Mathematical applications for the management life and social sciences 10th reynolds 1

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Business and Management

Accounting, 209, 520

Advertising, 9, 28, 46, 77, 94, 172–173, 267, 296, 337, 361, 365, 458, 484,

520, 555, 565, 597, 604, 611, 616, 644–645, 649, 662, 681, 725, 763,

773, 817, 826, 915, 918 Agribusiness, 237–239, 263–264, 280, 368, 889

Cost-benefit, 50, 178, 549–550, 555, 566, 590, 597, 691

Customer service, 818, 853

Diminishing returns, 661, 662, 695

Egg production, 207

Employee age and gender, 438–439, 447, 448

Employee benefits and compensation, 87, 256, 378, 475, 497, 905

Organizational growth, 355–356, 361 Package design, 35, 39

Parking costs, 555 Parts delivery, 448 Parts manufacturing, 244–245 Population and labor force, 187–188 Postal restrictions and rates, 77, 163, 164–165, 567, 677–678 Pricing, 83, 105, 207, 278, 323–324, 338, 509, 604, 649, 681, 699, 789, 919 Printing, 696, 919

Product design, 696 Product reliability, 336, 359, 436, 497, 508, 533, 815, 818, 850–851, 853 Production, 197–198, 208–209, 251, 273–274, 305, 314, 318–319, 611,

649, 662, 721, 725, 731, 763, 799–800, 818, 861, 875, 877–878, 889–890, 904, 910, 912–913, 915

Production costs, 165, 649, 681, 687, 698 Productivity, 487, 555, 565, 630, 649, 661, 680, 691, 695, 696, 763, 793, 795 Profit, 34, 50, 58–59, 63, 64, 76, 77, 106–107, 119–120, 121, 134, 135,

144, 256, 360, 378, 391, 554, 580, 630, 631, 691, 730–731, 780–781,

793, 826, 878 Profit maximization, 148–149, 177, 268–269, 279, 307, 319, 668–670, 672–674, 680, 695, 776–778, 899, 903, 904, 915

Property and real estate, 445, 508, 679–680, 766–767 Purchasing, 64

Purchasing electrical power, 869 Quality control, 436, 440, 457, 458, 460, 470, 473, 475, 476, 487, 495, 497,

498, 520, 529, 536, 853, 865, 866 Rentals, 34, 64, 105, 145, 221, 297, 519–520 Return to sales, 633, 881

Revenue, 34, 40, 50, 77, 106–107, 121, 145, 207, 297, 355, 360, 533, 536,

554, 555, 579, 586–587, 589, 597, 604, 608–609, 610, 616, 630, 631,

649, 712, 734–735, 777–778, 792, 793, 828, 839, 866, 913 Revenue maximization, 141–143, 177, 271, 619, 665–666, 671, 681, 682,

691, 694, 695, 737–738, 742 Rewards for employees, 475 Safety, 440, 731, 741–742 Salaries, 15, 64, 378, 422, 447, 509 Sales, 28, 46, 196, 337, 365, 438, 471, 475, 509, 520, 533, 554–555, 565,

597, 604, 611, 616, 649, 662, 680, 681, 715, 725, 773, 789, 817,

826, 915 Sales decay, 353, 359, 365 Sales growth, 360, 361 Scheduling, 277, 295, 296, 304–305, 311–312, 315, 487 Shadow prices, 278, 323–324

Shipping, 278, 296, 306–307, 322 Starbucks stores, 367, 616–617 Surplus, 865

Telecommunications, 490–491, 836, 839 Ticket sales, 221

Total income, 817, 826–827 Training, 554

Transportation, 28, 222, 236, 252, 322, 440 Unemployment rates, 507

Index of Selected Applications

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Unions, 146, 197, 556, 650–651

Utilities, 164, 209, 239, 245–246, 248, 487, 876, 886

Warranties, 853

Wilson’s lot size formula, 876

Wireless service spending, 64

Workers and output, 589, 610

Consumer (personal) spending, 39, 87, 94, 104, 337, 580, 715, 877

Consumer price index, 334, 338, 360, 663, 716–717, 755, 773

Dow Jones averages, 8–9, 66, 67, 75, 556, 674–675

Earnings and race, 87

Economic dependency ratio, 617, 650

Economic order quantity, 698

Economy models, 252

Elasticity of demand, 732–734, 738–739

Gasoline mileage and prices

Gini coefficient of income, 819, 827

Gompertz curves, 355–356, 360–361, 790

Gross domestic product, 164, 173, 591, 605, 790

Gross national product, 663

Annuities due, 397, 406, 410–412 APY, 384–385

ATM transaction, 475 Banks, 84, 279, 471, 475 Bond pricing, 405–406, 412 Bonds, 510

