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BRIEF EDITION HOFFMANN | BRADLEY | SOBECKI | PRICE FOR BUSINESS, ECONOMICS, AND THE SOCIAL AND LIFE SCIENCES BRIEF EDITION ISBN 978 0 07 353238 7 MHID 0 07 353238 X www mhhe com Eleventh Edition McGra.
BRIEF EDITION BRIEF EDITION FOR BUSINESS, ECONOMICS, AND THE SOCIAL AND LIFE SCIENCES McGraw-Hill Connect® Mathematics McGraw-Hill conducted in-depth research to create a new and improved learning experience that meets the needs of today’s students and instructors The result is a reinvented learning experience rich in information, visually engaging, and easily accessible to both instructors and students McGraw-Hill’s Connect is a Web-based assignment and assessment platform that helps students connect to their coursework and prepares them to succeed in and beyond the course Connect Mathematics enables math instructors to create and share courses and assignments HOFFMANN with colleagues and adjuncts with only a few clicks of the mouse All exercises, learning objectives, videos, and activities are directly tied to text-specific material BRADLEY SOBECKI PRICE Integrated Media-Rich eBook Eleventh Edition ▶ A Web-optimized eBook is seamlessly integrated within ConnectPlus® Mathematics for ease of use ▶ Students can access videos, images, and other media in context within each chapter or subject area to enhance their learning experience ▶ Students can highlight, take notes, or even access shared instructor highlights/notes to learn the course material MD DALIM #1167952 10/24/11 CYAN MAG YELO BLK ▶ The integrated eBook provides students with a cost-saving alternative to traditional textbooks McGraw-Hill Tegrity® records and distributes your class lecture, with just a click of a button Students can view anytime/anywhere via computer, iPod, or mobile device It indexes as it records your PowerPoint® presentations and anything shown on your computer so students can use keywords to find exactly what they want to study Tegrity is available as an integrated feature of McGraw-Hill Connect and Connect Plus www.mcgrawhillconnect.com Eleventh Edition ISBN 978-0-07-353238-7 MHID 0-07-353238-X HOFFMANN | BRADLEY | SOBECKI | PRICE www.mhhe.com hof3238x_fm_i-xxiv.qxd 11/25/11 7:27 PM Page i Calculus For Business, Economics, and the Social and Life Sciences hof3238x_fm_i-xxiv.qxd 11/25/11 7:27 PM Page ii hof3238x_fm_i-xxiv.qxd 11/25/11 7:27 PM Page iii BRIEF Eleventh Edition Calculus For Business, Economics, and the Social and Life Sciences Laurence Hoffmann Morgan Stanley Smith Barney Gerald Bradley Claremont McKenna College Dave Sobecki Miami University of Ohio Michael Price University of Oregon TM hof3238x_fm_i-xxiv.qxd 11/25/11 7:27 PM Page iv TM CALCULUS FOR BUSINESS, ECONOMICS, AND THE SOCIAL AND LIFE SCIENCES: BRIEF EDITION, ELEVENTH EDITION Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020 Copyright © 2013 by The McGraw-Hill Companies, Inc All rights reserved Printed in the United States of America Previous editions © 2010, 2007, and 2004 No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning Some ancillaries, including electronic and print components, may not be available to customers outside the United States This book is printed on acid-free paper RJE/RJE ISBN 978–0–07–353238–7 MHID 0–07–353238–X Vice President, Editor-in-Chief: Marty Lange Vice President, EDP: Kimberly Meriwether David Senior Director of Development: Kristine Tibbetts Editorial Director: Michael Lange Developmental Editor: Eve L Lipton Marketing Manager: Alexandra Coleman Senior Project Manager: Vicki Krug Senior Buyer: Kara Kudronowicz Lead Media Project Manager: Judi David Senior Designer: Laurie B Janssen Cover Designer: Ron Bissell Cover Image: Jillis van Nes, Gettyimages Senior Photo Research Coordinator: Lori Hancock Compositor: Aptara®, Inc Typeface: 10/12 Times Printer: R R Donnelley All credits appearing on page or at the end of the book are considered to be an extension of the copyright page CO 1, CO 2: © Corbis RF; p 195(right): © Nigel Cattlin/Photo Researchers, Inc.; p 195(left): Courtesy of Ricardo Bessin; CO 3: © Getty RF; CO 4: © The McGraw-Hill Companies, Inc./Jill Braaten, photographer; p 373: © Getty RF; CO 5: © Richard Klune/Corbis; p 477: © Corbis RF; CO 6: © AFP/Getty Images; p 538: © Alamy RF; CO 7(right): US Geological Survey; CO 7(left): Courtesy of Trails.com; p 663: © Getty RF Library of Congress Cataloging-in-Publication Data Calculus for business, economics, and the social and life sciences / Laurence Hoffmann [et al.] — Brief 11th ed p cm Includes index ISBN 978–0–07–353238–7 — ISBN 0–07–353238–X (hard copy: alk paper) Calculus—Textbooks I Hoffmann, Laurence D., 1943– QA303.2.H64 2013 515—dc23 2011016379 www.mhhe.com hof3238x_fm_i-xxiv.qxd 11/25/11 7:27 PM Page v In memory of our parents Doris and Banesh Hoffmann and Mildred and Gordon Bradley hof3238x_fm_i-xxiv.qxd 11/25/11 7:27 PM Page vi hof3238x_fm_i-xxiv.qxd 11/25/11 7:27 PM Page vii CONTENTS Preface xi CHAPTER Functions, Graphs, and Limits 1.1 1.2 1.3 1.4 1.5 1.6 CHAPTER Functions The Graph of a Function 16 Lines and Linear Functions 30 Functional Models 45 Limits 63 One-Sided Limits and Continuity 78 Chapter Summary 91 Important Terms, Symbols, and Formulas 91 Checkup for Chapter 91 Review Exercises 92 Explore! Update 97 Think About It 99 Differentiation: Basic Concepts 2.1 2.2 2.3 2.4 2.5 2.6 The Derivative 104 Techniques of Differentiation 119 Product and Quotient Rules; Higher-Order Derivatives 132 The Chain Rule 146 Marginal Analysis and Approximations Using Increments 160 Implicit Differentiation and Related Rates 172 Chapter Summary 185 Important Terms, Symbols, and Formulas 185 Checkup for Chapter 186 Review Exercises 186 Explore! Update 193 Think About It 195 vii hof3238x_fm_i-xxiv.qxd viii 12/9/11 10:55 AM Page viii CONTENTS CHAPTER Additional Applications of the Derivative 3.1 3.2 3.3 3.4 3.5 CHAPTER Exponential and Logarithmic Functions 4.1 4.2 4.3 4.4 CHAPTER Increasing and Decreasing Functions; Relative Extrema 198 Concavity and Points of Inflection 215 Curve Sketching 233 Optimization; Elasticity of Demand 248 Additional Applied Optimization 266 Chapter Summary 285 Important Terms, Symbols, and Formulas 285 Checkup for Chapter 285 Review Exercises 287 Explore! Update 292 Think About It 294 Exponential Functions; Continuous Compounding 298 Logarithmic Functions 314 Differentiation of Exponential and Logarithmic Functions 330 Additional Applications; Exponential Models 345 Chapter Summary 362 Important Terms, Symbols, and Formulas 362 Checkup for Chapter 363 Review Exercises 364 Explore! Update 370 Think About It 372 Integration 5.1 5.2 5.3 5.4 5.5 Indefinite Integration and Differential Equations 376 Integration by Substitution 392 The Definite Integral and the Fundamental Theorem of Calculus 407 Applying Definite Integration: Distribution of Wealth and Average Value 423 Additional Applications of Integration to Business and Economics 442 hof3238x_fm_i-xxiv.qxd 11/25/11 7:27 PM Page ix CONTENTS 5.6 CHAPTER Additional Applications of Integration to the Life and Social Sciences 453 Chapter Summary 467 Important Terms, Symbols, and Formulas 467 Checkup for Chapter 468 Review Exercises 469 Explore! Update 474 Think About It 477 Additional Topics in Integration 6.1 6.2 6.3 6.4 CHAPTER ix Integration by Parts; Integral Tables 480 Numerical Integration 494 Improper Integrals 508 Introduction to Continuous Probability 517 Chapter Summary 530 Important Terms, Symbols, and Formulas 530 Checkup for Chapter 531 Review Exercises 532 Explore! Update 535 Think About It 538 Calculus of Several Variables 7.1 7.2 7.3 7.4 7.5 7.6 Functions of Several Variables 546 Partial Derivatives 561 Optimizing Functions of Two Variables 577 The Method of Least-Squares 594 Constrained Optimization: The Method of Lagrange Multipliers 606 Double Integrals 621 Chapter Summary 638 Important Terms, Symbols, and Formulas 638 Checkup for Chapter 639 Review Exercises 640 Explore! Update 645 Think About It 647 hof3238x_ans_759-770.qxd ANSWERS 764 11/23/11 7:34 PM Page 764 ANSWERS y dy x ϭ 57 dx S-78 Ϫ (1 ϩ xy2)exy Ϫ ln(x ϩ y) Ϫ 2x2yexy ϩ x ϩ x xϩy 59 f(2.1623, 1.5811) ϭ 1.6723 61 f(0.9729, Ϫ0.1635) ϭ 2.9522 x xϩy 31 ln 33 35 2(e Ϫ 2) 37 y CHAPTER Section y ϭ Ϫ x2 Ϫ1 ln ϭ ln 16 ln 32 13 32 15 e2 Ϫ 17 11 x ͵ ͵ xϭ 24Ϫy yϭ4 yϭ0 f (x, y) dx dy xϭ0 y 39 y ϭ ͙ළx y ϭ x3 x 19 Vertical cross sections: Յ x Յ x2 Յ y Յ 3x Horizontal cross sections: Յ y Յ y Յ x Յ 1y 21 Vertical cross sections: Ϫ1 Յ x Յ 1ՅyՅ2 Horizontal cross sections: Յ y Յ Ϫ1 Յ x Յ 23 Vertical cross sections: Յ x Յ e Յ y Յ ln x Horizontal cross sections: Յ y Յ ey Յ x Յ e 25 27 29 44 15 ͵ ͵ xϭy1ր3 yϭ1 yϭ0 f (x, y) dx dy xϭy2 y 41 y ϭ ln x x e2 ͵ ͵ yϭ2 yϭ0 xϭey xϭ1 f (x, y) dx dy 11/23/11 7:35 PM Page 765 S-79 y 43 ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ xϭ1 67 Area ϭ y ϭ x2 ϩ yϭ2 dy dx ϭ xϭ0 yϭ0 yϭ2 Average ϭ Ϫ2 ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ yϭ2x xϭ3 49 xϭ1 xϭe 51 xϭ1 yϭ3 53 yϭ0 yϭ2 55 yϭ0 xϭ2 57 xϭ1 xϭ1 59 xϭ0 yϭ1 61 yϭ0 63 dy dx ϭ 18 xϭ4 yϭ 12 x2 dy dx ϭ 16 ͵ ͵ xϭ3 73 xϭ1 dy dx ϭ yϭx2 Ϫ4xϩ3 dy dx ϭ ͵ ͵ xϭ0 dx dy ϭ yϭ2 xϭ1 75 yϭ0 xϭ 24Ϫy yϭ5 19 yϭ2 xeϪy dy dx ϭ yϭ0 (2x ϩ y) dx dy ϭ xϭy xϭϪ2 (x ϩ 1) dy dx ϭ yϭx xϭ3 Average ϭ 15 64 yϭ2 xϭϪ2 yϭϪ1 ϩ (ln 5)2 Ϫ (ln 2)2 xy(x Ϫ 2y) dy dx ϭ eϪ2 yϭ7 dy dx ϭ 35 yϭ0 35 943 xϭ5 xϭ0 yϭ7 (2x3 ϩ 3x2y ϩ y3) dy dx yϭ0 ͵ ͵ ͵ ͵ xϭ1 yϭ1 xϭϪ1 yϭϪ1 0.01 (300 ϩ x ϩ y)eϪ0.01x dy dx ϭ 79,800e Ϫ 80,200eϪ0.01 Ϸ 1,200 That is, $1,200,000 83 (5 ln Ϫ 4) Ϸ 18.212 (18,212 people) 85 Average ϭ yϭϪ1 xϭ3 x dy dx ϭ yϭ0 xϭϪ2 ͵ ͵ ͵ ͵ 81 Value ϭ dy dx ϭ 15 xϭϪ2 32 xϭ125 yϭ89 [(x Ϫ 30)(70 ϩ 5x Ϫ 4y) 25 ؒ19 xϭ100 yϭ70 ϩ (y Ϫ 40)(80 Ϫ 6x ϩ 7y)] dy dx ϭ 24,896.5 ($2,489,650) yϭ2 65 Area ϭ yϭ4Ϫx2 xϭ2 79 Average ϭ yϭ8Ϫx ϭ 1 a1 Ϫ b e xϭ2Ϫy xϭ2 yϭ0 Average ϭ dy dx ϭ (ln 3)(ln 2) xy dy dx ϭ x3ex y dy dx ϭ xϭ0 xϭ0 yϭ1 yϭ4Ϫx2 xϭ5 (6 Ϫ 2x Ϫ 2y) dx dy ϭ yϭxր3 yϭ0 xϭ1 xy dy dx ϭ ln(xy) dy dx y yϭ1 77 Area ϭ yϭ1 xϭ0 ϭ (3 ln Ϫ 2) ln yϭln x yϭ3 xϭ3 Average ϭ 32 xϭyր3 yϭxր3 xϭϪ2 yϭxϩ4 yϭ0 xϭ0 71 Area ϭ xϭ0 yϭ1 xϭ2 xϭϪ2yϪ1 xϭϪ4 Average ϭ f (x, y) dx dy xϭ2 47 xϭ 2yϪ1 yϭ2 yϭ1 e2 Ϫ xyex y dx dy ϭ dy dx ϭ xϭ0 x Ϫ1 xϭ1 yϭ0 xϭ3 69 Area ϭ 45 765 ANSWERS ͵ ͵ xϭ4 12 xϭ0 ϭ 630 feet yϭ3 yϭ0 90(2x ϩ y2) dy dx ANSWERS hof3238x_ans_759-770.qxd hof3238x_ans_759-770.qxd ANSWERS 766 11/23/11 7:35 PM Page 766 ANSWERS 17,408 Ϸ 166 m3 105 304 Ϸ 11.26 89 27 87 91 62,949 people 93 64 (64,000 people) 95 a 0.991 square meters b No, it can only be considered the average surface area from birth until the time at which the person reached adulthood Ϫ6 7e 17 ϩ Ϸ 1.891 cubic units 97 9 S-80 b Parabolas with vertices on the x axis and opening to the left a Relative maximum: (0, 0); relative minimum: (1, 4); saddle points: (1, 0), (0, 4) b Saddle point: (Ϫ1, 0) c Relative minimum: (Ϫ1, Ϫ1) 16 at a , b 5 b Maximum value of at (1, 2) or (1, Ϫ2); minimum value of Ϫ4 at (Ϫ1, 2) or (Ϫ1, Ϫ2) a 16 a b CHAPTER Checkup a Domain: all ordered pairs (x, y) of real numbers fx ϭ 3x2 ϩ 2y2 fy ϭ 4xy Ϫ 12y3 fxx ϭ 6x fyx ϭ 4y b Domain: all ordered pairs (x, y) of real numbers for which x y Ϫ3y fx ϭ (x Ϫ y)2 3x fy ϭ (x Ϫ y)2 6y fxx ϭ (x Ϫ y)3 Ϫ3(x ϩ y) fyx ϭ (x Ϫ y)3 c Domain: all ordered pairs (x, y) of real numbers for which y2 Ͼ 2x fx ϭ 2e2xϪy Ϫ y Ϫ 2x 2y fy ϭ Ϫe2xϪy ϩ y Ϫ 2x fxx ϭ 4e2xϪy Ϫ (y Ϫ 2x)2 4y fyx ϭ Ϫ2e2xϪy ϩ (y Ϫ 2x)2 a Circles centered at the origin and the single point (0, 0) (e ϩ 3eϪ2) c ln Ϫ d Ϫ 10 e2 QK ϭ 180; QL ϭ 3.75 20 DVDs and video games 30 units of drug A and 25 units of drug B, which results in an equivalent dosage of E(30, 25) ϭ 83.75 units Since the total number of units is 55, which is less than 60, there is no risk of side effects, and since E(30, 25) Ͼ 70, the combination is effective (1 ϩ eϪ2) Ϸ 2.84ЊC a Profit 2 b y ϭ 0.45x ϩ 0.61 c 3.31 million dollars Year 11/23/11 7:35 PM Page 767 S-81 ANSWERS 21 Relative minimum at aϪ CHAPTER Review Exercises fx ϭ 6x2y ϩ 3y2 Ϫ y x ; fy ϭ 2x3 ϩ 6xy ϩ fx ϭ 3x Ϫ y2 ; fy ϭ Ϫ2y1x 21x fx ϭ 1y 1 1x Ϫ ; fy ϭ Ϫ 32 21xy 21xy 2x ր 2y ր fx ϭ fx ϭ 2x3 ϩ 3x2y Ϫ y2 (x ϩ y)2 2(x2 ϩ xy ϩ y2) (2x ϩ y)2 11 fxx ϭ (4x2 ϩ 2)ex fxy ϭ 4xy ex ϩy ϩy ; fy ϭ 5 23 Saddle points at a , Ϫ b and aϪ , b 6 40 25 Points (Ϫ1, 2), (Ϫ1, 0), aϪ , b, a , b; 2 17 17 max of 20 at (Ϫ4, 0); of at (Ϫ1, 2) Ϫx2(x ϩ 1) (x ϩ y)2 2 ; fyx ϭ 4xy ex ϩy2 ; ϩy 31 33 35 f = –2 37 (0, 2) f=2 x y Ϫ0.5 Ϫ1 Ϫ1.5 fy ϭ 213 413 , 2b, a , 5b; max of 52 at 3 (4, 2); of at (2, 4) Point (Ϫ2, 0); max of at any point on the boundary; of eϪ4 Ϸ 0.018 at (Ϫ2, 0) Maximum value of 12 at (1, Ϯ13); minimum value of at (Ϫ2, 0) Maximum value of 17 at (1, 8); minimum value of Ϫ17 at (Ϫ1, Ϫ8) Daily output will increase by approximately 16 units The level of unskilled labor should be decreased by approximately workers Maximize area A ϭ xy subject to fixed perimeter P ϭ 2x ϩ 2y Lagrange conditions are y ϭ (x), x ϭ (y), and 2x ϩ 2y ϭ C We must have Ͼ since x and y are positive, so x ϭ y and the optimum rectangle is actually a square Development x ϭ $4,000; promotion y ϭ $7,000 43 We have fx ϭ y Ϫ 1.5 0.5 39 41 Ϫ3 Ϫ2 Ϫ1 29 x 1 13 fxx ϭ 0; fyy ϭ Ϫ ; fxy ϭ ; fyx ϭ y y y y 15 a b 27 Points (2, 4), a (2x ϩ y)2 ; fyy ϭ (4y2 ϩ 2)ex 23 , 5b; saddle point at a , 1b Ϫx2 Ϫ 4xy Ϫ y2 ; fy ϭ x 767 x f=2 f=1 f=0 x ϭ Ϫ y y(x ϩ 3y) x ϩ 3y 17 Saddle point at (6, Ϫ6) 19 Relative maximum at (Ϫ2, 0); relative minimum at (0, 2); saddle points at (0, 0) and (Ϫ2, 2) 12 and fy ϭ x Ϫ 18 , x y2 so fx ϭ fy ϭ when x ϭ and y ϭ Since f(x, y) is large when either x or y is large or small, a relative minimum is indicated at (2, 3), and the minimum value is f(2, 3) ϭ 18 To verify this claim, note that 24 24 24 D ϭ a ba b Ϫ and fxx ϭ x y x so that D(2, 3) Ͼ and fxx(2, 3) Ͼ e3 Ϫ e2 Ϫ e ϩ 45 Ϸ 0.5466 e3 47 (e Ϫ eϪ2) 49 2e Ϫ 2 ANSWERS hof3238x_ans_759-770.qxd hof3238x_ans_759-770.qxd ANSWERS 768 51 53 55 57 59 61 11/23/11 7:36 PM Page 768 ANSWERS S-82 Ϫ3 Յ x Յ Ϫ6 Յ x Յ Ϫ2 x Յ Ϫ7 or x Ն 125 4 25 27 29 (e Ϫ 1) 2 Ϫ2 (e Ϫ eϪ3) cubic units 20 xϭyϭzϭ 110; at (0, Ϯ110, 0) y a 13 15 17 19 21 23 160 140 120 100 31 33 80 60 40 63 65 67 69 71 10 x 12 b y ϭ 11.54x ϩ 44.45 c Approximately $102,150 5.94; demand is increasing at the rate of about quarts per month Ϫ3; demand is decreasing at the rate of pies per week The amount of pollution is decreasing by about 113 units per day About 7.056 units Q(x, y) ϭ xayb Qx ϭ axaϪ1yb; Qy ϭ bxaybϪ1 xQx ϩ yQy ϭ x(axaϪ1yb) ϩ y(bxaybϪ1) ϭ (a ϩ b)xayb ϭ (a ϩ b)Q If a ϩ b ϭ 1, then xQx ϩ yQy ϭ Q Appendix Section A.1 1 Ͻ x Յ x Ͼ Ϫ5 x –2 11 x 35 n ϭ 10 37 n ϭ 39 n ϭ 13 41 n ϭ 43 a5b8c8 a8c12 45 b4 a10 47 14 bc a18b12 49 c6 1 51 ϩ ϩ abc a bc abc4 Ϫ1 2 53 a b ϩ a b 55 Ϫ2 57 1,3501 900 59 3812 61 916 63 a3b4c7 5b 65 7a 2 a2 ab 67 3 bc 69 a bc 11/23/11 7:36 PM Page 769 S-83 71 ANSWERS a2c3 15 b 73 a Ϫ 1b a5b3 ac 75 c 77 x (x Ϫ 4) 79 Ϫ25(x Ϫ 7) 81 2(x ϩ 1)2(x Ϫ 2)2(7x Ϫ 2) 83 xϪ1ր2(6x ϩ 1) 85 2(x ϩ 3) 2(x ϩ 3)3(5 Ϫ x) 87 (1 Ϫ x)3 89 5(13 ϩ 12) 7(3 ϩ 13) 91 93 3(15 Ϫ 2) 5(15 Ϫ 1) 95 97 1x ϩ h Ϫ 1x ϭ ( 1x ϩ h Ϫ 1x)(1x ϩ h ϩ 1x) 1x ϩ h ϩ 1x xϩhϪx 1x ϩ h ϩ 1x h ϭ 1x ϩ h ϩ 1x 99 a Surface area is approximately 5.212 ϫ 108 km2; mass of the atmosphere is 5.212 ϫ 1018 kg b 127,400 years ϭ Appendix Section A.2 11 13 3x2 Ϫ 27x x2 Ϫ 5x Ϫ 14 Ϫ6x2 ϩ 26x Ϫ 28 x3 ϩ x2 Ϫ 5x ϩ x5 Ϫ 3x4 Ϫ x3 ϩ 13x2 Ϫ 18x ϩ 2x2 ϩ 3x ϩ x2 Ϫ x2 x2 ϩ 2x Ϫ 3 2x Ϫ 7x Ϫ 15 x ϩ 10 17 Ϫ x ϩxϪ2 19 x ϩ 7x ϩ 12 Ϫx ϩ 21 xϩ3 23 Ϫ2 x 25 3x Ϫ x2 ϩ x Ϫ 27 Ϫ 3x Ϫ 29 (x ϩ 2)(x Ϫ 1) 31 (x Ϫ 3)(x Ϫ 4) 33 (x Ϫ 1)2 35 (4x ϩ 5)(4x Ϫ 5) 37 (x Ϫ 1)(x2 ϩ x ϩ 1) 39 x5(x ϩ 1)(x Ϫ 1) 41 2x(x Ϫ 5)(x ϩ 1) 43 (x ϩ 4)(x Ϫ 3) 45 (2x ϩ 5)(x Ϫ 3) 47 (x ϩ 2)(x Ϫ 9) 49 2(2x ϩ 1)(7x Ϫ 3) 51 x(x ϩ 5)(x Ϫ 3) 53 (x ϩ 3)(x2 Ϫ 3x ϩ 9) 55 x2(x ϩ 1)(x2 Ϫ x ϩ 1) 57 (3x ϩ 1)(x ϩ 2)2 59 x ϭ 4; x ϭ Ϫ2 61 x ϭ Ϫ5 63 x ϭ 4; x ϭ Ϫ4 65 x ϭ Ϫ ; x ϭ Ϫ1 67 x ϭ Ϫ 69 x ϭ 1; x ϭ Ϫ5 71 x ϭ 1; x ϭ Ϫ2 73 x ϭ Ϫ1 75 x ϭ 1; x ϭ Ϫ3 77 x ϭ ; x ϭ 769 ANSWERS hof3238x_ans_759-770.qxd hof3238x_ans_759-770.qxd ANSWERS 770 11/23/11 7:36 PM ANSWERS S-84 79 No real solutions Ϫ17 ϩ 1385 Ϫ17 Ϫ 1385 81 x ϭ ;xϭ 12 12 83 x ϭ Ϫ ; x ϭ Ϫ1 85 No real solutions 87 89 91 93 Page 770 xϭϪ x ϭ 3; y ϭ x ϭ 4, y ϭ x ϭ Ϫ7; y ϭ Ϫ5 and x ϭ 1, y ϭ Ϫ1 11 13 15 17 19 2ՅxՅ4 243 4 161 73 21 n ϭ 18 23 n ϭ Ϫ1 25 21 27 95 29 ϩ a Appendix Section A.3 3 31 0 0 e2 Appendix Section A.4 34 a j jϭ1 x ϭ Ϫ2; y ϭ x ϭ 1, y ϭ and x ϭ 15, y ϭ Ϫ26 53 x ϭ Ϫ2; x ϭ a 2xj 55 57 jϭ1 33 35 37 39 41 43 45 47 49 x ϭ Ϫ ; x ϭ 51 No real solutions (Ϫ1)k k kϭ2 x (x ϩ 3)(x Ϫ 3) x4(x6 ϩ 4)(x3 ϩ 2)(x3 Ϫ 2) x(x Ϫ 1)2 (x ϩ 5)(x Ϫ 3) (2x ϩ 3)2 (x ϩ 1)(x Ϫ 1)(x ϩ 3) x ϭ Ϫ4; x ϭ x ϭ Ϫ7 x ϭ Ϫ1; x ϭ Ϫ 11 13 15 x jϩ1 a (Ϫ1) j jϭ1 59 Appendix Review Exercises Ϫ2 Յ x Ͻ 3 –3 x 61 63 hof3238x_ndx_771-776.qxd 11/25/11 8:19 PM Page 771 INDEX A Absolute extrema See also Extrema explanation of, 248–249 extreme value property and, 249–251, 586–588 method to find, 249–253 second derivative test for, 253–255 Absolute maxima See also Maxima explanation of, 248–249 for function of two variables, 585 method to find, 249–253 Absolute minima See also Minima explanation of, 248–249 for function of two variables, 585 method to find, 249–253 Absolute value, 655–656 Absolute value function, 113 Absolute value inequalities, 656 Absorption coefficient, 313 Acceleration explanation of, 140 finding velocity and position from, 382–383 method to find, 140 rectilinear motion and, 125, 126 Addition, of polynomials, 663 Additive property of inequality, 653 Algebraic expressions, 659–660 Algebraic rules for indefinite integration, 380–381 Algebra review absolute value, 655–656 completing the square, 669–671 exponents and roots, 656–660 factoring polynomials with integer coefficients, 664–666 inequalities, 652–654 intervals, 654–655 L’Hôpital’s Rule, 674–679 