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College Mathematics for Business, Economics, Life Sciences, and Social Sciences For these Global Editions, the editorial team at Pearson has collaborated with educators across the world to address a wide range of subjects and requirements, equipping students with the best possible learning tools This Global Edition preserves the cutting-edge approach and pedagogy of the original, but also features alterations, customization, and adaptation from the North American version Global edition Global edition Global edition THIRTEENTH edition Barnett • Ziegler • Byleen This is a special edition of an established title widely used by colleges and universities throughout the world Pearson published this exclusive edition for the benefit of students outside the United States and Canada If you purchased this book within the United States or Canada, you should be aware that it has been imported without the approval of the Publisher or Author College Mathematics f or Business, Economics, Life Sciences, a nd Social Sciences THIRTEENTH edition R aymond A Barnett • Michael R Ziegler • Karl E Byleen Pearson Global Edition BARNETT_1292057661_mech.indd 16/07/14 11:29 am College Mathematics For Business, Economics, Life Sciences, and Social sciences Thirteenth Edition Global Edition Raymond A Barnett Michael R Ziegler Karl E Byleen Merritt College Marquette University Marquette Universit y Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montréal Toronto Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo A01_BARN7668_13_GE_FM.indd 7/18/14 7:09 PM Editor in Chief: Deirdre Lynch Executive Editor: Jennifer Crum Project Manager: Kerri Consalvo Editorial Assistant: Joanne Wendelken Senior Managing Editor: Karen Wernholm Senior Production Supervisor: Ron Hampton Head of Learning Asset Acquisition, Global Edition: Laura Dent Acquisitions Editor, Global Edition: Subhasree Patra 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page 594, Vario Images/Alamy; page 651, P Amedzro/ Alamy; page 733, Anonymous Donor/Alamy; page 795, Shime/Fotolia; page 838, Aurora Photos/Alamy Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsonglobaleditions.com © Pearson Education Limited 2015 The rights of Raymond A Barnett, Michael R Ziegler, and Karl E Byleen to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988 Authorized adaptation from the United States edition, entitled College Mathematics for Business, Economics, Life Sciences and Social Sciences, 13th edition, ISBN 978-0-321-94551-8, by Raymond A Barnett, Michael R Ziegler, and Karl E Byleen, published by Pearson Education © 2015 All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a license permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks Where those designations appear in this book, and the publisher was aware of a trademark claim, the designations have been printed in initial caps or all caps All trademarks used herein are the property of their respective owners The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners ISBN 10: 1-292-05766-1 ISBN 13: 978-1-292-05766-8 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library 10 9 8 7 6 5 4 3 2 1 Typeset in 11 TimesTen-Roman by Integra Publishing Services Printed and bound by Courier Kendallville in The United States of America A01_BARN7668_13_GE_FM.indd 7/18/14 7:09 PM Contents Preface Diagnostic Prerequisite Test 19 Part Chapter A Library of Elementary Functions Linear Equations and Graphs 22 1.1 Linear Equations and Inequalities 23 1.2 Graphs and Lines 32 1.3 Linear Regression 46 Chapter Summary and Review 58 Review Exercises 59 Chapter 2 Functions and Graphs 62 Functions 63 Elementary Functions: Graphs and Transformations 77 Quadratic Functions 89 Polynomial and Rational Functions 104 Exponential Functions 115 Logarithmic Functions 126 Chapter Summary and Review 137 Review Exercises 140 2.1 2.2 2.3 2.4 2.5 2.6 Part Finite Mathematics Chapter 3 Mathematics of Finance 146 Simple Interest 147 Compound and Continuous Compound Interest 154 Future Value of an Annuity; Sinking Funds 167 Present Value of an Annuity; Amortization 175 Chapter Summary and Review 187 Review Exercises 189 3.1 3.2 3.3 3.4 Chapter 4 Systems of Linear Equations; Matrices 193 Review: Systems of Linear Equations in Two Variables 194 Systems of Linear Equations and Augmented Matrices 207 Gauss–Jordan Elimination 216 Matrices: Basic Operations 230 Inverse of a Square Matrix 242 Matrix Equations and Systems of Linear Equations 254 Leontief Input–Output Analysis 262 Chapter Summary and Review 270 Review Exercises 271 4.1 4.2 4.3 4.4 4.5 4.6 4.7 A01_BARN7668_13_GE_FM.indd 7/18/14 7:09 PM Contents Chapter Linear Inequalities and Linear Programming 275 5.1 Linear Inequalities in Two Variables 276 5.2 Systems of Linear Inequalities in Two Variables 283 5.3 Linear Programming in Two Dimensions: A Geometric Approach 290 Chapter Summary and Review 302 Review Exercises 303 Chapter Linear Programming: The Simplex Method 305 6.1 The Table Method: An Introduction to the Simplex Method 306 6.2 The Simplex Method: Maximization with Problem Constraints of the Form … 317 6.3 The Dual Problem: Minimization with Problem Constraints of the Form Ú 333 6.4 Maximization and Minimization with Mixed Problem Constraints 346 Chapter Summary and Review 361 Review Exercises 362 Chapter Logic, Sets, and Counting 365 Logic 366 Sets 374 Basic Counting Principles 381 Permutations and Combinations 389 Chapter Summary and Review 400 Review Exercises 402 7.1 7.2 7.3 7.4 Chapter Probability 405 Sample Spaces, Events, and Probability 406 Union, Intersection, and Complement of Events; Odds 419 Conditional Probability, Intersection, and Independence 431 Bayes’ Formula 445 Random Variable, Probability Distribution, and Expected Value 452 Chapter Summary and Review 461 Review Exercises 463 8.1 8.2 8.3 8.4 8.5 Chapter 9 Markov Chains 