Mathematical apprlications for the management life and social sciences

0.7 Algebraic Fractions 42Key Terms and Formulas 49Review Exercises 51Chapter Test 54Extended Applications & Group Projects 56Campaign Management • Pricing for Maximum Profit 1 Warm-up 5

Trang 2

Index of Selected Applications

Electrical power use, 182, 614, 951

Employee age and gender, 475, 476, 481, 484, 486

Minimizing average cost, 727, 733, 734 Minimizing costs, 333, 334, 335, 344, 345, 348, 349, 351, 691, 738,

744, 745, 756, 760, 763, 764 Oil reﬁneries, 229

Operating leverage, 203 Organizational growth, 388, 393, 394 Package design, 36, 84, 181, 739, 763, 764 Parking costs, 601

Parts delivery, 485 Parts listings, 267, 272 Postal rates, 182, 614 Pricing, 90, 117, 118, 228, 658, 709, 765, 862, 1003 Pricing for maximum proﬁt, 57

Printing, 761 Product design, 761 Product display, 511 Product reliability, 367, 392, 551, 834, 890, 894 Production, 219, 228, 230, 235, 272, 274, 275, 344, 349, 665, 709,

723, 790, 795, 834, 868, 872, 893, 944, 955, 958, 961, 971, 973,

978, 988, 989, 994, 997, 999, 1000, 1001 Production costs, 220, 222, 277, 298, 304, 348, 709, 743, 750, 764 Productivity, 528, 600, 612, 630, 642, 665, 687, 709, 722, 743, 756,

760, 761, 834, 866 Proﬁt, 35, 54, 64, 70, 83, 84, 119, 121, 126, 134, 135, 136, 149, 150,

159, 196, 244, 280, 393, 411, 600, 630, 678, 679, 687, 688, 711,

715, 743, 756, 800, 813, 866, 867, 903, 961, 971, 988, 989,

1000, 1001 Proﬁt maximization, 57, 154, 166, 168, 197, 200, 292, 303, 305, 306,

307, 314, 323, 324, 325, 336, 340, 344, 348, 349, 350, 351, 353,

426, 683, 686, 729, 734, 753, 760, 761, 762, 765, 848, 849, 852,

979, 983 Project teams, 509 Property development, 482, 741, 945 Publicity, 576

Purchasing, 70, 329 Purchasing electric power, 951 Quality control, 472, 476, 477, 491, 495, 496, 499, 513, 516, 517, 528,

529, 538, 539, 540, 563, 564, 571, 576, 579, 936, 948, 949 Rentals, 35, 70, 117, 118, 160, 244, 447, 475, 484, 496, 530 Revenue, 35, 42, 54, 84, 85, 119, 126, 135, 158, 160, 199, 228, 387,

393, 562, 576, 579, 600, 620, 629, 639, 641, 642, 650, 651, 658,

661, 663, 665, 671, 672, 674, 686, 687, 688, 689, 710, 743, 745,

756, 760, 812, 813, 818, 821, 824, 831, 834, 843, 849, 866,

949, 997 Revenue maximization, 156, 197, 295, 325, 345, 724, 725, 733, 761,

763, 765 Rewards for employees, 515 Safety, 192, 478, 801, 812 Salaries, 70, 411, 412, 459, 484, 551, 552 Sales, 217, 456, 511, 516, 552, 563, 600, 612, 613, 756, 862 Sales decay, 382, 384, 391, 392, 398, 399, 782, 813 Sales growth, 390, 393, 394, 398

Sales promotion, 475 Scheduling, 290, 303, 304, 323, 333, 334, 335, 336, 340, 344 Shadow prices, 304, 319, 353

Shipping, 305, 323 Starbucks stores, 401

Trang 3

Wilson’s lot size formula, 960, 971

Wireless service spending, 71, 882, 893

Gasoline mileage and prices, 552, 571

Gini coefﬁcient of income, 895, 899, 904, 927, 944, 950

440, 445, 446, 447, 448, 459, 460, 613 Annuities due, 433, 435, 442, 443, 446 APY, 418, 424, 459, 461

ATM transactions, 515 Banks, 85, 91, 305, 511, 516 Bond pricing, 440

Bonds, 441, 446, 447, 460, 461, 462, 552 Budgeting, 323, 563

Capital value, 929, 935, 949 College savings, 412, 414, 418, 423, 424, 435, 436, 448, 459, 461, 462 Compound interest, 19, 33, 367, 368, 393, 398, 412, 414, 415, 417,

423, 458, 459, 460, 461, 462, 777, 779, 811, 812, 844, 861 Consumer credit, 397

Court settlements, 443, 447, 461 CPA exam, 564

Credit cards, 69, 517, 518 Debt, 218, 450, 454, 455, 458, 461 Debt reﬁnancing, 82

Deferred annuities, 444, 448 Delinquent accounts, 486, 490 Depreciation, 69, 86, 94, 95, 136, 426, 892 Doubling time, 371, 381, 419

Earnings, 660, 709 Future value, 53, 458 Future value of annuities, 427, 428, 429, 430, 431, 434, 435, 436, 459 Future value of income stream, 907, 914

Future value of investments, 33, 53, 62, 357, 359, 367, 393, 398, 410,

414, 423, 424, 425, 459, 460, 782, 959 Home equity loans, 463

Income levels, 529 Income stream, 913, 914, 920, 921, 924, 927, 943, 948, 949 Investing, 26, 33, 35, 41, 62, 65, 70, 110, 118, 134, 136, 239, 244,

245, 246, 259, 275, 277, 278, 325, 357, 359, 367, 368, 369, 371,

381, 393, 398, 404, 405, 410, 415, 417, 419, 423, 424, 425, 435,

436, 446, 459, 460, 462, 659, 775, 862, 866, 959 Loans, 53, 82, 118, 244, 410, 411, 450, 453, 454, 455, 456, 457, 458,

459, 460, 463, 1001 Mortgages, 76, 81, 133, 449, 450, 456, 457, 462, 464, 960, 970 Mutual funds, 394, 516

Net worth, 579 Perpetuities, 613 Personal income, 19 Present value, 19, 410, 458, 906, 914, 927, 943 Present value of annuities, 437, 438, 445, 446, 448 Purchasing a home, 464

Real estate, 117, 551, 836, 837 Retirement planning, 95, 160, 424, 425, 435, 436, 446, 447, 448, 461,

462, 951, 989, 999 Savings, 410, 425, 435, 436, 448, 459, 460, 948, 999 Simple interest, 41, 404, 405, 410, 411, 458, 459, 460, 461 Sinking funds, 431, 435, 436, 459

Stock market, 5, 7, 19, 368, 393, 410, 415, 424, 425, 460, 529,

601, 813 Taxes, 15, 70, 71, 73, 94, 107, 124, 130, 279, 280, 601, 602, 606, 613,

660, 904

Trang 6

Clarion University of Pennsylvania

University of South Carolina Beaufort

Trang 7

Associate Editor: Jeannine Lawless

Media Editor: Catie Ronquillo

Marketing Specialist: Ashley Pickering

Marketing Communications Manager: Mary

Anne Payumo

Project Manager, Editorial Production: Shelley

Dickerson

Art and Design Manager: Jill Haber

Manufacturing Buyer: Diane Gibbons Permissions Editor: Craig Mertens Text Designer: Jean Hammond Photo Manager: Jennifer Meyer Dare Photo Researcher: Stacey Dong Cover Designer: Anne Katzeff Cover Image: © Stuttgart Pforzheim/Getty Images Compositor: Aptara® , Inc.

