CALCULUS CALCULUS Tenth Edition BRIEF EDITION Tools for Success in Calculus PDF

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BRI EF EDI TI ON Tent h Edit ion CALCULUS For Business, Economics, and the Social and Life Sciences LAUREN CE D. HOFFMANN * GERALD L. BRADLEY Calculus For Business, Economics, and the Social and Life Sciences BRIEF Tenth Edition Calculus For Business, Economics, and the Social and Life Sciences Laur...

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BRI EF EDI TI ON

Tent h Edit ion

CALCULUS For Business, Economics, and the Social and Life Sciences

LAUREN CE D. HOFFMANN

* GERALD L. BRADLEY

Calculus For Business, Economics, and the Social and Life Sciences

BRIEF Tenth Edition

Calculus For Business, Economics, and the Social and Life Sciences

Laurence D. Hoffmann Smith Barney

Gerald L. Bradley Claremont McKenna College

CALCULUS FOR BUSINESS, ECONOMICS, AND THE SOCIAL AND LIFE SCIENCES, BRIEF EDITION, TENTH EDITION Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020. Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. Previous editions © 2007, 2004, and 2000. 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. 1 2 3 4 5 6 7 8 9 0 VNH/VNH 0 9 ISBN 978–0–07–353231–8 MHID 0–07–353231–2 Editorial Director: Stewart K. Mattson Senior Sponsoring Editor: Elizabeth Covello Director of Development: Kristine Tibbetts Developmental Editor: Michelle Driscoll Marketing Director: Ryan Blankenship Senior Project Manager: Vicki Krug Senior Production Supervisor: Kara Kudronowicz Senior Media Project Manager: Sandra M. Schnee Designer: Laurie B. Janssen Cover/Interior Designer: Studio Montage, St. Louis, Missouri (USE) Cover Image: ©Spike Mafford/Gettyimages Senior Photo Research Coordinator: Lori Hancock Supplement Producer: Mary Jane Lampe Compositor: Aptara®, Inc. Typeface: 10/12 Times Printer: R. R. Donnelley, Jefferson City, MO Chapter Opener One, Two: © Corbis Royalty Free; p. 188(left): © Nigel Cattlin/Photo Researchers, Inc.; p. 188(right): © Runk/Schoenberger/Grant Heilman; Chapter Opener Three: © Getty Royalty Free; Chapter Opener Four: © The McGraw-Hill Companies, Inc./Jill Braaten, photographer; p. 368: © Getty Royalty Free; Chapter Opener Five: © Richard Klune/Corbis; p. 472: © Gage/Custom Medical Stock Photos; Chapter Opener Six: © AFP/Getty Images; p. 518: © Alamy RF; Chapter Opener Seven(right): US Geological Survey; (left): Maps a la carte, Inc.; Chapter Opener Eight: © Mug Shots/Corbis; p. 702: © Corbis Royalty Free; Chapter Opener Nine, p. 755: © Getty Royalty Free; Chapter Opener Ten: © Corbis Royalty Free; p. 829: Courtesy of Zimmer Inc.; Chapter Opener Eleven, p. 890, Appendix Opener: Getty Royalty Free. Library of Congress Cataloging-in-Publication Data Hoffmann, Laurence D., 1943Calculus for business, economics, and the social and life sciences — Brief 10th ed. / Laurence D. Hoffmann, Gerald L. Bradley. p. cm. Includes index. ISBN 978–0–07–353231–8 — ISBN 0–07–353231–2 (hard copy : alk. paper) 1. Calculus—Textbooks. I. Bradley, Gerald L., 1940- II. Title. QA303.2.H64 2010 515—dc22 2008039622

www.mhhe.com

CONTENTS Preface

CHAPTER

1

Functions, Graphs, and Limits 1.1 1.2 1.3 1.4 1.5 1.6

CHAPTER

2

vii

Functions 2 The Graph of a Function 15 Linear Functions 29 Functional Models 45 Limits 63 One-Sided Limits and Continuity 78 Chapter Summary 90 Important Terms, Symbols, and Formulas 90 Checkup for Chapter 1 90 Review Exercises 91 Explore! Update 96 Think About It 98

