Calculus for Business

CALCULUS

FOR BUSINESS, ECONOMICS, LIFE SCIENCES,

AND SOCIAL SCIENCES

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.

Copyright 2010 by Pearson Education, Inc.

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CALCULUS


AND SOCIAL SCIENCES

TWELFTH EDITION

RAYMOND A. BARNETT Merritt College

MICHAEL R. ZIEGLER Marquette University

KARL E. BYLEEN Marquette University

Prentice Hall



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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

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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



Dedicated to the memory of Michael R. Ziegler,

trusted author, colleague, and friend.



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


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


CONTENTS

vii



viii Contents

Chapter 5 Graphing and Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .266

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

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

5-3 L Hopital s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .301

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

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

5-6 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .331

Chapter 5 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .344


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


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


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


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


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

Contents of Additional Calculus Topics to Accompany Calculus, 12e and

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 College Mathematics for Business, Economics,

Life Sciences and Social Sciences, 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

Applications Index



xi

PREFACE

The twelfth edition of Calculus for Business, Economics, Life Sciences, and SocialSciences is designed for a one- or two-term course in calculus for students who havehad one to two years of high school algebra or the equivalent. The book s overallapproach, 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. Notonly is it critical that the text be accurate and readable but also, in order for a bookto 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 incredibleimpact on how easily students comprehend the material. Here are some of the waysthis 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 exampleshelp students gain solid knowledge of the basic topics and assess their own levelof understanding before moving on.

Review material (Appendix A and Chapters 1 and 2) can be used judiciously tohelp 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 studentswith the content they need to remediate those skills.

Explore & Discuss problems lead the discussion into new concepts or buildupon a current topic. They help students of all levels gain better insight into themathematical concepts through thought-provoking questions that are effectivein 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 crafthomework assignments that best meet the needs of their students.

The MyMathLab course for this text is designed to help students help themselvesand provide instructors with actionable information about their progress.

In addition to the above, all students get substantial experience in modeling andsolving real-world problems through application exercises chosen from businessand economics, life sciences, and social sciences. Great care has been taken to writea 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). Thistext is one of three books in the authors college mathematics series. The othersare Finite Mathematics for Business, Economics, Life Sciences, and SocialSciences, and College Mathematics for Business, Economics, Life Sciences, andSocial Sciences; the latter contains selected content from the other two books.Additional Calculus Topics, a supplement written to accompany theBarnett/Ziegler/Byleen series, can be used in conjunction with these books.



New to This Edition

Fundamental to a book s growth and effectiveness is classroom use and feedback.Now in its twelfth edition, Calculus for Business, Economics, Life Sciences, andSocial Sciences has had the benefit of a substantial amount of both. Improvementsin this edition evolved out of the generous response from a large number of usersof 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 toprovide 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. (Seepages 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 sectionfor easy student reference.

The Self-Test on Basic Algebra has been renamed Diagnostic Algebra Test andhas moved from Appendix A to the front of the book just prior to Chapter 1 tobetter 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 StyleAs was stated earlier, this text is written for student comprehension. To thatend, the focus has been on making the book both mathematically correct andaccessible to students. Most derivations and proofs are omitted except where theirinclusion adds significant insight into a particular concept as the emphasis is oncomputational skills, ideas, and problem solving rather than mathematical theory.General concepts and results are typically presented only after particular caseshave been discussed.

DesignOne of the hallmark features of this text is the clean, straightforward design of itspages. 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 a2-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 easilyreference the content, examples, and Conceptual Insights they need to understandthe 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 (seepages 27, 100, 107).

