Calculus for Business

CALCULUS

FOR BUSINESS, ECONOMICS, LIFE SCIENCES,

AND SOCIAL SCIENCES

CALCULUS

FOR BUSINESS, ECONOMICS, LIFE SCIENCES,

AND SOCIAL SCIENCES

TWELFTH EDITION

RAYMOND A. BARNETT

Merritt College

MICHAEL R. ZIEGLER

Marquette University

KARL E. BYLEEN

Marquette University

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Library of Congress Cataloging-in-Publication Data

Barnett, Raymond A. Calculus for business, economics, life sciences, and social sciences / Raymond A. Barnett, Michael R. Ziegler. 12th ed. / Karl E. Byleen.

p. cm. Includes index. ISBN 0-321-61399-6

1. Calculus Textbooks. 2. Social sciences Mathematics Textbooks. 3. Biomathematics Textbooks. I. Ziegler, Michael R. II. Byleen, Karl. III. Title.

QA303.2.B285 2010 515 dc22

2009041541

Copyright © 2011, 2008, 2005 Pearson Education, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechan-ical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. For information on obtaining permission for use of material in this work, please submit a written request to Pearson Education, Inc., Rights and Contracts Department, 501 Boylston Street, Suite 900, Boston, MA 02116, fax your request to 617-671-3447, or e-mail at http://www.pearsoned.com/ legal/permissions.htm.

1 2 3 4 5 6 7 8 9 10 EB 14 13 12 11 10

ISBN 10: 0-321-61399-6 ISBN 13: 978-0-321-61399-8

Preface

. . . .xi

Supplements

. . . .xvii

Acknowledgments

. . . .xix

Diagnostic Algebra Test

. . . .xx

PART 1

A LIBRARY OF ELEMENTARY FUNCTIONS

Chapter 1

Linear Equations and Graphs

. . . .2

1-1 Linear Equations and Inequalities . . . .3

1-2 Graphs and Lines . . . .13

1-3 Linear Regression . . . .27

Chapter 1 Review . . . .39

Review Exercises . . . .40

Chapter 2

Functions and Graphs

. . . .43

2-1 Functions . . . .44

2-2 Elementary Functions: Graphs and Transformations . . . .58

2-3 Quadratic Functions . . . .70

2-4 Polynomial and Rational Functions . . . .85

2-5 Exponential Functions . . . .95

2-6 Logarithmic Functions . . . .106

Chapter 2 Review . . . .117

Review Exercises . . . .120

PART 2

CALCULUS

Chapter 3

Limits and the Derivative

. . . .126

3-1 Introduction to Limits . . . .127

3-2 Infinite Limits and Limits at Infinity . . . .141

3-3 Continuity . . . .154

3-4 The Derivative . . . .165

3-5 Basic Differentiation Properties . . . .178

3-6 Differentials . . . .187

3-7 Marginal Analysis in Business and Economics . . . .194

Chapter 3 Review . . . .204

Review Exercises . . . .205

Chapter 4

Additional Derivative Topics

. . . .210

4-1 The Constant e and Continuous Compound Interest . . . .211

4-2 Derivatives of Exponential and Logarithmic Functions . . . .217

4-3 Derivatives of Products and Quotients . . . .225

4-4 The Chain Rule . . . .233

4-5 Implicit Differentiation . . . .243

4-6 Related Rates . . . .250

4-7 Elasticity of Demand . . . .255

Chapter 4 Review . . . .263

Review Exercises . . . .264

CONTENTS

viii Contents

5-1 First Derivative and Graphs . . . .267

5-2 Second Derivative and Graphs . . . .284

5-3 LHopital s Rule . . . .301

5-4 Curve-Sketching Techniques . . . .310

5-5 Absolute Maxima and Minima . . . .323

5-6 Optimization . . . .331

Chapter 5 Review . . . .344

Review Exercises . . . .345

Chapter 6

Integration

. . . .349

6-1 Antiderivatives and Indefinite Integrals . . . .350

6-2 Integration by Substitution . . . .361

6-3 Differential Equations; Growth and Decay . . . .372

6-4 The Definite Integral . . . .383

6-5 The Fundamental Theorem of Calculus . . . .393

Chapter 6 Review . . . .405

Review Exercises . . . .407

Chapter 7

Additional Integration Topics

. . . .410

7-1 Area Between Curves . . . .411

7-2 Applications in Business and Economics . . . .421

7-3 Integration by Parts . . . .432

7-4 Integration Using Tables . . . .439

Chapter 7 Review . . . .445

Review Exercises . . . .447

Chapter 8

Multivariable Calculus

. . . .449

8-1 Functions of Several Variables . . . .450

8-2 Partial Derivatives . . . .459

8-3 Maxima and Minima . . . .467

8-4 Maxima and Minima Using Lagrange Multipliers . . . .476

8-5 Method of Least Squares . . . .485

8-6 Double Integrals over Rectangular Regions . . . .495

8-7 Double Integrals over More General Regions . . . .505

Chapter 8 Review . . . .514

Review Exercises . . . .516

Chapter 9

Trigonometric Functions

. . . .519

9-1 Trigonometric Functions Review . . . .520

9-2 Derivatives of Trigonometric Functions . . . .527

9-3 Integration of Trigonometric Functions . . . .533

Chapter 9 Review . . . .537

Review Exercises . . . .538

Appendix A Basic Algebra Review

. . . .541

Appendix B Special Topics

. . . .583

Appendix C Tables

. . . .598

Answers

. . . .A-1

Subject Index

. . . .I-1

Index of Applications

. . . .I-9

Contents ix

College Mathematics, 12e (available separately)

Chapter 1

Differential Equations

1-1 Basic Concepts

1-2 Separation of Variables

1-3 First-Order Linear Differential Equations

Chapter 1 Review

Review Exercises

Chapter 2

Taylor Polynomials and Infinite Series

2-1 Taylor Polynomials

2-2 Taylor Series

2-3 Operations on Taylor Series

2-4 Approximations Using Taylor Series

Chapter 2 Review

Review Exercises

Chapter 3

Probability and Calculus

3-1 Improper Integrals

3-2 Continuous Random Variables

3-3 Expected Value, Standard Deviation, and Median

3-4 Special Probability Distributions

Chapter 3 Review

Review Exercises

Appendices A and B

are found in the following publications:

Calculus for Business, Economics, Life Sciences and Social Sciences,

12e

(0-321-61399-6) and

12e (0-321-61400-3).

Appendix C Tables

Table III Area Under the Standard Normal Curve

Appendix D Special Calculus Topic

D-1 Interpolating Polynomials and Divided Differences

Answers

Solutions to Odd-Numbered Exercises

Index

xi

PREFACE

The twelfth edition of Calculus for Business, Economics, Life Sciences, and Social

Sciencesis designed for a one- or two-term course in calculus for students who have had one to two years of high school algebra or the equivalent. The book s overall approach, refined by the authors experience with large sections of college fresh-men, addresses the challenges of teaching and learning when prerequisite knowl-edge varies greatly from student to student.