467 9.1 Properties of Markov Chains 468 9.2 Regular Markov Chains 479 9.3 Absorbing Markov Chains 489 Chapter Summary and Review 503 Review Exercises 504 A01_BARN7668_13_GE_FM.indd 7/18/14 7:09 PM Contents Part Chapter 10 Calculus Limits and the Derivative 508 Introduction to Limits 509 Infinite Limits and Limits at Infinity 523 Continuity 535 The Derivative 546 Basic Differentiation Properties 561 Differentials 570 Marginal Analysis in Business and Economics 577 Chapter 10 Summary and Review 588 Review Exercises 589 10.1 10.2 10.3 10.4 10.5 10.6 10.7 Chapter 11 Additional Derivative Topics 594 The Constant e and Continuous Compound Interest 595 Derivatives of Exponential and Logarithmic Functions 601 Derivatives of Products and Quotients 610 The Chain Rule 618 Implicit Differentiation 628 Related Rates 634 Elasticity of Demand 640 Chapter 11 Summary and Review 647 Review Exercises 649 11.1 11.2 11.3 11.4 11.5 11.6 11.7 Chapter 12 Graphing and Optimization 651 First Derivative and Graphs 652 Second Derivative and Graphs 668 L’Hôpital’s Rule 685 Curve-Sketching Techniques 694 Absolute Maxima and Minima 707 Optimization 715 Chapter 12 Summary and Review 728 Review Exercises 729 12.1 12.2 12.3 12.4 12.5 12.6 Chapter 13 Integration 733 Antiderivatives and Indefinite Integrals 734 Integration by Substitution 745 Differential Equations; Growth and Decay 756 The Definite Integral 767 The Fundamental Theorem of Calculus 777 Chapter 13 Summary and Review 789 Review Exercises 791 13.1 13.2 13.3 13.4 13.5 A01_BARN7668_13_GE_FM.indd 7/18/14 7:09 PM Contents Chapter 14 Additional Integration Topics 795 Area Between Curves 796 Applications in Business and Economics 805 Integration by Parts 817 Other Integration Methods 823 Chapter 14 Summary and Review 834 Review Exercises 835 14.1 14.2 14.3 14.4 Chapter 15 Multivariable Calculus 838 Functions of Several Variables 839 Partial Derivatives 848 Maxima and Minima 857 Maxima and Minima Using Lagrange Multipliers 865 Method of Least Squares 874 Double Integrals over Rectangular Regions 884 Double Integrals over More General Regions 894 Chapter 15 Summary and Review 902 Review Exercises 905 15.1 15.2 15.3 15.4 15.5 15.6 15.7 Appendix A Basic Algebra Review 908 A.1 A.2 A.3 A.4 A.5 A.6 A.7 Real Numbers 908 Operations on Polynomials 914 Factoring Polynomials 920 Operations on Rational Expressions 926 Integer Exponents and Scientific Notation 932 Rational Exponents and Radicals 936 Quadratic Equations 942 Appendix B Special Topics 951 B.1 Sequences, Series, and Summation Notation 951 B.2 Arithmetic and Geometric Sequences 957 B.3 Binomial Theorem 963 Appendix C Tables 967 Answers 971 Index 1027 Index of Applications 1038 Available separately: Calculus Topics to Accompany Calculus, 13e, and College Mathematics, 13e Chapter A01_BARN7668_13_GE_FM.indd Differential Equations 1.1 Basic Concepts 1.2 Separation of Variables 1.3 First-Order Linear Differential Equations Chapter Review Review Exercises 7/18/14 7:09 PM Contents Chapter Taylor Polynomials and Infinite Series Chapter Probability and Calculus 2.1 Taylor Polynomials 2.2 Taylor Series 2.3 Operations on Taylor Series 2.4 Approximations Using Taylor Series Chapter Review Review Exercises 3.1 Improper Integrals 3.2 Continuous Random Variables 3.3 Expected Value, Standard Deviation, and Median 3.4 Special Probability Distributions Chapter Review Review Exercises Appendixes A and B (Refer to back of College Mathematics for Business, Economics, Life Sciences, and Social Sciences, 13e) Appendix C Tables Appendix D Special Calculus Topic Table III Area Under the Standard Normal Curve D.1 Interpolating Polynomials and Divided Differences Answers Solutions to Odd-Numbered Exercises Index Applications Index A01_BARN7668_13_GE_FM.indd 7/18/14 7:09 PM Preface The thirteenth edition of College Mathematics for Business, Economics, Life Sciences, and Social Sciences is designed for a two-term (or condensed one-term) course in finite mathematics and calculus for students who have had one to two years of high school algebra or the equivalent The book’s overall approach, refined by the authors’ experience with large sections of college freshmen, addresses the challenges of teaching and learning when prerequisite knowledge varies greatly from student to student The authors had three main goals when writing this text: ▶ To write a text that students can easily comprehend ▶ To make connections between what students are learning and how they may apply that knowledge ▶ To give flexibility to instructors to tailor a course to the needs of their students Many elements play a role in determining a book’s effectiveness for students Not only is it critical that the text be accurate and readable, but also, in order for a book to be e ffective, aspects such as the page design, the interactive nature of the presentation, and the ability to support and challenge all students have an incredible impact on how easily students comprehend the material Here are some of the ways this text addresses the needs of students at all levels: ▶ Page layout is clean and free of potentially distracting elements ▶ Matched Problems that accompany each of the completely worked examples help students gain solid knowledge of the basic topics and assess their own level of understanding before moving on ▶ Review material (Appendix A and Chapters and 2) can be used judiciously to help remedy gaps in prerequisite knowledge ▶ A Diagnostic Prerequisite Test prior to Chapter helps students assess their skills, while the Basic Algebra Review in Appendix A provides students with the content they need to remediate those skills ▶ Explore and Discuss problems lead the discussion into new concepts or build upon a current topic They help students of all levels gain better insight into the mathematical concepts through thought-provoking questions that are effective in both small and large classroom