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About the Cover For Mark van der Laan, the beauty of mathematical problems and the journeys he takes on the road to solving them are the driving force behind his work in the field of biostatistics. In particular, he is intrigued by the variety of approaches there may be to solve a problem and the fact that it requires a large diversity of scientists and people working in the field to find the most elegant and satisfying solutions. As he moved from an M.A. in mathematics to a Ph.D. in mathematical statistics from the University of Utrecht to his current position in the Department of Biostatistics at the University of California, Berkeley, Mark has found that the most interesting and creative mathematical problems are present in real-life applications. He says, “I have always realized, and have been told by experienced researchers, that solving these applied problems requires a thorough education in mathematics and that probability theory is fundamental. However, as in real life, the approach taken toward the solution is often by far the most important step and requires philosophical and abstract thinking.” MARK VAN DER LAAN Every day Mark is engaged in creatively solving mathematical problems that have implications Biostatistician in the fields of medical research, biology, and public health. For example, in collaboration with medical researchers at the University of California, San Francisco, Mark is investigating the effects of antiretroviral treatment (ART) on HIV/AIDS progression. As represented by the images on the cover, he is also involved in establishing the causal effect of air pollution on asthma in children, the causal effect of leisure-time activity and lean-to-fat ratio on health outcomes in the elderly, as well as the identification of regulatory networks in basic biology. Recognized for his progressive work in these fields, Mark van der Laan has received numerous awards. In April 2005, he was awarded the van Dantzig Award for his theoretical and practical contributions made to the fields of operation research and statistics. In August 2005, he received the COPSS (Committee of Presidents of Statistical Societies) Award, which is presented annually to a young researcher in recognition of outstanding contributions to the statistics profession. Mark currently holds the UC Berkeley Chancellor Endowed Chair 2005–2008, as well as the long-term Jiann-Ping Hsu/Karl E. Peace Endowed Chair in Biostatistics at University of California, Berkeley. Look for other featured applied researchers in forthcoming titles in the Tan applied mathematics series: PETER BLAIR HENRY International Economist Stanford University CHRIS SHANNON Mathematical Economist University of California, Berkeley JONATHAN D. FARLEY Applied Mathematician California Institute of Technology NAVIN KHANEJA Applied Scientist Harvard University Now that you’ve bought the textbook . . . GET THE BEST GRADE IN THE SHORTEST TIME POSSIBLE! Visit www.iChapters.com to view over 10,000 print, digital, and audio study tools that allow you to: • Study in less time to get the grade you want . . . using online resources such as chapter pre- and post-tests and personalized study plans. • Prepare for tests anywhere, anytime . . . using chapter review audio files that are downloadable to your MP3 player. • Practice, review, and master course concepts . . . using printed guides and manuals that work hand-in-hand with each chapter of your textbook. Join the thousands of students who have benefited from www.iChapters.com. Just search by author, title, or ISBN, then filter the results by “Study Tools” and select the format best suited for you. www.iChapters.com. Your First Study Break EDITION 5 APPLIED MATHEMATICS FOR THE MANAGERIAL, LIFE, AND SOCIAL SCIENCES SOO T. TAN STONEHILL COLLEGE Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States ©2010, 2007 Brooks/Cole, Cengage Learning Applied Mathematics: For the Managerial, Life, and Social Sciences, Fifth Edition Soo T. Tan ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher. 