高等微积分第二册真题及答案

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 max =1/4。

2

∫D∫(x^2+y^2)dxdy x=ρcosθ，y =ρsinθx^2+y^2=ρ^2 dxdy =(1/2)(dρ)(θdθ)

x^2+y^2=1,ρ=1 x^2+y^2=4ρ= 20 & lt；=θ& lt；=π/2

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

三

(cosnπ/3)^2 =(1/2)[1+cos(2nπ/3)]

-1/2 & lt；= cos2nπ/3 & lt；=1是周期为3的函数，不收敛。

级数(Cosn π/3) 2不收敛。

四

(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

五

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)