微积分学中辅助函数的构造
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f (b) f (a ) [ F ( x) F (a)] 转化为利用已知 F (b) F (a )
的罗尔定理加以证明; 在牛顿- 莱布尼兹公式的证明中也是构造辅 助函数 ( x) f (t )dt 利用积分上限函数 ( x) 的性质得到证明的.
a x
1.2 将复杂化为简单 一些命题较为复杂, 直接构造辅助函数往往较困难, 可通过恒 等变形, 由复杂转化为简单, 从中探索辅助函数的构造, 以达到解 决问题的目的, 这种通过巧妙的数学变换, 将一般化为特殊, 将复 杂问题化为简单问题的论证思想, 是一元微积分学中的重要而常用 的数学思维体现. 例 1 设函数 f ( x) ， g ( x) 都在 [a, b] 上连续, 在 a, b 内可导, 且
g ( x ) 0, f (a ) g (b) g (a ) f (b) . 证明在开区间 a, b 内存在一点 , 使得 f '( ) g ( ) f ( ) g '( ) ．
分析 将证明的结论变形为 f '( ) g ( ) f ( ) g '( ) 0 , 直接思考哪 个函数求导后为 f '( ) g ( ) f ( ) g '( ) 0 ,发现不易找到这个函数.进一 步考虑除以一个非零因子，不难发现所证结论可变形为
f ( x) g ( x)
因为 f ( x ) ，g ( x ) 都在 [ a, b] 上连续， 在 a, b
g ( x ) 0, f (a ) g (b) g (a ) f (b) .
f ( x) 在 [a, b] 上连续，在 a, b 内可导，并且 ( x) 0 ， g ( x) f (a ) f (b) ，所以 (a) (b) ． g (a ) f (b)
[1-2]
．
微积分学中辅助函数的构造是在一定条件下利用微积分中值定 理求解数学问题的方法．通过查阅现有的大量资料发现，现在国内外 对微积分学中辅助函数构造法的研究比较多， 其中有一部分研究的是 辅助函数构造法的思路 ，但大部分研究的是辅助函数的构造在微积 分学解题中的应用 ． 通过构造辅助函数，可以解决数学分析中众多难题，尤其是在微 积分学证明题中应用颇广，且可达到事半功倍的效果．
[4] [3]
1 构造辅助函数的原则
辅助函数的构造是有一定的规律的.当某些数学问题使用通常的 方法按定势思维去考虑很难奏效时，可根据题设条件和结论的特征,
第 1 页 (共 11 页)
性质展开联想，进而构造出解决问题的特殊模式，这就是构造辅助函 数解题的一般思路. 1.1 将未知化为已知 在一元微积分学中许多定理的证明都是在分析所给命题的条件、 结论的基础上构造一个函数将要证的问题转化为可利用的已知结论 来完成的. 比如, 柯西中值定理的证明就是通过对几何图形的分析, 构造辅助函数 ( x) f ( x) f (a)
作 者：司玉会 指导教师：葛玉丽 摘要：构造辅助函数是数学分析中解决问题的重要方法，在解决实际问题 中有广泛应用． 通过研究微积分学中辅助函数构造法，构造与问题相关的辅助函 数，从而得出欲证明的结论．本文介绍了构造辅助函数的概念及其重要性，分析 了构造辅助函数的原则， 归纳了构造辅助函数的几种方法，并研究了构造辅助函 数在微积分学中的重要作用和应用. 关键词：原函数法； 辅助函数；常数变易法；函数增量法
bf (b) af (a ) f ( ) f '( ) ． ba bf (b) af (a ) k ，我们来证明 k f ( ) f '( ) 记 ba
[ f ( x) f '( x) g ( x) f ( x) g '( x) f ( x) .对 ]' 0 .因此, 找到了辅助函数 ( x) 2 g ( x) g ( x) g ( x)
此函数在 [a, b] 上应用罗尔定理即得要证的结论.
第 2 页 (共 11 页)
证明 作辅助函数 ( x) 内可导, 且 所以 ( x)
2.4 积分法···························································································································· (6) 2.5 函数增量法··················································································································· (7) 2.6 参数变易法··················································································································· (7)
利用几何图形直观形象的特点构造辅助函数. 在各种版本的“高 等数学” 和“数学分析”的书中, 微分中值定理的证明大多是利用 对几何图形的分析,探索辅助函数的构造, 然后加以证明的.
