Brief Calculus: An Applied Approach, 7th Edition

709 Pages·2005·15.08 MB·English

by  Ron Larson

APPLICATIONS
Business and Economics Fuel cost,361,364
Future value,271,405
Account balances,267 Home mortgage,298,314
Advertising costs,161 Income distribution,364
Annual salary,34 Income and expenses,316
Annuity,0-18,352,355,383 Inflation rate,263,315
Average cost,0-24,199,217,226,230,350 Installment loan,0-32
Average cost and profit,254 Interest on a loan,80
Average profit,230,537 Inventory,0-32
Average retail price of 1 pound of 100% ground beef,168 cost,254
Average revenue,539 management,71,119
Average weekly profit,539 replenishment,130
Break-even analysis,22,77 Investment,482
Break-even point,17 Rule of 70,307
Budget variance,0-12 Investment strategy,513
Least-Cost Rule,513
Capital accumulation,355
Linear depreciation,35,77
Capital campaign,404
Lorenz curve,364
Capitalized cost,445,453
Manufacturing,0-12,385
Cash flow,336
Marginal analysis,242,243,247,355,434
Cash flow per share for Ruby Tuesday,30
Marginal cost,117,118,119,168,325,327,492,545
Cobb-Douglas production function,478,481
Marginal productivity,493
Complementary and substitute products,493
Marginal profit,111,115,117,118,119,168,349
Compound interest,0-18,60,68,71,139,267,271,280,288,
Marginal revenue,114,117,118,168,492,546
302,306,355,383
Marginal utility,493
Construction costs,513
Market equilibrium,48
Consumer and producer surplus,363,364,384,425
Marketing,414
Cost,26,47,66,104,130,161,180,190,230,239,328,336,
Maximum production level,506,507,546
355,382,383,481,503,512
Maximum profit,188,213,217,218,499,509
Cost and revenue,178
Maximum revenue,210,212,218,277
Cost,revenue,and profit,48,160,168,364
Median price of new privately owned U.S. homes in
Pixar,76
the south,146
Demand,47,77,112,113,129,130,151,153,219,247,255,
Minimum average cost,211,298
271,288,312,327,336,343,404,487,522
Minimum cost,218,219,253,512,546
Depreciation,26,32,139,263,280,315,355
Monthly payment,479
Diminishing returns,197
National debt,79
Doubling time,286
Owning
Dow Jones Industrial Average,9,200
a business,47
Ear infections treated at HMO clinics,10
a franchise,71
Earnings per share for Starbucks,482
Point of diminishing returns,197,199,253
Economics,10,118 Present value,269,271,313,401,402,405,434,445,451,453
marginal benefits and costs,328 Production,0-12,153,478,513
present value,451 Production level,0-6,159,382,506,507
revenue,255 Productivity,200
Effective rate,268,271,313 Profit,0-7,0-24,35,48,104,118,119,129,130,158,161,
Effective yield,306 168,169,180,190,239,243,253,254,307,328,349,
Elasticity of demand,215,218 383,482,503,546
Elasticity and revenue,215 Affiliated Computer Services,315
Equation of exchange,545 California Pizza Kitchen,312
Equilibrium point,18 Hershey Foods,425
Finance,0-24 Home Depot,450
median income,523 MBNA,315
Profit analysis,178 Life Sciences
Property value,263
Qualitycontrol,0-11,129,445 Biology
Real estate,47,547 cell division,272
Reimbursed expenses,34 coyote population,355
Reimbursement,0-7 deer population,412
Retail price,168 endangered species,312
Revenue,48,219,253,255,307,343,363,382,395, fertility rates,190
405,425,502,503,546 fish population,313,377
AT&T Wireless,Nextel,and Western Wireless,384 gestation period of rabbits,71
Earthlink,523 growth of a red oak tree,253
Microsoft,101 growth rate of a bacterial culture,139,265,272,343,414
Papa John’s,219,313 invertebrate species,76
Polo Ralph Lauren,90,104 pH values,0-7
Sonic,307 population growth,119,129,301,306,415,425
Symantec,415 preparing a culture medium,513
of symphony orchestras,312 strains of corn,79
Time Warner,384 trout population,343
Revenue and profit weights of male collies,0-12
The Yankee Candle,10 wildlife management,230,247,412,414,415,452,503
Walgreen,10 Blood pressure,127
Revenue per share Capitalized cost,453
McDonald’s,101 Environment pollutant removal,60,71
U.S. Cellular,136 Forestry,169,280,307,482
Walt Disney,255 Hardy-Weinberg Law,503,512
Salary contract,71 Health
Sales,0-7,161,304,307,451 body temperature,118
Avon Products,272,384 cancer deaths,256
Best Buy,76 epidemic,364,414
Dillard’s,19,20 exposure to sun,254
Dollar General,19 U.S. AIDS epidemic,153
Home Depot,166,167 Height of a population,0-12
for in-line skating and wheel sports,307 Medicine
Kohl’s,19,20 drug absorption,435
Lowe’s,219 drug concentration in bloodstream,106,117,166,435
Maytag,6 drug testing,503,546
Scotts,91,104 effectiveness of a pain-killing drug,117,255,451
Starbucks,6,263 kidney transplants,23
Target,545 Poiseuille’s Law,253
The Yankee Candle,343 spread of a virus,200,313,415
Sales analysis,130 temperature of a patient,48
Sales commission,35 velocity of blood,355
Sales growth,200 Physiology,0-7
Sales per share
Clorox,30
Dollar Tree,136 Social and Behavioral Sciences
Stock price,0-12
Supply and demand,22,77,360 Average salary for superintendents,343
