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1、 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
2、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 deri
3、vative, 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
4、 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 i
5、s 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 applic
6、ation 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
7、条件：（i）在闭区间a,b上连续；(ii) 在开区间内(a,b)可导；(iii) ，则在（a,b）内至少存在一点，使得.2 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=x0xkxn=b,使每个闭区间xk-1,xk(i=1,2,n)都属于S.证明:用反证法.设S不包含a,b的任何划分,则通过对a,b重复使用二等分法可得a,b的闭子区间列In,使得InIn+k (n=1,2,),
8、|In|0(n),且S不包含任何一个In的任何一个划分,其中|In|代表区间In的长度.由闭区间套定理, 存在唯一的xIn (n=1,2,)若x如定义1所述,则因|In|0(n),故存在自然数n0 ,使得In00,使得(x-x,x +x)0,使得(x-x,x+x)x,且函数f(x)在(x-x,x+x)上严格单调.设I是含有x且长度小于x的的任一闭子区间,则I(x-x,x+x),所以函数f(x)在I上的严格单调,即IS.由引理1,在S中必存在的一个划分I1,I2,Im,不妨设这些小区间是按自左到右顺序编号的.于是对k1 , 2 , , m,函数f(x)在Ik上严格单调.不妨设函数f(x)在I1
9、上严格增加,若I1的右端点为1,则1为I2的左端点,而对于1 ,必存在某个IS,使得1I且(II1 )-1与(II2 )-1都非空,于是,根据函数f(x)在1 严格增加就会得到函数f(x)在I2 上也严格增加,依次类推函数f(x)在I3 、I4 、Im 上严格增加,即函数f(x)在每个Ik( k = 1 , 2 , ,m)都严格增加,因此函数f(x)在上严格增加,对于、,且,有:一方面,f(an)f()f()f(bn),另一方面,函数f(x)在x=a右连续,在x =b左连续,故有从而有：即f(a)f(b),这与已知f(a)=f(b)矛盾,故有(a,b),使得.3 罗尔中值定理的应用例1（根的存
10、在性的证明）设a,b,c为三个实数，证明：方程的根不超过三个.证明：令,则，.用反证法.设原方程的根超过3个，那么F（x）至少有4个零点，不妨设为x1x2x3x4,那么有罗尔定理，存在x11x22x33x4，使.用罗尔定理，存在，使=0.再用罗尔定理，存在,使，因为，所以.矛盾，所以命题得证！二 拉格朗日中值定理的证明及应用1 拉格朗日中值定理若函数满足如下条件：（i）在闭区间a,b上连续；(ii) 在开区间内(a,b)可导；则在（a,b）内至少存在一点，使得.2 拉格朗日中值定理的证明（1） 两个引理引理1 设f(x)在c,d上连续，则在c,d上必存在两点与，使得-d=(d-c)/2, .证
11、明：设函数,由（x）在c,(c+d)/2上应用连续函数根存在性定理有xc,(c+d)/2使得.取即可.证毕！引理2 设（i）f(x)在区间I的内点x可微；（ii）数列n与n满足：nxn,n0,b0,所以0a/(a+b)1,根据介值定理，至少存在一点x0(0,1),使得，在（0，x0）上根据Lagrange中值定理，存在（0,x0）,使得。又由于f(0)=0,所以=x0（1）.从而在（x0,1）上，根据Lagrange中值定理，存在（x0,1）,使得。又由于f(1)=1,所以（2），（1）式加（2）式得（3）.再将代入（3）式即得.三 柯西中值定理的证明及应用1 柯西中值定理设函数和g满足（i）
12、在a,b上都连续；(ii)在(a,b)内都可导；（iii）f(x)和g(x)不同时为零；（iv）g(a)g(b). 则存在（a,b），使得.2、柯西中值定理的证明（1） 三个引理引理1 设函数f(x)在a,b上有定义，且在x0 （a,b）处可导，由2,2为一闭区间套，且，则.引理2 设函数f(x)在a,b上连续，则存在a1,b1 a,b,且,使得.引理3（引理2的推广） 设函数f(x)，g(x)在a,b上连续,且g(x)是单射，则存在a1,b1 a,b，且，使.（2） 柯西中值定理的证明证明：当，a,b，且时，有g() g().若g() =g()，由引理2，存在1,1 ,，且，使，从而g(1)
13、 =g(1).在1,1上再次应用引理2有，存在2,2 ,，且，使，从而又有g(2) =g(2).反复利用引理2，最终可得一个闭区间套n,n,满足，且g(n) =g(n),由闭区间套定理，存在, a,b，使.由引理1得：，这与条件g(x)0 (x（a,b）)相矛盾.再根据引理3有，存在a1,b1 a,b，且，使.反复利用引理3，类似于前面的证明，可得闭区间套n,n,满足，且.由闭区间套定理存在ca,b，使.再由引理1有：.3 柯西中值定理的应用例1（用柯西中值定理证明不等式）若0x1x20.证明：只需证.令g(x)=x2, 则f(x)、g(x)满足柯西中值定理条件,所以 c(a,b)，使，即.四
14、 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
15、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
16、you think that you have been dismissed unfairly. For more informu, take advice from one of the organisations listed underFur 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 p
17、eople 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 underFurther help. ation about dismissal and unfair dismissal, seeDismissal. You can make a clai
18、m to an employment tribunal, even if you haventappealedagainst 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 appli
19、cation 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
20、 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 adamages-based agreement. In England and Wales, your solicitor cant charge you more than 35% of your compensationif you win the case. You are cle
21、ar 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 onnearest CAB. For more information about making a claim to an employment tribunal, s
22、eeEmployment 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 anniversary celebrations and will attend Citys Premier League match at Chelsea on Sunday. I am going to be at the match tom
23、orrow and I have asked to meet Ya ya Toure, he told BBC Sport. For me its about how he felt and I would like to speak to him first to find out what his experience was. Uefa hasopened disciplinary proceedings against CSKAfor the racist behaviour of their fans duringCitys 2-1 win. Michel Platini, pres