Examples and Matched ProblemsMore than 300 completely worked examples are used to introduce concepts and todemonstrate 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 areclearly identified. To give students extra help in working through examples, dashedboxes 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 notneed these additional steps, many will appreciate the fact that the authors do notassume too much in the way of prior knowledge.

xii Preface



Preface xiii

EXAMPLE 5 Solving Logarithmic Equations Find x so that

SOLUTION

Property 7

Properties 5 and 6

Property 8x = 4

logb 4 = logb x

logb8 # 2

4= logb x

logb 8 - logb 4 + logb 2 = logb x

logb 43/2

- logb 82/3

+ logb 2 = logb x

32 logb 4 -

23 logb 8 + logb 2 = logb x

32 logb 4 -

23 logb 8 + logb 2 = logb x

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

EXPLORE & DISCUSS 2How 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 workwhile 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 foreasy reference.

Explore & DiscussEvery section contains Explore & Discuss problems at appropriate places toencourage 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. Thisserves to foster critical thinking and communication skills. The Explore & Discussmaterial can be used as in-class discussions or out-of-class group activities and iseffective in both small and large class settings.

Exercise SetsThe 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 ofthe same type and difficulty level. Each exercise set is designed to allow instructorsto craft just the right assignment for students. Exercise sets are categorized as A(routine, easy mechanics), B (more difficult mechanics), and C (difficult mechanicsand some theory) to make it easy for instructors to create assignments that areappropriate for their classes. The writing exercises, indicated by the icon , providestudents 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.

ApplicationsA major objective of this book is to give the student substantial experience inmodeling and solving real-world problems. Enough applications are included toconvince even the most skeptical student that mathematics is really useful (see theIndex of Applications at the back of the book). Almost every exercise set containsapplication problems, including applications from business and economics, life sci-ences, and social sciences. An instructor with students from all three disciplinescan let them choose applications from their own field of interest; if most studentsare 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 letrepresent 1940, for example, we would obtain a different logarithmic regres-

sion equation, but the prediction for 2015 would be the same. We would not letrepresent 1950 (the first year in Table 1) or any later year, because logarith-

mic functions are undefined at 0.x = 0

x = 0x = 0

CONCEPTUAL INSIGHT

The notation (2, 7) has two common mathematical interpretations: the ordered pair withfirst coordinate 2 and second coordinate 7, and the open interval consisting of all realnumbers 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 writtenfirst. So, is correct interval notation, but is not.(2, -7)(-7, 2)

(2, -7)

xiv Preface

of the applications are simplified versions of actual real-world problems inspiredby professional journals and books. No specialized experience is required to solveany of the application problems.

TechnologyAlthough access to a graphing calculator or spreadsheets is not assumed, it islikely that many students will want to make use of this technology. To assist thesestudents, optional graphing calculator and spreadsheet activities are included inappropriate places. These include brief discussions in the text, examples or por-tions of examples solved on a graphing calculator or spreadsheet, and exercisesfor the student to solve. For example, linear regression is introduced in Section1-3, and regression techniques on a graphing calculator are used at appropriatepoints to illustrate mathematical modeling with real data. All the optional graph-ing calculator material is clearly identified with the icon and can be omittedwithout loss of continuity, if desired. Optional spreadsheet material is identifiedwith the icon . All graphs are computer-generated to ensure mathematicalaccuracy. Graphing calculator screens displayed in the text are actual outputfrom a graphing calculator.

Additional Pedagogical FeaturesThe 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 morechallenging or just less frequent. These features provide much-needed guidance forstudents 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.TheBasic Algebra Review, in Appendix A, provides students with seven sections ofcontent to help them remediate in specific areas of need. Answers to theDiagnostic 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

Chapter ReviewsOften 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, andconcepts, keyed to completely worked examples, followed by a comprehensive setof review exercises.Answers to most review exercises are included at the back of thebook; each answer contains a reference to the section in which that type of problem isdiscussed so students can remediate any deficiencies in their skills on their own.

Content

The text begins with the development of a library of elementary functions inChapters 1 and 2, including their properties and uses. We encourage students toinvestigate mathematical ideas and processes graphically and numerically, as well asalgebraically. This development lays a firm foundation for studying mathematicsboth in this book and in future endeavors. Depending on the course syllabus and thebackground of students, some or all of this material can be covered at the beginningof a course, or selected portions can be referenced as needed later in the course.