Supply function,336 Center of population,119
Surpluses,360 College enrollment,35
Trade deficit,116 Consumer awareness
Unemployed workers,76 cab charges,71
Union negotiation,34 car buying options,44
Weekly salary,23 cellular phone charges,79
continued on back endsheets
S e v e n t h E d i t i o n
Brief Calculus
An Applied Approach
R O N L A R S O N
The Pennsylvania State University
The Behrend College
B R U C E H . E D W A R D S
University of Florida
with the assistance of
DAVID C. FALVO
The Pennsylvania State University
The Behrend College
HOUGHTON MIFFLIN COMPANY
Boston New York
Publisher:Jack Shira
Associate Sponsoring Editor:Cathy Cantin
Development Manager:Maureen Ross
Development Editor:David George
Editorial Assistant:Elizabeth Kassab
Supervising Editor:Karen Carter
Senior Project Editor:Patty Bergin
Editorial Assistant:Julia Keller
Production Technology Supervisor:Gary Crespo
Senior Marketing Manager:Danielle Potvin Curran
Marketing Coordinator:Nicole Mollica
Senior Manufacturing Coordinator:Marie Barnes
We have included examples and exercises that use real-life data as well as technology output from a variety
of software. This would not have been possible without the help of many people and organizations. Our
wholehearted thanks goes to all for their time and effort.
Trademark Acknowledgments:TI is a registered trademark of Texas Instruments,Inc. Mathcad is a regis-
tered trademark of MathSoft,Inc. Windows,Microsoft,Excel,and MS-DOS are registered trademarks of
Microsoft,Inc. Mathematica is a registered trademark of Wolfram Research,Inc. DERIVE is a registered
trademark of Soft Warehouse,Inc. IBM is a registered trademark of International Business Machines
Corporation. Maple is a registered trademark of the University of Waterloo. Graduate Record Examinations
and GRE are registered trademarks of Educational Testing Service. Graduate Management Admission Test
and GMAT are registered trademarks of the Graduate Management Admission Council.
Cover credit:©Ryan McVay/Getty Images
Copyright ©2006 by Houghton Mifflin Company. All rights reserved.
No part of this work may be reproduced or transmitted in any form or by any means,electronic or
mechanical,including photocopying and recording,or by any information storage or retrieval system,
without the prior written permission of Houghton Mifflin Company unless such use is expressly permitted
by federal copyright law. Address inquiries to College Permissions,Houghton Mifflin Company,
222 Berkeley Street,Boston,MA 02116-3764.
Printed in the United States of America
Library of Congress Catalog Number:2004116466
ISBN 0-618-54719-3
123456789-DOW-09 08 07 06 05
Contents iii
Contents
A Word from the Authors (Preface) vii
Features xii
A Plan for You as a Student (Study Strategies) xx
0
A Precalculus Review 0-1
0.1 The Real Number Line and Order 0-2
0.2 Absolute Value and Distance on the Real Number Line 0-8
0.3 Exponents and Radicals 0-13
0.4 Factoring Polynomials 0-19
0.5 Fractions and Rationalization 0-25
1
Functions, Graphs, and Limits 1
1.1 The Cartesian Plane and the Distance Formula 2
1.2 Graphs of Equations 11
1.3 Lines in the Plane and Slope 24
1.4 Functions 36
1.5 Limits 49
1.6 Continuity 61
Chapter 1 Algebra Review 72
Chapter Summary and Study Strategies 74
Review Exercises 76
Sample Post-Graduation Exam Questions 80
2
Differentiation 81
2.1 The Derivative and the Slope of a Graph 82
2.2 Some Rules for Differentiation 93
2.3 Rates of Change: Velocity and Marginals 105
2.4 The Product and Quotient Rules 120
2.5 The Chain Rule 131
2.6 Higher-Order Derivatives 140
2.7 Implicit Differentiation 147
2.8 Related Rates 154
Chapter 2 Algebra Review 162
Chapter Summary and Study Strategies 164
Review Exercises 166
Sample Post-Graduation Exam Questions 170
iv Contents
3
Applications of the Derivative 171
3.1 Increasing and Decreasing Functions 172
3.2 Extrema and the First-Derivative Test 181
3.3 Concavity and the Second-Derivative Test 191
3.4 Optimization Problems 201
3.5 Business and Economics Applications 210
3.6 Asymptotes 220
3.7 Curve Sketching: A Summary 231
3.8 Differentials and Marginal Analysis 240
Chapter 3 Algebra Review 248
Chapter Summary and Study Strategies 250
Review Exercises 252
Sample Post-Graduation Exam Questions 256
4
Exponential and Logarithmic Functions 257
4.1 Exponential Functions 258
4.2 Natural Exponential Functions 264
4.3 Derivatives of Exponential Functions 273
4.4 Logarithmic Functions 281
4.5 Derivatives of Logarithmic Functions 290
4.6 Exponential Growth and Decay 299
Chapter 4 Algebra Review 308
Chapter Summary and Study Strategies 310
Review Exercises 312
Sample Post-Graduation Exam Questions 316
5
Integration and Its Applications 317
5.1 Antiderivatives and Indefinite Integrals 318
5.2 The General Power Rule 329
5.3 Exponential and Logarithmic Integrals 337
5.4 Area and the Fundamental Theorem of Calculus 344
5.5 The Area of a Region Bounded by Two Graphs 356
