柯西中值定理的注记

Notes on Cauchy's Theorem

柯西中值定理是数学中非常重要的定理之一,它被广泛的应用在相关数学问题的证明当中。柯西中值定理认为,两个不同的函数在相关条件满足的情况下,存在一个点ξ,使得这两个函数在该点处的导数之比等于其在区间端点函数值的差之比。但是柯西中值定理并没有明确给出计算点ξ的方法以及相关极限和导数的求法。本文将柯西中值定理中的ξ看作是定义区间端点的函数,通过一系列的推导过程,给出了ξ的函数表达式,并求出了ξ在区间端点处的一、二阶导数值以及θ在区间端点处的极限和导数,为解柯西中值定理中ξ值的相关问题提供了新的思路和角度.

The ξ of Cauchy＇s Theorem, which is one of the most important theorems in the field of mathematics, is widely applied in the process of proving relevant mathematic problems. It holds that when proper and necessary conditions permit, there is a point ξ, which makes the proportion of the derivatives of the two functions equal to that of the dispatches of the two boundary values at that point of the span. However, The ξ of Cauchy＇s Theorem doesnt give any definite methods to calculate ξ or the relevant limit or the derivative. The article regards ξ as the function at the boundary point of the definition interval, gives the formula of ξ by means of a series of reasoning, and provides the first and second derivative of ξ and the limit and the derivative of θ at the boundary point of the definition interval, which offers new angles and methods to solve problems relevant to the ξ of Cauchy＇s Theorem.