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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 3)(a2 b2 3) 75 a17b9 B c11 (x 2)8(1 x)2 13 12 90 17 91 13 92 15 12 93 15 94 15 111 95 15 96 17 97 Show that 3 62 31 15 120 51 405 64 (1 x)4 6(x 2) (1 x)4 4(x 2)6(1 x)3 89 b 61 3124 2154 1486 15 20 35 63 a b c (x 2)4 4(1 x)2(x 3)3 2(1 x)(x 3)4 In Exercises 89 through 96, rationalize the root (or roots) in the given expression 56 118 162(27) 3(x 2)2(x 1)2 2(x 2)(x 1)3 54 (a1 b2)2 57 5 86 76 (a24b8c11)4 In Exercises 77–88, factor and simplify the given expression as much as possible 77 x5 4x4 78 3x3 12x4 79 100 25(x 3) 80 60 20(4 x) 81 8(x 1)3(x 2)2 6(x 1)2(x 2)3 h 1x h 1x where x and h are positive numbers 1x h 1x 98 Simplify the expression 1 1x h 1x where x and h are positive constants 99 ECOLOGY The atmosphere above each square centimeter of Earth’s surface weighs kilogram (kg) a Assuming Earth is a sphere of radius R 6,440 km, use the formula S 4R2 to calculate the surface area of Earth and then find the total mass of the atmosphere b Oxygen occupies approximately 22% of the total mass of the atmosphere, and it is estimated that plant life produces approximately 0.9 1013 kg of oxygen per year If none of this oxygen were used up by plants or animals (or combustion), how long would it take to build up the total mass of oxygen in the atmosphere [part (a)]?* n n 100 Show that (1 x)m xm in the case where m is a negative integer 82 12(x 3)5(x 1)3 8(x 3)6(x 1)2 83 x1 2(2x 1) 4x1 84 x1 4(3x 5) 4x3 (x 3)3(x 1) (x 3)2(x 1)2 85 (x 3)(x 1) *Adapted from a problem in E Batschelet, Introduction to Mathematics for Life Scientists, 2nd ed., New York: Springer-Verlag, 1979, p 31 hof3238x_appa_663-675.qxd 2:14 PM Page 663 SECTION A.2 FACTORING POLYNOMIALS AND SOLVING SYSTEMS OF EQUATIONS 663 SECTION A.2 Factoring Polynomials and Solving Systems of Equations A polynomial is an expression of the form a0 a1 x a x # # # an x n where n is a nonnegative integer and an, an1, , a0 are real numbers known as the coefficients of the polynomial Polynomials appear in a variety of mathematical contexts, and the first goal of this section is to examine some important algebraic properties of polynomials If an 0, n is said to be the degree of the polynomial A nonzero constant is said to be a polynomial of degree (Technically, the number is also a polynomial, but it has no degree.) For example, 3x5 7x 12 is a polynomial of degree 5, with terms 3x5, 7x, and 12 Similar terms in two polynomials in the variable x are terms with the same degree Thus, in the fifth-degree polynomial 3x5 5x2 and the third-degree polynomial 2x3 2x2 7x 9, the terms 5x2 and 2x2 are similar terms Polynomials are multiplied by constants and added and subtracted by combining similar terms, as illustrated in Example A.2.1 EXAMPLE A.2.1 Combining Polynomials Let p(x) 3x2 5x and q(x) 4x2 Find the polynomials 2p(x) and p(x) q(x) Solution We have 2p(x) 2(3)x2 2(5)x 2(7) 6x2 10x 14 and p(x) q(x) [3 (4)] x2 [5 0] x [7 9] x2 5x 16 F First product O Outer product I Inner product L Last product e e e A convenient way to remember how to multiply two first-degree polynomials p(x) ax b and q(x) cx d is the “FOIL” method: e A-13 11/14/11 (ax b)(cx d ) (ac) x2 (ad ) x (bc) x (bd ) Example A.2.2 demonstrates this EXAMPLE A.2.2 Multiplying Using FOIL Find (3x 5)(2x 7) hof3238x_appa_663-675.qxd 664 11/14/11 APPENDIX A 2:14 PM Page 664 Algebra Review A-14 Solution F First product O Outer product I Inner product L Last product e e e e Applying the FOIL method, we get (3x 5)(2x 7) (3)(2)x2 (3)(7)x (5)(2)x (5)(7) 6x2 11x 35 To multiply two polynomials that are not both of degree one, we use the distributive laws of real numbers, namely, a(b c) ab ac and (a b)c ac bc Example A.2.3 illustrates this procedure EXAMPLE A.2.3 Multiplying Polynomials Find (x2 3x 5)(x2 2x 4) Solution To find the required product, we must multiply each term of x2 3x by each term of x2 2x 4, and then combine