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

作　　者：杨雄[1] YANG Xiong(Department of Public T eaching,Loudi Vocational and Technical College,Loudi Hunan 417000)

机构地区：[1]娄底职业技术学院公共课教学部,湖南娄底417000

出　　处：《辽宁师专学报（自然科学版）》2024年第2期7-12,35,共7页Journal of Liaoning Normal College(Natural Science Edition)

基　　金：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)。

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.In order to promote higher vocational students to have a deeper understanding of the content and ap plication 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 

分 类 号：O172[理学—数学]