Question

几道微积分题目！求过程！

题如图。谢谢~

Answer 1

1
lim(x->0) [(1+cosx)^x -2^x]/sinx^2
=lim(x->0)2^x[(cos(x/2)^2x-1]/(sinx)^2
=lim(x->0)2^x[(cos(x/2)^x/sinx)^2-1/(sinx)^2]
lim( x->0) ((cosx/2)^x/sinx)^2-1/sinx^2=0
=0

2 B
x<x^2 , x>1
x>x^2, x<1

3
=ln2
f(x)=2^x f'(x)=ln2*2^x f'(0)=ln2

4
y=a^x+√(1-a^(2x))arccosa^x
y'=lna*a^x + [-lnaa^(2x)/√(1-a^2x)]arccosa^x -lna*a^x
dy=y'dx

5
x^2+y^2=arctan(y/x)
2x+2yy'=(y'/x-y/x^2)/(1+y^2/x^2)
2x+y/(x^2+y^2)=y'(x/(x^2+y^2)-2y)
y'=[2x+y/(x^2+y^2)]/ [ x/(x^2+y^2)-2y ]

Answer 2

第2题 F（x）=x (0<x<=1) F(x)=x^2 (1<x<2)
F'(x)=1(0<x<1) F'(x)=2x (1<x<2) x=1时连续但不可导，所以x取不到1 选D