Question

O gradiente da função f(x,y,z)=3x2+2y2+z2 no ponto P(1;2;3) é:

a) (6;4;2)
b) (1;2;3)
c) (3;2;1)
d) (2;4;6)
e) (6;8;6)

Answer 1

$\displaystyle \vec{\nabla}f(x,y,z)=\frac{\partial f}{\partial x}\hat{i}+\frac{\partial f}{\partial y}\hat{j}+\frac{\partial f}{\partial z}\hat{k}\\\\\\ \vec{\nabla}f=\frac{\partial}{\partial x}(3x^2+2y^2+z^2)\hat{i}+\frac{\partial}{\partial y}(3x^2+2y^2+z^2)\hat{j}+\frac{\partial }{\partial z}(3x^2+2y^2+z^2)\hat{k}\\\\\\ \vec{\nabla}f(x,y,z)=(6x\,\hat{i}+4y\,\hat{j}+2z\,\hat{k})\\\\\\ \vec{\nabla}f(1,2,3)=(6(1)\hat{i}+4(2)\hat{j}+2(3)\hat{k})=\boxed{6\hat{i}+8\hat{j}+6\hat{k}}$

resposta, letra e)