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Marketing Communications Manager: Mary Anne Payumo Senior Project Manager, Editorial Production: Cheryll Linthicum Creative Director: Rob Hugel Library of Congress Control Number: 2008937041 Senior Art Director: Vernon Boes Print Buyer: Judy Inouye ISBN-13: 978-0-495-55967-2 Permissions Editor: Bob Kauser ISBN-10: 0-495-55967-9 Production Service: Martha Emry Text Designer: Diane Beasley Brooks/Cole 10 Davis Drive Belmont, CA 94002-3098 USA Photo Researcher: Terri Wright Copy Editor: Betty Duncan Illustrator: Jade Myers, Matrix Art Services Compositor: Graphic World Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local office at www.cengage.com/international. Cover Designer: Irene Morris Cover Images: Mark Van Der Laan © Cengage Learning; Jogging People, Sally Ho/UpperCut Images/Getty Images; Woman with Microscope, Eric Audras/PhotoAlto/Getty Images; Spiral DNA in Lab, Mario Tama/Getty Images News/Getty Images Cengage Learning products are represented in Canada by Nelson Education, Ltd. To learn more about Brooks/Cole, visit www.cengage.com/brookscole Purchase any of our products at your local college store or at our preferred online store www.ichapters.com. Printed in Canada 1 2 3 4 5 6 7 12 11 10 09 08 TO PAT, BILL, AND MICHAEL This page intentionally left blank CONTENTS Preface xi CHAPTER 1 Fundamentals of Algebra 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 CHAPTER 2 1 Real Numbers 2 Polynomials 7 Factoring Polynomials 14 Rational Expressions 20 Integral Exponents 26 Solving Equations 30 Rational Exponents and Radicals 36 Quadratic Equations 44 Inequalities and Absolute Value 53 Chapter 1 Summary of Principal Formulas and Terms 62 Chapter 1 Concept Review Questions 63 Chapter 1 Review Exercises 63 Chapter 1 Before Moving On 65 Functions and Their Graphs 2.1 2.2 2.3 2.4 2.5 The Cartesian Coordinate System and Straight Lines 68 Equations of Lines 74 Using Technology: Graphing a Straight Line 84 Functions and Their Graphs 87 Using Technology: Graphing a Function 100 The Algebra of Functions 103 Linear Functions 111 PORTFOLIO: Esteban Silva 2.6 67 115 Using Technology: Linear Functions 120 Quadratic Functions 123 PORTFOLIO: Deb Farace 128 2.7 Using Technology: Finding the Points of Intersection of Two Graphs 132 Functions and Mathematical Models 134 Using Technology: Constructing Mathematical Models from Raw Data 144 Chapter 2 Summary of Principal Formulas and Terms 148 Chapter 2 Concept Review Questions 149 Chapter 2 Review Exercises 149 Chapter 2 Before Moving On 151 vi CONTENTS CHAPTER 3 Exponential and Logarithmic Functions 3.1 3.2 3.3 CHAPTER 4 4.2 4.3 4.4 CHAPTER 5 Exponential Functions 154 Using Technology 160 Logarithmic Functions 162 Exponential Functions as Mathematical Models 171 Using Technology: Analyzing Mathematical Models 180 Chapter 3 Summary of Principal Formulas and Terms 181 Chapter 3 Concept Review Questions 182 Chapter 3 Review Exercises 182 Chapter 3 Before Moving On 183 Mathematics of Finance 4.1 153 185 Compound Interest 186 Using Technology: Finding the Accumulated Amount of an Investment, the Effective Rate of Interest, and the Present Value of an Investment 201 Annuities 204 Using Technology: Finding the Amount of an Annuity 212 Amortization and Sinking Funds 215 Using Technology: Amortizing a Loan 225 Arithmetic and Geometric Progressions 228 Chapter 4 Summary of Principal Formulas and Terms 236 Chapter 4 Concept Review Questions 237 Chapter 4 Review Exercises 238 Chapter 4 Before Moving On 239 Systems of Linear Equations and Matrices 5.1 5.2 5.3 5.4 5.5 5.6 241 Systems of Linear Equations: An Introduction 242 Systems of Linear Equations: Unique Solutions 249 Using Technology: Systems of Linear Equations: Unique Solutions 263 Systems of Linear Equations: Underdetermined and Overdetermined Systems 265 Using Technology: Systems of Linear Equations: Underdetermined and Overdetermined Systems 274 