Photo Credits: p 1, Gail Shumway/Getty Images; p 58, © Natalie Behring/Digital Railroad;

p 139, © Patrick Brice/Arcaid/Corbis; p 201, NASA; p 204, © Klaus Leidorf/zefa/Corbis;

p 281, © Wang Hoof/epa/Corbis; p 355, © Roy Botterell/Corbis; p 402, Ted Russell/Getty

Images; p 465, © Mike Denton/Alamy; p 534, © Darell Gulin/CORBIS; p 584, Al Bello/Getty Images; p 693, © Nakheel Corporation; p 766, © Pat Bennett/Alamy; p 816, © John Edward Linden/Arcaid/Corbis; p 870, © 2008 Jupiterimages Corporation; p 953, © PCL/Alamy

ALL RIGHTS RESERVED No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108

or the 1976 United States Copyright Act, without the prior written permission of the publisher For product information and technology assistance, contact us at

Cengage Learning Academic Resource Center, 1-800-423-0563

For permission to use material from this text or product,

submit all requests online at www.cengage.com/permissions.

Further permissions questions can be e-mailed to

Cengage Learning products are represented in Canada by Nelson Education, Ltd.

For your course and learning solutions, visit

academic.cengage.com.

Purchase any of our products at your local college store

or at our preferred online store www.ichapters.com.

Printed in the United States of America

1 2 3 4 5 6 7 12 11 10 09 08

Trang 8

0.7 Algebraic Fractions 42Key Terms and Formulas 49Review Exercises 51Chapter Test 54Extended Applications & Group Projects 56

Campaign Management • Pricing for Maximum Profit

1

Warm-up 591.1 Solution of Linear Equations and Inequalities in One Variable 601.2 Functions 71

1.3 Linear Functions 851.4 Graphs and Graphing Utilities 96

Graphical Solutions of Equations

1.5 Solutions of Systems of Linear Equations 107

Three Equations in Three Variables

1.6 Applications of Functions in Business and Economics 119

Total Cost, Total Revenue, and Profit • Break-Even Analysis • Supply, Demand, and Market Equilibrium

Key Terms and Formulas 130Review Exercises 132Chapter Test 135Extended Applications & Group Projects 137

Hospital Administration • Fundraising

Trang 9

Inequalities and Linear Programming 281

Factoring Methods • The Quadratic Formula

2.2 Quadratic Functions: Parabolas 1512.3 Business Applications of Quadratic Functions 161

Supply, Demand, and Market Equilibrium • Break-Even Points and Maximization

2.4 Special Functions and Their Graphs 169

Polynomial and Rational Functions • Piecewise Defined Functions

2.5 Modeling; Fitting Curves to Data with Graphing Utilities (optional) 183Key Terms and Formulas 194

Review Exercises 195Chapter Test 198Extended Applications & Group Projects 201

An Inconvenient Truth • Body-Mass Index • Operating Leverage and Business Risk

Warm-up 2053.1 Matrices 2063.2 Multiplication of Matrices 2203.3 Gauss-Jordan Elimination: Solving Systems of Equations 230

Systems with Unique Solutions • Systems with Nonunique Solutions

3.4 Inverse of a Square Matrix; Matrix Equations 246

Matrix Equations • Determinants

3.5 Applications of Matrices: Leontief Input-Output Models 260Key Terms and Formulas 273

Taxation • Company Profits After Bonuses and Taxes

Warm-up 2824.1 Linear Inequalities in Two Variables 2834.2 Linear Programming: Graphical Methods 2924.3 The Simplex Method: Maximization 306

Nonunique Solutions: Multiple Solutions and No Solution • Shadow Prices

4.4 The Simplex Method: Duality and Minimization 3264.5 The Simplex Method with Mixed Constraints 336

Trang 10

Contents ● viiKey Terms 345

Transportation • Slack Variables and Shadow Prices

Modeling with Exponential Functions

5.2 Logarithmic Functions and Their Properties 369

Logarithmic Functions and Graphs • Modeling with Logarithmic Functions • Properties of Logarithms • Change of Base

5.3 Solution of Exponential Equations: Applications of Exponential and LogarithmicFunctions 382

Solving Exponential Equations Using Logarithmic Properties • Growth and Decay • Economic and Management Applications • Gompertz Curves and Logistic Functions

Starbucks Stores • Agricultural Business Management

Warm-up 4036.1 Simple Interest; Sequences 404

Simple Interest • Sequences • Arithmetic Sequences

6.2 Compound Interest; Geometric Sequences 412

Compound Interest • Geometric Sequences

6.3 Future Value of Annuities 427

Ordinary Annuities • Annuities Due

6.4 Present Values of Annuities 437

Ordinary Annuities • Annuities Due • Deferred Annuities

6.5 Loans and Amortization 449

Unpaid Balance of a Loan

Mail Solicitation Home Equity Loan: Is This a Good Deal? • Profit Reinvestment • Purchasing a Home

Trang 11

Introduction to Probability 465

7

Warm-up 4667.1 Probability; Odds 4677.2 Unions and Intersections of Events: One-Trial Experiments 4787.3 Conditional Probability: The Product Rule 486

7.4 Probability Trees and Bayes’ Formula 497

Probability Trees • Bayes’ Formula

7.5 Counting: Permutations and Combinations 5057.6 Permutations, Combinations, and Probability 5127.7 Markov Chains 517

Phone Numbers • Competition in the Telecommunications Industry

8

Warm-up 5358.1 Binomial Probability Experiments 5368.2 Data Description 540

8.3 Discrete Probability Distributions; The Binomial Distribution 553

Discrete Probability Distributions • Measures of Dispersion • The Binomial Distribution

8.4 Normal Probability Distribution 5648.5 The Normal Curve Approximation to the Binomial Distribution 572Key Terms and Formulas 577

Lotteries • Statistics in Medical Research; Hypothesis Testing

9

Warm-up 5859.1 Limits 5869.2 Continuous Functions; Limits at Infinity 602

Trang 12

Contents ● ix9.8 Higher-Order Derivatives 666

9.9 Applications of Derivatives in Business and Economics 673Key Terms and Formulas 683

Marginal Return to Sales • Tangent Lines and Optimization in Business and Economics

11

Warm-up 76711.1 Derivatives of Logarithmic Functions 76811.2 Derivatives of Exponential Functions 77711.3 Implicit Differentiation 785