Budgeting, 520 Capital value, 849, 853 College fund, 422, 423, 508, 509 Compound interest, 20, 336, 338, 360, 365, 383–384, 385, 386, 389–390,

741, 772, 788–789 Consumer credit and credit cards, 63, 364, 427, 477

Debt refinancing and payment, 75, 196–197 See also Loans

Deferred annuities, 407–408, 412–413 Delinquent accounts, 452–453 Depreciation, 63, 79, 86, 87, 391, 817 Doubling time, 349–350, 385–386 Earnings, 94, 196, 605, 649, 905 Future value of annuities, 392–401 Future value of income stream, 829–830 Future value of investments, 329–330, 372, 379–381, 383–384, 423, 712, 714–715

Income stream, 815, 829–830 Investing, 20, 28, 35, 59, 64, 98–99, 105, 120, 218–219, 221–222, 223,

236, 251, 297–298, 327, 329–330, 340–341, 372–373, 384, 393–394,

395, 423, 427–428, 589, 604, 695, 708–709, 731, 789, 793, 876 Loans, 50, 105, 221, 413–422, 423

Mortgages and home loans, 19, 69–70, 74–75, 414–415, 427, 428, 876–877 Mutual funds, 475

Perpetuities, 565–566, 849, 853 Present value, 372, 381 Present value of annuities, 401–404, 408 Property and real estate, 445, 508, 679–680, 766–767 Purchasing power, 716

Retirement planning, 87, 475, 868, 905, 915 Savings, 779, 781, 865, 914–915

Simple interest, 39, 50, 371–373, 376–377 Sinking funds, 396, 423

Stock market, 6–7, 20–21, 338, 360, 487, 522 Taxes, 15, 16, 63, 64, 66, 67, 69, 94, 165–166, 168–169, 174, 255–256,

555, 559–560, 566, 736–738, 739–740, 742, 915–916 Trusts, 411, 412, 425, 847, 849

Venture capital, 204–205, 231–232

Life Science

Adrenalin response, 566, 598 Age-sleep relationship, 87 AIDS cases, 177, 391, 447 Allometric relationships, 164, 590, 601, 731, 788, 793 Atmospheric pressure, 790

Bacterial growth, 105, 237, 327, 336, 391, 789, 793 Bee ancestry and reproduction, 237, 377

Biology, 533 Birth control, 458, 520, 540 Birth defects, 494, 496 Birth weights, 509 Blood flow, 726, 727–728, 731 Blood pressure, 529, 540, 715–716, 773

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Mathematical

Applications

for the Management, Life,

and Social Sciences

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Mathematical

Applications

and Social Sciences

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ISBN-13: 978-1-133-10623-4 ISBN-10: 1-133-10623-4 Instructor’s Edition:

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Mathematical Applications for the

Management, Life, and Social Sciences,

Tenth Edition

Ronald J Harshbarger

James J Reynolds

Publisher: Richard Stratton

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1 2 3 4 5 6 7 15 14 13 12 11

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0.7 Algebraic Fractions 40

Key Terms and Formulas 46 Review Exercises 48 Chapter Test 50

Extended Applications & Group Projects 52

Campaign Management

1 Linear Equations and Functions 53

Warm-up 54 1.1 Solutions of Linear Equations and Inequalities in One Variable 551.2 Functions 65

1.3 Linear Functions 781.4 Graphs and Graphing Utilities 881.5 Solutions of Systems of Linear Equations 95

1.6 Applications of Functions in Business and Economics 106

Hospital Administration • Fundraising

2 Quadratic and Other Special Functions 125

Warm-up 1262.1 Quadratic Equations 1272.2 Quadratic Functions: Parabolas 1362.3 Business Applications Using Quadratics 146

2.4 Special Functions and Their Graphs 1532.5 Modeling; Fitting Curves to Data with Graphing Utilities (optional) 165

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

3 Matrices 184

Warm-up 1853.1 Matrices 1863.2 Multiplication of Matrices 1993.3 Gauss-Jordan Elimination: Solving Systems

of Equations 2093.4 Inverse of a Square Matrix; Matrix Equations 224

3.5 Applications of Matrices: Leontief Output Models 237

Input-Key Terms and Formulas 249 Review Exercises 250 Chapter Test 252

Taxation • Company Profits after Bonuses and Taxes

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vi contents

4 Inequalities and Linear Programming 257

Warm-up 2584.1 Linear Inequalities in Two Variables 259

4.2 Linear Programming: Graphical

Methods 2684.3 The Simplex Method: Maximization 279

4.4 The Simplex Method: Duality and

Minimization 2984.5 The Simplex Method with Mixed

Constraints 307

Transportation • Slack Variables and Shadow Prices

5 Exponential and Logarithmic Functions 325

Warm-up 3265.1 Exponential Functions 327

5.2 Logarithmic Functions and Their

Properties 3395.3 Solutions of Exponential Equations:

Applications of Exponential and Logarithmic Functions 351

Starbucks Stores (Modeling) • Agricultural Business Management

6 Mathematics of Finance 369

Warm-up 3706.1 Simple Interest; Sequences 371

6.2 Compound Interest; Geometric

Sequences 3786.3 Future Values of Annuities 392

6.4 Present Values of Annuities 401

6.5 Loans and Amortization 413

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

7 Introduction to Probability 429

Warm-up 4307.1 Probability; Odds 4317.2 Unions and Intersections of Events: One-Trial Experiments 441

7.3 Conditional Probability: The Product Rule 449

7.4 Probability Trees and Bayes’ Formula 4597.5 Counting: Permutations and

Combinations 4667.6 Permutations, Combinations, and Probability 472

7.7 Markov Chains 476

Phone Numbers • Competition in the Telecommunications Industry

8 Further Topics in Probability; Data Description 492

Warm-up 4938.1 Binomial Probability Experiments 4948.2 Data Description 498

8.3 Discrete Probability Distributions; The Binomial Distribution 511

8.4 Normal Probability Distribution 5218.5 The Normal Curve Approximation to the Binomial Distribution 529

Lotteries • Statistics in Medical Research;

Hypothesis Testing

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Inflation • Knowledge Workers (Modeling)

12 Indefinite Integrals 746

Warm-up 747 12.1 Indefinite Integrals 748 12.2 The Power Rule 755 12.3 Integrals Involving Exponential and

Logarithmic Functions 765 12.4 Applications of the Indefinite Integral in

Business and Economics 774 12.5 Differential Equations 782

Employee Production Rate • Supply and Demand

13 Definite Integrals: Techniques

of Integration 797

Warm-up 798 13.1 Area Under a Curve 799 13.2 The Definite Integral: The Fundamental

Theorem of Calculus 809 13.3 Area Between Two Curves 819 13.4 Applications of Definite Integrals in

Business and Economics 827 13.5 Using Tables of Integrals 836 13.6 Integration by Parts 841 13.7 Improper Integrals and Their

Applications 847 13.8 Numerical Integration Methods: The

Trapezoidal Rule and Simpson’s Rule 854

Retirement Planning • Purchasing Electrical Power (Modeling)

9 Derivatives 541

Warm-up 5429.1 Limits 5439.2 Continuous Functions; Limits at Infinity 556

9.3 Rates of Change and Derivatives 5689.4 Derivative Formulas 581

9.5 The Product Rule and the Quotient Rule 591

9.6 The Chain Rule and the Power Rule 5999.7 Using Derivative Formulas 606

9.8 Higher-Order Derivatives 6119.9 Applications: Marginals and Derivatives 618

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

10 Applications of Derivatives 636

Warm-up 637 10.1 Relative Maxima and Minima: Curve

Sketching 638 10.2 Concavity; Points of Inflection 651

10.3 Optimization in Business and

Economics 664 10.4 Applications of Maxima and Minima 675

10.5 Rational Functions: More Curve

Sketching 683

Production Management • Room Pricing in the Off-Season (Modeling)

11 Derivatives Continued 700

Warm-up 701 11.1 Derivatives of Logarithmic Functions 702

11.2 Derivatives of Exponential Functions 710

11.3 Implicit Differentiation 717

11.4 Related Rates 726

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14.5 Maxima and Minima of Functions Subject

to Constraints: Lagrange Multipliers 906

Advertising • Competitive Pricing

Appendix A Financial Tables AP-1

Table 1 Future Value of an Ordinary Annuity of $1 AP-1

Table 2 Present Value of an Ordinary Annuity of $1 AP-3

Appendix B Areas Under the Standard Normal

Curve AP-5 Appendix C Graphing Calculator Guide AP-7 Appendix D Excel Guide AP-33

Part 2 Excel 2007 and 2010 AP-51

Answers A-1 Index I-1

www.freebookslides.com

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ix

Preface

To paraphrase English mathematician, philosopher, and educator Alfred North Whitehead,

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 is not 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 mathematical skills and

con-cepts, and to apply them to ideas that are important to students in the management, life,

and social sciences We hope that this look at the relevance of mathematical ideas to a

broad range of fields will help inspire the imaginative thinking and excitement for learning

that Whitehead spoke of The applications included allow students to view mathematics

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

sec-tion contains a number of applicasec-tion examples and problems An index of these

applica-tions 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

edi-tions have been retained They are as follows

concepts and problem solving rather than on mathematical theory Yet each topic is

care-fully developed and explained, and examples illustrate the techniques involved

may vary according to the purpose of the course and the nature of the student audience

