柯西中值定理的证明及其应用探索

Exploration of the proof of Cauchy's mean value theorem and its application

摘要

为促进高职学生深入理解柯西中值定理的内容及应用,探索了两种证明柯西中值定理的方法,并阐释了三个中值定理之间的关系.介绍了柯西中值定理在等式证明、不等式证明、函数单调性判断及极限计算中的应用,并进一步应用柯西中值定理证明了洛必达法则、积分中值定理及泰勒定理.对柯西中值定理的证明、内涵及其在解题中的应用进行探索,为微分中值定理的学习和应用提供参考.

Abstract

In order to promote higher vocational students to have a deeper understanding of the content and application of Cauchy's mean value theorem, two methods were explored to prove Cauchy's mean value theorem, and the relationships between three mean value theorems were explained. The application of Cauchy's mean value theorem in equality proof, inequality proof, function monotonicity judgment and extreme limit calculation was introduced, and Cauchy's mean value theorem was further applied to prove the L'Hospital's rule, integral mean value theorem and Taylor's theorem. The exploration of the proof, connotation and application of Cauchy's mean value theorem in problem-solving provides reference for the learning and application of differential mean value theorem.

关键词

柯西中值定理 / 证明 / 内涵 / 应用

Key words

Cauchy mean value theorem / proof / connotation / application

参考文献

[1] 同济大学数学教研室. 高等数学:上册:4版[M]. 北京: 高等教育出版社, 1996: 165.

[2] 曹绳武, 王振中, 于远许. 高等数学重要习题集:第四版[M]. 大连: 大连理工大学出版社, 2003: 86.

[3] 孙清华, 孙昊. 数学分析内容、方法与技巧:上[M]. 武汉: 华中科技大学出版社, 2003: 204.

[4] 舒阳春. 高等数学中的若干问题解析[M]. 北京: 科学出版社, 2005: 39.

[5] 薛利敏. 高等数学[M]. 北京: 教育科学出版社, 2012: 61.

基金资助

2023年娄底职业技术学院职教高地建设理论与实践课题(LDZYZJGD202302)

2022年娄底职业技术学院教学改革研究项目(LZJY22BZC01)

2022年湖南省社会科学成果评审委员会课题(XSP22YBC054)