摘要不等式证明作为数学学习中的一个重要问题,是中学数学教学以及高等数学中非常重要的一部分.并且此类问题在近几年成为高考数学中的一大重点以及难点问题,本文首先对中学数学中不等式证明的一些初等方法进行了归纳总结,并通过分析得出在解决此类问题中需要注意的地方.然后做进一步分析,通过函数的性质,如函数的凹凸性、单调性、最值性、微分中值定理、泰勒公式来创建或证明不等式,在创建或者证明不等式过程中加深对函数与不等式关系的理解,从而巩固所学习的知识,以及发现一些不等式,并在这些活动中培养学生的自主思考意识和数学应用意识,提高学生数学建模能力和自主解题能力,激发学生学习难点的积极性和解决难题的意志力.58282
As an issue of mathematics learning,proof of inequality methods are very important in high school mathematics teaching and higher mathematics,and these questions became a major focus and difficult issue in the college entrance examination of mathematics.In this article ,we summarize some proof of inequality elementary methods in high school mathematics first,then obtain some details we need to pay attention to by analyzing. Then we do further analysis,we creating and proving inequality by the nature of the function such as the function of sex,the monotonicity of function,the most value of function,differential mean value theorem and Taylor formula.In the process of creating and proving inequality we deep the understanding of the relationship between function and inequality,then to consolidate the knowledge acquired, as well as find some inequality.I hope it can cultivate the students' independent thinking and mathematics application consciousness, improve students' mathematical modeling ability and independent problem-solving ability, and arouse the enthusiasm of students learning difficulties and solving the difficult problems of will power.
毕业论文关键词:不等式证明; 初等方法; 函数性质; 培养能力;
Keyword: proof of Inequality; elementary method; function property; developing capacity;
目 录
摘 要 2
1.问题引入 4
1.1不等式的定义和基本性质 4
1.2不等式证明问题介绍 4
1.3问题的提出 4
2.不等式证明的初等方法 4
2.1比较法 5
2.2综合法 6
2.3放缩法 7
2.4换元法 9
2.5数学归纳法 13
2.6反证法 14
3.函数的性质与不等式证明 14
3.1利用函数的凹凸性发现或证明不等式 14
3.2利用函数的单调性发现或证明不等式 16
3.3利用函数的最大值最小值发现或者证明不等式 17
3.4利用微分中值定理发现或者证明不等式 18
3.5利用泰勒公式发现或者证明不等式 19
4.课题总结 20
参考文献 21
致谢 21
1.问题引入
1.1不等式的定义
一般来说,用纯粹的大于号“ ”、小于号“ ”连接的不等式称为严格不等式,用大于或等于号“ ”小于或等于号“ ”连接的不等式称为非严格不等式,或广义不等式.总而言之的话,用不等号( 、 、 、 )连接的式子就叫做不等式.