Introductory Mathematical Analysis

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5 Introductory Mthemticl Anlysis For Business, Economics, nd the Life nd Socil Sciences Arb World Edition Ernest F. Heussler, Jr. The Pennsylvni Stte University Richrd S. Pul The Pennsylvni Stte University Richrd J. Wood Dlhousie University Sdi Khouyibb Americn University of Shrjh

6 Acquisitions Editor: Rsheed Roussn Senior Development Editor: Sophie Bulbrook Project Editor: Nicole Elliott Copy-editor: Alice Yew Proofreder: John King nd XXXX Design Mnger: Srh Fch Permissions Editor: XXXX Picture Resercher: XXXX Indeer: XXXX Mrketing Mnger: Sue Miney Senior Mnufcturing Controller: Christopher Crow Cover Designer: XXXX Typesetter: Integr Typefce: Tir Printed in Chin. Person Eduction Limited Edinburgh Gte Hrlow Esse CM JE Englnd nd Associted Compnies throughout the world c Person Eduction Limited Authorized for sle only in the Middle Est nd North Afric. The rights of Ernest Heussler, Richrd Pul, Richrd Wood, nd Sdi Khouyibb to be identified s uthors of this work hve been sserted by them in ccordnce with the Copyright, Designs nd Ptents Act 988. All rights reserved. No prt of this publiction my be reproduced, stored in retrievl system, or trnsmitted in ny form or by ny mens, electronic, mechnicl, photocopying, recording or otherwise, without either the prior written permission of the publisher or licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Sffron House, 6 Kirby Street, London ECN 8TS. All trdemrks used herein re the property of their respective owners. The use of ny trdemrk in this tet does not vest in the uthor or publisher ny trdemrk ownership rights in such trdemrks, nor does the use of such trdemrks imply ny ffilition with or endorsement of this book by such owners. Person Eduction is not responsible for the content of third prty internet sites. First published IMP ISBN:

7 About the Adpting Author Sdi Khouyibb, Ph. D., is n instructor of mthemtics in the Deprtment of Mthemtics nd Sttistics, Americn University of Shrjh, UAE. She received Mster s degree in Grph Theory from Montrel University, Cnd, nd Ph. D. degree in History of Mthemtics from Lvl University, Quebec, Cnd. Her reserch interests re relted to the history of mthemtics nd mthemticl eduction, though her first voction is teching mthemtics. At AUS since 6, she hs tught severl courses including Preclculus, Algebr, Clculus, nd Mthemtics for Business. Her dediction nd pssion for the teching profession mkes her ecellent instructor who lwys mnges to find the best wy to communicte her knowledge, cpture students interest, nd stimulte their curiosity. When not teching, Dr. Khouyibb enjoys the compny of her husbnd Guillume nd their two kids Yssine nd Skin, with whom she shres some very enjoyble nd rewrding moments. v

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9 Contents Foreword v Prefce vii Acknowledgments i CHAPTER Review of Bsic Algebr. Sets of Rel Numbers. Some Properties of Rel Numbers. Eponents nd Rdicls 9. Opertions with Algebric Epressions 6.5 Fctoring Polynomils.6 Rtionl Epressions Chpter Review 9 Importnt Terms nd Symbols 9 Review Problems 9 Chpter Test CHAPTER Equtions nd Inequlities. Equtions, in Prticulr Liner Equtions. Qudrtic Equtions. Applictions of Equtions 9. Liner Inequlities 58.5 Applictions of Inequlities 6.6 Absolute Vlue 66 Chpter Review 7 Importnt Terms nd Symbols 7 Summry 7 Review Problems 7 Chpter Test 7 Vrible-Qulity Recording 7 CHAPTER Functions, Grphs nd Lines 75. Functions 76. Specil Functions 85. Combintions of Functions 9. Inverse Functions 95.5 Grphs in Rectngulr Coordintes 98.6 Lines 7.7 Liner Functions nd Applictions 6.8 Qudrtic Functions nd Prbols Chpter Review Importnt Terms nd Symbols Summry Review Problems Chpter Test Mobile Phones 5 CHAPTER Eponentil nd Logrithmic Functions 7. Eponentil Functions 8. Logrithmic Functions 5. Properties of Logrithms 58. Logrithmic nd Eponentil Equtions 6 vii

10 viii Contents Chpter Review 69 Importnt Terms nd Symbols 69 Summry 69 Review Problems 7 Chpter Test 7 Drug Dosges 7 CHAPTER Mthemtics of Finnce 76. Summtion Nottion nd Sequences 76. Simple nd Compound Interest 9. Present Vlue 97. Interest Compounded Continuously.5 Annuities 5.6 Amortiztion of Lons.7 Perpetuities Chpter Review Importnt Terms nd Symbols Summry Review Problems 5 Chpter Test 5 Tresury Securities 7 CHAPTER 5 Mtri Algebr 9 5. Systems of Liner Equtions 5. Applictions of Systems of Liner Equtions 5. Mtrices 9 5. Mtri Addition nd Sclr Multipliction Mtri Multipliction Solving Systems of Liner Equtions by the Guss Jordn Method Inverses Leontief s Input Output Anlysis 99 Chpter 5 Review 5 Importnt Terms nd Symbols 5 Summry 6 Review Problems 6 Chpter Test 8 Insulin Requirements s Liner Process 9 CHAPTER 6 Liner Progrmming 6. Liner Inequlities in Two Vribles 6. Liner Progrmming: Grphicl Approch 7 6. The Simple Method: Mimiztion 9 6. The Simple Method: Nonstndrd Mimiztion Problems Minimiztion The Dul 66 Chpter 6 Review 76 Importnt Terms nd Symbols 76 Summry 76 Review Problems 77 Chpter Test 79 Drug nd Rdition Therpies 8 CHAPTER 7 Introduction to Probbility nd Sttistics 8 7. Bsic Counting Principle nd Permuttions 8 7. Combintions nd Other Counting Principles 9 7. Smple Spces nd Events 7. Probbility Conditionl Probbility nd Stochstic Processes

