Smith R., Minton R. Calculus

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xx Seeing the Beauty and Power of Mathematics

SUPPLEMENTS

ONLINE INSTRUCTOR’S SOLUTIONS MANUAL

An invaluable, timesavingresource, the Instructor’s Solutions Manual contains comprehen-

sive, worked-out solutions to the odd- and even-numbered exercises in the text.

STUDENT SOLUTIONS MANUAL (ISBN 978-0-07-7256968)

The Student Solutions Manual is a helpful reference that contains comprehensive, worked-

out solutions to the odd-numbered exercises in the text.

ONLINE TESTBANK AND PREFORMATTED TESTS

Brownstone Diploma

testing software offers instructors a quick and easy way to cre-

ate customized exams and view student results. Instructors may use the software to sort

questions by section, difﬁculty level, and type; add questions and edit existing questions;

create multiple versions of questions using algorithmically-randomized variables; prepare

multiple-choice quizzes; and construct a grade book.

ONLINE CALCULUS CONCEPTS VIDEOS

Students will see essential concepts explained and brought to life through dynamic ani-

mations in this new video series available on DVD and on the Smith/Minton website. The

twenty-ﬁve key concepts, chosen after consultation with calculus instructors across the

country, are the most commonly taught topics that students need help with and that also

lend themselves most readily to on-camera demonstration.

CALCULUS AND TECHNOLOGY

It is our conviction that graphing calculators and computer algebra systems must not

be used indiscriminately. The focus must always remain on the calculus. We have en-

sured that each of our exercise sets offers an extensive array of problems that should

be worked by hand. We also believe, however, that calculus study supplemented with an

intelligent use of technology gives students an extremely powerful arsenal of problem-

solving skills. Many passages in the text provide guidance on how to judiciously use—

and not abuse—graphing calculators and computers. We also provide ample opportunity

for students to practice using these tools. Exercises that are most easily solved with the aid

of a graphing calculator or a computer algebra system are easily identiﬁed with a

icon.

IMPROVEMENTS IN THE FOURTH EDITION

Building upon the success of the Third Edition of Calculus, we have made the following

revisions to produce an even better Fourth Edition:

Presentation

A key goal of the Fourth Edition revision was to offer a clearer presentation of

calculus. With this goal in mind, the authors were able to reduce the amount of

material by nearly 150 pages.

The level of rigor has been carefully balanced to ensure that concepts are presented

in a rigorously correct manner without allowing technical details to overwhelm

beginning calculus students. For example, the sections on continuity, sum rule, chain

rule, the deﬁnite integral and Riemann sums, introductory vectors, and advanced

multivariable calculus (including the Green’s Theorem section) have been revised to

improve the theorems, deﬁnitions, and/or proofs.

The exercise sets were redesigned in an effort to aid instructors by allowing them to

more easily identify and assign problems of a certain type.

The derivatives of hyperbolic functions are developed in Section 6.6, giving this

important class of functions a full development. Separating these functions from the

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Seeing the Beauty and Power of Mathematics xxi

exponential and trigonometric functions allows for early and comprehensive

exploration of the relationship between these functions, exponential functions,

trigonometric functions, and their derivatives and integrals.

Exercises

More than 1,000 new classic calculus problems were added, covering topics

from polynomials to multivariable calculus, including optimization, related rates,

integration techniques and applications, parametric and polar equations, vectors,

vector calculus, and differential equations.

A reorganization of the exercise sets makes the range of available exercises more

transparent. Earlier exercises focus on fundamentals, as developed in examples in the

text. Later exercises explore interesting extensions of the material presented in the

text.

Multi-step exercises help students make connections among concepts and require

students to become more critical readers. Closely related exercises are different

parts of the same numbered exercise, with follow-up questions to solidify lessons

learned.

Application exercises have been separated out in all appropriate sections. A new

header identiﬁes the location of applied exercises which are designed to show

students the connection between what they learn in class, other areas of study, and

outside life. This differentiates the applications from exploratory exercises that allow

students to discover connections and extensions for themselves.

ACKNOWLEDGMENTS

A project of this magnitude requires the collaboration of an incredible number of talented

and dedicated individuals. Our editorial staff worked tirelessly to provide us with countless

surveys, focus group reports, and reviews, giving us the best possible read on the current

state of calculus instruction. First and foremost, we want to express our appreciation to

our sponsoring editor John Osgood and our developmental editor Eve Lipton for their

encouragement and support to keep us on track throughout this project. They challenged

us to make this a better book. We also wish to thank our editorial director Stewart Mattson,

and director of development Kris Tibbets for their ongoing strong support.