24、ident of European footballs governing body, has also ordered an immediate investigation into the referees actions. CSKA said they were surprised and disappointed by Toures complaint. In a statement the Russian side added: We found no racist insults from fans of CSKA. Age has reached the end of the b
25、eginning of a word. May be guilty in his seems to passing a lot of different life became the appearance of the same day; May be back in the past, to oneself the paranoid weird belief disillusionment, these days, my mind has been very messy, in my mind constantly. Always feel oneself should go to do
26、something, or write something. Twenty years of life trajectory deeply shallow, suddenly feel something, do it.一字开头的年龄已经到了尾声。或许是愧疚于自己似乎把转瞬即逝的很多个不同的日子过成了同一天的样子；或许是追溯过去，对自己那些近乎偏执的怪异信念的醒悟，这些天以来，思绪一直很凌乱，在脑海中不断纠缠。总觉得自己似乎应该去做点什么，或者写点什么。二十年的人生轨迹深深浅浅，突然就感觉到有些事情，非做不可了。The end of our life, and can meet many th
27、ings really do?而穷尽我们的一生，又能遇到多少事情是真正地非做不可？ During my childhood, think lucky money and new clothes are necessary for New Year, but as the advance of the age, will be more and more found that those things are optional; Junior high school, thought to have a crush on just means that the real growth, but
28、over the past three years later, his writing of alumni in peace, suddenly found that isnt really grow up, it seems is not so important; Then in high school, think dont want to give vent to out your inner voice can be in the high school children of the feelings in a period, but was eventually infarct
29、ion when graduation party in the throat, later again stood on the pitch he has sweat profusely, looked at his thrown a basketball hoops, suddenly found himself has already cant remember his appearance. Baumgartner the disappointing news: Mission aborted. r plays an important role in this mission. St
30、arting at the ground, conditions have to be very calm - winds less than 2 mph, with no precipitation or humidity and limited cloud cover. The balloon, with capsule attached, will move through the lower level of the atmosphere (the troposphere) where our day-to-day weather lives. It will climb higher
31、 than the tip of Mount Everest (5.5 miles/8.85 kilometers), drifting even higher than the cruising altitude of commercial airliners (5.6 miles/9.17 kilometers) and into the stratosphere. As he crosses the boundary layer (called the tropopause),e can expect a lot of turbulence. We often close ourselv
32、es off when traumatic events happen in our lives; instead of letting the world soften us, we let it drive us deeper into ourselves. We try to deflect the hurt and pain by pretending it doesnt exist, but although we can try this all we want, in the end, we cant hide from ourselves. We need to learn t
33、o open our hearts to the potentials of life and let the world soften us.生活发生不幸时，我们常常会关上心门；世界不仅没能慰藉我们，反倒使我们更加消沉。我们假装一切仿佛都不曾发生，以此试图忘却伤痛，可就算隐藏得再好，最终也还是骗不了自己。既然如此，何不尝试打开心门，拥抱生活中的各种可能，让世界感化我们呢？ Whenever we start to let our fears and seriousness get the best of us, we should take a step back and re-evalua
34、te our behavior. The items listed below are six ways you can open your heart more fully and completely.当恐惧与焦虑来袭时，我们应该退后一步，重新反思自己的言行。下面六个方法有助于你更完满透彻地敞开心扉。Whenever a painful situation arises in your life, try to embrace it instead of running away or trying to mask the hurt. When the sadness strikes, t
35、ake a deep breath and lean into it. When we run away from sadness thats unfolding in our lives, it gets stronger and more real. We take an emotion thats fleeting and make it a solid event, instead of something that passes through us.当生活中出现痛苦的事情时，别再逃跑或隐藏痛苦，试着拥抱它吧；当悲伤来袭时，试着深呼吸，然后直面它。如果我们一味逃避生活中的悲伤，悲伤只会变得更强烈更真实悲伤原本只是稍纵即逝的情绪，我们却固执地耿耿于怀By utilizing our breath we soften our experiences. If we dam them up, our lives will stagnate, but when we keep them flowing, we allow more newness and greater experiences to blossom.深呼吸能减缓我们的感受。屏住呼吸，生活停滞；呼出呼吸，更多新奇与经历又将拉开序幕。