## 微分中值定理的证明及应用+文献综述

The provation and the application of the differential mean value theorem
Abstract: The differential mean value theorem, as the important and the core  theorem of the differential,is the theoretical basis of the application of differential. It plays an important role in mathematical analysis. This paper describes the historical background of the Rolle mean value theorem、Lagrange mean value theorem and Cauchy mean value theorem firstly.Then gives a variety methods of the provation of them. Lastly, the paper introduces some examples of the application of the theorems. So that people can get deeply understanding.
Key words:  Rolle mean value theorem；Lagrange mean value theorem；Cauchy mean value theorem；The structure method of auxiliary function

1.微分中值定理的背景    3
2.微分中值定理的证明    5
2.1罗尔(Rolle)定理及其证明    5
2.2拉格朗日(Lagrange)中值定理及其证明    6
2.2.1构造辅助函数证明拉格朗日中值定理    6
2.2.2 用区间套定理证明拉格朗日中值定理    8

2.3柯西(Cauchy)中值定理及其证明    10
2.3.1构造辅助函数证明柯西中值定理    10
2.3.2利用单调性判定定理证明柯西(Cauchy)中值定理    12
2.3.3利用积分中值定理证明柯西中值定理    13
3.微分中值定理的应用    13

1.微分中值定理的背景
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.古希腊数学家在几何研究中得到如下结论:“过抛物线弓形的顶点的切线必平行于抛物线弓形的底”,这正是拉格朗日定理的特殊情况.希腊著名数学家阿基米德(Archimedes)正是巧妙地利用这一结论,求出了抛物弓形的面积. 微分中值定理的证明及应用+文献综述:http://www.lwfree.cn/shuxue/20170420/5384.html
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