We are indebted to the McGraw-Hill production team, especially project manager

Peggy Selle and design coordinator Laurie Janssen, for (among other things) producing a

beautifullydesignedtext.TheteamatMRCChasprovideduswithnumeroussuggestionsfor

clarifying and improving the exercise sets and ensuring the text’s accuracy. Our marketing

manager Kevin Ernzen has been instrumental in helping to convey the story of this book to

a wider audience, and media project manager Sandy Schnee created an innovative suite of

media supplements.

Our work on this project beneﬁted tremendously from the insightful comments we

received from many reviewers, survey respondents and symposium attendees. We wish to

thank the following individuals whose contributions helped to shape this book:

REVIEWERS OF THE FOURTH EDITION

Andre Adler, Illinois Institute of Technology

Daniel Balaguy, Sierra College

Frank Bauerle, University of California–

Santa Cruz

Fred Bourgoin, Laney College

Kris Chatas, Washtenaw Community College

Raymond Clapsadle, University of Memphis

Dan Edidin, University of Missouri–

Columbia

Timothy Flaherty, Carnegie Mellon University

Gerald Greivel, Colorado School of Mines

Jerrold Grossman, Oakland University

Murli Gupta, George Washington University

Ali Hajjafar, University of Akron

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xxii Seeing the Beauty and Power of Mathematics

Donald Hartig, California Polytechnic State

University–San Luis Obispo

John Heublein, Kansas State University–Salina

Joseph Kazimir, East Los Angeles College

Harihar Khanal, Embry-Riddle Aeronautical

University–Daytona Beach

Todd King, Michigan Technological University

Donna Latham, Sierra College

Rick Leborne, Tennessee Technological

University

Hong Liu, Embry-Riddle Aeronautical

University–Daytona Beach

Paul Loya, Binghamton University

Laurie Pieracci, Sierra College

Michael Price, University of Oregon

Michael Quail, Washtenaw Community College

Jayakumar Ramanatan, Eastern Michigan

University

Louis Rossi, University of Delaware

Mohammad Saleem, San Jose State University

Angela Sharp, University of Minnesota–Duluth

Greg Spradlin, Embry-Riddle Aeronautical

University–Daytona Beach

Kelly Stady, Cuyahoga Community College

Richard Swanson, Montana State

University–Bozeman

Fereja Tahir, Illinois Central College

Marie Vitulli, University of Oregon

Patrick Ward, Illinois Central College

Jay Zimmerman, Towson University

WITH MANY THANKS TO OUR PREVIOUS REVIEWER PANEL:

Kent Aeschliman, Oakland Community College

Stephen Agard, University of Minnesota

Charles Akemann, University of California,

Santa Barbara

Tuncay Aktosun, University of Texas–

Arlington

Gerardo Aladro, Florida International

University

Dennis Bila, Washtenaw Community College

Ron Blei, University of Connecticut

Joseph Borzellino, California Polytechnic State

University

Timmy Bremer, Broome Community College

Qingying Bu, University of Mississippi

Katherine Byler, California State University–

Fresno

Roxanne Byrne, University of Colorado–

Denver

Fengxin Chen, University of Texas at San

Antonio

Youn-Min Chou, University of Texas at San

Antonio

Leo G. Chouinard, University of Nebraska–

Lincoln

Si Kit Chung, The University of Hong Kong

Donald Cole, University of Mississippi

David Collingwood, University of Washington

Tristan Denley, University of Mississippi

Judith Downey, University of Nebraska–Omaha

Linda Duchrow, Regis University

Jin Feng, University of Massachusetts, Amherst

Carl FitzGerald, University of California, San

Diego

Mihail Frumosu, Boston University

John Gilbert, University of Texas

Rajiv Gupta, University of British Columbia

Guershon Harel, University of California, San

Diego

Richard Hobbs, Mission College

Shun-Chieh Hsieh, Chang Jung Christian

University

Joseﬁna Barnachea Janier, University Teknologi

Petronas

Jakub Jasinski, University of Scranton

George W. Johnson, University of South

Carolina

Nassereldeen Ahmed Kabbashi, International

Islamic University

G. P. Kapoor, Indian Institute of Technology

Kanpur

Jacob Kogan, University of Maryland, Baltimore

Carole King Krueger, University of Texas at

Arlington

Kenneth Kutler, Brigham Young University

Hong-Jian Lai, West Virginia University

John Lawlor, University of Vermont

Richard Le Borne, Tennessee Technological

University

Glenn Ledder, University of Nebraska–Lincoln

Sungwook Lee,

University of Southern

Mississippi

Mary Legner, Riverside Community College

Steffen Lempp, University of Wisconsin–

Madison

Barbara MacCluer, University of Virginia

William Margulies, California State University,

Long Beach

Mary B. Martin, Middle Tennessee State

University

Mike Martin, Johnson Community College

James Meek, University of Arkansas

Carrie Muir, University of Colorado

Michael M. Neumann, Mississippi State

University

Sam Obeid, University of North Texas

Iuliana Oprea, Colorado State University

Anthony Peressini, University of Illinois

Greg Perkins, Hartnell College

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Seeing the Beauty and Power of Mathematics xxiii

Tan Ban Pin, National University of Singapore

Linda Powers, Virginia Polytechnic Institute and

State University

Mohammad A. Rammaha, University of

Nebraska–Lincoln

Richard Rebarber, University of Nebraska–

Lincoln

Kim Rescorla, Eastern Michigan University

Edgar Reyes, Southeastern Louisiana

University

Mark Smith, University of Illinois

Donald Solomon, University of

Wisconsin–Milwaukee

Rustem Suncheleev, Universiti Putra Malaysia

Anthony Thomas, University of

Wisconsin–Platteville

Anthony Vance, Austin Community College

P. Veeramani, Indian Institute of Technology

Madras

Anke Walz, Kutztown University of

Pennsylvania

Scott Wilde, Baylor University

James Wilson, Iowa State University

Raymond Wong, University of California, Santa

Barbara

Bernardine R. Wong Cheng Kiat, University of

Malaya

Teri Woodington, Colorado School of Mines

Gordon Woodward, University of

Nebraska–Lincoln

Haidong Wu, University of Mississippi

Adil Yaqub, University of California, Santa

Barbara

Hong-Ming Yin, Washington State University,

Pullman

Paul Yun, El Camino College

Jennifer Zhao, University of Michigan at

Dearborn

In addition, a number of our colleagues graciously gave their time and energy to help

create or improve portions of the manuscript. We would especially like to thank Richard

Grant, Bill Ergle, Jack Steehler, Ben Huddle, Chris Lee, Dave Taylor, Dan Larsen and Jan

Minton of Roanoke College for sharing their expertise in calculus and related applications;

student assistants Danielle Shiley and Hannah Green for their insight and hard work; Tim

Pennings and Art Benjamin for inspirations both mathematical and personal; Tom Burns

for help with an industrial application; Gregory Minton and James Albrecht for suggesting

severalbrilliantproblems;Dorothee BlumofMillersville Universityforhelpingtoclass-test

an early version of the manuscript; Bruce Ikenaga of Millersville University for generously

sharing his expertise in TeX and Corel Draw and Pam Vercellone-Smith, for lending us her

expertise in many of the biological applications. We also wish to thank Dorothee Blum,

Bob Buchanan, Antonia Cardwell, Roxana Costinescu, Chuck Denlinger, Bruce Ikenaga,

Zhoude Shao, Ron Umble and Zenaida Uy of Millersville University for offering numerous

helpful suggestions for improvement. In addition, we would like to thankall of our students

throughout the years, who have (sometimes unknowingly) ﬁeld-tested innumerable ideas,

some of which worked and the rest of which will not be found in this book.

Ultimately, this book is for our families. We simply could not have written a book of

this magnitude without their strong support. We thank them for their love and inspiration

throughout our growth as textbook authors. Their understanding, in both the technical

and the personal sense, was essential. They provide us with the reason why we do all of the

things we do. So, it is ﬁtting that we especially thank our wives, Pam Vercellone-Smith and

Jan Minton and our children, Katie and Michael Smith and Kelly and Greg Minton; and

our parents, George and Anne Smith and Paul and Mary Frances Minton.