11.4 Related Rates 79511.5 Applications in Business and Economics 802

Elasticity of Demand • Taxation in a Competitive Market

Inflation • Knowledge Workers

10

Warm-up 69410.1 Relative Maxima and Minima: Curve Sketching 69510.2 Concavity: Points of Inflection 711

Second-Derivative Test

10.3 Optimization in Business and Economics 724

Maximizing Revenue • Minimizing Average Cost • Maximizing Profit

10.4 Applications of Maxima and Minima 73710.5 Rational Functions: More Curve Sketching 745

Asymptotes • More Curve Sketching

Production Management • Room Pricing in the Off-Season

Trang 13

Definite Integrals: Techniques of Integration 870

13

Warm-up 87113.1 Area Under a Curve 87213.2 The Definite Integral: The Fundamental Theorem of Calculus 88313.3 Area Between Two Curves 894

13.4 Applications of Definite Integrals in Business and Economics 904

Continuous Income Streams • Consumer’s Surplus • Producer’s Surplus

13.5 Using Tables of Integrals 91513.6 Integration by Parts 92113.7 Improper Integrals and Their Applications 92713.8 Numerical Integration Methods: Trapezoidal Rule and Simpson’s Rule 936Key Terms and Formulas 945

Retirement Planning • Purchasing Electrical Power

12

Warm-up 81712.1 The Indefinite Integral 81812.2 The Power Rule 82612.3 Integrals Involving Exponential and Logarithmic Functions 83612.4 Applications of the Indefinite Integral in Business and Economics 845

Total Cost and Profit • National Consumption and Savings

12.5 Differential Equations 854

Separable Differential Equations • Applications of Differential Equations

Employee Production Rate • Supply and Demand

Trang 14

Contents ● xi

14

Warm-up 95414.1 Functions of Two or More Variables 95514.2 Partial Differentiation 962

First-Order Partial Derivatives • Higher-Order Partial Derivatives

14.3 Applications of Functions of Two Variables in Business and Economics 972

Joint Cost and Marginal Cost • Production Functions • Demand Functions

14.4 Maxima and Minima 979

Linear Regression

14.5 Maxima and Minima of Functions Subject to Constraints: Lagrange Multipliers 990Key Terms and Formulas 998

Advertising • Competitive Pricing

Appendix AP-1Table I: Future Value of an Ordinary Annuity of $1 ( ) AP-1Table II: Present Value of an Ordinary Annuity of $1 ( ) AP-3Table III: Areas Under the Standard Normal Curve AP-5Answers A-1

Index I-1

a n 0i

s n 0i

Trang 15

To paraphrase English mathematician, philosopher, and educator Alfred North

White-head, the purpose of education is not to ﬁll a vessel but to kindle a ﬁre In particular,Whitehead encouraged students to be creative and imaginative in their learning and

to continually form ideas into new and more exciting combinations This desirable goal isnot always an easy one to realize in mathematics with students whose primary interests are

in areas other than mathematics The purpose of this text, then, is to present mathematicalskills and concepts, and to apply them to ideas that are important to students in the man-agement, life, and social sciences We hope that this look at the relevance of mathematicalideas to a broad range of ﬁelds will help inspire the imaginative thinking and excitementfor learning that Whitehead spoke of The applications included allow students to viewmathematics in a practical setting relevant to their intended careers Almost every chapter

of this book includes a section or two devoted to the applications of mathematical topics,and every section contains a number of application examples and problems An index ofthese applications on the front and back inside covers demonstrates the wide variety used

in examples and exercises Although intended for students who have completed two years

of high school algebra or its equivalent, this text begins with a brief review of algebra which,

if covered, will aid in preparing students for the work ahead

Pedagogical Features

In this new edition, we have incorporated many suggestions that reﬂect the needs and wishes

of our users Important pedagogical features that have characterized previous editions havebeen retained They are as follows

Intuitive Viewpoint. The book is written from an intuitive viewpoint, with emphasis onconcepts and problem solving rather than on mathematical theory Yet each topic is care-fully developed and explained, and examples illustrate the techniques involved

Flexibility. At different colleges and universities, the coverage and sequencing of topicsmay vary according to the purpose of the course and the nature of the student audience Toaccommodate alternate approaches, the text has a great deal of ﬂexibility in the order inwhich topics may be presented and the degree to which they may be emphasized

Chapter Warm-ups. With the exception of Chapter 0, a Warm-up appears at the beginning

of each chapter and invites students to test themselves on the skills needed for that ter The Warm-ups present several prerequisite problem types that are keyed to the appro-priate sections in the upcoming chapter where those skills are needed Students who havedifficulty with any particular skill are directed to specific sections of the text for review.Instructors may also find the Warm-ups useful in creating a course syllabus that includes

chap-an appropriate scope chap-and sequence of topics

Application Previews. Each section begins with an Application Preview that establishesthe context and direction for the concepts that will be presented Each of these Previewscontains an example that motivates the mathematics in the section and is then revisited in

a completely worked Application Preview example appearing later in the section

Comprehensive Exercise Sets. The overall variety and grading of drill and application cises offer problems for different skill levels, and there are enough challenging problems tostimulate students in thoughtful investigations Many exercise sets contain critical thinkingand thought-provoking multistep problems that extend students’ knowledge and skills.xii

Trang 16

exer-Preface ● xiii

Applications. We have found that integrating applied topics into the discussions and cises helps provide motivation within the sections and demonstrates the relevance of eachtopic Numerous real-life application examples and exercises represent the applicability ofthe mathematics, and each application problem is identified, so the instructor or student canselect applications that are of special interest In addition, we have found that offeringseparate lessons on applied topics such as cost, revenue, and profit functions brings the pre-ceding mathematical discussions into clear, concise focus and provides a thread of conti-nuity as mathematical sophistication increases There are ten such sections in the book, andentire chapters devoted to linear programming and financial applications Of the more than5,500 exercises in the book, over 2,000 are applied

exer-Extended Applications and Group Projects. Each chapter ends with at least two of these

32 case studies, which further illustrate how mathematics can be used in business and sonal decision making In addition, many applications are cumulative in that solutionsrequire students to combine the mathematical concepts and techniques they learned in some

per-of the preceding chapters

Graphical, Numerical, and Symbolic Methods. A large number of real data and modelingapplications are included in the examples and exercises throughout the text and are denoted

by the header Modeling Many sections include problems with functions that are modeledfrom real data and some problems that ask students to model functions from the data given.These problems are solved by using one or more graphical, numerical, or symbolic methods

