1. Nájdite rovnicu dotykovej roviny ku grafu funkcie f(x, y) = 1 - (x^2 + y^2)^(1/3)  v bode T = [-2, 2, ?].

f[x_, y_] = 1 - (x^2 + y^2)^(1/3)

f[2, -2]

fx[x_, y_] = D[f[x, y], x]

fy[x_, y_] = D[f[x, y], y]

fd[x_, y_] = 
      f[2, -2] + fx[2, -2]*(x - 2) + 
        fy[2, -2]*(y + 2)

Plot3D[f[x, y], {x, -2, 2}, {y, -2, 2}]

Plot3D[fd[x, y], {x, -2, 2}, {y, -2, 2}]

Show[%, %%]

2. Nájdite rovnicu dotykovej roviny ku grafu funkcie g(x, y) = x^2 - y^2 v bode T = [0, 0, ?].

g[x_, y_] = x^2 - y^2

gx[x_, y_] = D[g[x, y], x]

gy[x_, y_] = D[g[x, y], y]

gd[x_, y_] = 
      g[0, 0] + gx[0, 0]*(x - 0) + gy[0, 0]*(y + 0)

Plot3D[g[x, y], {x, -2, 2}, {y, -2, 2}]

Plot3D[gd[x, y], {x, -2, 2}, {y, -2, 2}]

Show[%, %%]

3. Nájdite rovnicu dotykovej roviny ku grafu funkcie h(x, y) = x^2 + 2y^2 v bode lokálneho extrému.

h[x_, y_] = x^2 + 2*y^2

hx[x_, y_] = D[h[x, y], x]

hy[x_, y_] = D[h[x, y], y]

Solve[{hx[x, y] == 0, hy[x, y] == 0}, {x, y}]

hxx[x_, y_] = D[h[x, y], x, x]

hxy[x_, y_] = D[h[x, y], x, y]

hyy[x_, y_] = D[h[x, y], y, y]

H = {{2, 0}, {0, 4}}

MatrixForm[%]

hes = Det[H]

hd[x_, y_] = 
      h[0, 0] + hx[0, 0]*(x - 0) + hy[0, 0]*(y + 0)

Plot3D[h[x, y], {x, -3, 3}, {y, -3, 3}]

Plot3D[hd[x, y], {x, -3, 3}, {y, -3, 3}]

Show[%, %%]

4. Nájdite rovnicu dotykovej roviny ku grafu funkcie k(x,y)=(x^2+2y^2)*e^(-x^2-y^2) v bodoch lokálnych extrémov.

k[x_, y_] = Exp[-x^2 - y^2]*(x^2 + 2*y^2)

kx[x_, y_] = D[k[x, y], x] // Simplify

ky[x_, y_] = D[k[x, y], y] // Simplify

Solve[{kx[x, y] == 0, ky[x, y] == 0}, {x, y}]

he1[x_, y_] = 
      Det[{{D[k[x, y], x, x], D[k[x, y], x, y]}, {D[k[x, y], y, x], 
              D[k[x, y], y, y]}}] // Simplify

he2[x_, y_] = D[k[x, y], x, x] // Simplify

he1[-1, 0]

he2[-1, 0]

Funkcia nemá v bode lokálny extrém.

he1[0, 0]

he2[0, 0]

Funkcia má v bode [0, 
        0]  lokálne minimum.

k[0, 0]

he1[1, 0]

he2[1, 0]

Funkcia nemá v bode lokálny extrém.

he1[0, -1]

he2[0, -1]

Funkcia má v bode [
        0, -1] lokálne maximum.

k[0, -1]

he1[0, 1]

he2[0, 1]

Funkcia má v bode [0, 
        1]  lokálne maximum.

k[0, 1]

Plot3D[k[x, y], {x, -2, 2}, {y, -2, 2}]

Plot3D[2/E, {x, -2, 2}, {y, -2, 2}]

Plot3D[0, {x, -2, 2}, {y, -2, 2}]

Show[%, %%, %%%, 
      ViewPoint -> {-2.273,  2.403,  0.713}]