11 Contents i 7.6 Independent Events Byes s Formul 6 Chpter 7 Review 5 Importnt Terms nd Symbols 5 Summry 55 Review Problems 56 Chpter Test 58 Probbility nd Cellulr Automt 6 CHAPTER 8 Additionl Topics in Probbility 6 8. Discrete Rndom Vribles nd Epected Vlue 6 8. The Binomil Distribution 7 8. Mrkov Chins 78 Chpter 8 Review 88 Importnt Terms nd Symbols 88 Summry 88 Review Problems 89 Chpter Test 9 Mrkov Chins in Gme Theory 9 CHAPTER 9 Limits nd Continuity 9 9. Limits 9 9. One-Sided Limits nd Limits t Infinity 5 9. Continuity 5 9. Continuity Applied to Inequlities 57 Chpter 9 Review 5 Importnt Terms nd Symbols 5 Summry 5 Review Problems 5 Chpter Test 55 Public Debt 55 CHAPTER Differentition 57. The Derivtive 58. Rules for Differentition 56. The Derivtive s Rte of Chnge 5. The Product Rule nd the Quotient Rule The Chin Rule 565 Chpter Review 57 Importnt Terms nd Symbols 57 Summry 57 Review Problems 57 Chpter Test 576 Mrginl Propensity to Consume 577 CHAPTER Additionl Differentition Topics 579. Derivtives of Logrithmic Functions 58. Derivtives of Eponentil Functions 586. Elsticity of Demnd 59. Implicit Differentition Logrithmic Differentition 6.6 Higher-Order Derivtives 67 Chpter Review 6 Importnt Terms nd Symbols 6 Summry 6 Review Problems 6 Chpter Test 6 Economic Order Quntity 65

12 Contents CHAPTER Curve Sketching 67. Reltive Etrem 68. Absolute Etrem on Closed Intervl 6. Concvity 6. The Second-Derivtive Test 6.5 Asymptotes 6.6 Applied Mim nd Minim 65 Chpter Review 66 Importnt Terms nd Symbols 66 Summry 66 Review Problems 66 Chpter Test 667 Popultion Chnge over Time 668 CHAPTER Integrtion 67. Differentils 67. The Indefinite Integrl 676. Integrtion with Initil Conditions 68. More Integrtion Formuls Techniques of Integrtion 69.6 The Definite Integrl The Fundmentl Theorem of Integrl Clculus 75.8 Are between Curves 7.9 Consumers nd Producers Surplus 7 Chpter Review 76 Importnt Terms nd Symbols 76 Summry 77 Review Problems 78 Chpter Test 7 Delivered Price 7 CHAPTER Methods nd Applictions of Integrtion 7. Integrtion by Prts 75. Integrtion by Tbles 79. Averge Vlue of Function 76. Differentil Equtions 78.5 More Applictions of Differentil Equtions Improper Integrls 76 Chpter Review 765 Importnt Terms nd Symbols 765 Summry 765 Review Problems 766 Chpter Test 768 Dieting 769 CHAPTER 5 Continuous Rndom Vribles Continuous Rndom Vribles The Norml Distribution The Norml Approimtion to the Binomil Distribution 78 Chpter 5 Review 787 Importnt Terms nd Symbols 787 Summry 787 Review Problems 788 Chpter Test 788 Cumultive Distribution from Dt 789 CHAPTER 6 Multivrible Clculus Functions of Severl Vribles Prtil Derivtives 8

13 Contents i 6. Applictions of Prtil Derivtives Higher-Order Prtil Derivtives Mim nd Minim for Functions of Two Vribles Lgrnge Multipliers Lines of Regression 8 Chpter 6 Review 86 Importnt Terms nd Symbols 86 Summry 86 Review Problems 87 Chpter Test 88 Dt Anlysis to Model Cooling 8 APPENDIX A Compound Interest Tbles 8 APPENDIX B Tble of Selected Integrls 85 APPENDIX C Ares Under the Stndrd Norml Curve 85 English Arbic Glossry of Mthemticl Terms G- Answers to Odd-Numbered Problems AN- Inde I- Photo Credits P-

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15 Prefce The Arb World edition of Introductory Mthemticl Anlysis for Business, Economics, nd the Life nd Socil Sciences is built upon one of the finest books of its kind. This edition hs been dpted specificlly to meet the needs of students in the Arb world, nd provides mthemticl foundtion for students in vriety of fields nd mjors. It begins with preclculus nd finite mthemtics topics such s functions, equtions, mthemtics of finnce, mtri lgebr, liner progrmming, nd probbility. Then it progresses through both single vrible nd multivrible clculus, including continuous rndom vribles. Technicl proofs, conditions, nd the like re sufficiently described but re not overdone. Our guiding philosophy led us to include those proofs nd generl clcultions tht shed light on how the corresponding clcultions re done in pplied problems. Informl intuitive rguments re often given s well. Approch The Arb World Edition of Introductory Mthemticl Anlysis for Business, Economics, nd the Life nd Socil Sciences follows unique pproch to problem solving. As hs been the cse in erlier editions of this book, we estblish n emphsis on lgebric clcultions tht sets this tet prt from other introductory, pplied mthemtics books. The process of clculting with vribles builds skill in mthemticl modeling nd pves the wy for students to use clculus. The reder will not find definition-theorem-proof tretment, but there is sustined effort to imprt genuine mthemticl tretment of rel world problems. Emphsis on developing lgebric skills is etended to the eercises, in which mny, even those of the drill type, re given with generl coefficients. In ddition to the overll pproch to problem solving, we im to work through emples nd eplntions with just the right blend of rigor nd ccessibility. The tone of the book is not too forml, yet certinly not lcking precision. One might sy the book reds in reled tone without scrificing opportunities to bring students to higher level of understnding through strongly motivted pplictions. In ddition, the content of this edition is presented in more logicl wy for those teching nd lerning in the Arb region, in very mngeble portions for optiml teching nd lerning. Wht s New in the Arb World Edition? A number of dpttions nd new fetures hve been dded to the Arb World Edition. Additionl Emples nd Problems: Hundreds of rel life emples nd problems bout the Arb World hve been incorported. Additionl Applictions: Mny new Apply It fetures from cross the Arb region hve been dded to chpters to provide etr reinforcement of concepts, nd to provide the link between theory nd the rel world. Chpter test: This new feture hs been dded to every chpter to solidify the lerning process. These problems do not hve solutions provided t the end of the book, so cn be used s clss tests or homework. Biogrphies: These hve been included for prominent nd importnt mthemticins. This historicl ccount gives its rightful plce to both Arb nd interntionl contributors of this gret science. English-Arbic Glossry: Mthemticl, finncil nd economic terms with trnsltion to Arbic hs been dded to the end of the book. Any instructor with eperience in the Arb World knows how helpful this is for the students who studied in high school in Arbic. v

