Considere a função f(x, y)=3 x^2-5 y^2+3 x y-2 x+9 y-1+cos (x)-sen(y). Calcule:
(a) partial fpartial x(x, y) e partial fpartial y(x, y)
(b) partial fpartial x(0,0) e partial fpartial y(0,0)
(c) partial^2 fpartial x^2(x, y), partial^2 fpartial y^2(x, y), partial^2 fpartial x partial y(x, y) e partial^2 fpartial y partial x(x, y)
(d) partial^2 fpartial x^2(0,0), partial^2 fpartial y^2(0,0), partial^2 fpartial x partial y(0,0) e partial^2 fpartial y partial x(0,0)

Passo 1

O enunciado dá a função
f(x,y)=3x^2-5y^2+3xy-2x+9y-1+cos(x)-sin(y)
e pede (a) as derivadas parciais de primeira ordem, (b) esses valores em (0,0), (c) as derivadas parciais de segunda ordem, e (d) esses valores em (0,0). Vamos usar regras básicas de diferenciação termo a termo.

Passo 2

Para encontrar partial fpartial x e partial fpartial y, derivamos cada termo de f em relação à variável indicada:
partial fpartial x = partialpartial xl(3x^2r) + partialpartial xl(3xyr) + partialpartial x(-2x) + partialpartial xl(cos(x)r)
=6x +3y -2 -sin(x),
partial fpartial y = partialpartial yl(-5y^2r) + partialpartial yl(3xyr) + partialpartial y(9y) + partialpartial yl(-sin(y)r)
=-10y +3x +9 -cos(y)

Passo 3

Agora avaliamos em (0,0):
partial fpartial x(0,0) =6cdot0+3cdot0-2-sin(0)=-2,
partial fpartial y(0,0) =-10cdot0+3cdot0+9-cos(0)=9-1=8

Passo 4

Para as derivadas de segunda ordem, derivamos novamente:
partial^2fpartial x^2 = partialpartial xl(6x+3y-2-sin(x)r) =6 -cos(x),
partial^2fpartial y^2 = partialpartial yl(-10y+3x+9-cos(y)r) =-10 +sin(y),
partial^2fpartial xpartial y = partialpartial yl(6x+3y-2-sin(x)r) =3,
partial^2fpartial ypartial x = partialpartial xl(-10y+3x+9-cos(y)r) =3

Passo 5

Finalmente, avaliamos essas segundas derivadas em (0,0):
partial^2fpartial x^2(0,0) =6-cos(0)=6-1=5,
partial^2fpartial y^2(0,0) =-10+sin(0)=-10+0=-10,
partial^2fpartial xpartial y(0,0)=3, partial^2fpartial ypartial x(0,0)=3

Resposta

(a) partial fpartial x(x,y)=6x+3y-2-sin(x), partial fpartial y(x,y)=-10y+3x+9-cos(y).
(b) partial fpartial x(0,0)=-2, partial fpartial y(0,0)=8.
(c) partial^2fpartial x^2(x,y)=6-cos(x), partial^2fpartial y^2(x,y)=-10+sin(y), partial^2fpartial xpartial y(x,y)=3, partial^2fpartial ypartial x(x,y)=3.
(d) partial^2fpartial x^2(0,0)=5, partial^2fpartial y^2(0,0)=-10, partial^2fpartial xpartial y(0,0)=3, partial^2fpartial ypartial x(0,0)=3.

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