学 年 论 文题 目： 微分中值定理的证明及应用 学 院： 数学与信息科学学院 专 业： 数学与应用数学 学生姓名： * 学 号： * 指导教师： * 微分中值定理的证明及应用*摘要：微分中值定理是数学分析中很重要的基本定理,在数学分析中有着广泛的应用.它是沟通函数及其导数之间的桥梁，是应用导数研究函数在某点的局部性质和在某个区间上的整体性质的重要工具.利用微分中值定理可以论证方程的根的存在问题、方程根的个数问题以及根的存在区间问题，也经常用于证明一些含有导数的等式.微分中值定理是罗尔中值定理，拉格朗日中值定理，柯西中值定理的统称，它是微分中值定理学中重要的理论基础.拉格朗日中值定理可视为中心定理，以它为中心展开，罗尔中值定理是拉格朗日中值定理的一个特值，而柯西中值定理可视为拉格朗日中值定理在应用上的推广.关键词：罗尔中值定理 拉格朗日中值定理 柯西中值定理 证明 应用Abstract: the differential mean value theorem in mathematical analysis is very important basic theorem in the mathematical analysis, is widely used. It is a communication bridge between a function and its derivative, is the application of derivative of function at a certain point of the local nature and in a certain interval on the overall properties of the important tools. The use of differential mean value theorem can be proved equation for the root of the problem, the problem of the number of roots of equations and existence of root interval problems, are also frequently used to prove some containing derivative equation. The differential mean value theorem is the Rolle mean value theorem, Lagrange mean value theorem, Cauchy mean value theorem of differential mean value theorem collectively, it is of important theoretical basis. Lagrange mean value theorem can be regarded as the center in the center of its expansion theorem, Rolle mean value theorem, Lagrange mean value theorem is a special value, and the Cauchy mean value theorem can be regarded as the Lagrange mean value theorem in application promotion.Key words: Rolle mean value theorem Lagrange mean value theorem Cauchy mean value theorem Prove Application微分中值定理是数学分析中很重要的定理，它是罗尔中值定理、拉格朗日中值定理、柯西中值定理的统称，微分中值定理在数学分析中有广泛的应用.一般教科书中都是通过构造辅助函数，利用罗尔定理来证明拉格朗日中值定理和柯西中值定理的.下面我将利用不同于教科书的方法来证明这三个中值定理，并列举每个中值定理的应用.一 罗尔中值定理的证明和应用1 罗尔中值定理若函数 满足如下条件：f（i） 在闭区间a,b上连续；(ii) 在开区间内(a,b)可导；f(iii) ，)(bfa则在（a,b）内至少存在一点 ，使得 .0)(f2 罗尔中值定理的证明（1） 预备知识和两个引理定义1 闭区间a,b的闭子区间族S称为a,b的一个完全覆盖,是指对任意xa,b,存在x0,使得a,b的每个含有x且长度小于x的闭子区间都属于S.引理1 若S是闭区间a,b的一个完全覆盖,则S包含a,b的一个划分,即存在a=x 00,使得(x-x,x +x)0,使得(x-x,x+x)0,b0,所以00.证明：只需证 .令g(x)=x 2, 则f(x)、g(x)满cfabf2)()(2足柯西中值定理条件,所以 c(a,b)，使 ，)()(cgfabgf即 .cfabf2)()(2四 总结通过以上内容，说明微分中值定理有很多证法并且三个中值定理之间有内在联系，而且每个中值定理都有极其广泛的应用.所以学好微分中值定理对数学分析及其他学科的学习有很大帮助.五 参考文献1吴泽礼.Lagrange中值定理的两个新证法J.韩山师专学报.1991(03).2李国辉.崔媛.Lagrange中值定理的一个应用J.高等职业教育天津职业大学学报.2005(06).3李万军.Rolle中值定理的一个新证明J.宜宾学院学报.2004(03).4荆天.柯西中值定理及其应用J.高校理科研究.2008.5余后强.微分学中值定理的证明及其应用J.咸宁学院学报.2006(06).6黄德丽.用五种方法证明柯西中值定理J.湖州师范学院学报.2003.ag an employment tribunal clai Emloyment tribunals sort out disagreements between employers and employees. You may need to make a claim to an employment tribunal if: you dont agree with the disciplinary action your employer has taken against you your employer dismisses you and you think that you have been dismissed unfairly. For more informu, take advice from one of the organisations listed under Fur ther help. Employment tribunals are less formal than some other courts, but it is still a legal process and you will need to give evidence under an oath or affirmation. Most people find making a claim to an employment tribunal challenging. If you are thinking about making a claim to an employment tribunal, you should get help straight away from one of the organisations listed under Further help. ation about dismissal and unfair dismissal, see Dismissal. You can make a claim to an employment tribunal, even if you havent appealed against the disciplinary action your employer has taken against you. However, if you win your case, the tribunal may reduce any compensation awarded to you as a result of your failure to appeal. Remember that in most cases you must make an application to an employment tribunal within three months of the date when the event you are complaining about happened. If your application is received after this time limit, the tribunal will not usually accept i. If you are worried about how the time limits apply to you If you are being represented by a solicitor at the tribunal, they may ask you to sign an agreement where you pay their fee out of your compensation if you win the case. This is known as a damages-based agreement. In England and Wales, your solicitor cant charge you more than 35% of your compensation if you win the case. You are clear about the terms of the agreement. It might be best to get advice from an experienced adviser, for example, at a Citizens Advice Bureau. To find your nearest CAB, including those that give advice by e-mail, click on nearest CAB. For more information about making a claim to an employment tribunal, see Employment tribunals. The (lack of) air up there Watch m Cay man Islands-based Webb, the head of Fifas anti-racism taskforce, is in London for the Football Associations 150th a