微积分教学资料-cha

Chapter14PartialDerivatives14.1FunctionsofSeveralVariables14.2LimitsandContinuity14.3PartialDerivatives14.4TangentPlanesandLinearApproximations14.5TheChainRule14.6DirectionalDerivativesandtheGradientVector14.7MaximumandMinimumValues14.8LagrangeMultipliers,Sofarwehavedealtwiththecalculusoffunctionsofasinglevariable.But,intherealworld,physicalquantitiesoftendependontwoormorevariables,sointhischapterweturnourattentiontofunctionsofseveralvariablesandextendthebasicideasofdifferentialcalculustosuchfunctions.,14.1FunctionsofSeveralVariables,Inthissectionwestudyfunctionsoftwoormorevariablesfromfourpointsofview:Verbally(byadescriptioninwords)Numerically(byatableofvalues)Algebraically(byanexplicitformula)Visually(byagraphorlevelcurves),FunctionsofTwoVariables,DefinitionAfunctionoftwovariablesisarulethatassignstoeachorderedpairofrealnumbers(x,y)inasetDauniquerealnumberdenotedbyf(x,y).ThesetDisthedomainoffanditsrangeisthesetofvaluesthatftakeson,thatis,f(x,y)|(x,y)D.Weoftenwritez=f(x,y)tomakeexplicitthevaluetakenonbyfatthegeneralpoint(x,y).Thevariablesxandyareindependentvariablesandzisthedependentvariable.,（x,y）,f(x,y),x,y,z,f,Example1Findthedomainsofthefollowingfunctionsandevaluatef(3,2).(a)(b)Solution(a)(b),Example,Findthedomainof,Solution,yy,yy,y,x,oo,oo,o,o,o,0,0,0o,Oo0o,o,Example,Solution,GraphsAnotherwayofvisualizingthebehaviorofafunctionoftwovariablesistoconsideritsgraph.,DefinitionIffisafunctionoftwovariableswithdomainD,thenthegraphoffisthesetofallpoints(x,y,z)insuchthatz=f(x,y)and(x,y)isinD.,x,y,z,D,S,Levelcurves,DefinitionThelevelcurvesofafunctionoftwovariablesarethecurveswithequationsf(x,y)=kwherekisaconstant(intherangeoff).,Sketchsomelevelcurvesofthefunction,Example,FunctionsofThreeorMoreVariables,Afunctionofthreevariables,f,isarulethatassignstoeachorderedtriple(x,y,z)inadomainDauniquerealnumberdenotedbyf(x,y,z).Afunctionofnvariables,f,isarulethatassignsanumbertoann-tupleofrealnumber.Wedenotedbythesetofallsuchn-tuples.,14.2LimitsandContinuity,DefinitionLetfbeafunctionoftwovariableswhosedomainDincludespointsarbitrarilycloseto(a,b).Thenwesaythatthelimitoff(x,y)as(x,y)approaches(a,b)isLandwewriteifforeverynumberthereisacorrespondingnumbersuchthatwheneverand,ExampleFindifitexists.,Solution,Let,Since,Ifwechoose,then,whenever,Hence,Ifasalongapathandasalongapath,where,thendoesnotexist.Exampledoesnotexist.Solutionasalongthex-axisasalongthey-axis,Thusthegivenlimitdoesnotexist.,Example,doesnotexist?,Solution,Thusthegivenlimitdoesnotexist.,Since,ExampleFind,Solution,ExampleFind,Solution,ExampleFind,Solution,So,Continuity,DefinitionAfunctionfoftwovariablesiscalledcontinuousat(a,b)ifWesayfiscontinuousonDiffiscontinuousatEverypoint(a,b)inD.Example,Examplewhereisthefollowingfunctioncontinuous.,Solution,Thefunctionfiscontinuousfor,sinceitisequaltoarationalfunctionthere.,Thefunctionfiscontinuousat(0,0)because,Sothefunctioniscontinuousin,Examplewhereisthefollowingfunctiondiscontinuous.,Solution,sincefarationalfunction,itiscontinuousonitsdomain,whichistheset,Sothefunctionisdiscontinuousonthecircle.,机动目录上页下页返回结束,14.3PartialDerivatives,Iffisafunctionoftwovariables,itspartialderivativesarethefunctionsanddefinedbyNotationsforpartialderivativesIf,wewrite,Ruleforfindingpartialderivativesof1.Tofindregardasaconstantanddifferentiatewithrespectto2.TofindregardasaconstantanddifferentiatewithrespecttoExampleIf,findand.Solution,Solution1:,Solution2:,机动目录上页下页返回结束,ExampleIffindand,ExampleIf,proof:,机动目录上页下页返回结束,，,）,and,1,0,(,=,x,x,x,z,y,provethat,Solution:,机动目录上页下页返回结束,ExampleIffindand,Interpretationsofpartialderivatives,canbeinterpretedgeometricallyastheslopeoftangentthroughofcurve,机动目录上页下页返回结束,canbeinterpretedgeometricallyastheslopeoftangentthroughofcurve,Solution,doesnotexist,and,bothexistbutfisnotcontinuous,because,Iffisafunctionofthreevariablesx,y,z,itspartialderivativeswithrespecttoxisdefindas,ExampleIf,Solution:,find,and