Graphing Utilities and Spreadsheets. In the Ninth Edition, we have increased the overallpresence of our treatment of graphing utilities and especially spreadsheets More examples,applications, Technology Notes, Calculator Notes, and Spreadsheet Notes, denoted by the icon, are scattered throughout the text After the introduction to Excel in Section 0.5 and to graph-ing calculators in Section 1.4, discussions of the use of technology are placed in subsectionsand examples in many sections, so that they can be emphasized or de-emphasized at the option

of the instructor

The discussions of graphing utility technology highlight its most common features anduses, such as graphing, window setting, trace, zoom, Solver, tables, finding points of inter-section, numerical derivatives, numerical integration, matrices, solving inequalities, and mod-eling (curve fitting) While technology never replaces the mathematics, it does supplement andextend it by providing opportunities for generalization and alternative ways of understanding,doing, and checking Some exercises that are better worked with the use of technology, includ-ing graphing calculators, computer programs, and computer spreadsheets, are highlighted withthe technology icon Of course, many additional exercises can benefit from the use of tech-nology, at the option of the instructor Technology can be used to graph functions and todiscuss the generalizations, applications, and implications of problems being studied

In addition, the basics of spreadsheet operations are introduced and opportunities tosolve problems with spreadsheets are provided The Ninth Edition now includes more Excelinformation speciﬁcally relating to solving equations, systems of equations, quadratic equa-

tions, matrices, linear programming, output comparisons of f (x), f (x), and f (x), and the

maxima and minima of functions subject to constraints Excel is also a particularly useful

problem-solving tool when studying the Mathematics of Finance in Chapter 6 A separate

Excel Guide offering examples and user information is available for purchase Please see

the Resources for the Student section on page xv of this Preface for more information

Checkpoints. The Checkpoints ask questions and pose problems within each section’sdiscussion, allowing students to check their understanding of the skills and concepts underdiscussion before they proceed Solutions to these Checkpoints appear before the sectionexercises

Objective Lists. Every section begins with a brief list of objectives that outline the goals

of that section for the student

Trang 17

Procedure/Example and Property/Example Tables. Appearing throughout the text, thesetables aid student understanding by giving step-by-step descriptions of important proceduresand properties with illustrative examples worked out beside them.

Boxed Information. All important information is boxed for easy reference, and key termsare highlighted in boldface

Key Terms and Formulas. At the end of each chapter, just before the Chapter ReviewExercises, there is a section-by-section listing of that chapter’s key terms and formulas Thisprovides a well-organized core from which a student can build a review, both to consultwhile working the Review Exercises and to identify quickly any section needing additionalstudy

Review Exercises and Chapter Tests. At the end of each chapter, a set of Review Exercisesoffers students extra practice on topics in that chapter These Reviews cover each chapter’stopics primarily in their section order, but without section references, so that students get atrue review but can readily ﬁnd a section for further review if difﬁculties occur A ChapterTest follows each set of Review Exercises All Chapter Tests provide a mixture of problemsthat do not directly mirror the order of topics found within the chapter This organization

of the Chapter Test ensures that students have a ﬁrm grasp of all material in the chapter

Changes in the Ninth Edition

In the Ninth Edition, we continue to offer a text characterized by complete and accuratepedagogy, mathematical precision, excellent exercise sets, numerous and varied applications,and student-friendly exposition There are many changes in the mathematics, prose, and art.The more signiﬁcant ones are as follows

■ A new section, “Normal Approximation to the Binomial,” which discusses binomialprobability applications involving large numbers of trials, has been added

■ Three new Extended Applications/Group Projects have been added, two in Chapter 2and one in Chapter 5 The two added in Chapter 2, “An Inconvenient Truth” and

“Body Mass Index,” address important health, environmental, and economic issues.The “Inconvenient Truth” project is based on and utilizes Al Gore’s Oscar-winningﬁlm of the same title to investigate some of the mathematics behind Gore’s NobelPeace Prize-winning work on global warming

■ Extended Applications/Group Projects have been updated as appropriate These includeHospital Administration in Chapter 1, Phone Area Code Numbers in Chapter 7, andPowerball Lotteries in Chapter 8

■ Expanded applications of matrix inverses explore the sensitivity of solutions in cations of systems of linear equations and in Input-Output models

appli-■ Drill Exercises throughout the text have been revised and reorganized to improvetheir grading and variety

■ Solution of exponential and logarithmic equations (in Chapter 5) boasts expandedexposition, additional examples, and more comprehensive drill exercises

■ Bond Pricing has been added to Section 6.4, Present Values of Annuities This new

application utilizes calculations of simple interest and present values of both pound interest investments and ordinary annuities and combines these to determinemarket prices for bonds

com-■ Exercise Sets for Mathematics of Finance in Chapter 6 have been reorganized and

revised to provide adequate drills as well as to expand the sets of problems thatinclude miscellaneous ﬁnance applications

Trang 18

Preface ● xv

■ The section reference labels have been removed from the Chapter Review Exercisesthroughout the text This improves the effectiveness of these exercises because stu-dents now must establish a context for each problem as well as solve it

■ Data-driven and modeling examples and exercises have been updated or replacedwith current applications

■ An increased number of optional Excel discussions have been inserted whereappropriate

Resources for the Student

Student Solutions Manual. Need a leg up on your homework or help to prepare for an exam?The Student Solutions Manual contains worked-out solutions for all odd-numbered exercises

in the text It is a great resource to help you understand how to solve those tough problems

Microsoft Excel Guide. This guide provides a list of exercises from the text that can becompleted after each step-by-step Excel example No prior knowledge of Excel is necessary

WebAssign. WebAssign, the most widely used homework system in higher education,offers instant feedback and repeatable problems—everything you could ask for in an onlinehomework system WebAssign’s homework system lets you practice and submit homework

via the web It is easy to use and loaded with extra resources With this edition of

Mathe-matical Applications for the Management, Life, and Social Sciences there are over 1,500

algorithmic homework exercises to use for practice and review

DVD Lecture Series. Comprehensive, instructional lecture presentations serve a number

of uses They are great if you need to catch up after missing a class, need to supplementonline or hybrid instruction, or need material for self-study or review

Resources for the Instructor

WebAssign. Instant feedback, grading precision, and ease of use are just three reasons whyWebAssign is the most widely used homework system in higher education WebAssign’shomework delivery system lets instructors deliver, collect, grade, and record assignments

via the web With this edition of Mathematical Applications for the Management, Life, and

Social Sciences there are over 1,500 algorithmic homework exercises to choose from These

algorithmic exercises are based on the section exercises from the textbook to ensurealignment with course goals

Instructor’s Complete Solutions Manual. The Instructor’s Complete Solutions Manualcontains worked solutions for all exercises in the text It also contains solutions to the specialfeatures in the text such as the Extended Applications and Group Projects It is available atthe Instructor’s Resource Center on the book’s website

Power Lecture. This comprehensive CD-ROM includes the Instructor’s Solutions Manual,PowerPoint slides, and the computerized test bank featuring algorithmically created ques-tions to create, deliver, and customize tests

Computerized Test Bank. Create, deliver, and customize tests and study guides in utes with this easy-to-use assessment software on CD The thousands of algorithmic ques-tions in the test bank are derived from the textbook exercises, ensuring consistency betweenexams and the book

min-JoinIn on TurningPoint. Enhance how your students interact with you, your lecture, andeach other Cengage Learning is now pleased to offer you book-speciﬁc content for

