罗尔中值定理

罗尔中值定理新证

New Proof of the Method of Rolle Intermediate Value Theorem

文章给出罗尔中值定理的一个推论及给出辅助函数新的构造方法，来证明拉格朗日中值定理和柯西中值定理。

This paper provides an inference of Rolle mean value theorem and a new structure method of auxiliary function so as to prove Lagrange mean value theorem and Cauchy mean value theorem .

其证明方法关键在于构造一个辅助函数，再应用罗尔中值定理推出拉格朗日中值定理的结论。

Its key proof is to construct an auxiliary function , which is used by Roll 's theorem to reach a conclusion of Lagrange 's theorem .

利用罗尔中值定理给出了函数与其拉格朗日插值函数间的关系，得到中值定理的另外一种形式，并给出了它的应用。

Another form about the median theory and its application is put forward according to the relation between the function and its Lagrange interpolation function in accordance with Rolls Median Theory .

根据罗尔中值定理的理论，用分析的基本方法，推广了定理的三个条件，得到相应的结论，并通过例子说明推广后的罗尔中值定理具有更广泛的应用。

On the basis of Rolle 's Mid-value Theorem , this paper uses the fundamental methods of calculus to generalize the three conditions of the theorem of Rolle , and we can obtain the corresponding conclusions , which have more extensive application shown by some examples .

由此可见罗尔微分中值定理可以是实数的完备性的直接推论。

This implies that Rolles Theorem is the direct consequence of completeness of real numbers .

给出罗尔定理与微分中值定理在较弱条件下的结论，使之适用范围更广泛一点。

The paper gains some conclusion under the weak condition about Rolle theorem and differential mean theorem and makes it be used more widely .

本文将微积分中的罗尔定理从有限闭区间推广到了半无限区间和无限区间，有助于对罗尔中值定理的理解。

The author generalizes the theorem of Rolle in the calculus from closed interval to semi-infinite interval and infinite interval , which is useful for students to understand theorem of Rolle better .