微分学中值定理的证明及其应用

微分学中值定理是微分学中的重要的基本定理,它一般包括三个定理:罗尔(Rolle)定理,拉格朗日(Lagrange)中值定理与柯西(Cauchy)中值定理.在证明后两个定理时,通常的教科书是采用构造一个辅助函数,使它满足罗尔定理的条件,利用罗尔定理的结论来证明的.在本文中,将对微分学中值定理给出新的证法,然后归纳介绍微分学中值定理的几种推广形式及一些常见的应用.

The Prove and Application of Theorem of Differential Mean Value

The theorem of differential mean value is one of the most important and basically theorem in differential,including three theorems: Rolle theorem,Lagrange mean value theorem and Cauchy mean value theorem.These latter two theorems are demonstrated by employing the conclusion of Rolle theorem,with auxiliary function meeting the conditions of Rolle theorem in general textbooks.This article will provide new proving methods to theorem of differential mean value,and summarize several kinds of popularize forms and its applications.