Matrices 275 Using Technology: Matrix Operations 284 Multiplication of Matrices 287 Using Technology: Matrix Multiplication 299 The Inverse of a Square Matrix 301 Using Technology: Finding the Inverse of a Square Matrix 313 Chapter 5 Summary of Principal Formulas and Terms 316 Chapter 5 Concept Review Questions 316 Chapter 5 Review Exercises 317 Chapter 5 Before Moving On 319 CONTENTS CHAPTER 6 Linear Programming 6.1 6.2 6.3 6.4 321 Graphing Systems of Linear Inequalities in Two Variables 322 Linear Programming Problems 330 Graphical Solution of Linear Programming Problems 338 The Simplex Method: Standard Maximization Problems 351 PORTFOLIO: Morgan Wilson 352 6.5 CHAPTER 7 Using Technology: The Simplex Method: Solving Maximization Problems 372 The Simplex Method: Standard Minimization Problems 376 Using Technology: The Simplex Method: Solving Minimization Problems 387 Chapter 6 Summary of Principal Terms 390 Chapter 6 Concept Review Questions 390 Chapter 6 Review Exercises 391 Chapter 6 Before Moving On 393 Sets and Probability 395 7.1 7.2 7.3 Sets and Set Operations 396 The Number of Elements in a Finite Set 405 The Multiplication Principle 411 7.4 Permutations and Combinations 417 Using Technology: Evaluating n!, P(n, r), and C(n, r) Experiments, Sample Spaces, and Events 431 Definition of Probability 439 Rules of Probability 449 PORTFOLIO: Stephanie Molina 413 7.5 7.6 7.7 430 PORTFOLIO: Todd Good 451 Chapter 7 Summary of Principal Formulas and Terms 459 Chapter 7 Concept Review Questions 460 Chapter 7 Review Exercises 460 Chapter 7 Before Moving On 462 CHAPTER 8 Additional Topics in Probability 8.1 8.2 8.3 8.4 8.5 463 Use of Counting Techniques in Probability 464 Conditional Probability and Independent Events 471 Bayes’ Theorem 485 Distributions of Random Variables 494 Using Technology: Graphing a Histogram 502 Expected Value 504 PORTFOLIO: Ann-Marie Martz 510 8.6 Variance and Standard Deviation 516 Using Technology: Finding the Mean and Standard Deviation 527 Chapter 8 Summary of Principal Formulas and Terms 528 Chapter 8 Concept Review Questions 529 Chapter 8 Review Exercises 530 Chapter 8 Before Moving On 531 vii viii CONTENTS CHAPTER 9 The Derivative 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 533 Limits 534 Using Technology: Finding the Limit of a Function 552 One-Sided Limits and Continuity 554 Using Technology: Finding the Points of Discontinuity of a Function 568 The Derivative 570 Using Technology: Graphing a Function and Its Tangent Line 586 Basic Rules of Differentiation 589 Using Technology: Finding the Rate of Change of a Function 600 The Product and Quotient Rules; Higher-Order Derivatives 602 Using Technology: The Product and Quotient Rules 615 The Chain Rule 618 Using Technology: Finding the Derivative of a Composite Function 629 Differentiation of Exponential and Logarithmic Functions 630 Using Technology 640 Marginal Functions in Economics 642 PORTFOLIO: Richard Mizak 647 Chapter 9 Summary of Principal Formulas and Terms 651 Chapter 9 Concept Review Questions 653 Chapter 9 Review Exercises 653 Chapter 9 Before Moving On 656 CHAPTER 10 Applications of the Derivative 10.1 10.2 10.3 10.4 10.5 657 Applications of the First Derivative 658 Using Technology: Using the First Derivative to Analyze a Function 675 Applications of the Second Derivative 678 Using Technology: Finding the Inflection Points of a Function 695 Curve Sketching 697 Using Technology: Analyzing the Properties of a Function 709 Optimization I 711 Using Technology: Finding the Absolute Extrema of a Function 724 Optimization II 726 PORTFOLIO: Gary Li 731 Chapter 10 Summary of Principal Terms 738 Chapter 10 Concept Review Questions 738 Chapter 10 Review Exercises 738 Chapter 10 Before Moving On 740 CHAPTER 11 Integration 11.1 11.2 11.3 11.4 741 Antiderivatives and the Rules of Integration 742 Integration by Substitution 755 Area and the Definite Integral 765 The Fundamental Theorem of Calculus 774 Using Technology: Evaluating Definite Integrals 784 CONTENTS 11.5 11.6 11.7 