高数微分证明题感觉不难,就是那里卡住了,/>
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高数微分证明题
![](http://img.wesiedu.com/upload/8/ca/8cafebfe3a9bfd4f0360c1c626e56370.jpg)
感觉不难,就是那里卡住了,/>
![](http://img.wesiedu.com/upload/8/ca/8cafebfe3a9bfd4f0360c1c626e56370.jpg)
感觉不难,就是那里卡住了,/>
![高数微分证明题感觉不难,就是那里卡住了,/>](/uploads/image/z/8536606-70-6.jpg?t=%E9%AB%98%E6%95%B0%E5%BE%AE%E5%88%86%E8%AF%81%E6%98%8E%E9%A2%98%E6%84%9F%E8%A7%89%E4%B8%8D%E9%9A%BE%2C%E5%B0%B1%E6%98%AF%E9%82%A3%E9%87%8C%E5%8D%A1%E4%BD%8F%E4%BA%86%2C%2F%3E)
F(x) = (x-1)^2f(x)
F(1)=0
F(2) = f(2)=0
F'(x) = (x-1)^2 f'(x) + 2(x-1)f(x)
F'(1) = 0
expands F(x) about 1
F(x) = F(1)+ F'(1)(x-1) +F''(a)(x-1)^2/2!
= F''(a)(x-1)^2/2!
put x=2
F(2) = F''(a)/2 =0
F''(a)=0
再问: F(x) = F(1)+ F'(1)(x-1) +F''(a)(x-1)^2/2! = F''(a)(x-1)^2/2! 没看懂,哪儿出来的,f(x)去哪儿了
再答: 这是Taylor expansion(泰勒展开) of F(x) about x=1 F(x) 在【0,1】上有二阶导数 =>存在ξ∈(1,2) F(x) = F(1)+ F'(1)(x-1)/1! +F''(ξ)(x-1)^2/2! 代入 x=2 F(2) = F(1)+ F'(1) +F''(ξ)/2 =0 =>F''(ξ)=0
F(1)=0
F(2) = f(2)=0
F'(x) = (x-1)^2 f'(x) + 2(x-1)f(x)
F'(1) = 0
expands F(x) about 1
F(x) = F(1)+ F'(1)(x-1) +F''(a)(x-1)^2/2!
= F''(a)(x-1)^2/2!
put x=2
F(2) = F''(a)/2 =0
F''(a)=0
再问: F(x) = F(1)+ F'(1)(x-1) +F''(a)(x-1)^2/2! = F''(a)(x-1)^2/2! 没看懂,哪儿出来的,f(x)去哪儿了
再答: 这是Taylor expansion(泰勒展开) of F(x) about x=1 F(x) 在【0,1】上有二阶导数 =>存在ξ∈(1,2) F(x) = F(1)+ F'(1)(x-1)/1! +F''(ξ)(x-1)^2/2! 代入 x=2 F(2) = F(1)+ F'(1) +F''(ξ)/2 =0 =>F''(ξ)=0