settings ▶ Instructors are able to easily craft homework assignments that best meet the needs of their students by taking advantage of the variety of types and difficulty levels of the exercises Exercise sets at the end of each section consist of a Skills Warm-up (four to eight problems that review prerequisite knowledge specific to that section) followed by problems of varying levels of difficulty ▶ The MyMathLab course for this text is designed to help students help themselves and provide instructors with actionable information about their progress The immediate feedback students receive when doing homework and practice in MyMathLab is invaluable, and the easily accessible e-book enhances student learning in a way that the printed page sometimes cannot Most important, all students get substantial experience in modeling and solving real-world problems through application examples and exercises chosen from business and economics, life sciences, and social sciences Great care has been taken to write a book that is mathematically correct, with its emphasis on computational skills, ideas, and problem solving rather than mathematical theory A01_BARN7668_13_GE_FM.indd 7/18/14 7:09 PM Preface Finally, the choice and independence of topics make the text readily adaptable to a ariety of courses (see the chapter dependencies chart on page 13) This text is one of v three books in the authors’ college mathematics series The others are Finite Mathematics for Business, Economics, Life Sciences, and Social Sciences, and Calculus for Business, Economics, Life Sciences, and Social Sciences Additional Calculus Topics, a supplement written to accompany the Barnett/Ziegler/Byleen series, can be used in conjunction with any of these books New to This Edition Fundamental to a book’s effectiveness is classroom use and feedback Now in its thirteenth edition, College Mathematics for Business, Economics, Life Sciences, and Social Sciences has had the benefit of a substantial amount of both Improvements in this edition evolved out of the generous response from a large number of users of the last and previous editions as well as survey results from instructors, mathematics departments, course outlines, and college catalogs In this edition, ▶ The Diagnostic Prerequisite Test has been revised to identify the specific deficiencies in prerequisite knowledge that cause students the most difficulty with finite mathematics and calculus ▶ Most exercise sets now begin with a Skills Warm-up—four to eight problems that review prerequisite knowledge specific to that section in a just-in-time approach References to review material are given for the benefit of students who struggle with the warm-up problems and need a refresher ▶ Section 6.1 has been rewritten to better motivate and introduce the simplex method and associated terminology ▶ Section 14.4 has been rewritten to cover the trapezoidal rule and Simpson’s rule ▶ Examples and exercises have been given up-to-date contexts and data ▶ Exposition has been simplified and clarified throughout the book ▶ MyMathLab for this text has been enhanced greatly in this revision Most notably, a “Getting Ready for Chapter X” has been added to each chapter as an optional resource for instructors and students as a way to address the prerequisite skills that students need, and are often missing, for each chapter Many more improvements have been made See the detailed description on pages 17 and 18 for more information Trusted Features Emphasis and Style As was stated earlier, this text is written for student comprehension To that end, the focus has been on making the book both mathematically correct and accessible to students Most derivations and proofs are omitted, except where their inclusion adds significant insight into a particular concept as the emphasis is on computational skills, ideas, and problem solving rather than mathematical theory General concepts and results are typically presented only after particular cases have been discussed Design One of the hallmark features of this text is the clean, straightforward design of its pages Navigation is made simple with an obvious hierarchy of key topics and a judicious use of call-outs and pedagogical features We made the decision to maintain a two-color d esign to A01_BARN7668_13_GE_FM.indd 7/18/14 7:09 PM SECTION 2.2 Elementary Functions: Graphs and Transformations 85 Matched Problem Graph the piecewise-defined function h1x2 = e -2x + x - if … x … if x As the next example illustrates, piecewise-defined functions occur naturally in many applications Example Natural Gas Rates Easton Utilities uses the rates shown in Table to compute the monthly cost of natural gas for each customer Write a piecewise definition for the cost of consuming x CCF (cubic hundred feet) of natural gas and graph the function Table Charges per Month $0.7866 per CCF for the first CCF $0.4601 per CCF for the next 35 CCF $0.2508 per CCF for all over 40 CCF Solution If C(x) is the cost, in dollars, of using x CCF of natural gas in one month, then the first line of Table implies that C1x2 = 0.7866x if … x … Note that C152 = 3.933 is the cost of CCF If x … 40, then x - represents the amount of gas that cost $0.4601 per CCF, 0.46011x - 52 represents the cost of this gas, and the total cost is C(x) If x 40, then $30 $20 0.7866x C1x2 = • 3.933 + 0.46011x - 52 20.0365 + 0.25081x - 402 $10 (5, 3.933) 20 30 40 50 60 x Figure 12 Cost of purchasing x CCF of natural gas C1x2 = 20.0365 + 0.25081x - 402 where 20.0365 = C1402, the cost of the first 40 CCF Combining all these equations, we have the following piecewise definition for C(x): (40, 20.0365) 10 C1x2 = 3.933 + 0.46011x - 52 if … x … if x … 40 if x 40 To graph C, first note that each rule in the definition of C represents a transformation of the identity function f1x2 = x Graphing each transformation over the indicated interval produces the