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 máx =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; p>∫∫(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 es una función con período 3 y no converge.
La serie (Cosn π/3)2 no converge.
Cuatro
(x^2-3y^2)dx 3xdy=0
Si es (x 2-3y) dx 3xdy = 0.
3xdy-3ydx= -x^2dx
3dy/x-3ydx/x^2=-dx
d(3y/x)=-dx
Solución general 3y/x=-x C
Cinco
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)