换元法在数学解题中的应用.docVIP

换元法在数学解题中的应用 摘 要 换元法通过引入新的变量，将题目移至新对象的知识背景中去研究，从而使非标准题目标准化，繁琐题目简单化，变得容易解决.因此换元法是数学解题中一种十分重要而且应用非常广泛的思想方法.本文研究了换元法在代数式、因式分解、方程、函数、证明和微积分方面的应用及解题技巧，并且给出了一些典型例子.用换元法解数学题有助于培养学生灵活解决数学问题的能力，把理论知识和实际应用结合起来，培养人的数学思维品格,提高解题效率. 关键字：换元法;构造新元;标准化;应用 Application of the Method of Exchange Element Method in Solving Mathematics Problem Abstract: The method of substitution put the topic into the background in the new object to study by introducing a new variable, so that the non-standard topics of standardization, simplification complicated topic, become easier to solve. Therefore the method of substitution is a very important and widely used method of thinking in mathematical problem solving. This paper will give a simple analysis about the basic concepts of substitution method, theoretical basis, and the basic principles of using the method of substitution. And it also discussed about the using of the method of substitution in Algebraic expressions, factorization, equations, functions, proven and calculus and gives some typical examples. This paper’s content will help for training the problem-solving skills in the mathematics, just like how to transform the higher degree into the low degree, the fraction into the integral expression, the irrational formula into the rational expression, and the transcendental expression into the algebraic expression. And this paper’s skills help solve complex and complicated mathematical problems.? Key words: method of substitution; structures; standardization; application 目 录 1 引言 1 2 文献综述 1 2.1 国内外研究现状 1 2.2 国内外研究现状评价 2 2.3 提出问题 2 3 预备知识 2 3.1 换元法的基本概念 3 3.2 换元法的理论依据 3 3.3 换元法的两种基本类型 3 4 换元法在数学解题中的应用 3 4.1 换元法在代数方面的应用 4 4.1.1换元法在计算中的应用 4 4.1.2换元法在因式分解中的应用 5 4.1.3换元法在解方程中的应用 6 4.1.4换元法在函数中的应用 10 4.1.5 换元法在证明中的应用 14 4.2 换元法在微积分中的应用 16 4.2.1 换元法在微分中的应用 16 4.2.2 换元法在积分中的应用 17 5 结论 21 5.1 主要发现 21 5.2 启示 21 5.3 局限性 21 5.4 努力方向 21 参考文献 22 1引言 在数学学习过程中，我们往往会遇到各种