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| 罗尔中值定理在构造辅助函数中的方法及应用
摘要
罗尔中值定理是微分学基本定理中的一个重要的定理，同时在解决构造辅助函数一类问题中起着重要作用。因此，本文首先对罗尔中值定理的内容、几何意义及证明过程等基础知识进行了回顾；然后介绍了罗尔中值定理在解决构造辅助函数一类问题中的方法：凑微分法、分组构造法、积分因子法、积分上限函数构造法；最后通过具体的例子用上述的方法介绍罗尔中值定理在此类问题中的应用。本文同时对于学生后续学好拉格朗日中值定理和柯西中值定理起到打好基础的作用，并且对学生应用罗尔定理解决好数学问题提供很大的帮助。
关键词：罗尔中值定理；构造辅助函数；凑微分法；分组法；积分因子法；积分上限函数构造法
Abstract
Rolle's mean value theorem is an important theorem in the basic theorem of differential calculus, and plays an important role in solving the problem of constructing auxiliary function. Therefore, this paper first reviews the basic knowledge of Rolle's mean value theorem,such as the content, geometric meaning and proof process;then introduces the methods of Rolle's mean value theoremin solving the problems of constructing auxiliary functions: the method of making up differentiation, the method of grouping and the method of integral factor, integral upper limit function method; finally, through specific examples, it introduces the application of Rolle's mean value theorem in such problems. the same time, this paper will lay a good foundation for students to learn Lagrange mean value theorem and Cauchy mean value theorem,and provide great help for students to apply Rolle theorem to solve mathematical problems.
Keywords: Rolle's mean value theorem; construction of auxiliary function;approximation differentiation method; grouping method; integral factor method；integral upper limit function method
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