CHAPTER 12 Evaluating Definite Integrals 785 Using Technology: Evaluating Definite Integrals for Piecewise-Defined Functions 795 Area between Two Curves 797 Using Technology: Finding the Area between Two Curves 807 Applications of the Definite Integral to Business and Economics 808 Using Technology: Business and Economic Applications/Technology Exercises 820 Chapter 11 Summary of Principal Formulas and Terms 821 Chapter 11 Concept Review Questions 822 Chapter 11 Review Exercises 823 Chapter 11 Before Moving On 826 Calculus of Several Variables 12.1 12.2 12.3 827 Functions of Several Variables 828 Partial Derivatives 837 Using Technology: Finding Partial Derivatives at a Given Point 849 Maxima and Minima of Functions of Several Variables 850 PORTFOLIO: Kirk Hoiberg 852 Chapter 12 Summary of Principal Terms 860 Chapter 12 Concept Review Questions 860 Chapter 12 Review Exercises 861 Chapter 12 Before Moving On 862 Answers to Odd-Numbered Exercises 863 Index 915 How-To Technology Index 920 ix This page intentionally left blank PREFACE M ath is an integral part of our increasingly complex daily life. Applied Mathematics for the Managerial, Life, and Social Sciences, Fifth Edition, attempts to illustrate this point with its applied approach to mathematics. Our objective for this Fifth Edition is threefold: (1) to write an applied text that motivates students while providing the background in the quantitative techniques necessary to better understand and appreciate the courses normally taken in undergraduate training, (2) to lay the foundation for more advanced courses, such as statistics and operations research, and (3) to make the text a useful tool for instructors. Since the book contains more than enough material for the usual two-semester or three-semester course, the instructor may be flexible in choosing the topics most suitable for his or her course. The following chart on chapter dependency is provided to help the instructor design a course that is most suitable for the intended audience. 1 2 3 7 Fundamentals of Algebra Functions and Their Graphs Exponential and Logarithmic Functions Sets and Probability 4 5 9 8 Mathematics of Finance Systems of Linear Equations and Matrices The Derivative Additional Topics in Probability 6 10 Linear Programming Applications of the Derivative 11 12 Integration Calculus of Several Variables xii PREFACE THE APPROACH Level of Presentation My approach is intuitive, and I state the results informally. However, I have taken special care to ensure that this approach does not compromise the mathematical content and accuracy. Problem-Solving Approach A problem-solving approach is stressed throughout the book. Numerous examples and applications illustrate each new concept and result. Special emphasis is placed on helping students formulate, solve, and interpret the results of the problems involving applications. Because students often have difficulty setting up and solving word problems, extra care has been taken to help students master these skills: ■ ■ ■ ■ Very early on in the text students are given practice in solving word problems (see Example 7, Section 1.8). Guidelines are given to help students formulate and solve word problems (see Section 2.7). One entire section is devoted to modeling and setting up linear programming problems (see Section 6.2). Optimization problems are covered in two sections. In Section 10.4 students are asked to solve problems in which the objective function to be optimized is given, and in Section 10.5 students are asked to solve problems in which the optimization problems must first be formulated. Intuitive Introduction to Concepts Mathematical concepts are introduced with concrete, real-life examples wherever appropriate. An illustrative list of some of the topics introduced in this manner follows: ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ The algebra of functions: The U.S. Budget Deficit Mathematical modeling: Social Security Trust Fund Assets Limits: The Motion of a Maglev The chain