Our main goal was to write a text that students can easily comprehend.

Many elements play a role in determining a book s effectiveness for students. Not only is it critical that the text be accurate and readable but also, in order for a book to be effective, aspects such as the page design, the interactive nature of the presen-tation, and the ability to support and challenge all students have an incredible impact on how easily students comprehend the material. Here are some of the ways this text addresses the needs of students at all levels:

Page layout is clean and free of potentially distracting elements.

Matched Problems that accompany each of the completely worked examples help students gain solid knowledge of the basic topics and assess their own level of understanding before moving on.

Review material (Appendix A and Chapters 1 and 2) can be used judiciously to help remedy gaps in prerequisite knowledge.

A Diagnostic Algebra Test prior to Chapter 1 helps students assess their prereq-uisite skills, while the Basic Algebra Review in Appendix A provides students with the content they need to remediate those skills.

Explore & Discuss problems lead the discussion into new concepts or build upon a current topic. They help students of all levels gain better insight into the mathematical concepts through thought-provoking questions that are effective in both small and large classroom settings.

Exercise sets are very purposely and carefully broken down into three cate-gories by level of difficulty: A, B, and C. This allows instructors to easily craft homework assignments that best meet the needs of their students.

The MyMathLab course for this text is designed to help students help themselves and provide instructors with actionable information about their progress. In addition to the above, all students get substantial experience in modeling and solving real-world problems through application exercises chosen from business and economics, life sciences, and social sciences. Great care has been taken to write a book that is mathematically correct, with its emphasis on computational skills, ideas, and problem solving rather than mathematical theory.

Finally, the choice and independence of topics make the text readily adapt-able to a variety of courses (see the chapter dependencies chart on page xvi). This text is one of three books in the authors college mathematics series. The others are Finite Mathematics for Business, Economics, Life Sciences, and Social

Sciences, and College Mathematics for Business, Economics, Life Sciences, and

Social Sciences; the latter contains selected content from the other two books.

Additional Calculus Topics, a supplement written to accompany the Barnett/Ziegler/Byleen series, can be used in conjunction with these books.

Fundamental to a book s growth and effectiveness is classroom use and feedback. Now in its twelfth edition, Calculus for Business, Economics, Life Sciences, and

Social Scienceshas had the benefit of a substantial amount of both. Improvements in this edition evolved out of the generous response from a large number of users of the last and previous editions as well as survey results from instructors, mathe-matics departments, course outlines, and college catalogs. In this edition:

Chapter 2 contains a new Section (2-4) on polynomial and rational functions to provide greater flexibility in the use of the review chapter.

Continuous compound interest appears as a minor topic in Section 2-5. In Chapter 3, a discussion of vertical and horizontal asymptotes (Section 3-2) now precedes the treatment of continuity (Section 3-3).

Examples and exercises have been given up-to-date contexts and data. (See pages 101, 104 5).

Exposition has been simplified and clarified throughout the book.

Answers to the Matched Problems are now included at the end of each section for easy student reference.

The Self-Test on Basic Algebra has been renamed Diagnostic Algebra Test and has moved from Appendix A to the front of the book just prior to Chapter 1 to better encourage students to make use of this helpful assessment.

Exercise coverage within MyMathLab has been expanded, including a com-plete chapter of prerequisite skills exercises labeled Getting Ready.

Trusted Features

Emphasis and Style

As was stated earlier, this text is written for student comprehension. To that end, the focus has been on making the book both mathematically correct and accessible to students. Most derivations and proofs are omitted except where their inclusion adds significant insight into a particular concept as the emphasis is on computational skills, ideas, and problem solving rather than mathematical theory. General concepts and results are typically presented only after particular cases have been discussed.

Design

One of the hallmark features of this text is the clean, straightforward design of its

pages. Navigation is made simple with an obvious hierarchy of key topics and a judi-cious use of call-outs and pedagogical features. We made the decision to maintain a 2-color design to help students stay focused on the mathematics and applications. Whether students start in the chapter opener or in the exercise sets, they can easily reference the content, examples, and Conceptual Insights they need to understand the topic at hand. Finally, a functional use of color improves the clarity of many illus-trations, graphs, and explanations, and guides students through critical steps (see pages 27, 100, 107).

Examples and Matched Problems

More than 300 completely worked examples are used to introduce concepts and to

demonstrate problem-solving techniques. Many examples have multiple parts, sig-nificantly increasing the total number of worked examples. The examples are anno-tated using blue text to the right of each step, and the problem-solving steps are clearly identified.To give students extra help in working through examples, dashed

boxes are used to enclose steps that are usually performed mentally and rarely men-tioned in other books (see Example 2 on page 4). Though some students may not need these additional steps, many will appreciate the fact that the authors do not assume too much in the way of prior knowledge.

Preface xiii SOLUTION Property 7 Properties 5 and 6 Property 8 x_{= 4} logb 4 = logbx logb 8

#

logb 8- logb 4 + logb 2= logbx

logb 43/2- logb 82/3+ logb 2= logbx 3

2logb 4- 23logb 8 + logb 2= logbx 2logb 4- 3logb 8+ logb 2 = logbx

Matched Problem 5 Find x so that 3 logb 2+ 12logb 25- logb 20 = logbx.

EXPLORE& DISCUSS2 How many x intercepts can the graph of a quadratic function have? How many y

intercepts? Explain your reasoning.

Each example is followed by a similar Matched Problem for the student to work while reading the material.This actively involves the student in the learning process. The answers to these matched problems are included at the end of each section for easy reference.

Explore & Discuss

Every section contains Explore & Discuss problems at appropriate places to encourage students to think about a relationship or process before a result is stat-ed, or to investigate additional consequences of a development in the text. This serves to foster critical thinking and communication skills. The Explore & Discuss material can be used as in-class discussions or out-of-class group activities and is effective in both small and large class settings.

Exercise Sets

The book contains over 4,300 carefully selected and graded exercises. Many prob-lems have multiple parts, significantly increasing the total number of exercises. Exercises are paired so that consecutive odd and even numbered exercises are of the same type and difficulty level. Each exercise set is designed to allow instructors to craft just the right assignment for students. Exercise sets are categorized as A (routine, easy mechanics), B (more difficult mechanics), and C (difficult mechanics and some theory) to make it easy for instructors to create assignments that are appropriate for their classes. The writing exercises, indicated by the icon , provide students with an opportunity to express their understanding of the topic in writing. Answers to all odd-numbered problems are in the back of the book.