graph of C shown in Figure 12 Matched Problem Trussville Utilities uses the rates shown in Table to compute the monthly cost of natural gas for residential customers Write a piecewise definition for the cost of consuming x CCF of natural gas and graph the function Table 2 Charges per Month $0.7675 per CCF for the first 50 CCF $0.6400 per CCF for the next 150 CCF $0.6130 per CCF for all over 200 CCF Exercise 2.2 In Problems 1–8, find the domain and range of the function 5 f1x2 = 2∙ x ∙ + 7 6 f1x2 = - 5∙ x ∙ + f1x2 = 5x - 10 2 f1x2 = - 4x + 12 3 f1x2 = x + 100 8 f1x2 = 20 - 102 x f1x2 = 15 - 1x 4 f1x2 = + 1x M02_BARN7668_13_GE_C02.indd 85 16/07/14 6:16 PM 86 CHAPTER Functions and Graphs In Problems 9–24, graph each of the functions using the graphs of functions f and g below f(x) g(x) 5 37 38 y 5 Ϫ5 Ϫ5 x Ϫ5 y x x x Ϫ5 Ϫ5 Ϫ5 39 Ϫ5 Ϫ5 40 y y 5 y = f1x2 + 10 y = g1x2 - 11 y = f1x + 22 12 y = g1x - 12 13 y = g1x - 32 14 y = f1x + 32 Ϫ5 x Ϫ5 x 15 y = g1x2 - 16 y = f1x2 + 17 y = - f1x2 18 y = - g1x2 Ϫ5 Ϫ5 19 y = 0.5g1x2 20 y = 2f1x2 In Problems 41–46, the graph of the function g is formed by applying the indicated sequence of transformations to the given function f Find an equation for the function g and graph g using - … x … and - … y … 21 y = 2f1x2 + 22 y = - 0.5g1x2 + 23 y = 21 f1x2 + 12 24 y = - 10.5g1x2 + 32 In Problems 25–32, indicate verbally how the graph of each function is related to the graph of one of the six basic functions in Figure on page 79 Sketch a graph of each function 41 The graph of f1x2 = 1x is shifted units to the right and units down 42 The graph of f1x2 = x is shifted units to the left and units up h1x2 = - ∙ x - ∙ 25 g1x2 = - ∙ x + ∙ 26 27 f1x2 = 1x - 42 - 28 m1x2 = 1x + 32 + 43 The graph of f1x2 = ∙ x ∙ is reflected in the x axis and shifted to the left units 29 f1x2 = - 1x 30 g1x2 = - + 2x 31 h1x2 = - 3∙ x ∙ 32 m1x2 = - 0.4x2 44 The graph of f1x2 = ∙ x ∙ is reflected in the x axis and shifted to the right unit Each graph in Problems 33–40 is the result of applying a sequence of transformations to the graph of one of the six basic functions in Figure on page 79 Identify the basic function and describe the transformation verbally Write an equation for the given graph 45 The graph of f1x2 = x3 is reflected in the x axis and shifted units to the right and down unit 33 y 34 y x 36 50 h1x2 = e 5 M02_BARN7668_13_GE_C02.indd 86 49 h1x2 = e y Ϫ5 48 g1x2 = e Ϫ5 y Ϫ5 47 f1x2 = e x Ϫ5 Ϫ5 35 Graph each function in Problems 47–52 5 Ϫ5 x 46 The graph of f1x2 = x2 is reflected in the x axis and shifted to the left units and up units Ϫ5 Ϫ5 x - 2x x - if x if x Ú x + + 2x if x - if x Ú - + 0.5x - 10 + 2x if … x … 10 if x 10 10 + 2x 40 + 0.5x if … x … 20 if x 20 2x 51 h1x2 = • x + 20 0.5x + 40 if … x … 20 if 20 x … 40 if x 40 4x + 20 52 h1x2 = • 2x + 60 - x + 360 if … x … 20 if 20 x … 100 if x 100 16/07/14 6:17 PM SECTION 2.2 Elementary Functions: Graphs and Transformations Each of the graphs in Problems 53–58 involves a reflection in the x axis and/or a vertical stretch or shrink of one of the basic functions in Figure on page 79 Identify the basic function, and describe the transformation verbally Write an equation for the given graph y y 54 53 66 Price-supply. The manufacturers of the DVD players in Problem 65 are willing to supply x players at a price of p(x) as given by the equation p1x2 = 41x 9 … x … 289 (A) Describe how the graph of function p can be obtained from the graph of one of the basic functions in Figure on page 79 (B) Sketch a graph of function p using part (A) as an aid 5 Ϫ5 x Ϫ5 x 87 67 Hospital costs. Using statistical methods, the financial department of a hospital arrived at the cost equation C1x2 = 0.000481x - 5002 + 60,000 100 … x … 1,000 Ϫ5 55 Ϫ5 56 y y 5 Ϫ5 x Ϫ5 Ϫ5 57 x 68 Price-demand. A company manufactures and sells in-line skates Its financial department has established the pricedemand function Ϫ5 58 y p1x2 = 190 - 0.0131x - 102 2 10 … x … 100 y 5 Ϫ5 x Ϫ5 x Ϫ5 Ϫ5 Changing the order in a sequence of transformations may change the final result Investigate each pair of transformations in Problems 59–64 to determine if reversing their order can produce a different result Support your conclusions with specific examples and/or mathematical arguments 59 Vertical shift; horizontal shift 60 Vertical shift; reflection in y axis 61 62 63 64 65 Price-demand. A retail chain sells DVD players The retail price p(x) (in dollars) and the weekly demand x for a particular model are related by Table 3 Summer (July–October) Base charge, $8.50 First 700 kWh or less at 0.0650/kWh Over 700 kWh at 0.0900/kWh 70 Electricity rates. Table shows the electricity rates charged by Monroe Utilities in the winter months (A) Write a piecewise definition of the monthly charge W(x) for a customer who uses x kWh in a winter month Table 4 Winter (November–June) p1x2 = 115 - 41x … x … 289 (A) Describe how the graph of function p can be obtained from the graph of one of the basic functions in Figure on page 79 (B) Sketch a graph of function p using part (A) as an aid M02_BARN7668_13_GE_C02.indd 87 where p(x) is the price at which x thousand pairs of in-line skates can be sold (A) Describe how the graph of function p can be obtained from the graph of one of the basic functions in Figure on page 79 (B) Sketch a graph of function p using part (A) and a graphing calculator as aids 69 Electricity rates. Table shows the electricity rates charged by Monroe Utilities in the summer months The base is a fixed monthly charge, independent of the kWh (kilowatt-hours) used during the month (A) Write a piecewise definition of the monthly charge S(x) for a customer who uses x kWh in a summer month (B) Graph S(x) Vertical shift; reflection in x axis Vertical shift; vertical stretch Horizontal shift; reflection in y axis Horizontal shift; vertical shrink Applications where C(x) is the cost in dollars for handling x cases per month (A) Describe how the graph of function C can be obtained from the graph of one of the basic functions in Figure on page 79 (B) Sketch a graph of function C using part (A) and a graphing calculator as aids Base charge, $8.50 First 700 kWh or less at 0.0650/kWh Over 700 kWh at 0.0530/kWh (B) Graph W(x) 16/07/14 6:17 PM 88 CHAPTER Functions and Graphs 71 State income tax. Table shows a recent state income tax schedule for married couples filing a joint return in Kansas (A) Write a piecewise definition for the tax due T(x) on an income of x dollars (B) Graph T(x) (C) Find the tax due on a taxable income of $40,000 Of $70,000 Table 5 Kansas State Income Tax Schedule I—Married Filing Joint If taxable income is Over $0 $30,000 But Not Over $30,000 $60,000 $60,000 Tax Due Is 3.50% of taxable income $1,050 plus 6.25% of excess over $30,000 $2,925 plus 6.45% of excess over $60,000 72 State income tax. Table shows a recent state income tax schedule for individuals filing a return in Kansas Table 6 Kansas State Income Tax Schedule II—Single, Head of Household, or Married Filing Separate If taxable income is Over $0 $15,000 But Not Over $15,000 $30,000 $30,000 Tax Due Is 3.50% of taxable income $525 plus 6.25% of excess over $15,000 $1,462.50 plus 6.45% of excess over $30,000 (A) Write a piecewise definition for the tax due T(x) on an income of x dollars (B) Graph T(x) (C) Find the tax due on a taxable income of $20,000 Of $35,000 (D) Would it be better for a married couple in Kansas with two equal incomes to file jointly or separately? Discuss 73 Human weight. A good approximation of the normal weight of a person 60 inches or taller but not taller than 80 inches is given by w1x2 = 5.5x - 220, where x is height in inches and w(x) is weight in pounds (A) Describe how the graph of function w can be obtained from the graph of one of the basic functions in Figure 1, page 79 (B) Sketch a graph of function w using part (A) as an aid 74 Herpetology. The average weight of a particular species of snake is given by w1x2 = 463x3, 0.2 … x … 0.8, where x is length in meters and w(x) is weight in grams (A) Describe how the graph of function w can be obtained from the graph of one of the basic functions in Figure 1, page 79 (B) Sketch a graph of function w using part (A) as an aid 75 Safety research. Under ideal conditions, if a person driving a vehicle slams on the brakes and skids to a stop, the speed of the vehicle v(x) (in miles per hour) is given approximately by v1x2 = C1x, where x is the length of skid marks (in feet) and C is a constant that depends on the road conditions and the weight of the vehicle For a particular vehicle, v1x2 = 7.081x and … x … 144 (A) Describe how the graph of function v can be obtained from the graph of one of the basic functions in Figure 1, page 79 (B) Sketch a graph of function v using part (A) as an aid 76 Learning. A production analyst has found that on average it takes a new person T(x) minutes to perform a particular assembly operation after x performances of the operation, where T1x2 = 10 - 2x, … x … 125 (A) Describe how the graph of function T can be obtained from the graph of one of the basic functions in Figure 1, page 79 (B) Sketch a graph of function T using part (A) as an aid Answers to Matched Problems (A) f17292 = 729, h17292 = 531,441, m17292 = 387,420,489, n17292 = 27, g17292 = 729 (B) f1 - 5.252 = - 5.25, p17292 = 9, h1 - 5.252 = 27.5625, m1- 5.252 = - 144.7031, n1- 5.252 is not a real number, p1 - 5.252 = - 1.7380, g1 - 5.252 = 5.25 (A) The graph of y = 1x + is the same as the graph of y = 1x shifted upward units, and the graph of y = 1x - is the same as the graph of y = 1x shifted downward units The figure confirms these conclusions y y ϭ ͙x ϩ 10 y ϭ ͙x Ϫ10 Ϫ5 x y ϭ ͙x Ϫ Ϫ5 Ϫ10 M02_BARN7668_13_GE_C02.indd 88 16/07/14 6:17 PM SECTION 2.3 Quadratic Functions (B) The graph of y = 1x + is the same as the graph of y = 1x shifted to the left units, and the graph of y = 1x - is the same as the graph of y = 1x shifted to the right units The figure confirms these conclusions y Ϫ10 Ϫ5 10 Ϫ5 10 y ϭ ͙x ϩ Ϫ10 yϭx 10 y 89 x y ϭ Ϫ0.5x Ϫ10 y ϭ ͙x x Ϫ5 y ϭ ͙x Ϫ Ϫ5 The graph of function G is a reflection in the x axis and a horizontal translation of units to the left of the graph of y = x3 An equation for G is G1x2 = - 1x + 22 6 Ϫ10 h(x) 3 3 H1x2 = x + 3, G1x2 = x - 2, M1x2 = x + 2, N1x2 = x - 4 (A) The graph of y = 2x is a vertical stretch of the graph of y = x, and the graph of y = 0.5x is a vertical shrink of the graph of y = x The figure confirms these conclusions y y ϭ 2x yϭx x 10 y ϭ 0.5x Ϫ10 Ϫ5 10 x if … x … 50 if 50 x … 200 if 200 x 0.7675x C1x2 = • 38.375 + 0.641x - 502 134.375 + 0.6131x - 2002 C(x) Ϫ5 200 Ϫ10 (B) The graph of y = - 0.5x is a vertical shrink and a reflection in the x axis of the graph of y = x The figure confirms this conclusion (200, 134.375) 100 (50, 38.375) 100 200 300 x 2.3 Quadratic Functions • Quadratic Functions, Equations, and Inequalities • Properties of Quadratic Functions and Their Graphs • Applications h(x) 5 Ϫ5 x Ϫ5 If the degree of a linear function is increased by one, we obtain a second-degree function, usually called a quadratic function, another basic function that we will need in our library of elementary functions We will investigate relationships between quadratic functions and the solutions to quadratic equations and inequalities Other important properties of quadratic functions will also be investigated, including maxima and minima We will then be in a position to solve important practical problems such as finding production levels that will generate maximum revenue or maximum profit Quadratic Functions, Equations, and Inequalities The graph of the square function h1x2 = x2 is shown in Figure Notice that the graph is symmetric with respect to the y axis and that 10, 02 is the lowest point on the graph Let’s explore the effect of applying a sequence of basic transformations to the graph of h Figure Square function h(x) ∙ x2 M02_BARN7668_13_GE_C02.indd 89 16/07/14 6:17 PM 90 CHAPTER Functions and Graphs Explore and Discuss Indicate how the graph of each function is related to the graph of the function h1x2 = x2 Find the highest or lowest point, whichever exists, on each graph (A) f1x2 = 1x - 32 - = x2 - 6x + 2 (B) g1x2 = 0.51x + 22 + = 0.5x2 + 2x + 5 (C) m1x2 = - 1x - 42 + = - x2 + 8x - 8 (D) n1x2 = - 31x + 12 - = - 3x2 - 6x - 4 Graphing the functions in Explore and Discuss produces figures similar in shape to the graph of the square function in Figure These figures are called parabolas.