Question

Obtain the gradient vector for the function 𝑓(𝑥,𝑦)=5𝑥²𝑦³, at the point (𝑥,𝑦)=(3,2)

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Obtenha o vetor gradiente para a função f(x,y) = 5x^2 y^3 , no ponto (x,y) = (3,2)

[Solução]

1. Encontre a derivada parcial def(x,y) em relação a x :

\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} (5x^2 y^3) = 10x y^3

2. Encontre a derivada parcial def(x,y) em relação a y :

\frac{\partial f}{\partial y} = \frac{\partial}{\partial y} (5x^2 y^3) = 15x^2 y^2

3. Avalie as derivadas parciais no ponto(3,2) :

\frac{\partial f}{\partial x}(3,2) = 10 \cdot 3 \cdot 2^3 = 10 \cdot 3 \cdot 8 = 240

\frac{\partial f}{\partial y}(3,2) = 15 \cdot 3^2 \cdot 2^2 = 15 \cdot 9 \cdot 4 = 540

4. Assim, o vetor gradiente no ponto(3,2) é:

\nabla f(3,2) = (240, 540)

[Solução]

1. Encontre a derivada parcial de

2. Encontre a derivada parcial de

3. Avalie as derivadas parciais no ponto

4. Assim, o vetor gradiente no ponto

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