limits, 69–70, 674–679 natural logarithm table, 686 polynomials, 663–664 powers of e table, 685 quadratic formula, 671–672 rational expressions, 667–668 rationalizing, 660–661 real numbers, 652 solving equations by factoring, 668–669 summation notation, 680 system of equations, 672–674 Allometric models, 99–100 Allometry explanation of, 99, 605 law of, 391 Amortization of debt, 311 Annuities applications involving, 442–445 explanation of, 442 Antiderivatives See also Indefinite integrals explanation of, 376 of function, 377–378 fundamental property of, 377–378 graphs of, 377–378 indefinite integrals and, 378–382 method to verify, 376 substitution and, 483 Antidifferentiation See also Indefinite integrals explanation of, 376 relationship between differentiation and, 378–379 Approximation of area under curve, 407–408 best-fit, 295–296 of definite integrals, 494–501 differentials and, 167 by increments, 164–167 by rectangles, 494–495 Simpson’s rule and, 498–501 tangent line, 171 Approximation formula explanation of, 164, 166 incremental, 570–571 Area under curves, 104, 407–408 as definite integral, 411 as limit of sum, 407–410 numerical integration to find, 501–502 of region in plane, 629, 630 between two curves, 425–428 use of double integral to find, 629, 630 use of fundamental theorem of calculus to find, 412–413 Area formula, 629 Arrhenius equation, 368 Asymptotes explanation of, 233 horizontal, 70, 235–236 vertical, 233–234 Average cost, 256 Average cost functions, Average rate of change, 106, 107 Average value formula, 631 Average value of function applications involving, 434 explanation of, 433–434, 631 geometric interpretation of, 436–437 rate interpretation of, 436–437 B Base e, natural exponential, 303–304 Bell, Alexander Graham, 328 Benford’s law, 372–373 Best-fit approximation, 295–296 Best-fit equations, 598 Best-fit polynomials, 294–295 Best-fitting line, 39 Binomial theorem, 109 Bouguer-Lambert law, 312–313 Boundary, 585 Boundary point, 585 Break-even analysis examples of, 54–56 explanation of, 53 Break-even point explanation of, 53 intermediate value property to estimate, 86–87 C Calculators See Graphing calculators Calculus See also Fundamental theorem of calculus historical background of, 104 integral, 477–478 Capitalized cost, 516 Carbon dating, 324–325 Cardiac output, 538–541 Carrying capacity, 351 Cartesian coordinate system See Rectangular coordinate system Catenary, 361 Centroid, 493 Certain events, 518 Chain rule differentiating using, 393 for eu, 332 explanation of, 147 to find tangent line, 149–150 for partial derivatives, 569–571 rate of change and, 150–151 root functions and, 151 for ln u, 334–335 use of, 147–149, 153, 331–332, 334 Change See also Rate of change net, 417–418 percentage, 166–167 relative, 166 Change of variables See Substitution Circle, equation of, 552 Circular paraboloid, 552 Closed, bounded region, 585–586 Closed intervals, 654 Cobb-Douglas production function, 548–549, 611, 632 Cobb-Douglas utility function, 609–610 Coefficients absorption, 313 diffusion, 648 dispersion, 648 integer, 664–666 of polynomial, 663 Complementary commodities, 566–567 Completing the square, 669–671 Complex fractions, 121 Composite functions construction of, explanation of, expressing cost as, 9–10 inverse relationship between, 319 outer and inner functions of, 10–11 to study air pollution, 11 Compound fractions, 667–668 Compound interest applications involving, 321–323 explanation of, 304–306 present value and, 306–307 Computer graphics, 554–555 Concavity explanation of, 216 in graphing, 222–224 inflection points and, 218–221 intervals of, 217–218 Cone, 550 Constant elasticity of substitution (CES) production function, 558, 618 Constant function, derivative of, 120 Constant multiple rule for definite integrals, 414, 415 explanation of, 122, 146 for indefinite integrals, 301, 380 use of, 122 Constant of integration, 378 Constant rule explanation of, 119 for integrating common functions, 379, 380 use of, 121 Constrained optimization See also Lagrange multipliers applications involving, 607–612, 615 explanation of, 606–607 Consumers’ surplus, 447–449 Consumer willingness to spend, 445–447 Continuity differentiability and, 113–114 on interval, 85–86 limits and, 82–83 of polynomials and rational functions, 83–85 Continuous compounding applications of, 306–307, 321–323, 348–350 explanation of, 304–306 using differential equations, 386–387 Continuous compounding formula, 386–387 Continuous functions on closed, bounded region, 585, 586 example of, 113–114 explanation of, 79, 82, 83, 199 771 hof3238x_ndx_771-776.qxd 772 11/25/11 8:19 PM Page 772 INDEX Continuous functions—Cont use of derivative to sketch graph of, 205–206 Continuous probability continuous random variables and probability density functions and, 517–521 expected value of random variables and, 524–525 exponential density functions and, 522–523 normal density functions and, 523–524 uniform density functions and, 521–522 Continuous random variables expected value of, 524–525 explanation of, 518 Pareto distribution and, 530 probability density function for, 518–519 Convergence, 509 Conversion formula for logarithms, 321 Coordinates, 17, 652 Cost functions evaluation of, 7–8 explanation of, linear, 35–36 total, 273–274 Costs average, 256 capitalized, 516 fixed, 278 marginal, 161–163, 382 minimal, 613–614 net change in, 417–418 related rate of, 177–178 total, 276–278, 382 variable, 278 Critical numbers explanation of, 202 finding and classifying, 204 Critical points explanation of, 202–203, 578 method to classify, 225, 226, 580–583 relative extrema and, 578–579 second partials test and, 579, 580 Cross sections horizontal, 626–628 vertical, 624–625 Cubes, factoring difference of, 666 Curve-fitting, nonlinear, 598–600 Curves area between two, 425–428 area of region under, 104, 407–408 of constant product C, 553 of constant temperature, 560 demand, 338 exponential, 346–347 Gompertz, 360 indifference, 553, 610 learning, 350–351, 390 level, 550–554 logarithmic, 345–346 logistic, 351–353, 487 Lorenz, 430–432 Phillips, 116 revenue, 338 I-2 Curve sketching applications for, 242–243 cusps and vertical tangents and, 240–241 exponential graphs, 346–347 horizontal asymptotes and, 235–236 logarithmic graphs, 345–346 procedure for, 236 for rational function, 236–240 with second derivative, 221–224 use of, 202, 205, 206, 208 vertical asymptotes and, 233–234 Cusps, graphs with, 240–241 Cylinders, area and volume of, 270 D Decibels, 328 Definite integrals See also Integrals; Integration applications involving, 423–424 approximation of, 494–501 area as limit of sum, 407–410 area between two curves and, 425–430 average value of function and, 433–437 evaluation of, 413 explanation of, 378, 407, 410 fundamental theorem of calculus and, 411–412, 418–419 integration by parts and, 483, 484 Lorenz curves and, 430–432 net change and, 417–418 net excess profit and, 428–430 rules for, 414–415 substitution and, 416–417 symbol for, 410 Degree, of polynomial, 23, 663 Delta notation, 31, 108, 164 Demand elasticity of, 256–258, 263, 337–338 rate of change of, 154–155 Demand curve, applications involving, 338 Demand function explanation of, 6, 51 exponential, 598–599 graph of, 338 Denominator, rationalizing the, 660–661 Dependent variables, Derivatives See also Differentiation; Instantaneous rate of change; specific types of derivatives of constant function, 120 to determine intervals of increase and decrease of function, 200–201 difference quotient and, 11, 108 differentiability and continuity and, 113–114 estimating change in cost using, 164–165 of ex, 330, 331 explanation of, 104, 108 of exponential functions, 330–331 first, 138, 139 formulas for, 330–336 to graph functions, 205–208 on graphing calculators, 111, 112, 120, 139, 155 graphs of, 223–227, 331 higher-order, 140–142, 153 of implicitly defined functions, 173–174 of logarithmic functions, 333–336 method to find, 108–109 notation for, 109, 111–112 of order n or nth, 140–141 partial, 561–571, 578–579 second, 138–140, 154 sign of, 111 slope and rate of change and, 104–110, 112 Descartes, René, 17 Difference of cubes formula, 666 Difference of squares formula, 666 Difference quotient explanation of, 108 method to compute, 11–12 Difference rule for definite integrals, 414, 415 for indefinite integrals, 380, 381 Differentiable functions continuity and, 113–114 explanation of, 108 integration by parts and, 480 Differential equations applications of, 385 continuous compounding using, 386–387 explanation of, 383–384 separable, 384–385, 387–388 use of substitution to solve, 399–400 Differentials, 167 Differentiation See also Derivatives chain rule and, 146–156 explanation of, 104, 108 of exponential and logarithmic functions, 330–340 implicit, 172–180 logarithmic, 339 marginal analysis