利用常数变易法构造辅助函数在微分学中值定理证明题中的应用研究  

The application of variation of parameters to the construction of auxiliary functions in the proof of differential mean value theorem

作　　者：朱存斌[1] ZHU Cunbin(School of Statistics and Applied Mathematics,Anhui University of Finance and Economics,Bengbu 233030,China)

机构地区：[1]安徽财经大学统计与应用数学学院,安徽蚌埠233030

出　　处：《渤海大学学报（自然科学版）》2023年第3期243-249,共7页Journal of Bohai University:Natural Science Edition

基　　金：安徽财经大学研究生联合培养基地项目(No:cxjhlhpy2001);安徽省高等学校省级质量工程项目(No:2021jxtd002,No:2021xxkc002);安徽财经大学“六卓越、一拔尖”卓越人才培养创新项目(No:aclzy2022007).

摘　　要：利用微分学中值定理构造辅助函数来证明一类中值问题,关键环节就是构造出适当的辅助函数,对辅助函数验证相关定理的条件,辅助函数得到微分学中值定理的结论,从而恒等变形得到需要证明的结论.就微分学中值定理相关证明题型,给出构造辅助函数的一种通用方法——常数变易法.首先给出常数变易法思想和具体推导过程,通过一阶、二阶和三阶具体问题给出推导过程,最后,给出一般问题的构造辅助函数的结论.To prove some median problems by using differential mean value theorem to construct auxiliary functions,the key step is to construct appropriate auxiliary functions and to verify the conditions of relevant theorems for auxiliary functions,the auxiliary function obtains the conclusion of the theorem,thus the identical deformation obtains the conclusion which needs to prove.In this paper,we give a general method of constructing auxiliary functions——variation of parameters for the proof of differential mean value theorem.In this paper,we give the variation of parameters thought and the specific derivation process,and the derivation process through the first-order,second-order and third-order concrete problems.Finally,we give the conclusion of constructing auxiliary functions for general problems.

关 键 词：常数变易法 辅助函数 罗尔定理 线性齐次微分方程 

分 类 号：O172[理学—数学]