数学竞赛中的不等式应用

摘要在整个数学知识体系中，不等式是一个庞大的家族。不等关系在各个学习阶段都不不同体现。小学数学中就有比较两个数的大小，初中数学则介绍到不等式具体概念和性质，以及简单的一元一次不等式（组）的解法，而高中更是在初中的基础上，进一步深入的学习了不等式，并且接触到了柯西不等式。所以不等式在整个数学学习的过程中具有至关重要的作用。作为解决数学难题的一个重要工具，不等式在数学竞赛中体现出其独一无二的作用。本文详细介绍了排序不等式、均值不等式、柯西不等式三种不等式的表现形式，并给出了相关证明。此外，本文结合历年数学竞赛题，应用相关不等式。44274

所以，不等式教学研究显得尤其重要。最后，结合不等式在教材中的位置与地位，给出了不等式教学方法以及建议。

Throughout the mathematical knowledge system, inequality is a huge family. Unequal relations are not reflected in all different stages of learning. Primary Mathematics in there size comparison of two numbers, the junior high school mathematics is introduced to the concept and the specific nature of inequality, as well as a simple one dollar inequality (group) of the solution, however it is the basis of the junior high school on the further study of the inequality, and exposed to the Cauchy inequality. So inequality has a crucial role in the whole process of learning mathematics.

As an important tool for solving mathematical problems, inequality reflects its unique role in the mathematics competition. This paper describes the sort inequality, mean inequality, Cauchy inequality three kinds of manifestations of inequality, and gives the relevant proof. In addition, this paper calendar mathematics competition problems, application-related inequalities.

Therefore, inequality is particularly important in teaching and research. Finally, combined with the location and status of inequality in textbooks, having the inequality of teaching methods and suggestions.

毕业论文关键词：不等式竞赛题；排序不等式；均值不等式；柯西不等式；不等式教学

Keyword:Competition Problems of Inequality；Rearrangement inequality；Mean Inequality；Cauchy inequality；Teaching Inequality

1.1本学位论文研究的内容和目的 5

1.1.1研究的内容 5

1.1.2研究的目的 5

1.2不等式教学 5

1.2.1不等式的重要性 5

1.2.2.《普通高中数学课程标准》对不等式内容提出了新要求 5

1.2.3.数学课程改革对不等式教学提出了新要求 5

1. 3研究现状 5

1.3.1有关《不等式选讲》教材内容设置方式及教材编写理念的研究 6

1.3.2有关《不等式选讲》的教学参考 6

2.1排序不等式 6

2.1.1排序不等式表现形式 7

2.1.2排序不等式用于证明不等式 7

2.1.3排序不等式用于求最值 8

2.1.4排序不等式用于比较大小 8

2.1.5排序不等式用于解应用题 9

2.2均值不等式 9

2.2.1均值不等式表现形式 9

2.2.2均值不等式竞赛题数学竞赛中的不等式应用:http://www.751com.cn/shuxue/lunwen_45511.html