Question

微积分题，求解答

一共5道，虽然看不懂，但是感觉会的人，应该不难。

Answer 1

Answer 2

(1)
y=xe^(-x^2)
y'=(1 -2x^2).e^(-x^2)
y'=0
(1 -2x^2).e^(-x^2) =0
x= √2/2 or -√2/2
y'|x=√2/2+ <0 , y'|x=√2/2- >0
x=√2/2 ( max)
y'|x=-√2/2+ >0 , y'|x=-√2/2- <0
x=-√2/2 ( min)
y=xe^(-x^2)
max y = y(√2/2) = (√2/2)e^(-1/2)
min y = y(-√2/2) = (-√2/2)e^(-1/2)
(2)
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
=1
=f'(0-)
ie
f'(x)
=(1-x) e^(x) ; x<0
=1 ; x=0
=cosx. cos(sinx) ; x>0
(3)
e^(x+y) +y=x^2
(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) ]
(4)
lim(x->0)∫(0->x) (1-cost) dt / x^3 (0/0 分子分母分别求导)
=lim(x->0) (1-cosx)/ (3x^2)
=lim(x->0) (1/2)x^2/ (3x^2)
=1/6
(5)
lim(x->0)∫(0->x) (e^(t^2)-1) dt / x^3 (0/0 分子分母分别求导)
=lim(x->0) (e^(x^2)-1) / (3x^2)
=lim(x->0) x^2 / (3x^2)
=1/3

Answer 3

😁

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