rule: The Population of Americans Age 55 and Older Increasing and decreasing functions: The Fuel Economy of a Car Concavity: U.S. and World Population Growth Inflection points: The Point of Diminishing Returns Curve sketching: The Dow Jones Industrial Average on “Black Monday” Exponential functions: Income Distribution of American Families Area between two curves: Petroleum Saved with Conservation Measures Connections One example (the maglev) is used as a common thread throughout the development of calculus—from limits through integration. The goal here is to show students the connections between the concepts presented—limits, continuity, rates of change, the derivative, the definite integral, and so on. Motivation Illustrating the practical value of mathematics in applied areas is an important objective of my approach. Many of the applications are based on mathematical models (functions) that I have constructed using data drawn from various sources, including current newspapers, magazines, and the Internet. Sources are given in the text for these applied problems. xiii PREFACE Modeling I believe that one of the important skills that a student should acquire is the ability to translate a real problem into a mathematical model that can provide insight into the problem. In Section 2.7, the modeling process is discussed, and students are asked to use models (functions) constructed from real-life data to answer questions. Students get hands-on experience constructing these models in the Using Technology sections. NEW TO THIS EDITION Algebra Review Where Students Need It Most Well-placed algebra review notes, keyed to the early algebra review chapters, appear where students often need help in the calculus portion of the text. These are indicated by the (x ) icon. See this feature in action on pages 542 and 577. 2 EXAMPLE 6 Evaluate: lim hS0 11 ⫹ h ⫺ 1 h Letting h approach zero, we obtain the indeterminate form 0/0. Next, we rationalize the numerator of the quotient by multiplying both the numerator and the denominator by the expression 1 11 ⫹ h ⫹ 12 , obtaining Solution 1 11 ⫹ h ⫺ 12 1 11 ⫹ h ⫹ 12 11 ⫹ h ⫺ 1 ⫽ * (x ) See page 41. h h1 11 ⫹ h ⫹ 12 1⫹h⫺1 ⫽ 1 1a ⫺ 1b 2 1 1a ⫹ 1b 2 ⫽ a ⫺ b h1 11 ⫹ h ⫹ 12 2 h h1 11 ⫹ h ⫹ 12 1 ⫽ 11 ⫹ h ⫹ 1 ⫽ Motivating Real-World Applications More than 220 new applications have been added to the Applied Examples and Exercises. Among these applications are global warming, depletion of Social Security trust fund assets, driving costs for a 2008 medium-sized sedan, hedge fund investments, mobile instant messaging accounts, hiring lobbyists, Web conferencing, the autistic brain, the revenue of Polo Ralph Lauren, U.S. health-care IT spending, and consumption of bottled water. APPLIED EXAMPLE 1 Global Warming The increase in carbon dioxide (CO2) in the atmosphere is a major cause of global warming. The Keeling curve, named after Charles David Keeling, a professor at Scripps Institution of Oceanography, gives the average amount of CO2, measured in parts per million volume (ppmv), in the atmosphere from the beginning of 1958 through 2007. Even though data were available for every year in this time interval, we’ll construct the curve based only on the following randomly selected data points. Year Amount 1958 1970 1974 1978 1985 1991 1998 2003 2007 315 325 330 335 345 355 365 375 380 The scatter plot associated with these data is shown in Figure 54a. A mathematical model giving the approximate amount of CO2 in the atmosphere during this period is given by A(t) ⫽ 0.010716t 2 ⫹ 0.8212t ⫹ 313.4 (1 ⱕ t ⱕ 50) xiv PREFACE Modeling with Data Modeling with Data exercises are now found in many of the Using Technology sections throughout the text. Students can actually see how some of the functions found in the exercises are constructed. (See Internet Users in China, Exercise 40, page 159, and the corresponding exercise where the model is derived in Exercise 14, page 161.) 