* The functions that produce these parabolas are examples of the important class of quadratic functions Definition Quadratic Functions If a, b, and c are real numbers with a ∙ 0, then the function f1x2 = ax2 + bx + c Standard form is a quadratic function and its graph is a parabola Conceptual I n s i g h t If x is any real number, then ax2 + bx + c is also a real number According to the agreement on domain and range in Section 2.1, the domain of a quadratic function is R, the set of real numbers We will discuss methods for determining the range of a quadratic function later in this section Typical graphs of quadratic functions are illustrated in Figure f(x) 10 10 Ϫ5 h(x) g(x) x 10 Ϫ5 x Ϫ10 Ϫ10 (A) f(x) ϭ x2 Ϫ (B) g(x) ϭ 3x2 Ϫ 12x ϩ 14 Ϫ5 x Ϫ10 (C) h(x) ϭ Ϫ 2x Ϫ x2 Figure Graphs of quadratic functions Conceptual I n s i g h t An x intercept of a function is also called a zero of the function The x intercept of a linear function can be found by solving the linear equation y = mx + b = for x, m ∙ (see Section 1.2) Similarly, the x intercepts of a quadratic function can be found by solving the quadratic equation y = ax2 + bx + c = for x, a ∙ Several methods for solving quadratic equations are discussed in Appendix A, Section A.7 The most popular of these is the quadratic formula If ax2 + bx + c = 0, a ∙ 0, then x = - b { 2b2 - 4ac , provided b2 - 4ac Ú 2a *The arc of a basketball shot is a parabola Reflecting telescopes, solar furnaces, and automobile headlights are some of the many applications of parabolas M02_BARN7668_13_GE_C02.indd 90 16/07/14 6:17 PM SECTION 2.3 Quadratic Functions 91 Example Intercepts, Equations, and Inequalities (A) Sketch a graph of f1x2 = - x2 + 5x + in a rectangular coordinate system (B) Find x and y intercepts algebraically to four decimal places (C) Graph f1x2 = - x2 + 5x + in a standard viewing window (D) Find the x and y intercepts to four decimal places using trace and zero on your graphing calculator (E) Solve the quadratic inequality - x2 + 5x + Ú graphically to four decimal places using the results of parts (A) and (B) or (C) and (D) (F) Solve the equation - x2 + 5x + = graphically to four decimal places using INTERSECT on your graphing calculator Solution (A) Hand-sketching a graph of f: f(x) x y -1 -3 9 -3 10 Ϫ10 Ϫ5 10 x (B) Finding intercepts algebraically: y intercept: f102 = - 102 + 5102 + = x intercepts: f1x2 = -x2 + 5x + = 0 Quadratic equation -b { 2b2 - 4ac Quadratic formula (see Appendix A.7) 2a - 152 { 252 - 41 -12132 x = 21 -12 x = = -5 { 237 = -0.5414 or 5.5414 -2 (C) Graph in a graphing calculator: 10 10 Ϫ10 Ϫ10 M02_BARN7668_13_GE_C02.indd 91 16/07/14 6:17 PM 92 CHAPTER Functions and Graphs (D) Finding intercepts graphically using a graphing calculator: x intercept: - 0.5414 x intercept: 5.5414 y intercept: (E) Solving - x + 5x + Ú graphically: The quadratic inequality -x2 + 5x + Ú holds for those values of x for which the graph of f1x2 = - x2 + 5x + in the figures in parts (A) and (C) is at or above the x axis This happens for x between the two x intercepts [found in part (B) or (D)], including the two x intercepts The solution set for the quadratic inequality is - 0.5414 … x … 5.5414 or - 0.5414, 5.54144 (F) Solving the equation - x2 + 5x + = using a graphing calculator: 10 10 10 Ϫ10 10 Ϫ10 Ϫ10 - x2 + 5x + = at x = 0.2087 Ϫ10 - x2 + 5x + = at x = 4.7913 Matched Problem (A) Sketch a graph of g1x2 = 2x2 - 5x - in a rectangular coordinate system (B) Find x and y intercepts algebraically to four decimal places (C) Graph g1x2 = 2x2 - 5x - in a standard viewing window (D) Find the x and y intercepts to four decimal places using trace and the zero command on your graphing calculator (E) Solve 2x2 - 5x - Ú graphically to four decimal places using the results of parts (A) and (B) or (C) and (D) (F) Solve the equation 2x2 - 5x - = - graphically to four decimal places using intersect on your graphing calculator Explore and Discuss How many x intercepts can the graph of a quadratic function have? How many y intercepts? Explain your reasoning. Properties of Quadratic Functions and Their Graphs Many useful properties of the quadratic function can be uncovered by transforming f1x2 = ax2 + bx + c a ∙ into the vertex form f1x2 = a1x - h2 + k The process of completing the square (see Appendix A.7) is central to the transformation We illustrate the process through a specific example and then generalize the results M02_BARN7668_13_GE_C02.indd 92 16/07/14 6:17 PM SECTION 2.3 Quadratic Functions 93 Consider the quadratic function given by f1x2 = -2x2 + 16x - 24 (1) We use completing the square to transform this function into vertex form: f1x2 = -2x2 + 16x - 24 Factor the coefficient of x 2 out of the first two terms = -21x2 - 8x2 - 24 = -21x2 - 8x + ?2 - 24 Therefore, f(x) Maximum: f(4) ϭ 10 = -21x2 - 8x + 162 - 24 + 32 Add 16 to complete the square inside the parentheses Because of the -2 outside the parentheses, we have actually added -32, so we must add 32 to the outside = -21x - 42 + The transformation is complete and can be checked by multiplying out f1x2 = -21x - 42 + 8 (2) If x = 4, then - 21x - 42 = and f142 = For any other value of x, the negative number - 21x - 42 is added to 8, making it smaller Therefore, f142 = 10 x f(x) ϭ Ϫ2x2 ϩ 16x Ϫ 