College mathematics for business economics life science and social sciences 13th by barneet 1

College mathematics for business economics life science and social sciences 13th by barneet 1 College mathematics for business economics life science and social sciences 13th by barneet 1 College mathematics for business economics life science and social sciences 13th by barneet 1 College mathematics for business economics life science and social sciences 13th by barneet 1 College mathematics for business economics life science and social sciences 13th by barneet 1 College mathematics for business economics life science and social sciences 13th by barneet 1 College mathematics for business economics life science and social sciences 13th by barneet 1

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 www.downloadslide.net SECTION 11.6  Related Rates 635 need in equation (2) to solve for dx>dt, except y When x = 10, y can be found from equation (1): 102 + y2 = 262 y = 2262 - 102 = 24 feet Substitute dy>dt = - 2, x = 10, and y = 24 into (2) Then solve for dx>dt: 21102 dx + 212421 -22 = dt -212421 -22 dx = = 4.8 feet per second dt 21102 The bottom of the ladder is moving away from the wall at a rate of 4.8 feet per second Conceptual I n s i g h t In the solution to Example 1, we used equation (1) in two ways: first, to find an equation relating dy>dt and dx>dt, and second, to find the value of y when x = 10 These steps must be done in this order Substituting x = 10 and then differentiating does not produce any useful results: Substituting 10 for x has the effect of x2 + y2 = 262  100 + y2 = 262 ­stopping the ladder The rate of change of a stationary object + 2yy′ =   y′ = is always 0, but that is not the rate of change of the moving ladder Matched Problem Again, a 26-foot ladder is placed against a wall (Fig 1) If the bottom of the ladder is moving away from the wall at feet per second, at what rate is the top moving down when the top of the ladder is 24 feet above ground? Explore and Discuss (A) For which values of x and y in Example is dx>dt equal to (i.e., the same rate that the ladder is sliding down the wall)? (B) When is dx>dt greater than 2? Less than 2? Definition Suggestions for Solving Related-Rates Problems Step Sketch a figure if helpful Step Identify all relevant variables, including those whose rates are given and Step Step Step Step those whose rates are to be found Express all given rates and rates to be found as derivatives Find an equation connecting the variables identified in step Implicitly differentiate the equation found in step 4, using the chain rule where appropriate, and substitute in all given values Solve for the derivative that will give the unknown rate Example Related Rates and Motion  Suppose that two motorboats leave from the same point at the same time If one travels north at 15 miles per hour and the other travels east at 20 miles per hour, how fast will the distance between them be changing after hours? M11_BARN7668_13_GE_C11.indd 635 16/07/14 10:55 PM 636 CHAPTER 11 www.downloadslide.net  Additional Derivative Topics Solution First, draw a picture, as shown in Figure All variables, x, y, and z, are changing with time They can be considered as functions of time: x = x1t2, y = y1t2, and z = z1t2, given implicitly It now makes sense to find derivatives of each variable with respect to time From the Pythagorean theorem, N z z2 = x2 + y2(3) y E x We also know that dx = 20 miles per hour dt Figure dy = 15 miles per hour dt and We want to find dz>dt at the end of hours—that is, when x = 40 miles and y = 30 miles To this, we differentiate both sides of equation (3) with respect to t and solve for dz>dt: 2z dy dz dx = 2x + 2y (4) dt dt dt We have everything we need except z From equation (3), when x = 40 and y = 30, we find z to be 50 Substituting the known quantities into equation (4), we obtain 21502 dz = 214021202 + 213021152 dt dz = 25 miles per hour dt The boats will be separating at a rate of 25 miles per hour Matched Problem Repeat Example for the same situation at the end of hours Example Related Rates and Motion Suppose that a point is moving along the graph of x2 + y2 = 25 (Fig 3) When the point is at - 3, 42, its x coordinate is increasing at the rate of 0.4 unit per second How fast is the y coordinate changing at that moment? y (Ϫ3, 4) Solution Since both x and y are changing with respect to time, we can consider each as a function of time, namely, Ϫ4 Ϫ4 x2 ϩ y2 ϭ 25 Figure x but restricted so that x = x1t2 and y = y1t2 x2 + y2 = 25(5) We want to find dy>dt, given x = - 3, y = 4, and dx>dt = 0.4 Implicitly differentiating both sides of equation (5) with respect to t, we have x2 + y2 dy dx 2x + 2y dt dt dy dx x + y dt dt dy -3210.42 + dt dy dt = 25 = =   Divide both sides by   Substitute x = -3, y = 4, and dx>dt = 0.4, and solve for dy>dt = = 0.3 unit per second Matched Problem A point is moving on the graph of y3 = x When the point is at - 8, 42, its y coordinate is decreasing by units per second How fast is the x coordinate changing at that moment? M11_BARN7668_13_GE_C11.indd 