Applications

A major objective of this book is to give the student substantial experience in modeling and solving real-world problems. Enough applications are included to convince even the most skeptical student that mathematics is really useful (see the Index of Applications at the back of the book). Almost every exercise set contains application problems, including applications from business and economics, life sci-ences, and social sciences. An instructor with students from all three disciplines can let them choose applications from their own field of interest; if most students are from one of the three areas, then special emphasis can be placed there. Most

CAUTION Note that in Example 11 we let represent 1900. If we let represent 1940, for example, we would obtain a different logarithmic regres-sion equation, but the prediction for 2015 would be the same. We would not let represent 1950 (the first year in Table 1) or any later year, because logarith-mic functions are undefined at 0.

x = 0 x _{= 0}

x= 0

CONCEPTUAL INSIGHT

The notation (2, 7) has two common mathematical interpretations: the ordered pair with first coordinate 2 and second coordinate 7, and the open interval consisting of all real numbers between 2 and 7. The choice of interpretation is usually determined by the con-text in which the notation is used. The notation could be interpreted as an or-dered pair but not as an interval. In interval notation, the left endpoint is always written first. So,(-7, 2)is correct interval notation, but (2,-7)is not.

(2,-7)

xiv Preface

by professional journals and books. No specialized experience is required to solve any of the application problems.

Technology

Although access to a graphing calculator or spreadsheets is not assumed, it is likely that many students will want to make use of this technology. To assist these students, optional graphing calculator and spreadsheet activities are included in appropriate places. These include brief discussions in the text, examples or por-tions of examples solved on a graphing calculator or spreadsheet, and exercises for the student to solve. For example, linear regression is introduced in Section 1-3, and regression techniques on a graphing calculator are used at appropriate points to illustrate mathematical modeling with real data. All the optional graph-ing calculator material is clearly identified with the icon and can be omitted without loss of continuity, if desired. Optional spreadsheet material is identified with the icon . All graphs are computer-generated to ensure mathematical accuracy. Graphing calculator screens displayed in the text are actual output from a graphing calculator.

Additional Pedagogical Features

The following features, while helpful to any student, are particularly helpful to stu-dents enrolled in a large classroom setting where access to the instructor is more challenging or just less frequent. These features provide much-needed guidance for students as they tackle difficult concepts.

Call-out boxes highlight important definitions, results, and step-by-step processes

(see pages 90, 96 97).

Caution statements appear throughout the text where student errors often occur.

Conceptual Insights, appearing in nearly every section, make explicit

connec-tions to students previous knowledge.

Boldface type is used to introduce new terms and highlight important comments.

The Diagnostic Algebra Test, now located at the front of the book, provides

stu-dents with a tool to assess their prerequisite skills prior to taking the course.The

Basic Algebra Review, in Appendix A, provides students with seven sections of

content to help them remediate in specific areas of need. Answers to the Diagnostic Algebra Test are at the back of the book and reference specific sec-tions in the Basic Algebra Review for students to use for remediation.

Preface xv

Often it is during the preparation for a chapter exam that concepts gel for students, making the chapter review material particularly important. The chapter review sec-tions in this text include a comprehensive summary of important terms, symbols, and concepts, keyed to completely worked examples, followed by a comprehensive set of review exercises.Answers to most review exercises are included at the back of the book; each answer contains a reference to the section in which that type of problem is

discussedso students can remediate any deficiencies in their skills on their own.

Content

The text begins with the development of a library of elementary functions in

Chapters 1 and 2, including their properties and uses. We encourage students to

investigate mathematical ideas and processes graphically and numerically, as well as algebraically. This development lays a firm foundation for studying mathematics both in this book and in future endeavors. Depending on the course syllabus and the background of students, some or all of this material can be covered at the beginning of a course, or selected portions can be referenced as needed later in the course.

The material in Part Two (Calculus) consists of differential calculus (Chapters 3 5), integral calculus (Chapters 6 7), multivariable calculus (Chapter 8), and a brief discussion of differentiation and integration of trigonometric functions (Chapter 9). In general, Chapters 3 6 must be covered in sequence; however, cer-tain sections can be omitted or given brief treatments, as pointed out in the discus-sion that follows (see chart on next page).

Chapter 3 introduces the derivative. The first three sections cover limits

(includ-ing infinite limits and limits at infinity), continuity, and the limit properties that are essential to understanding the definition of the derivative in Section 3-4. The remaining sections of the chapter cover basic rules of differentiation, dif-ferentials, and applications of derivatives in business and economics. The inter-play between graphical, numerical, and algebraic concepts is emphasized here and throughout the text.

In Chapter 4 the derivatives of exponential and logarithmic functions are obtained before the product rule, quotient rule, and chain rule are introduced. Implicit dif-ferentiation is introduced in Section 4-5 and applied to related rates problems in Section 4-6. Elasticity of demand is introduced in Section 4-7. The topics in these last three sections of Chapter 4 are not referred to elsewhere in the text and can be omitted.

Chapter 5 focuses on graphing and optimization. The first two sections cover

first-derivative and second-derivative graph properties. L Hôpital s rule is dis-cussed in Section 5-3.A graphing strategy is presented and illustrated in Section 5-4. Optimization is covered in Sections 5-5 and 5-6, including examples and problems involving end-point solutions.

Chapter 6 introduces integration. The first two sections cover

antidifferentia-tion techniques essential to the remainder of the text. Secantidifferentia-tion 6-3 discusses some applications involving differential equations that can be omitted. The definite integral is defined in terms of Riemann sums in Section 6-4 and the fun-damental theorem of calculus is discussed in Section 6-5. As before, the inter-play between graphical, numerical, and algebraic properties is emphasized. These two sections are also required for the remaining chapters in the text.

Chapter 7 covers additional integration topics and is organized to provide

maxi-mum flexibility for the instructor. The first section extends the area concepts intro-duced in Chapter 6 to the area between two curves and related applications. Section 7-2 covers three more applications of integration, and Sections 7-3 and 7-4 deal with additional techniques of integration.Any or all of the topics in Chapter 7 can be omitted.

xvi

Chapter 8 deals with multivariable calculus. The first five sections can be

cov-ered any time after Section 5-6 has been completed. Sections 8-6 and 8-7 require the integration concepts discussed in Chapter 6.

Chapter 9 provides brief coverage of trigonometric functions that can be

incor-porated into the course, if desired. Section 9-1 provides a review of basic trigonometric concepts. Section 9-2 can be covered any time after Section 5-3 has been completed. Section 9-3 requires the material in Chapter 6.

Appendix A contains a concise review of basic algebra that may be covered as

part of the course or referenced as needed.As mentioned previously,Appendix B

contains additional topics that can be covered in conjunction with certain sections in the text, if desired.

