浅谈数学分析中反例的作用

单位代码： 005

分类号：O1

延安大学西安创新学院

本科毕业论文（设计）

题目：浅谈数学分析中反例的作用

专业：数学与应用数学

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学号： ********** 指导教师：**

职称：讲师

毕业时间：二〇一五年六月

浅谈数学分析中反例的作用

摘要：数学分析中，反例常被用于证明之中.有许多数学猜想或命题的叙述时全称命题，声称所有的一类事物都有某种性质，或者是只要满足某个条件，就会得出某种结果.当证明这样的数学猜想遇到困难时，人们会趋向于寻找一个反例，以说明这个猜想是错误的.证明在数学分析中有着重要的作用.

本文主要总结了反例在数学分析中起到的作用.首先对反例进行了认识，主要是对反例和反证法在概念和运用上的一个区别；其次是总结反例加强对概念的认识，主要是从无界函数、函数在一点的连续、二元函数的偏导和可微这几个方面来说明；再其次是对定理的理解，主要介绍了罗尔中值定理和拉格朗日中值定理这两个定理；再是说明反例对概念之间关系的把握，主要是分别对可导与连续、无穷大与无界量等概念之间进行了区别联系；最后简单总结了反例能有培养逆向思维的能力.

关键词：数学分析；反例；作用；归纳总结

The Effect of Counter Example in Mathematical Analysis Abstract：In mathematical analysis,a counterexample is often used in proofs.There are many mathematical conjectureor proposition describes universal proposition,that kind of thing all have certain properties,or as long as acondition is met,will come tosome sort of conclusion.When that mathematical conjecture this difficulty,a mathematician would tend to look for a a counter example,to show that this conjecture is false.That plays an important role in mathematical analysis.

This paper mainly summarizes the counterexample to play in mathematical analysis. The first is the exceptions to the recognition, mainly to the counterexample and reduction to absurdity in concept and use them to prove a difference step on; This is followed by a summary of the counterexample to enhance understanding of the concept, mainly from the unbounded function, function and Er Yuan functionfor a partial derivative and differentiability of several aspects of this example; then to understand theorem, mainly introduced the Rollemean value theorem and Lagrange mean value theorem and the two theorem; then explains the concept ofthe relationship between the example grasp, mainly on between the concept of derivative and thecontinuous, infinite with an unbounded amount of difference; summarizes the counterexample can have theability of reverse thinking.

Key words:Mathematical analysis ;The counterexample ;Effect;For example

目录

1绪论 (1)

1.1 引言 (1)

1.2 课题的背景及目的 (1)

1.3 国内外研究状况 (2)

1.4 课题研究方法 (2)

1.5 论文构成及研究内容 (2)

2认识反例 (2)

2.1 反例的概念 (2)

2.2 区别举反例与反证法 (3)

2.2.1举反例及其运用反例的证明步骤 (3)

2.2.2 反证法及其原理 (3)

2.2.3运用反证法的证明步骤 (3)

3反例精确对概念的认识 (4)

3.1 无界函数 (4)

3.2 连续函数 (4)

3.3 二元函数偏导数与可微 (5)

4反例加深对定理的理解 (6)

4.1 罗尔中值定理 (6)

4.2 拉格朗日中值定理 (7)

5反例准确把握概念之间的关系 (8)

5.1 可导与连续 (8)

5.2 无穷大与无界量 (9)

5.3 函数极大（小）值与最大（小）值 (9)

5.4 可积函数 (10)

6运用反例培养逆向思维能力 (10)

7总结 (12)

参考文献 (14)

致谢 (15)