分段函数的原函数与导函数论文

数学与统计学学院

中期报告

学院: 数学与统计学学院

专业: 统计学年级:2009 题目: 分段函数的原函数与导函数学生姓名: 马颖峰学号:09063207 指导教师姓名:俞诗秋职称:副教授

2011年6月1日

目录

摘要 ........................................................................ 1 关键词 ...................................................................... 1 引言 ........................................................................ 2 1. 分段函数 ............................................................... 2 2.

分段函数的原函数 ....................................................... 2 2.1 分段函数()f x 在区间[,]a b 上连续的情形下的原函数问题 ................ 2 2.2 分段函数()f x 在区间[,]a b 上x c =点处具有第一类间断点的情形 ......... 3 2.3 分段函数()f x 在[,]a b , 上x c =处具有第二类间断点的情形[3]

(4)

3.

关于分段函数求导问题的探究 ............................................. 5 3.1 函数在分界点处不连续的情况 ......................................... 5 3.2

函数在分界点处连续的情况 (5)

结论 ........................................................................ 9 参考文献 (9)

分段函数的原函数与导函数

摘要:文章主要讨论了什么是分段函数。分段函数的原函数与导数问题。在讨论分段函数原函数

时又分为连续分段函数、具有第一类间断点的分段函数、具有第二类间断点的分段函数三种情形的原函数来分类讨论。分段函数在高等数学中既是重点也是难点,“难”在于讨论它的可导性,尤其是在求一个分段函数导函数时对分界点处的可导性判断上。由于高等数学中涉及的分段函数通常是除去分界点外各段上都是初等函数,故分段函数的导数是先按段求导,后单独讨论分界点处的导数(可能存在,也可能不存在) 。对分界点可导性的讨论,一般采用导数“增量比”形式的定义。虽然这种方法具有一般性,但是遇到解析式复杂的函数往往就不方便了。下文将分几种情况,对分段函数的求导问题进行探讨,从而得出一种通用的解题方法。

关键词：分段函数；原函数；分段函数求导

Original Function and Derived Function of Segmentation

Function

Abstract：The article mainly talked about what is a segmentation function.The original function of

segmentation function with lead few questions.Is again divided into continuous segmentation function, has the segmentation function that no 1 interrupteds a dot and have the original function of segmentation function three kinds of situations that second interrupteds a dot to categorize a discussion while talking about original function of segmentation function.The segmentation function is a point and is a crux in Gao Deng's mathematics, "difficult" lie in talk about it of can lead sex, particularly is begging a segmentation function derived function vs canning lead sex criterion of cut - off point up. Because the segmentation function involved in Gao Deng mathematics usually is remove outside each all of tops of cut - off point are elementary functions, past leading of segmentation function number is to press the segment first to beg to lead, behind talk about a leading of cut - off point number alone.(possible existence, also may nonexistent)Can lead a sexual discussion to the cut - off point, generally adopt the definition of leading the number "quotient of difference" form.Although this method has general,this method's meeting the function of analysis type complications is getting more inconvenient. The continuation will divide a few conditions, lead a question progress study to the segmentation function's begging and get the solution of a kind of universal method thus.

Key words：segmentation function；Original function；derived function