微分中值定理的另类证明与推广

[]2006206226243Vol.24,.320086COLLEGEMATHEMATICSJun.2008(,311300)[],(Rolle).,.,.[];;;[]O171[]C[]167221454(2008)0320169203.,.,(Rolle),[1,2].,.,(Lagrange).,..1,:f(x)[a,b],f(a)=f(b),[,][a,b],-=12(b-a),f()=f().(x)=fx+12(b-a)-f(x),xa,a+b2.,(x)a,a+b2,(a)=fa+12(b-a)-f(a)=fb+a2-f(a),a+b2=fa+b2+12(b-a)-fb+a2=f(b)-fb+a2=f(a)-fb+a2=-(a).,a,a+b2,©1994-2009ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.()=f+12(b-a)-f()=0.,=,=+12(b-a),f()=f(),-a=12(b-a).,f(b)=fa+b2,==a=a+b2.1(Lagrange)f(x)[a,b],(a,b),(a,b),f(b)-f(a)=f()(b-a).(1)F(x)=f(x)-x,x[a,b],=f(b)-f(a)b-a.F(x)[a,b],F(a)=bf(a)-af(b)b-a=F(b).,[1,1][a,b],F(1)=F(1),1-1=12(b-a),f(1)-1=f(1)-1.=f(1)-f(1)1-1.F(x),,[2,2][1,1],F(2)=F(2),2-2=12(1-1)=122(b-a).=f(2)-f(2)2-2.,{[n,n],n1},[n,n]=[n+1,n+1],n1,n-n=12n(b-a)0,n;=f(n)-f(n)n-n,n1.,[n,n][a,b],n1,limnn=limnn=.F(n)-F(n)F()-F(n)F(n)-F(),f(n)-f(n)n-nf()-f(n)-nf(n)-f()n-.,limnf()-f(n)-n=f(),limnf(n)-f()n-=f().limnf(n)-f(n)n-n=f().,=f(b)-f(a)b-a,f(n)-f(n)n-n,n1.07124©1994-2009ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.=f(b)-f(a)b-a=f(),(1).2f(x)[a,b],(a,b),,[3].1f(x)[a,b],(a,b),(a,b),|f(b)-f(a)||f()|(b-a).(2)f(x)c(a,b),[a,c][c,b]1,f(c)-f(a)=f(1)(c-a),1(a,c),f(b)-f(c)=f(2)(b-c),2(c,b).|f()|=max{|f(1)|,|f(2)|},.2f(x)[a,b],(a,b)n,n+1{k,1kn+1}(a,b)n+1{k,1kn+1},n+1k=1k=1,f(b)-f(a)=n+1k=1kf(k)(b-a).f(x)c(a,b),[a,c][c,b]1,f(c)-f(a)=f(1)(c-a),1(c,b)f(a)-f(c)=f(2)(a-c),2(a,c).1,2,1(b-a)=c-a,2(b-a)=b-c,1+2=1,10,20,f(b)-f(a)=[1f(1)+2f(2)](b-a).nn+1,.[][1],.()()[M].:,2003.[2],.()()[M].:,2003.[3],.()[M].:,1995.TheDifferentProofandSpreadoftheMeanValueTheoremWANGJia2jun(SchoolofSciencesZhejiangForestryUniversity,LinanZhejiang311300,China)Abstract:Inthegeneraltextbooks,theproofofthedifferentialmeanvaluetheoremisbuiltontheRolleTheorem.ThispaperisbasedontheTheoremofwhichisveryimportantinthecontinuityofthesystemofrealnumbers.AlsoitgivesadifferentwayofprovingtheLagrangeMeanValueTheorem.BesidesitdrawsseveralgeneralizingformsoftheMeanValueTheorem.Keywords:themeanvaluetheorem;theoremofnestedintervals;continuity;differentiable1713:©1994-2009ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.