To accommodate alternate approaches, the text has a great deal of flexibility in the order in

which topics may be presented and the degree to which they may be emphasized

exercises helps provide motivation within the sections and demonstrates the relevance of

each topic Numerous real-life application examples and exercises represent the

applica-bility of the mathematics, and each application problem is identified, so the instructor or

student can select applications that are of special interest In addition, we have found that

offering separate lessons on applied topics such as cost, revenue, and profit functions brings

the preceding mathematical discussions into clear, concise focus and provides a thread

of continuity as mathematical sophistication increases There are ten such sections in the

book, and entire chapters devoted to linear programming and financial applications Of the

more than 5500 exercises in the book, over 2000 are applied

begin-ning of each chapter and invites students to test themselves on the skills needed for that

chapter The Warm-ups present several prerequisite problem types that are keyed to the

appropriate sections in the upcoming chapter where those skills are needed Students who

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x PrefAce

have difficulty 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 an appropriate scope and sequence of topics

the context and direction for the concepts that will be presented Each of these Previews contains 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

exercises offer problems for different skill levels, and there are enough challenging problems

to stimulate students in thoughtful investigations Many exercise sets contain ing and thought-provoking multistep problems that extend students’ knowledge and skills

with at least two case studies, which further illustrate how mathematics can be used in business and personal decision making In addition, many applications are cumulative in that solutions require students to combine the mathematical concepts and techniques they learned in some of the preceding chapters

applications 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 modeled from real data, and some problems ask students to model functions from the data given

These problems are solved by using one or more graphical, numerical, or symbolic methods

Calculator Notes, and Spreadsheet Notes, denoted by the icon, are scattered out the text Many of these notes reference detailed step-by-step instructions in the new Appendix C (Graphing Calculator Guide) and Appendix D (Excel Guide) and in the Online Guide for Excel Discussions of the use of technology are placed in subsections and exam-ples in many sections, so that they can be emphasized or de-emphasized at the option of the instructor

through-The discussions of graphing calculator technology highlight its most common features and uses, such as graphing, window setting, Trace, Zoom, Solver, tables, finding points of intersection, numerical derivatives, numerical integration, matrices, solving inequalities, and modeling (curve fitting) While technology never replaces the mathematics, it does supplement and extend the mathematics by providing opportunities for generalization and alternative ways of understanding, doing, and checking Some exercises that are better worked with the use of technology—including graphing calculators and Excel—are high-lighted with the technology icon Of course, many additional exercises can benefit from the use of technology, at the option of the instructor Technology can be used to graph func-tions and to discuss the generalizations, applications, and implications of problems being studied

Excel is useful in solving problems involving linear equations; systems of equations;

quadratic equations; matrices; linear programming; output comparisons of f(x), f 9(x), and

f 0(x); and maxima and minima of functions subject to constraints Excel is also a useful

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

dis-cussion, allowing students to check their understanding of the skills and concepts under discussion before they proceed Solutions to these Checkpoints appear before the section exercises

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At the end of each chapter, just before the Chapter Review Exer-vides a well-organized core from which a student can build a review, both to consult while working the Review Exercises and to identify quickly any section needing additional study

offers students extra practice on topics in that chapter These Reviews cover each chapter’s topics primarily in their section order, but without section references, so that students get a true review but can readily find a section for further review if difficulties occur A Chapter Test follows each set of Review Exercises All Chapter Tests provide a mixture of problems that 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 Tenth Edition

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

■ Two new Appendices have been added

• tions and examples for operating TI-83 and TI-84 Plus calculators

Appendix C Graphing Calculator Guide, containing detailed step-by-step instruc-• ples for Excel 2003, and for Excel 2007 and 2010

Appendix D Excel Guide, containing detailed step-by-step instructions and exam-■ References to specific calculator and Excel steps in Appendix C and/or Appendix D are given each time a new technology process is introduced

■ The exposition and example discussions have been streamlined to eliminate repetitions and redundancies

■ Section 2.5 “Modeling; Fitting Curves to Data with Graphing Utilities (optional)” has been extensively rewritten to improve logic and flow

■ ganized to clarify the decisions at each step and how the method extends beyond

Section 3.3 “Gauss-Jordan Elimination: Solving Systems of Equations” has been reor-3 3 3 matrices

■ Most of the real-data examples and exercises that are time related have been updated or replaced with current applications

■ Drill Exercises thoughout the text have been revised and reorganized to improve their grading and variety

■ Additional multistep applications have been added to the exercises

■ Many images and illustrations that relate to the mathematics topics have been added

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xii PrefAce

Resources for the Student

Student Solutions Manual (978-1-133-10852-8)

This manual provides complete worked-out solutions to all odd-numbered exercises in the text, giving you a chance to check your answers and ensure you took the correct steps to arrive at an answer

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Resources for the Instructor

Complete Solutions Manual (978-1-133-36435-1)

The Complete Solutions Manual provides worked-out solutions of all exercises in the text

In addition, it contains the solutions of the special features in the text, such as Extended

Applications and Group Projects.