The material in Part Two (Calculus) consists of differential calculus (Chapters3 5), integral calculus (Chapters 6 7), multivariable calculus (Chapter 8), and abrief 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 thatare 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 hereand throughout the text.

In Chapter 4 the derivatives of exponential and logarithmic functions are obtainedbefore 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 inSection 4-6. Elasticity of demand is introduced in Section 4-7. The topics in theselast three sections of Chapter 4 are not referred to elsewhere in the text and canbe omitted.

Chapter 5 focuses on graphing and optimization. The first two sections coverfirst-derivative and second-derivative graph properties. L Hpital s rule is dis-cussed in Section 5-3.A graphing strategy is presented and illustrated in Section5-4. Optimization is covered in Sections 5-5 and 5-6, including examples andproblems involving end-point solutions.

Chapter 6 introduces integration. The first two sections cover antidifferentia-tion techniques essential to the remainder of the text. Section 6-3 discussessome applications involving differential equations that can be omitted. Thedefinite 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 7-4deal with additional techniques of integration.Any or all of the topics in Chapter 7can 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

*Selected topics from Part One may be referred to as needed in Part Two or reviewed systematically before starting Part Two.



STUDENT SUPPLEMENTS

Student s Solutions Manual

By Garret J. Etgen, University of Houston

This manual contains detailed, carefully worked-outsolutions to all odd-numbered section exercises and allChapter Review exercises. Each section begins withThings 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 CollegeMathematics, 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

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These Worksheets provide students with a structuredplace to take notes, define key concepts and terms, andwork through unique examples to reinforce what istaught in the lecture.

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The video lectures with optional captioning for this textmake it easy and convenient for students to watchvideos from a computer at home or on campus. Thecomplete digitized set, affordable and portable forstudents, is ideal for distance learning or supplementalinstruction. There is a video for every text example.

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This manual contains detailed solutions to all even-numbered section problems.

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Mini Lectures are provided for the teaching assistant,adjunct, part-time, or even full-time instructor for lecturepreparation by providing learning objectives, examples(and answers) not found in the text, and teaching notes.

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TestGen (www.pearsoned.com/testgen) enables instructorsto build, edit, print, and administer tests using a computer-ized bank of questions developed to cover all the objectivesof the text. TestGen is algorithmically based, allowinginstructors to create multiple but equivalent versions of thesame question or test with the click of a button. Instructorscan also modify test bank questions or add new questions.The software and testbank are available for download fromPearson Education s online catalog.

PowerPoint Lecture Slides

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

xvii

Accuracy Check

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 ifnotification were sent to Karl E. Byleen, 9322 W. Garden Court, Hales Corners, WI53130; or by e-mail, to [email protected].



xviii

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Homework and Test Manager lets you assign home-work, quizzes, and tests that are automatically graded.Select just the right mix of questions from theMyMathLab exercise bank, instructor-created customexercises, and/or TestGen test items.

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Personalized Study Plan, generated when studentscomplete a test or quiz, indicates which topics havebeen 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 matchtheir course concepts.

Multimedia learning aids, such as videos for every

example in the text, provide help for students whenthey need it. Other student-help features include HelpMe Solve This and Additional Examples. These areassignable as homework, to further encourage their use.

Gradebook, designed specifically for mathematics andstatistics, automatically tracks students results, letsyou stay on top of student performance, and gives youcontrol over how to calculate final grades.

MathXL Exercise Builder allows you to create staticand algorithmic exercises for your online assignments.You can use the library of sample exercises as an easystarting point or use the Exercise Builder to edit anyof the course-related exercises.