40. INTERNET USERS IN CHINA The number of Internet users in China is projected to be N(t) ⫽ 94.5e0.2t (1 ⱕ t ⱕ 6) where N(t) is measured in millions andDtATA is measured in of Internet users in 14. MODELING WITH The number years, with t ⫽ 1 corresponding the beginning of 2005. China (in to millions) from the beginning of 2005 through a. How many Internet 2010 usersare were thereinat the the following beginningtable: of shown 2005? At the beginning of 2006? b. How many Internet users are there expected to be at the 2005 2006 2007 2008 2009 2010 beginning of 2010? Year Number 116.1 141.9 169.0 209.0 258.1 314.8 c. Sketch the graph of N. Source: C. E. Unterberg a. Use ExpReg to find an exponential regression model for the data. Let t ⫽ 1 correspond to the beginning of 2005. Hint: a x ⫽ e x ln a. b. Plot the scatter diagram and the graph of the function f found in part (a). Making Connections with Technology Many Using Technology sections have been updated. A new example—Market for CholesterolReducing Drugs—has been added to Using Technology 2.5. Using Technology 3.3 includes a new example in which an exponential model is constructed—Internet Gaming Sales—using the logistic function of a graphing utility. Another new example—TV Mobile Phones—has been added to Using Technology 10.3. Additional graphing calculator screens in some sections and Exploring with Technology examples illustrating the use of the graphing calculator to solve inequalities and to generate random numbers have been added throughout. APPLIED EXAMPLE 2 Market for Cholesterol-Reducing Drugs In a study conducted in early 2000, experts projected a rise in the market for cholesterol-reducing drugs. The U.S. market (in billions of dollars) for such drugs from 1999 through 2004 is approximated by M(t) ⫽ 1.95t ⫹ 12.19 where t is measured in years, with t ⫽ 0 corresponding to 1999. a. Plot the graph of the function M in the viewing window [0, 5] ⫻ [0, 25]. b. Assuming that the projection held and the trend continued, what was the market for cholesterol-reducing drugs in 2005 (t ⫽ 6)? APPLIED EXAMPLE 1 Internet Gaming Sales The estimated growth c. What was the rate of increase of the market for cholesterol-reducing drugs in global Internet-gaming revenue (in billions of dollars), as predicted by over the period in question? industry analysts, is given in the following table: Source: S. G. Cowen Year Revenue 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 3.1 3.9 5.6 8.0 11.8 15.2 18.2 20.4 22.7 24.5 a. Use Logistic to find a regression model for the data. Let t ⫽ 0 correspond to 2001. b. Plot the scatter diagram and the graph of the function f found in part (a) using the viewing window [0, 9] ⫻ [0, 30]. Source: Christiansen Capital/Advisors PREFACE Variety of Problem Types Additional rote questions, true or false questions, and concept questions have been added throughout the text to enhance the exercise sets. (See, for example, the graphical questions added to Concept Questions 2.3, page 95.) 2.3 Concept Questions 1. a. What is a function? b. What is the domain of a function? The range of a function? c. What is an independent variable? A dependent variable? c. d. y 2. a. What is the graph of a function? Use a drawing to illustrate the graph, the domain, and the range of a function. b. If you are given a curve in the xy-plane, how can you tell if the graph is that of a function f defined by y ⫽ f(x)? y y x 0 x 0 3. Are the following graphs of functions? Explain. a. b. y 4. What are the domain and range of the function f with the following graph? y 4 0 x 0 x 3 2 1 (2, 12 ) 1 Action-Oriented Study Tabs Convenient color-coded study tabs, similar to Post-it® flags, make it easy for students to tab pages that they want to return to later, whether it be for additional review, exam preparation, online exploration, or identifying a topic to be discussed with the instructor. xv 2 3 x 4 5 xvi PREFACE Specific Content Changes More applications have been added to the algebra review chapters. The discussion of mortgages has been enhanced with a new example on adjustable-rate mortgages and the