Compute the gradient of the function at the given point.
-$f ( x , y , z ) = - 7 x - 7 y - 6 z , \quad ( 3 , - 8,5 )$
A) ${\nabla} f = - 7 \mathbf { i } - 7 \mathbf { j } - 6 \mathbf { k }$
B) ${\nabla} f = - 21 \mathbf { i } - 56 \mathbf { j } - 30 \mathbf { k }$
C) ${\nabla} f = \mathbf { i } - 8 \mathbf { j } + 5 \mathbf { k }$
D) ${\nabla} f = - 21 \mathbf { i } + 56 \mathbf { j } - 30 \mathbf { k }$

Question

Compute the gradient of the function at the given point.
-$f ( x , y , z ) = - 7 x - 7 y - 6 z , \quad ( 3 , - 8,5 )$
A) ${\nabla} f  = - 7 \mathbf { i } - 7 \mathbf { j } - 6 \mathbf { k }$
B) ${\nabla} f = - 21 \mathbf { i } - 56 \mathbf { j } - 30 \mathbf { k }$
C) ${\nabla} f  = \mathbf { i } - 8 \mathbf { j } + 5 \mathbf { k }$
D) ${\nabla} f = - 21 \mathbf { i } + 56 \mathbf { j } - 30 \mathbf { k }$

Quizplus · Accepted Answer

The gradient of a function $f(x, y, z)$ is given by $\nabla f = \frac{\partial f}{\partial x} \mathbf{i} + \frac{\partial f}{\partial y} \mathbf{j} + \frac{\partial f}{\partial z} \mathbf{k}$. For $f(x, y, z) = -7x - 7y - 6z$, the partial derivatives are constant: $\frac{\partial f}{\partial x} = -7$, $\frac{\partial f}{\partial y} = -7$, and $\frac{\partial f}{\partial z} = -6$, making the gradient $-7\mathbf{i} - 7\mathbf{j} - 6\mathbf{k}$, independent of the point of evaluation.

Compute the Gradient of the Function at the Given Point f(x,y,z)=−7x−7y−6z,(3,−8,5)f ( x , y , z ) = - 7 x - 7 y - 6 z , \quad ( 3 , - 8,5 )f(x,y,z)=−7x−7y−6z,(3,−8,5)

Compute the Gradient of the Function at the Given Point $f ( x , y , z ) = - 7 x - 7 y - 6 z , \quad ( 3 , - 8,5 )$