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Acknowledgments

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

Christine Cosgrove, Fitchburg State CollegeDarryl Egley, North Harris CollegeLauren Fern, University of MontanaGregory Goeckel, Presbyterian CollegeGarland Guyton, Montgomery CollegeVirginia Hanning, San Jacinto CollegeBruce Hedman, University of ConnecticutYvette Hester, Texas A&M UniversityFritz Keinert, Iowa State UniversitySteven Klassen, Missouri Western State UniversityWesley W. Maiers, Valparaiso UniversityJames Martin, Christopher Newport UniversityGary R. Penner, Richland CollegeJon Prewett, University of WyomingCynthia Schultz, Illinois Valley Community CollegeMaria Terrell, Cornell UniversityFred M. Wright, Iowa State UniversityAmy 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, LouD 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, RobertH. 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, CarolineWoods, Charles W. Zimmerman, Pat Zrolka, and Cathleen A. Zucco-Tevelot.

We also express our thanks to:

Caroline Woods, Anthony Gagliardi, Damon Demas, John Samons, TheresaSchille, 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 thesupplemental 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.



Diagnostic Algebra Test

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 numberproperty:

(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 anirrational 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 - y1>2)(2x1>2 + 3y1>2)

(x1>2 + y1>2)250

32+

3-2

2-2

(9a4b-2)1>2u5>3u2>3

(x-3y2)-2(2 * 105)(3 * 10-3)

9u8v6

3u4v86(xy3)5

4.06 * 10-42.55 * 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 + 82 - 3x2x + 23x - 4

(2 + 3)x = ?

(+): 2 + (x + y) = ?( # ): x(y + z) = ?

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 followingreal number properties or definitions. Indicate which one.

x-1 + y-1

x-2 - y-2

1

7 + h-

1

7

h

(x + y)2 - x2

y

x

x2 - 16-

x + 4

x2 - 4x

3x

3x2 - 12x+

1

6x

2

5b-

4

3a3-

1

6a2b2

6x(2x + 1)2 - 15x2(2x + 1)(4x - y)2 - 9x2

6n3 - 9n2 - 15nt2 - 4t - 6

8x2 - 18xy + 9y212x2 + 5x - 3

Commutative (+ , # ) Associative (+ , # ) Distributive

Identity (+ , # ) Inverse (+ , # ) Subtraction

Division Negatives Zero

(A)

(B)

(C)

(D)

(E)

(F)

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 numbers:

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. 49. -6x2 + 7x - 1 = 0x2 - x - 20 = 0

3x2 - 21 = 0x2 = 5x

1u + h - 1u

h

1x - 5

x - 5

x - 5

1x - 15

3x

13x

41x - 3

21x

axp + bxq,

2x1>2 - 3x2>3

62x25

- 72(x - 1)34

(x - y) + 0 = (x - y)

u

-(v - w)= -

u

v - w

9 # (4y) = (9 # 4)y

(5m - 2)(2m + 3) =

(5m - 2)2m + (5m - 2)3

5u + (3v + 2) = (3v + 2) + 5u

(-7) - (-5) = (-7) + 3-(-5)4

xx



CALCULUS


AND SOCIAL SCIENCES



A LIBRARY

OF ELEMENTARY

FUNCTIONS

PART



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 inone 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.

A solution of an equation (or inequality) involving a single variable is a numberthat when substituted for the variable makes the equation (or inequality) true. Theset 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. Westart by recalling the idea of equivalent equations and equivalent inequalities. If weperform an operation on an equation (or inequality) that produces another equa-tion (or inequality) with the same solution set, then the two equations (or inequali-ties) are said to be equivalent. The basic idea in solving equations or inequalities isto perform operations that produce simpler equivalent equations or inequalitiesand 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 givenequation.