addition of many new applied exercises. Section 2.7, on functions and mathematical models, has been reorganized and new models have been introduced. Here, students are now asked to use a model describing global warming to predict the amount of carbon dioxide (CO2) that will be present in the atmosphere in 2010, and a model describing the assets of the Social Security trust fund to determine when those assets are expected to be depleted. The chain rule in Section 9.6 is now introduced with an application: the population of Americans aged 55 years and older. A How-To Technology Index has been added for easy reference. New Using Technology Excel sections for Microsoft Office 2007 are now available on the Web. ■ ■ ■ ■ ■ ■ TRUSTED FEATURES In addition to the new features, we have retained many of the following hallmarks that have made this series so usable and well-received in past editions: ■ ■ ■ ■ Self-Check Exercises Offering students immediate feedback on key concepts, these exercises begin each end-ofsection exercise set. Fully workedout solutions can be found at the end of each exercise section. Section exercises to help students understand and apply concepts Optional technology sections to explore mathematical ideas and solve problems End-of-chapter review sections to assess understanding and problem-solving skills Features to motivate further exploration 2.5 Self-Check Exercises A manufacturer has a monthly fixed cost of $60,000 and a production cost of $10 for each unit produced. The product sells for $15/unit. 1. What is the cost function? 2. What is the revenue function? Concept Questions 2.5 Concept Questions Designed to test students’ understanding of the basic concepts discussed in the section, these questions encourage students to explain learned concepts in their own words. 1. a. What is a linear function? Give an example. b. What is the domain of a linear function? The range? c. What is the graph of a linear function? Exercises Each exercise section contains an ample set of problems of a routine computational nature followed by an extensive set of application-oriented problems. 2. What is the general form of a linear cost function? A linear revenue function? A linear profit function? 3. What is the profit function? 4. Compute the profit (loss) corresponding to production levels of 10,000 and 14,000 units/month. Solutions to Self-Check Exercises 2.5 can be found on page 120. 3. Explain the meaning of each term: a. Break-even point b. Break-even quantity c. Break-even revenue 2.5 Exercises In Exercises 1–10, determine whether the equation defines y as a linear function of x. If so, write it in the form y ⴝ mx ⴙ b. 1. 2x ⫹ 3y ⫽ 6 2. ⫺2x ⫹ 4y ⫽ 7 3. x ⫽ 2y ⫺ 4 4. 2x ⫽ 3y ⫹ 8 5. 2x ⫺ 4y ⫹ 9 ⫽ 0 6. 3x ⫺ 6y ⫹ 7 ⫽ 0 7. 2 x 2 ⫺ 8y ⫹ 4 ⫽ 0 9. 2x ⫺ 3y2 ⫹ 8 ⫽ 0 8. 3 1x ⫹ 4y ⫽ 0 10. 2x ⫹ 1y ⫺ 4 ⫽ 0 11. A manufacturer has a monthly fixed cost of $40,000 and a production cost of $8 for each unit produced. The product sells for $12/unit. In Exercises 15–20, find the point of intersection of each pair of straight lines. 15. y ⫽ 3 x ⫹ 4 y ⫽ ⫺2 x ⫹ 14 16. y ⫽ ⫺4x ⫺ 7 ⫺y ⫽ 5x ⫹ 10 17. 2x ⫺ 3y ⫽ 6 3x ⫹ 6y ⫽ 16 18. 2 x ⫹ 4y ⫽ 11 ⫺5x ⫹ 3y ⫽ 5 19. 1 x⫺5 4 3 2x ⫺ y ⫽ 1 2 y⫽ 20. y ⫽ 2 x⫺4 3 x ⫹ 3y ⫹ 3 ⫽ 0 PREFACE Using Technology These optional features appear after the section exercises. They can be used in the classroom if desired or as material for self-study by the student. Here, the graphing calculator and Microsoft Excel 2003 are used as a tool to solve problems. (Instructions for Microsoft Excel 2007 are given at the Companion Website.) These sections are written in the traditional example–exercise format, with answers given at the back of the book. Illustrations showing graphing calculator screens are extensively used. In keeping with the theme of motivation through real-life examples, many sourced applications are again included. Students can construct their own models using real-life data in many of the Using Technology sections. These include models for the growth