推广的积分中值定理及其应用.doc

推广的积分中值定理及其应用 摘要:定积分是微积分的重要组成部分,而积分中值定理是定积分的重要性质之一,所以积分中值定理在微积分中占了很重要的地位,本文系统的叙述了推广的积分中值定理包括:必可以在开区间中取得,导函数的积分中值定理等多个方面,我们所学知识中积分中值定理与微分中值定理的中间点的存在区间是不统一的,但推广后的积分中值定理能够与微分中值定理的存在区间从形式上统一起来,使与其相关的理论得以联系和应用.同时,在本篇论文中以实例的形式列举了推广的积分中值定理在确定零点分布、证明积分不等式、求极限等方面的应用,显然,推广的积分中值定理的优点就在于此,它可以解决原积分中值定理无法解决的问题,这表明了积分中值定理在推广后更具有应用性. 关键词:积分中值定理;导函数;微分中值定理 Promotion of Integral Mean Value Theorem and Its Application Abstract: Definite integral is an important component of calculus, the mean value theorem is one of the important properties of the definite integral, so integral mean value theorem in calculus plays a very important position .This paper describes the system to promote the integral mean value theorem, including: will be achieved in the open interval ,of the derivatives and other integral mean value theorem, we have the knowledge of the differential mean value theorem and the Intermediate Value Theorem Existence interval is not uniform, but after the promotion of integral mean value theorem and the Mean Value Theorem to the presence of range from the formal unity, so that contact can be associated with the theory and application. Meanwhile, in this paper an example to cite a form of integral mean value theorem in determining the zeros to prove inequality, such as the application of limit, obviously, to promote the advantages of integral mean value theorem in this, it Can solve the original integral mean value theorem can not solve the problem, suggesting that the integral mean value theorem in the promotion of a more applied after. Keywords: Integral mean value theorem, derivative, mean value theorem 1预备知识 在本部分中具体叙述了这篇论文中所需要的相关知识,包括导函数介值性定理、拉格朗日中值定理以及变上限积分函数的定义和性质等,这些理论知识为第二部分的定理推导以及证明做了铺垫,所以起了重要的作用. 1.1设在上非负可积,且则存在使得 1.2 设在上连续,,,若,,则存在,使得 1.3若函数在上可导,且,k为介于,之间的任意数,则在内至少存在一点,使得 1.4若为上的非负导函数,且存在,使,则必有 1.5（拉格朗日中值定理）若函数满足如下条件: （1）在闭区间上连续; （2）在开区间内可导; 则在内至少存在一点使得 1.6变上限积分函数:设在上可积,为内任意一点,则称函数为变上限积分函数 1.7变上限积分函数有以下若干性质 （1）有界性 命题1 设函数在上可积,则在上有界 （2）连续性 命题2 设函数在上可