636 16/07/14 10:55 PM www.downloadslide.net SECTION 11.6  Related Rates 637 Example Related Rates and Business  Suppose that for a company manufac- turing flash drives, the cost, revenue, and profit equations are given by   Cost equation R = 10x - 0.001x   Revenue equation P = R - C   Profit equation C = 5,000 + 2x where the production output in week is x flash drives If production is increasing at the rate of 500 flash drives per week when production is 2,000 flash drives, find the rate of increase in (A) Cost    (B)  Revenue    (C)  Profit Solution If production x is a function of time (it must be, since it is changing with respect to time), then C, R, and P must also be functions of time These functions are given implicitly (rather than explicitly) Letting t represent time in weeks, we differentiate both sides of each of the preceding three equations with respect to t and then substitute x = 2,000 and dx>dt = 500 to find the desired rates (A) C = 5,000 + 2x     Think: C = C1t2 and x = x1t2 dC d d = 15,0002 + 12x2   Differentiate both sides with respect to t dt dt dt dC dx dx = + = dt dt dt Since dx>dt = 500 when x = 2,000, dC = 215002 = $1,000 per week dt Cost is increasing at a rate of $1,000 per week (B) R = 10x - 0.001x2 dR d d = 110x2 - 0.001x2 dt dt dt dR dx dx = 10 - 0.002x dt dt dt dR dx = 110 - 0.002x2 dt dt Since dx>dt = 500 when x = 2,000, dR = 310 - 0.00212,0002415002 = $3,000 per week dt Revenue is increasing at a rate of $3,000 per week (C) P = R - C dP dR dC =   Results from parts (A) and (B) dt dt dt = $3,000 - $1,000 = $2,000 per week Profit is increasing at a rate of $2,000 per week Matched Problem Repeat Example for a production level of 6,000 flash drives per week M11_BARN7668_13_GE_C11.indd 637 16/07/14 10:55 PM www.downloadslide.net Exercises 11.6 Skills Warm-up Exercises W For Problems 1–8, review the geometric formulas in Appendix C, the distance between the dock and the boat decreasing when it is 30 feet from the dock? if necessary A circular flower bed has an area of 300 square feet Find its diameter to the nearest tenth of a foot A central pivot irrigation system covers a circle of radius 400 meters Find the area of the circle to the nearest square meter The hypotenuse of a right triangle has length 50 meters, and another side has length 20 meters Find the length of the third side to the nearest meter The legs of a right triangle have lengths 54 feet and 69 feet Find the length of the hypotenuse to the nearest foot A person 69 inches tall stands 40 feet from the base of a streetlight The streetlight casts a shadow of length 96 inches How far above the ground is the streetlight? The radius of a spherical balloon is meters Find its volume to the nearest tenth of a cubic meter A right circular cylinder and a sphere both have radius 12 feet If the volume of the cylinder is twice the volume of the sphere, find the height of the cylinder The height of a right circular cylinder is twice its radius If the volume is 1,000 cubic meters, find the radius and height to the nearest hundredth of a meter  In Problems 9–14, assume that x = x1t2 and y = y1t2 Find the indicated rate, given the other information y = x2 + 2; dx>dt = when x = 5; find dy>dt 10 y = x3 - 3; dx>dt = - when x = 2; find dy>dt 11 x2 + y2 = 1; dy>dt = - when x = - 0.6 and y = 0.8; find dx>dt  12 x2 + y2 = 4; dy>dt = when x = 1.2 and y = - 1.6; find dx>dt 13 x2 + 3xy + y2 = 11; dx>dt = when x = and y = 2; find dy>dt 14 x2 - 2xy - y2 = 7; dy>dt = - when x = and y = - 1; find dx>dt 15 A point is moving on the graph of xy = 36 When the point is at (4, 9), its x coordinate is increasing by units per second How fast is the y coordinate changing at that moment? 16 A point is moving on the graph of 4x2 + 9y2 = 36 When the point is at (3, 0), its y coordinate is decreasing by units per second How fast is its x coordinate changing at that moment? 17 A boat is being pulled toward a dock as shown in the figure If the rope is being pulled in at feet per second, how fast is Rope ft Figure for 17 and 18 18 Refer to Problem 17 Suppose that the distance between the boat and the dock is decreasing by 3.05 feet per second How fast is the rope being pulled in when the boat is 10 feet from the dock? 19 A rock thrown into a still pond causes a circular ripple If the radius of the ripple is increasing by feet per second, how fast is the area changing when the radius is 10 feet? 20 Refer to Problem 19 How fast is the circumference of a ­circular ripple changing when the radius is 10 feet? 21 The radius of a spherical balloon is increasing at the rate of centimeters per minute How fast is the volume changing when the radius is 10 centimeters? 22 Refer to Problem 21 How fast is the surface area of the sphere increasing when the radius is 10 centimeters? 