and approximations and, 160–167 partial, 562 (See also Partial derivatives) product and quotient rules and, 132–142, 480 relationship between antidifferentiation and, 379 techniques of, 119–128 Diffusion coefficient, 648 Diffusion equation, 647–649 Directly proportional, 50 Discriminant, 671, 672 Dispersion coefficient, 648 Distance formula, 17–18 Distributive laws to multiply polynomials, 664 Divergence, 509 Division rule, 302 Domain of function, 2, 4, 546 natural, Double decline balance formula, 357 Double integrals applications involving, 629–633 evaluation of, 623–624, 627–628 limits of integration for, 627–628 over nonrectangular region, 624–628 over rectangular region, 622 Doubling time, investment, 321–322 Dye dilution, 538, 539 E Economic order quantity (EOQ), 281 Economic production quantity (EPQ), 281 Economics functions used in, 6–8 implict differentiation in, 175–176 level curves in, 553–554 use of derivative to approximate change in, 161 Effective interest rate, 307–308 Elastic demand, 258, 260 Elasticity of demand, 256–258, 263, 337–338 revenue levels and, 259–261 Ellipsoid, 550 Empty set, 518 Equality rule explanation of, 301 for logarithms, 315 Equations Arrhenius, 368 of circle, 552 differential, 383–388, 399 diffusion, 647–649 exponential, 302 Laplace’s, 576 least-squares, 596 of line, 32–35 logarithmic, 320 logistic, 487 partial differential, 647 quadratic, 669–673 rational, 669 system of, 672–674 of tangent line, 335–336 Equilibrium, 52, 53 Equilibrium price, 52 Error, propagated, 165 Error estimates explanation of, 165 for Simpson’s rule, 499–501 for trapezoid rule, 497–498 Events, 518 Excess profit explanation of, 428–429 net, 428–430 Expected value, 524–525 Explicit form, 172–173 Exponential decay, 387 Exponential demand function, 598–599 Exponential density function, 522–523 Exponential equations, 302 Exponential expressions, 302, 656 Exponential functions applications for, 299, 336–338 curve sketching and, 346–347 hof3238x_ndx_771-776.qxd 11/25/11 8:19 PM Page 773 I-3 derivatives of, 330, 331 differentiation of, 330–340 evaluation of, 303–304 explanation of, 299–304 finding extreme values of, 332–333 graphs of, 300–301 inverse relationship between logarithmic and, 319 method to solve, 320 natural, 303, 330 population, 320–321 properties of, 301, 316 rules for, 302 substitution used on, 395–396 Exponential growth/exponential decay, 348–350, 387 Exponentially distributed random variables, 522–523 Exponential notation, 299, 657 Exponential rule, 379, 380 Exponents evaluating expressions with, 657–658 explanation of, 656–657 fractional, 657 laws of, 658 simplifying expressions with, 659 Extrema See also Maxima; Minima absolute, 248–255 relative, 202–209, 224, 225 Extreme value property applications involving, 587–588 explanation of, 249 to find absolute extrema of continuous functions, 249–251, 586–587 for functions of two variables, 585, 586 F Factoring difference of cubes, 666 with integer coefficients, 664–666 to solve equations, 668–669 Factorization formulas, 666 Fad, 209 First derivative See also Derivatives explanation of, 138, 139 of production function, 215 First derivative test, for relative extreme, 203–204 Fixed-budget problem, 618 Fixed costs, 278 FOIL method, 663–664 Fractional exponents, 657 Fractions complex, 121 compound, 667–668 properties of, 667 rule to simplify, 121 Functional composition, 8–12 Functional notation, Functions See also Functions of two or more variables; specific types of functions absolute value, 113 average cost, average value of, 433–437 composition of, 8–11 INDEX constant, 120 continuous, 79, 82–86, 113–114, 205–206 cost, 6–8 demand, 6, 51 derivative of, 108–109 differentiable, 108, 113–114, 173, 480 domain of, 2, evaluation of, 3–5 explanation of, 2–4 in explicit form, 172 exponential, 299–304, 316, 320–321, 330–340, 395–396 general probability density, 361 on graphing calculators, 3, 7, 19, 35, 120 graphs of, 16–26 implicitly defined, 173–174 increasing and decreasing, 198–203 inflection points of, 218–221 limit of, 64–66 (See also Limits) linear, 30, 35–37 logarithmic, 314–325, 330–341, 397 piecewise-defined, 5, 19–20, 49–50 power, 23, 301 probability density, 518–524 profit, 6, 47 quadratic, 24–26 range of, rational, 23, 68, 83–85, 236–240, 396–397 renewal, 454–456 root, 151 standard normal probability density, 347 supply, 6, 51 surge, 360 survival, 454–456 used in economics, 6–8 value of, zeros of, 20 Functions of two or more variables applications involving, 546, 548–549 computer-generated graphs for, 554–555 double integrals and, 621–633 evaluation of, 547 explanation of, 546–547 extreme value property for, 585, 586 graphs of, 549–550 incremental approximation formula and, 570–571 Lagrange multipliers and, 607–615 level curves and, 550–554 optimization of, 577–588 partial derivatives and, 561–571 Fundamental glottochronology equation, 313 Fundamental property of antiderivatives, 377–378 Fundamental theorem of calculus area justification of, 418–419 definite integrals and, 411–413 explanation of, 378, 407 partial integrals and, 621 Future value of annuity, 442–443 of investment, 305, 442–443, 484–485 method to compute, 306 G Gas constant, 576 General power rule explanation of, 151 to find rate of change, 154–156 to find second derivative, 154 use of, 152–154 General probability density function, 361 Geometric interpretation of average value of function, 436–437 of expected value, 524 of partial derivatives, 564–565 Gini index (GI), 431–433 Glottochronology, 313 Gompertz curve, 360 Graphing calculators absolute value function on, 113 antiderivatives on, 378 average value on, 433 best-fit equations on, 598 compound interest on, 307, 322 derivatives on, 111, 112, 120, 139, 155, 331 functions on, 3, 7, 19, 35, 120 indifference curves on, 553 least-squares equation on, 596 logarithms on, 318, 320–321 multivariate functions of, 547 numerical integration on, 496, 509 partial derivatives on, 563 probability density functions of, 522, 523 roots on, 193–194 secant lines on, 107 Simpson’s rule on, 499, 500 slope on, 32 Graphs of antiderivatives, 377–378 computer-generated, 554–555 concavity on, 216–221 of continuous functions, 205–206 of demand function, 338 of derivatives, 223–227, 331 of exponential functions, 300–301, 346–347 exponential growth and decay, 348–349 of functions, 16–26 of functions of two variables, 549–552 of increasing and decreasing functions, 198–203 of inequalities, 655 inflection points on, 218–223 intersections of, 20–21 of logarithmic functions, 317, 318, 345–346 of polynomials, 23 of rational functions, 236–240 Growth rate, relative, 339–340 773 H Haldane equation, 281 Half-life, 323–325 Half-open intervals, 654 Higher-order derivatives chain rule and, 153 explanation of, 140–141 product rule and, 141–142 Histograms, 518 Hoorweg’s law, 358 Horizontal asymptotes explanation of, 70, 235 method to find, 235–236 Horizontal cross sections, 626–628 Horizontal lines, 32 I Ideal gas law, 576 Identity law of exponents, 658 Implicit differentiation computing slope of tangent line by, 174–175 economics applications for, 175–176 examples of, 173–174 explanation of, 172–173 related rates and, 177–180 tangent line and, 175–176 Implicit function, 174 Impossible events, 518 Improper integrals See also Integrals applications involving, 512–515 evaluation of, 509–512 explanation of, 508–509 Income stream future value of, 442–443 present value of, 444–445, 502–503, 512–513 Incremental approximation formula for functions of two variables, 570–571 Indefinite integrals See also Integrals; Integration applications involving, 381–383 constant multiple rule and, 301, 380 explanation of, 378–379 method to compute, 380 rules for, 379–381 sum rule and, 380, 381 Indefinite integration See Antiderivatives Independent variables in difference quotients, 108 explanation of, functions of two or more, 546–552 Indeterminate forms, 675–679 Index of income inequality See Gini index (GI) Indifference curves, 553, 610 Inelastic demand, 258, 260 Inequalities absolute value, 656 to describe intervals, 654–655 explanation of, 562–563 graphs of, 655 method to solve, 653–654 properties of, 653 Infinite intervals, 654 hof3238x_ndx_771-776.qxd 774 11/25/11 8:19 PM Page 774 INDEX Infinite limits example of, 73–74 explanation of, 66, 73 Infinity, limits involving, 70–74, 233 Infinity symbol, 70 Inflection points explanation of, 218–219 graphical possibilities of, 221 method to find, 219–221 Initial value problem, 385 Instantaneous rate of change See also Derivatives as derivative, 109 explanation of, 106 method to compute, 106–107 Instantaneous velocity, 106–107 Integer coefficients, 664–666 Integer powers, 657 Integers, 652 Integral formulas, 480, 524–525 Integrals See also Definite integrals; Indefinite integrals applications involving, 