16 vi Prefce CAUTION Other Fetures nd Pedgogy Applictions: An bundnce nd vriety of new nd dditionl pplictions for the Arb udience pper throughout the book; students continully see how the mthemtics they re lerning cn be used in fmilir situtions, providing rel-world contet. These pplictions cover such diverse res s business, economics, biology, medicine, sociology, psychology, ecology, sttistics, erth science, nd rcheology. Mny of these rel-world situtions re drwn from literture nd re documented by references, sometimes from the Web. In some, the bckground nd contet re given in order to stimulte interest. However, the tet is self-contined, in the sense tht it ssumes no prior eposure to the concepts on which the pplictions re bsed. (See, for emple, pge XXX, Emple X in X.X) Apply It: The Apply It eercises provide students with further pplictions, with mny of these covering compnies nd trends from cross the region. Locted in the mrgins, these dditionl eercises give students rel-world pplictions nd more opportunities to see the chpter mteril put into prctice. An icon indictes Apply It problems tht cn be solved using grphing clcultor. Answers to Apply It problems pper t the end of the tet nd complete solutions to these problems re found in the Solutions Mnuls. (See, for emple, pge XXX, Apply It X in X.X) Now Work Problem N: Throughout the tet we hve retined the populr Now Work Problem N feture. The ide is tht fter worked emple, students re directed to n end of section problem (lbeled with blue eercise number) tht reinforces the ides of the worked emple. This gives students n opportunity to prctice wht they hve just lerned. Becuse the mjority of these keyed eercises re odd-numbered, students cn immeditely check their nswer in the bck of the book to ssess their level of understnding. The complete solutions to these eercises cn be found in the Student Solutions Mnul. (See, for emple, pge XXX, Emple X in XX.X) Cutions: Throughout the book, cutionry wrnings re presented in very much the sme wy n instructor would wrn students in clss of commonly-mde errors. These Cutions re indicted with n icon to help students prevent common misconceptions. (See, for emple, pge XXX, Emple X in XX.X) Definitions, key concepts, nd importnt rules nd formuls re clerly stted nd displyed s wy to mke the nvigtion of the book tht much esier for the student. (See, for emple, pge XXX, Definition of Derivtive in XX.X) Eplore & Etend Activities: Strtegiclly plced t the end of the chpter, these to bring together multiple mthemticl concepts studied in the previous sections within the contet of highly relevnt nd interesting ppliction. Where pproprite, these hve been dpted to the Arb World. These ctivities cn be completed in or out of clss either individully or within group. (See, for emple, pge XXX, in Chpter XX) Review Mteril: Ech chpter hs review section tht contins list of importnt terms nd symbols, chpter summry, nd numerous review problems. In ddition, key emples re referenced long with ech group of importnt terms nd symbols. (See, for emple, pge XXX, in Chpter XX) Bck-of-Book Answers: Answers to odd-numbered problems pper t the end of the book. For mny of the differentition problems, the nswers pper in both unsimplified nd simplified forms. This llows students to redily check their work. (See, for emple, pge AN-XX, in Answers for XX.X) Emples nd Eercises Most instructors nd students will gree tht the key to n effective tetbook is in the qulity nd quntity of the emples nd eercise sets. To tht end, hundreds emples re worked out in detil. Mny of these re new nd bout the Arb World, with rel regionl dt nd sttistics included wherever possible. These problems tke the reder from the popultion growth of Ciro, to the Infnt Mortlity rte in Tunisi, the life epectncy in Morocco, the

17 Prefce vii divorce rte in Algeri, the unemployment rte in Sudi Arbi, the eports nd imports of Kuwit, the oil production in Tunisi nd Sudi Arbi, Lbor Force in Morocco, the CPI of Liby, the GDC of Lebnon, the popultion of Bhrin in the ge group of 5 to 6, nd the number of doctors in Jordn. They lso include populr products from the region, nd locl compnies like Air Arbi, Royl Jordnin Airline, Emirtes, oil compnies such s Armco, postl compnies like Arme, telecommuniction providers such s Etislt or Mentel, the stocks of Emr. Regionl trends re lso covered in these problems, such s internet users in Yemen, mobile subscriptions in Syri, the emission of CO in Qtr, the number of shops in Dubi, the production of oil nd nturl gs in Omn, the production of electricity nd fresh ornge in Morocco, the prticiption to the Olympic gmes by the Arb ntions, nd the concept of Murbh in Islmic finnce. Some emples include strtegy bo designed to guide students through the generl steps of the solution before the specific solution is obtined (See pges XXX XXX, XX.X emple X). In ddition, n bundnt number of digrms nd eercises re included. In ech eercise set, grouped problems re given in incresing order of difficulty. In most eercise sets the problems progress from the bsic mechnicl drill-type to more interesting thought-provoking problems. The eercises lbeled with blue eercise number correlte to Now Work Problem N sttement nd emple in the section. A gret del of effort hs been put into producing proper blnce between the drilltype eercises nd the problems requiring the integrtion nd ppliction of the concepts lerned. (see pges XXX XXX, Eplore nd Etend for Chpter X; XXX, Eplore nd Etend for Chpter X; XXX XXX, Emple X in XX.X on Lines of Regression) Technology In order tht students pprecite the vlue of current technology, optionl grphing clcultor mteril ppers throughout the tet both in the eposition nd eercises. It ppers for vriety of resons: s mthemticl tool, to visulize concept, s computing id, nd to reinforce concepts. Although clcultor displys for TI-8 Plus ccompny the corresponding technology discussion, our pproch is generl enough so tht it cn be pplied to other grphing clcultors. In the eercise sets, grphing clcultor problems re indicted by n icon. To provide fleibility for n instructor in plnning ssignments, these problems re typiclly plced t the end of n eercise set. Course Plnning One of the obvious ssets of this book is tht considerble number of courses cn be served by it. Becuse instructors pln course outline to serve the individul needs of prticulr clss nd curriculum, we will not ttempt to provide detiled smple outlines. Introductory Mthemticl Anlysis is designed to meet the needs of students in Business, Economics, nd Life nd Socil Sciences. The mteril presented is sufficient for two semester course in Finite Mthemtics nd Clculus, or three semester course tht lso includes College Algebr nd Core Preclculus topics. The book consists of three importnt prts: Prt I: College Algebr The purpose of this prt is to provide students with the bsic skills of lgebr needed for ny subsequent work in Mthemtics. Most of the mteril covered in this prt hs been tught in high school. Prt II: Finite Mthemtics The second prt of this book provides the student with the tools he needs to solve rel-world problems relted to Business, Economic or Life nd Socil Sciences. Prt III: Applied Clculus In this lst prt the student will lern how to connect some Clculus topics to rel life problems.

18 viii Prefce Supplements The Student Solutions Mnul includes worked solutions for ll odd-numbered problems nd ll Apply It problems. ISBN XXXXX XXXXX The Instructor s Solution Mnul hs worked solutions to ll problems, including those in the Apply It eercises nd in the Eplore & Etend ctivities. It is downlodble from the Instructor s Resource Center t XXXXX. TestGen ( enbles instructors to build, edit, nd print, nd dminister tests using computerized bnk of questions developed to cover ll the objectives of the tet. TestGen is lgorithmiclly bsed, llowing instructors to crete multiple but equivlent versions of the sme question or test with the click of button. Instructors cn lso modify test bnk questions or dd new questions. The softwre nd testbnk re vilble for downlod from Person Eduction s online ctlog nd from the Instructor s Resource Center t XXXXXX. MyMthLb, gretly pprecited by instructors nd students, is powerful online lerning nd ssessment tool with interctive eercises nd problems, uto-grding, nd ssignble sets of questions tht cn be ssigned to students by the click of mouse.