2. Each side of a given equation is multiplied by or divided by the samenonzero 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

,6 , 7 , ,= ,

Standard form: ax + b , 0 a - 0

x

2+ 2(3x - 1) 5

3 - 2(x + 3) =x

3- 5

SECTION 1-1 Linear Equations and Inequalities 3

1-1 Linear Equations and Inequalities

Linear Equations

Linear Inequalities

Applications



4 CHAPTER 1 Linear Equations and Graphs

Matched Problem 1 Solve and check:

EXPLORE & DISCUSS 1According to equality property 2, multiplying both sides of an equation by anonzero 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 toproduce 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 divisibleby each denominator, we can use the multiplication property of equality to clearthe denominators. The LCD (least common denominator) of the fractions, 6, isexactly what we are looking for! Actually, any common denominator will do, butthe LCD results in a simpler equivalent equation. So, we multiply both sides ofthe equation by 6:

*

Use the distributive property.

Combine like terms.


CHECK

Matched Problem 2 Solve and check:

In many applications of algebra, formulas or equations must be changed toalternative 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 theaccount 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 #(x + 2)

2

1

-

2

6 #x

3

1

= 30

6ax + 22

-x

3b = 6 # 5

x + 2

2-x

3= 5

x + 2

2-x

3= 5

x + 1

3-x

4=

1

2

3x - 2(2x - 5) = 2(x + 3) - 8

*Dashed boxes are used throughout the book to denote steps that are usually performed mentally.




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 isgiven 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 (lessthan) 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 expectthat if a positive number was added to any real number, the sum would be largerthan the original.That is essentially what the definition states. If we may alsowrite

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 realnumber line. If then a is to the left of b on the number line; if then c isto the right of d on the number line (Fig. 1). Check this geometric property with theinequalities in Example 4.

EXPLORE & DISCUSS 2Replace ? with or in each of the following:

(A)

(B)

(C)

(D)

Based on these examples, describe verbally the effect of multiplying both sides of aninequality 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)

76

c 7 d,a 6 b,

-3? -30-20?02?8

7 .6

- 10 + 10 = 0- 10 6 00 7 -10

-6 + 4 = -2-6 6 -2

3 + 2 = 53 6 5

b 7 a

a 6 b,

a + p = b.

a 6 b

7

6

S = 2LW + 2LH + 2WH

P =A

1 + r t

P(1 + r t) = A

P + Prt = A

A = P + Prt

r =A - P

Pt

Prt = A - P

P + Prt = A

A = P + Prt

a d 0 cb

Figure 1 a 6 b, c 7 d




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

THEOREM 2 Inequality Properties

An equivalent inequality will result, and the sense or direction will remain thesame 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 eachside 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 thatwe perform on equations, with the exception that the sense of the inequality revers-es if we multiply or divide both sides by a negative number. Otherwise, the sense ofthe 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 ofby 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 byinequalities, as shown in Table 1.

The numbers a and b in Table 1 are called the endpoints of the interval. Aninterval is closed if it contains all its endpoints and open if it does not contain anyof 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 thatstarts 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 numberline.

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

q-qq

q

q

qq-q

q-q

x


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.

EXPLORE & DISCUSS 3The solution to Example 5B shows the graph of the inequality What is thegraph of What is the corresponding interval? Describe the relationship be-tween these sets.

EXAMPLE 6 Solving a Linear Inequality Solve and graph:

SOLUTION Remove parentheses.

Combine like terms.

Subtract 6x 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 thepoint 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 -5.

x 6 3

(-7, 4]

[-5, q).x -5

-2 x 6 3.-

x -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 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

ba

ba

ba

ba

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, including9 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 problems.

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 willindicate the best way to introduce this variable.

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

3. Write a verbal statement using the conditions stated in the problem and thenwrite 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 priceof 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

26

2x

2

6

2

-6 6 2x 6

-3 - 3 6 2x + 3 - 3 9 - 3

-3 6 2x + 3 9

-3.-32x + 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.


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 newcar 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, whichis 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 dependenton the number of items produced at a certain cost per item.

