crosses the y-axis is called the y-intercept

of the line The y-intercept can be given either as the y-coordinate b or as the point (0, b) Such a line can be expressed very simply in terms of its slope and

y-intercept, representing the points by variable coordinates (or “variables”)

For lines through the origin, the equation takes the particu-

larly simple form, y  mx (since b  0), as illustrated

on the left.

Point-Slope Form of a Line

(x1, y1) point on the line

y1 2,

y  y1 m(x  x1)

x2and y2by x and y, and then multiplying each side by It is most useful when you know the slope of the line and a point on it.

Find an equation of the line through (6,  2) with slope

–4 –3 –2

–4

x y

y  2x (slope 2)

y  3x (slope 3)

The most useful equation for a line is the point-slope form.

withslope 1 and point (4, 1)

x

horizontal line

y  b

to evaluate b.

Find an equation for the line through the points (4, 1) and (7,  2).

with a being the

Graph the lines x  2 and y  6.

Solution

a. Find an equation for the vertical line through (3, 2).

b. Find an equation for the horizontal line through (3, 2).

slopes of vertical and horizontal lines:

Vertical line: slope is undefined.

Horizontal line: slope is defined, and is zero.

There is one form that covers all lines, vertical and nonvertical.

x y

General Linear Equation

For constants a, b, c, with

a and b not both zero

ax  by  c