微分中值定理及其应用(大学毕业论文)

毕业论文(设计)题目名称：微分中值定理的推广及应用题目类型：理论研究型学生姓名：邓奇峰院(系)：信息与数学学院专业班级：数学10903班指导教师：熊骏辅导教师：熊骏时间：2012年12月至2013年6月目录毕业设计任务书..........................................................................................................................I开题报告.......................................................................................................................................II指导老师审查意见...................................................................................................................III评阅老师评语............................................................................................................................IV答辩会议记录..............................................................................................................................V中文摘要.....................................................................................................................................VI外文摘要....................................................................................................................................VII1引言.............................................................................................................................................12题目来源....................................................................................................................................13研究目的和意义......................................................................................................................14国内外现状和发展趋势与研究的主攻方向....................................................................15微分中值定理的发展过程....................................................................................................26微分中值定理的基本内容....................................................................................................36.1罗尔(Rolle)中值定理..............................................................................................36.2拉格朗日(Lagrange)中值定理..............................................................................46.3柯西（Cauchy）中值定理.......................................................................................46.4泰勒（Taylor）定理................................................................................................47微分中值定理之间的联系....................................................................................................58微分中值定理的应用.............................................................................................................58.1根的存在性证明..........................................................................................................68.2利用微分中值定理求极限.......................................................................................88.3利用微分中值定理证明函数的连续性...............................................................108.4利用微分中值定理解决含高阶导数的中值问题............................................108.5利用微分中值定理求近似值.................................................................................108.6利用微分中值定理解决导数估值问题...............................................................108.7利用微分中值定理证明不等式............................................................................119微分中值定理的推广...........................................................................................................149.1微分中值定理的推广定理.....................................................................................159.2微分中值定理的推广定理的应用........................................................................17参考文献......................................................................................................................................18致谢...........................................................................................................................................19VI微分中值定理的推广及应用学生：邓奇峰，信息与数学学院指导老师：熊骏，信息与数学学院【摘要】微分中值定理，是微积分的基本定理，是沟通函数与其导数之间的桥梁，是应用导数的局部性研究函数整体性的重要数学工具，在微积分中起着极其重要的作用。本文首先介绍了微分中值定理的发展过程、微分中值定理的内容和微分中值定理之间的内在联系，接着再看微分中值定理在解题中的应用,如:“讨论方程根（零点）的存在性”,“求极限”和“证明不等式”等方面的应用。由于微分中值定理及有关命题的证明方法中往往出现的形式并非这三个定理中的某个直接结论，这就需要借助于一个适当的辅助函数，来实现数学问题的等价转换，但是，怎样构造适当的辅助函数往往是比较困难的。在此重点给出如何通过构造辅助函数来解决中值定理问题，从理论和实际的结合上阐明微分中值定理的重要性。拉格朗日中值定理及柯西中值定理都是罗尔中值定理的推广。本文从其它角度归纳、推导了几个新的形式，拓宽了罗尔中值定理的使用范围。同时，用若干实例说明了微分中值定理在导数极限、导数估值、方程根的存在性、不等式的证明、以及计算函数极限等方面的一些应用。【关键词】微分中值定理罗尔中值定理拉格朗日中值定理柯西中值定理联系推广应用VIITheExtensionandApplicationoftheDifferentialMeanValueTheoremStudent:DengQifeng,SchoolofInformationandMathematicsTutor:XiongJun,SchoolofInformationandMathematics【Abstract】Thedifferentialmeanvaluetheorem,isthefundamentaltheoremofcalculus,isthecommunicationbridgebetweenfunctionanditsderivative,isanimportantmathematicaltoolintegratedlocalresearchapplicationfunctionderivative,playsaveryimportantroleinCalculus.Thispaperdescribesthedevelopprogress，thecontentsandtheintrinsiclinkbetweenthedifferentialmeanvaluetheorem;Thenlookatthedifferentialmeanvaluetheoreminsolvingproblems,suchas:thediscussionoftheroots(zero)inexistence,limitandproofofinequality.Becauseoftenproofofdifferentialmeanvaluetheoremandrelatedpropositionsintheformisnotthethreetheoremsofadirectconclusion,thisrequiresthehelpofasuitableauxiliaryfun