PART ONE A LIBRARY OF ELEMENTARY FUNCTIONS*

PART TWO CALCULUS

APPENDICES

A Basic Algebra Review B Special Topics

3 Limits and the Derivative 5 Graphing and Optimization 9 Trigonometric Functions 8 Multivariable Calculus 6 Integration 7 Additional Integration Topics 4 Additional Derivative Topics 1 Linear Equations and Graphs

2 Functions and Graphs

CHAPTER DEPENDENCIES

STUDENT

SUPPLEMENTS

Student s Solutions Manual

By Garret J. Etgen, University of Houston

This manual contains detailed, carefully worked-out solutions to all odd-numbered section exercises and all Chapter Review exercises. Each section begins with Things to Remember, a list of key material for review. ISBN 13: 978-0-321-65498-4; ISBN 10: 0-321-65498-6

Additional Calculus Topics to

Accompany Calculus, 12e and College

Mathematics, 12e

This separate book contains three unique chapters: Differential Equations, Taylor Polynomials and Infinite Series, and Probability and Calculus. ISBN 13: 978-0-321-65509-7; ISBN 10: 0-321-65509-5

Worksheets for Classroom or Lab

Practice

These Worksheets provide students with a structured place to take notes, define key concepts and terms, and work through unique examples to reinforce what is taught in the lecture.

ISBN 13: 978-0-321-65398-7; ISBN 10: 0-321-65398-X

Videos on DVD-ROM with Optional

Captioning

The video lectures with optional captioning for this text make it easy and convenient for students to watch videos from a computer at home or on campus. The complete digitized set, affordable and portable for students, is ideal for distance learning or supplemental instruction. There is a video for every text example. ISBN 13: 0-978-0-321-70869-4; ISBN 10: 0-321-70869-5

INSTRUCTOR

SUPPLEMENTS

Instructor s Edition

This book contains answers to all exercises in the text. ISBN 13: 978-0-321-64543-2; ISBN 10: 0-321-64543-X

Online Instructor s Solutions Manual

(downloadable)

By Jason Aubrey, University of Missouri Columbia This manual contains detailed solutions to all even-numbered section problems.

Available in MyMathLab or through http://www.pearson highered.com.

Mini Lectures (downloadable)

Mini Lectures are provided for the teaching assistant, adjunct, part-time, or even full-time instructor for lecture preparation by providing learning objectives, examples (and answers) not found in the text, and teaching notes. Available in MyMathLab or through http://www.pearson highered.com.

TestGen®

TestGen®_{(www.pearsoned.com/testgen) enables instructors}

to build, edit, print, and administer tests using a computer-ized bank of questions developed to cover all the objectives of the text. TestGen is algorithmically based, allowing instructors to create multiple but equivalent versions of the same question or test with the click of a button. Instructors can also modify test bank questions or add new questions. The software and testbank are available for download from Pearson Education s online catalog.

PowerPoint® Lecture Slides

These slides present key concepts and definitions from the text. They are available in MyMathLab or at http:// www.pearsonhighered.com/educator.

xvii

Because of the careful checking and proofing by a number of mathematics instruc-tors (acting independently), the authors and publisher believe this book to be sub-stantially error free. If an error should be found, the authors would be grateful if notification were sent to Karl E. Byleen, 9322 W. Garden Court, Hales Corners, WI 53130; or by e-mail, to [email protected].

xviii

TECHNOLOGY

RESOURCES

MyMathLab® Online Course

(access code required)

MyMathLab is a text-specific, easily customizable online course that integrates interactive multimedia instruction with textbook content. MyMathLab gives you the tools you need to deliver all or a portion of your course online, whether your students are in a lab setting or working from home.

Interactive homework exercises, correlated to your

textbook at the objective level, are algorithmically gen-erated for unlimited practice and mastery. Most exercis-es are free-rexercis-esponse and provide guided solutions, sam-ple problems, and tutorial learning aids for extra help.

Personalized Study Plan, generated when students

complete a test or quiz, indicates which topics have been mastered and links to tutorial exercises for top-ics students have not mastered. You can customize the Study Plan so that the topics available match your course contents or so that students homework results also determine mastery.

Multimedia learning aids, such as videos for every example in the text, provide help for students when

they need it. Other student-help features include Help Me Solve This and Additional Examples. You can assign these multimedia learning aids as home-work to help your students grasp the concepts.

Homework and Test Manager lets you assign

home-work, quizzes, and tests that are automatically graded. Select just the right mix of questions from the

MyMathLab exercise bank, instructor-created custom exercises, and/or TestGen®test items.

Gradebook, designed specifically for mathematics and

statistics, automatically tracks students results, lets you stay on top of student performance, and gives you control over how to calculate final grades. You can also add offline (paper-and-pencil) grades to the gradebook.

MathXL Exercise Builder allows you to create static

and algorithmic exercises for your online assign-ments. You can use the library of sample exercises as an easy starting point, or you can edit any course-related exercise.

Pearson Tutor Center (www.pearsontutorservices.com)

access is automatically included with MyMathLab. The Tutor Center is staffed by qualified math instruc-tors who provide textbook-specific tutoring for stu-dents via toll-free phone, fax, email, and interactive Web sessions.

Students do the assignments in the new Flash®-based MathXL Player, which is compatible with almost any browser (Firefox®, Safari , or Internet Explorer®) on almost any platform (Macintosh®or Windows®). MyMathLab is powered by CourseCompass , Pearson Education s online teaching and learning environment, and by MathXL®, our online homework, tutorial, and assessment system. MyMathLab is available to qualified adopters. For more information, visit www.mymathlab.com or contact your Pearson representative.

MathXL® Online Course

(access code required)

MathXL®_{is an online homework, tutorial, and}

assess-ment system that accompanies Pearson s textbooks in mathematics or statistics.

Interactive homework exercises, correlated to your

textbook at the objective level, are algorithmically gen-erated for unlimited practice and mastery. Most exer-cises are free-response and provide guided solutions, sample problems, and learning aids for extra help.

Personalized Study Plan, generated when students

complete a test or quiz, indicates which topics have been mastered and links to tutorial exercises for top-ics students have not mastered. Instructors can cus-tomize the available topics in the study plan to match their course concepts.

Multimedia learning aids, such as videos for every example in the text, provide help for students when

they need it. Other student-help features include Help Me Solve This and Additional Examples. These are assignable as homework, to further encourage their use.

Gradebook, designed specifically for mathematics and

statistics, automatically tracks students results, lets you stay on top of student performance, and gives you control over how to calculate final grades.

MathXL Exercise Builder allows you to create static

and algorithmic exercises for your online assignments. You can use the library of sample exercises as an easy starting point or use the Exercise Builder to edit any of the course-related exercises.

Homework and Test Manager lets you create online

homework, quizzes, and tests that are automatically graded. Select just the right mix of questions from the MathXL exercise bank, instructor-created custom exercises, and/or TestGen test items.

The new Flash®_{-based MathXL Player is compatible}

xix

Windows®). MathXL is available to qualified adopters. For more information, visit our website at www.mathxl. com, or contact your Pearson sales representative.

InterAct Math Tutorial Website:

www.interactmath.com

Get practice and tutorial help online! This interactive tutorial website provides algorithmically generated

prac-textbook. Students can retry an exercise as many times as they like with new values each time for unlimited prac-tice and mastery. Every exercise is accompanied by an interactive guided solution that provides helpful feedback for incorrect answers, and students can also view a worked-out sample problem that steps them through an exercise similar to the one they re working on.