Differentiation: Basic Concepts 101 2.1 2.2 2.3 2.4 2.5 2.6

The Derivative 102 Techniques of Differentiation 117 Product and Quotient Rules; Higher-Order Derivatives 129 The Chain Rule 142 Marginal Analysis and Approximations Using Increments 156 Implicit Differentiation and Related Rates 167 Chapter Summary 179 Important Terms, Symbols, and Formulas 179 Checkup for Chapter 2 180 Review Exercises 181 Explore! Update 187 Think About It 189

v

vi

CONTENTS

CHAPTER

3

Additional Applications of the Derivative 3.1 3.2 3.3 3.4 3.5

CHAPTER

4

Exponential and Logarithmic Functions 4.1 4.2 4.3 4.4

CHAPTER

5

Increasing and Decreasing Functions; Relative Extrema 192 Concavity and Points of Inflection 208 Curve Sketching 225 Optimization; Elasticity of Demand 240 Additional Applied Optimization 259 Chapter Summary 277 Important Terms, Symbols, and Formulas 277 Checkup for Chapter 3 278 Review Exercises 279 Explore! Update 285 Think About It 287

Exponential Functions; Continuous Compounding 292 Logarithmic Functions 308 Differentiation of Exponential and Logarithmic Functions 325 Applications; Exponential Models 340 Chapter Summary 357 Important Terms, Symbols, and Formulas 357 Checkup for Chapter 4 358 Review Exercises 359 Explore! Update 365 Think About It 367

Integration 371 5.1 5.2 5.3 5.4 5.5 5.6

Antidifferentiation: The Indefinite Integral 372 Integration by Substitution 385 The Definite Integral and the Fundamental Theorem of Calculus 397 Applying Definite Integration: Area Between Curves and Average Value 414 Additional Applications to Business and Economics 432 Additional Applications to the Life and Social Sciences 445 Chapter Summary 462 Important Terms, Symbols, and Formulas 462 Checkup for Chapter 5 463 Review Exercises 464 Explore! Update 469 Think About It 472

CONTENTS

CHAPTER

6

Additional Topics in Integration 6.1 6.2 6.3 6.4

CHAPTER

7

A

TEXT SOLUTIONS

Functions of Several Variables 558 Partial Derivatives 573 Optimizing Functions of Two Variables 588 The Method of Least-Squares 601 Constrained Optimization: The Method of Lagrange Multipliers 613 Double Integrals 624 Chapter Summary 644 Important Terms, Symbols, and Formulas 644 Checkup for Chapter 7 645 Review Exercises 646 Explore! Update 651 Think About It 653

Algebra Review A.1 A.2 A.3 A.4

TA B L E S

Integration by Parts; Integral Tables 476 Introduction to Differential Equations 490 Improper Integrals; Continuous Probability 509 Numerical Integration 526 Chapter Summary 540 Important Terms, Symbols, and Formulas 540 Checkup for Chapter 6 541 Review Exercises 542 Explore! Update 548 Think About It 551

Calculus of Several Variables 7.1 7.2 7.3 7.4 7.5 7.6

APPENDIX

vii

A Brief Review of Algebra 658 Factoring Polynomials and Solving Systems of Equations 669 Evaluating Limits with L'Hôpital's Rule 682 The Summation Notation 687 Appendix Summary 668 Important Terms, Symbols, and Formulas 668 Review Exercises 689 Think About It 692

I Powers of e 693 II The Natural Logarithm (Base e) 694 Answers to Odd-Numbered Excercises, Chapter Checkup Exercises, and Odd-Numbered Chapter Review Exercises 695 Index 779

P R E FA C E Overview of the Tenth 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 they pursue careers in business, economics, and the life and social sciences. Students achieve success using this text as a result of the author’s applied and real-world orientation to concepts, problem-solving approach, straightforward and concise writing style, and comprehensive exercise sets. More than 100,000 students worldwide have studied from this text!

Improvements to This Edition

Enhanced Topic Coverage Every section in the text underwent careful analysis and extensive review to ensure the most beneficial and clear presentation. Additional steps and definition boxes were added when necessary for greater clarity and precision, and discussions and introductions were added or rewritten as needed to improve presentation. Improved Exercise Sets Almost 300 new routine and application exercises have been added to the already extensive problem sets. A wealth of new applied problems has been added to help demonstrate the practicality of the material. These new problems come from many fields of study, but in particular more applications focused on economics have been added. Exercise sets have been rearranged so that odd and even routine exercises are paired and the applied portion of each set begins with business and economics questions. Just-in-Time Reviews More Just-in-Time Reviews have been added in the margins to provide students with brief reminders of important concepts and procedures from college algebra and precalculus without distracting from the material under discussion. Graphing Calculator Introduction The Graphing Calculator Introduction can now be found on the book’s website at www.mhhe.com/hoffmann. This introduction includes instructions regarding common calculator keystrokes, terminology, and introductions to more advanced calculator applications that are developed in more detail at appropriate locations in the text. Appendix A: Algebra Review The Algebra Review has been heavily revised to include many new examples and figures, as well as over 75 new exercises. The discussions of inequalities and absolute value now include property lists, and there is new material on factoring and rationalizing expressions, completing the square, and solving systems of equations. New Design The Tenth Edition design has been improved with a rich, new color palette; updated writing and calculator exercises; and Explore! box icons, and all figures have been revised for a more contemporary and visual aesthetic. The goal of this new design is to provide a more approachable and student-friendly text. Chapter-by-Chapter Changes Chapter-by-chapter changes are available on the book’s website, www.mhhe.com/hoffmann.