Response Systems tailored to Mathematical Applications for the Management, Life, and

Social Sciences, allowing you to transform your classroom and assess your students’

progress with instant in-class quizzes and polls

Trang 19

We would like to thank the many people who helped us at various stages of revising thistext The encouragement, criticism, contributions, and suggestions that were offered wereinvaluable to us Special thanks go to Victoria Sapko of Framingham State College for hercontribution to the Inconvenient Truth Extended Application We are also especially grate-ful to Dave Hipfel for his insights regarding bond pricing and his help with applicationexercises and Extended Applications

Once again we have been fortunate to have Helen Medley’s assistance with accuracychecking of the entire text and answer section during manuscript preparation and on thepage proofs As always, we continue to be impressed by and most appreciative of her workethic, attention to detail, accuracy, and skill

For their reviews of draft manuscript and the many helpful comments that were offered,

we would like to thank

Jerry Allison Black Hawk CollegeJoy Beverly University of MiamiElsie Campbell Angelo State UniversityVladimir Fomichov Southern New Hampshire UniversityMallie Hood Paris Junior College

Lance Lana University of Colorado DHSCGabriel Mendoza El Paso Community CollegeSam Obeid Richland College

Elaine Russell Angelina CollegeCatherine Stanley Acadia UniversityMargaret Jeanne Trubek Emmanuel College

Dr Henry L Wyzinski Indiana University Northwest

Ronald J Harshbarger James J Reynolds

Trang 20

This chapter provides a brief review of the algebraic

concepts that will be used throughout the text You may

be familiar with these topics, but it may be helpful to

spend some time reviewing them In addition, each chapter

after this one opens with a warm-up page that identifies

prerequisite skills needed for that chapter If algebraic

skills are required, the warm-up cites their coverage in this

chapter Thus you will find that this chapter is a useful

reference as you study later chapters

The topics and applications studied in this chapter

include the following

Sections Applications

Venn diagrams 0.2 The Real Numbers Income taxes, health

Inequalities and intervals statistics

Absolute value 0.3 Integral Exponents Personal income,

Expressions

0.6 Factoring Simple interest, revenue

Common factors Factoring trinomials 0.7 Algebraic Fractions Average cost, advertising

Complex fractions

Algebraic Concepts

0

Trang 21

A set is a well-deﬁned collection of objects We may talk about a set of books, a set of

dishes, a set of students, or a set of individuals with a certain blood type There are two

ways to tell what a given set contains One way is by listing the elements (or members)

of the set (usually between braces) We may say that a set A contains 1, 2, 3, and 4 by

writ-ing To say that 4 is a member of set A, we write Similarly, wewrite to denote that 5 is not a member of set A.

If all the members of the set can be listed, the set is said to be a ﬁnite set.

and are examples of ﬁnite sets When we do not wish to list all theelements of a ﬁnite set, we can use three dots to indicate the unlisted members of the set.For example, the set of even integers from 8 to 8952, inclusive, could be written as

For an inﬁnite set, we cannot list all the elements, so we use the three dots For example,

is an inﬁnite set This set N is called the set of natural numbers.

Another way to specify the elements of a given set is by description For example, wemay write to describe the set of all Ford automobiles Fur-thermore, is read “F is the set of all y such that y is an

odd natural number.”

● EXAMPLE 1 Describing Sets

Write the following sets in two ways

(a) The set A of natural numbers less than 6 (b) The set B of natural numbers greater than 10 (c) The set C containing only 3

Solution

(a)(b)(c)

Note that set C of Example 1 contains one member, 3; set A contains ﬁve members; and set B contains an inﬁnite number of members It is possible for a set to contain no

members Such a set is called the empty set or the null set, and it is denoted by or by{ } The set of living veterans of the War of 1812 is empty because there are no livingveterans of that war Thus

Special relations that may exist between two sets are deﬁned as follows

5x: x is a living veteran of the War of 18126 

C 536 or C 5x: x 36

B 511, 12, 13, 14, 6 or B 5x: x is a natural number greater than 106

A 51, 2, 3, 4, 56 or A 5x: x is a natural number less than 66

F D 5y: y is an odd natural number6 5x: x is a Ford automobile6

2 A is called a subset of B, which is written if

every element of A is an element of B The empty set

is a subset of every set Each set A is a subset of itself.

3 If C and D have no elements in common, they are

Trang 22

Set Intersection

0.1 Sets ● 3

In the discussion of particular sets, the assumption is always made that the sets under

discussion are all subsets of some larger set, called the universal set U The choice of the

universal set depends on the problem under consideration For example, in discussing theset of all students and the set of all female students, we may use the set of all humans asthe universal set

We may use Venn diagrams to illustrate the relationships among sets We use a

rec-tangle to represent the universal set, and we use closed ﬁgures inside the recrec-tangle to resent the sets under consideration Figures 0.1–0.3 show such Venn diagrams

X and Y are not disjoint.

The shaded portion of Figure 0.3 indicates where the two sets overlap The set containing the

members that are common to two sets is said to be the intersection of the two sets.

The intersection of A and B, written is deﬁned by

● EXAMPLE 2 Set Intersection

(b) Which of A, B, and is a subset of A?

Solution

(a) because 3 and 5 are the common

elements of A and B Figure 0.4 shows the sets

and their intersection

(b) and A are subsets of A.

answer the following

1 (a) Of which sets is 6 an element? (b) Of which sets is {6} an element?

2 Which of the following are true?

3 Which pair of A, B, and C is disjoint?

4 Which of A, B, and C are subsets of

(a) the set P of all prime numbers? (b) the set M of all multiples of 2?

5 Which of A, B, and C is equal to

for natural numbers 1 n 56?

9

11 7 5

3 4

Figure 0.4

Trang 23

Set Union

Set Complement

The union of two sets is the set that contains all members of the two sets.

The union of A and B, written is deﬁned by

*

We can illustrate the intersection and union of two sets by the use of Venn diagrams

Figures 0.5 and 0.6 show Venn diagrams with universal set U represented by the rectangles and with sets A and B represented by the circles The shaded region in Figure 0.5 repre-

sents the intersection of A and B, and the shaded region in Figure 0.6—which sists of all parts of both circles—represents A ´ B.

Solution

All elements of the universal set that are not contained in a set A form a set called the

complement of A.

The complement of A, written is deﬁned by

We can use a Venn diagram to illustrate the complement of a set The shaded region

of Figure 0.7 represents A¿, and the unshaded region of Figure 0.5 represents (A B)¿.

A¿ 5x: x U and x A6 A¿,

X 5a, b, c, f 6 and Y 5e, f, a, b6

*In mathematics, the word or means “one or the other or both.”