381–383, 423–424, 512–515 definite, 378, 407–419, 423–437, 483, 484 double, 622–633 improper, 508–515 indefinite, 378–383 iterated, 622 partial, 621, 622 Integral symbol, 378 Integral tables example of, 488–489 explanation of, 486 use of, 486–487, 489–490 Integrand, 378, 410 Integration See also Definite integrals; Indefinite integrals; Integrals applications of, 412–413, 417–418, 442–449, 453–463 constant of, 378 definite, 410–419 explanation of, 104 limits of, 627–628 lower and upper limits of, 410 numerical, 494–503 variable of, 378 Integration by parts applications of, 484–485 definite, 483–485 explanation of, 480–481 formula for, 480, 482, 483 procedure for, 481–483 repeated application of, 485–486 use of integral tables and, 486–490 Integration by substitution differential equations and, 399 explanation of, 393–394 failure of, 398 linear, 394–395 on logarithmic function, 397 price-adjustment model and, 401–402 procedure for, 394 quadratic, 395–396 on rational function, 396–397 using algebra before, 397–398 I-4 Intercepts, 33–34 Interest compound, 304–307, 321–323 continuous compounding, 304–306, 321, 323, 348–350, 386–387 Interest rate in decimal form, 386 effective, 307–308 method to compute, 323 nominal rate of, 307 Intermediate value property continuous functions and, 199 to estimate break-even point, 86–87 explanation of, 86 Intervals of concavity, 217–218 continuity on, 85–86 explanation of, 654 of increase and decrease, 198–201 inequalities to describe, 654–655 open, 224 unbounded, 508, 511 Inversely proportional, 50 Inverse relationship between functions, 319 Inversion rule for logarithms, 315 Investments doubling time of, 321–322 future value of, 305, 442–443, 484–485 Irrational numbers, 652 Isoquant, 176, 553 Isotherms, 560 Iterated integrals, 622 J Jointly proportional, 50 Just-in-time inventory management, 277 L Lagrange equations, 607, 611 Lagrange multipliers allocation of resources and, 611–612 applications involving, 608–609 explanation of, 607–608, 614–615 for functions of three variables, 613–614 maximization of utility and, 609–610 significance of, 612–613 Laplace’s equation, 576 Law of allometry, 391 Law of diminishing returns, 573 Law of supply and demand, 52 Learning curves, 350–351, 390 Learning rate, 390 Least-cost combination of inputs, 617 Least-squares equation, 596 Least-squares line, 596–597 Least-squares linear approximation of data, 38, 39 Least-squares method explanation of, 594–595 least-squares line and, 596–597 nonlinear curve-fitting and, 598–600 predictions and, 597–598 use of, 595–596, 599, 600 Least-squares prediction, 597–598 Leibniz, Gottfried, 104 Level curves economics applications involving, 553–554 explanation of, 550–552 L’Hopital’s Rule, 676–679 Libby, W F., 324 Limits computation of, 67–70 estimation of, 64–65 existence of, 81 explanation of, 64, 675 improper intervals and, 510–511 indeterminate forms of, 675–679 infinite, 66, 73–74 of integration for double integrals, 627–628 intuitive introduction to, 63–66 involving infinity, 70–74, 233 L’Hopital’s Rule, 676–679 one-sided, 79–83 of polynomials and rational functions, 68 properties of, 66–67 quotient rule for, 68 of rational functions, 68, 72 of two linear functions, 67 two-sided, 81–83 Linear cost functions, 35–36 Linear functions See also Functions applications of, 35–36 explanation of, 30 Linear price functions, 36–37 Linear substitution, 394–395 Lines See also Tangent lines best-fitting, 39 equations of, 33–35 horizontal, 32 parallel, 37–38, 377 perpendicular, 37–38 secant, 106, 107 slope of, 30–33 use of intercepts to graph, 33–34 vertical, 32 Logarithmic differentiation, 339 Logarithmic equations, 320 Logarithmic functions applications for, 336–338 curve sketching and, 345–346 derivatives of, 333–336 differentiation of, 330–340 explanation of, 314 graphs of, 317, 318 inverse relationship between exponential and, 319 properties of, 318 substitution used on, 397 Logarithmic rule, 379 Logarithms applications for, 314 conversion formula for, 321 evaluation of, 314 explanation of, 314, 319 natural, 318–319, 333, 686 product rule for, 315, 316, 599 rules for, 315–317 solving equations involving, 314–315 Logistic curves, 351–353, 487 Logistic equation, 487 Logistic formula, 353 Logistic models, 298 Log-linear regression, 600 Lorenz curves, 430–432 M Malthus, Thomas R., 298 Marginal analysis approximation by increments and, 164–167 explanation of, 161, 565 in labor management, 176 for maximum profit, 255–256 for minimal average cost, 256 partial derivatives and, 565–566 use of, 162–163 Marginal cost explanation of, 161–163 finding total cost from, 382 Marginal productivity of capital, 565–566 Marginal productivity of labor, 565–566 Marginal profit, 162 Marginal propensity to consume, 390 Marginal rate of technical substitution (MRTS), 176 Marginal revenue applications of, 336–337 explanation of, 162–163 Marginal utility of money, 617 Mathematical models examples of, 46–50, 294–296 explanation of, 45–46 probability and, 517 proportionality and, 50–51 stages of, 45 Maxima absolute, 248–255, 585 applications of, 268–270, 274–276 method to find, 208–209 relative, 202, 204, 577 Maximum attainable yield, 73 Maximum likelihood estimate, 264 Mean, 518 Median, 529 Method of least-squares See Leastsquares method Minima absolute, 248–255, 585 applications of, 267–268, 270–274, 276–278 relative, 202, 204, 224–226, 577 Minimum-cost problem, 617 Mitscherlich model, 406 Mixed second-order partial derivatives, 568 Models allometric, 99–100 logistic, 298 mathematical, 45–50, 294–296 population change, 298–299 Motion of projectile, 127–128 rectilinear, 125–127 hof3238x_ndx_771-776.qxd 11/25/11 8:19 PM Page 775 I-5 mth root, 657 Multiplication, of polynomials, 663–664 Multiplication rule, 302 Multiplicative property of inequality, 653 N Natural domain, Natural exponential base e, 302–303 Natural exponential functions, 303, 330 Natural logarithms See also Logarithms derivative formula for, 333 evaluation of, 318–319 explanation of, 318 inverse relationship between exponential and, 319 properties of, 319 table of, 686 Negative exponents, 120–121 Negative integer powers, 657 Net change, 417–418 Net excess profit, 428–430 Newton, Isaac, 104 Newton’s method for approximating roots of equations, 171 Nonlinear curve fitting, 598–600 Normal density function, 523–524 Notation/symbols definite integral, 410 delta, 31, 108, 164 derivative, 109, 111–112 exponential, 299, 657 function, inequality, 653 infinity, 70 integral, 378 summation, 424, 680 nth derivative, 140–141 Numbers critical, 202, 204 irrational, 652 rational, 652 real, 652–654 Numerator, rationalizing the, 661 Numerical integration applications of, 502–503 approximation by rectangles and, 494–495 approximation by trapezoids and, 495–498 approximation using parabolas and, 498–501 explanation of, 494 interpreting data with, 501–503 O One-sided limits evaluation of, 80–81 explanation of, 79–80 to find two-sided limits, 81–82 Open intervals, 224 Optimal holding time, 347–348 Optimization absolute extrema and, 248–253 constrained, 606–615 elasticity and, 256–261 INDEX examples of problems involving, 267–278 of functions of two variables, 577–588 guidelines to solve problems of, 267 marginal analysis and, 255–256 Osmotic pressure, 560 Outcome, 518 P Packing fraction, 291 Parabolas approximation using, 498–501 explanation of, 24 vertex of, 24 Paraboloids, 550, 552 Parallel lines explanation of, 37, 377 method to find, 37–38 Pareto distribution, 530 Parkinson’s law, 359 Partial derivatives chain rule for, 569–571 explanation of, 562 geometric interpretation of, 564–565 marginal analysis and, 565–566 method to find, 563–564 relative extrema and, 578–579 second-order, 567–568, 582 substitute commodities and complementary commodities and, 566–567 Partial differential equations, 647 Partial differentiation, 379 Partial integrals, 621, 622 Peak efficiency, 390 Percentage change, 166–167 Percentage rate of change explanation of, 124 method to find, 124–125 Perception, 477–478 Perpendicular lines, 37–38 Phillips, A W., 116 Phillips curve, 116 Piecewise-defined functions continuous, 85 evaluation of, explanation of, graphs of, 19–20 modeling with, 49–50 Plotting points, 18–19 Point of diminishing returns explanation of, 215 method to find, 226–227 Point-slope form explanation of, 33–34 use of, 35, 134 Polynomial regression, 294–295 Polynomials addition and subtraction of, 663 coefficients of, 663 continuity of, 83–85 degree of, 23, 663 explanation of, 23, 663 limits of, 67, 68 method to differentiate, 123 method to factor, 664–666 multiplication of, 663–664 Population density double integration and, 632–633 estimation of, 514–515 explanation of, 456–458 finding population from, 633 Population diffusion modeling, 647–649 Population/population change