Question

急！考试用！麻烦大家了！

Answer 1

15 lim(x->2)∫[2,x] f(t)dt/(1-4/x^2)
=lim(x->2)f(x)/(8/x^3) =f(2)

16 lim(x->0) xsin(1/x)
-1<=sin(1/x)<=1
lim(x->0)-x <= lim(x->0) xsin(1/x) <= lim(x->0)x
lim(x->0)xsin(1/x)=0
lim(x->0)(x+a)=0 a=0

17 f(a^(1/3)-1)=a-1
=(a^(1/3)-1)(a^(2/3)+a^(1/3)+1)
=(a^(1/3)-1)[(a^(1/3)-1)^2-3a^(1/3)+3-3]
=(a^(1/3)-1)^3-3(a^(1/3)-1)^2-3(a^(1/3)-1)
f(x)=x^3-3x^2-3x

1 S=∫[0,π/2] (2-cosx)dx= 2x+sinx|[0,π/2]=π+1
2 交于A(-1,1) B(2,2)
S=∫[-1,2] (x+2-x^2) dx=(1/2)x^2+2x-(1/3)x^3|[-1,2]=4/3+3/2+1/3=5/3+3/2=19/6

3 y'=e^x y'|x=0 =1 y=x+1
y=e^x和y=x+1交于A(0,1) x>0 e^x>x+1,无解

4 xy=1和x=1交于A(1,1)
和x=2交于B(2,1/2)
dS=2πydx=(2π/x)dx
V=∫[1,2] x*(2π/x)dx=3*2π=6π

1
y=ln(1+x)-(x-(1/2)x^2))
y'=(1/(1+x))-1+x=x^2/(1+x) x>0,y'>0
y>=0

2 y=e^x-(1+x)
y'=e^x-1 x>0,y'>0
y>=0

3 y=e^x-ex
y'=e^x-e x>1,y'>0
y>0