Acknowledgments

In addition to the authors many others are involved in the successful publication of a book. We wish to thank the following reviewers of the 11th and 12th editions:

Christine Cosgrove, Fitchburg State College Darryl Egley, North Harris College

Lauren Fern, University of Montana Gregory Goeckel, Presbyterian College Garland Guyton, Montgomery College Virginia Hanning, San Jacinto College Bruce Hedman, University of Connecticut Yvette Hester, Texas A&M University Fritz Keinert, Iowa State University

Steven Klassen, Missouri Western State University Wesley W. Maiers, Valparaiso University

James Martin, Christopher Newport University Gary R. Penner, Richland College

Jon Prewett, University of Wyoming

Cynthia Schultz, Illinois Valley Community College Maria Terrell, Cornell University

Fred M. Wright, Iowa State University

Amy Ann Yielding, Washington State University

We also wish to thank our colleagues who have provided input on previous editions: Chris Boldt, Bob Bradshaw, Bruce Chaffee, Robert Chaney, Dianne Clark, Charles E. Cleaver, Barbara Cohen, Richard L. Conlon, Catherine Cron, Lou D Alotto, Madhu Deshpande, Kenneth A. Dodaro, Michael W. Ecker, Jerry R. Ehman, Lucina Gallagher, Martha M. Harvey, Sue Henderson, Lloyd R. Hicks, Louis F. Hoelzle, Paul Hutchins, K. Wayne James, Jeffrey Lynn Johnson, Robert H. Johnston, Robert Krystock, Inessa Levi, James T. Loats, Frank Lopez, Roy H. Luke, Wayne Miller, Mel Mitchell, Linda M. Neal, Ronald Persky, Kenneth A. Peters, Jr., Dix Petty, Tom Plavchak, Bob Prielipp, Thomas Riedel, Stephen Rodi, Arthur Rosenthal, Sheldon Rothman, Elaine Russell, John Ryan, Daniel E. Scanlon, George R. Schriro, Arnold L. Schroeder, Hari Shanker, Joan Smith, J. Sriskandarajah, Steven Terry, Beverly Vredevelt, Delores A. Williams, Caroline Woods, Charles W. Zimmerman, Pat Zrolka, and Cathleen A. Zucco-Tevelot. We also express our thanks to:

Caroline Woods, Anthony Gagliardi, Damon Demas, John Samons, Theresa Schille, Blaise DeSesa, and Debra McGivney for providing a careful and thor-ough accuracy check of the text, problems and answers.

Garret Etgen, Jason Aubrey, Dale R. Buske, and Karla Neal for developing the supplemental materials so important to the success of a text.

All the people at Pearson Education who contributed their efforts to the produc-tion of this book.

Work through all the problems in this self-test and check your answers in the back of the book. Answers are keyed to relevant sections in Appendix A. Based on your results, review the appro-priate sections in Appendix A to refresh your algebra skills and better prepare yourself for this course.

1. Replace each question mark with an appropriate

expres-sion that will illustrate the use of the indicated real number property:

(A) Commutative (B) Associative (C) Distributive:

Problems 2 6 refer to the following polynomials:

(A) (B)

(C) (D)

2. Add all four.

3. Subtract the sum of (A) and (C) from the sum of (B) and (D). 4. Multiply (C) and (D).

5. What is the degree of each polynomial?

6. What is the leading coefficient of each polynomial?

In Problems 7 12, perform the indicated operations and simplify.

7. 8. 9. 10.

11. 12.

13. Write in scientific notation:

(A) 4,065,000,000,000 (B) 0.0073

14. Write in standard decimal form:

(A) (B)

15. Indicate true (T) or false (F):

(A) A natural number is a rational number.

(B) A number with a repeating decimal expansion is an irrational number.

16. Give an example of an integer that is not a natural number.

Simplify Problems 17 25 and write answers using positive expo-nents only. All variables represent positive real numbers.

17. 18. 19. 20. 21. 22. 23. 24. 25. (3x1>2 _{- y}1>2_)(2x1>2 _{+ 3y}1>2₎ (x1>2 _{+ y}1>2₎2 50 32 + 3-2 2-2 (9a4_b-2₎1>2 u5>3u2>3 (x-3_y2₎-2 (2 * 105_{)(3 * 10}-3₎ 9u8_v6 3u4_v8 6(xy3₎5 4.06 * 10-4 2.55* 108 (x - 2y)3 (3x3 _{- 2y)}2 (2x - y)(2x + y) - (2x - y)2 (2a - 3b)2 (2x + y)(3x - 4y) 5x2 _{- 3x34 - 3(x - 2)4} x3 + 8 2 - 3x2 x + 2 3x - 4 (2 + 3)x = ? (+): 2 + (x + y) = ? (

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Write Problems 26 31 in completely factored form relative to the integers. If a polynomial cannot be factored further relative to the integers, say so.

26. 27.

28. 29.

30. 31.

In Problems 32 37, perform the indicated operations and reduce to lowest terms. Represent all compound fractions as simple frac-tions reduced to lowest terms.

32. 33.

34. 35.

36. 37.

38. Each statement illustrates the use of one of the following

real number properties or definitions. Indicate which one. x-1 + y-1 x-2 - y-2 1 7 + h -1 7 h (x + y)2 _{- x}2 y x x2 - 16 -x + 4 x2 - 4x 3x 3x2_{- 12x} + 1 6x 2 5b -4 3a3 -1 6a2_b2 6x(2x + 1)2 _{- 15x}2_{(2x + 1)} (4x- y)2 _{- 9x}2 6n3_{- 9n}2 _{- 15n} t2 - 4t - 6 8x2 _{- 18xy + 9y}2 12x2 _{+ 5x - 3}

Commutative (+,

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39. Change to rational exponent form: 40. Change to radical form:

41. Write in the form where a and b are real

num-bers and p and q are rational numnum-bers:

In Problems 42 and 43, rationalize the denominator.

42. 43.

In Problems 44 and 45, rationalize the numerator.

44. 45.

Solve Problems 46 49 for x.

46. 47. 48. _x2_{- x - 20 = 0} 49. -6x2_{+ 7x - 1 = 0} 3x2 _{- 21 = 0} x2= 5x 1u + h - 1u h 1x - 5 x - 5 x - 5 1x - 15 3x 13x 41x - 3 21x axp _{+ bx}q_, 2x1>2 - 3x2>3 62x5 2 _{- 72(x - 1)}4 3 (x - y) + 0 = (x - y) u -(v - w) = -u v - w 9

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CALCULUS

FOR BUSINESS, ECONOMICS, LIFE SCIENCES,

AND SOCIAL SCIENCES

CALCULUS

FOR BUSINESS, ECONOMICS, LIFE SCIENCES,

AND SOCIAL SCIENCES

A LIBRARY

OF ELEMENTARY

FUNCTIONS

PART

Calculus for Business, Economics, Life Sciences and Social Sciences, Twelfth Edition, by Raymond A. Barnett, Michael R. Ziegler, and Karl E. Byleen. Published by Pearson.

