数学微积分的题目求解答，过程要详细，答案要正确，好快的加分！

Question

如题所述

Answer 1

1
z=xy x+y=1
z=x(1-x)=-x^2+x=-(x-1/2)^2+1/4
z'x=-2x+1
x=1/2,z'x=0, z最大=1/4

2
∫D∫(x^2+y^2)dxdy x=ρcosθ,y=ρsinθ x^2+y^2=ρ^2 dxdy=(1/2)(dρ)(θdθ)
D x^2+y^2=1,ρ=1 x^2+y^2=4 ρ=2 0<=θ<=π/2
∫∫(x^2+y^2)dxdy=∫[1,2]ρ^2dρ∫[0,π/2] (1/2)θdθ=∫[1,2](π^2/4)ρ^2dρ=(7π^2/12)

3
(cosnπ/3)^2 =(1/2)[1+cos(2nπ/3)]
-1/2<=cos2nπ/3<=1 周期为3函数，不收敛
级数(cosnπ/3)^2不收敛

4
(x^2-3y^2)dx +3xdy=0
如果是(x^2-3y)dx+3xdy=0
3xdy-3ydx= -x^2dx
3dy/x-3ydx/x^2=-dx
d(3y/x)=-dx
通解3y/x=-x+C
5
xy=xf(z)+yg(z)
x(y-f(z))=yg(z)
(y-f(z))-xf'x=yg'x
y(x-g(z))=xf(z)
x-g(z)-yg'y=xf'y
(y-f(z))=xdf/dx)+ydg/dx
(y-f(z))dx/dz=xdf/dz+ydg/dz
(x-g(z))=ydg/dy+xdf/dy
(x-g(z))dy/dz=ydg/dz+xdf/dz

[y-f(z)] dx/dz=[x-g(z)]dy/dz

[x-g(z)] (dz/dx) =[y-f(z)] (dz/dy)