The gradient of $f ( x , y , z ) = x ^ { 2 } y + y ^ { 2 } z + z ^ { 2 } x$ at (1, 2, -1) is

A) $- 5 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$
B) $5 \mathbf { i } - 3 \mathbf { j } - 2 \mathbf { k }$
C) $5 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$
D) $- 5 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }$
E) $5 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }$

Question

The gradient of $f ( x , y , z ) = x ^ { 2 } y + y ^ { 2 } z + z ^ { 2 } x$ at (1, 2, -1) is

A) $- 5 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$
B) $5 \mathbf { i } - 3 \mathbf { j } - 2 \mathbf { k }$
C) $5 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$
D) $- 5 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }$
E) $5 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }$

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 $f(x, y, z) = x^2y + y^2z + z^2x$, we have $\frac{\partial f}{\partial x} = 2xy + z^2$, $\frac{\partial f}{\partial y} = x^2 + 2yz$, and $\frac{\partial f}{\partial z} = y^2 + 2zx$. Substituting $x=1$, $y=2$, and $z=-1$ gives $\nabla f = (2*1*2 + (-1)^2)\mathbf{i} + (1^2 + 2*2*(-1))\mathbf{j} + (2^2 + 2*1*(-1))\mathbf{k} = 5\mathbf{i} - 3\mathbf{j} + 2\mathbf{k}$.

The Gradient Of f(x,y,z)=x2y+y2z+z2xf ( x , y , z ) = x ^ { 2 } y + y ^ { 2 } z + z ^ { 2 } xf(x,y,z)=x2y+y2z+z2x

The Gradient Of $f ( x , y , z ) = x ^ { 2 } y + y ^ { 2 } z + z ^ { 2 } x$