Linear Equations

and Graphs

1-1

Linear Equations and

Inequalities

1-2

Graphs and Lines

1-3

Linear Regression

Chapter 1 Review

Review Exercises

2

Introduction

We begin by discussing some algebraic methods for solving equations and inequal-ities. Next, we introduce coordinate systems that allow us to explore the relation-ship between algebra and geometry. Finally, we use this algebraic geometric relationship to find equations that can be used to describe real-world data sets. For example, in Section 1-3 you will learn how to find the equation of a line that fits data on winning times in an Olympic swimming event (see Problems 27 and 28 on page 38). We also consider many applied problems that can be solved using the concepts discussed in this chapter.

The equation

and the inequality

are both first degree in one variable. In general, a first-degree, or linear, equation in

one variable is any equation that can be written in the form

(1) If the equality symbol, in (1) is replaced by or the resulting ex-pression is called a first-degree, or linear, inequality.

Asolution of an equation (or inequality) involving a single variable is a number

that when substituted for the variable makes the equation (or inequality) true. The set of all solutions is called the solution set. When we say that we solve an equation

(or inequality), we mean that we find its solution set.

Knowing what is meant by the solution set is one thing; finding it is another. We start by recalling the idea of equivalent equations and equivalent inequalities. If we perform an operation on an equation (or inequality) that produces another equa-tion (or inequality) with the same soluequa-tion set, then the two equaequa-tions (or inequali-ties) are said to be equivalent. The basic idea in solving equations or inequalities is

to perform operations that produce simpler equivalent equations or inequalities and to continue the process until we obtain an equation or inequality with an obvi-ous solution.

Linear Equations

Linear equations are generally solved using the following equality properties.

THEOREM 1 Equality Properties An equivalent equation will result if

1. The same quantity is added to or subtracted from each side of a given equation.

2. Each side of a given equation is multiplied by or divided by the same nonzero quantity.

EXAMPLE 1 Solving a Linear Equation Solve and check:

SOLUTION Use the distributive property. Combine like terms.

Subtract 3x from both sides. Subtract 12 from both sides. Divide both sides by 2.

CHECK -33 =* -33 -72 - 3(-13) * 3(-13) + 6 8(*9) - 3[(*9) - 4] * 3[(*9) - 4] + 6 8x _{- 3(x - 4) = 3(x - 4) + 6} x _{= -9} 2x _{= -18} 2x _{+ 12 = -6} 5x _{+ 12 = 3x - 6} 8x _{- 3x + 12 = 3x - 12 + 6} 8x _{- 3(x - 4) = 3(x - 4) + 6} 8x _{- 3(x - 4) = 3(x - 4) + 6} Ú, 6, 7, , =, Standard form: ax + b , 0 a - 0 x 2 + 2(3x - 1) Ú 5 3 _{- 2(x + 3) =} x 3 - 5

1-1

Linear Equations and Inequalities

Linear Equations Linear Inequalities Applications

Matched Problem 1 Solve and check:

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1

nonzero number always produces an equivalent equation. What is the smallest pos-itive number that you could use to multiply both sides of the following equation to produce an equivalent equation without fractions?

EXAMPLE 2 Solving a Linear Equation Solve and check:

SOLUTION What operations can we perform on

to eliminate the denominators? If we can find a number that is exactly divisible by each denominator, we can use the multiplication property of equality to clear the denominators. The LCD (least common denominator) of the fractions, 6, is exactly what we are looking for! Actually, any common denominator will do, but the LCD results in a simpler equivalent equation. So, we multiply both sides of the equation by 6:

*

Use the distributive property. Combine like terms.

Subtract 6 from both sides.

CHECK

Matched Problem 2 Solve and check:

In many applications of algebra, formulas or equations must be changed to alternative equivalent forms. The following example is typical.

EXAMPLE 3 Solving a Formula for a Particular Variable If you deposit a principle P in an ac-count that earns simple interest at an annual rate r, then the amount A in the account after t years is given by . Solve for

(A) r in terms of A, P, and t (B) P in terms of A, r, and t A _{= P + Prt} x _{+ 1} 3 -x 4 = 1 2 5 =* 5 13 _{- 8 * 5} 24 + 2 2 -24 3 * 5 x _{+ 2} 2 -x 3 = 5 x _{= 24} x _{+ 6 = 30} 3x _{+ 6 - 2x = 30} 3(x + 2) - 2x = 30 3 6

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SOLUTION (A) Reverse equation.

Subtract P from both sides. Divide both members by Pt.

(B) Reverse equation.

Factor out P (note the use of the distributive property).

Divide by (1 + rt).

Matched Problem 3 If a cardboard box has length L, width W, and height H, then its surface area is given by the formula . Solve the formula for

(A) L in terms of S, W, and H (B) H in terms of S, L, and W

Linear Inequalities

Before we start solving linear inequalities, let us recall what we mean by (less than) and (greater than). If a and b are real numbers, we write

a is less than b

if there exists a positive number p such that Certainly, we would expect that if a positive number was added to any real number, the sum would be larger than the original. That is essentially what the definition states. If we may also write b is greater than a. EXAMPLE 4 Inequalities (A) Since (B) Since (C) Since (because )

Matched Problem 4 Replace each question mark with either or

(A) (B) (C)

The inequality symbols have a very clear geometric interpretation on the real number line. If then a is to the left of b on the number line; if then c is to the right of d on the number line (Fig. 1). Check this geometric property with the inequalities in Example 4.

E

& D

2

(A) (B) (C) (D)

Based on these examples, describe verbally the effect of multiplying both sides of an inequality by a number. 12?-8 and 12 -4? -8 -4 12?-8 and 12₄ ?-8 4 -1?3 and _-2(-1)?-2(3) -1?3 and 2(_-1)?2(3) 7 6 c _{7 d,} a _{6 b,} -3?-30 -20?0 2?8 7. 6 -10 + 10 = 0 -10 6 0 0 _{7 -10} -6 + 4 = -2 -6 6 -2 3 + 2 = 5 3 _{6 5} b _{7 a} a _{6 b,} a _{+ p = b.} a _{6 b} 7 6 S _{= 2LW + 2LH + 2WH} P ₌ A 1 _{+ rt} P(1 _{+ rt) = A} P _{+ Prt = A} A _{= P + Prt} r ₌ A - P P t P r t _{= A - P} P _{+ Prt = A} A _{= P + Prt} a d b 0 c Figure 1 a6 b, c 7 d

The procedures used to solve linear inequalities in one variable are almost the same as those used to solve linear equations in one variable, but with one important exception, as noted in item 3 of Theorem 2.

THEOREM 2 Inequality Properties

An equivalent inequality will result, and the sense or direction will remain the same if each side of the original inequality

1. has the same real number added to or subtracted from it.

