凸函数、琴生不等式及其在中学数学中的应用
发布时间：2018-10-22 12:43
【摘要】：凸函数不仅是一类非常重要的函数，而且是聚集诸多优良性质的典型函数，对于函数凸性的研究，在数学的许多领域和分支中都有非常重要的应用，特别是有关不等式的推导问题、求最值和取值范围问题以及三角函数问题这三个方面，凸函数都有着十分重要的作用。除此之外，凸函数还涉及了许多数学概念、性质、理论、命题的证明和应用，而且凸函数在高考和数学竞赛中也是有着极其重要的理论价值和应用价值。本文首先简单介绍了凸函数的定义、几何意义、等价定义、性质、判定以及琴生不等式的证明、变形、推广；其次，阐述了凸函数、琴生不等式在中学教学中所体现的教育价值和功能；最后，对凸函数与琴生不等式在中学数学中的应用进行系统、全面、明确的划分。例如：对于相似的题型给出统一的解题方法，即简化了证明过程，又开拓了学生的解题思路；对于有些题型，还给出了相应的推广，如由“求解圆的内接、外切多边形的面积的最值问题”推广到“求解椭圆的内接、外切多边形面积的最值问题”。经过系统、全面、明确的整理，有助于学生对这部分知识所涉及的相关题型有更深入的了解，学会利用凸函数的性质和琴生不等式更加便捷简单的解决相关问题，同时为学生提供一些新的解题思路和技巧。
[Abstract]:Convex function is not only a kind of very important function, but also a typical function that gathers many excellent properties. For the research of function convexity, it has very important applications in many fields and branches of mathematics. In particular, convex functions play a very important role in the derivation of inequalities, the problem of finding the maximum value and the range of values, and the trigonometric function problem. In addition, convex functions also involve many mathematical concepts, properties, theories, propositions and applications, and convex functions also have extremely important theoretical and applied value in college entrance examination and mathematics competitions. In this paper, the definition of convex function, geometric meaning, equivalent definition, property, judgment and proof, deformation and generalization of Qin Sheng inequality are introduced briefly. Finally, the application of convex function and Qinsheng inequality in middle school mathematics is systematically, comprehensively and clearly divided. For example, a unified method for solving similar problem types is given, which simplifies the proof process and opens up the students' thinking of solving problems. For some problem types, it also gives corresponding generalizations, such as "solving the inner connection of the circle," The problem of the minimum value of the area of a tangent polygon is extended to "solving the problem of the interior connection of an ellipse and the area of an outer polygon". Through systematic, comprehensive and definite arrangement, it is helpful for students to have a deeper understanding of the related question types involved in this part of knowledge, and to learn to use the properties of convex functions and Qin Sheng's inequality to solve the related problems more conveniently and simply. At the same time for students to provide some new ideas and skills to solve problems.
【学位授予单位】：西北大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：G633.6
本文编号：2287192
[Abstract]:Convex function is not only a kind of very important function, but also a typical function that gathers many excellent properties. For the research of function convexity, it has very important applications in many fields and branches of mathematics. In particular, convex functions play a very important role in the derivation of inequalities, the problem of finding the maximum value and the range of values, and the trigonometric function problem. In addition, convex functions also involve many mathematical concepts, properties, theories, propositions and applications, and convex functions also have extremely important theoretical and applied value in college entrance examination and mathematics competitions. In this paper, the definition of convex function, geometric meaning, equivalent definition, property, judgment and proof, deformation and generalization of Qin Sheng inequality are introduced briefly. Finally, the application of convex function and Qinsheng inequality in middle school mathematics is systematically, comprehensively and clearly divided. For example, a unified method for solving similar problem types is given, which simplifies the proof process and opens up the students' thinking of solving problems. For some problem types, it also gives corresponding generalizations, such as "solving the inner connection of the circle," The problem of the minimum value of the area of a tangent polygon is extended to "solving the problem of the interior connection of an ellipse and the area of an outer polygon". Through systematic, comprehensive and definite arrangement, it is helpful for students to have a deeper understanding of the related question types involved in this part of knowledge, and to learn to use the properties of convex functions and Qin Sheng's inequality to solve the related problems more conveniently and simply. At the same time for students to provide some new ideas and skills to solve problems.
【学位授予单位】：西北大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：G633.6
【参考文献】
相关期刊论文 前10条
1 胡宇晨;牛松;;例谈加权琴生不等式的应用[J];中等数学;2015年09期
2 覃平阳;;凸函数与Hadamard不等式[J];学周刊;2015年04期
3 周晓晖;;凸函数的性质在不等式证明中的应用[J];牡丹江教育学院学报;2014年06期
4 王恒亮;;例析三类条件不等式的证明[J];中学数学研究;2013年03期
5 刘达;;椭圆中面积最大的内接三角形和平行四边形构造[J];数学教学;2011年06期
6 卫福山;;一类条件不等式的特征及证法[J];河北理科教学研究;2011年02期
7 王洪斌;;利用琴生不等式证明一类条件不等式[J];中学教研(数学);2011年04期
8 周子君;;高中数学中有关凸函数的命题探究[J];福建中学数学;2010年10期
9 王磊;;凸函数及其相关的一些重要不等式[J];民营科技;2010年06期
10 杨丽娟;;凸函数与几个重要不等式的证明[J];吉林农业科技学院学报;2010年01期
相关硕士学位论文 前3条
1 余池增;柯西不等式在高中数学中的应用研究[D];广州大学;2012年
2 周弋林;高考数学命题中的竞赛数学背景研究[D];广州大学;2012年
3 刘丽红;凸函数的Jensen不等式及其应用[D];重庆理工大学;2010年
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