of the Indian gaming industry, TIVO owners, nicotine content of cigarettes, computer security, and online gaming, among others. USING TECHNOLOGY xvii Finding the Accumulated Amount of an Investment, the Effective Rate of Interest, and the Present Value of an Investment Graphing Utility Some graphing utilities have built-in routines for solving problems involving the mathematics of finance. For example, the TI-83/84 TVM SOLVER function incorporates several functions that can be used to solve the problems that are encountered in Sections 4.1–4.3. To access the TVM SOLVER on the TI-83 press 2nd , press FINANCE , and then select 1: TVM Solver . To access the TVM Solver on the TI-83 plus and the TI-84, press APPS , press 1: Finance , and then select 1: TVM Solver . Step-by-step procedures for using these functions can be found on our Companion Web site. EXAMPLE 1 Finding the Accumulated Amount of an Investment Find the accumulated amount after 10 years if $5000 is invested at a rate of 10% per year compounded monthly. Solution Using the TI-83/84 TVM SOLVER with the following inputs, N ⫽ 120 N = 120 I% = 10 PV = −5000 PMT = 0 FV = 13535.20745 P/Y = 12 C/Y = 12 PMT : END BEGIN I% ⫽ 10 PV ⫽ ⫺5000 PMT ⫽ 0 (10)(12) Recall that an investment is an outflow. FV ⫽ 0 P/Y ⫽ 12 The number of payments each year C/Y ⫽ 12 The number of conversion periods each year PMT:END BEGIN FIGURE T1 The TI-83/84 screen showing the future value (FV) of an investment we obtain the display shown in Figure T1. We conclude that the required accumulated amount is $13,535.21. EXAMPLE 2 Finding the Effective Rate of Interest Find the effective rate of interest corresponding to a nominal rate of 10% per year compounded quarterly. Eff (10, 4) Excel 10.38128906 Excel has many built-in functions for solving problems involving the mathematics of SolutiontheHere functionvalue), of the TI-83/84 calculator to obtain the finance. Here we illustrate usewe of use thethe FVEff(future EFFECT (effective rate), result shown in Figure T2. The required effective rate is approximately 10.38% per and the PV (presentyear. value) functions to solve problems of the type that we have encountered in Section 4.1. FIGURE T2 EXAMPLE 3 Finding the Present Value of an Investment Find the present value the of $20,000 due in 5 years if the interest rateInvestment is 7.5% per year compounded EXAMPLE 4 Finding Accumulated Amount of an Find the accumulated amountdaily. after 10 years if $5000 is invested at a rate of 10% per year compounded monthly.l i Ui h TI 83/84 i h h f ll i i The TI-83/84 screen showing the effective rate of interest (Eff) Exploring with Technology Designed to explore mathematical concepts and to shed further light on examples in the text, these optional questions appear throughout the main body of the text and serve to enhance the student’s understanding of the concepts and theory presented. Often the solution of an example in the text is augmented with a graphical or numerical solution. Complete solutions to these exercises are given in the Instructor’s Solutions Manual. Exploring with TECHNOLOGY In the opening paragraph of Section 3.1, we pointed out that the accumulated amount of an account earning interest compounded continuously will eventually outgrow by far the accumulated amount of an account earning interest at the same nominal rate but earning simple interest. Illustrate this fact using the following example. Suppose you deposit $1000 in account I, earning interest at the rate of 10% per year compounded continuously so that the accumulated amount at the end of t years is A1(t) ⫽ 1000e0.1t. Suppose you also deposit $1000 in account II, earning simple interest at the rate of 10% per year so that the accumulated amount at the end of t years is A2(t) ⫽ 1000(1 ⫹ 0.1t). Use a graphing utility to sketch the graphs of the functions A1 and A2 in the viewing window [0, 20] ⫻ [0, 10,000] to see the accumulated amounts A1(t) and A2(t) over a 20-year period.
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