alevel資源-純數(shù)學課件

Chapter7

Applicationsofdifferentiationxyf(x)=x2–2DerivativesasfunctionsThegradientisalsoafunctionofx;itissometimescalledthederivedfunction.xyf(x)=x2–2xyf’(x)=2xIncreasinganddecreasingfunctions

Forsimplicity,theword‘function’inthischapterwillmeanfunctionswhicharecontinuouswithintheirdomains.Youcanusetheideathatthederivativeofafunctiontoinvestigatetheshapeofagraphfromitsequation.ExampleFindtheintervalinwhichf(x)=x2–6x+4isincreasing,andtheintervalinwhichitisdecreasing.f(x)x3x<3x>3x=3f’(x)ispositiveinanintervalp<x<qyxf(x2)f(x1)x1x2pqIfx1andx2aretwovaluesofxintheintervalp≤x≤q,andifx2>x1,thenf(x2)>f(x1).Afunctionwiththispropertyissaidtobeincreasingovertheintervalp≤x≤q

yxf(x1)f(x2)x1x2pqf’(x)isnegativeinanintervalp<x<q,thefunctionhastheoppositeproperty;ifx2>x1,thenf(x2)<f(x1).Afunctionwiththispropertyissaidtobedecreasingovertheintervalp≤x≤qExample

Findthederivedfunctionoff(x)=x2–1/3x3f(x)f’(x)x<0f(x)f’(x)x=0f(x)f’(x)0<x<2f(x)f’(x)x>2f(x)f’(x)x=2f(x)f’(x)Iff’(x)>0inanintervalp<x<q,thenf(x)isincreasingovertheintervalp≤x≤q

Iff’(x)<0inanintervalp<x<q,thenf(x)isdecreasingovertheintervalp≤x≤q

Noticethat,forf(x)tobeincreasingforp≤x≤q,thegradientf’(x)doesnothavetobepositiveattheendsoftheinterval,wherex=porx=q.Atthesepointsitmaybe0,orevenundefined.Thismayseemaminordistinction,butithasimportantconsequences.Itisapay-offfromthedecisiontoworkonlywithcontinuousfunctions.Example

Forthefunctionf(x)=x4–4x3,findtheintervalsinwhichf(x)isincreasingandthoseinwhichitisdecreasing.xy3Iff’(x)>0inanintervalp<x<qexceptatisolatedpointswheref’(x)=0,thenf(x)isincreasingintheintervalintheintervalp≤x≤q.Iff’(x)<0inanintervalp<x<qexceptatisolatedpointswheref’(x)=0,thenf(x)isdecreasingintheintervalintheintervalp≤x≤q.Examp