PowerLecture (978-1-133-10920-4)

This comprehensive CD contains all art from the text in both jpeg and PowerPoint formats, key equations and tables from the text, complete pre-built PowerPoint lectures, Solution Builder, and Diploma Computerized Testing

Solution Builder

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Diploma Computerized Testing

Create, deliver, and customize tests in print and online formats with Diploma, an use assessment software containing hundreds of algorithmic questions derived from the text exercises Diploma is available on the PowerLecture CD

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PrefAce xiii

Enhanced WebAssign www.webassign.net

Enhanced WebAssign’s homework delivery system lets you deliver, collect, grade, and

record assignments via the web Enhanced WebAssign for Harshbarger/Reynolds’

Math-ematical Applications for the Management, Life, and Social Sciences includes: Cengage

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Acknowledgments

We would like to thank the many people who helped us at various stages of revising this text The encouragement, criticism, contributions, and suggestions that were offered were invaluable to us Once again we have been fortunate to have Helen Medley’s assistance with accuracy checking of the entire text and answer section during manuscript preparation and

on the page proofs As always, we continue to be impressed by and most appreciative of her work ethic, attention to detail, accuracy, and skill

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

we would like to thankPriscilla Chaffe-Stengel California State University–Fresno

Ronald J Harshbarger James J Reynolds

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06234_fm_hr_i-1.indd 14 9/27/11 4:15 PM

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Mathematical

Applications

and Social Sciences

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through-The topics and applications studied in this chapter include the following.

0.1 Sets Dow Jones average,

0.5 operations with Algebraic expressions revenue, profit

0.6 Factoring Simple interest, revenue

Common factors

0.7 Algebraic Fractions average cost, advertising

Complex fractions

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Section 0.1 Sets 3

Sets

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 writing

A 5 {1, 2, 3, 4} To say that 4 is a member of set A, we write 4 [ A Similarly, we write

5 o A 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

A 5 {1, 2, 3, 4} and B 5 {x, y, z} are examples of finite sets When we do not wish to list all

the elements 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

{8, 10, 12, 14, , 8952}

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

N 5 {1, 2, 3, 4, } 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, we

may write D 5 {x: x is a Ford automobile} to describe the set of all Ford automobiles thermore, F 5 { y: y is an odd natural number} is read “F is the set of all y such that y is an

Fur-odd natural number.”

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) A 5 {1, 2, 3, 4, 5} or A 5 {x: x is a natural number less than 6}

(b) B 5 {11, 12, 13, 14, } or B 5 {x: x is a natural number greater than 10}

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 living veterans of that war Thus

{x: x is a living veteran of the War of 1812} 5 [

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

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

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4 chapter 0 algebraic concepts

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 A d B, is defined by

(a) A d B 5 {3, 5} because 3 and 5 are the common

elements of A and B Figure 0.4 shows the sets

and their intersection

(b) A d B and A are subsets of A ■

Let A 5 {2, 3, 5, 7, 11}, B 5 {2, 4, 6, 8, 10}, and C 5 {6, 10, 14, 18, 22} Use these sets to

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?

(a) 2 [ A (b) 2 [ B (c) 2 [ C (d) 5 o A (e) 5 o B

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

9

11 7 5

3 4

2 A

Figure 0.4

U A

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 the set of all students and the set of all female students, we may use the set of all humans as the universal set

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

rectan-gle to represent the universal set, and we use closed figures inside the rectanrectan-gle to represent the sets under consideration Figures 0.1–0.3 show such Venn diagrams

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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 D 5 {x: x 5 4n 1 2 for natural numbers 1 # n # 5}?

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

The union of A and B, written A ∪ B, is defined by

A ∪ B 5 {x: x ∈ A or x ∈ B (or both)}*

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 represents

A d B, the intersection of A and B, and the shaded region in Figure 0.6—which consists of

all parts of both circles—represents A ∪ B.

Set intersection and Union

Let A 5 {x: x is a natural number less than 6} and B 5 {1, 3, 5, 7, 9, 11}.

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

Figure 0.7 represents A9, and the unshaded region of Figure 0.5 represents ( AdB)′.

exAmple 3

exAmple 4

Set complement

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operations with Sets

If U 5 {x ∈ N: x < 10}, A = {1, 3, 6}, and B 5 {1, 6, 8, 9}, find the following.

(a) A9 (b) B9 (c) (A d B)9 (d) A9 ∪ B9

Given the sets U 5 {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A 5 {1, 3, 5, 7, 9}, B 5 {2, 3, 5, 7}, and

C 5 {4, 5, 6, 7, 8, 9, 10}, find the following.

