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. This page intentionally left blank 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.
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