19 Acknowledgments We epress our pprecition to the following collegues who contributed comments nd suggestions tht were vluble to us in the evolution of this tet: Nizr Bu Fkhreeddine, Deprtment of Mthemtics nd Sttistics, Notre Dme University Zouk Mousbeh, Lebnon Dr. Mged Isknder, Fculty of Business Administrtion, Economics nd Politicl Science, British University in Egypt Dr. Fud A. Kittneh, Deprtment of Mthemtics, University of Jordn, Jordn Hithm S. Solh, Deprtment of Mthemtics, Americn University in Dubi, UAE Michel M. Zlzli, Deprtment of Mthemtics, UAE University, UAE Mny reviewers nd contributors hve provided vluble contributions nd suggestions for previous editions of Introductory Mthemticl Anlysis. Mny thnks to them for their insights, which hve informed our work on this dpttion. Sdi Khouyibb i

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21 Introductory Mthemticl Anlysis For Business, Economics, nd the Life nd Socil Sciences Arb World Edition

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23 Integrtion. Differentils. The Indefinite Integrl. Integrtion with Initil Conditions. More Integrtion Formuls.5 Techniques of Integrtion.6 The Definite Integrl.7 The Fundmentl Theorem of Integrl Clculus.8 Are between Curves.9 Consumers nd Producers Surplus Chpter Review Delivered Price Anyone who runs business knows the need for ccurte cost estimtes. When jobs re individully contrcted, determining how much job will cost is generlly the first step in deciding how much to bid. For emple, pinter must determine how much pint job will tke. Since gllon of pint will cover certin number of squre meters, the key is to determine the re of the surfces to be pinted. Normlly, even this requires only simple rithmetic wlls nd ceilings re rectngulr, nd so totl re is sum of products of bse nd height. But not ll re clcultions re s simple. Suppose, for instnce, tht the bridge shown below must be sndblsted to remove ccumulted soot. How would the contrctor who chrges for sndblsting by the squre meter clculte the re of the verticl fce on ech side of the bridge? A D The re could be estimted s perhps three-qurters of the re of the trpezoid formed by points A, B, C, nd D. But more ccurte clcultion which might be desirble if the bid were for dozens of bridges of the sme dimensions (s long stretch of rilrod) would require more refined pproch. If the shpe of the bridge s rch cn be described mthemticlly by function, the contrctor could use the method introduced in this chpter: integrtion. Integrtion hs mny pplictions, the simplest of which is finding res of regions bounded by curves. Other pplictions include clculting the totl deflection of bem due to bending stress, clculting the distnce trveled underwter by submrine, nd clculting the electricity bill for compny tht consumes power t differing rtes over the course of month. Chpters delt with differentil clculus. We differentited function nd obtined nother function, its derivtive. Integrl clculus is concerned with the reverse process: We re given the derivtive of function nd must find the originl function. The need for doing this rises in nturl wy. For emple, we might hve mrginl-revenue function nd wnt to find the revenue function from it. Integrl clculus lso involves concept tht llows us to tke the limit of specil kind of sum s the number of terms in the sum becomes infinite. This is the rel power of integrl clculus! With such notion, we cn find the re of region tht cnnot be found by ny other convenient method. C B 67

24 67 Chpter Integrtion Objective To define the differentil, interpret it geometriclly, nd use it in pproimtions. Also, to restte the reciprocl reltionship between d/dy nd dy/d.. Differentils We will soon give reson for using the symbol dy/d to denote the derivtive of y with respect to. To do this, we introduce the notion of the differentil of function. Definition Let y = f () be differentible function of, nd let denote chnge in, where cn be ny rel number. Then the differentil of y, denoted dy or d( f ()), is given by dy = f () Note tht dy depends on two vribles, nmely, nd. In fct, dy is function of two vribles. EXAMPLE Computing Differentil Find the differentil of y = +, nd evlute it when = nd =.. Solution: The differentil is dy = d d ( + ) = ( + ) When = nd =., Isc Newton Isc Newton (6 77) is considered to be one of the most influentil physicists ever. His groundbreking findings, published in 687 in Philosophie Nturlis Principi Mthemtic ( Mthemticl Principles of Nturl Philosophy ), form the foundtion of clssicl mechnics. He nd Leibniz independently developed wht could be clled the most importnt discovery in mthemtics: the differentil nd integrl clculus. dy = [() () + ](.) =.8 Now Work Problem If y =, then dy = d() = =. Hence, the differentil of is. We bbrevite d() by d. Thus, d =. From now on, it will be our prctice to write d for when finding differentil. For emple, d( + 5) = d d ( + 5) d = d Summrizing, we sy tht if y = f () defines differentible function of, then dy = f () d where d is ny rel number. Provided tht d, we cn divide both sides by d: dy d = f () Tht is, dy/d cn be viewed either s the quotient of two differentils, nmely, dy divided by d, or s one symbol for the derivtive of f t. It is for this reson tht we introduced the symbol dy/d to denote the derivtive. EXAMPLE Finding Differentil in Terms of d. If f () =, then d( ) = d d ( ) d = / d = d b. If u = ( + ) 5, then du = 5( + ) () d = ( + ) d. Now Work Problem

25 Section. Differentils 67 y y f( ) f ( d) Q f ( d) f( ) f( ) P R S L dy y d d FIGURE. Geometric interprettion of dy nd. The differentil cn be interpreted geometriclly. In Figure., the point P(, f ()) is on the curve y = f (). Suppose chnges by d, rel number, to the new vlue + d. Then the new function vlue is f ( + d), nd the corresponding point on the curve is Q( + d, f ( + d)). Pssing through P nd Q re horizontl nd verticl lines, respectively, tht intersect t S. A line L tngent to the curve t P intersects segment QS t R, forming the right tringle PRS. Observe tht the grph of f ner P is pproimted by the tngent line t P. The slope of L is f () but it is lso given by SR/PS so tht Since dy = f () d nd d = PS, f () = SR PS dy = f () d = SR PS = SR PS Thus, if d is chnge in t P, then dy is the corresponding verticl chnge long the tngent line t P. Note tht for the sme d, the verticl chnge long the curve is y = SQ = f (+d) f (). Do not confuse y with dy. However, from Figure., the following is pprent: When d is close to, dy is n pproimtion to y. Therefore, y dy APPLY IT. The number of personl computers in Kuwit from 995 to 5 cn be pproimted by N(t) = where t = corresponds to the yer 995. Use differentils to pproimte the chnge in the number of computers s t goes from 995 to 5. Source: Bsed on dt from the United Ntions Sttistics Division. This fct is useful in estimting y, chnge in y, s Emple shows. EXAMPLE Using the Differentil to Estimte Chnge in Quntity A governmentl helth gency in the Middle Est emined the records of group of individuls who were hospitlized with prticulr illness. It ws found tht the totl proportion P tht re dischrged t the end of t dys of hospitliztion is given by ( ) P = P(t) = + t Use differentils to pproimte the chnge in the proportion dischrged if t chnges from to 5. Solution: The chnge in t from to 5 is t = dt = 5 = 5. The chnge in P is P = P(5) P(). We pproimte P by dp: ( ) ( P dp = P (t) dt = ) dt = + t ( + t) ( + t) dt