viii

KEY FEATURES OF THIS TEXT Applications Throughout the text great effort is made to ensure that topics are applied to practical problems soon after their introduction, providing methods for dealing with both routine computations and applied problems. These problem-solving methods and strategies are introduced in applied examples and practiced throughout in the exercise sets.

EXAMPLE 5.1.3 Find the following integrals: a. b. c.

冕 冕冢 冕

(2x 5  8x 3  3x 2  5) dx

冣

x 3  2x  7 dx x

(3e 5t  兹t) dt

Solution a. By using the power rule in conjunction with the sum and difference rules and the multiple rule, you get

冕

冕

冕

冕

冕

(2x 5  8x 3  3x 2  5) dx  2 x 5 dx  8 x 3 dx  3 x 2 dx  5 dx

EXPLORE! Refer to Example 5.1.4. Store the function f (x )  3x2  1 into Y1. Graph using a bold graphing style and the window [0, 2.35]0.5 by [2, 12]1. Place into Y2 the family of antiderivatives

2 

y

1 6 x  2x 4  x 3  5x  C 3

b. There is no “quotient rule” for integration, but at least in this case, you can still divide the denominator into the numerator and then integrate using the method in part (a):

冕冢

F(x )  x3  x  L1 where L1 is the list of integer values 5 to 5. Which of these antiderivatives passes through the point (2, 6)? Repeat this exercise for f (x )  3x 2  2.

冢 冣 冢 冣 冢 冣

x6 x4 x3 8 3  5x  C 6 4 3

冣

x 3  2x  7 dx  x 

c.

冕

冕

冕冢

x2  2 

冣

7 dx x

Integration Rules

1 3 x  2x  7 ln |x|  C 3

Rules for Definite Integrals Let f and g be any functions continuous on a  x  b. Then,

(3e 5t  兹t) dt  (3e 5t  t 1/2) dt 3

冢5 e 冣  3/2 t 1

5t

1

3/2

This list of rules can be used to simplify the computation of definite integrals.

1. Constant multiple rule:

3 2  C   e5t  t3/2  C 5 3

2. Sum rule:

冕

冕

b

k f (x) dx  k

a

b

[ f (x)  g(x)] dx 

a

冕

冕

f(x) dx 

Procedural Examples and Boxes Each new topic is approached with careful clarity by providing step-by-step problem-solving techniques through frequent procedural examples and summary boxes.

4.

冕

f (x) dx  

冕

b

g(x) dx

a

f(x) dx

冕

b

Q(x) dx

冕

f(x) dx 

a

f (x) dx 

a

b

g(x) dx

b

a

6. Subdivision rule:

Net Change ■ If Q(x) is continuous on the interval a  x  b, then the net change in Q(x) as x varies from x  a to x  b is given by

a

b

f (x) dx  0 a

b

5.1.5 through 5.1.8). However, since Q(x) is an antiderivative of Q(x), the fundamental theorem of calculus allows us to compute net change by the following definite integration formula.

Q(b)  Q(a) 

冕 冕

冕

冕

b

a

a

5.

[ f (x)  g(x)] dx 

for constant k

f(x) dx

b

a

a

b

a

b

3. Difference rule:

冕

冕

c

f (x) dx 

a

冕

b

f(x) dx

c

Definitions Definitions and key concepts are set off in shaded boxes to provide easy referencing for the student.

a

Here are two examples involving net change.

EXAMPLE 5.3.9 At a certain factory, the marginal cost is 3(q  4)2 dollars per unit when the level of production is q units. By how much will the total manufacturing cost increase if the level of production is raised from 6 units to 10 units?

4 b. We want to find a time t  ta with 2  ta  11 such that T(ta)   . Solving 3 this equation, we find that 3 

Just-In-Time REVIEW

Just-In-Time Reviews These references, located in the margins, are used to quickly remind students of important concepts from college algebra or precalculus as they are being used in examples and review.

Since there are 60 minutes in an hour, 0.61 hour is the same as 0.61(60) ⬇ 37 minutes. Thus, 7.61 hours after 6 A.M. is 37 minutes past 1 P.M. or 1.37 P.M.