EXAMPLE 9 Break-Even Analysis A multimedia company produces DVDs. One-time fixedcosts 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 aretailer. The DVD is sold to retail outlets at $17.40 each. How many DVDs mustbe 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 # 755 = $39.26 Shipping charges = $57.00

Price = $755.00

x = 755

1.052x = 794.26

1.052x + 57 = 851.26

x + 57 + 0.052x = 851.26

x + 57 + 0.052x = 851.26

Price + Shipping Charges + Sales Tax = Total Order Cost




Step 5 Check:

Matched Problem 9 How many DVSs would a multimedia company have to make and sell to breakeven if the fixed costs are $36,000, variable costs are $10.40 per DVD, and theDVDs 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 equals100. The index is based on prices of basic consumer goods and services. Table 2lists the CPI for several years from 1960 to 2005. What net annual salary in 2005would 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 LaborStatistics)

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 ofthe 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 tothe nearest dollar.

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

Exercises 1-1

195.3

29.6= 6.59797

85,774

13,000= 6.598

CPI RatioSalary Ratio

= $85,774 per year

x = 13,000 #195.3

29.6

x

13,000=

195.3

29.6

CPI in 2005 = 195.3

CPI in 1960 = 29.6

1960 = $13,000

x =

= $167,040= $167,040

17.4(9,600)48,000 + 12.4(9,600)

RevenueCosts

Table 2 CPI (1982 1984 = 100)

Year Index

1960 29.6

1975 53.8

1990 130.7

2005 195.3

A

Solve Problems 1 6.

1. 2.

3. 4.

5. 6. -4x 8-3x -12

5x + 2 7 12x + 3 6 -4

3y - 4 = 6y - 192m + 9 = 5m - 6

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 92 x + 3 5

-2x + 8 6 4-4x - 7 7 5




13. 14.

15. 16.

B

17. 18.

19. 20.

21. 22.

23. 24.

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 ineach case?

(A) (B)

(C) (D)

38. What can be said about the signs of the numbers a, b, and cin each case?

(A) (B)

(C) (D)

39. If both a and b are positive numbers and b/a is greater than1, then is positive or negative?

40. If both a and b are negative numbers and b/a is greater than1, 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, thentheir intersection is an open interval.

42. If the intersection of two closed intervals is nonempty, thentheir 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, thentheir union is an open interval.

a - b

a - b

a2

bc6 0

a

bc7 0

ab

c6 0abc 7 0

a

b6 0

a

b7 0

ab 6 0ab 7 0

-10 8 - 3u -6-3 4 - 7x 6 18

C = 59 (F - 32);

F =95C + 32;y = mx + b;

y (B Z 0)Ax + By = C;

y = -23x + 8;3x - 4y = 12;

-1 23 t + 5 11-495C + 32 68

-4 5x + 6 6 212 3x - 7 6 14

u

2-

2

36u

3+ 2

m

5- 3 6

3

5-m

2

y

4-

y

3=

1

2

x

5-x

6=

6

5

x - 2 2(x - 5)3 - y 4(y - 3)

-3(4 - x) = 5 - (x + 1)10x + 25(x - 3) = 275

-3y + 9 + y = 13 - 8y2u + 4 = 5u + 1 - 7u

x

-46

5

6

y

-57

3

2

46. If the intersection of two closed intervals is nonempty, thentheir union is a closed interval.

Applications

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

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

49. IRA. You have $500,000 in an IRA (Individual RetirementAccount) 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% annually. How should you divideyour 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 yourmoney between Fund A and Fund B to produce an annualinterest income of $30,000?

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

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

53. Retail and wholesale prices. Retail prices in a departmentstore are obtained by marking up the wholesale price by40%. That is, retail price is obtained by adding 40% of thewholesale 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 areobtained by marking down the retail price by 15%. That is,sale price is obtained by subtracting 15% of the retail pricefrom 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 ofgolf using a set of their clubs, and $44 if you have your ownclubs. If you buy a set of clubs for $270, how many roundsmust you play to recover the cost of the clubs?

