函数单调性的应用毕业论文

学号：函数单调性的应用系别专业班级姓名指导教师2013年5月8日安阳师范学院人文管理学院本科毕业论文（设计）安阳师范学院人文管理学院本科毕业论文（设计）摘要函数单调性是函数的重要性质之一，同时也是解决实际问题求最值的重要方法。本课题从函数单调性的概念与定义入手，主要介绍函数单调性的若干性质和判别方法，然后深入探讨和总结单调性在数学领域的相关应用，继而联系实际，分析单调性在解决实际问题中的重要作用，从而总结出函数单调性所适用的条件，应用的范围等。所以，无论是从研究教学来讲，还是实际应用来讲，研究函数的单调性都具有重要理论意义和现实意义。关键词：函数单调性，判别，导数，应用AbstractMonotonicfunctionnotonlyisoneoftheimportantnaturesofthefunction,butalsoisanimportantmethodforthepracticalproblems.Thisprojectplantostartwiththeconceptanddefinitionofthefunctionmonotonicity,mainlyintroducessomepropertiesofmonotonefunctionsanddiscriminantmethods,andthenfurtherdiscussedandsummarizedmonotonicrelatedapplicationsinthefieldofmathematics,andthencontactwithpractice,analysiswhat’stheimportantroleofmonotonicinsolvingpracticalproblems,thussummedtheconditionsapplied,theapplicationscopeandsoon.So,whetheritisfromresearchandteaching,orfromitspracticalapplication,monotonicityalsohasimportanttheoreticalandpracticalsignificance.Keywords：Monotonicfunction,Distinguish,Derivative,Application安阳师范学院人文管理学院本科毕业论文（设计）目录1、前言··································································································12、函数单调性的基础理论···········································································12.1函数单调性的基本概念······································································12.2函数单调性的常用定理与性质·····························································33、函数单调性的判别·················································································73.1初等数学中函数单调性的判别·····························································73.2高等数学中利用导数判别函数单调性····················································84、函数单调性的解题应用···········································································84.1单调性在求极值、最值中的应用··························································84.2单调性在不等式中的应用··································································144.3单调性在求方程解问题中的应用·························································154.4单调性在化简求值方面的应用····························································164.5单调性在比较大小方面的应用····························································175、函数单调性在实际生活中的应用······························································175.1单调性在材料合理利用中的应用·························································175.2单调性在生产利润中的应用·······························································185.3单调性在结构工程中的应用·······························································205.4单调性在优化路径中的应用········································································216、结论·······························································································22致谢·······································································································23参考文献·································································································24安阳师范学院人文管理学院本科毕业论文（设计）11、前言单调性是近代数学的重要基础，是联系初等数学与高等数学的重要纽带。研究函数在无限变化中的变化趋势，从有限认识无限，从近似中认识精确，从量变中认识质变，都要用到单调性。它的引入为解决相关数学问题提供了新的视野，为研究函数的性质、证明不等式、求解方程、比较大小等方面提供了有力的工具。本文将在已有文献的基础之上，总结单调性在解决数学问题中的相关应用，并且探讨单调性在利润最大化、材料优化、资源整合和路径选择等方面的应用。2、函数单调性的基础理论2.1函数单调性的基本概念2.1.1函数单调性的定义一般地，设函数()fx的定义域为I：如果对属于I内某个区间上的任意两个自变量12,xx,当12xx时,都有12fxfx，那么就说()fx在这个区间上是增函数。如果对属于I内某个区间上的任意两个自变量12,xx,当12xx时,都有12fxfx，那么就说()fx在这个区间上是减函数。若函数在这一区间具有(严格的)单调性,则就说函数()yfx在某个区间是增函数或减函数，这一区间叫做函数的单调区间，此时也说函数是这一区间上的单调函数。2.1.2函数单调性的意义在单调区间上，增函数的图像是上升的，减函数的图像是下降的。函数的这一性质在解决函数求极值、比较大小、求解方程的根、解不等式等问题时都有很大的帮助，在现实生活中，例如在经济领域中如何实现利润最大化，在工程领域中如何计算材料的极限强度，在航空领域中计算航空器回收落地时间等等，函数单调性都有很重要的应用。2.1.3函数单调性的理解(1)图形理解在区间D上，()fx的图像上升（或下降）()fx是区间D上的增函数（或减函数）。安阳师范学院人文管理学院本科毕业论文（设计）2例1证明函数]1,01)(在区间（xxxf上是减函数。证明：设12,xx是区间]1,0(上的任意实数，且12xx，则121212121221121212121111()()()()()1()()(1)fxfxxxxxxxxxxxxxxxxxxx12121212121212121010,01,10,01,1()(1)0,()()0.xxxxxxxxxxxxfxfxxx又即1212()()1(0),xxfxfxfx因为当时，有，在区间上所以单调递减。图像如下:(2)正向理解（定义理解）)(xf在区间D上单调递增，Dxx21,,且)()(2121xfxfxx；)(xf在区间D上单1x2f(x2)()(x2)))(1xf图1.1.1OxX1X2y②减函数图像OxX1X2y①增函数图像001x1安阳师范学院人文管理学院本科毕业论文（设计）3调递减，Dxx21,,且)()(2121xfxfxx。例2设函数)(xfy在2,0上是增函数，函数)2(xfy是偶函数，确定)27(),25(),1(fff的大小关系。解：函数)2(xfy是偶函数，)2()2(xfxf，)23()25()21()27(ffff，，又因为)(xf在）（2,0上是增函数，且2312113()(1)(),22fff即75()(1)()22fff(3)逆向理解)(xf在区间D上单调递增，Dxx21,,且2121)()(xxxfxf；)(xf在区间D上单调递减，Dxx21,,且2121)()(xxxfxf。例3已知奇函数)(xf是定义在11，上的减函数，若0)1()1(2afaf，求实数a的取值范围。解：由已知可知，)1()1(2afaf，又)(xf是奇函数2(1)(1)fafa。)(xf是定义在11，上的减函数，11112aa，解得10a。(4)导数理解设函数)(xf在区间D内可导，若'()0fx，则)(xf是减函数；若'()0fx，则)(xf是增函数。反之，若函数)(xf是增函数，则'()0fx；若函数)(xf是减函数，则'()0fx。例4函数xaxxf3)(在,是减函数，求a的取值范围。解：)(xf在R上递减，'2()310fxax恒成立，则(1)当0a时，'()10fx，满足条件。(2)当0a时，只须满足0120aa且即可。综上所述得0a.2.2函数单调性的常用定理和性质2.2.1最值定理对于在区间I上有定义的函数)(xf,如果有Ix0,使得对于0xI,都有安阳师范学院人文管理学院本科毕业论文（设计）4)()(0xfxf(或)()(0xfxf)，则称)(0xf是函数)(xf在区间I上的最大值(或最小值)。例1求函数xxfsin1)(在区间2,0上的最大值和最小值。解：由三角函数xsin的性质可知,当2x时，函数xsin取得最大值12sin；当23x时，函数xsin取得最小值12sin.故函数)(xf的最大值为2，最小值为0。定理1（最大、最小值定理）若函数)(xf在闭区间ba,上连续，则)(xf在ba,上有最大值与最小值。如果函数)(xf在闭区