26 67 Chpter Integrtion When t = nd dt = 5, dp = 6 5 = 5 6 = =. For comprison, the true vlue of P is P(5) P() = =.7 (to five deciml plces). Now Work Problem Formul () is used to pproimte function vlue, wheres the formul y dy is used to pproimte chnge in function vlues. so tht We sid tht if y = f (), then y dy if d is close to zero. Thus, y = f ( + d) f () dy f ( + d) f () + dy () This formul gives us wy of estimting function vlue f ( + d). For emple, suppose we estimte ln(.6). Letting y = f () = ln, we need to estimte f (.6). Since d(ln ) = (/) d, we hve, from Formul (), f ( + d) f () + dy ln ( + d) ln + d We know the ect vlue of ln, so we will let = nd d =.6. Then + d =.6, nd d is close to zero. Therefore, ln ( +.6) ln () + (.6) ln (.6) +.6 =.6 The true vlue of ln(.6) to five deciml plces is.587. EXAMPLE Using the Differentil to Estimte Function Vlue A shoe mnufcturer in Sudn estblished tht the demnd function for its sports shoes is given by p = f (q) = q where p is the price per pir of shoes in dollrs for q pirs. By using differentils, pproimte the price when 99 pirs of shoes re demnded. Solution: We wnt to pproimte f (99). By Formul (), where f (q + dq) f (q) + dp dp = q dq dp dq = q / We choose q = nd dq = becuse q + dq = 99, dq is smll, nd it is esy to compute f () = =. We thus hve f (99) = f [ + ( )] f () ( ) f (99) +.5 =.5 Hence, the price per pir of shoes when 99 pirs re demnded is pproimtely $.5. Now Work Problem 5 The eqution y = defines y s function of. We could write f () = However, the eqution lso defines implicitly s function of y. In fct,

27 Section. Differentils 675 if we restrict the domin of f to some set of rel numbers so tht y = f () is one-toone function, then in principle we could solve for in terms of y nd get = f (y). [Actully, no restriction of the domin is necessry here. Since f () = + >, for ll, we see tht f is strictly incresing on (, ) nd is thus one-to-one on (, ).] As we did in Section., we cn look t the derivtive of with respect to y, d/dy nd we hve seen tht it is given by d dy = dy d provided tht dy/d Since d/dy cn be considered quotient of differentils, we now see tht it is the reciprocl of the quotient of differentils dy/d. Thus d dy = + It is importnt to understnd tht it is not necessry to be ble to solve y = for in terms of y, nd the eqution d dy = holds for ll. + EXAMPLE 5 Find dp dq if q = 5 p. Solution: Finding dp/dq from dq/dp Strtegy There re number of wys to find dp/dq. One pproch is to solve the given eqution for p eplicitly in terms of q nd then differentite directly. Another pproch to find dp/dq is to use implicit differentition. However, since q is given eplicitly s function of p, we cn esily find dq/dp nd then use the preceding reciprocl reltion to find dp/dq. We will tke this pproch. We hve Hence, dq dp = (5 p ) / p ( p) = 5 p dp dq = dq dp = 5 p p Now Work Problem 5 PROBLEMS. In Problems 9, find the differentil of the function in terms of nd d.. y = + b. y =. f () = 9. f () = ( 5 + ) 5. u = 6. u = 7. p = ln ( + 7) 8. p = e y = ln + In Problems, find y nd dy for the given vlues of nd d.. y = 5 ; =, d =.. y = + b; for ny nd ny d. y = + 5 7; =, d =.. y = ( + ) ; =, d =.. y = ; =, d =.5 Round your nswer to three deciml plces. 5. Let f () = () Evlute f (). (b) Use differentils to estimte the vlue of f (.). In Problems 6, pproimte ech epression by using differentils (Hint: 7 = 89.)

28 676 Chpter Integrtion. ln.97. ln.. e.. e. In Problems 9, find d/dy or dp/dq.. y = y = 6. q = (p + 5) 7. q = p q = p 9. q = e p. If y = 7 6 +, find the vlue of d/dy when =.. If y = ln, find the vlue of d/dy when =. In Problems nd, find the rte of chnge of q with respect to p for the indicted vlue of q.. p = 5 q + ; q = 8. p = 6 q; q = 5. Profit Suppose tht the profit (in dollrs) of producing q units of product is P = 97q.q Using differentils, find the pproimte chnge in profit if the level of production chnges from q = 9 to q = 9. Find the true chnge. 5. Revenue Given the revenue function r = 5q + 5q q use differentils to find the pproimte chnge in revenue if the number of units increses from q = to q =. Find the true chnge. 6. Demnd The demnd eqution for product is p = q Using differentils, pproimte the price when units re demnded. 7. Demnd Given the demnd function p = q + 8 use differentils to estimte the price per unit when units re demnded. 8. If y = f (), then the proportionl chnge in y is defined to be y/y, which cn be pproimted with differentils by dy/y. Use this lst form to pproimte the proportionl chnge in the cost function c = f (q) = q + 5q + when q = nd dq =. Round your nswer to one deciml plce. 9. Sttus/Income Suppose tht S is numericl vlue of sttus bsed on person s nnul income I (in thousnds of dollrs). For certin popultion, suppose S = I. Use differentils to pproimte the chnge in S if nnul income decreses from $5, to $,5.. Biology The volume of sphericl cell is given by V = πr, where r is the rdius. Estimte the chnge in volume when the rdius chnges from 6.5 cm to 6.6 cm.. Muscle Contrction The eqution (P + )(v + b) = k is clled the fundmentl eqution of muscle contrction. Here P is the lod imposed on the muscle, v is the velocity of the shortening of the muscle fibers, nd, b, nd k re positive constnts. Find P in terms of v, nd then use the differentil to pproimte the chnge in P due to smll chnge in v.. Profit The demnd eqution for monopolist s product is nd the verge-cost function is p = q 66q + 7 c = 5 q + 8, q () Find the profit when units re demnded. (b) Use differentils nd the result of prt () to estimte the profit when units re demnded. Objective To define the ntiderivtive nd the indefinite integrl nd to pply bsic integrtion formuls.. The Indefinite Integrl Given function f, if F is function such tht F () = f () () then F is clled n ntiderivtive of f. Thus, An ntiderivtive of f is simply function whose derivtive is f. Multiplying both sides of Eqution () by the differentil d gives F () d = f () d. However, becuse F () d is the differentil of F, we hve df = f () d. Hence, we cn think of n ntiderivtive of f s function whose differentil is f () d. R. W. Stcy et l., Essentils of Biologicl nd Medicl Physics (New York: McGrw-Hill, 955).