56. Equipment rental. The local supermarket rents carpetcleaners for $20 a day. These cleaners use shampoo in a spe-cial cartridge that sells for $16 and is available only from thesupermarket. 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 twoshampoo cartridges to clean the 10-foot-by-12- foot carpet




in your living room with the rented cleaner. Cleaning thesame area with the home cleaner will consume three bottlesof shampoo. If you buy the home cleaner, how many timesmust you clean the living-room carpet to make buyingcheaper than renting?

57. Sales commissions. One employee of a computer store ispaid a base salary of $2,000 a month plus an 8% commissionon all sales over $7,000 during the month. How much mustthe employee sell in one month to earn a total of $4,000 forthe month?

58. Sales commissions. A second employee of the computerstore in Problem 57 is paid a base salary of $3,000 a monthplus a 5% commission on all sales during the month.

(A) How much must this employee sell in one month toearn a total of $4,000 for the month?

(B) Determine the sales level at which both employees re-ceive the same monthly income.

(C) If employees can select either of these payment meth-ods, how would you advise an employee to make thisselection?

59. Break-even analysis. A publisher for a promising newnovel figures fixed costs (overhead, advances, promotion,copy editing, typesetting) at $55,000, and variable costs(printing, paper, binding, shipping) at $1.60 for each bookproduced. If the book is sold to distributors for $11 each,how many must be produced and sold for the publisher tobreak even?

60. Break-even analysis. The publisher of a new book figuresfixed costs at $92,000 and variable costs at $2.10 for eachbook produced. If the book is sold to distributors for $15each, how many must be sold for the publisher to breakeven?

61. Break-even analysis. The publisher in Problem 59 findsthat rising prices for paper increase the variable costs to$2.10 per book.

(A) Discuss possible strategies the company might use todeal with this increase in costs.

(B) If the company continues to sell the books for $11, howmany books must they sell now to make a profit?

(C) If the company wants to start making a profit at thesame production level as before the cost increase, howmuch should they sell the book for now?

62. Break-even analysis. The publisher in Problem 60 findsthat rising prices for paper increase the variable costs to$2.70 per book.

(A) Discuss possible strategies the company might use todeal with this increase in costs.

(B) If the company continues to sell the books for $15, howmany books must they sell now to make a profit?

(C) If the company wants to start making a profit at thesame production level as before the cost increase, howmuch should they sell the book for now?

63. Wildlife management. A naturalist estimated the totalnumber of rainbow trout in a certain lake using thecapture mark recapture technique. He netted, marked, andreleased 200 rainbow trout. A week later, allowing for thor-ough mixing, he again netted 200 trout, and found 8 markedones among them. Assuming that the proportion of marked

fish in the second sample was the same as the proportion ofall marked fish in the total population, estimate the numberof rainbow trout in the lake.

64. Temperature conversion. If the temperature for a 24-hourperiod at an Antarctic station ranged between and14F (that is, ), what was the range in de-grees Celsius?

65. Psychology. The IQ (intelligence quotient) is found by di-viding the mental age (MA), as indicated on standard tests,by the chronological age (CA) and multiplying by 100. Forexample, if a child has a mental age of 12 and a chronologi-cal age of 8, the calculated IQ is 150. If a 9-year-old girl hasan IQ of 140, compute her mental age.

66. Psychology. Refer to Problem 65. If the IQ of a group of12-year-old children varies between 80 and 140, what is therange of their mental ages?

67. Anthropology. In their study of genetic groupings, anthro-pologists use a ratio called the cephalic index. This is theratio of the breadth B of the head to its length L (lookingdown from above) expressed as a percentage.A study of theGurung community of Nepal published in the KathmanduUniversity Medical Journal in 2005 found that the averagehead length of males was 18 cm, and their head breadthsvaried between 12 and 18 cm. Find the range of the cephalicindex for males. Round endpoints to one decimal place.

68. Anthropology. Refer to Problem 67. The same study foundthat the average head length of females was 17.4 cm, andtheir head breadths varied between 15 and 20 cm. Find therange of the cephalic index for females. Round endpoints toone decimal place.