• 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 the following

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

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to a single region, namely that two stocks belonged to all three sets (see Figure 0.8(b)) Because the region common to all three sets is also common to any pair, we can next use the information about stocks that belonged to two of the sets (see Figure 0.8(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 ChP That is,

18 stocks 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

U P

D C

(a)

U P

D C

(b) 2

U P

D C

(c)

2

8 1 5

U P

D C

(d)

2 0

8 1 1

1 5 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; 2 o C (d) False; 5 [ A

(e) True

3 A and C are disjoint.

4 (a) [ # P and A # P (b) [ # M, B # M, C # M

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8 Chapter 0 algebraic Concepts

Use ∈ or ∉ to indicate whether the given object is an

ele-ment of the given set in the following problems.

In Problems 17–20, indicate whether the following pairs

of sets are equal.

17 A 5 {a, b, p, !3}, B 5 {a, p, !3, b}

and that U is the universal set of natural numbers less

than 11 Find the following.

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Dow Jones Industrial Average

Source: Dow Jones & company, 2010

projected in 2025, and the projected annual growth rate for jobs in some cities are shown in the following table

Source: nPA Data Services, inc.

summarizes the opinions of various groups on the issue

of national health care Use this table for Problems 49

Identify the number of individuals in each of the follow-(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

50 lowing sets

Identify the number of individuals in each of the fol-(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

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 E d F ? (c) How many aides are in EhF ?

(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?

the following about the enrollments of 100 first-year students in mathematics, fine arts, and economics

(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?

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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

Draw a Venn diagram representing this survey and

label all the areas Use this diagram to answer the

following

(a) How many liked PT or DH?

(b) How many liked all three?

(c) How many liked only DH?

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 cation scheme

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

of the Venn diagram (such as A1 to indicate type

Thus, two real numbers are said to be equal whenever they are represented by the same

point on the real number line The equation a 5 b (a equals b) means that the symbols a and b represent the same real number Thus, 3 1 4 5 7 means that 3 1 4 and 7 represent

the same number Table 0.1 lists special subsets of the real numbers

0.2

| tAble 0.1 |

SUbSetS oF the Set oF ReAl nUmbeRS

natural numbers {1, 2, 3, } the counting numbers

integers { , 22, 21, 0, 1, 2, } the natural numbers, 0,

and the negatives of the natural numbers

rational numbers all numbers that can be written as the ratio of

two integers, a∙b, with b ≠ 0 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)

5 4 3 2 1 0 –1 –2 –3

4 3 2 1 0 –1 –2 –3 –4

2 1

0 –1

π 2

3

−

3 2 1 0 –1 –2 –3

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Section 0.2 the real numbers 11

The following properties of the real numbers are fundamental to the study of algebra

Let a, b, and c denote real numbers.

1 (Commutative Property) Addition and multiplication are commutative

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

6 (Multiplicative Inverse) Each nonzero element a has a multiplicative inverse, denoted

by a2 1

Note that a21 5 1∙a.

7 (Distributive Law) Multiplication is distributive over addition

a(b 1 c) 5 ab 1 ac

Note that Property 5 provides the means to subtract by defining a 2 b 5 a 1 (2b) and Property 6 provides a means to divide by defining a 4 b 5 a ? (1∙b) The number 0 has no

multiplicative inverse, so division by 0 is undefined

We say that a is less than b (written a , b) if the point representing a is to the left of the point representing b on the real number line For example, 4 , 7 because 4 is to the left of

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

indicate that the number x is less than or equal to another number y by writing x # y We may also indicate that p is greater than or equal to 4 by writing p $ 4.

inequalities

Use , or notation to write the following

(a) 6 is greater than 5 (b) 10 is less than 15

(c) 3 is to the left of 8 on the real number line (d) x is at most 12.

Solution

(a) 6 5 (b) 10 , 15 (c) 3 , 8

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

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 a , x , b 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

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

intervals.

We can use [a, 1`) to represent the inequality x $ a and (2`, a) to represent x , a In

each of these cases, the symbols 1` and 2` are not real numbers but represent the fact that

properties

of the Real numbers

inequalities and intervals

exAmple 1

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x increases without bound (1`) or decreases without bound (2`) Table 0.2 summarizes

three types of intervals

| tAble 0.2 | inteRvAlS

a

b

b a

line is the absolute value of a, denoted |a| The absolute value of any nonzero number is

positive, and the absolute value of 0 is 0

Note that if a is a nonnegative number, then |a| 5 a, but if a is negative, then |a| is the

positive number (2a) Thus

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In performing computations with real numbers, it is important to remember the rules for computations

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

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

order of operations

1 perform operations within parentheses

2 Find indicated powers (23 5 2 ? 2 ? 2 5 8)

3 perform multiplications and divisions from left to right

4 perform additions and subtractions from left to right

order of operations

Evaluate the following

(a) 24 1 3(b) 242 1 3(c) (24 1 3)2 1 3(d) 6 4 2(2 1 1)