2 构造辅助函数的方法探讨
2.1 常数变易法 常数变易法的思想就是，将于证明题中的某个常量用变量代替而 构成辅助函数，对Hale Waihona Puke Baidu助函数进行讨论，使欲证明题得到证明. 2.1.1 罗尔定理应用举例 在微分学等式证明中，我们通过引入辅助函数来证明，而辅助函 数构造是关键，一般情况下可以用常数变易发来构造辅助函数. 例 2 函数 f ( x) 在区间 a, b 上可微， 证明在区间 a, b 内至少存在一 点 ，使得 证明
编号：08005110137
南阳师范学院 2012 届毕业生
毕业论文（设计）
题 目： 微 积 分 学 中 辅 助 函 数 的 构 造 司玉会 2008-01 4 年 数学与应用数学 葛玉丽 2012-03-31 完 成 人： 班 学 专 级： 制： 业：
指导教师： 完成日期：
目
录
摘要············································································································································ (1) 0 引言······································································································································· (1) 1 构造辅助函数的原则····························································································· (1)
2 构造辅助函数的方法探讨················································································· (3)
2.1 常数变易法··················································································································· (3) 2.1.1 罗尔定理应用举例··························································································· (3) 2.1.2 构造辅助函数证明积分不等式··································································· (4) 2.2 原函数法························································································································(4) 2.3 微分方程法··················································································································· (6)
3 构造辅助函数在微分中值定理证明中的应用分析····················· (8)
3.1 辅助函数构造在拉格朗日定理中应用························································· (8) 3.1.1 应用举例·············································································································· (9)
0 引言
当某些数学问题使用通常办法按定势思维去考虑而很难奏效时， 可根据题设条件和结论特征、性质展开联想，进而构造出解决问题的 特殊模式——构造辅助函数． 辅助函数构造法是数学分析中一个重要 的思想方法，在数学分析中具有广泛的应用．构造辅助函数是把复杂 问题转化为已知的容易解决问题的一种方法，在解题时，常表现为不 对问题本身求解，而是构造一个与问题有关的辅助问题进行求解
4 结束语······························································································································· (10) 参考文献······························································································································(10)
Abstract···································································································································(11)
微积分学中辅助函数的构造
1.1 将未知化为已知········································································································· (2) 1.2 将复杂化为简单········································································································ (2) 1.3 利用几何特征············································································································· (3)
有罗尔定理知存在一点 ，使得 '( ) 0 即
'( ) [
f ( ) f '( ) g ( ) f ( ) g '( ) ]' 0 ． g ( ) g 2 ( ) f '( ) g ( ) f ( ) g '( ) ．
所以 1.3 利用几何特征
的罗尔定理加以证明; 在牛顿- 莱布尼兹公式的证明中也是构造辅 助函数 ( x) f (t )dt 利用积分上限函数 ( x) 的性质得到证明的.
a x
1.2 将复杂化为简单 一些命题较为复杂, 直接构造辅助函数往往较困难, 可通过恒 等变形, 由复杂转化为简单, 从中探索辅助函数的构造, 以达到解 决问题的目的, 这种通过巧妙的数学变换, 将一般化为特殊, 将复 杂问题化为简单问题的论证思想, 是一元微积分学中的重要而常用 的数学思维体现. 例 1 设函数 f ( x) ， g ( x) 都在 [a, b] 上连续, 在 a, b 内可导, 且
g ( x ) 0, f (a ) g (b) g (a ) f (b) . 证明在开区间 a, b 内存在一点 , 使得 f '( ) g ( ) f ( ) g '( ) ．
分析 将证明的结论变形为 f '( ) g ( ) f ( ) g '( ) 0 , 直接思考哪 个函数求导后为 f '( ) g ( ) f ( ) g '( ) 0 ,发现不易找到这个函数.进一 步考虑除以一个非零因子，不难发现所证结论可变形为
f ( x) g ( x)
因为 f ( x ) ，g ( x ) 都在 [ a, b] 上连续， 在 a, b
g ( x ) 0, f (a ) g (b) g (a ) f (b) .