1 4 (ta  4)2   3 3 1 4 13 (ta  4)2    3   3 3 3

冢 133 冣  13

(ta  4)2  (3) 

ta  4   兹13

subtract 3 from both sides

multiply both sides by 3 take square roots on both sides

ta  4  兹13 ⬇ 0.39

or 7.61

Since t  0.39 is outside the time interval 2  ta  11 (8 A.M. to 5 P.M.), it follows that the temperature in the city is the same as the average temperature only when t  7.61, that is, at approximately 1:37 P.M.

ix

x

KEY FEATURES OF THIS TEXT

■

Exercise Sets Almost 300 new problems have been added to increase the effectiveness of the highly praised exercise sets! Routine problems have been added where needed to ensure students have enough practice to master basic skills, and a variety of applied problems have been added to help demonstrate the practicality of the material.

5.5 11. S(q)  0.3q2  30; q0  4 units

CONSUMERS’ WILLINGNESS TO SPEND For the consumers’ demand functions D(q) in Exercises 1 through 6: (a) Find the total amount of money consumers are willing to spend to get q0 units of the commodity. (b) Sketch the demand curve and interpret the consumers’ willingness to spend in part (a) as an area.

12. S(q)  0.5q  15; q0  5 units 13. S(q)  10  15e0.03q; q0  3 units 14. S(q)  17  11e0.01q; q0  7 units CONSUMERS’ AND PRODUCERS’ SURPLUS AT EQUILIBRIUM In Exercises 15 through 19, the demand and supply functions, D(q) and S(q), for a particular commodity are given. Specifically, q thousand units of the commodity will be demanded (sold) at a price of p  D(q) dollars per unit, while q thousand units will be supplied by producers when the price is p  S(q) dollars per unit. In each case: (a) Find the equilibrium price pe (where supply equals demand). (b) Find the consumers’ surplus and the producers’ surplus at equilibrium.

1. D(q)  2(64  q2) dollars per unit; q0  6 units 2. D(q) 

300 dollars per unit; q0  5 units (0.1q  1)2

3. D(q) 

400 dollars per unit; q0  12 units 0.5q  2

4. D(q) 

300 dollars per unit; q0  10 units 4q  3

5. D(q)  40e0.05q dollars per unit; q0  10 units

364

1 2 15. D(q)  131  q2; S(q)  50  q2 3 3

6. D(q)  50e0.04q dollars per unit; q0  15 units CONSUMERS’ SURPLUS In Exercises 7 through 10, p  D(q) is the price (dollars per unit) at which q units of a particular commodity will be demanded by the market (that is, all q units will be sold at this price), and q0 is a specified level of production. In each case, find the price p0  D(q0) at which q0 units will be demanded and compute the corresponding consumers’ surplus CS. Sketch the demand curve y  D(q) and shade the region whose area represents the consumers’ surplus.

CHAPTER SUMMARY

EXERCISES

1 16. D(q)  65  q2; S(q)  q2  2q  5 3 17. D(q)  0.3q 2  70; S(q)  0.1q2  q  20 18. D(q)  245  2q; S(q)  5  q 19. D(q) 

16 1  3; S(q)  (q  1) q2 3

20. PROFIT OVER THE USEFUL LIFE OF A MACHINE Suppose that when it is t years old, a particular industrial machine generates revenue at the rate R(t)  6,025  8t 2 dollars per year and that operating and servicing costs accumulate at the rate C(t)  4,681  13t2 dollars per year. a. How many years pass before the profitability of the machine begins to decline? b. Compute the net profit generated by the machine over its useful lifetime. c. Sketch the revenue rate curve y  R(t) and the cost rate curve y  C(t) and shade the region whose area represents the net profit computed in part (b).

7. D(q)  2(64  q2); q0  3 units 8. D(q)  150  2q  3q2; q0  6 units 9. D(q)  40e0.05q; q0  5 units 10. D(q)  75e0.04q; q0  3 units PRODUCERS’ SURPLUS In Exercises 11 through 14, p  S(q) is the price (dollars per unit) at which q units of a particular commodity will be supplied to the market by producers, and q0 is a specified level of production. In each case, find the price p0  S(q0) at which q0 units will be supplied and compute the corresponding producers’ surplus PS. Sketch the supply curve y  S(q) and shade the region whose area represents the producers’ surplus.

84. RADIOLOGY The radioactive isotope gallium-67 (67Ga), used in the diagnosis of malignant tumors, has a half-life of 46.5 hours. If we start with 100 milligrams of the isotope, how many milligrams will be left after 24 hours? When will there be only 25 milligrams left? Answer these questions by first using a graphing utility to graph an appropriate exponential...