29 Section. The Indefinite Integrl 677 Definition An ntiderivtive of function f is function F such tht Equivlently, in differentil nottion, F () = f () df = f () d For emple, becuse the derivtive of is, is n ntiderivtive of. However, it is not the only ntiderivtive of : Since d d ( + ) = nd d d ( 5) = both + nd 5 re lso ntiderivtives of. In fct, it is obvious tht becuse the derivtive of constnt is zero, +C is lso n ntiderivtive of for ny constnt C. Thus, hs infinitely mny ntiderivtives. More importntly, ll ntiderivtives of must be functions of the form + C, becuse of the following fct: Any two ntiderivtives of function differ only by constnt. Since + C describes ll ntiderivtives of, we cn refer to it s being the most generl ntiderivtive of, denoted by d, which is red the indefinite integrl of with respect to. Thus, we write d = + C The symbol is clled the integrl sign, is the integrnd, nd C is the constnt of integrtion. The d is prt of the integrl nottion nd indictes the vrible involved. Here is the vrible of integrtion. More generlly, the indefinite integrl of ny function f with respect to is written f () d nd denotes the most generl ntiderivtive of f. Since ll ntiderivtives of f differ only by constnt, if F is ny ntiderivtive of f, then f () d = F() + C, where C is constnt To integrte f mens to find f () d. In summry, f () d = F() + C if nd only if F () = f () Thus we hve ( d d ) f () d = f () nd d (F()) d = F() + C d which shows the etent to which differentition nd indefinite integrtion re inverse procedures.

30 678 Chpter Integrtion APPLY IT. Suppose tht the mrginl cost for Mhrn Co. is f (q) = 8., find 8. dq, which gives the form of the cost function. CAUTION A common mistke is to omit C, the constnt of integrtion. EXAMPLE 6 Find 5 d. Solution: Finding n Indefinite Integrl Strtegy First we must find (perhps better words re guess t) function whose derivtive is 5. Then we dd the constnt of integrtion. Since we know tht the derivtive of 5 is 5, 5 is n ntiderivtive of 5. Therefore, 5 d = 5 + C Now Work Problem Abu Ali Ibn l-hythm Abu Ali l-hsn ibn l-hythm (965 ), born in Irq, ws one of the most fmousarb scientists, who left importnt works in stronomy, mthemtics, medicine nd physics. More thn 6 yers before integrls were known; he developed in his mnuscript Kitāb l-mnāzir ( Book of Optics ) method to clculte wht were, in fct, integrls of fourth-degree polynomils. Tble. Elementry Integrtion Formuls. k d = k + C k is constnt. d = C. d = d =. e d = e + C kf () d = k f () d (f () ± g()) d = d = = ln + C for > f () d ± k is constnt g() d Using differentition formuls from Chpters nd, we hve compiled list of elementry integrtion formuls in Tble.. These formuls re esily verified. For emple, Formul () is true becuse the derivtive of + /( + ) is for =. (We must hve = becuse the denomintor is when =.) Formul () sttes tht the indefinite integrl of power of, other thn, is obtined by incresing the eponent of by, dividing by the new eponent, nd dding constnt of integrtion. The indefinite integrl of will be discussed in Section.. To verify Formul (5), we must show tht the derivtive of k f () d is kf (). Since the derivtive of k f () d is simply k times the derivtive of f () d, nd the derivtive of f () d is f (), Formul (5) is verified. The reder should verify the other formuls. Formul (6) cn be etended to ny number of terms. EXAMPLE 7. Find d. Indefinite Integrls of Constnt nd of Power of Solution: By Formul () with k = d = + C = + C Usully, we write d s d. Thus, d = + C.

31 b. Find 5 d. Section. The Indefinite Integrl 679 Solution: By Formul () with n = 5, 5 d = C = C Now Work Problem APPLY IT. If the rte of chnge of Hossm Compny s revenues cn be modeled by dr =.t, then find.t dt, dt which gives the form of the compny s revenue function. CAUTION Only constnt fctor of the integrnd cn pss through n integrl sign. EXAMPLE 8 Find 7 d. Indefinite Integrl of Constnt Times Function Solution: By Formul (5) with k = 7 nd f () =, 7 d = 7 d Since is, by Formul () we hve d = C = + C where C is the constnt of integrtion. Therefore, ( ) 7 d = 7 d = 7 + C = 7 + 7C Since 7C is just n rbitrry constnt, we will replce it by C for simplicity. Thus, 7 d = 7 + C It is not necessry to write ll intermedite steps when integrting. More simply, we write 7 d = (7) + C = 7 + C Now Work Problem 5 EXAMPLE 9 Indefinite Integrl of Constnt Times Function Find 5 e d. Solution: 5 e d = e d Formul (5) 5 = 5 e + C Formul () Now Work Problem APPLY IT. Suppose tht due to new competition, the number of subscriptions toarb World mgzine is declining t rte of ds = 8 subscriptions per month, dt t where t is the number of months since the competition entered the mrket. Find the form of the eqution for the number of subscribers to the mgzine. EXAMPLE Finding Indefinite Integrls. Find dt. t Solution: Here t is the vrible of integrtion. We rewrite the integrnd so tht bsic formul cn be used. Since / t = t /, pplying Formul () gives dt = t / dt = t( /)+ t / t + C = + C = t + C +