23 Boyle’s law for enclosed gases states that if the volume is kept constant, the pressure P and temperature T are related by the equation P = k T where k is a constant If the temperature is increasing at 3 kelvins per hour, what is the rate of change of pressure when the temperature is 250 kelvins and the pressure is 500 pounds per square inch? 24 Boyle’s law for enclosed gases states that if the temperature is kept constant, the pressure P and volume V of a gas are ­related by the equation VP = k where k is a constant If the volume is decreasing by cubic inches per second, what is the rate of change of pressure when the volume is 1,000 cubic inches and the pressure is 40 pounds per square inch? 25 A 10-foot ladder is placed against a vertical wall Suppose that the bottom of the ladder slides away from the wall at a constant rate of feet per second How fast is the top of the ladder sliding down the wall when the bottom is feet from the wall? 26 A weather balloon is rising vertically at the rate of meters per second An observer is standing on the ground 300 meters from where the balloon was released At what rate is the distance between the observer and the balloon changing when the balloon is 400 meters high? 638 M11_BARN7668_13_GE_C11.indd 638 16/07/14 10:55 PM www.downloadslide.net 27 A streetlight is on top of a 20-foot pole A person who is feet tall walks away from the pole at the rate of feet per second At what rate is the tip of the person’s shadow moving away from the pole when he is 20 feet from the pole? 28 Refer to Problem 27 At what rate is the person’s shadow growing when he is 20 feet from the pole? 29 Helium is pumped into a spherical balloon at a constant rate of cubic feet per second How fast is the radius increasing after minute? After minutes? Is there any time at which the radius is increasing at a rate of 100 feet per second? Explain 30 A point is moving along the x axis at a constant rate of units per second At which point is its distance from (0, 1) increasing at a rate of units per second? At units per second? At 5 units per second? At 10 units per second? Explain 31 A point is moving on the graph of y = ex + x + in such a way that its x coordinate is always increasing at a rate of units per second How fast is the y coordinate changing when the point crosses the x axis? 32 A point is moving on the graph of x3 + y2 = in such a way that its y coordinate is always increasing at a rate of units per second At which point(s) is the x coordinate increasing at a rate of unit per second? Applications SECTION 11.6  Related Rates The current weekly advertising costs are $2,000, and these costs are increasing at the rate of $300 per week Find the current rate of change of sales 36 Advertising.  Repeat Problem 35 for s = 50,000 - 20,000e-0.0004x 37 Price–demand.  The price p (in dollars) and demand x for a product are related by 2x2 + 5xp + 50p2 = 80,000 (A) If the price is increasing at a rate of $2 per month when the price is $30, find the rate of change of the demand (B) If the demand is decreasing at a rate of units per month when the demand is 150 units, find the rate of change of the price 38 Price–demand.  Repeat Problem 37 for x2 + 2xp + 25p2 = 74,500 39 Pollution.  An oil tanker aground on a reef is forming a circular oil slick about 0.1 foot thick (see the figure) To estimate the rate dV>dt (in cubic feet per minute) at which the oil is leaking from the tanker, it was found that the radius of the slick was increasing at 0.32 foot per minute 1dR>dt = 0.322 when the radius R was 500 feet Find dV>dt Tanker R 33 Cost, revenue, and profit rates.  Suppose that for a company manufacturing calculators, the cost, revenue, and profit equations are given by C = 90,000 + 30x R = 300x - P = R - C x2 30 where the production output in week is x calculators If production is increasing at a rate of 500 calculators per week when production output is 6,000 calculators, find the rate of increase (decrease) in (A) Cost    (B)  Revenue    (C) Profit 34 Cost, revenue, and profit rates.  Repeat Problem 33 for C = 72,000 + 60x 639 R = 200x - x2 30 P = R - C where production is increasing at a rate of 500 calculators per week at a production level of 1,500 calculators 35 Advertising.  A retail store estimates that weekly sales s and weekly advertising costs x (both in dollars) are related by Oil slick A ϭ R2 V ϭ 0.1 A 40 Learning.  A person who is new on an assembly line performs an operation in T minutes after x performances of the operation, as given by T = 6a1 + b 1x If dx>dt = operations per hours, where t is time in hours, find dT>dt after 36 performances of the operation Answers to Matched Problems dy>dt = - 1.25 ft>sec    2. dz>dt = 25 mi>hr dx>dt = units>sec (A) dC>dt = $1,000>wk   (B) dR>dt = - $1,000>wk (C) dP>dt = - $2,000>wk s = 60,000 - 40,000e-0.0005x M11_BARN7668_13_GE_C11.indd 639 16/07/14 10:55 PM 640 CHAPTER 11 www.downloadslide.net  Additional Derivative Topics 11.7 Elasticity of Demand • Relative Rate of Change • Elasticity of Demand When will a price increase lead to an increase in revenue? To answer this question and study relationships among price, demand, and revenue, economists use the notion of elasticity of demand In this section, we define the concepts of relative rate of change, percentage rate of change, and elasticity of demand Relative Rate of Change Explore and Discuss A broker