,Higherderivatives,Ifthenthesecondpartialderivativesoff,Ifthenthepartialderivativesoforder3,ExamplefindthesecondpartialderivativesofClairautsTheoremSupposefisdefinedonadiskDthatcontainsthepoint(a,b).IfthefunctionsandarebothcontinuousonD,then,Solution,Solution,Examplefind,proof,ExampleShowthatthefunction,isasolutionoftheequation,LinearApproximationsandDifferentials,DifferentialsLety=f(x)beadifferentiablefunctionwithindependentvariablex.Thenthedifferential,dx,oftheindependentvariablexisanarbitraryincrementofx;thatis,dx=x;thedifferential,dy,ofdependentvariableyatthepointxisdy=f(x)dx.,14.4TangentPlanesandLinearApproximations,Supposefhascontinuouspartialderivatives.Anequationofthetangentplanetothesurfaceatthepointis.,Theapproximationiscalledthelinearapproximation.,DefinitionIf,thenfisdifferentiableat(a,b),ifcanbeexpressedintheformwhereandas.TheoremIfthepartialderivativesandexistnear(a,b)andarecontinuousat(a,b),thenfisdifferentiableat(a,b).,Differentials,Foradifferentiablefunctionoftwovariables,wedefinethedifferentialsandtobeindependentvariables;thatis,theycanbegivenanyvalues.Thenthedifferential,alsocalledthetotaldifferential,isdefinedby,Ifwetakethenthedifferentialofis,thelinearapproximationcanbewrittenas,Solution,and,butisnotcontinuous,bothexist,and,Wehave,But,atallpointsontheline,Sofisnotdifferentiableat(0,0).,Foradifferentiablefunctionofthreevariables,wedefinethedifferentials,andtobeindependentvariables;thatis,theycanbegivenanyvalues.Thenthetotaldifferentialdu,isdefinedby,ExampleFindthedifferentialofthefunction:,Solution(1).,14.5TheChainRule,Example,If,where,Find,Solution,Because,So,TheChainRule(Case1)Supposethatisadifferentiablefunctionofand,whereandarebothdifferentiablefunctionsof.Thenisadifferentiablefunctionofandor,proof,Achangeofinproduceschangesofinandin.these,duceachangeofin.sincefisdifferentiable,so,Dividingbothsidesofthisequationby,weget,Thus,Example,where,Find,Solution,ApplyingCase1oftheChainRule,weget,Let,So,ExampleLetz=excosy,x=sintandy=t2.FindSolutionWefindthat,TheChainRule(Case2)Supposethatisadifferentiablefunctionofand,whereandaredifferentiablefunctionsofand.Then,ExampleLetz=xy,x=3u2+v2,andy=4u+2v.FindandSolution,ApplyingCase2oftheChainRule,weget,m,Let,Then,Example,If,Find,Solution,Example,If,Find,Solution,Example,If,andFisdifferentiable,showthat,Solution,where,Example,If,find,Solution,where,Let,hascontinuoussecond,Let,then,-orderpartialderivatives,find,Example,If,find,Solution,where,ImplicitDifferentiation,Wesupposethatanequationoftheformdefinesimplicitlyasadifferentiablefunctionof,thatis,whereforallinthedomainoff.IfFisdifferentiable.Then,Wesupposethatanequationoftheformdefinesimplicitlyasadifferentiablefunctionofand,thatis,whereforallinthedomainoff.IfFisdifferentiable.Then,Example1FindifSolutionLetthenExample2FindandifSolutionLetthen,Example1FindifSolutionLetthenExample2FindandifSolutionLetthen,Example1,Findandif,Solution,ExampleFindandifSolution,14.6DirectionalDerivativesandtheGradientVector,DefinitionThedirectionalderivativeoffatinthedirectionofaunitvectoris,ifthislimitexists.,Thepartialderivativeofwithrespecttoandarespecialcasesofthedirectionalderivative.,ofatthepointinthedirectionof,istherateofchange,Theoremifisadifferentiablefunctionofand,thenhasadirectionalderivativeinthedirectionofanyunitvectorand,Proof,Ifwedefineafunctionofthesinglevariable,by,then,Ontheotherhand,wecanwrite,BytheChainRule,wehave,Itfollowsthat,Therefore,Example,Findthedirectionalderivativeofatthe,givenpointinthedirectionindicatedbytheangle,Solution,TheGradientVector,Theoremifisafunctionoftwovariablesand,thenthegradientofisthevectorfuntiondefinedby,Wehave,greidint,DefinitionThedirectionalderivativeoffatinthedirectionofaunitvectoris,ifthislimitexists.,Ifweusevectornotation,then,WhereIfn=3.,Wehave,TheoremSupposeisadifferentiablefunctionoftwoorthreevariables.Themaximumvalueofthedirec-tionalderivativeisanditoccurswhenhasthesamedirectionasthegradientvector.,proof,whereistheanglebetweenand,Themaximumvalueofthedirectionalderivativeisanditoccurswhenhasthesamedirectionasthegradientvector.,Example,(a)Iffindtherateofchange,ofatthepointinthedirectionfrom,Solution,to,Inwhatdirectiondoeshavethemaximumrateofchange？