f ( x) 在 [a, b] 上连续，在 a, b 内可导，并且 ( x) 0 ， g ( x) f (a ) f (b) ，所以 (a) (b) ． g (a ) f (b)
[1-2]
．
微积分学中辅助函数的构造是在一定条件下利用微积分中值定 理求解数学问题的方法．通过查阅现有的大量资料发现，现在国内外 对微积分学中辅助函数构造法的研究比较多， 其中有一部分研究的是 辅助函数构造法的思路 ，但大部分研究的是辅助函数的构造在微积 分学解题中的应用 ． 通过构造辅助函数，可以解决数学分析中众多难题，尤其是在微 积分学证明题中应用颇广，且可达到事半功倍的效果．
[4] [3]
1 构造辅助函数的原则
辅助函数的构造是有一定的规律的.当某些数学问题使用通常的 方法按定势思维去考虑很难奏效时，可根据题设条件和结论的特征,
第 1 页 (共 11 页)
性质展开联想，进而构造出解决问题的特殊模式，这就是构造辅助函 数解题的一般思路. 1.1 将未知化为已知 在一元微积分学中许多定理的证明都是在分析所给命题的条件、 结论的基础上构造一个函数将要证的问题转化为可利用的已知结论 来完成的. 比如, 柯西中值定理的证明就是通过对几何图形的分析, 构造辅助函数 ( x) f ( x) f (a)
作 者：司玉会 指导教师：葛玉丽 摘要：构造辅助函数是数学分析中解决问题的重要方法，在解决实际问题 中有广泛应用． 通过研究微积分学中辅助函数构造法，构造与问题相关的辅助函 数，从而得出欲证明的结论．本文介绍了构造辅助函数的概念及其重要性，分析 了构造辅助函数的原则， 归纳了构造辅助函数的几种方法，并研究了构造辅助函 数在微积分学中的重要作用和应用. 关键词：原函数法； 辅助函数；常数变易法；函数增量法
bf (b) af (a ) f ( ) f '( ) ． ba bf (b) af (a ) k ，我们来证明 k f ( ) f '( ) 记 ba
[ f ( x) f '( x) g ( x) f ( x) g '( x) f ( x) .对 ]' 0 .因此, 找到了辅助函数 ( x) 2 g ( x) g ( x) g ( x)
此函数在 [a, b] 上应用罗尔定理即得要证的结论.
第 2 页 (共 11 页)
证明 作辅助函数 ( x) 内可导, 且 所以 ( x)
2.4 积分法···························································································································· (6) 2.5 函数增量法··················································································································· (7) 2.6 参数变易法··················································································································· (7)
利用几何图形直观形象的特点构造辅助函数. 在各种版本的“高 等数学” 和“数学分析”的书中, 微分中值定理的证明大多是利用 对几何图形的分析,探索辅助函数的构造, 然后加以证明的.
2 构造辅助函数的方法探讨
2.1 常数变易法 常数变易法的思想就是，将于证明题中的某个常量用变量代替而 构成辅助函数，对Hale Waihona Puke Baidu助函数进行讨论，使欲证明题得到证明. 2.1.1 罗尔定理应用举例 在微分学等式证明中，我们通过引入辅助函数来证明，而辅助函 数构造是关键，一般情况下可以用常数变易发来构造辅助函数. 例 2 函数 f ( x) 在区间 a, b 上可微， 证明在区间 a, b 内至少存在一 点 ，使得 证明
编号：08005110137
南阳师范学院 2012 届毕业生
毕业论文（设计）
题 目： 微 积 分 学 中 辅 助 函 数 的 构 造 司玉会 2008-01 4 年 数学与应用数学 葛玉丽 2012-03-31 完 成 人： 班 学 专 级： 制： 业：
指导教师： 完成日期：
目
录
摘要············································································································································ (1) 0 引言······································································································································· (1) 1 构造辅助函数的原则····························································································· (1)
2 构造辅助函数的方法探讨················································································· (3)
2.1 常数变易法··················································································································· (3) 2.1.1 罗尔定理应用举例··························································································· (3) 2.1.2 构造辅助函数证明积分不等式··································································· (4) 2.2 原函数法························································································································(4) 2.3 微分方程法··················································································································· (6)
3 构造辅助函数在微分中值定理证明中的应用分析····················· (8)
3.1 辅助函数构造在拉格朗日定理中应用························································· (8) 3.1.1 应用举例·············································································································· (9)
0 引言
当某些数学问题使用通常办法按定势思维去考虑而很难奏效时， 可根据题设条件和结论特征、性质展开联想，进而构造出解决问题的 特殊模式——构造辅助函数． 辅助函数构造法是数学分析中一个重要 的思想方法，在数学分析中具有广泛的应用．构造辅助函数是把复杂 问题转化为已知的容易解决问题的一种方法，在解题时，常表现为不 对问题本身求解，而是构造一个与问题有关的辅助问题进行求解
4 结束语······························································································································· (10) 参考文献······························································································································(10)
Abstract···································································································································(11)
微积分学中辅助函数的构造
1.1 将未知化为已知········································································································· (2) 1.2 将复杂化为简单········································································································ (2) 1.3 利用几何特征············································································································· (3)
有罗尔定理知存在一点 ，使得 '( ) 0 即
'( ) [
f ( ) f '( ) g ( ) f ( ) g '( ) ]' 0 ． g ( ) g 2 ( ) f '( ) g ( ) f ( ) g '( ) ．
所以 1.3 利用几何特征