exponential, 298–299 on graphs, 242–243 models for, 298–299 quotient rule and, 136–137 survival and renewal functions and, 455–456 use of derivative to study, 123 Position, method to find velocity from, 382–383 Power functions, 23, 301 Power laws of exponents, 658 Power rule applications of, 121, 392, 497 explanation of, 120, 302 general, 151–156 for integrating common functions, 379–381 for logarithms, 315, 334 proof for, 121 used with quotients, 137–138 verification of, 120–121 Powers of e table, 685 Prediction, least-squares, 597–598 Present value for compound interest, 306–307 explanation of, 306 as function of four variables, 549 of income stream, 444–445, 502–503, 512–513 optimal holding time and, 347–348 Price, equilibrium, 52 Price-adjustment model, 401–402 Price elasticity of demand explanation of, 256–257 method to find, 257–258 Probability continuous, 517–525 continuous random variables and probability density functions and, 517–521 expected value of random variables and, 524–525 explanation of, 517, 518 exponential density functions and, 522–523 normal density functions and, 523–524 uniform density functions and, 521–522 Probability density function explanation of, 518–520 exponential, 522–523 formation of, 520–521 general, 361 normal, 523–524 standard normal, 347 uniform, 521–522 Probability theory, 517 Producers’ surplus, 447–449 Production, as function of two variables, 548–549 775 Production function Cobb-Douglas, 548–549 constant elasticity of substitution, 558, 618 first derivative of, 215 second derivative of, 215 Product law of exponents, 658 Product rule derivation of, 141–143 explanation of, 133, 146, 302, 667 integration by parts and, 480 for logarithms, 315, 316, 599 rate of change of revenue and, 134–135 use of, 133–135, 332 Profit applications involving, 268–270 marginal, 162 maximum, 46–47, 255–256, 583–584 net excess, 428–430 rate of change in, 110–111 Profit formula, 268 Profit function, 6, 47, 269 Projectile, motion of, 127–128 Propagated error, 165 Proportionality, 50, 51 Q Quadrants, 17 Quadratic equations explanation of, 669 method to solve, 669–672 Quadratic formula explanation of, 21, 671 use of, 175, 671–672 Quadratic functions explanation of, 24 graphs of, 24–26 revenue, 24–26 Quadratic substitution, 395 Quotient law of exponents, 658 Quotient rule derivation of, 141, 146 explanation of, 135, 302, 667 for limits, 68 for logarithms, 315, 334 use of, 135–138, 154, 332 Quotients, difference, 11–12, 108 R Radioactive decay, 323–325 Random experiments, 517–518 Random variables continuous, 518–519, 524–525 expected value of, 524–525 explanation of, 518 exponentially distributed, 522–523 mean of, 529 uniformly distributed, 521 Range, of function, 2, 546 Rate of change average, 107 chain rule and, 150 estimation of, 105 instantaneous, 106–107, 109 method to find, 112, 154–156, 417–418 hof3238x_ndx_771-776.qxd 776 11/25/11 8:19 PM Page 776 INDEX Rate of change—Cont obtaining price from, 399–400 percentage, 124–125 of population, 136–137 in profit, 110–111 relative, 124, 339–340 of revenue, 134–135 second derivative and, 138–140 slope and, 104–112 Rational equations, 669 Rational expressions, 667–668 Rational functions continuity of, 83–85 explanation of, 23 graphs of, 236–240 limit of, 68, 72 substitution used on, 396–397 Rationalizing, 660–661 Rational numbers, 652 Real number line distance between numbers on, 655–656 explanation of, 652 Real numbers absolute value of, 655–656 explanation of, 652 intervals of, 654 Reciprocal integer powers, 657 Reciprocal limit rules, 70 Rectangles, approximation by, 494–495 Rectangular coordinate system, 17 Rectilinear motion, 125–127 Regression log-linear, 600 polynomial, 294–295 Regression analysis See Leastsquares method Regression line, 596–597 Related rates applications for, 178–180 explanation of, 177 methods to find, 177–178 Relative change, 166 Relative extrema See also Extrema applications of, 205–209, 583–584 critical points and, 578–583 explanation of, 202–203, 577 extreme value property and, 585–588 first derivative test and, 203–204 second derivative test and, 224, 225 Relative maxima See also Maxima critical points and, 580–583 explanation of, 202, 577 first derivative test and, 204 second derivative test and, 224–226 Relative minima See also Minima critical points and, 580–583 explanation of, 202, 577 first derivative test and, 204 second derivative test and, 224–226 Relative rate, of growth, 339–340 Relative rate of change, 124, 339–340 I-6 Renewal function, 454–456, 514 Resource allocation, 611–612 Revenue applications involving, 385 elasticity and, 259–261 explanation of, as function of two variables, 548 marginal, 162–163, 336–337 rate of change of, 134–135 Revenue curve, 338 Revenue function, 208, 209, 275 Reverse osmosis, 560 Riemann sums applications involving, 478 average value and, 433, 434 explanation of, 410, 424 Root function, 151 Roots on graphing calculators, 193–194 method to simplify, 659 mth, 657 S Saddle points, 579, 580 Saddle surface, 550, 579 Sample points, 518 Sample space, 518 Scatter diagrams, 594, 599 Secant line, 106–107 Second derivative curve sketching with, 221–224 explanation of, 138 general power rule and, 154 intervals of concavity and, 217–218 method to find, 138–139, 154 of production function, 215 rate of change and, 138–140 Second derivative test for absolute extrema, 253–255 application of, 225–226 explanation of, 224–225 for relative extrema, 224, 225 two-variable version of, 579–583 Second-order partial derivatives, 567–568, 582 Second partials test explanation of, 579–580 use of, 580–584 Sensitivity, 265 Separable differential equations, 384–385, 387 Shortage, market, 52 Similar terms, of two polynomials, 663 Simple events, 518 Simpson’s rule accuracy of, 499–501 explanation of, 498–499 use of, 499 Skellam, J G., 648–649 Slope as derivative, 109 explanation of, 30–31 method to find, 31–32, 112 rate of change and, 104–112 of secant line, 107 of tangent line, 107, 109, 174–175, 381–382 use of derivative to find, 109 Slope-intercept form, 33, 35 Solution of differential equations, 383, 384 of equations, 668 Solution set, 653 Square of difference formula, 666 Square of sum formula, 666 Standard normal probability density function, 347 Subdivision rule for definite integrals, 414, 415 Substitute commodities, 566–567 Substitution algebra used before, 397–398 antiderivatives and, 483 in definite integrals, 415–417 differential equations and, 399–400 failure of, 398 integration by, 393–402 linear, 394–395 quadratic, 395 used on logarithmic functions, 397–398 used on rational functions, 396–397 Subtraction, of polynomials, 663 Summation notation, 424, 680 Sum of cubes formula, 666 Sum rule for definite integrals, 414 explanation of, 122, 667 for indefinite integrals, 380, 381 use of, 122–123 Supply, 180 Supply and demand law, 52 Supply function, 6, 51 Surge function, 360 Surplus, market, 52 Survival function, 454–456 Survival/renewal problems, 454–456, 513–514 Symbols See Notation/symbols System of equations explanation of, 672 method to solve, 672–674 T Table of integrals See Integral tables Tangent lines approximation of, 171 horizontal, 152–153, 175 method to find equation of, 335–336 product rule and, 133–134 slope of, 107, 109, 174–175, 381–382 use of chain rule to find, 149–150 Temperature, 435 Thermodilution, 540 Three-dimensional coordinate system, 549–550 Topographical map, 552 Transitive property of inequality, 653 Trapezoid rule error estimates for, 497–498 explanation of, 496 use of, 496–497 Trapezoids approximation by, 495–498 explanation of, 409 Trials, 517 Two-sided limits, 81–83 U Unbounded intervals, 508, 511 Uniformly distributed random variables, 521 Uniform probability density function, 521–522 Utility function, 553–554, 609–610 V Value absolute, 655–656 expected, 524 future, 305–306, 442–444 present, 306–307, 444–445, 549 van der Waals’ equation, 282, 560 van’t Hoff equation, 560 Variable costs, 278 Variables See also Functions of two or more variables; Random variables dependent, explanation of, independent, 3, 546–552 of integration, 378 method of separation of, 387–388 Velocity average, 106 formula for, 109 instantaneous, 106–107 method to find, 382–383 Vertex, of parabola, 24 Vertical asymptotes, 233–234 Vertical cross sections, 624–625 Vertical lines, 32 Vertical line test, 21–22 Vertical tangent lines, 241 Volume as double integral, 630–631 estimation of, 462–463 of solid, 461–462 Volume formula, 461 X x axis, 17 x coordinates, 17, 32 x intercepts, 22 Y y axis, 17 y coordinates, 17, 32 y intercepts, 22 Z Zeros, 20 hof3238x_es.indd Page 11/24/11 9:25 PM user-f462 Volume/202/MHDQ282/hof3238X_disk1of1/007353238X/hof3238X_pagefiles Index of Applications Business Problems Accounting, 41, 357 Admission fees, 58 Advertising, 129, 143–144, 