2. is multiplied or divided by the same positive number.

An equivalent inequality will result, and the sense or direction will reverse if each

side of the original inequality

3. is multiplied or divided by the same negative number.

NOTE: Multiplication by 0 and division by 0 are not permitted.

Therefore, we can perform essentially the same operations on inequalities that we perform on equations, with the exception that the sense of the inequality revers-es if we multiply or divide both sidrevers-es by a negative number. Otherwise, the sense of

the inequality does not change. For example, if we start with the true statement

and multiply both sides by 2, we obtain

and the sense of the inequality stays the same. But if we multiply both sides of by the left side becomes 6 and the right side becomes 14, so we must write

to have a true statement. The sense of the inequality reverses.

If the double inequality means that and ; that is,

x is between a and b. Interval notation is also used to describe sets defined by

inequalities, as shown in Table 1.

The numbers a and b in Table 1 are called the endpoints of the interval. An

interval is closed if it contains all its endpoints and open if it does not contain any

of its endpoints. The intervals [a, b], ( , a], and [b, ) are closed, and the inter-vals (a, b), ( , a), and (b, ) are open. Note that the symbol (read infinity) is not a number. When we write [b, ), we are simply referring to the interval that starts at b and continues indefinitely to the right. We never refer to as an end-point, and we never write [b, ]. The interval ( , ) is the entire real number line.

Note that an endpoint of a line graph in Table 1 has a square bracket through it if the endpoint is included in the interval; a parenthesis through an endpoint indi-cates that it is not included.

q - q q q q q q - q - q q x<b a<x a _{6 x 6 b} a _{6 b,} 6 _{6 14} -2, -3 7 -7 -6 7 -14 -3 7 -7 CONCEPTUAL INSIGHT

The notation (2, 7) has two common mathematical interpretations: the ordered pair with first coordinate 2 and second coordinate 7, and the open interval consisting of all real numbers between 2 and 7. The choice of interpretation is usually determined by the con-text in which the notation is used. The notation could be interpreted as an or-dered pair but not as an interval. In interval notation, the left endpoint is always written first. So,(-7, 2)is correct interval notation, but (2,-7)is not.

EXAMPLE 5 Interval and Inequality Notation, and Line Graphs

(A) Write [ 2, 3) as a double inequality and graph. (B) Write in interval notation and graph.

SOLUTION (A) [ 2, 3) is equivalent to

(B) is equivalent to

Matched Problem 5 (A) Write as a double inequality and graph. (B) Write in interval notation and graph.

E

& D

3

tween these sets.

EXAMPLE 6 Solving a Linear Inequality Solve and graph:

SOLUTION Remove parentheses. Combine like terms.

Subtract 6x from both sides. Subtract 6 from both sides.

Divide both sides by 2 and reverse the sense of the inequality.

Notice that in the graph of we use a parenthesis through 4, since the point 4 is not included in the graph.

x 7 4, x _{7 4 or (4, q)} --2x 6 -8 -2x + 6 6 -2 4x _{+ 6 6 6x - 2} 4x _{+ 6 6 6x - 12 + 10} 2(2x _{+ 3) 6 6(x - 2) + 10} 2(2x _{+ 3) 6 6(x - 2) + 10} x _{6 -5?} x _{Ú -5.} x _{6 3} (_{-7, 4]} [_{-5, q).} x _{Ú -5} -2 x _{6 3.} -x _{Ú -5}

-Table 1 Interval Notation

Interval Notation Inequality Notation Line Graph [a, b] [a, b) (a, b] (a, b) x _{7 b} (b,q) x Ú b [b,q) x _{6 a} (- q, a) x a (- q, a] a _{6 x 6 b} a 6 x b a x _{6 b} a x b [ ( [ ( ( ( [ [ [ [ ( ( a a b b b a b a b a b a x x x x x x x x [ ( *2 3 x [ *5 x ( 4 x

Matched Problem 6 Solve and graph:

EXAMPLE 7 Solving a Double Inequality Solve and graph:

SOLUTION We are looking for all numbers x such that is between and 9, including 9 but not We proceed as before except that we try to isolate x in the middle:

Matched Problem 7 Solve and graph:

Note that a linear equation usually has exactly one solution, while a linear in-equality usually has infinitely many solutions.

Applications

To realize the full potential of algebra, we must be able to translate real-world prob-lems into mathematics. In short, we must be able to do word probprob-lems.

Here are some suggestions that will help you get started:

Procedure for Solving Word Problems

1. Read the problem carefully and introduce a variable to represent an un-known quantity in the problem. Often the question asked in a problem will indicate the best way to introduce this variable.

2. Identify other quantities in the problem (known or unknown), and whenever possible, express unknown quantities in terms of the variable you introduced in Step 1.

3. Write a verbal statement using the conditions stated in the problem and then write an equivalent mathematical statement (equation or inequality).

4. Solve the equation or inequality and answer the questions posed in the problem.

5. Check the solution(s) in the original problem.

EXAMPLE 8 Purchase Price John purchases a computer from an online store for $851.26, in-cluding a $57 shipping charge and 5.2% state sales tax.What is the purchase price of the computer?

SOLUTION Step 1 Introduce a variable for the unknown quantity. After reading the

prob-lem, we decide to let x represent the purchase price of the computer.

Step 2 Identify quantities in the problem.

Total cost: $851.26 Sales tax: 0.052x Shipping charges: $57 -8 3x _{- 5 6 7} -3 6 x 3 _{or (-3, 3]} -6 2 6 2x 2 6 2 -6 6 2x 6 -3 - 3 6 2x + 3 - 3 9 - 3 -3 6 2x + 3 9 -3. 2x + 3 -3 -3 6 2x + 3 9 3(x _{- 1)} 5(x _{+ 2) - 5} ( [ *3 3 x

Step 3 Write a verbal statement and an equation.

Step 4 Solve the equation and answer the question.

Combine like terms.

Subtract 57 from both sides. Divide both sides by 1.052.

The price of the computer is $755.

Step 5 Check the answer in the original problem.

Matched Problem 8 Mary paid 8.5% sales tax and a $190 title and license fee when she bought a new car for a total of $28,400. What is the purchase price of the car?

The next example involves the important concept of break-even analysis, which

is encountered in several places in this text. Any manufacturing company has costs, C, and revenues, R. The company will have a loss if will break even if

and will have a profit if Costs involve fixed costs, such as plant

over-head, product design, setup, and promotion, and variable costs, which are dependent

on the number of items produced at a certain cost per item.

EXAMPLE 9 Break-Even Analysis A multimedia company produces DVDs. One-time fixed costs for a particular DVD are $48,000, which include costs such as filming, edit-ing, and promotion. Variable costs amount to $12.40 per DVD and include manu-facturing, packaging, and distribution costs for each DVD actually sold to a retailer. The DVD is sold to retail outlets at $17.40 each. How many DVDs must be manufactured and sold in order for the company to break even?

