2014_AP微积分CALCULUS知识点总结

APCALCULUSDerivative1ADERIVATIVEFUNCTION1.Thederivativefunctionorsimplythederivativeisdefinedas)(xf＝y＝xxfxxfxyxx)()(limlim002.Findthederivativefunctiona)Findy,b)Findtheaveragerateofchangexy,c)Findthelimitxyx0lim.3.GeometricsignificanceConsiderageneralfunctiony=f(x),afixedpointA(a,f(a))andavariablepointB(x,f(x)).TheslopeofchordAB=axafxf)()(.NowasBA,xaandtheslopeofchordABslopeoftangentatA.So,axafxfax)()(limis)(af.Thus,wecanknowthederivativeatx=aistheslopeofthetangentatx=a.4.Rules)(xf)(xfC(aconstant)0nx1nnxxsinxcosxcosxsinxtanxx22cos1secxarcsin2-11xAPCALCULUSDerivative2xlnx1xalogexalog1xexexaaaxln)()(xvxu)()(xvxu)()(xvxu)()()()(xvxuxvxu)()(xvxu2vvuvu)0(v5.ThechainruleIf)(ufywhere)(xuuthendxdududydxdy.)()(xgexf)()()(xgexfxg)(ln)(xgxf)()()(xgxgxf)(ln)()(ln)()()()(xuxvxuxveexuxfxv,])()()()(ln)([)()(ln)(xuxuxvxuxvexfxuxv6.Inversefunction,ParametricfunctionandImplicitfunctionInversefunction:dydxdxdy1,])([1)(1xfxf,i.e.,xyarcsin,yxsinAPCALCULUSDerivative3Parametricfunction:dtdxdtdydxdy,i.e.,)(ty,)(tx→)(1xt,)]([1xy)()(ttdtdxdtdydxdtdtdydxdyImplicitfunction:0))(,(xyxF,0))(,(xfxF.0-222ayx,taytaxsincos,t]2,0[ttatadxdyxycotsincos)(7.Highderivativexxfxxfdxydxfx)()(lim)(022tatatdtdxdtyddxydxyxyx32sin1sincsc])([)(xxfxxfxfnnxn)()(lim)()1()1(0)(y=sinx)2sin(cosxxy,)22sin()2cos(xxy)2sin()(nxynBAPPLICATIONSOFDIFFERENTIALCALCULUS1.Monotonicitya)IfSisanintervalofrealnumbersandf(x)isdefinedforallxinS,then：APCALCULUSDerivative4f(x)isincreasingonS0)(xfforallxinS,andf(x)isdecreasingonS0)(xfforallxinS.b)FindthemonotoneintervalFinddomainofthefunction,Find)(xf,andxwhichmake0)(xf,Drawsigndiagram,findthemonotoneinterval.2.Maxima/Minima,Horizontalinflection,StationarypointCINTEGRAL1.TheideaofdefiniteintegralWedefinetheuniquenumberbetweenallloweranduppersumsasbadxxf)(andcallit“thedefiniteintegralof)(xffromatob”,APCALCULUSDerivative5i.e.,niinibaixxfdxxfxxf110)()()(wherenabx.Wenotethatasn,10)()(nibaidxxfxxfandbaniidxxfxxf)()(1Wewritebaniindxxfxxf)()(lim1.If0)(xfforallxon[a,b]thenbadxxf)(istheshadedarea.2.Propertiesofdefiniteintegralsbabadxxfdxxf)()]([babadxxfcdxxcf)()(,cisanyconstantcabacbdxxfdxxfdxxf)()()(bababadxxgdxxfdxxgxf)()()]()([APCALCULUSDerivative6)()()()(aFbFxFdxxfbaba,wheredxxfxF)()(aadxxf0)(（f(x)odd）,aaadxxfdxxf0)(2)((f(x)even)If0)(xfonbxathenbadxxf0)(If)()(xgxfonbxathenbabadxxgdxxf)()(Theaveragevalueofafunctiononaninterval[a,b]baavedxxfabf)(13.TheinfiniteintegralIf)()(xfxF,thenCxFdxxf)()(Formulas:Cxndxxnn111,Caadxaxxln1Cxinxdxcoss,Cxxdxsincos,Cxxdxcoslntan,CxxdxsinlncotCxxdxarcsin12(12x),Cxxdxarctan12USubstitutiondxxgxgf)())((substitutionu=g(x)duuf)(IntegrationbyPartsvduuvudvAPCALCULUSDerivative7