数学毕业论文-微分中值定理的证明及应用

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ABSTRACTTHEDIFFERENTIALMEANVALUETHEOREMINMATHEMATICALANALYSISISVERYIMPORTANTBASICTHEOREMINTHEMATHEMATICALANALYSIS,ISWIDELYUSEDITISACOMMUNICATIONBRIDGEBETWEENAFUNCTIONANDITSDERIVATIVE,ISTHEAPPLICATIONOFDERIVATIVEOFFUNCTIONATACERTAINPOINTOFTHELOCALNATUREANDINACERTAININTERVALONTHEOVERALLPROPERTIESOFTHEIMPORTANTTOOLSTHEUSEOFDIFFERENTIALMEANVALUETHEOREMCANBEPROVEDEQUATIONFORTHEROOTOFTHEPROBLEM,THEPROBLEMOFTHENUMBEROFROOTSOFEQUATIONSANDEXISTENCEOFROOTINTERVALPROBLEMS,AREALSOFREQUENTLYUSEDTOPROVESOMECONTAININGDERIVATIVEEQUATIONTHEDIFFERENTIALMEANVALUETHEOREMISTHEROLLEMEANVALUETHEOREM,LAGRANGEMEANVALUETHEOREM,CAUCHYMEANVALUETHEOREMOFDIFFERENTIALMEANVALUETHEOREMCOLLECTIVELY,ITISOFIMPORTANTTHEORETICALBASISLAGRANGEMEANVALUETHEOREMCANBEREGARDEDASTHECENTERINTHECENTEROFITSEXPANSIONTHEOREM,ROLLEMEANVALUETHEOREM,LAGRANGEMEANVALUETHEOREMISASPECIALVALUE,ANDTHECAUCHYMEANVALUETHEOREMCANBEREGARDEDASTHELAGRANGEMEANVALUETHEOREMINAPPLICATIONPROMOTIONKEYWORDSROLLEMEANVALUETHEOREMLAGRANGEMEANVALUETHEOREMCAUCHYMEANVALUETHEOREMPROVEAPPLICATION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；FIII，BFA则在（A,B）内至少存在一点，使得0F2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AX00,使得XX,XX0,使得XX,XX0,B0,所以00证明只需证令GXX2,则FX、GX满CFABF22足柯西中值定理条件,所以CA,B，使，CGFABGF即CFABF22四总结通过以上内容，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1991032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2005063李万军ROLLE中值定理的一个新证明J宜宾学院学报2004034荆天柯西中值定理及其应用J高校理科研究20085余后强微分学中值定理的证明及其应用J咸宁学院学报2006066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2003AGANEMPLOYMENTTRIBUNALCLAIEMLOYMENTTRIBUNALSSORTOUTDISAGREEMENTSBETWEENEMPLOYERSANDEMPLOYEESYOUMAYNEEDTOMAKEACLAIMTOANEMPLOYMENTTRIBUNALIFYOUDONTAGREEWITHTHEDISCIPLINARYACTIONYOUREMPLOYERHASTAKENAGAINSTYOUYOUREMPLOYERDISMISSESYOUANDYOUTHINKTHATYOUHAVEBEENDISMISSEDUNFAIRLYFORMOREINFORMU,TAKEADVICEFROMONEOFTHEORGANISATIONSLISTEDUNDERFURTHERHELPEMPLOYMENTTRIBUNALSARELESSFORMALTHANSOMEOTHERCOURTS,BUTITISSTILLALEGALPROCESSANDYOUWILLNEEDTOGIVEEVIDENCEUNDERANOATHORAFFIRMATIONMOSTPEOPLEFINDMAKINGACLAIMTOANEMPLOYMENTTRIBUNALCHALLENGINGIFYOUARETHINKINGABOUTMAKINGACLAIMTOANEMPLOYMENTTRIBUNAL,YOUSHOULDGETHELPSTRAIGHTAWAYFROMONEOFTHEORGANISATIONSLISTEDUNDERFURTHERHELPATIONABOUTDISMISSALANDUNFAIRDISMISSAL,SEEDISMISSALYOUCANMAKEACLAIMTOANEMPLOYMENTTRIBUNAL,EVENIFYOUHAVENTAPPEALEDAGAINSTTHEDISCIPLINARYACTIONYOUREMPLOYERHASTAKENAGAINSTYOUHOWEVER,IFYOUWINYOURCASE,THETRIBUNALMAYREDUCEANYCOMPENSATIONAWARDEDTOYOUASARESULTOFYOURFAILURETOAPPEALREMEMBERTHATINMOSTCASESYOUMUSTMAKEANAPPLICATIONTOANEMPLOYMENTTRIBUNALWITHINTHREEMONTHSOFTHEDATEWHENTHEEVENTYOUARECOMPLAININGABOUTHAPPENEDIFYOURAPPLICATIONISRECEIVEDAFTERTHISTIMELIMIT,THETRIBUNALWILLNOTUSUALLYACCEPTIIFYOUAREWORRIEDABOUTHOWTHETIMELIMITSAPPLYTOYOUIFYOUAREBEINGREPRESENTEDBYASOLICITORATTHETRIBUNAL,THEYMAYASKYOUTOSIGNANAGREEMENTWHEREYOUPAYTHEIRFEEOUTOFYOURCOMPENSATIONIFYOUWINTHECASETHISISKNOWNASADAMAGESBASEDAGREEMENTINENGLANDANDWALES,YOURSOLICITORCANTCHARGEYOUMORETHAN35OFYOURCOMPENSATIONIFYOUWINTHECASEYOUARECLEARABOUTTHETERMSOFTHEAGREEMENTITMIGHTBEBESTTOGETADVICEFROMANEXPERIENCEDADVISER,FOREXAMPLE,ATACITIZENSADVICEBUREAUTOFINDYOURNEARESTCAB,INCLUDINGTHOSETHATGIVEADVICEBYEMAIL,CLICKONNEARESTCABFORMOREINFORMATIONABOUTMAKINGACLAIMTOANEMPLOYMENTTRIBUNAL,SEEEMPLOYMENTTRIBUNALSTHELACKOFAIRUPTHEREWATCHMCAYMANISLANDSBASEDWEBB,THEHEADOFFIFASANTIRACISMTASKFORCE,ISINLONDONFORTHEFOOTBALLASSOCIATIONS150THANNIVERSARYCELEBRATIONSANDWILLATTENDCITYSPREMIERLEAGUEMATCHATCHELSEAONSUNDAY“IAMGOINGTOBEATTHEMATCHTOMORROWANDIHAVEASKEDTOMEETYAYATOURE,“HETOLDBBCSPORT“FORMEITSABOUTHOWHEFELTANDIWOULDLIKETOSPEAKTOHIMFIRSTTOFINDOUTWHATHISEXPERIENCEWAS“UEFAHASOPENEDDISCIPLINARYPROCEEDINGSAGAINSTCSKAFORTHE“RACISTBEHAVIOUROFTHEIRFANS“DURINGCITYS21WINMICHELPLATINI,PRESIDENTOFEUROPEANFOOTBALLSGOVERNINGBODY,HASALSOORDEREDANIMMEDIATEINVESTIGATIONINTOTHEREFEREESACTIONSCSKASAIDTHEYWERE“SURPRISEDANDDISAPPOINTED“BYTOURESCOMPLAINTINASTATEMENTTHERUSSIANSIDEADDED“WEFOUNDNORACISTINSULTSFROMFANSOFCSKA“AGEHASREACHEDTHEENDOFTHEBEGINNINGOFAWORDMAYBEGUILTYINHISSEEMSTOPASSINGALOTOFDIFFERENTLIFEBECAMETHEAPPEARANCEOFTHESAMEDAYMAYBEBACKINTHEPAST,TOONESELFTHEPARANOIDWEIRDBELIEFDISILLUSIONMENT,THESEDAYS,MYMINDHASBEENVERYMESSY,INMYMINDCONSTANTLYALWAYSFEELONESELFSHOULDGOTODOSOMETHING,ORWRITESOMETHINGTWENTYYEARSOFLIFETRAJECTORYDEEPLYSHALLOW,SUDDENLYFEELSOMETHING,DOIT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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