关于微分学中值定理的一些注解和新证法

Some Notes and New Proofs on Mean Value Theorem in Differential Calculus

目的为培养高素质人才,在各个阶段、各个层次的数学教学中,都应十分重视对学生学习能力、创新能力和应用能力的培养。方法以"微分学中值定理"为例,说明在教与学中,对抽象的较难的定理,要把它"掰开"看,要把它形象化、浅显化;在深刻理解的基础上,对别人叙述的不足之处,给以纠正和弥补;思考研究新方法。结果举出了很多定理成立的充分条件,而不是必要条件的例子。关于辅助函数的理解与构造,也有一些新想法,并与已知知识相联系。结论在教育教学改革中,对教材的科学性、先进性的研究,是一个重要方面。在数学教学中,不仅要传知识,还要传思想、方法,要从点点滴滴做起,开发学生的智力,培养学生的能力。

Objective To train high-quality talents,great importance should be attached to the cultivation of students’ learning ability,innovation ability and application ability in all stages and levels of mathematics teaching.Methods Taking"the theorem of mean in differential calculus"as an example,this paper illustrates that in teaching and learning,the more difficult abstract theorem should be"broken apart",visualized and superficially manifested;on the basis of profound understanding,the inadequacies of other people’s narration should be corrected and remedied;and new methods should be considered and studied.Results The students gave many examples of sufficient conditions,without examples of not necessary conditions,for the establishment of the theorem.There were also some new ideas about the understanding and construction of auxiliary functions,which were related to known knowledge.Some students thought that they had new understanding and consideration to the proof of the mean value theorem and other theorems in differential calculus.Conclusion In the reform of education and teaching,it is an important aspect to study the scientificity and advancement of textbooks.In mathematics teaching,we should not only impart knowledge,but also ideas and methods,starting from bit by bit,developing students’intelligence and cultivating students’ability.