5.6 The Definite Integral as the Limit of a Sum 365
5.7 Volumes of Solids of Revolution 371
Chapter 5 Algebra Review 378
Chapter Summary and Study Strategies 380
Review Exercise 382
Sample Post-Graduation Exam Questions 386
Contents v
6
Techniques of Integration 387
6.1 Integration by Substitution 388
6.2 Integration by Parts and Present Value 396
6.3 Partial Fractions and Logistic Growth 406
6.4 Integration Tables and Completing the Square 416
6.5 Numerical Integration 426
6.6 Improper Integrals 436
Chapter 6 Algebra Review 446
Chapter Summary and Study Strategies 448
Review Exercises 450
Sample Post-Graduation Exam Questions 454
7
Functions of Several Variables 455
7.1 The Three-Dimensional Coordinate System 456
7.2 Surfaces in Space 464
7.3 Functions of Several Variables 474
7.4 Partial Derivatives 483
7.5 Extrema of Functions of Two Variables 494
7.6 Lagrange Multipliers 504
7.7 Least Squares Regression Analysis 514
7.8 Double Integrals and Area in the Plane 524
7.9 Applications of Double Integrals 532
Chapter 7 Algebra Review 540
Chapter Summary and Study Strategies 542
Review Exercises 544
Sample Post-Graduation Exam Questions 548
Appendices A1
Appendix A: Alternate Introduction to the
Fundamental Theorem of Calculus A2
Appendix B: Formulas A12
Appendix C: Differential Equations*
C.1 Solutions of Differential Equations
C.2 Separation of Variables
C.3 First-Order Linear Differential Equations
C.4 Applications of Differential Equations
Appendix D: Properties and Measurement*
D.1 Review of Algebra, Geometry, and Trigonometry
D.2 Units of Measurements
Appendix E: Graphing Utility Programs*
E.1 Graphing Utility Programs
Answers to Selected Exercises A21
Answers to Try Its A85
Index A97
*Available at the text-specific website at college.hmco.com.
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A Word from the Authors vii
A Word from the Authors
Welcome to Brief Calculus: An Applied Approach, Seventh Edition. In this revision, we
have focused on making the text even more student-oriented. To encourage mastery and
understanding, we have outlined a straightforward program of study with continual
reinforcement and applicability to the real world.
Student-Oriented Approach
Each chapter begins with “What you should learn” and “Why you should learn it.” The
“What you should learn” is a list of Objectives that students will examine in the chapter.
The “Why you should learn it”lists sample applications that appear throughout the chap-
ter. Each section begins with a list of learning Objectives,enabling students to identify and
focus on the key points of the section.
Following every example is a Try It exercise. The new problem allows for students to
immediately practice the concept learned in the example.
It is crucial for a student to understand an algebraic concept before attempting to master a
related calculus concept. To help students in this area,Algebra Reviewtips appear at point
of use throughout the text. A two-page Algebra Reviewappears at the end of each chapter,
which emphasizes key algebraic concepts discussed in the chapter.
Before students are exposed to selected topics, Discovery projects allow them to explore
concepts on their own,making them more likely to remember the results. These optional
boxed features can be omitted, if the instructor desires, with no loss of continuity in the
coverage of the material.
Throughout the text, Study Tips address special cases, expand on concepts, and help stu-
dents avoid common errors. Side Commentshelp explain the steps of a solution. State-of-
the-art graphics help students with visualization,especially when working with functions
of several variables.
Advances in Technologyare helping to change the world around us. We have updated and
increased technology coverage to be even more readily available at point of use. Students
are encouraged to use a graphing utility, computer program, or spreadsheet software as
a tool for exploration, discovery, and problem solving. Students are not required to
have access to a graphing utility to use this text effectively. In addition to describing the
benefits of using technology, the text also pays special attention to its possible misuse
or misinterpretation.
Just before each section exercise set,the Take Another Lookfeature asks students to look
back at one or more concepts presented in the section,using questions designed to enhance
understanding of key ideas.
Each chapter presents many opportunities for students to assess their progress,both at the
end of each section (Prerequisite Review and Section Exercises) and at the end of each
chapter (Chapter Summary,Study Strategies,Study Tools,and Review Exercises). The test
items in Sample Post-Graduation Exam Questionsshow the relevance of calculus. The test
questions are representative of types of questions on several common post-graduation
exams.