Robert T. Smith

Lancaster, Pennsylvania

Roland B. Minton

Salem, Virginia

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Applications Index

Biology

Alligator sex, 77

Anthills, 344

Bacterial growth, 491–492, 499

Biological oscillations, 668

Bird ﬂight, 246

Birth rates, 283

Cell age, 484

Cell division, 398

Circulatory system, 232

Competitive exclusion principle,

526

Coyote, 358

Cricket chirping, 20

Decay, 491–496

Dinosaurs, 491

Dog swimming, 886

E. coli growth, 499

Fireﬂy ﬂashes, 203

Fish population, 509

Fish speed, 246

Fossils, 491

Genetics, 886

Growth, 491–496

Height of person, 485

Human speed, 117

Kangaroo legs, 315

Leg width and body weight, 134

Paleontology, 491

Pollutant density, 947

Pollution, 169

Predator-prey system, 293, 527

Pupil dilation, 47, 246

Pupil size, 84, 466

Reproductive rate, 184

Shark prey tracking, 873

Spruce budworm, 184, 222

Tree infestation, 520

Zebra stripes, 520

Chemistry

Acid-base titration, 383

Autocatalytic reaction, 242–243

Bimolecular reactions, 509

Concentration analysis, 396, 397

Methane, 713

Product, 246

Rate of reaction, 242–243

Reactants, 246

Second-order chemical reaction, 246

Temperature, entropy, and Gibbs free

energy, 843

Construction

Building height, 237

Sand pile height, 238

Sliding ladder, 234

Demographics

Birthrate, 323

Critical threshold, 512–513

Maximum sustainable population, 506

Population and time, 20

Population carrying capacity, 505

Population estimation, 920

Population growth maximum, 244

Population prediction, 12

Rumor spread, 398

Urban population growth, 126

Economics

Advertising costs, 211, 238

Airline ticket sales, 35

Annual percentage yield, 496

Asset depreciation, 497–498

Bank account balance, 210

Barge costs, 246

Capital expenditure, 863

Complementary commodities, 843

Compound interest, 496–498, 499

Consumer surplus, 291

Cost minimization, 229–230, 231

Coupon collectors’ problem, 564

Demand, 244

Diminishing returns, 233

Economic Order Quantity, 283, 291

Elasticity of demand, 241–242, 244

Endowment, 508

Future value, 501

Gini index, 272

Gross domestic product, 265, 272

Ice cream sales, 713

Income calculation, 862

Income stream, 500

Income tax, 68, 127

Initial investment, 508

Investing, 210

Investment strategies, 506

Investment value, 843

Just-in-time inventory, 251, 291

Manufacturing costs, 211

Marginal cost, 239

Marginal proﬁt, 239

Mortgages, 508

Multiplier effect, 553

National debt, 135

Oil consumption, 323

Oil prices, 925

Packing, 544

Parking fees, 57

Present value, 500, 553

Product sales, 202

Production costs, 238, 240–241, 324

Production optimization, 891–892

Proﬁt maximization, 324

Rate of change, 236, 239–244

Relative change in demand, 240–241

Relative change in price, 240

Resale value, 509

Retirement fund, 508

Revenue, 139

Revenue maximization, 233

Rule of 72, 501

Salary increase, 77, 391

Stock investing, 873

Substitute commodities, 843

Supply and demand, 323

Tax rates, 500

Tax tables, 73

xxiv

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Applications Index xxv

Unit price, 141

Worker productivity, 211

Engineering

Airplane design, 227

Beam sag, 851

Capacitors, 1090

Car design, 977

Catenary, 411

Ceiling shape, 676

Circuit charge, 1091

Circuitry, 35, 1096

Cooling towers, 745

Electric dipole, 586

Falling ladder, 646

Flashlight design, 670

Highway construction costs, 229–230,

231

Hydrostatic force on dam, 368

International Space Station, 375

Light pole height, 35

Metal sheet gauge, 851

Norman window, 232

Oil pipeline, 231

Oil rig beam, 713

Oil tank capacity, 404

Parabolic dish, 741, 742

Pyramids, 335

Reliability testing, 421

RoboCup, 749

Robot vision, 239

Rocket fuel, 369

Rocket height, 86

Rocket launch, 35

Rocket thrust, 890

Saddle Dome, 744

Screen door closing, 1081

Series circuit, 1096

Soda can design, 227–228

Spacecraft launch, 238

Spiral staircase, 758

St. Louis Gateway Arch, 411, 418

Telegraph cable, 383

Thrust-time curve, 370

Tower height, 33–34

Useful life phase, 421

Voltage, 1091–1092

Water ﬂow, 323

Water pumping, 239, 363

Water reservoir, 211

