finding the stationary point找到固定点

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1、LearnerDevelopmentUnitMoreDifferentiation:StationaryPointsYouneedtobeabletofindastationarypointonacurveanddecidewhetheritisaturningpoint(maximumorminimum)orapointofinflexion.MaximumturningpointPointofinflexionMinimumturningpointFindingtheStationaryPoint:Lookingatthe3diagramsaboveyoushouldbe

2、abletoseethatateachofthepointsshownthegradientis0(i.e.thecurvegoesflat).Inlesson6wesaidthatdifferentiationgivesusthegradientfunction.Sotofindanystationarypointsweneedtodifferentiatefirstthenputtheresultinggradientfunctionequaltozero.Whendydx=0(i.e.thegradientiszero)youhaveastationarypoint.I

3、nshort:findthe1stdifferential,putitequaltozeroandsolve.Examples:·Findthestationarypointonthecurvey=3x2-18x-7:Step1:Differentiatetogive:dydx=6x-18Step2:Putthisequaltozero:6x-18=0Step3:Solve:6x=18x=3Wenowknowthatthestationarypointiswhenx=3.Ifwerequirethefullco-ordinateweneedtofindthevalueofyw

4、henx=3.(Wearelookingforysoweneedtousetheoriginalequation,y=3x2-18x-7)Originalequation:y=3x2-18x-7Substitutex=3:y=3(3)2-183-7Solve:y=-34Sothestationarypointisat(3,-34)©HJackson11/LDU4·Findthestationarypointonthecurvey=13x3+2x2-12x+5:Step1:Differentiatetogive:dydx=x2+4x-12Step2:Putthisequalto

5、zero:x2+4x-12=0Step3:Solve:x+6x-2=0x=-6orx=2Wenowknowthatthereare2stationarypoints:oneatx=-6andoneatx=2.Nextfindthecorrespondingyvalues,usingtheequation,y=13x3+2x2-12x+5Whenx=-6:y=13(-6)3+2(-6)2-12-6+5y=77Whenx=2:y=13(2)3+2(2)2-122+5y=-253Sothe2stationarypointsareat(-6,77)and2,-253Intheabov

6、eexampleswehavefoundoutwherethestationarypointsarebutwedon’tknowwhattypetheyare(i.e.maximum,minimumorpointofinflexion).Wenowneedtogoastepfurther(anddifferentiateagain)tofindthetype.TypeofStationarypoint:Tofindthetypeofstationarypointweneedtodifferentiateagain(i.e.findthe2nddifferential,deno

7、tedbyd2ydx2).Thismeasuresthechangeingradientandcanhelptodecidewhetherastationarypointisamaximum,minimumorpointofinflexion.Afterfindingwherethestationarypointis,findthe2nddifferentialandif:d2ydx2>0thenyouhaveaminimum(positiveisminimum).d2ydx2<0thenyouhave