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1、APCALCULUSDerivativeADERIVATIVEFUNCTION1.Thederivativefunctionorsimplythederivativeisdefinedas==2.Findthederivativefunctiona)Find,b)Findtheaveragerateofchange,c)Findthelimit.3.GeometricsignificanceConsiderageneralfunctiony=f(x),afixedpointA(a,f(a))an
2、davariablepointB(x,f(x)).TheslopeofchordAB=.NowasBA,xaandtheslopeofchordABslopeoftangentatA.So,is.Thus,wecanknowthederivativeatx=aistheslopeofthetangentatx=a.4.RulesC(aconstant)07APCALCULUSDerivative3.ThechainruleIfwherethen.,6.Inversefunction,Parame
3、tricfunctionandImplicitfunctionInversefunction:,,i.e.,,7APCALCULUSDerivativeParametricfunction:,i.e.,,→,Implicitfunction:,.,,t7.Highderivativey=sinx,BAPPLICATIONSOFDIFFERENTIALCALCULUS1.Monotonicitya)IfSisanintervalofrealnumbersandf(x)isdefinedforall
4、xinS,then:7APCALCULUSDerivativef(x)isincreasingonSforallxinS,andf(x)isdecreasingonSforallxinS.a)FindthemonotoneintervallFinddomainofthefunction,lFind,andxwhichmake,lDrawsigndiagram,findthemonotoneinterval.1.Maxima/Minima,Horizontalinflection,Stationa
5、rypointCINTEGRAL1.TheideaofdefiniteintegralWedefinetheuniquenumberbetweenallloweranduppersumsasandcallit“thedefiniteintegraloffromatob”,7APCALCULUSDerivativei.e.,where.Wenotethatas,andWewrite.Ifforallxon[a,b]thenistheshadedarea.1.Propertiesofdefinite
6、integrals,cisanyconstant7APCALCULUSDerivative,where(f(x)odd),(f(x)even)IfonthenIfonthenTheaveragevalueofafunctiononaninterval[a,b]3.TheinfiniteintegralIf,thenFormulas:,,,,(),USubstitutionsubstitutionu=g(x)IntegrationbyParts7APCALCULUSDerivative7