一道AP微积分题Which of the following functions is not odd?有个选项我知道它是odd function,但只有感觉,无法从式子上看出来:f(x)=x/(x^2+1),怎么把它化成多项式的样子再直接通过次数判断奇偶?

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一道AP微积分题Which of the following functions is not odd?有个选项我知道它是odd function,但只有感觉,无法从式子上看出来:f(x)=x/(x^2+1),怎么把它化成多项式的样子再直接通过次数判断奇偶?

一道AP微积分题Which of the following functions is not odd?有个选项我知道它是odd function,但只有感觉,无法从式子上看出来:f(x)=x/(x^2+1),怎么把它化成多项式的样子再直接通过次数判断奇偶?
一道AP微积分题
Which of the following functions is not odd?
有个选项我知道它是odd function,但只有感觉,无法从式子上看出来:
f(x)=x/(x^2+1),怎么把它化成多项式的样子再直接通过次数判断奇偶?

一道AP微积分题Which of the following functions is not odd?有个选项我知道它是odd function,但只有感觉,无法从式子上看出来:f(x)=x/(x^2+1),怎么把它化成多项式的样子再直接通过次数判断奇偶?
这个好办,不就是想把f(x)=x/(x^2+1),展开成幂级数吗.
积分f(x)=1/2ln(1+x~2)+c=1/2(x~2-1/2x~4+.)
后面是taylor展开,不过把x换成x~2而已.
之后再两边求导,发现右边的幂级数所有的x的指数一定是奇数.
因此是奇函数

为什么一定要多项式才能判断奇偶呢?
首先显然f(x)在R上有定义,其次f(-x) = -x / ((-x)^2 + 1) = -( x/(x^2 + 1) ) = -f(x),所以f(x)是odd function。

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