similar terms We have (x2 3x 5)(x2 2x 4) x2(x2 2x 4) 3x(x2 2x 4) 5(x2 2x 4) [x4 2x3 4x2] [3x3 6x2 12x] [5x2 10x 20] x4 (2 3)x3 (4 5)x2 (12 10)x 20 x4 x3 15x2 2x 20 The computation can also be done “vertically”: x2 3x x2 2x 4x 12x 20 2x 6x2 10x x 3x3 5x2 x4 x3 15x2 2x 20 Factoring Polynomials with Integer Coefficients Many of the polynomials that arise in practice have integer coefficients (or are closely related to polynomials that do) Techniques for factoring polynomials with integer coefficients are illustrated in Examples A.2.4 and A.2.5 In each, the goal is to rewrite the given polynomial as a product of polynomials of lower degree that also have integer coefficients EXAMPLE A.2.4 Factoring a Polynomial Factor the polynomial x2 2x using integer coefficients hof3238x_appa_663-675.qxd A-15 11/14/11 2:14 PM Page 665 SECTION A.2 FACTORING POLYNOMIALS AND SOLVING SYSTEMS OF EQUATIONS 665 Solution The goal is to write the polynomial as a product of the form x2 2x (x a)(x b) where a and b are integers The distributive law gives us (x a)(x b) x2 (a b)x ab Hence, we must find integers a and b such that x2 2x x2 (a b)x ab or, equivalently, such that a b 2 ab 3 and From the list 1, 3 1, and of pairs of integers whose product is 3, choose a 3 and b as the only pair whose sum is 2 It follows that x2 2x (x 3)(x 1) which you should check by multiplying out the right-hand side EXAMPLE A.2.5 Factoring a Polynomial Factor the polynomial 12x2 11x 15 using integer coefficients Solution We wish to write the polynomial as a product of the form 12x2 11x 15 (ax b)(cx d ) Expanding the product on the left by the FOIL method, we get 12x2 11x 15 (ac)x2 (bc ad )x bd Our goal is to find integers a, b, c, d such that ac 12 bc ad 11 and bd 15 Since ac is to be positive, there is no harm in assuming that a and c are both positive (What happens if both are negative?) The factors of the coefficients 12 and –15 are as follows: 15 12 a c b d 12 15 1 3 3 5 15 12 hof3238x_appa_663-675.qxd 666 11/14/11 APPENDIX A 2:14 PM Page 666 Algebra Review A-16 We try each pair on the left with each pair on the right, with a goal of finding a combination that produces the middle term bc ad 11 By trial and error, we find that a and c matched with b and d 5 gives the correct middle term We obtain the following factorization: 12x2 11x 15 (4x 3)(3x 5) Certain polynomial types occur so often that it is useful to have the following formulas for factoring them: Factorization Formulas Square of sum: A2 2AB B2 (A B)2 Square of difference: A2 2AB B2 (A B)2 Difference of squares: A2 B2 (A B)(A B) Difference of cubes: A3 B3 (A B)(A2 AB B2) Sum of cubes: A3 B3 (A B)(A2 AB B2) EXAMPLE A.2.6 Factoring a Difference of Cubes Factor the polynomial x3 using integer coefficients Solution Since 23, we can use the difference of cubes formula, with A x and B 2, to obtain the factorization x3 x3 23 (x 2)(x2 2x 4) Sometimes a polynomial can be factored by grouping terms strategically Example A.2.7 illustrates this EXAMPLE A.2.7 Factoring Polynomials Factor the following polynomials: a p(x) 4(x 2)3 3(x 2)2 b q(x) 9x2 49 Solution a By factoring out the common term (x 2)2, we find that 4(x 2)3 3(x 2)2 (x 2)2[4(x 2) 3] (x 2)2(4x 5) b The polynomial q(x) 9x2 49 can be written as a difference of squares A2 B2, with A 3x and B Thus, we have 9x2 49 (3x)2 72 (3x 7)(3x 7) hof3238x_appa_663-675.qxd A-17 11/14/11 2:15 PM Page 667 SECTION A.2 FACTORING POLYNOMIALS AND SOLVING SYSTEMS OF EQUATIONS Rational Expressions 667 The quotient of two polynomials is called a rational expression For instance, x 2x3 7x 2x2 5x2 3x and x3 x are all rational expressions One of our goals in working with rational expressions is to reduce such an expression to lowest terms, that is, to eliminate all common factors from the numerator and denominator The following properties of fractions will be useful in this process Properties of Fractions c ad bc a Sum rule: b d bd a c ac Product rule: a b a b b d bd a