Water system, 117

Water tank, 288, 290

Water tower, 369

Environment

Oil spill, 234, 237

Hobbies

Jewelry making, 344

Photography, 20

Home

Garden construction, 223–224

Medicine

Achilles tendon, 315

Allosteric enzyme, 142

Antibiotics, 500

Antidepressants, 500

Brain neurons, 398

Computed tomography, 327

Drug concentration, 398

Drug dosage, 553

Drug half-life, 500

Drug injection, 86

Drug sensitivity, 246

Enzymatic reaction, 194

Foot arch, 322

Glucose concentration, 1097

HIV, 283

Infection, 237

Pneumotachograph, 310

Tendon force, 322

Music

Digital, 531

Guitar string, 238, 843

Octaves, 8

Piano tuning, 35, 619

Synthesizers, 617–618

Timbre, 617

Tuning, 8

Physics

AC circuit, 232, 293

Acceleration, 135

Air resistance, 356

Atmospheric pressure, 398

Ball motion, 353

Black body radiation, 606

Boiling point of water at

elevation, 20

Change in position, 277

Distance fallen, 288

Electric potential, 607

Electromagnetic ﬁeld, 511

Electromagnetic radiation, 606

Falling object position, 257

Falling object velocity, 418

Freezing point, 117

Frequency modulation, 222

Friction, 77

Gas laws, 238

Gravitation, 184

Half-life, 499

Hydrostatic forces, 371

Hyperbolic mirrors, 674

Impulse-momentum equation, 283

Light path, 211, 231

Light reﬂection, 232

Newton’s Law of Cooling, 494, 499

Object distance fallen, 6

Object launch, 358

Object velocity, 310

Pendulum, 1097

Planck’s law, 398, 606

Planetary orbits, 640

Projectiles, 265

Radio waves, 27–28

Radioactive decay, 494

Raindrop evaporation, 237

Relativity, 86, 184

Rod density, 243–244

Solar and Heliospheric

Observatory, 173

Sound waves, 632

Spring motion, 154, 1081

Spring-mass system, 142

Terminal velocity, 509

Thrown ball, 231

Velocity, 57

Velocity required to reach height, 354

Voltage, 293

Volume and pressure, 157

Weightlessness, 360

Sports/

Entertainment

Auto racing, 687

Badminton, 283

Ball height, 233

Baseball bat corking, 371

Baseball bat hit, 238, 321

Baseball bat mass, 366

Baseball bat sweet spot, 367

Baseball batting average, 863

Baseball impulse, 365

Baseball knuckleball, 57

Baseball outﬁelding, 404

Baseball pitching, 57, 359

Baseball player gaze, 406–407

Baseball spin, 724

Baseball statistics analysis, 8

Baseball velocity, 383

Basketball free throws, 359, 486

Basketball perfect swish, 86

Bicycling, 20, 553

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xxvi Applications Index

Catch, 369

Circus act, 359, 634

Coin toss, 479, 564

Computer game plays, 20

Curveball, 725

Diver velocity, 352, 358

Equipment design, 901

Football kick, 410

Football points scored, 885

Football punt, 778

Football spiral pass, 725

Goal probability, 486

Golf ball aim, 404

Golf ball distance, 809

Golf ball motion, 972

Golf ball on moon, 360

Golf ball spin, 20

Golf club impact, 141

Golf hook shot, 725

Golf put, 655

Golf shot analysis, 140

High jumping, 315

Hockey shot, 410

Home run, 778

Juggling, 360

Jumping, 358

Kayaking, 696

Keno, 486

Knuckleball, 357, 466

Marathon, 107

Merry-go-round, 775

Mountaineering, 873

Movie theater design, 404

Olympic Games, 676

Racetrack design, 713

Record player, 35

Roller coaster, 194, 778

Rugby ball, 344

Sailing, 895

Scrambler ride, 637, 641

Skateboarding, 360

Skiing, 410, 644, 901

Skydiving, 132–133, 418,

466, 694, 696

Soccer goal probability, 398

Soccer kick, 359

Stadium design, 798

Stadium wave, 798

Tennis ball energy lost, 320

Tennis ball work done, 369

Tennis game win, 553

Tennis matches, 194

Tennis serve, 355, 778

Tennis serve error margin, 126

Tennis serve speed, 873

Tennis slice serve, 725

Tie probability, 203

Torque, 722

Track construction, 233

Trading cards, 500

Weightlifting, 362

Travel

Aircraft steering, 694

Airline ticket sales, 35

Airplane engine thrust, 696, 704

Car engine force, 369

Car speed, 291

Car velocity, 265

Car weight, 713

Commuting, 246

Crash test, 370

Distance from airport, 237

Fuel efﬁciency, 126, 142

Gas costs, 884

Gas mileage, 126, 863

Jet speed, 633

Jet tracking, 236

Plane altitude, 135

Speed of sound, 625

Speed trap, 235

Stopped car, 77

Walking, 571

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CHAPTER

Preliminaries

In this chapter, we present a collection of familiar topics, primarily those

that we consider essential for the study of calculus. While we do not

intend this chapter to be a comprehensive review of precalculus math-

ematics, we have tried to hit the highlights and provide you with some

standard notation and language that we will use throughout the text.

As it grows, a chambered nautilus creates a spiral shell. Behind this

beautiful geometry is a surprising amount of mathematics. The nautilus

grows in such a way that the overall proportions of its shell remain

constant. That is, if you draw a rectangle to circumscribe the shell, the

ratio of height to width of the rectangle remains nearly constant.

There are several ways to represent this property mathematically. In

polar coordinates (which we present in Chapter 10), we study logarith-

mic spirals that have the property that the angle of growth is constant,

corresponding to the constant proportions of a nautilus shell. Using basic

geometry, you can divide the circumscribing rectangle into a sequence

of squares as in the ﬁgure. The relative sizes of the squares form the

famousFibonacci sequence 1, 1, 2, 3, 5, 8, .. ., where each number in the sequence

is the sum of the preceding two numbers.

A nautilus shell

The Fibonacci sequence has an amazing list of interesting properties. (Search

on the Internet to see what we mean!) Numbers in the sequence have a surprising

habit of showing up in nature, such as the number of petals on a lily (3), buttercup

(5), marigold (13), black-eyed Susan (21) and pyrethrum (34). Although we have

a very simple description of how to generate the Fibonacci sequence, think about

howyoumightdescribeitasafunction.Aplot

of the ﬁrst several numbers in the sequence

(shown in Figure 0.1) should give you the

impression of a graph curving up, perhaps a

parabola or an exponential curve.

Two aspects of this problem are impor-

tant themes throughout the calculus. One of

these is the importance of looking for pat-

terns to help us better describe the world. A

second theme is the interplay between graphs

and functions. By connecting the techniques

of algebra with the visual images provided

by graphs, you will signiﬁcantly improve

your ability to solve real-world problems

mathematically.

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2 CHAPTER 0

Preliminaries 0-2

12345678

FIGURE 0.1

The Fibonacci sequence

0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE

The Real Number System and Inequalities

Our journey into calculus begins with the real number system, focusing on those properties

that are of particular interest for calculus.

The set of integers consists of the whole numbers and their additive inverses: 0,

±1, ±2, ±3,.... A rational number is any number of the form

, where p and q are

integers and q = 0. For example,

, −

and

125

are all rational numbers. Notice that every

integer n is also a rational number, since we can write it as the quotient of two integers:

n =

Theirrational numbers are allthosereal numbersthatcannotbewritten intheform

where p and q are integers.Recall that rational numbershave decimal expansionsthat either

terminate or repeat. For instance,

= 0.5,

= 0.3333

= 0.125 and

= 0.16666

6 are

all rational numbers. By contrast, irrational numbers have decimal expansions that do

not repeat or terminate. For instance, three familiar irrational numbers and their decimal

expansions are

√

2 = 1.4142135623...,

π = 3.1415926535...

and

e = 2.7182818284....

We picture the real numbers arranged along the number line displayed in Figure 0.2

(the real line). The set of real numbers is denoted by the symbol R.

05432112345

兹2 兹3 p

FIGURE 0.2

The real line

For real numbers a and b, where a < b, we deﬁne the closed interval [a, b]tobethe

set of numbers between a and b, including a and b (the endpoints). That is,

[a, b] ={x ∈ R | a ≤ x ≤ b},

as illustrated in Figure 0.3, where the solid circles indicate that a and b are included in

[a, b].

a b

FIGURE 0.3

A closed interval