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1、微积分英文版4三、其他未定式二、型未定式一、型未定式机动目录上页下页返回结束洛必达法则第三章一、存在(或为)定理1.型未定式(洛必达法则)机动目录上页下页返回结束例2.求解:原式机动目录上页下页返回结束二、型未定式存在(或为∞)定理2.证:仅就极限存在的情形加以证明.(洛必达法则)机动目录上页下页返回结束1)的情形从而机动目录上页下页返回结束2)的情形.取常数可用1)中结论机动目录上页下页返回结束3)时,结论仍然成立.(证明略)说明:定理中换为之一,条件2)作相应的修改,定理仍然成立.定理2目录上页下页返回结束例3.求解:原式机动
2、目录上页下页返回结束例3.求解:原式机动目录上页下页返回结束例3.例4.说明:例如,而用洛必达法则在满足定理条件的某些情况下洛必达法则不能解决计算问题.机动目录上页下页返回结束3.1Maxima&MinimaMaxima:pointwhosefunctionvalueisgreaterthanorequaltofunctionvalueofanyotherpointintheintervalMinima:pointwhosefunctionvalueislessthanorequaltofunctionvalueofanyothe
3、rpointintheintervalExtrema:EitheramaximaoraminimaWheredoextremaoccur?Peaksorvalleys(eitheronasmoothcurve,oratacusporcorner)f’(c)=0orf’(c)isundefinedDiscontinutiesEndpointsofanintervalTheseareknownasthecriticalpointsofthefunctionOnceyouknowyouhaveacriticalpoint,youcant
4、estapointoneithersidetodetermineifit’samaxormin(ormaybeneither…justalevelingoffpoint)3.2MonotonicityandConcavityLetfbedefinedonanintervalI(open,closed,orneither).ThenfisINCREASINGonIif,DECREASINGonIif,MONTONIConIifitisetherincreasingordecreasingMonotonicityTheoremLetf
5、becontinuousonanintervalIanddifferentiableateveryinteriorpointofI.Iff’(x)>0forallxinteriortoI,thenfisincreasingonIIff’(x)<0forallxinteriortoI,thenfisdecreasingonI.ConcaveUPvs.ConcaveDOWNLetfbedifferentiableonanopenintervalI.Iff’isincreasingonI,fisconcaveup(thegraphapp
6、earstobecurvedup,asacontainerthatwouldholdwater)Iff’isdecreasingonI,fisconcavedown(thegraphappearstobecurveddown,asifacontainerweredumpingwaterout)PointofInflectionWhereconcavitychanges:goesfromconcaveuptoconcavedown(orviceversa)f’isneitherincreasingordecreasing,thech
7、angeinf’=0,thusf’’=0Findinflectionpoints&determineconcavityforf(x)Inflectionpts:x=-2,0,1Concaveup:(-2,0),(1,infinity)Concavedn:(-inf.,-2),(0,1)3.3LocalExtremaandExtremaonOpenIntervalsFirstDerivativeTestLetfbecontinuousonanopeninterval(a,b)thatcontainsacriticalpointc.I
8、ff’(x)>0forallxin(1,c)andf’(x)<0forallxin(c,b),thenf(c)isalocalmax.value.Iff’(x)<0forallxin(1,c)andf’(x)>0forallxin(c,b),the
