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](/uploads/image/z/8426219-59-9.jpg?t=%E4%B8%80%E9%81%93%E8%8B%B1%E8%AF%AD%E7%9A%84%E5%BE%AE%E7%A7%AF%E5%88%86%E9%A2%98f%28x%29%3D%7B%28cosx-1%29%2Fx%5E2+for+x%E2%89%A00+-1%2F2for+x%3D0%7Dthe+function+f%2Cdefined+above%2Chas+derivatives+of+all+orders.Let+g+be+the+function+defined+by+g%28x%29%3D1%2B%E2%88%ABf%28t%29dt%2Ct%2C0%2CxWrite+the+fifth-degree+Taylor+polynomial+for+g+about+x%3D0%E8%A6%81%E8%AE%B2%E5%8E%9F%E5%9B%A0)
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.