Math 281 Intermediate Calculus数学281中级微积分

Math281

IntermediateCalculusInthiscourse,wewillstudy

CalculusoffunctionsofseveralvariablesandAnalyticgeometryof3dimensionalobjects.WhatcanCalculusdo?InCalculusI,wesawthatdifferentiationcanbeusedtofindtheslopeofacurveatany(smooth)point.Wecanalsousedifferentiationtofindthelocalmaxandlocalminimumofafunction.Inthiscourse,wewilllearnhowtofindtheslopeofasurfaceatanysmoothpoint,andinanydirection.Wewillalsolearnhowtofindthelocalmaximumandlocalminimumofasurface.AbsolutemaxlocalmaxabsoluteminlocalminInCalculusI,wesawthatIntegrationcanbeusedtofindtheareabetweenacurveandthex-axis(from

x=atox=b)Inthiscourse,wewilllearnhowtofindthevolumeoftheregionboundedbyseveralsurfaces.Andthevolumesofmanyothersolids,suchasabumpysphere!z10.50-0.5-1y10.50-0.5-1x10.50-0.5-1Ofcourse,youwillneedtolearnmanyotherusefulmathematicaltechniquesinthisprogram.Thereforeyouneedtoworkhardandlearnfast.Plantostudyatleast4hourseachday.ReviewRecallthatforfunctionsofonevariablesuchasf

(x)=2exsinx,

wecandrawitsgraphonapieceofpaperbecauseweneedonlytwoperpendicularaxes,onefortheinput(i.e.thexvalue)andonefortheoutput(i.e.theyvalue)xyHowever,forfunctionsoftwovariables,suchasf(x,y)=2x2y+xy2thegraphwillnolongerbeacurveandweneed3mutuallyperpendicularaxestorepresentthetwoinputsandoneoutput.Theoutputvalueisusuallydenotedbytheletterz.i.e.z=2x2y+xy2Inordertoaccommodatethezaxisthatisperpendiculartoboththexandyaxes,weneedtomakeitcomingoutfromthescreen(seethenextslide).Sincethezaxisiscomingoutfromthexy-plane,wecannotliterallydrawthisonapieceofpaper.Youneedtouseabitofimaginationtoseethatthezaxisisreallycomingout.Wealsoneedtorotatetheaxessothatthezaxisispointingup,becauseweareusedtoseeingtheoutputontheverticalaxis.xzyThiswillthenbethestandardorientationofthe3axes,andmostofourdiagramswillbedrawnthisway.Thecoordinatesofanypointinspacewillbeanorderedtriple(x,y,z).Thelastcomponentzwilltellushowfarthepointisaboveorbelowthehorizontalplane.

xzyInadditiontothe3coordinateaxes,wealsohave3coordinateplanesnamely(1)thexy-planexzyInadditiontothe3coordinateaxes,wealsohave3coordinateplanesnamely(1)thexy-plane(2)thexz-plane(3)theyz-planexzy(1)thexy-plane(2)thexz-planexInadditiontothe3coordinateaxes,wealsohave3coordinateplanesnamelyxzyThese3coordinateplaneswilldividethespaceinto