Trang 24

■ their closing price on the previous day was less than $50/share (set C)

■ their price-to-earnings ratio was less than 20 (set P)

■ their dividend per share was at least $1.50 (set D)

Of these 23 stocks,

16 belonged to set P 10 belonged to both C and P

12 belonged to set C 7 belonged to both D and P

8 belonged to set D 2 belonged to all three sets

3 belonged to both C and D

Draw a Venn diagram that represents this information Use the diagram to answer thefollowing

(a) How many stocks had closing prices of less than $50 per share or price-to-earningsratios of less than 20?

(b) How many stocks had none of the characteristics of set C, P, or D?

(c) How many stocks had only dividends per share of at least $1.50?

Trang 25

we can next use the information about stocks that belonged to two of the sets (see Figure0.8(c)) Finally, we can complete the Venn diagram (see Figure 0.8(d)).

(a) We need to add the numbers in the separate regions that lie within That is, 18stocks closed under $50 per share or had price-to-earnings ratios of less than 20

(b) There are 5 stocks outside the three sets C, D, and P.

(c) Those stocks that had only dividends of at least $1.50 per share are inside D but outside both C and P There are no such stocks.

C ´ P.

● Checkpoint Solutions

U P

D

C

2

U P

D C

U P

D

C

2 0

1 1 5 5

U P

D

C

2 8 1 5

1 (a) Sets B and C have 6 as an element.

(b) None of A, B, or C has {6} as an element; {6} is itself a set, and the elements of

A, B, and C are not sets.

2 (a) True (b) True (c) False; (d) False; (e) True

3 A and C are disjoint.

element of the given set in the following problems.

3

4 5 {x: x is a natural number greater than 5}

5 6 {x: x is a natural number less than 6}

6 3

12 51, 2, 3, 4, 6 3 51, 2, 3, 4, 5, 66

x 5x, y, z, a6

7 {x: x is a natural number less than 8}

8 {x: x is a natural number greater than 6, less than 10}

Trang 26

0.1 Sets ● 7

other in the following problems.

15

16

17

18

In Problems 19–22, indicate whether the following pairs

of sets are equal.

19

20

21

22

23 From the following list of sets, indicate which pairs of

sets are disjoint

24 If A and B are disjoint sets, what does equal?

In Problems 33–44, assume that

and that U is the universal set of natural numbers less

than 11 Find the following.

containing all elements of A except those in B That is,

centage changes for the years 2000 to 2006 Let L be

the set of years where the low was greater than 8000

Let H be the set of years where the high was greater than 11,000 Let C be the years when the percentage

change (from low to high) exceeded 35%

(a) List the elements of L, H, and C.

(b) Is any of L, H, or C a subset of one of the others

Dow Jones Industrial Average

Source: Dow Jones & Company, 2007

50 Job growth The number of jobs in 2000, the numberprojected in 2025, and the projected annual growth ratefor jobs in some cities are shown in the table on thenext page Consider the following sets

set of cities with at least 2,000,000 jobs in 2000

or in 2025set of cities with at least 1,500,000 jobs in 2000set of cities with projected annual growth rate of

at least 2.5%

C

B

A

Trang 27

(a) List A, B, and C (using the letters to represent the

cities)

(b) Is any of A, B, or C a subset of the other?

(c) Find and describe the set in words

(d) Give a verbal description of

Projected Jobs Annual Jobs in 2000 in 2025 Rates of Cities (thousands) (thousands) Increase (%)

Source: NPA Data Services, Inc.

National health care Suppose that the table below

summarizes the opinions of various groups on the issue

of national health care Use this table for Problems 51

(a) Republicans and those who favor national health care

(b) Republicans or those who favor national health care

(c) White Republicans or those who oppose national

health care

52 Identify the number of individuals in each of the

fol-lowing sets

(a) Whites and those who oppose national health care

(b) Whites or those who oppose national health care

(c) Nonwhite Democrats and those who favor national

health care

53 Languages A survey of 100 aides at the United

Nations revealed that 65 could speak English, 60 could

speak French, and 40 could speak both English and

French

(a) Draw a Venn diagram representing the 100 aides

Use E to represent English-speaking aides and F to

represent French-speaking aides

(b) How many aides are in

(c) How many aides are in

(d) How many aides are in E E ´ F? F¿?

E F?

B¿

A C

54 Advertising Suppose that a survey of 100 advertisers in

U.S News, These Times, and World found the following.

14 advertised in all three

30 advertised in These Times and U.S News

26 advertised in World and U.S News

27 advertised in World and These Times

60 advertised in These Times

52 advertised in U.S News

50 advertised in World

Draw a Venn diagram representing this information anduse it to answer the following

(a) How many advertised in none of these publications?

(b) How many advertised only in These Times? (c) How many advertised in U.S News or These Times?

55 College enrollments Records at a small college showthe following about the enrollments of 100 ﬁrst-yearstudents in mathematics, ﬁne arts, and economics

38 take math

42 take ﬁne arts

20 take economics

4 take economics and ﬁne arts

15 take math and economics

9 take math and ﬁne arts

12 take math and economics but not ﬁne artsDraw a Venn diagram representing this information andlabel all the areas Use this diagram to answer the fol-lowing

(a) How many take none of these three courses?(b) How many take math or economics?

(c) How many take exactly one of these three courses?

56 Survey analysis In a survey of the dining preferences of

110 dormitory students at the end of the spring semester,the following facts were discovered about Adam’s Lunch(AL), Pizza Tower (PT), and the Dining Hall (DH)

30 liked AL but not PT

(b) How many liked all three?

(c) How many liked only DH?

Trang 28

0.2 The Real Numbers ● 9

57 Blood types Blood types are determined by the

pres-ence or abspres-ence of three antigens: A antigen, B antigen,

and an antigen called the Rh factor The resulting blood

types are classiﬁed as follows:

type A if the A antigen is present

type B if the B antigen is present

type AB if both the A and B antigens are present

type O if neither the A nor the B antigen is present

These types are further classiﬁed as Rh-positive if the Rh-factor antigen is present and Rh-negative otherwise.

(a) Draw a Venn diagram that illustrates this tion scheme

classiﬁca-(b) Identify the blood type determined by each region

of the Venn diagram (such as to indicate type

A, Rh-positive)

(c) Use a library or another source to ﬁnd what centage of the U.S population has each blood type

per-A

The Real Numbers

This text uses the set of real numbers as the universal set We can represent the real bers along a line called the real number line This number line is a picture, or graph, of

the real numbers Each point on the real number line corresponds to exactly one real ber, and each real number can be located at exactly one point on the real number line Thus,two real numbers are said to be equal whenever they are represented by the same point onthe real number line The equation (a equals b) means that the symbols a and b rep-

num-resent the same real number Thus, means that and 7 represent the samenumber Table 0.1 lists special subsets of the real numbers

3 4

3 4 7

a b

0.2

TABLE 0.1 Subsets of the Set of Real Numbers

Natural numbers {1, 2, 3, } The counting numbers.

and the negatives of the natural numbers.

Rational numbers All numbers that can be written as the ratio of

two integers, These numbers have decimal representations that either terminate or repeat.