Solution

(a) 21(b) Note that with 242 the power 2 is applied only to 4, not to 24 (which would be writ-ten (24)2) Thus 242 1 3 5 2(42) 1 3 5 216 1 3 5 213

(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

6 5 2

12 (b) 1242 1 1132 5 21

1152 1 1232 5 12 a2117 b 1 1112 5 247

2 1292 2 1282 5 1292 1 1182 5 21

16 2 182 5 16 1 1282 5 18

3 (a) 1232 1242 5 112 a134b1142 5 13 (b) 51232 5 215

1232 1142 5 212

4 (a) 12142 4 1222 5 17 136y4 5 19

(b) 12282y4 5 27

45 4 1252 5 29

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True or false:

4 2(25)2 5 25

5 |4 2 6| 5 |4| 2 |6|

6 9 2 2(2)(210) 5 7(2)(210) 5 2140

We will assume that you have a scientific or graphing calculator Discussions of some

of the capabilities of graphing calculators and graphing utilities will be found throughout the text

Most scientific and graphing calculators use standard algebraic order when ing arithmetic expressions Working outward from inner parentheses, calculations are per-formed from left to right Powers and roots are evaluated first, followed by multiplications and divisions, and then additions and subtractions

1 (a) Meaningless A denominator of zero means division by zero, which is undefined

2 (a) 1 , x , 3 (b) 0 , x # 3 (c) 21 # x , ` or x $ 21 (d) 2` , x , 2 or x , 2

3 (a) [3, 6]; closed interval(b) [26, 4); half-open interval

4 False 2(25)2 5 (21)(25)2 5 (21)(25) 5 225 Exponentiation has priority and applies only to 25

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

5 (a) 2e ? 1 5 2e

(b) 4 1 (24) 5 0

6 (a) a32b a23b 51(b) 1122 a34b5 a34b1122

Insert the proper sign ,, 5, or to replace u in lems 7–14.

13 0230 1 050 u 023 1 50

14 029 2 30 u 0290 1 030

| exeRciSeS | 0.2

3 2 1 0

1 0

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In Problems 15–26, evaluate each expression.

28 Write the interval corresponding to x $ 0.

In Problems 29–32, express each inequality or graph

using interval notation, and name the type of interval.

29 1 , x # 3

30 24 # x # 3

In Problems 33–36, write an inequality that describes

each interval or graph.

In Problems 37–44, graph the subset of the real numbers

that is represented by each of the following and write

your answer in interval notation.

In Problems 45–50, use your calculator to evaluate each

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

pay is found by subtracting all taxes and retirement contributions from gross pay (which consists of salary plus commission) Given the following information, complete parts (a)–(c)

Salary 5 $300.00 Commission 5 $788.91Retirement 5 5% of gross pay

Taxes: State 5 5% of gross pay

Local 5 1% of gross pay Federal withholding 5 25% of (gross pay less retirement) Federal social security and Medicare 5 7.65% of gross pay

(a) Find the gross pay

(b) Find the amount of federal withholding

(c) Find the take-home pay

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

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

Cen-ters for Medicare and Medicaid Services)

(a) What t-value represents the year 2010?

(b) Actual expenditures for 2010 were $2879 lion What does the formula give as the 2010 approximation?

bil-(c) Predict the expenditures for 2015

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

(1) P 5 0.3179t 1 13.85 or by (2) P 5 0.0194t3 2 0.1952t2 1 0.8282t 1 13.63 where t is the number of years past 2000 (Source: U.S

Census Bureau)

(a) Both (1) and (2) closely approximate the data, but which is more accurate for 2006, when 15.8% of the population had no health insurance?

(b) Use both formulas to estimate the percentage of the U.S population not covered in 2012

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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

H 5 2.31A 1 31.26

To account for normal variability among boys, normal

height for a given age is ±5% of the height obtained

from the equation

(a) Find the normal height range for a boy who is 10.5

years old, and write it as an inequality

(b) Find the normal height range for a boy who is 5.75

years old, and write it as an inequality

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

Source: internal revenue Service

(a) Write the last three taxable income ranges as inequalities

(b) If an individual has a taxable income of $34,000, calculate the tax due Repeat this calculation for a taxable income of $82,400

(c) Write an interval that represents the amount of tax due for a taxable income between $34,000 and

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

multiplications For example,

(1.1)5 5 (1.1)(1.1)(1.1)(1.1)(1.1)

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

For any real number a,

a2 5 a ? a, a3 5 a ? a ? a, and a n 5 a ? a ? a ? ? a (n factors)

for any positive integer n The positive integer n is called the exponent, the number a is called the base, and a n is read “a to the nth power.”