212, 231, 246, 310–311, 326, 355–356, 366, 389, 422, 439, 636 Allocation of funds, 590, 617, 641–642 Allocation of labor, 574 Allocation of resources, 574, 611–612 Allocation of unrestricted funds, 617 Annual earnings, 129, 157 Auction buyer’s premium, 58 Average costs, 76, 212, 245, 439 Average monthly output, 632 Average monthly sales, 434 Average price, 472 Average production, 439 Average profit, 73–74, 262, 492, 636 Average supply, 438 Balance of trade, 469 Break-even analysis, 53–55, 59, 86, 95 Business training, 356 Construction costs, 48, 59–60, 62, 95, 270–271, 273–274, 283, 288, 290 Construction plans decision, 451 Consumer demand, 572–574, 642–643 Consumer expenditure, 342 Consumers’ and producers’ surplus, 448–450 Consumers’ surplus, 450, 452, 469, 470, 491, 504, 505 Consumer willingness to spend, 446–447, 450 Corporate organization, 360–361 Cost analysis, 28, 88, 130, 246, 279 Cost-efficient design, 95 Customer service, 4–5, 527 Data transfer, 14 Demand analysis, 143, 154–155, 182, 190, 303, 310, 337–338, 404, 504, 569–570, 572–574, 598–601 Depreciation, 158, 213, 342, 355, 366, 404, 421, 471 Dimensions to minimize construction cost, 620 Distribution costs, 13, 246 Earnings, 76, 89 Equilibrium price, 450 Exponential demand, 303 Exponential sales rates, 349–350 Fair price, 502–503 Fixed budget, 618 Food prices, 438 Fundraising, 451, 492 Future revenue, 472 Installation costs, 279, 283 Inventory control, 89, 95, 276–281, 290, 438, 558 Inventory cost estimates, 245–246, 276, 290 Labor management, 356 Law of diminishing returns, 573 Linear depreciation, 41 Lumber production, 183–184 Magazine subscription growth, 533 Maintenance shed location, 591 Management principles, 76 Manufacturing costs, 7–8, 27, 41, 95, 157, 168, 169, 191 Manufacturing efficiency, 94, 169 Manufacturing output, 28, 116, 169, 181–182 Manufacturing overhead, 95 Marginal analysis, 356, 357, 390, 473, 565, 572–574, 617, 618, 641 Marginal cost, 162, 389, 404, 421, 471, 491 Marginal productivity, 573, 639 Marginal profit, 389, 404 Marginal propensity to consume, 390 Marginal revenue, 336–337, 389 Market research, 357 Maximizing profit, 583–584, 587–588 Maximizing revenue, 364 Maximizing sales, 590 Maximum production level, 612–613 Maximum profit and average cost, 253–255, 261–262 Minimizing cost, 590, 613–614, 617–618, 620 National productivity, 572 Net asset value, 471 Net change in cost, 417 Net change in revenue, 469 Net excess profit, 429–430 Net profit from data, 505 Newspaper circulation, 94, 130, 169 Optimal holding time, 347–348, 356, 357, 367 Optimal location for a warehouse, 584–585 Optimal selling price, 94 Optimal setup costs, 280 Packaging calculations, 56, 62, 283, 290 Price adjustments, 401–402, 405 Price analysis, 364 Front: Page Price elasticity, 259–260 Pricing calculations, 36–37, 93–94, 399–400, 589 Printing costs, 41–42 Producers’ surplus, 450, 505 Production control, 263 Production costs, 6–7, 9–10, 13, 14, 35–36, 57, 58, 116, 150, 164, 177–178, 186, 190, 279, 280, 289, 290, 439, 556 Production output, 76, 95, 139–140, 158, 166, 169, 182, 186, 188, 189, 246, 389, 421, 439, 548–549, 556–558, 565–566, 568, 570–571, 636, 644 Production units, 290–291 Product reliability, 310, 355, 516, 526–527, 531 Profit calculations, 13, 28, 46–48, 57, 73–74, 110, 116, 143, 144, 262, 268–270, 279, 286, 288, 367, 452–453, 590–591, 616, 636, 640 Profit from an invention, 452–453 Profit over the useful life of a machine, 450–451 Profit under monopoly, 589–590 Property tax, 94–95, 129, 169, 186 Property value, 636 Publishing costs and revenues, 60, 574 Real estate, 28, 288 Real estate inventory, 472 Relative growth rate, 339–340 Retail prices, 404 Retail sales, 27, 57, 279, 355, 557, 575, 589 Revenue calculations, 25, 76, 134–135, 143, 144, 208–209, 274–276, 303, 336–337, 339–340, 385, 404, 453, 472, 548 Revenue growth, 342 Salary increases, 130 Sales decrease over time, 355 Sales revenue, 57, 58, 143, 230, 246, 389, 421, 601, 616–617, 642 Sorting mail efficiently, 350–351, 438 Storage costs, 421, 453 Substitute and complementary commodities, 567, 572, 573 Supply and demand, 59, 60, 92, 311, 326–327, 404–405 Supply calculations, 180, 182, 311, 438 Survival and renewal problem, 454–455, 513–514 Total cost from marginal cost, 382, 534 Total future revenue, 385, 453, 504 Truth in lending, 311 Useful life of a machine, 527 Warranty protection, 533 Worker efficiency, 326, 350–351, 355, 366, 438, 439, 491, 531 Economics Problems Accounting, 41, 357 Agricultural production, 60, 73, 279 Amortization of debt, 558 Balance of trade, 469 Car rental, 41, 55–56 Checking account, 58 Consumer demand, 14, 157 Consumer expenditures, 27–28, 94, 116–117 Consumers’ and producers’ surplus, 448–450 Consumers’ surplus, 450, 452, 472 Consumer willingness to spend, 446–447, 450 Cost-benefit analysis, 89 Credit card debt, 41 Debt repayment, 366 Disposable income and consumption, 602 Distribution of income, 432–433, 440, 473 Elasticity of demand, 257–260, 262, 264, 289, 337–338, 343 Elasticity of revenue, 259–260, 263, 289 Equipment rental, 28, 42 Farm crop value, 421 Fund raising, 451 Gasoline prices, 92, 602 Government spending, 231 Gross domestic product, 124, 130, 166–167, 188, 213, 311, 327, 369, 602–603 Housing starts, 231 Income distribution, 492 Income elasticity of demand, 263 Income tax, 57 Inflation, 190, 310 Land value, 405, 421, 439 Lottery payout, 452 Marginal analysis, 161–163, 168, 169, 176, 212, 230, 255, 262–264, 336–337, 342 Market equilibrium, 51–52, 59 Mathematical modeling, 45–46 Maximizing utility level of consumer, 609–610 Minimum costs, 289 Monopoly profit, 212 Monopoly taxation, 280 National consumption, 263 Optimal holding time, 347–348, 356, 357, 367 Postage rates, 29 Price elasticity of demand, 256–261 Producers’ surplus, 450 Profit over the useful life of a machine, 450–451 Ranking investments, 310 Real estate evaluation, 439 Retirement income, 405, 451 Revenue from demand data, 143 Sports contracts, 452 Supply and demand, 59, 60, 92, 311, 326–327 Transportation costs, 57, 58, 280 Unemployment, 39, 42, 105, 116 Utility level of consumer, 553–554, 557, 609–610, 617, 640 Value of dirt from construction site, 505–506 Worker efficiency, 14, 129, 144, 190, 230–231, 263, 280, 286 Finance and Investment Problems Asset appreciation, 42 Balance of trade, 469 Capitalized cost of an asset, 516 Comparing investments, 308 Compound interest, 158, 310, 322–323, 326, 342, 365–366 Construction plans decision, 451 Continuous compounding interest, 76–77, 326, 386, 438, 451, 452 Debt repayment, 366 Depreciation, 158, 213, 342, 366 Doubling time, 321–322 Effective rate of interest, 307–308, 366 Endowment, 516 Expected value, 529 Finance payments, 311 Future value, 306, 364, 442–443, 453, 469, 471–472, 484–485, 491, 504–505 Income stream amount, 451 Interest rate computations, 323 Investing in a down market, 473 Investment analysis, 451, 601, 636 Investment evaluation, 491 Investment goals, 322–323, 405 Investment growth, 531 Investment rates of profitability, 438–439 Investment satisfaction, 575 Front: Page Investment value over time, 317, 421, 438 Market equilibrium modeling, 52–53, 59 Mortgage payments, 311 Mortgage refinancing, 212 Present value, 307, 364, 444–445, 451–453, 472, 491, 504, 512–513, 516, 531, 533, 549 Ranking investments, 310 Real estate investment, 311 Retirement income, 405, 451 Rule of 70, 367 Savings accounts, 290 Stock market average, 602 Stock prices, 5, 14, 42 Stock speculation, 356 Tripling time, 326 Life, Health, and Environmental Sciences Problems Aerobic rating, 357, 440 Age of cells, 528 Agriculture, 279, 406 AIDS epidemic, 188–189, 294–296 Air pollution, 11, 15, 28–29, 44, 89, 94, 131, 144, 155–156, 158, 169, 188, 272–273, 367, 406, 422, 466, 559–560, 643 Alcohol abuse, 44, 343 Allometry, 100–101, 327, 391, 605 Animal behavior, 78, 118, 159 Animal demography, 367 Aquatic life, survival of, 265, 312–313 Archaeology, 369 Area of real estate, 439 Arteriosclerosis, 170 Auditory perception of humans, 477–478 Average temperature, 435, 507 Avian behavior, 281 Bacterial growth, 77, 92, 136–137, 144, 189, 246, 286, 303–304, 312, 328, 364, 366, 390, 440, 455–456, 466–467, 604 Basal energy expenditure of a human, 559 Basal metabolic rate, 100, 183 Biomass net change, 125, 390, 471 Blood cell production, 145, 265 Blood circulation, 170, 264, 575 Blood flow, 15, 28, 183, 189, 391, 458–459, 464, 560, 575 Blood pressure, 118 Blood type alleles, 619 Botany, 282 hof3238x_es.indd Page 11/24/11 9:25 PM user-f462 Volume/202/MHDQ282/hof3238X_disk1of1/007353238X/hof3238X_pagefiles Index of Applications (continued) Life, Health, and Environmental Sciences Problems (continued) Butterfly wing patterns, 592 Cancer research, 361 Cancer therapy, 391, 592 Cardiac output, 170, 538–543 Cardiology, 118, 575–576 