一道英语的微积分题f(x)={(cosx-1)/x^2 for x≠0 -1/2for x=0}the function f,defined above,has derivatives of all orders.Let g be the function defined by g(x)=1+∫f(t)dt,t,0,xWrite the fifth-degree Taylor polynomial for g about x=0要讲原因
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一道英语的微积分题f(x)={(cosx-1)/x^2 for x≠0 -1/2for x=0}the function f,defined above,has derivatives of all orders.Let g be the function defined by g(x)=1+∫f(t)dt,t,0,xWrite the fifth-degree Taylor polynomial for g about x=0要讲原因
一道英语的微积分题
f(x)={(cosx-1)/x^2 for x≠0 -1/2for x=0}
the function f,defined above,has derivatives of all orders.Let g be the function defined by g(x)=1+∫f(t)dt,t,0,x
Write the fifth-degree Taylor polynomial for g about x=0
要讲原因和过程
一道英语的微积分题f(x)={(cosx-1)/x^2 for x≠0 -1/2for x=0}the function f,defined above,has derivatives of all orders.Let g be the function defined by g(x)=1+∫f(t)dt,t,0,xWrite the fifth-degree Taylor polynomial for g about x=0要讲原因
本题是求g(x)在x=0处的泰勒展开.
首先f(x)在其定义域内连续,(cosx-1)/x^2 在 x=0 处的极限就是-1/2,所以f(x)是连续函数,处处可导可积.
g(x)在x=0处的泰勒展开为g(x)=g(0)+g'(0)*x+g''(0)*x^2/2!+...+g'''''(0)*x^5/5!,但完全没有必要将g(x)的原函数求出：x=0时,g(0)=1,g的第n阶导数=f的第n-1阶导数（n=1时即为f（x））,然后求f的第n-1阶导数在x=0处的极限值即可得g'(0)到g'''''(0),然后将其带入g的泰勒展开即可.
问题到这就结束了,结果就不给出了.