24 ϭ Ϫ2(x Ϫ 4)2 ϩ Ϫ5 Ϫ10 Line of symmetry: x ϭ Figure 3 Graph of a quadratic function is the maximum value of f1x2 for all x Furthermore, if we choose any two x values that are the same distance from 4, we will obtain the same function value For example, x = and x = are each one unit from x = and their function values are f132 = -213 - 42 + = f152 = -215 - 42 + = Therefore, the vertical line x = is a line of symmetry That is, if the graph of equation (1) is drawn on a piece of paper and the paper is folded along the line x = 4, then the two sides of the parabola will match exactly All these results are illustrated by graphing equations (1) and (2) and the line x = simultaneously in the same coordinate system (Fig 3) From the preceding discussion, we see that as x moves from left to right, f1x2 is increasing on (- ∞, 44, and decreasing on 34, ∞ ), and that f1x2 can assume no value greater than Thus, Range of f : y … or (- ∞, 84 In general, the graph of a quadratic function is a parabola with line of symmetry parallel to the vertical axis The lowest or highest point on the parabola, whichever exists, is called the vertex The maximum or minimum value of a quadratic function always occurs at the vertex of the parabola The line of symmetry through the vertex is called the axis of the parabola In the example above, x = is the axis of the parabola and 14, 82 is its vertex Conceptual I n s i g h t Applying the graph transformation properties discussed in Section 2.2 to the transformed equation, f1x2 = - 2x2 + 16x - 24 = - 21x - 42 + M02_BARN7668_13_GE_C02.indd 93 16/07/14 6:17 PM 94 CHAPTER Functions and Graphs we see that the graph of f1x2 = -2x2 + 16x - 24 is the graph of g1x2 = x2 vertically stretched by a factor of 2, reflected in the x axis, and shifted to the right units and up units, as shown in Figure y 10 g(x) ϭ x2 f(x) ϭ Ϫ2x2 ϩ 16x Ϫ 24 ϭ Ϫ2(x Ϫ 4)2 ϩ Ϫ10 10 Ϫ5 x Ϫ5 Ϫ10 Figure 4 Graph of f is the graph of g transformed Note the important results we have obtained from the vertex form of the quadratic function f: • The vertex of the parabola • The axis of the parabola • The maximum value of f1x2 • The range of the function f • The relationship between the graph of g1x2 = x2 and the graph of f1x2 = - 2x2 + 16x - 24 The preceding discussion is generalized to all quadratic functions in the following summary: Summary Properties of a Quadratic Function and Its Graph Given a quadratic function and the vertex form obtained by completing the square f1x2 = ax2 + bx + c a ∙ Standard form Vertex form = a1x - h2 + k we summarize general properties as follows: The graph of f is a parabola: f(x) f(x) Axis xϭh Axis xϭh Vertex (h, k) k Max f(x) Vertex (h, k) k Min f(x) h aϾ0 Opens upward x h x aϽ0 Opens downward Vertex: 1h, k2 (parabola increases on one side of the vertex and decreases on the other) M02_BARN7668_13_GE_C02.indd 94 16/07/14 6:17 PM SECTION 2.3 Quadratic Functions 95 Axis (of symmetry): x = h (parallel to y axis) f1h2 = k is the minimum if a and the maximum if a Domain: All real numbers Range: (- ∞ , k4 if a or 3k, ∞) if a The graph of f is the graph of g1x2 = ax translated horizontally h units and vertically k units Example Analyzing a Quadratic Function Given the quadratic function f1x2 = 0.5 x2 - 6x + 21 (A) Find the vertex form for f (B) Find the vertex and the maximum or minimum State the range of f (C) Describe how the graph of function f can be obtained from the graph of g1x2 = x2 using transformations (D) Sketch a graph of function f in a rectangular coordinate system (E) Graph function f using a suitable viewing window (F) Find the vertex and the maximum or minimum using the appropriate graphing calculator command Solution ( A) Complete the square to find the vertex form: f1x2 = = = = 0.5 x2 - 6x + 21 0.51x2 - 12x + ?2 + 21 0.51x2 - 12x + 362 + 21 - 18 0.51x - 62 + (B) From the vertex form, we see that h = and k = Thus, vertex: 16, 32; minimum: f162 = 3; range: y Ú or 33, ∞2 (C) The graph of f1x2 = 0.51x - 62 + is the same as the graph of g1x2 = x2 vertically shrunk by a factor of 0.5, and shifted to the right units and up units (D) Graph in a rectangular coordinate system: f(x) 10 5 10 x (E) Graph in a graphing calculator: 10 10 Ϫ10 Ϫ10 M02_BARN7668_13_GE_C02.indd 95 16/07/14 6:17 PM 96 CHAPTER Functions and Graphs (F) Find the vertex and minimum using the minimum command: 10 10 Ϫ10 Ϫ10 Vertex: 16, 32; minimum: f162 = Matched Problem Given the quadratic function f1x2 = - 0.25x - 2x + (A) Find the vertex form for f (B) Find the vertex and the maximum or minimum State the range of f (C) Describe how the graph of function f can be obtained from the graph of g1x2 = x2 using transformations (D) Sketch a graph of function f in a rectangular coordinate system (E) Graph function f using a suitable viewing window (F) Find the vertex and the maximum or minimum using the appropriate graphing calculator command Applications Example Maximum Revenue This is a continuation of Example in Section 2.1 Recall that the financial department in the company that produces a digital camera arrived at the following price–demand function and the corresponding revenue function: p1x2 = 94.8 - 5x Price–demand function R1x2 = xp1x2 = x194.8 - 5x2 Revenue function where p1x2 is the wholesale price per camera at which x million cameras can be sold and R(x) is the corresponding revenue (in millions of dollars) Both functions have domain … x … 15 (A) Find the value of x to the nearest thousand cameras that will generate the maximum revenue What is the maximum revenue to the nearest thousand dollars? Solve the problem algebraically by completing the square (B) What is the wholesale price per camera (to the nearest dollar) that generates the maximum revenue? (C) Graph the revenue function using an appropriate viewing window (D) Find the value of x to the nearest thousand cameras that will generate the maximum revenue What is the maximum