Note that 4a n means 4(a n ), which is different from (4a) n The 4 is the coefficient of

a n in 4a n Note also that 2x n is not equivalent to (2x) n when n is even For example,

234 5 281, but (23)4 5 81

Some of the rules of exponents follow

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

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Section 0.3 integral exponents 17 positive integer exponents

Use rules of positive integer exponents to rewrite the following Assume all denominators are nonzero

(a) 56

54

(b) x x25(c) ax yb4

For any nonzero real number a, we define a0 5 1 We leave 00 undefined

In Section 0.2, we defined a2 1 as 1∙a for a 2 0, so we define a2n as (a2 1)n

a n 1a 2 02

aa bb2n5 c aa bb21dn5 ab ab n 1a 2 0, b 2 02

negative and Zero exponents

Write the following without exponents

(a) 6 ? 30

(b) 62 2

(c) a13b21(d) 2a3b2 24(e) (24)22

Solution

(a) 6 ? 30 5 6 ? 1 5 6(b) 6225 1

625 136(c) a13b215 3

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(d) 2a3b2 245 2a32b45 281

16(e) 1242225 1

124225 1

As we’ll see in the chapter on the mathematics of finance (Chapter 6), negative nents arise in financial calculations when we have a future goal for an investment and want to know how much to invest now For example, if money can be invested at 9%, com-pounded annually, then the amount we must invest now (which is called the present value)

expo-in order to have $10,000 expo-in the account after 7 years is given by $10,000(1.09)27 tions such as this are often done directly with a calculator

Calcula-Using the definitions of zero and negative exponents enables us to extend the rules of exponents to all integers and to express them more simply

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

1 a m ? a n 5 a m1n 2 a m∙a n 5 a m2n (a 2 0)

3 (ab) m 5 a m b m 4 (a m)n 5 a mn

5 (a∙b) m 5 a m∙b m (b 2 0) 6 a0 5 1 (a 2 0)

7 a2n 5 1∙a n (a 2 0) 8 (a∙b)2n 5 (b∙a) n (a, b 2 0)

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

operations with exponents

Use the rules of exponents and the definitions of a0 and a2n to simplify the following with positive exponents

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Section 0.3 integral exponents 19

3 Evaluate the following, if possible For any that are meaningless, so state Assume x 0.

Rewriting with positive exponents

Simplify the following so all exponents are positive

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In Problems 9–18, use rules of exponents to simplify the

expressions Express answers with positive exponents.

In Problems 23–36, use the rules of exponents to simplify

so that only positive exponents remain.

In Problems 37–48, compute and simplify so that only

positive exponents remain.

In many applications it is often necessary to write

inte-ger In Problems 49–56, write the expressions in this form.

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

earned is I 5 S 2 P In Problems 61–64, find S and I for the given P, n, and i.

61 $1200 for 5 years at 12%

62 $1800 for 7 years at 10%

63 $5000 for 6 years at 11.5%

64 $800 for 20 years at 10.5%

of $S after n years, invested at interest rate i (as a mal), compounded annually, then the present value P

Prob-lems 65 and 66, find P for the given S, n, and i.

65 $15,000 after 6 years at 11.5%

66 $80,000 after 20 years at 10.5%

total U.S personal income I (in billions of dollars) can

be approximated by the formula

where t is the number of years past 1980.

(a) What t-values correspond to the years 1985, 1995,

What does the formula estimate for these years?

(c) What does the formula estimate for the total sonal income in 2012?

(d) Does this formula seem to indicate that total sonal income doubles almost every 12 years?

(NYSE) for 1970–2006, the average daily shares traded

S (in millions of shares) can be approximated by the

formula

Trang 40

SeCtion 0.4 radicals and rational exponents 21

156.777 1041.58 2343.16 What does the formula predict for these years?

(c) Suppose in 2015 that a stock market average (such

as the Dow Jones Industrial Average) dramatically soared or tumbled; do you think this formula’s pre-dictions would be accurate, too low, or too high?

Fish and Wildlife Service)

(a) The actual numbers of endangered species for selected years were as follows

gered species predicted by the formula Round your answer to the nearest integer

For each of these years, find the number of endan-(b) How many more species does the formula estimate will be added to the endangered list for 2020 than the actual number given for 2007?

(c) Why do you think the answer to (b) is smaller than the number of species added from 1990 to 2003?

Department of Commerce)

(a) The percents of U.S households with Internet service for selected years were as follows

For each of these years, use the equation to find the predicted percent of households with Internet service

(b) From 2001 to 2004, the percent of households with Internet service increased by 18.8 percentage points What 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?

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 x3

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

"3V units

When we seek the cube root of a number such as 8 (written "38), we are looking for

a real number whose cube equals 8 Because 23 5 8, we know that "38 5 2 Similarly,

"3227 5 23 because (23)3 5 227 The expression "n a is called a radical, where " is

roots

0.4

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