is trying to sell you two stocks: Biotech and Comstat The broker estimates that Biotech’s price per share will increase $2 per year over the next several years, while Comstat’s price per share will increase only $1 per year Is this sufficient information for you to choose between the two stocks? What other information might you request from the broker to help you decide? Interpreting rates of change is a fundamental application of calculus In Explore and Discuss 1, Biotech’s price per share is increasing at twice the rate of Comstat’s, but that does not automatically make Biotech the better buy The obvious information that is missing is the current price of each stock If Biotech costs $100 a share and Comstat costs $25 a share, then which stock is the better buy? To answer this question, we introduce two new concepts: relative rate of change and percentage rate of change Definition Relative and Percentage Rates of Change f = 1x2 The relative rate of change of a function f1x2 is , or equivalently, f1x2 d ln f1x2 dx f = 1x2 d , or equivalently, 100 * ln f1x2 The percentage rate of change is 100 * f1x2 dx d The alternative form for the relative rate of change, ln f1x2, is called the l­ ogarithmic dx derivative of f1x2 Note that f = 1x2 d ln f1x2 = dx f1x2 by the chain rule So the relative rate of change of a function f1x2 is its logarithmic derivative, and the percentage rate of change is 100 times the logarithmic derivative Returning to Explore and Discuss 1, the table shows the relative rate of change and percentage rate of change for Biotech and Comstat We conclude that Comstat is the better buy M11_BARN7668_13_GE_C11.indd 640 Relative rate of change Percentage rate of change Biotech = 0.02 100 2% Comstat = 0.04 25 4% 16/07/14 10:55 PM www.downloadslide.net SECTION 11.7  Elasticity of Demand 641 Example Percentage Rate of Change  Table lists the GDP (gross domes- tic product expressed in billions of 2005 dollars) and U.S population from 2000 to 2012 A model for the GDP is f1t2 = 209.5t + 11,361 where t is years since 2000 Find and graph the percentage rate of change of f1t2 for … t … 12 Table 1  Year Real GDP (billions of 2005 dollars) Population (in millions) 2000 2004 2008 2012 $11,226 $12,264 $13,312 $13,670 282.2 292.9 304.1 313.9 p(t) Solution If p1t2 is the percentage rate of change of f1t2, then d ln 1209.5t + 11,3612 dx 20,950 = 209.5t + 11,361 p1t2 = 100 * Figure 12 t The graph of p1t2 is shown in Figure (graphing details omitted) Notice that p1t2 is decreasing, even though the GDP is increasing Matched Problem A model for the population data in Table is f1t2 = 2.7t + 282 where t is years since 2000 Find and graph p1t2, the percentage rate of change of f1t2 for … t … 12 Conceptual I n s i g h t If $10,000 is invested at an annual rate of 4.5% compounded continuously, what is the relative rate of change of the amount in the account? The answer is the logarithmic derivative of A1t2 = 10,000e0.045t, namely 10,000e0.045t 10.0452 d ln 110,000e0.045t = = 0.045 dx 10,000e0.045t So the relative rate of change of A1t2 is 0.045, and the percentage rate of change is just the annual interest rate, 4.5% Elasticity of Demand Explore and Discuss In both parts below, assume that increasing the price per unit by $1 will decrease the demand by 500 units If your objective is to increase revenue, should you increase the price by $1 per unit? (A) At the current price of $8.00 per baseball cap, there is a demand for 6,000 caps (B) At the current price of $12.00 per baseball cap, there is a demand for 4,000 caps In Explore and Discuss 2, the rate of change of demand with respect to price was assumed to be - 500 units per dollar But in one case, part (A), you should increase the price, and in the other, part (B), you should not Economists use the concept of M11_BARN7668_13_GE_C11.indd 641 16/07/14 10:55 PM 642 CHAPTER 11 www.downloadslide.net  Additional Derivative Topics elasticity of demand to answer the question “When does an increase in price lead to an increase in revenue?” Definition Elasticity of Demand Let the price p and demand x for a product be related by a price–demand equation of the form x = f1p2 Then the elasticity of demand at price p, denoted by E1p2, is E1p2 = - relative rate of change of demand relative rate of change of price Using the definition of relative rate of change, we can find a formula for E1p2: d ln f1p2 relative rate of change of demand dp E1p2 = = relative rate of change of price d ln p dp f = 1p2 f1p2 = p = - pf = 1p2 f1p2 Theorem 1  Elasticity of Demand If price and demand are related by x = f1p2, then the elasticity of demand is given by E1p2 = - pf = 1p2 f1p2 Conceptual I n s i g h t Since p and f1p2 are nonnegative and f = 1p2 is negative (demand is usually a ­decreasing function of price), E1p2 is nonnegative This is why elasticity of demand is ­defined as the negative of a ratio Example Elasticity of Demand  The price p and the demand x for a product are related by the price–demand equation x + 500p = 10,000(1) Find the elasticity of demand, E1p2, and interpret each of the following: (A) E142      (B) E1162      (C) E1102 Solution To find E1p2, we first express the demand x as a function of the price p by solving (1) for x: x = 10,000 - 500p = 500120 - p2   Demand as a function of price or M11_BARN7668_13_GE_C11.indd 642 x = f1p2 = 500120 - p2 … p … 20(2) 16/07/14 10:55 PM www.downloadslide.net SECTION 11.7  Elasticity of Demand 643 Since x and p both represent nonnegative quantities, we must restrict p so that … p … 20 Note that the demand is a decreasing function of price That is, a price increase results in lower demand, and a price decrease results in higher demand (see Figure 2) x 10,000 Price increases Demand Demand decreases x ϭ f (p) ϭ 500(20 Ϫ p) Demand increases Price decreases 10 Price Figure E1p2 = - 20 p pf = 1p2 p1 -5002 p = = f1p2 500120 - p2 20 - p In order to interpret values of E1p2, we must recall the definition of elasticity: E1p2 = - relative rate of change of demand relative rate of change of price or -a relative rate of relative rate of b ≈ E1p2 a b change of demand change of price (A) E142 = 16 = 0.25 If the $4 price changes by 10%, then the demand will change by approximately 0.25110%2 = 2.5% (B) E1162 = 16 = If the $16 price changes by 10%, then the demand will change by approximately 4110%2 = 40% (C) E1102 = 10 10 = If the $10 price changes by 10%, then the demand will also change by approximately 10% Matched Problem Find E1p2 for the price–demand equation x = f1p2 = 1,000140 - p2 Find and interpret each of the following: (A) E182      (B)  E1302      (C)  E1202 The three cases illustrated in the solution to Example are referred to as ­inelastic demand, elastic demand, and unit elasticity, as indicated in Table Table 2  E p2 Demand Interpretation Revenue E1p2 Inelastic A price increase will increase revenue E1p2 Elastic E1p2 = Unit Demand is not sensitive to changes in price, that is, percentage change in price produces a smaller percentage change in demand Demand is sensitive to changes in price, that is, a percentage change in price produces a larger percentage change in demand A percentage change in price produces the same percentage change in demand A price increase will decrease revenue To justify the connection between elasticity of demand and revenue as given in the fourth column of Table 2, we recall that revenue R is the demand x (number of M11_BARN7668_13_GE_C11.indd 643 16/07/14 10:55 PM 644 CHAPTER 11 www.downloadslide.net  Additional Derivative Topics items sold) multiplied by p (price per item) Assume that the price–demand equation is written in the form x = f1p2 Then R1p2 = xp = f1p2p R 1p2 = f1p2 # + pf 1p2 = = R= 1p2 = f1p2 + pf = 1p2 R= 1p2 = f1p2 c +   Use the product rule   Multiply and divide by f1p2 f1p2   Factor out f1p2 f1p2 pf = 1p2 d   Use Theorem f1p2 R= 1p2 = f1p231 - E1p24 Since x = f1p2 0, it follows that R= 1p2 and - E1p2 have the same sign So if E1p2 1, then R= 1p2 is positive and revenue is increasing (Fig 3) Similarly, if E1p2 1, then R= 1p2 is negative, and revenue is decreasing (Fig 3) R(p) Inelastic demand E(p) Ͻ Increasing revenue RЈ(p) Ͼ Revenue increases Revenue increases Price decreases Revenue decreases Elastic demand E(p) Ͼ Decreasing revenue RЈ(p) Ͻ Price increases Price decreases Price increases Revenue decreases p Figure Revenue and elasticity Example Elasticity and Revenue  A manufacturer of sunglasses currently sells one type for $15 a pair The price p and the demand x for these glasses are related by x = f1p2 = 9,500 - 250p If the current price is increased, will revenue increase or decrease? Solution          E1p2 = - pf = 1p2 f1p2 p1 -2502 9,500 - 250p p = 38 - p 15 E1152 = ≈ 0.65 23 = - At the $15 price level, demand is inelastic and a price increase will increase revenue Matched Problem Repeat Example if the current price for sunglasses is $21 a pair In summary, if demand is inelastic, then a price increase will increase revenue But if demand is elastic, then a price increase will decrease revenue M11_BARN7668_13_GE_C11.indd 644 16/07/14 10:55 PM www.downloadslide.net Exercises 11.7 Skills Warm-up Exercises W In Problems 1–8, use the given equation, which expresses price p as a function of demand x, to find a function f1p2 that expresses demand x as a function of price p Give the domain of f1p2 (If necessary, review Section 2.1) p = 42 - 0.4x, … x … 105  p = 125 - 0.02x, … x … 6,250  p = 50 - 0.5x2, … x … 10  p = 180 - 0.8x2, … x … 15  p = 25e-x>20, … x … 20  p = 45 - e x>4 , … x … 12  In Problems 33–38, use the price–demand equation to find E1p2, the elasticity of demand 33 x = f1p2 = 25,000 - 450p  34 x = f1p2 = 10,000 - 190p  35 x = f1p2 = 4,800 - 4p2  36 x = f1p2 = 8,400 - 7p2  37 x = f1p2 = 98 - 0.6ep  38 x = f1p2 = 160 - 35 ln p  In Problems 39–46, find the logarithmic derivative 39 A1t2 = 500e0.07t 40 A1t2 = 2,000e0.052t p = 80 - 10 ln x, … x … 30  41 A1t2 = 3,500e0.15t p = ln 1500 - 5x2, … x … 90  43 f1x2 = xex 44 f1x2 = x2ex 45 f1x2 = ln x 46 f1x2 = x ln x In Problems 9–14, find the relative rate of change of f1x2 In Problems 47–50, use the price–demand equation to determine whether demand is elastic, is