加入VIP
  • 专属下载特权
  • 现金文档折扣购买
  • VIP免费专区
  • 千万文档免费下载

上传资料

关闭

关闭

关闭

封号提示

内容

首页 美国大学微积分教材James Stewart课本以及习题解答

美国大学微积分教材James Stewart课本以及习题解答.pdf

美国大学微积分教材James Stewart课本以及习题解答

dawncoming
2010-08-08 0人阅读 举报 0 0 暂无简介

简介:本文档为《美国大学微积分教材James Stewart课本以及习题解答pdf》,可适用于其他资料领域

FunctionsandModelsAgraphicalrepresentationofafunction––herethenumberofhoursofdaylightasafunctionofthetimeofyearatvariouslatitudes––isoftenthemostnaturalandconvenientwaytorepresentthefunctionThefundamentalobjectsthatwedealwithincalculusarefunctionsThischapterpreparesthewayforcalculusbydiscussingthebasicideasconcerningfunctions,theirgraphs,andwaysoftransformingandcombiningthemWestressthatafunctioncanberepresentedindifferentways:byanequation,inatable,byagraph,orinwordsWelookatthemaintypesoffunctionsthatoccurincalculusanddescribetheprocessofusingthesefunctionsasmathematicalmodelsofrealworldphenomenaWealsodiscusstheuseofgraphingcalculatorsandgraphingsoftwareforcomputers||||FourWaystoRepresentaFunctionFunctionsarisewheneveronequantitydependsonanotherConsiderthefollowingfoursituationsATheareaofacircledependsontheradiusofthecircleTherulethatconnectsandisgivenbytheequationWitheachpositivenumberthereisassociatedonevalueof,andwesaythatisafunctionofBThehumanpopulationoftheworlddependsonthetimeThetablegivesestimatesoftheworldpopulationattimeforcertainyearsForinstance,ButforeachvalueofthetimethereisacorrespondingvalueofandwesaythatisafunctionofCThecostofmailingafirstclassletterdependsontheweightoftheletterAlthoughthereisnosimpleformulathatconnectsand,thepostofficehasarulefordeterminingwhenisknownDTheverticalaccelerationofthegroundasmeasuredbyaseismographduringanearthquakeisafunctionoftheelapsedtimeFigureshowsagraphgeneratedbyseismicactivityduringtheNorthridgeearthquakethatshookLosAngelesinForagivenvalueofthegraphprovidesacorrespondingvalueofFIGUREVerticalgroundaccelerationduringtheNorthridgeearthquake{cms}(seconds)CalifDeptofMinesandGeologyatat,tawCCwwCtPP,tP���,,,t,P�t�tPrAArA��rArrAPopulationYear(millions)Eachoftheseexamplesdescribesarulewhereby,givenanumber(,,,or),anothernumber(,,,or)isassignedIneachcasewesaythatthesecondnumberisafunctionofthefirstnumberAfunctionisarulethatassignstoeachelementinasetexactlyoneelement,called,inasetWeusuallyconsiderfunctionsforwhichthesetsandaresetsofrealnumbersThesetiscalledthedomainofthefunctionThenumberisthevalueofatandisread“of”TherangeofisthesetofallpossiblevaluesofasvariesthroughoutthedomainAsymbolthatrepresentsanarbitrarynumberinthedomainofafunctioniscalledanindependentvariableAsymbolthatrepresentsanumberintherangeofiscalledadependentvariableInExampleA,forinstance,ristheindependentvariableandAisthedependentvariableIt’shelpfultothinkofafunctionasamachine(seeFigure)Ifisinthedomainofthefunctionthenwhenentersthemachine,it’sacceptedasaninputandthemachineproducesanoutputaccordingtotheruleofthefunctionThus,wecanthinkofthedomainasthesetofallpossibleinputsandtherangeasthesetofallpossibleoutputsThepreprogrammedfunctionsinacalculatoraregoodexamplesofafunctionasamachineForexample,thesquarerootkeyonyourcalculatorcomputessuchafunctionYoupressthekeylabeled(or)andentertheinputxIf,thenisnotinthedomainofthisfunctionthatis,isnotanacceptableinput,andthecalculatorwillindicateanerrorIf,thenanapproximationtowillappearinthedisplayThus,thekeyonyourcalculatorisnotquitethesameastheexactmathematicalfunctiondefinedbyAnotherwaytopictureafunctionisbyanarrowdiagramasinFigureEacharrowconnectsanelementoftoanelementofThearrowindicatesthatisassociatedwithisassociatedwith,andsoonThemostcommonmethodforvisualizingafunctionisitsgraphIfisafunctionwithdomain,thenitsgraphisthesetoforderedpairs(Noticethattheseareinputoutputpairs)Inotherwords,thegraphofconsistsofallpointsinthecoordinateplanesuchthatandisinthedomainofThegraphofafunctiongivesusausefulpictureofthebehavioror“lifehistory”ofafunctionSincethecoordinateofanypointonthegraphis,wecanreadthevalueoffromthegraphasbeingtheheightofthegraphabovethepoint(seeFigure)ThegraphofalsoallowsustopicturethedomainofontheaxisanditsrangeontheaxisasinFigurexy�ƒ(x)domainrangeyFIGURE{x,ƒ}ƒf()f()xyxFIGUREyxffxf�x�y�f�x��x,y�yffxy�f�x��x,y�f��x,f�x���x�A�Afaf�a�x,f�x�BAf�x��sxfsxsxx�xxx�sxsf�x�xf,xffxf�x�fxfxff�x�ABABf�x�AxfaCPAtwtr❙❙❙❙CHAPTERFUNCTIONSANDMODELSFIGUREMachinediagramforafunctionƒx(input)ƒ(output)ffABƒf(a)axFIGUREArrowdiagramforƒSECTIONFOURWAYSTOREPRESENTAFUNCTION❙❙❙❙EXAMPLEThegraphofafunctionisshowninFigure(a)Findthevaluesofand(b)WhatarethedomainandrangeofSOLUTION(a)WeseefromFigurethatthepointliesonthegraphof,sothevalueofatis(Inotherwords,thepointonthegraphthatliesabovex�isunitsabovethexaxis)Whenx�,thegraphliesaboutunitbelowthexaxis,soweestimatethat(b)Weseethatisdefinedwhen,sothedomainofistheclosedintervalNoticethattakesonallvaluesfrom�to,sotherangeofisEXAMPLESketchthegraphandfindthedomainandrangeofeachfunction(a)(b)SOLUTION(a)Theequationofthegraphis,andwerecognizethisasbeingtheequationofalinewithslopeandyintercept�(Recalltheslopeinterceptformoftheequationofaline:SeeAppendixB)ThisenablesustosketchthegraphofinFigureTheexpressionisdefinedforallrealnumbers,sothedomainofisthesetofallrealnumbers,whichwedenoteby�Thegraphshowsthattherangeisalso�(b)Sinceand,wecouldplotthepointsand,togetherwithafewotherpointsonthegraph,andjointhemtoproducethegraph(Figure)Theequationofthegraphis,whichrepresentsaparabola(seeAppendixC)Thedomainoftis�Therangeoftconsistsofallvaluesof,thatis,allnumbersoftheformButforallnumbersxandanypositivenumberyisasquareSotherangeoftisThiscanalsobeseenfromFigure(, )(, )yxy=≈FIGURE�y�y����,��x�xt�x�y�x��,��,�t��������t����fx�fy�mx�by�x�t�x��xf�x��x��y���y�����,ff�,f�x�f�x�f����f���ff�,�FIGURExyff��f��f||||ThenotationforintervalsisgiveninAppendixAFIGURExy=xy❙❙❙❙CHAPTERFUNCTIONSANDMODELSRepresentationsofFunctionsTherearefourpossiblewaystorepresentafunction:��verbally(byadescriptioninwords)��numerically(byatableofvalues)��visually(byagraph)��algebraically(byanexplicitformula)Ifasinglefunctioncanberepresentedinallfourways,itisoftenusefultogofromonerepresentationtoanothertogainadditionalinsightintothefunction(InExample,forinstance,westartedwithalgebraicformulasandthenobtainedthegraphs)ButcertainfunctionsaredescribedmorenaturallybyonemethodthanbyanotherWiththisinmind,let’sreexaminethefoursituationsthatweconsideredatthebeginningofthissectionAThemostusefulrepresentationoftheareaofacircleasafunctionofitsradiusisprobablythealgebraicformula,thoughitispossibletocompileatableofvaluesortosketchagraph(halfaparabola)Becauseacirclehastohaveapositiveradius,thedomainis,andtherangeisalsoBWearegivenadescriptionofthefunctioninwords:isthehumanpopulationoftheworldattimetThetableofvaluesofworldpopulationonpageprovidesaconvenientrepresentationofthisfunctionIfweplotthesevalues,wegetthegraph(calledascatterplot)inFigureIttooisausefulrepresentationthegraphallowsustoabsorballthedataatonceWhataboutaformulaOfcourse,it’simpossibletodeviseanexplicitformulathatgivestheexacthumanpopulationatanytimetButitispossibletofindanexpressionforafunctionthatapproximatesInfact,usingmethodsexplainedinSection,weobtaintheapproximationandFigureshowsthatitisareasonablygood“fit”ThefunctioniscalledamathematicalmodelforpopulationgrowthInotherwords,itisafunctionwithanexplicitformulathatapproximatesthebehaviorofourgivenfunctionWewillsee,however,thattheideasofcalculuscanbeappliedtoatableofvaluesanexplicitformulaisnotnecessaryFIGUREFIGUREx'Ptx'PtfP�t��f�t������tP�t�P�t�P�t��,���r�r���,��A�r���rSECTIONFOURWAYSTOREPRESENTAFUNCTION❙❙❙❙ThefunctionistypicalofthefunctionsthatarisewheneverweattempttoapplycalculustotherealworldWestartwithaverbaldescriptionofafunctionThenwemaybeabletoconstructatableofvaluesofthefunction,perhapsfrominstrumentreadingsinascientificexperimentEventhoughwedon’thavecompleteknowledgeofthevaluesofthefunction,wewillseethroughoutthebookthatitisstillpossibletoperformtheoperationsofcalculusonsuchafunctionCAgainthefunctionisdescribedinwords:isthecostofmailingafirstclassletterwithweightTherulethattheUSPostalServiceusedasofisasfollows:Thecostiscentsforuptooneounce,pluscentsforeachsuccessiveounceuptoouncesThetableofvaluesshowninthemarginisthemostconvenientrepresentationforthisfunction,thoughitispossibletosketchagraph(seeExample)DThegraphshowninFigureisthemostnaturalrepresentationoftheverticalaccelerationfunctionIt’struethatatableofvaluescouldbecompiled,anditisevenpossibletodeviseanapproximateformulaButeverythingageologistneedstoknowamplitudesandpatternscanbeseeneasilyfromthegraph(Thesameistrueforthepatternsseeninelectrocardiogramsofheartpatientsandpolygraphsforliedetection)FiguresandshowthegraphsofthenorthsouthandeastwestaccelerationsfortheNorthridgeearthquakewhenusedinconjunctionwithFigure,theyprovideagreatdealofinformationabouttheearthquakeInthenextexamplewesketchthegraphofafunctionthatisdefinedverballyEXAMPLEWhenyouturnonahotwaterfaucet,thetemperatureofthewaterdependsonhowlongthewaterhasbeenrunningDrawaroughgraphofasafunctionofthetimethathaselapsedsincethefaucetwasturnedonSOLUTIONTheinitialtemperatureoftherunningwaterisclosetoroomtemperaturebecauseofthewaterthathasbeensittinginthepipesWhenthewaterfromthehotwatertankstartscomingout,increasesquicklyInthenextphase,isconstantatthetemperatureoftheheatedwaterinthetankWhenthetankisdrained,decreasestothetemperatureofthewatersupplyThisenablesustomaketheroughsketchofasafunctionofinFiguretTTTTtTTFIGURENorthsouthaccelerationfortheNorthridgeearthquake{cms}at(seconds)CalifDeptofMinesandGeologyFIGUREEastwestaccelerationfortheNorthridgeearthquakeat{cms}(seconds)CalifDeptofMinesandGeologya�t�wC�w�P||||Afunctiondefinedbyatableofvaluesiscalledatabularfunction(ounces)(dollars)�w��w��w��w��w�C�w�wtTFIGURE❙❙❙❙CHAPTERFUNCTIONSANDMODELSAmoreaccurategraphofthefunctioninExamplecouldbeobtainedbyusingathermometertomeasurethetemperatureofthewateratsecondintervalsIngeneral,scientistscollectexperimentaldataandusethemtosketchthegraphsoffunctions,asthenextexampleillustratesEXAMPLEThedatashowninthemargincomefromanexperimentonthelactonizationofhydroxyvalericacidatTheygivetheconcentrationofthisacid(inmolesperliter)afterminutesUsethesedatatodrawanapproximationtothegraphoftheconcentrationfunctionThenusethisgraphtoestimatetheconcentrationafterminutesSOLUTIONWeplotthefivepointscorrespondingtothedatafromthetableinFigureThecurvefittingmethodsofSectioncouldbeusedtochooseamodelandgraphitButthedatapointsinFigurelookquitewellbehaved,sowesimplydrawasmoothcurvethroughthembyhandasinFigureThenweusethegraphtoestimatethattheconcentrationafterminutesismoleliterInthefollowingexamplewestartwithaverbaldescriptionofafunctioninaphysicalsituationandobtainanexplicitalgebraicformulaTheabilitytodothisisausefulskillinsolvingcalculusproblemsthataskforthemaximumorminimumvaluesofquantitiesEXAMPLEArectangularstoragecontainerwithanopentophasavolumeofmThelengthofitsbaseistwiceitswidthMaterialforthebasecosts$persquaremetermaterialforthesidescosts$persquaremeterExpressthecostofmaterialsasafunctionofthewidthofthebaseSOLUTIONWedrawadiagramasinFigureandintroducenotationbylettingandbethewidthandlengthofthebase,respectively,andbetheheightTheareaofthebaseis,sothecost,indollars,ofthematerialforthebaseisTwoofthesideshaveareaandtheothertwohavearea,sothecostofthematerialforthesidesisThetotalcostisthereforeToexpressasafunctionofalone,weneedtoeliminateandwedosobyusingthefactthatthevolumeismThuswhichgivesh�w�ww�w�h�hwCC��w����wh���wh��w�wh��wh���wh�whwh�w��w�w�whwwC���FIGUREC(t)ttC(t)FIGUREtC�t��CtC�t�wwhFIGURESECTIONFOURWAYSTOREPRESENTAFUNCTION❙❙❙❙Substitutingthisintotheexpressionfor,wehaveTherefore,theequationexpressesasafunctionofEXAMPLEFindthedomainofeachfunction(a)(b)SOLUTION(a)Becausethesquarerootofanegativenumberisnotdefined(asarealnumber),thedomainofconsistsofallvaluesofxsuchthatThisisequivalentto,sothedomainistheinterval(b)Sinceanddivisionbyisnotallowed,weseethatisnotdefinedwhenorThus,thedomainofiswhichcouldalsobewritteninintervalnotationasThegraphofafunctionisacurveintheplaneButthequestionarises:WhichcurvesintheplanearegraphsoffunctionsThisisansweredbythefollowingtestTheVerticalLineTestAcurveintheplaneisthegraphofafunctionofifandonlyifnoverticallineintersectsthecurvemorethanonceThereasonforthetruthoftheVerticalLineTestcanbeseeninFigureIfeachverticallineintersectsacurveonlyonce,at,thenexactlyonefunctionalvalueisdefinedbyButifalineintersectsthecurvetwice,atand,thenthecurvecan’trepresentafunctionbecauseafunctioncan’tassigntwodifferentvaluestoFIGURExay