revenue to the nearest thousand dollars? Solve the problem graphically using the maximum command Solution ( A) Algebraic solution: R(x) = = = = = x194.8 - 5x2 -5x2 + 94.8x -51x2 - 18.96x + ?2 -51x2 - 18.96x + 89.87042 + 449.352 -51x - 9.482 + 449.352 The maximum revenue of 449.352 million dollars 1$449,352,0002 occurs when x = 9.480 million cameras (9,480,000 cameras) M02_BARN7668_13_GE_C02.indd 96 16/07/14 6:17 PM SECTION 2.3 Quadratic Functions 97 (B) Finding the wholesale price per camera: Use the price-demand function for an output of 9.480 million cameras: p1x2 = 94.8 - 5x p19.4802 = 94.8 - 519.4802 = $47 per camera (C) Graph on a graphing calculator: 500 15 (D) Graphical solution using a graphing calculator: 500 15 The manufacture and sale of 9.480 million cameras (9,480,000 cameras) will generate a maximum revenue of 449.352 million dollars 1$449, 352, 0002 Matched Problem The financial department in Example 3, using statistical and analytical techniques (see Matched Problem in Section 2.1), arrived at the cost function C1x2 = 156 + 19.7x Cost function where C1x2 is the cost (in millions of dollars) for manufacturing and selling x million cameras (A) Using the revenue function from Example and the preceding cost function, write an equation for the profit function (B) Find the value of x to the nearest thousand cameras that will generate the maximum profit What is the maximum profit to the nearest thousand dollars? Solve the problem algebraically by completing the square (C) What is the wholesale price per camera (to the nearest dollar) that generates the maximum profit? (D) Graph the profit function using an appropriate viewing window (E) Find the output to the nearest thousand cameras that will generate the maximum profit What is the maximum profit to the nearest thousand dollars? Solve the problem graphically using the maximum command Example Break-Even Analysis Use the revenue function from Example and the cost function from Matched Problem 3: R1x2 = x194.8 - 5x2 Revenue function C1x2 = 156 + 19.7x Cost function Both have domain … x … 15 (A) Sketch the graphs of both functions in the same coordinate system (B) Break-even points are the production levels at which R1x2 = C1x2 Find the break-even points algebraically to the nearest thousand cameras M02_BARN7668_13_GE_C02.indd 97 16/07/14 6:17 PM 98 CHAPTER Functions and Graphs (C) Plot both functions simultaneously in the same viewing window (D) Use intersect to find the break-even points graphically to the nearest thousand cameras (E) Recall that a loss occurs if R1x2 C1x2 and a profit occurs if R1x2 C1x2 For what values of x (to the nearest thousand cameras) will a loss occur? A profit? Solution ( A) Sketch of functions: R(x) C(x) 500 Cost 250 Revenue 10 x 15 (B) Algebraic solution: Find x such that R1x2 = C1x2: x194.8 - 5x2 = 156 + 19.7x -5x + 75.1x - 156 = x = -75.1 { 275.12 - 41 -521 -1562 21 -52 -75.1 { 22,520.01 -10 x = 2.490 and 12.530 = The company breaks even at x = 2.490 million cameras (2,490,000 cameras) and at x = 12.530 million cameras (12,530,000 cameras) (C) Graph on a graphing calculator: Cost Revenue 500 15 (D) Graphical solution: Cost Revenue 500 M02_BARN7668_13_GE_C02.indd 98 500 15 Cost Revenue 15 16/07/14 6:17 PM SECTION 2.3 Quadratic Functions 99 The company breaks even at x = 2.490 million cameras (2,490,000 cameras) and at x = 12.530 million cameras (12,530,000 cameras) (E) Use the results from parts (A) and (B) or (C) and (D): Loss: … x 2.490 or 12.530 x … 15 Profit: 2.490 x 12.530 Matched Problem Use the profit equation from Matched Problem 3: P1x2 = R1x2 - C1x2 = -5x2 + 75.1x - 156 Profit function Domain: … x … 15 (A) Sketch a graph of the profit function in a rectangular coordinate system (B) Break-even points occur when P1x2 = Find the break-even points algebraically to the nearest thousand cameras (C) Plot the profit function in an appropriate viewing window (D) Find the break-even points graphically to the nearest thousand cameras (E) A loss occurs if P1x2 0, and a profit occurs if P1x2 For what values of x (to the nearest thousand cameras) will a loss occur? A profit? A visual inspection of the plot of a data set might indicate that a parabola would be a better model of the data than a straight line In that case, rather than using linear regression to fit a linear model to the data, we would use quadratic regression on a graphing calculator to find the function of the form y = ax2 + bx + c that best fits the data Example Outboard Motors Table gives performance data for a boat powered by an Evinrude outboard motor Use quadratic regression to find the best model of the form y = ax2 + bx + c for fuel consumption y (in miles per gallon) as a function of speed x (in miles per hour) Estimate the fuel consumption (to one decimal place) at a speed of 12 miles per hour Table 1 rpm mph mpg 2,500 10.3 4.1 3,000 18.3 5.6 3,500 24.6 6.6 4,000 29.1 6.4 4,500 33.0 6.1 5,000 36.0 5.4 5,400 38.9 4.9 Solution Enter the data in a graphing calculator (Fig 5A) and find the quadratic regression equation (Fig 5B) The data set and the regression equation are graphed in Figure 5C Using trace, we see that the estimated fuel consumption at a speed of 12 mph is 4.5 mpg 10 (A) (B) 50 (C) Figure M02_BARN7668_13_GE_C02.indd 99 16/07/14 6:17 PM ... Chapter 11 Summary and Review 647 Review Exercises 649 11 .1 11. 2 11 .3 11 .4 11 .5 11 .6 11 .7 Chapter 12 Graphing and Optimization 6 51 First... 11 x 1 + = 13 y -5 M 01_ BARN7668 _13 _GE_C 01. indd 30 m 12 . - = 3 x 14 . -4 15 2u + = 5u + - 7u 16 . - 3y + + y = 13 - 8y 17 10 x + 251x - 32 = 275 18 . - 314 - x2 = - 1x + 12 19 - y … 41y - 32... = - 13 44. - 12 + 13 A 01_ BARN7668 _13 _GE_FM.indd 20 7 /18 /14 7:09 PM Part A Library of Elementary Functions M 01_ BARN7668 _13 _GE_C 01. indd 21 16/07 /14 6:08 PM Linear Equations and Graphs 1. 1 Linear
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