inelastic, or has unit elasticity at the indicated values of p 9 f1x2 = 35x - 0.4x2  10 f1x2 = 60x - 1.2x2  11 f1x2 = + 4e-x  12 f1x2 = 15 - 3e-0.5x  13 f1x2 = 12 + ln x  14 f1x2 = 25 - ln x  In Problems 15–24, find the relative rate of change of f1x2 at the indicated value of x Round to three decimal places 15 f1x2 = 45; x = 100 16 f1x2 = 580; x = 300 17 f1x2 = 420 - 5x; x = 25 18 f1x2 = 500 - 6x; x = 40 19 f1x2 = 420 - 5x; x = 55 20 f1x2 = 500 - 6x; x = 75 21 f1x2 = 4x2 - ln x; x = 22 f1x2 = 9x - ln x; x = 23 f1x2 = 4x2 - ln x; x = 24 f1x2 = 9x - ln x; x = 42 A1t2 = 900e0.24t 47 x = f1p2 = 12,000 - 10p2 (A) p = 10 (B) p = 20 (C) p = 30 48 x = f1p2 = 1,875 - p2 (A) p = 15 (B) p = 25 (C) p = 40 49 x = f1p2 = 950 - 2p - 0.1p2 (A) p = 30 (B) p = 50 (C) p = 70 50 x = f1p2 = 875 - p - 0.05p2 (A) p = 50 (B) p = 70 (C) p = 100 51 Given the price–demand equation p + 0.005x = 30 In Problems 25–32, find the percentage rate of change of f1x2 at the indicated value of x Round to the nearest tenth of a percent (A) Express the demand x as a function of the price p.   25 f1x2 = 225 + 65x; x = (B) Find the elasticity of demand, E1p2.   26 f1x2 = 75 + 110x; x = (C) What is the elasticity of demand when p = $10? If this price is increased by 10%, what is the approximate percentage change in demand? (D) What is the elasticity of demand when p = $25? If this price is increased by 10%, what is the approximate percentage change in demand? (E) What is the elasticity of demand when p = $15? If this price is increased by 10%, what is the approximate percentage change in demand? 27 f1x2 = 225 + 65x; x = 15 28 f1x2 = 75 + 110x; x = 16 29 f1x2 = 5,100 - 3x2; x = 35 30 f1x2 = 3,000 - 8x2; x = 12 31 f1x2 = 5,100 - 3x2; x = 41 32 f1x2 = 3,000 - 8x2; x = 18 645 M11_BARN7668_13_GE_C11.indd 645 16/07/14 10:55 PM 646 CHAPTER 11  Additional Derivative Topics www.downloadslide.net 52 Given the price–demand equation p + 0.01x = 50 In Problems 69–72, use the price–demand equation to find E1x2 at the indicated value of x (A) Express the demand x as a function of the price p.  69 p = g1x2 = 50 - 0.1x, x = 200 (B) Find the elasticity of demand, E1p2.  70 p = g1x2 = 30 - 0.05x, x = 400 (C) What is the elasticity of demand when p = $10? If this price is decreased by 5%, what is the approximate change in demand? 71 p = g1x2 = 50 - 22x, x = 400 (D) What is the elasticity of demand when p = $45? If this price is decreased by 5%, what is the approximate change in demand? In Problems 73–76, use the price–demand equation to find the values of x for which demand is elastic and for which demand is inelastic (E) What is the elasticity of demand when p = $25? If this price is decreased by 5%, what is the approximate change in demand? 53 Given the price–demand equation 0.02x + p = 60 (A) Express the demand x as a function of the price p.  (B) Express the revenue R as a function of the price p.  (C) Find the elasticity of demand, E1p2.  (D) For which values of p is demand elastic? Inelastic?  (E) For which values of p is revenue increasing? Decreasing?  (F) If p = $10 and the price is decreased, will revenue ­increase or decrease? (G) If p = $40 and the price is decreased, will revenue ­increase or decrease? 54 Repeat Problem 53 for the price–demand equation  0.025x + p = 50 In Problems 55–62, use the price–demand equation to find the values of p for which demand is elastic and the values for which demand is inelastic Assume that price and demand are both positive 55 x = f1p2 = 210 - 30p  56. x = f1p2 = 480 - 8p  72 p = g1x2 = 20 - 2x, x = 100 73 p = g1x2 = 180 - 0.3x  74.  p = g1x2 = 640 - 0.4x  75 p = g1x2 = 90 - 0.1x2  76. p = g1x2 = 540 - 0.2x2  77 Find E1p2 for x = f1p2 = Ap-k, where A and k are positive constants 78 Find E1p2 for x = f1p2 = Ae-kp, where A and k are positive constants Applications 79 Rate of change of cost.  A fast-food restaurant can produce a hamburger for $2.50 If the restaurant’s daily sales are increasing at the rate of 30 hamburgers per day, how fast is its daily cost for hamburgers increasing? 80 Rate of change of cost.  The fast-food restaurant in Problem 79 can produce an order of fries for $0.80 If the restaurant’s daily sales are increasing at the rate of 45 orders of fries per day, how fast is its daily cost for fries increasing? 81 Revenue and elasticity.  The price–demand equation for hamburgers at a fast-food restaurant is x + 400p = 3,000 59 x = f1p2 = 2144 - 2p  60.  x = f1p2 = 2324 - 2p  Currently, the price of a hamburger is $3.00 If the price is increased by 10%, will revenue increase or decrease? 82 Revenue and elasticity.  Refer to Problem 81 If the current price of a hamburger is $4.00, will a 10% price increase cause revenue to increase or decrease? 62 x = f1p2 = 23,600 - 2p2  83 Revenue and elasticity.  The price–demand equation for an order of fries at a fast-food restaurant is 57 x = f1p2 = 3,125 - 5p2  58.  x = f1p2 = 2,400 - 6p2  61 x = f1p2 = 22,500 - 2p2  In Problems 63–68, use the demand equation to find the revenue function Sketch the graph of the revenue function, and indicate the regions of inelastic and elastic demand on the graph 63 x = f1p2 = 20110 - p2  64. x = f1p2 = 10116 - p2  65 x = f1p2 = 401p - 152 2  66.  