Rolle定理的推广及应用

Rolle Theorem and Applications

微分中值定理是数学分析中很重要的基本定理,在数学分析中有着广泛的应用。它是沟通函数及其导数之间的桥梁,是应用导数研究函数在某点的局部性质和在某个区间上的整体性质的重要工具。利用微分中值定理可以论证方程的根的存在问题、方程根的个数问题以及根的存在区间问题,也经常用于证明一些含有导数的等式。在形式结构上,Rolle定理是中值定理的基础,一方面它包含在其它中值定理之中,另一方面其它中值定理的证明又往往通过Rolle定理来实现,但该定理要求自变量的范围是闭区间,这就使某些问题的解决受到了限制。主要将Rolle定理推广到有限开区间和无穷区间,用两种方法进行证明,并且举例说明其应用。

Mean Value Theorem,a very important in mathematical analysis,fundamental theorem in mathematical analysis has a wide range of applications.It functions a bridge between the derivative of the function,which is applied at a point in a range of local properties and the overall nature of the important tools.Using the differential equation in the mean value theorem can be proved the existence of the root of the problem,the number of all roots of the problem and the root problem of the existence interval,a number often used to prove that the equation with derivative.In the form of structure,Rolle theorem is the basis of the mean value theorem,on the one hand it contains among the mean value theorem in the other,on the other hand other proof of the mean value theorem is often achieved through the Rolle theorem,but the theorem requiring the independent variable range is a closed interval,which makes some of the problems limited.In this paper,the Rolle theorem and the open interval are infinite interval,proved by two methods,and examples of its application.