(a, c)(a, b)x=axayx=a(a, b)a�a,c��a,b�x�af�a��b�a,b�x�axxyxyxy���,���,���,���x�x�,x��tx�x�t�x�t�x��x�x�x�x����,��x��x��ft�x��x�xf�x��sx�wCwC�w��w�wC�w�w�w��w�wC||||InsettingupappliedfunctionsasinExample,itmaybeusefultoreviewtheprinciplesofproblemsolvingasdiscussedonpage,particularlyStep:UnderstandtheProblem||||Ifafunctionisgivenbyaformulaandthedomainisnotstatedexplicitly,theconventionisthatthedomainisthesetofallnumbersforwhichtheformulamakessenseanddefinesarealnumber❙❙❙❙CHAPTERFUNCTIONSANDMODELSForexample,theparabolashowninFigure(a)isnotthegraphofafunctionofbecause,asyoucansee,thereareverticallinesthatintersecttheparabolatwiceTheparabola,however,doescontainthegraphsoftwofunctionsofNoticethattheequationimplies,soThus,theupperandlowerhalvesoftheparabolaarethegraphsofthefunctionsfromExample(a)andSeeFigures(b)and(c)Weobservethatifwereversetherolesofand,thentheequationdoesdefineasafunctionof(withastheindependentvariableandasthedependentvariable)andtheparabolanowappearsasthegraphofthefunctionPiecewiseDefinedFunctionsThefunctionsinthefollowingfourexamplesaredefinedbydifferentformulasindifferentpartsoftheirdomainsEXAMPLEAfunctionisdefinedbyEvaluate,,andandsketchthegraphSOLUTIONRememberthatafunctionisaruleForthisparticularfunctiontheruleisthefollowing:FirstlookatthevalueoftheinputIfithappensthat,thenthevalueofisOntheotherhand,if,thenthevalueofisHowdowedrawthegraphofWeobservethatif,then,sothepartofthegraphofthatliestotheleftoftheverticallinemustcoincidewiththeline,whichhasslopeandinterceptIf,then,sothepartofthegraphofthatliestotherightofthelinemustcoincidewiththegraphof,whichisaparabolaThisenablesustosketchthegraphinFigurelThesoliddotindicatesthatthepointisincludedonthegraphtheopendotindicatesthatthepointisexcludedfromthegraph�,��,�y�xx�ff�x��xxy�y��xx�ff�x���xx�fSince,wehavef����Since�,wehavef�����Since�,wehavef�����xf�x�x�xf�x��xxf��f��f��f�x���xxifx�ifxfFIGURE(, )(a)x=¥xy(c)y=œ„„„„xxy(b)y=œ„„„„xxyhxyyxx�h�y��y�yxt�x���sx�f�x��sx�y��sx�y�x�x�y�xxx�y�FIGURExySECTIONFOURWAYSTOREPRESENTAFUNCTION❙❙❙❙ThenextexampleofapiecewisedefinedfunctionistheabsolutevaluefunctionRecallthattheabsolutevalueofanumber,denotedby,isthedistancefromtoontherealnumberlineDistancesarealwayspositiveor,sowehaveforeverynumberForexample,Ingeneral,wehave(Rememberthatifisnegative,thenispositive)EXAMPLESketchthegraphoftheabsolutevaluefunctionSOLUTIONFromtheprecedingdiscussionweknowthatUsingthesamemethodasinExample,weseethatthegraphofcoincideswiththelinetotherightoftheaxisandcoincideswiththelinetotheleftoftheaxis(seeFigure)EXAMPLEFi

用户评价(4)

  • 94048204a 谢谢

    2013-01-29 20:38:07

  • xboxpsp 真的是好东西,很清楚,答案很全

    2011-07-31 21:51:45

  • 108.85.231.156 太有用了

    2011-06-21 20:38:15

  • 59.44.13.239 It probably is the Fifth Edition of Calculus, Early Transcendentals. Thank you very very much, indeed.

    2011-01-03 02:30:17

关闭

新课改视野下建构高中语文教学实验成果报告(32KB)

抱歉,积分不足下载失败,请稍后再试!

提示

试读已结束,如需要继续阅读或者下载,敬请购买!

文档小程序码

使用微信“扫一扫”扫码寻找文档

1

打开微信

2

扫描小程序码

3

发布寻找信息

4

等待寻找结果

我知道了
评分:

/49

美国大学微积分教材James Stewart课本以及习题解答

仅供在线阅读

VIP

在线
客服

免费
邮箱

爱问共享资料服务号

扫描关注领取更多福利