Irrational numbers Those real numbers that cannot be written as

the ratio of two integers Irrational numbers have decimal representations that neither terminate nor repeat.

Real numbers The set containing all rational and irrational numbers

(the entire number line).

a b, with b 0.

3 2 1 0 –1 –2 –3

3 2 1 0 –1 –2

π 2

0 –1

5 4 3 2 1 0 –1 –2 –3

The properties of the real numbers are fundamental to the study of algebra These erties follow

prop-Let a, b, and c denote real numbers.

1 Addition and multiplication are commutative

a b b a ab ba

Properties of the

Real Numbers

Trang 29

2 Addition and multiplication are associative.

3 The additive identity is 0

4 The multiplicative identity is 1

5 Each element a has an additive inverse, denoted by

Note that there is a difference between a negative number and the negative of anumber

6 Each nonzero element a has a multiplicative inverse, denoted by

Note that

7 Multiplication is distributive over addition

Note that Property 5 provides the means to subtract by deﬁning andProperty 6 provides a means to divide by deﬁning The number 0 has

no multiplicative inverse, so division by 0 is undeﬁned

We say that a is less than b (written ) if the point representing a is to the left of the point representing b on the real number line For example, because 4 is to the left

of 7 on the real number line We may also say that 7 is greater than 4 (written ) We

may indicate that the number x is less than or equal to another number y by writing

We may also indicate that p is greater than or equal to 4 by writing

(d) “x is at most 12” means it must be less than or equal to 12 Thus,

The subset of the real numbers consisting of all real numbers x that lie between a and

b, excluding a and b, can be denoted by the double inequality or by the open

interval (a, b) It is called an open interval because neither of the endpoints is included in

the interval The closed interval [a, b] represents the set of all real numbers x satisfying

Intervals containing one endpoint, such as (a, b] and [a, b), are called half-open

intervals.

We can use to represent the inequality and to represent

In each of these cases, the symbols and are not real numbers but represent the fact

that x increases without bound or decreases without bound Table 0.2 rizes three types of intervals

summa-(()

x a.

(

x a 3a, )

a 1 1 a a

a 0 0 a a (a b) c a (b c) (ab)c a(bc)

Inequalities and

Intervals

Trang 30

Absolute Value

1 Evaluate the following, if possible For any that are meaningless, so state

2 For (a)–(d), write the inequality corresponding to the given interval and sketch its graph

on a real number line

num-● EXAMPLE 2 Absolute Value

Evaluate the following

44

04

40

TABLE 0.2 Intervals

Type of Interval Inequality Notation Interval Notation Graph

( , b) (a, b)

( , b]

[a, b) (a, b]

b a

● Checkpoint

Absolute Value

Trang 31

When two or more operations with real numbers are indicated in an evaluation, it isimportant that everyone agree upon the order in which the operations are performed so that

a unique result is guaranteed The following order of operations is universally accepted.

Operations with Real (Signed) Numbers

1 (a) To add two real numbers with the same sign,

add their absolute values and afﬁx their common

sign

(b) To add two real numbers with unlike signs, ﬁnd

the difference of their absolute values and afﬁx

the sign of the number with the larger absolute

value

2 To subtract one real number from another, change

the sign of the number being subtracted and proceed

36>4 9(

(5(

a3

4b(4) 3(

16(

117

47

((

16

26

36

12(5) (6) 11

Order of Operations

1 Perform operations within parentheses

2 Find indicated powers

3 Perform multiplications and divisions from left to right

4 Perform additions and subtractions from left to right

(23 2 2 2 8)

● EXAMPLE 3 Order of Operations

Evaluate the following

Solution

(a)(b) Note that with the power 2 is applied only to 4, not to (which would bewritten ) Thus

2 3

Trang 32

1 (a) Meaningless A denominator of zero means division by zero, which is undeﬁned.(b) A numerator of zero (when the denominator is not zero) means the fractionhas value 0

(c)(d) Meaningless The denominator is zero

2 (a)(b)(c)(d)

3 (a) closed interval(b) half-open interval

4 False Exponentiation has priority and appliesonly to

In Problems 1–2, indicate whether the given expression

is one or more of the following types of numbers:

rational, irrational, integer, natural If the expression is

8 6 6 8

96

0

6

40

9310

8

(4 2)22

4 222

3

p ⵧ 3.14

ⵧ

Trang 33

21 22.

25 What part of the real number line corresponds to the

interval

26 Write the interval corresponding to

In Problems 27–30, express each inequality or graph

using interval notation, and name the type of interval.

27

28

29

30

In Problems 31–34, write an inequality that describes

each interval or graph.

31

32

33

34

In Problems 35–42, graph the subset of the real numbers

that is represented by each of the following and write

your answer in interval notation.

In Problems 43–48, use your calculator to evaluate each

of the following List all the digits on your display in the

49 Take-home pay A sales representative’s take-home

pay is found by subtracting all taxes and retirement

con-tributions from gross pay (which consists of salary plus

commission) Given the following information,

com-plete (a)–(c)

Retirement 5% of gross pay

Commission $788.91Salary $300.00

51.412127.0125916.8

x x

(

x x

10 8 6 4

(a) Find the gross pay

(b) Find the amount of federal withholding

(c) Find the take-home pay

50 Public health expenditures The expenditures E for

government public health activities (in billions of dollars)can be approximated by

where t is the number of years past 1960 (Source:

Cen-ters for Medicare and Medicaid Services)

(a) What t-value represents the year 1995?

(b) Actual expenditures for 1995 were $31.0 billion.What does the formula give as the 1995 approximation?(c) Predict the expenditures for 2012

51 Health insurance coverage The percentage P of the

U.S population with no health insurance can be imated quite accurately either by

(2)

where t is the number of years past 2000 (Source: U.S.

Census Bureau)(a) Both (1) and (2) closely approximate the data, butwhich is more accurate for 2006, when 15.8% ofthe population had no health insurance?

(b) Use both formulas to predict the percentage of theU.S population not covered in 2012 Which equa-tion’s result seems more accurate?

52 Health statistics From data adapted from the National

Center for Health Statistics, the height H in inches and age A in years for boys between 4 and 16 years of age

are related according to

To account for normal variability among boys, normalheight for a given age is of the height obtainedfrom the equation

(a) Find the normal height range for a boy who is10.5 years old, and write it as an inequality.(b) Find the normal height range for a boy who is5.75 years old, and write it as an inequality

Local 1% of gross payState 5% of gross pay,

Trang 34

Positive Integer Exponents

0.3 Integral Exponents ● 15

53 Income taxes The 2007 tax brackets for a single

per-son claiming one perper-sonal exemption are given in the

Source: Internal Revenue Service

(a) Write the last three taxable income ranges asinequalities

(b) If an individual has a taxable income of $31,850,calculate the tax due Repeat this calculation for ataxable income of $77,100

(c) Write an interval that represents the amount of taxdue for a taxable income between $31,850 and

$77,100

Integral Exponents

If $1000 is placed in a 5-year savings certiﬁcate that pays an interest rate of per year,compounded annually, then the amount returned after 5 years is given by

The 5 in this expression is an exponent Exponents provide an easier way to denote

cer-tain multiplications For example,

An understanding of the properties of exponents is fundamental to the algebra needed tostudy functions and solve equations Furthermore, the deﬁnition of exponential and loga-rithmic functions and many of the techniques in calculus also require an understanding ofthe properties of exponents

For any real number a,

for any positive integer n The positive integer n is called the exponent, the number a is

called the base, and is read “a to the nth power.”