Whatisthismaximumrateofch-ange?,since,Therateofchangeof,inthedirectionfromto,is,Themaximumvalueoftherateofchangeis,（b）increasesfastestinthedirectionofthegradientvector,14.7MaximumandMinimumValues,DefinitionAfunctionoftwovariableshasalocalmaximumat(a,b)ifwhen(x,y)isnear(a,b).Thismeansthatforallpoints(x,y)insomediskwithcenter(a,b).Thenumberf(a,b)iscalledalocalmaximumvalue.Ifwhen(x,y)isnear(a,b),thenf(a,b)iscalledalocalminimumvalue.Iftheinequalitiesholdforallpoints(x,y)inthedomainoff,thenfhasanabsolutemaxi-mum(orabsoluteminimum)at(a,b).,TheoremIffhasalocalmaximumorminimumat(a,b)andthefirst-orderpartialderivativesoffexistthere,thenand.,Proof,LetIfhasalocalmaximum,at,then,hasalocalmaximumata,so,but,Similaly,Apoint(a,b)iscalledacriticalpoint(orstationarypoint)offifand,oroneofpartialDerivativesdoesnotexist.Theoremsaysthatiffhasalocalmaximumormini-mumat(a,b),then(a,b)isacriticalpointoff.However,notallcriticalpointsgiverisetomaximumorminimum.,ExampleLetf(x,y)=x2y2.Showthattheoriginistheonlycriticalpointbutthatfhasnoextremevalueattheorigin.SolutionThepartialderivativesoffexistateverypointinthedomainoff,andwehavethatfx(x,y)=2x,fy(x,y)=-2y,sothat(0,0)istheonlycriticalpointoff.However,f(0,0)=0isnoaextremevalueoff,becausef(x,0)=x20forallx0andf(0,y)=-y20.SinceA0,weknowthatf(1,1)=-1isalocalminimum.For(0,0)wehaveA=0,B=-3,C=0,andD=-90,hencefhasnoextremevalueat(0,0).,Example,AbsoluteMaximumandMinimumValues,AboundarypointofD,Aclosedsetin,Aboundedsetin,AbsoluteMaximumandMinimumValues,ExtremevaluetheoremforfunctionsoftwovariablesIffiscontinuousonaclosed,boundedsetDin,thenfattainsanabsolutemaximumvalueandanabsoluteminimumvalueatsomepointsandinD.Tofindtheabsolutemaximumandminimumvaluesofacontinuousfunctionoffonaclosed,boundedsetD:1.FindthevaluesoffatthecriticalpointsoffinD.2.FindtheextremevaluesoffontheboundaryofD.3.Thelargestofthevaluesfromsteps1and2istheabsolutemaximumvalue;thesmallestofthesevaluesistheabsoluteminimumvalue.,Example,Findtheabsolutemaximumandminimum,Valuesofthefunction,Ontherectangle,solution,Step1wefirstfindthecriticalpoints.,since,Let,Wegettheonlycriticalpoint,andthevalueof,is,Step2WelookatthevaluesofontheboundaryofD,Whichconsistsofthefourlinesegments,showinfigure1.,Onwehaveand,Thisisaincreasingfunctionof,soitsminimumvalueisanditsmaximumvalueis,Onwehaveand,Thisisadecreasingfunctionof,soitsminimumvalueisanditsmaximumvalueis,Onwehaveand,Soitsminimumvalueisanditsmaximumvalueis,Onwehaveand,Thisisaincreasingfunctionof,soitsminimumvalueisanditsmaximumvalueis,Thusontheboundary,theminimumvalueofisoandthemaximumis9.,Step3Wecomparethesevaluesinstep1andstep2,WeconcludethattheabsolutemaximumvalueofonDisandtheabsoluteminimumvalueis,Example,solution,Step1wefirstfindthecriticalpoints.,since,Evaluate,Let,Wegettheonlycriticalpoint,andthevalueof,is,Step2WelookatthevaluesofontheboundaryofD,itisthecircle,Onthecircle,wehaveand,Soitsminimumvalueisanditsmaximumvalueis,Thusontheboundary,theminimumvalueofis22andthemaximumis94.,since,when,Wehave,Step3Wecomparethesevaluesinstep1andstep2,WeconcludethattheabsolutemaximumvalueofonDisandtheabsoluteminimumvalueis,14.8LagrangeMultipliers,Inpreviousdiscussiononextremevaluesoffunctionsoftwovariablesxandy,thevariablesxandyareindependenttoeachother,i.e.,theyarenotconstrainedbyanysideconditiononthevaluesofxandy.,However,wemayencounterproblemsoffindingtheextremevaluesoffunctionf(x,y)subjecttoasidecondition,alsocalledaconstraint,oftheformg(x,y)=k.Suchextremevaluesarecalledconditionalextremevalues.,Wewilldescribeageneralmethodoffindingtheextremevaluesofafunctionsubjecttoaconstraint,whichiscalledtheLagrangemultiplier