32 68 Chpter Integrtion b. Find Solution: 6 d. 6 d = 6 d = ( ) C = + C = + C Now Work Problem 9 APPLY IT 5. The growth of the popultion of Riydh for the yers 995 to is estimted to follow the lw dn dt = t where t is the number of yers fter 995 nd N is in thousnds of individuls. Find n eqution tht describes the popultion of Riydh. Source: Bsed on dt from mongby.com/popultion_estimtes/full/ Riydh-Sudi_Arbi. html (ccessed November 8, ). When integrting n epression involving more thn one term, only one constnt of integrtion is needed. APPLY IT 6. The growth rte of pssengers flown by Royl Jordnin Airlines from to 8 cn be modeled by dn dt =.68t +.55t.95t +.76 where t is the time in yers nd N is the number of pssengers in millions. Find the form of the eqution describing the number of pssengers flown by Royl Jordnin. Source: Bsed on dt from the 9 Royl Jordnin Annul Report. EXAMPLE Indefinite Integrl of Sum Find ( + ) d. Solution: By Formul (6), ( + ) d = Now, nd Thus, d = d + d d = C = + C d = () C = + C ( + ) d = + + C + C For convenience, we will replce the constnt C + C by C. We then hve ( + ) d = + + C Omitting intermedite steps, we simply integrte term by term nd write ( + ) d = + () + C = + + C Now Work Problem EXAMPLE Indefinite Integrl of Sum nd Difference Find ( e ) d. Solution: ( e ) d = = () 9/5 9 5 /5 d 7 d + e d d Formuls (5) nd (6) (7) + e + C Formuls (), (), nd () = 9 9/5 7 + e + C Now Work Problem 5

33 Section. The Indefinite Integrl 68 Sometimes, in order to pply the bsic integrtion formuls, it is necessry first to perform lgebric mnipultions on the integrnd, s Emple shows. EXAMPLE Using Algebric Mnipultion to Find n Indefinite Integrl ( Find y y + ) dy. CAUTION In Emple, we first multiplied the fctors in the integrnd. The nswer could not hve been found simply in terms of y dy nd (y + ) dy. There is not formul for the integrl of generl product of functions. Solution: The integrnd does not fit fmilir integrtion form. However, by multiplying the integrnd we get ( y y + ) dy = (y + ) y dy ( ) = y y + + C = y + y 9 + C Now Work Problem 9 EXAMPLE ( )( + ). Find d. 6 Using Algebric Mnipultion to Find n Indefinite Integrl Solution: By fctoring out the constnt nd multiplying the binomils, we get 6 ( )( + ) d = ( + 5 ) d 6 6 = ) (() 6 + (5) + C Another lgebric pproch to prt (b) is d = ( ) d = ( ) d nd so on. b. Find d. = C Solution: We cn brek up the integrnd into frctions by dividing ech term in the numertor by the denomintor: ( d = ) d = ( ) d = + C = + + C Now Work Problem 7 PROBLEMS. In Problems 5, find the indefinite integrls.. 7 d. d. 8 d. 5 d z d 6. dz d 8. d 7 9/ 9. dt t7/.. ( + t) dt.. (5 w 6w ) dw. 5. (t t + 5) dt 6. (7r 5 + r + ) dr (y 5 5y) dy ( + t + t + t 6 ) dt ( + e) d

34 68 Chpter Integrtion 7. (5 ) d 8. ( ) d.. πe d.. (.7y + + y ) dy. d d 7. ( 8. ) d 9. ( w. ) dw. w u. du. 5. (u e + e u ) du dt 7. ( 7 ) d (e + + ) d ( ) d dz () d ( ) d 7e s ds ( ) e d ( ) d ( ) d ( ) 8. u + u du 9. ( + 5)( ) d (. ( ) d. + ) d. (z + ) dz. (u + ) du ( ). 5 d 5. ( + 5) d 6. (6e u u ( z u + )) du + z 7. dz z e + e d 9. d 5 e 5. ( + ) d 5. If F() nd G() re such tht F () = G (), is it true tht F() G() must be zero? 5. () Find function F such tht F() d = e + C. (b) Is there only one function F stisfying the eqution given in prt (), or re there mny such functions? 5. Find d d ( + ) d. Objective To find prticulr ntiderivtive of function tht stisfies certin conditions. This involves evluting constnts of integrtion.. Integrtion with Initil Conditions If we know the rte of chnge, f, of the function f, then the function f itself is n ntiderivtive of f (since the derivtive of f is f ). Of course, there re mny ntiderivtives of f, nd the most generl one is denoted by the indefinite integrl. For emple, if f () = then f () = f () d = d = + C. () Tht is, ny function of the form f () = +C hs its derivtive equl to. Becuse of the constnt of integrtion, notice tht we do not know f () specificlly. However, if f must ssume certin function vlue for prticulr vlue of, then we cn determine the vlue of C nd thus determine f () specificlly. For instnce, if f () =, then, from Eqution (), Thus, f () = + C = + C C = f () = + Tht is, we now know the prticulr function f () for which f () = nd f () =. The condition f () =, which gives function vlue of f for specific vlue of, is clled n initil condition.

35 Section. Integrtion with Initil Conditions 68 APPLY IT 7. The rte of growth of species of bcteri is estimted by dn = 8+e t, dt where N is the number of bcteri (in thousnds) fter t hours. If N(5) =,, find N(t). EXAMPLE 5 Initil-Condition Problem Suppose tht the mrginl profit of plstics fctory in Qtr is given by the function P () = where is the number (in thousnds) of items produced nd P represents the profit in thousnds of dollrs. Find the profit function, ssuming tht selling no items results in loss of $,. Solution: The profit function is P() = + 5 d = 5 5 d = d 5 d + 5 d d + 5 d = C = C () We determine the vlue of C by using the initil condition: substitute = nd P() = into Eqution () to get Hence, 75 () + 5() + C = C = P() = () Now Work Problem APPLY IT 8. The ccelertion of n object fter t seconds is given by y = 8t +, the velocity t 8 seconds is given by y (8) = 89 m, nd the position t seconds is given by y() = 85 m. Find y(t). EXAMPLE 6 Initil-Condition Problem Involving y Given tht y = 6, y () =, nd y() =, find y. Solution: Strtegy To go from y to y, two integrtions re needed: the first to tke us from y to y nd the other to tke us from y to y. Hence, there will be two constnts of integrtion, which we will denote by C nd C. Since y = d d (y ) = 6, y is n ntiderivtive of 6. Thus, y = ( 6) d = 6 + C () Now, y () = mens tht y = when = ; therefore, from Eqution (), we hve Hence, C =, so = 6() + C y = 6 +

36 68 Chpter Integrtion By integrtion, we cn find y: ( ) y = 6 + d ( ) = (6) + + C so y = + + C (5) Now, since y = when =, we hve, from Eqution (5), Thus, C =, so = () + () + C y = + Now Work Problem 5 Integrtion with initil conditions is pplicble to mny pplied situtions, s the net three emples illustrte. EXAMPLE 7 Income nd Eduction Suppose tht for prticulr Arb group, sociologists studied the current verge yerly income y (in dollrs) tht person cn epect to receive with yers of eduction before seeking regulr employment. They estimted tht the rte t which income chnges with respect to eduction is given by where y = 8,7 when = 9. Find y. dy d = / 6 Solution: Here y is n ntiderivtive of /. Thus, y = / d = / d = () 5/ 5 + C y = 5/ + C (6) The initil condition is tht y = 8,7 when = 9. By putting these vlues into Eqution (6), we cn determine the vlue of C: 8,7 = (9) 5/ + C = () + C 8,7 = 97 + C Therefore, C = 9,, nd y = 5/ + 9, Now Work Problem 7