SOLUTION Step 1 Let

Step 2

Step 3 The company breaks even if that is, if

Step 4 Subtract 12.4x from both sides. Divide both sides by 5.

The company must make and sell 9,600 DVDs to break even. x _{= 9,600} 5x _{= 48,000} 17.4x _{= 48,000 + 12.4x} $17.40x _{= $48,000 + $12.40x} R _{= C;} R _{= $17.40x} = $48,000 + $12.40x

C _{= Fixed costs + variable costs} Variable costs = $12.40x

Fixed costs _{= $48,000}

R = revenue (return) on sales of x DVDs C _{= cost of producing x DVDs}

x _{= number of DVDs manufactured and sold.} R _{7 C.} R _{= C,} R _{6 C,} Total = $851.26 Tax 0.052

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Step 5 Check:

Matched Problem 9 How many DVSs would a multimedia company have to make and sell to break even if the fixed costs are $36,000, variable costs are $10.40 per DVD, and the DVDs are sold to retailers for $15.20 each?

EXAMPLE 10 Consumer Price Index The Consumer Price Index (CPI) is a measure of the aver-age change in prices over time from a designated reference period, which equals 100. The index is based on prices of basic consumer goods and services. Table 2 lists the CPI for several years from 1960 to 2005. What net annual salary in 2005 would have the same purchasing power as a net annual salary of $13,000 in 1960? Compute the answer to the nearest dollar. (Source: U.S. Bureau of Labor Statistics)

SOLUTION Step 1 Let the purchasing power of an annual salary in 2005.

Step 2 Annual salary in

Step 3 The ratio of a salary in 2005 to a salary in 1960 is the same as the ratio of the CPI in 2005 to the CPI in 1960.

Multiply both sides by 13,000.

Step 4

Step 5

Note: The slight difference in these ratios is due to rounding the 2005 salary to

the nearest dollar.

Matched Problem 10 What net annual salary in 1975 would have had the same purchasing power as a net annual salary of $100,000 in 2005? Compute the answer to the nearest dollar.

Exercises 1-1

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A

Solve Problems 7 10 and graph.

7. 8. 9. 10. Solve Problems 11 24. 11. 12. m 3 - 4= 2 3 x 4 + 1 2 = 1 8 -4 6 2y - 3 6 9 2 x + 3 5 -2x + 8 6 4 -4x - 7 7 5

13. 14. 15. 16.

B

Solve Problems 25 28 and graph.

25. 26.

27. 28.

C

Solve Problems 29 34 for the indicated variable.

29. for y 30. for x

31. for

32. for m 33. for C

34. for F

Solve Problems 35 and 36 and graph.

35. 36.

37. What can be said about the signs of the numbers a and b in

each case?

(A) (B)

(C) (D)

38. What can be said about the signs of the numbers a, b, and c

in each case?

(A) (B)

(C) (D)

39. If both a and b are positive numbers and b/a is greater than

1, then is positive or negative?

40. If both a and b are negative numbers and b/a is greater than

1, then is positive or negative?

In Problems 41 46, discuss the validity of each statement. If the statement is true, explain why. If not, give a counterexample.

41. If the intersection of two open intervals is nonempty, then

their intersection is an open interval.

42. If the intersection of two closed intervals is nonempty, then

their intersection is a closed interval.

43. The union of any two open intervals is an open interval. 44. The union of any two closed intervals is a closed interval. 45. If the intersection of two open intervals is nonempty, then

their union is an open interval. a - b a - b a2 bc 6 0 a bc 7 0 ab c 60 abc 7 0 a b 6 0 a b 7 0 ab 6 0 ab 7 0 -10 8 - 3u -6 -3 4 - 7x 6 18 C = 5 9(F - 32); F = 9 5C + 32; y = mx + b; y (B Z 0) Ax + By = C; y = -2 3x + 8; 3x - 4y = 12; -1 23t + 5 11 -4 95C + 32 68 -4 5x + 6 6 21 2 3x - 7 6 14 u 2 -2 3 6 u 3 + 2 m 5 - 3 6 3 5 -m 2 y 4 -y 3 = 1 2 x 5 -x 6 = 6 5 x - 2 Ú 2(x - 5) 3 - y 4(y - 3) -3(4 - x) = 5 - (x + 1) 10x + 25(x - 3) = 275 -3y + 9 + y = 13 - 8y 2u + 4 = 5u + 1 - 7u x -4 6 5 6 y -5 7 3 2

46. If the intersection of two closed intervals is nonempty, then

their union is a closed interval.

Applications

47. Ticket sales. A rock concert brought in $432,500 on the sale of 9,500 tickets. If the tickets sold for $35 and $55 each, how many of each type of ticket were sold?

48. Parking meter coins. An all-day parking meter takes only dimes and quarters. If it contains 100 coins with a total value of $14.50, how many of each type of coin are in the meter?

49. IRA. You have $500,000 in an IRA (Individual Retirement Account) at the time you retire. You have the option of in-vesting this money in two funds: Fund A pays 5.2% annual-ly and Fund B pays 7.7% annualannual-ly. How should you divide your money between Fund A and Fund B to produce an an-nual interest income of $34,000?

50. IRA. Refer to Problem 49. How should you divide your money between Fund A and Fund B to produce an annual interest income of $30,000?

51. Car prices. If the price change of cars parallels the change in the CPI (see Table 2 in Example 10), what would a car sell for (to the nearest dollar) in 2005 if a comparable model sold for $10,000 in 1990?

52. Home values. If the price change in houses parallels the CPI (see Table 2 in Example 10), what would a house val-ued at $200,000 in 2005 be valval-ued at (to the nearest dollar) in 1960?

53. Retail and wholesale prices. Retail prices in a department store are obtained by marking up the wholesale price by 40%. That is, retail price is obtained by adding 40% of the wholesale price to the wholesale price.

(A) What is the retail price of a suit if the wholesale price is $300?

(B) What is the wholesale price of a pair of jeans if the re-tail price is $77?

54. Retail and sale prices. Sale prices in a department store are obtained by marking down the retail price by 15%. That is, sale price is obtained by subtracting 15% of the retail price from the retail price.

(A) What is the sale price of a hat that has a retail price of $60?

(B) What is the retail price of a dress that has a sale price of $136?

55. Equipment rental. A golf course charges $52 for a round of golf using a set of their clubs, and $44 if you have your own clubs. If you buy a set of clubs for $270, how many rounds must you play to recover the cost of the clubs?

56. Equipment rental. The local supermarket rents carpet cleaners for $20 a day. These cleaners use shampoo in a spe-cial cartridge that sells for $16 and is available only from the supermarket. A home carpet cleaner can be purchased for $300. Shampoo for the home cleaner is readily available for $9 a bottle. Past experience has shown that it takes two shampoo cartridges to clean the 10-foot-by-12- foot carpet