(2.10)
F(0) = f(0)(1+|sin0)) = f(0)
F'(0+)
=lim(h->0+) [F(h)-F(0)]/h
=lim(h->0+) [f(h)(1+|sinh|) - f(0) ]/h
=lim(h->0+) [f(h)(1+sinh) - f(0) ]/h (0/0 分子分母分别求导)
=lim(h->0+) [ f(h) . cosh + f'(h)(1+sinh) ]
=f(0) + f'(0)
F'(0-)
=lim(h->0-) [F(h)-F(0)]/h
=lim(h->0-) [f(h)(1+|sinh|) - f(0) ]/h
=lim(h->0-) [f(h)(1-sinh) - f(0) ]/h (0/0 分子分母分别求导)
=lim(h->0-) [ -f(h) . cosh + f'(h)(1-sinh) ]
=-f(0) + f'(0)
F'(0+)= F'(0-)
f(0) + f'(0) = -f(0) + f'(0)
=> f(0) =0
ans : A
(2.11)
f(0)=0, f'(0) ≠0
F(x)=∫(0->x) (x^2-t^2) f(t) dt
x->0
F'(x) 与 x^k 是同价无穷小: k =?
solution:
F(x)
=∫(0->x) (x^2-t^2) f(t) dt
=x^2.∫(0->x) f(t) dt - ∫(0->x) t^2.f(t) dt
F'(x)
=2x.∫(0->x) f(t) dt + x^2.f(x) - x^2.f(x)
=2x.∫(0->x) f(t) dt
F''(x) =2xf(x)
F'''(x) = 2xf'(x) + 2f(x)
F'(x) 与 x^k 是同价无穷小
=> F'''(x) 与 k(k-1)x^(k-2) 是同价无穷小
=> 2xf'(x) + 2f(x) 与 k(k-1)x^(k-2) 是同价无穷小
=> 2xf'(x) 与 k(k-1)x^(k-2) 是同价无穷小
k-2 =1
k= 3
ans : C
