15Second derivatives A-level考试纯数学P1课件.pptVIP

SecondderivativesChapter15

ContentsInterpretingandsketchinggraphsSecondderivativesinpracticeMinimaandmaximarevistiedLogicaldistinctionsExtendingnotation

InterpretingandsketchinggraphsThederivativeofafunctioncanberegardedasafunctionf’(x)

InterpretingandsketchinggraphsInthegraphofy=f(x)=x3–3x2,identifytheintervalsinwhichf(x),f’(x)andf’’(x)arepositive,andinterpretthesegraphically.f(x)x(1,-2)(2,-4)f’(x)(1,-3)xf’’(x)x

BendingupwardsBendingdownwardsf’’(x)0f’’(x)0

ExampleInvestigatethegraphofy=f(x),wherewithdomainx0.

Apointofagraphwhichseparatesapartofthecurvewhichbendsonewayfromapartwhichbendstheotherwayiscalledapointofinflexionofthegraph.If(p,f(p))isapointofinflexionofthegraphofasmoothfunction,f’’(p)=0.

SecondderivativesinpracticeTherearemanyreal-worldsituationsinwhichsecondderivativesareimportant,becausetheygiveadvancewarningoffuturetrends.

MinimaandmaximarevisitedIff’(q)=0andf’’(q)0,thenf(x)hasaminimumatx=q.Iff’(q)=0andf’’(q)0,thenf(x)hasamaximumatx=q.

Theproceduredescribedinchapter7canthenbeamendedasfollowsTofindtheminimumandmaximumpointsonthegraphofy=f(x):Step1Decidethedomaininwhichyouareinterested.Step2Findanexpressionforf’(x)Step3Listthevaluesofxinthedomainforwhichf’(x)is0.(Iftherearevalueswheref’(x)isundefined,usetheoldprocedure)Step4Findanexpressionforf’’(x)Step5ForeachvalueofxinStep3,findthesignoff’’(x).Ifthesignis+,thegraphhasaminimumpoint;if-,amaximum.(Ifthevalueoff’’(x)is0,followtheoldprocedure.)Step6Foreachvalueofxwhichgivesaminimumoramaximum,calculatef(x).

ExampleFindtheminimumandmaximumpointsonthegraphoff(x)=x4+x5.

FindtheminimumandmaximumpointsonthegraphofWecanfindthattheminimumvalueisgr