Question

高数大佬看一下这30题怎么做呀

Answer 1

(30)

f(0)=0, f'(0) 存在, F(x) =∫(0->x) t^4. f(x^5-t^5) dt

To find : lim(x->0) F(x)/x^10

solution:

let

u=x^5-t^5

du=-5t^4 dt

t=0, u=x^5

t=x, u=0

 F(x) 

=∫(0->x) t^4 .f(x^5-t^5) dt

=(-1/5)∫(x^5->0)  f(u) du

=(1/5)∫(0->x^5)  f(t) dt

lim(x->0) F(x)/x^10

=lim(x->0) (1/5)∫(0->x^5)  f(t) dt /x^10                       (0/0 分子分母分别求导)

=lim(x->0) x^4. f(x^5)  /(10x^9)

=lim(x->0) f(x^5)  /(10x^5)                             (0/0 分子分母分别求导)

=lim(x->0) 5x^4.f'(x^5)  /(50x^4) 

=(1/10) lim(x->0) f'(x^5)  

=(1/10)f'(0)

 

Answer 2

全手写，给个采纳吧，楼上有个哥们算错了哦

Answer 3