Find the gradient of the function at the given point. $$w = 4 x ^ { 2 } y - 3 y z + z ^ { 2 } , \quad ( 1,1 , - 2 )$$
A) $8 \hat { \mathbf { i } } + 10 \hat { \mathbf { j } } - 5 \hat { \mathbf { k } }$
B) $8 \hat { \mathbf { i } } + 10 \hat { \mathbf { j } } - 7 \hat { \mathbf { k } }$
C) $4 \hat { \mathbf { i } } + 10 \hat { \mathbf { j } } - 7 \hat { \mathbf { k } }$
D) $4 \hat { \mathbf { i } } + 16 \hat { \mathbf { j } } - 5 \hat { \mathbf { k } }$
E) $12 \hat { \mathbf { i } } + 13 \hat { \mathbf { j } } - 7 \hat { \mathbf { k } }$

Question

Quizplus · Accepted Answer

The gradient of the function $ w = 4x^2y - 3yz + z^2 $ is calculated by finding the partial derivatives with respect to $ x $, $ y $, and $ z $. At the point $ (1, 1, -2) $, the partial derivatives are:  $\frac{\partial w}{\partial x} = 8xy$, $\frac{\partial w}{\partial y} = 4x^2 - 3z$, $\frac{\partial w}{\partial z} = -3y + 2z$.  Substituting the point into these derivatives gives:  $\frac{\partial w}{\partial x} = 8(1)(1) = 8$,  $\frac{\partial w}{\partial y} = 4(1)^2 - 3(-2) = 4 + 6 = 10$,  $\frac{\partial w}{\partial z} = -3(1) + 2(-2) = -3 - 4 = -7$.  Thus, the gradient is $8 \hat{\mathbf{i}} + 10 \hat{\mathbf{j}} - 7 \hat{\mathbf{k}}$.

Find the Gradient of the Function at the Given Point w=4x2y−3yz+z2,(1,1,−2)w = 4 x ^ { 2 } y - 3 y z + z ^ { 2 } , \quad ( 1,1 , - 2 )w=4x2y−3yz+z2,(1,1,−2)

Find the Gradient of the Function at the Given Point $w = 4 x ^ { 2 } y - 3 y z + z ^ { 2 } , \quad ( 1,1 , - 2 )$