x = f1p2 = 101p - 92 2  67 x = f1p2 = 30 - 102p  68. x = f1p2 = 30 - 52p  If a price–demand equation is solved for p, then price is expressed as p = g1x2 and x becomes the independent variable In this case, it can be shown that the elasticity of demand is given by E1x2 = - M11_BARN7668_13_GE_C11.indd 646 g1x2 x + 1,000p = 2,500 Currently, the price of an order of fries is $0.99 If the price is decreased by 10%, will revenue increase or decrease? 84 Revenue and elasticity.  Refer to Problem 83 If the current price of an order of fries is $1.49; will a 10% price decrease cause revenue to increase or decrease? 85 Maximum revenue.  Refer to Problem 81 What price will maximize the revenue from selling hamburgers? 86 Maximum revenue.  Refer to Problem 83 What price will maximize the revenue from selling fries? xg′1x2 16/07/14 10:56 PM www.downloadslide.net 87 Population growth.  A model for Canada’s population (Table 3) is  Summary and Review 647 90 Crime.  A model for the number of assaults in the United States (Table 4) is f1t2 = 0.31t + 18.5 a1t2 = 6.0 - 1.2 ln t where t is years since 1960 Find and graph the percentage rate of change of f1t2 for … t … 50 Table 3  Population Year Canada (millions) Mexico (millions) 1960 1970 1980 1990 2000 2010 18 22 25 28 31 34  39  53  68  85 100 112 where t is years since 1990 Find the relative rate of change for assaults in 2020 Answers to Matched Problems p1t2 = 270 2.7t + 282 p(t) 88 Population growth.  A model for Mexico’s population (Table 3) is  f1t2 = 1.49t + 38.8 where t is years since 1960 Find and graph the percentage rate of change of f1t2 for … t … 50 89 Crime.  A model for the number of robberies in the United States (Table 4) is r1t2 = 3.3 - 0.7 ln t where t is years since 1990 Find the relative rate of change for robberies in 2020 E1p2 = 12 t p 40 - p (A) E182 = 0.25; demand is inelastic (B) E1302 = 3; demand is elastic (C) E1202 = 1; demand has unit elasticity 21 ≈ 1.2; demand is elastic Increasing price will 17 decrease revenue 3 E1212 = Table 4  Number of Victimizations per 1,000 Population 1995 2000 2005 2010 Robbery Aggravated Assault 2.21 1.45 1.41 1.19 4.18 3.24 2.91 2.52 Chapter 11 Summary and Review Important Terms, Symbols, and Concepts 11.1  The Constant e and Continuous Compound Interest EXAMPLES • The number e is defined as lim a1 + S x ∞ n b = lim 11 + s2 1>s = 2.718 281 828 459 c xS0 n • If a principal P is invested at an annual rate r (expressed as a decimal) compounded continuously, then the amount A in the account at the end of t years is given by the compound interest formula A = Pert Ex Ex Ex Ex 1, 2, 3, 4, p p p p 597 597 598 598 Ex Ex Ex Ex Ex 1, 2, 3, 4, 5, p p p p p 602 604 605 606 607 11.2  Derivatives of Exponential and Logarithmic Functions • For b 0, b ∙ 1, d x e = ex dx d x b = bx ln b dx For b 0, b ∙ 1, and x 0, d ln x = x dx M11_BARN7668_13_GE_C11.indd 647 d 1 logb x = dx ln b x 16/07/14 10:56 PM 648 CHAPTER 11  Additional Derivative Topics www.downloadslide.net 11.2  Derivatives of Exponential and Logarithmic Functions (Continued) • The change-of-base formulas allow conversion from base e to any base b, b 0, b ∙ 1: bx = ex ln b log b x = ln x ln b 11.3  Derivatives of Products and Quotients • Product rule If y = f1x2 = F1x2 S1x2, then f = 1x2 = F1x2S′1x2 + S1x2F = 1x2, provided that both F = 1x2 and S′1x2 exist • Quotient rule If y = f1x2 = T1x2 B1x2 both T′1x2 and B′1x2 exist , then f = 1x2 = B1x2 T′1x2 - T1x2 B′1x2 B1x2 provided that Ex 1, p 610 Ex 2, p 611 Ex 3, p 611 Ex 4, p 613 Ex 5, p 614 Ex 6, p 614 11.4  The Chain Rule • A function m is a composite of functions f and g if m1x2 = f [g1x2] Ex 1, p 619 • The chain rule gives a formula for the derivative of the composite function m1x2 = E[I1x2]: Ex 2, p 619 m′1x2 = E′[I1x2]I′1x2 Ex 4, p 623 • A special case of the chain rule is called the general power rule: Ex 5, p 624 d [ f1x2]n = n[ f1x2]n - 1f = 1x2 dx • Other special cases of the chain rule are the following general derivative rules: Ex 3, p 620 Ex 6, p 624 d = f 1x2 ln [ f1x2] = dx f1x2 d f1x2 e = ef1x2f = 1x2 dx 11.5  Implicit Differentiation • If y = y1x2 is a function defined implicitly by the equation F1x, y2 = 0, then we use implicit differentiation to find an equation in x, y, and y′ Ex 1, p 629 Ex 2, p 631 Ex 3, p 632 11.6  Related Rates • If x and y represent quantities that are changing with respect to time and are related by the equation F1x, y2 = 0, then implicit differentiation produces an equation that relates x, y, dy>dt, and dx>dt Problems of this type are called related-rates problems • Suggestions for solving related-rates problems are given on page 241 Ex 1, p 634 Ex 2, p 635 Ex 3, p 636 11.7  Elasticity of Demand • The relative rate of change, or the logarithmic derivative, of a function f1x2 is f = 1x2 >f1x2, and the percentage rate of change is 100 * f = 1x2 >f1x24 • If price and demand are related by x = f1p2, then the elasticity of demand is given by = E1p2 = - p f 1p2 f1p2 = - Ex 