高等数学微积分课件拉格朗日乘法

LagrangeMultipliersByDr.JuliaArnoldProfessorofMathematicsTidewaterCommunityCollege,NorfolkCampus,Norfolk,VAWithAssistancefromaCPDPGrantInthislessonyouwillUnderstandthemethodofLagrangeMultipliersUseLagrangeMultiplierstosolveconstrainedoptimizationproblemsUsethemethodofLagrangemultiplierswithtwoconstraintsObjective1UnderstandthemethodofLagrangeMultipliersFig.13.77RecallFig.13.48Objective2UseLagrangeMultiplierstosolveconstrainedoptimizationproblemsBasically,LagrangeMultipliersareanotherwaytothinkaboutsolvingoptimizationproblemswithconstraints.Letslookataproblemwecansolveeitherway.Problem:SupposeIhave50feetoffencingforarectangularshapedgardenandwanttoenclosethemaximumarea.Theusualwaywewouldsolvethisproblemistolookatapictureandputinvariables.xyNextwewritetheconstraintequation2x+2y=50LastwedeterminewewanttomaximizetheAreaoftheGardenwhoseformulaisA=xyProblem:SupposeIhave50feetoffencingforarectangularshapedgardenandwanttoenclosethemaximumarea.xy2x+2y=50ConstraintA=xyObjectiveWewanttogettheAreaintermsofjustxorysoweusetheconstrainttoeliminateoneortheother.2y=50-2xY=25-xA=x(25-x)=25x–x2A’=25–2x0=25–2xX=12.5andy=12.5NowwewillusethemethodofLagrangeMultiplierstoSolvethesameproblem.2x+2y=50ConstraintA=xyObjectivef(x,y)=xyistheobjectiveequationandg(x,y)=2x+2yistheconstraintequation.FindNowlet’slookataproblemthatmightbedifficulttosolvetheoldway.Problem2:FindthemaximumandminimumvaluesforTheconstraintistheredgraph.Thebluegraphandgreengrapharelevelcurvesoftheparaboloid.Thebluegraphisat1andthegreengraphisatwhichistheexactvalueforthemax.Astheparaboloidextendsinthezdirectionitgetswiderthantheconstraint.Z=1Z=Objective3UsethemethodofLagrangemultiplierswithtwoconstraintsSupposewewanttofindthemaxandminvaluesoff(x,y,z)subjecttotwoconstraintsoftheformg(x,y,z)=pandh(x,y,z)=q.Geometrically,thismeansthatwearelookingfortheextremevaluesoffwhen(x,y,z)isrestrictedtolieonthecurveofintersectionofthelevelsurfacesgandh.Itcanbeshownthatifanextremevalueoccursat,thenthegradientvectorisintheplanedeterminedbyand.Weassumethesegradientvectorsarenot0orparallelandthustherearenumbers(Lagrangemultipliers)suchthat

Problem:Maximizef(x,y,z)=xyzsubjecttothetwoconstraintsSolvedeq2formuSolvedeq3forlambdaSubstitutedintoeq1formuandlambda.ContinuedIfoundthisveryinterestingwebsitewrittenbyastudentwhoissharinghistalentswitheveryone.ThetopicisLagrangeMultipliers.Checkitout./watch?v=ry9cgNx1QV8Hismethodisslight