Note that means which is different from The 4 is the coefﬁcient of

in Note also that is not equivalent to when n is even For example,

but Some of the rules of exponents follow

For any real numbers a and b and positive integers m and n,

1

2 For 3

Trang 35

Negative Exponents

Zero Exponent

● EXAMPLE 1 Positive Integer Exponents

Use properties of positive integer exponents to rewrite each of the following Assume alldenominators are nonzero

For any nonzero real number a, we deﬁne We leave undeﬁned

In Section 0.2, we deﬁned as for so we deﬁne as

● EXAMPLE 2 Negative and Zero Exponents

Write each of the following without exponents

6 30 6 1 6

(2

3b

a1

3b6

Trang 36

Throughout the remainder of the text, we will assume all expressions are deﬁned.

● EXAMPLE 3 Operations with Exponents

Use the rules of exponents and the deﬁnitions of and to simplify each of the lowing with positive exponents

● EXAMPLE 4 Rewriting a Quotient

Write with all factors in the numerator

Solution

● EXAMPLE 5 Rewriting with Positive Exponents

Simplify the following so all exponents are positive

4(x2y)0

(4x y)2(23x y5)

x x 2(x2)

Trang 37

(a)(b)

1

040

In Problems 9–16, use rules of exponents to simplify the

expressions Express answers with positive exponents.

In Problems 21–34, use the rules of exponents to simplify

so that only positive exponents remain.

3

( 5 6

3( 4

2x 2x2

(2x) (2x)2

2x (2x)2

Trang 38

Compound interest If $P is invested for n years at rate i

(as a decimal), compounded annually, the future value that

accrues is given by and the interest earned

is In Problems 59–62, ﬁnd S and I for the given

Present value If an investment has a goal (future value)

of $S after n years, invested at interest rate i (as a decimal),

compounded annually, then the present value P that must be

invested is given by In Problems 63 and

64, ﬁnd P for the given S, n, and i.

63 $15,000 after 6 years at 11.5%

64 $80,000 after 20 years at 10.5%

65 Personal income For selected years from 1960 to

2005, billions of dollars of total U.S personal income

I can be approximated by the formula

Department of Commerce)

(a) What t-values correspond to the years 1970, 1990,

and 2002?

(b) The actual total personal incomes (in billions of

dollars) for the years in part (a) were as follows

838.8 4878.6 8881.9

What does the formula predict for these years?

(c) What does the formula predict for the total personal

income in 2012?

(d) Does this formula seem to indicate that total

per-sonal income doubles almost every 10 years?

66 Stock shares traded On the New York Stock

Exchange (NYSE) for 1970–2006, the average daily

shares traded S (in millions of shares) can be

approxi-mated by the formula

where t is the number of years past 1950 (Source: New

York Stock Exchange)

( 31.24

67 Endangered species The total number of endangered

species y can be approximated by the formula

Fish and Wildlife Service)(a) The actual numbers of endangered species forselected years were as follows

For each of these years, ﬁnd the number of gered species predicted by the formula Round youranswer to the nearest integer

endan-(b) How many more species does the formula estimatewill be added to the endangered list for 2020 thanthe actual number given for 2007?

(c) Why do you think the answer to (b) is smaller thanthe number of species added from 1990 to 2003?(d) Why is it reasonable for a formula such as this tohave an upper limit that cannot be exceeded? Use

large t-values in the formula to discover this

for-mula’s upper limit

68 Internet users The percent p of U.S households with

Internet service can be approximated by the equation

Department of Commerce)(a) The percents of U.S households with Internet ser-vice for selected years were as follows

For each of these years, use the equation to ﬁndthe predicted percent of households with Internetservice

p 73.92

1 5.441(1.515)

1 7.892(1.097)

Trang 39

nth Root of a

(b) In the three years from 2001 to 2004, the percent of

households with Internet service increased by 18.8%

What percent increase does the equation predict from

2008 to 2011? Why do you think the 2008–2011

change is so different from the 2001–2004 change?

(c) Why is it reasonable for a formula such as this to

have an upper limit that cannot be exceeded? Use

large t-values in the formula to discover this

formula’s upper limit

69 Health care expenditures The national health care

expenditure H (in billions of dollars) can be modeled

(that is, accurately approximated) by the formula

Department of Health and Human Services)

(a) What t-value corresponds to 1970?

(b) Approximate the national health care expenditure in1970

(c) Approximate the national health care expenditure in2005

(d) Estimate the national health care expenditure in 2015

H 30.58(1.102)t

Radicals and Rational Exponents

A process closely linked to that of raising numbers to powers is that of extracting roots

From geometry we know that if an edge of a cube has a length of x units, its volume is cubic units Reversing this process, we determine that if the volume of a cube is V cubic units, the length of an edge is the cube root of V, which is denoted

When we seek the cube root of a number such as 8 (written ), we are looking for areal number whose cube equals 8 Because we know that Similarly,

because The expression is called a radical, where is

the radical sign, n the index, and a the radicand When no index is indicated, the index

is assumed to be 2 and the expression is called a square root; thus is the square root

of 4 and represents the positive number whose square is 4

Only one real number satisﬁes for a real number a and an odd number n; we call

that number the principal nth root or, more simply, the nth root.

For an even index n, there are two possible cases:

1 If a is negative, there is no real number equal to For example, there are no realnumbers that equal or because there is no real number b such that

or In this case, we say is not a real number

2 If a is positive, there are two real numbers whose nth power equals a For example,

and In order to have a unique nth root, we deﬁne the (principal)

nth root, as the positive number b that satisﬁes

We summarize this discussion as follows

The (principal) nth root of a real number is deﬁned as

subject to the following conditions:

Trang 40

(d) is not a real number because an even root of a negative number is not real.

In order to perform evaluations on a calculator or to perform calculus operations, it is times necessary to rewrite radicals in exponential form with fractional exponents

some-We have stated that for and

This means that or In order to extend the properties ofexponents to rational exponents, it is necessary to deﬁne

For a positive integer n, we deﬁne

2 if a is nonnegative when n is even.

Throughout the remaining discussion, we assume all expressions are real

● EXAMPLE 2 Radical Form

Write the following in radical form and simplify

Solution

(a)(b)

Fractional Exponents

Định dạng
Số trang	1.109
Dung lượng	11,64 MB