微分中值定理的理论及应用.doc

摘 要 微分中值定理是微分学的基本定理，也是微分学的核心定理。中值定理论是导数应用的基础理论。它把函数的变化与导数联系起来，从而把有型的无形的结合在一起，通过局部状态的导数的反应，研究函数在区间上的整体状态。在微分学里，由费马定理奠定理论基础，中值定理包括罗尔定理，拉格朗日中值定理与柯西中值定理。本文主要介绍在微分中值定理证明过程中，辅助函数的构造方法；利用微分中值定理证明等式，证明不等式，求极限，判断函数的单调性，证明方程根的存在性；同时给出两个中值定理的推广和逆向证明的方法。着重介绍了利用微分中值定理解决的实际问题，使其应用更加广泛，进而体现了微分中值定理在《数学分析》中的重要性. 关键词：中值定理，辅助函数，应用，推广 The theory and application of differential mean value theorem Abstract: The differential mean value theorem is the fundamental theorem of the calculus, the differential of the core theorem is also. The theoretical mean value theorem is the basic theory of derivative application. The function change and derivative link, the tangible and intangible unifies in together, through the derivative reaction of the local state, the research function in the interval of the whole state. In differential calculus, and lay the theoretical foundation from Fermats theorem, theorem of Roll theorem, Lagrange mean value theorem and Cauchy mean value theorem. This paper mainly introduced in the proof of the differential mean value theorem in the process, methods of constructing auxiliary function; the use of differential mean value theorem proving the equality, proving the inequality, limit, monotonic function, proof of Fang existence; at the same time, gives three median theorem and converse proof, The paper mainly introduces the practical problems in using differential mean value theorem to solve, so it is used widely, and shows the importance of the differential mean value theorem in mathematical analysis in. Keywords: Mean value theorem, auxiliary function, application, promotion 目录 一、引言 1 二、微分中值定理的理论 1 （一）微分中值定理内在联系 1 （二）微分中值定理在证明中辅助函数的构造方法 2 1.“K”值法 2 2.推理法 3 3.积分法 4 三、微分中值定理的应用 5 （一）利用微分中值定理证明等式 5 （二）利用微分中值定理证明不等式 7 （三）利用微分中值定理求极限 8 （四）讨论方程实根(或函数零点)的存在性 10 （五）利用微分中值定理判断函数的单调性 13 四、微分中值定理的推广 15 （一）两个微分中值定理的推广 15 1.罗尔中值定理的推广 15 2.拉格朗日中值定理的推广 16 （二）微分中值定理的逆向证明法 17 五、结束语 18 六、参考文献 19 一、引言 人们对微分中值定理的认识可以追溯到公元前古希腊时代。古希腊