Cardiovascular system, 189–190 Cell growth, 169 Center of a region, 493 Cholesterol regulation, 464–465 Cooling an animal’s body, 643 Depletion of energy resources, 452 Drug abuse, 603 Drug concentration, 77, 213, 246, 286, 312, 327, 360, 368, 406, 422, 441, 469, 492, 531 Drug dosage, 144, 640 Drug effectiveness evaluation, 465 Drug remaining in body over time, 517 Drug sensitivity, 265 Ecology, 662 Effect of a toxin, 358, 472 Endangered species, 343, 390, 422, 465 Energy consumption, 89, 464 Energy expended by bird in flight, 465–466 Entomology, 43 Epidemic modeling, 56, 130, 294–296 Epidemiology, 231, 246–247, 517 Exercise regimen, 464 Exposure to disease, 637 Fick’s law, 367 Fishery management, 213–214, 359 Fox and rabbit population on island, 619 Genetics, 591 Growth of child, 43 Hazardous waste management, 282, 534, 618 Health care, 516 House fly egg yolk volume, 328 Hummingbird visits to flower, 529 Immunization, 14, 247 Incidence of disease, 604 Influenza strain testing, 637 Insect growth, 159 Intelligence quotient, 560 Island ecology, 15 Life expectancy, 61, 465 Lizard speed, 183 Malignant tumor radiology, 369 Mammalian growth, 159 Marine biology, 282 Marine life, 643 Memory, 311, 328, 343, 359 Mental health care, 506 Metabolic rate, 440 Microbiology, 94 Mortality rates, 361 Mutation, 92 Nuclear waste, 464, 472, 517, 644 Nutrition, 43 Oil production, 390 Oil spills, 178–179 Optimal age for reproduction, 353–354, 358 Ornithology, 130–131, 264–265 Pediatric drug dosage, 61 Pediatric measurement, 186 Pharmacology, 144–145 Physiology, 60 Plant growth, 344 Poiseuille’s law, 464 Pollution control, 179–180, 182, 183, 213, 506 Population diffusion of muskrats, 647–650 Progress of an epidemic, 352–353 Psychological testing, 533 Radiology, 328, 369 Rat in maze, 528–529 Reaction to medication, 441 Recycling, 60 Respiration, 265, 466 Response to stimulus, 358–359, 391, 591, 637 Shopping mall security, 493 Solute concentration, 367 Sorting mail learning curve, 350–351 Species growth, 214, 264 Species population, 77 Spread of AIDS, 604 Spread of an epidemic, 358, 360, 368 Spread of disease, 358, 464, 492, 506 Sprinkler system, 473 Surface area of human body, 472, 558–559, 575, 618–619, 638 Survival and renewal of bacterial colony, 455–456 Survival time of cancer patients, 529 Survival time of patients receiving experimental drug, 528 Thermal effect of food, 440 Threshold of pain, 328–329 Thyroid gland radiology, 328 Tissue growth, 232 Total pollution of a community, 643 Back: Page Trachea measurement, 250–251, 265 Tree growth, 390, 405, 472 Trout population of lake, 473 Tumor growth, 61, 182, 186 Tumor size, 466 Volume of a bacterium, 619 Volume of a biomass, 630–631 Volume of a tumor, 462–463 Volume of blood during systole, 441 Volume of dirt from construction site, 505–506 Water consumption, 43, 49–50, 422 Water evaporation for ponderosa pine, 343 Water pollution, 89, 159, 182, 405, 422, 466 Weather, 89–90 Physical Sciences and Geometry Problems Acceleration, 140, 145, 160, 192 Acidity of a solution, 368 Aerodynamics, 266 Amplitude of oscillation, 266 Architectural design, 637 Area measurement, 189, 412–413, 576, 592, 593 Area of swimming pool, 501–502 Astronomy, 44 Auditory perception of humans, 477–478 Average elevation, 636 Average temperature, 247, 435, 441, 507, 640 Average velocity, 441 Biochemistry, 61 Body temperature of murder victim, 368 Boyle’s law, 184 Carbon dating, 325, 327, 364, 368, 369 Catenary curves, 361 Chemical reaction rate, 368 Cooling a hot drink, 328, 344, 361, 367 Cooling an animal’s body, 643 Crystallography, 291 Dead Sea Scrolls, 327 Defrosting, 391 Dimensions calculations, 283–284, 620 Dimensions of jewelry box with interior partitions, 620 Dimensions to minimize construction cost, 620 Distance, 186, 493 Distance and velocity, 391, 422, 473, 533 Distance between moving objects, 283 Earthquake magnitude, 329 Electric circuit, 576 Electric field intensity, 90 Electricity, 266 Energy consumption, 89 Fencing needed, 608, 619 Forensic science, 368 Fossil age, 327 Half-life calculations, 324, 328 Ice Age patterns of temperature, 561 Ideal temperature for drinking coffee, 328 Lens focal length, 560, 620 Material expansion, 170, 182 Newton’s law of cooling, 328, 344 Nuclear waste, 517, 534 Ozone depletion, 357 Particle physics, 593, 620 Physical chemistry, 131, 282 Position of moving object, 16 Probability density function, 361 Projectile motion, 29, 127–128, 132, 188, 422 Protein mass net change, 418, 422 Radiation, 170 Radioactive decay, 56, 313, 324, 328, 365, 367 Radiology, 328, 369 Rectilinear motion, 125–126, 131 Refrigeration, 184 Respiration measuring, 466 Reverse osmosis, 560 Seismology, 329 Shroud of Turin age, 327 Solute concentration, 367 Sound levels, 328–329 Speed and distance, 184, 190, 191, 473 Speed of flight, 265 Stopping distance of car, 382–383, 392 Structural design, 361 Surface area of tumor, 189 Temperature change, 14–15, 56, 472 Temperature conversion, 43 Temperature of a gas, 560 Temperatures of space probe, 620–621 Thermal inversion, 117 Threshold of pain, 328–329 Velocity, 106, 118, 126–127, 140, 145, 160, 186 Volume measurement, 165, 170, 184, 189–191, 232, 461–462, 467, 619, 637 Volume of a cone, 467 Volume of a cylinder, 576, 619–620 Volume of a gas, 576 Volume of a solid, 461–462 Volume of a sphere, 467 Volume of jewelry box, 637 Volume of rectangular package, 619 Volume of storage bin, 637 Wind power, 559 Wire displacement, 78 Social Sciences Problems Admission to events, 421 Adoption of technology, 247 Airplane arrivals, 528 Air travel, 62 Ancient fables, 44 Architecture, 29, 290 Area estimation, 56 Art forgery, 327 Average population, 440, 473 Broadcasting, 266 Bureaucratic growth, 359 Car pooling, 43 Childhood learning, 357 City planning, 591 College admissions, 43–44, 603 Commuting, 528 Comparative growth, 440 Computer dating, 464 Correctional facility management, 390 Corruption in government, 492 Course registration, 42–43 Demand for airline tickets, 262–263 Demand for art, 262 Demand for cruise tickets, 289–290 Distance estimated, 92 Distribution of income, 432–433, 440, 505 Drug abuse, 603 Duration of telephone calls, 523, 525, 533 Educational funding, 94 Educational testing, 130 Experimental psychology, 15, 77, 247 Fencing estimates, 56, 267–268, 278, 289 Firefighting, 291 Freeway traffic, 250 Grade point average, 597–598 Group membership, 264, 463, 464, 492 Income distribution, 492 Landscaping, 56 Learning theory, 95, 159, 264, 311, 328, 343, 344, 350–351, 359–360, 390, 422, 591 Legislative turnover, 529 Level of satisfaction, 557–558 Linguistics, 313 Livable space of a building, 592 Marginal propensity to consume, 390 Membership fees, 43 Back: Page Mental health care, 506 Mortality rates, 361, 368–369, 507 Movie show times, 529, 533 Net growth of population, 422, 463 Oil consumption, 471 Per capita growth, 342 Political corruption, 56–57 Political polling, 247 Political trends, 463–464 Population density, 15, 189, 312, 320, 456–458, 464, 507, 637, 643 Population derived from population density, 457–458, 633, 643 Population distribution, 213 Population estimation, 514–515, 532 Population growth, 15, 51, 56, 77, 117, 118, 123–124, 130, 145, 169, 188, 190, 231–232, 312, 327, 328, 343, 357, 359, 366, 368, 390, 464, 469, 492, 516, 549 Population prediction, 603 Population trends, 465 Postage functions, 90 Postal regulations, 284 Poster design, 62, 283 Property value, 636 Public health, 604 Public transportation, 130 Publishing decision, 58 Quality of life, 159 Rapid transit, 188 Rate of learning, 422 Recycling, 281–282 Redistricting, 637–638 Relay race travel time, 592–593 Renewable resources, 117 Road safety, 29 Shroud of Turin age, 327 Snacks at a bakery, 533 Social choices, 591 Spread of a rumor, 232, 358, 487 Spy story problems, 62, 131, 284, 329, 391, 492–493, 620 Structural design, 94 Ten-year census of U.S., 369 Ticket sales, 471 Traffic control, 289, 533 Traffic management, 441 Transportation waiting time, 531 Travel time minimization, 289 Urban planning, 281 Urban population, 517 Voter turnout, 264, 604 Waiting time at traffic light, 517, 522, 524–525, 527–528 Water consumption, 49–50 World population, 359 ... of the Brief Eleventh Edition Calculus for Business, Economics, and the Social and Life Sciences, Brief Edition, provides a sound, intuitive understanding of the basic concepts students need as... hof3238x_fm_i-xxiv.qxd 11/25/11 7:27 PM Page iv TM CALCULUS FOR BUSINESS, ECONOMICS, AND THE SOCIAL AND LIFE SCIENCES: BRIEF EDITION, ELEVENTH EDITION Published by McGraw- Hill, a business unit of The McGraw- Hill. .. function D( x) for the commodity is the price p ϭ D( x) that must be charged for each unit of the commodity if x units are to be sold (demanded) The supply function S(x) for the commodity is the unit