Answers to Matched Problems

1. 2.

3. (A) (B)

4. (A) (B) (C)

5. (A)

(B)

6. or

7. or

8. $26,000 9. 7,500 DVDs 10. $27,547

[-1, 4)-1 x 6 4

[-4, q)x -4

(-q , 3)

-7 6 x 4;

766

H =S - 2LW

2L + 2WL =

S - 2WH

2W + 2H

x = 2x = 4

[Note: F = 95C + 32.]-49 F 14

-49F

L

BTop of

Head

Figure for 67 68

( [*7 4

x

)3

x

[*4

x

[ )*1 4

x



1-2 Graphs and Lines

SECTION 1-2 Graphs and Lines 13

In this section, we will consider one of the most basic geometric figures a line.When we use the term line in this book, we mean straight line. We will learn how torecognize and graph a line, and how to use information concerning a line to find itsequation. Examining the graph of any equation often results in additional insightinto the nature of the equation s solutions.

Cartesian Coordinate System

Recall that to form a Cartesian or rectangular coordinate system, we select two realnumber lines one horizontal and one vertical and let them cross through theirorigins as indicated in Figure 1. Up and to the right are the usual choices for the pos-itive directions. These two number lines are called the horizontal axis and thevertical axis, or, together, the coordinate axes. The horizontal axis is usually referredto as the x axis and the vertical axis as the y axis, and each is labeled accordingly.Thecoordinate axes divide the plane into four parts called quadrants, which are num-bered counterclockwise from I to IV (see Fig. 1).

Cartesian Coordinate System

Graphs of

Slope of a Line

Equations of Lines: Special Forms

Applications

Ax + By = C

II

III IV

I

R+ (10, *10)

P + (a, b)

Q+ (*5, 5)b

a

Coordinates

Abscissa

Ordinate

Axis

Origin

x

y

50*5

5

10

*10

*10 10

*5

Figure 1 The Cartesian (rectangular) coordinate system

*Here we use (a, b) as the coordinates of a point in a plane. In Section 1-1, we used (a, b) to represent an

interval on a real number line. These concepts are not the same. You must always interpret the symbol

(a, b) in terms of the context in which it is used.

Now we want to assign coordinates to each point in the plane. Given an arbi-trary point P in the plane, pass horizontal and vertical lines through the point(Fig. 1). The vertical line will intersect the horizontal axis at a point with coordinatea, and the horizontal line will intersect the vertical axis at a point with coordinate b.These two numbers, written as the ordered pair (a, b)* form the coordinates of thepoint P. The first coordinate, a, is called the abscissa of P; the second coordinate, b,is called the ordinate of P.The abscissa of Q in Figure 1 is and the ordinate of Qis 5.The coordinates of a point can also be referenced in terms of the axis labels.Thex coordinate of R in Figure 1 is 10, and the y coordinate of R is The point withcoordinates (0, 0) is called the origin.

The procedure we have just described assigns to each point P in the plane aunique pair of real numbers (a, b). Conversely, if we are given an ordered pair ofreal numbers (a, b), then, reversing this procedure, we can determine a unique pointP in the plane. Thus,

There is a one-to-one correspondence between the points in a plane and the

elements in the set of all ordered pairs of real numbers.

This is often referred to as the fundamental theorem of analytic geometry.

-10.

-5,




Graphs of

In Section 1-1, we called an equation of the form a linear equa-tion in one variable. Now we want to consider linear equations in two variables:

DEFINITION Linear Equations in Two Variables

A linear equation in two variables is an equation that can be written in thestandard form

where A,B, and C are constants (A and B not both 0), and x and y are variables.

A solution of an equation in two variables is an ordered pair of real numbers thatsatisfies the equation. For example, (4, 3) is a solution of . The solutionset of an equation in two variables is the set of all solut

Calculus for Business

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