37 Section. Integrtion with Initil Conditions 685 EXAMPLE 8 Finding Revenue from Mrginl Averge Revenue Suppose tht the mrginl verge revenue in dollrs of Ali Bb Museum resulting from the sle of tickets is given by R () = + If the verge revenue from the sle of tickets is $5, wht is the revenue when 5 tickets re sold? Solution: To find the revenue function, we first find the verge revenue. We hve R() = + d = + ln + C To find C, we use the initil condition R() = 5. This gives R() = + ln() + C = 5 C = 5 ln() Therefore R() = + ln() + nd hence R() = ( + ln() + ). So the revenue from the sle of 5 tickets is R(5) = 5(5 + ln(5) + ) 796 dollrs EXAMPLE 9 Finding the Demnd Function from Mrginl Revenue If the mrginl-revenue function for mnufcturer s product is find the demnd function. Solution: dr = q q dq Strtegy By integrting dr/dq nd using n initil condition, we cn find the revenue function r. But revenue is lso given by the generl reltionship r = pq, where p is the price per unit. Thus, p = r/q. Replcing r in this eqution by the revenue function yields the demnd function. Since dr/dq is the derivtive of totl revenue r, r = ( q q ) dq so tht = q () q ()q + C Revenue is when q is. r = q q q + C (7) We ssume tht when no units re sold, there is no revenue; tht is, r = when q =. This is our initil condition. Putting these vlues into Eqution (7) gives Although q = gives C =, this is not true in generl. It occurs in this section = () () + C becuse the revenue functions re polynomils. In lter sections, evluting Hence, C =, nd t q = my produce nonzero vlue for C. r = q q q

38 686 Chpter Integrtion To find the demnd function, we use the fct tht p = r/q nd substitute for r: p = r q = q q q q p = q q Now Work Problem EXAMPLE Finding Cost from Mrginl Cost Suppose tht Al Hllb Resturnt s fied costs per week re $. (Fied costs re costs, such s rent nd insurnce, tht remin constnt t ll levels of production during given time period.) If the mrginl-cost function is dc dq =.(.q 5q) +. where c is the totl cost (in dollrs) of producing q mels per week, find the cost of producing mels in week. Solution: Since dc/dq is the derivtive of the totl cost c, c(q) = [.(.q 5q) +.] dq =. (.q 5q) dq +. dq When q is, totl cost is equl to fied cost. Although q = gives C vlue equl to fied costs, this is not true in generl. It occurs in this section becuse the cost functions re polynomils. In lter sections, evluting t q = my produce vlue for C tht is different from fied cost. (.q ) c(q) =. 5q +.q + C Fied costs re constnt regrdless of output. Therefore, when q =, c =, which is our initil condition. Putting c() = in the lst eqution, we find tht C =, so (.q ) c(q) =. 5q +.q + (8) From Eqution (8), we hve c() = Thus, the totl cost for producing mels in week is $88.7. Now Work Problem 5 PROBLEMS. In Problems nd, find y subject to the given conditions.. dy/d = ; y( ) =. dy/d = ; y() = 9 In Problems, if y stisfies the given conditions, find y() for the given vlue of.. y = 9 8, y(6) = ; = 9 In Problems 7, find y subject to the given conditions.. y = + ; y () =, y() = 5 5. y = + ; y () =, y() = 6. y = ; y ( ) =, y () =, y() = 7. y = e + ; y () = 7, y () = 5, y() = In Problems 8, dr/dq is mrginl-revenue function. Find the demnd function. 8. dr/dq =.7 9. dr/dq = 6 q. dr/dq = 5, (q+q ). dr/dq = 75 q.q

39 Section. More Integrtion Formuls 687 In Problems 5, dc/dq is mrginl-cost function nd fied costs re indicted in brces. For Problems nd, find the totl-cost function. For Problems nd 5, find the totl cost for the indicted vlue of q.. dc/dq =.7; {59}. dc/dq = q + 75; {}. dc/dq =.q.6q + 6; {5,}; q = 5. dc/dq =.8q.6q + 6.5; {8}; q = 5 6. Winter Moth A study of the winter moth ws mde in Nov Scoti, Cnd. The prepupe of the moth fll onto the ground from host trees. It ws found tht the (pproimte) rte t which prepupl density y (the number of prepupe per squre foot of soil) chnges with respect to distnce (in feet) from the bse of host tree is If y = 59.6 when =, find y. dy d = Diet for Rts A group of biologists studied the nutritionl effects on rts tht were fed diet contining % protein. The protein consisted of yest nd corn flour. If G = 8 when P =, find G. dg dp = P 5 + P 8. Fluid Flow In the study of the flow of fluid in tube of constnt rdius R, such s blood flow in portions of the body, one cn think of the tube s consisting of concentric tubes of rdius r, where r R. The velocity v of the fluid is function of r nd is given by v = (P P )r lη where P nd P re pressures t the ends of the tube, η ( Greek letter red et ) is fluid viscosity, nd l is the length of the tube. If v = when r = R, show tht dr v = (P P )(R r ) lη 9. Averge Cost Amrn mnufctures jens nd hs determined tht the mrginl-cost function is dc dq =.q.q + where q is the number of pirs of jens produced. If mrginl cost is $7.5 when q = 5 nd fied costs re $5, wht is the verge cost of producing pirs of jens? Over period of time, the group found tht the (pproimte) rte of chnge of the verge weight gin G (in grms) of rt with respect to the percentge P of yest in the protein mi ws. If f () = + nd f () =, evlute f (965.55) f ( ) Objective To lern nd pply the formuls for u du, e u du, nd u du.. More Integrtion Formuls Power Rule for Integrtion The formul d = + n + + C if = which pplies to power of, cn be generlized to hndle power of function of. Let u be differentible function of. By the power rule for differentition, if =, then ( (u()) +) d d + = ( + )(u()) u () + = (u()) u () Adpted from D. G. Embree, The Popultion Dynmics of the Winter Moth in Nov Scoti, 95 96, Memoirs of the Entomologicl Society of Cnd, no. 6 (965). Adpted from R. Bressni, The Use of Yest in Humn Foods, in Single-Cell Protein, eds. R. I. Mteles nd S. R. Tnnenbum (Cmbridge, MA: MIT Press, 968). R. W. Stcy et l., Essentils of Biologicl nd Medicl Physics (New York: McGrw-Hill, 955).