1, p 641 Ex 2, p 642 relative rate of change of demand relative rate of change of price • Demand is inelastic if E1p2 (Demand is not sensitive to changes in price; a percentage change in price produces a smaller percentage change in demand.) Demand is elastic if E1p2 (Demand is sensitive to changes in price; a percentage change in price produces a larger percentage change in demand.) Demand has unit elasticity if E1p2 = (A percentage change in price produces the same percentage change in demand.) Ex 3, p 644 • If R1p2 = pf1p2 is the revenue function, then R′1p2 and 31 - E1p2 always have the same sign If demand is inelastic, then a price increase will increase revenue If demand is elastic, then a price increase will decrease revenue M11_BARN7668_13_GE_C11.indd 648 16/07/14 10:56 PM www.downloadslide.net Review Exercises 649 Review Exercises Work through all the problems in this chapter review, and check your answers in the back of the book Answers to all review problems are there, along with section numbers in italics to indicate where each type of problem is discussed Where weaknesses show up, review appropriate sections of the text 23 Find y′ for y = y1x2 defined implicitly by x - y2 = ey, and evaluate at 11,02.  Use a calculator to evaluate A = 2,000e0.09t to the nearest cent for t = 5, 10, and 20 In Problems 25–27, find the logarithmic derivatives In Problems 2–4, find functions E(u) and I(x) so that f1x2 = E3I1x24 27 f1x2 = + x2 f1x2 = 16x + 52 3>2   3 f1x2 = ln 1x2 + 42  f1x2 = e0.02x  In Problems 5–8, find the indicated derivative d d 1ln x3 + 2e-x 2   6 e2x - 3  dx dx y′ for y = ln12x + 72 24 Find y′ for y = y1x2 defined implicitly by ln y = x2 - y2, and evaluate at 11, 12.  f1p2 = 100 - 3p  25 A1t2 = 400e0.049t  26 28 A point is moving on the graph of 3y2 + 40x2 = 16 so that its y coordinate is increasing by units per second when 1x, y2 = 12, 22 Find the rate of change of the x coordinate 29 A 17-foot ladder is placed against a wall If the foot of the ladder is pushed toward the wall at 0.5 foot per second, how fast is the top of the ladder rising when the foot is feet from the wall? f = 1x2 for f1x2 = ln 13 + ex 30 Water is leaking onto a floor The resulting circular pool has an area that is increasing at a rate of 30 square inches per minute How fast is the circumference C of the pool increasing when the radius R is 10 inches? 10 For y = 3x2 - 5, where x = x1t2 and y = y1t2, find dy/dt if dx>dt = when x = 12 31 Find the values of p for which demand is elastic and the ­values for which demand is inelastic if the price–demand equation is  Find y′ for y = y1x2 defined implicitly by the equation y4 - ln x + 2x + = 0, and evaluate at 1x, y2 = 11, 12 11 Given the demand equation 25p + x = 1,000, (A) Express the demand x as a function of the price p.  (B) Find the elasticity of demand, E1p2.  (C) Find E1152 and interpret.  (D) Express the revenue function as a function of price p.  (E) If p = $25, what is the effect of a small price cut on revenue?  12 Find the slope of the line tangent to y = ln1x + 22 + 5e-3x when x = 13 Use a calculator and a table of values to investigate n a1 + b lim nS ∞ n Do you think the limit exists? If so, what you think it is?  x = f1p2 = 201p - 152 … p … 15 32 Graph the revenue function as a function of price p, and ­indicate the regions of ­inelastic and elastic demand if the price–demand equation is  x = f1p2 = 5120 - p2 … p … 20 33 Let y = w2, w = e2u, and u = ln x (A) Express y in terms of x (B) Use the chain rule to find dy   dx Find the indicated derivatives in Problems 34–36 34 y′ for y = 72x +4 d 2ln1x2 + x2 dx d   35 log5 1x2 - x2  dx Find the indicated derivatives in Problems 14–19 36 d d 31ln z2 + ln z7 4  15 1x ln x2  dz dx d ex 16   17 y′ for y = ln12x6 + ex 2  dx x6 18 f = 1x2 for f1x2 = ex - x   19 dy>dx for y = e-2x ln 5x  37 Find y′ for y = y1x2 defined implicitly by the equation exy = x2 + y + 1, and evaluate at 10, 02.  14 20 Find the equation of the line tangent to the graph of y = f1x2 = + e-x at x = At x = - 1.  21 Find y′ for y = y1x2 defined implicitly by the equation x2 - 3xy + 4y2 = 23, and find the slope of the graph at - 1, 22.  22 Find x′ for x1t2 defined implicitly by 4t x2 - x2 + 100 = 0, and evaluate at 1t, x2 = - 1, - 12.  M11_BARN7668_13_GE_C11.indd 649 38 A rock thrown into a still pond causes a circular ripple The radius is increasing at a constant rate of feet per second Show that the area does not increase at a constant rate When is the rate of increase of the area the smallest? The largest? Explain.  39 A point moves along the graph of y = x3 in such a way that its y coordinate is increasing at a constant rate of units per second Does the x coordinate ever increase at a faster rate than the y coordinate? Explain 16/07/14 10:56 PM ... 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. .. 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... Act 19 88 Authorized adaptation from the United States edition, entitled College Mathematics for Business, Economics, Life Sciences and Social Sciences, 13 th edition, ISBN 978-0-3 21- 945 51- 8, by