d ad ab ⴢ Quotient rule: cd b c bc EXAMPLE A.2.8 Simplifying Rational Expressions Write each of the following as a rational expression in lowest terms: 2 x x3 7x2 10x x a b a ba b x1 x5 x 1 x 6x Solution a 2 x 1 b a x 2 x x1 x1 x1x1 x 1 2 x2 x x2 x x 1 x 1 x2 (x 2)(x 1) x2 for x 1, 1 (x 1)(x 1) x1 x3 7x2 10x x 6x ba x3 b x5 x(x2 7x 10)(x 3) (x 3)2(x 5) x(x 2)(x 5)(x 3) x2 2x (x 3)(x 5)(x 3) x3 for x 5, 3 A rational expression with fractions in both the numerator and the denominator is known as a compound fraction It is often useful to represent a compound fraction as the quotient of two polynomials This procedure is illustrated in Example A.2.9 hof3238x_appa_663-675.qxd 668 11/14/11 APPENDIX A 2:15 PM Page 668 Algebra Review A-18 EXAMPLE A.2.9 Simplifying a Compound Fraction Simplify the compound fraction 3x 4x2 4x 5x2 Solution Writing both the numerator and the denominator as rational expressions and then simplifying, we obtain 3x 4x2 4x 5x2 x2 3x x2 x2 4x x2 (x2 3x 4)x2 (x 4x 5)x (x 4)(x 1)x2 2 (x 5)(x 1)x2 x4 x5 Solving Equations by Factoring since ab ad cd bc for x 0, 1, 5 The solutions of an equation are the values of the variable that make the equation true For example, x is a solution of the equation x3 6x2 12x because substitution of for x gives 23 6(22) 12(2) 24 24 In Examples A.2.10 and A.2.11, you will see how factoring can be used to solve certain equations The technique is based on the fact that if the product of two (or more) terms is equal to zero, then at least one of the terms must be equal to zero For example, if ab 0, then either a or b (or both) EXAMPLE A.2.10 Solving an Equation Using Factoring Solve the equation x2 3x 10 Solution First subtract 10 from both sides to get x2 3x 10 and then factor the resulting polynomial on the left-hand side to get (x 5)(x 2) Since the product (x 5)(x 2) can be zero only if one (or both) of its factors is zero, it follows that the solutions are x (which makes the first factor zero) and x 2 (which makes the second factor zero) hof3238x_appa_663-675.qxd A-19 11/14/11 2:15 PM Page 669 SECTION A.2 FACTORING POLYNOMIALS AND SOLVING SYSTEMS OF EQUATIONS EXAMPLE A.2.11 Solve the equation 669 Solving a Rational Equation x x Solution Put the fractions on the left-hand side over the common denominator x2 and add to get x2 x x x x2 0 or x2 x x2 0 Now factor the polynomial in the numerator to get (x 1)(x 2) x2 0 A quotient is zero only if its numerator is zero and its denominator is not zero, so it follows that x 1 and x are the required solutions Completing the Square An equation of the form ax2 bx c for a is called a quadratic equation A quadratic equation can have at most two solutions As you have seen, one way to find the solutions is to factor the equation Another is by the algebraic procedure called completing the square, in which the equation is rewritten in the form (x r)2 s for real numbers r and s Here are the steps in the procedure Step Divide both sides of the given equation ax2 bx c by a (remember, a 0) to obtain b c x2 a b x a b a a Then subtract c from both sides: a b c x2 a b x a a Step Add the square of b a b to both sides: a b b c b x2 a b x a b a b a a 2a 2a hof3238x_appa_663-675.qxd 670 11/14/11 APPENDIX A 2:15 PM Page 670 Algebra Review A-20 Step Notice that the left side of the equation is ax b b Thus, the equation 2a can be written as ax b c b b a b a 2a 2a EXAMPLE A.2.12 Solving an Equation by Completing the Square Solve the quadratic equation x2 5x by completing the square Solution x2 5x subtract from both sides x2 5x 4 5 x2 5x a b 4 a b 2 ax b add the square of 12 (5) to both sides x2 5x (5兾2)2 (x 5兾2)2 So x B4 and x B4 and the solutions are x 1 2 and x 4 2 EXAMPLE A.2.13 Solving an Equation by Completing the Square Solve the quadratic equation 3x2 5x by completing the square Solution We have 3x2 5x x2 a b x a b 3 x2 a b x 3 5 x2 a b x a b a b 6 59 ax b 36 divide each term by from each side add the square of a b to both sides subtract