Question

三道高等数学微积分题，急

Answer 1

(1)
f(x)
=xe^(-x) ; x<0
=sin(sinx) ; x≥0
f(0-)=lim(x->0) xe^(-x) = 0
f(0+)=f(0) =lim(x->0) sin(sinx) =0
x=0, f(x) 连续
f'(0+)
=lim(h->0) he^(-h)/h
=lim(h->0) e^(-h)
=1
f'(0-)
=lim(h->0) sin(sinh)/h
=lim(h->0) h/h
=1
=>
f'(0) =1
ie
f'(x)
=(x-1)e^(-x) ; x<0
=1 ; x=0
=cosx. cos(sinx) ; x>0
(2)
∫ln√x/√x dx
=2∫ln√x d√x
=2√x .ln√x -∫dx/√x
=2√x .ln√x -2√x +C
(3)
e^(x+y)+y = x^2
两边求导
(1+y')e^(x+y)+y' = 2x
[1+ e^(x+y) ]y' = 2x - e^(x+y)
y' = [2x - e^(x+y)]/[1+ e^(x+y) ]