$\text { The temperature at the point } ( x , y ) \text { on a metal plate is modeled by }$ $T ( x , y ) = 200 e ^ { - \left( x ^ { 2 } + y \right) / 2 } , x \geq 0 , y \geq 0$. Find the direction of greatest increase in heat from the point $( 4,15 )$.
A) The greatest increase is in the direction of the gradient $- 4 \mathbf { i } - \frac { 1 } { 2 } \mathbf { j }$.
B) The greatest increase is in the direction of the gradient $- 4 \mathbf { i } - \frac { 1 } { 4 } \mathbf { j }$.
C) The greatest increase is in the direction of the gradient $- 6 \mathbf { i } - \frac { 1 } { 2 } \mathbf { j }$.
D) The greatest increase is in the direction of the gradient $- 8 \mathbf { i } - \frac { 1 } { 2 } \mathbf { j }$.
E) The greatest increase is in the direction of the gradient $- 6 \mathbf { i } - \frac { 1 } { 4 } \mathbf { j }$.

Question

$\text { The temperature at the point } ( x , y ) \text { on a metal plate is modeled by }$ $T ( x , y ) = 200 e ^ { - \left( x ^ { 2 } + y \right) / 2 } , x \geq 0 , y \geq 0$. Find the direction of greatest increase in heat from the point $( 4,15 )$. 
A) The greatest increase is in the direction of the gradient $- 4 \mathbf { i } - \frac { 1 } { 2 } \mathbf { j }$.
B) The greatest increase is in the direction of the gradient $- 4 \mathbf { i } - \frac { 1 } { 4 } \mathbf { j }$.
C) The greatest increase is in the direction of the gradient $- 6 \mathbf { i } - \frac { 1 } { 2 } \mathbf { j }$.
D) The greatest increase is in the direction of the gradient $- 8 \mathbf { i } - \frac { 1 } { 2 } \mathbf { j }$.
E) The greatest increase is in the direction of the gradient $- 6 \mathbf { i } - \frac { 1 } { 4 } \mathbf { j }$.

Quizplus · Accepted Answer

The direction of greatest increase in heat is given by the gradient of $T(x, y)$. Calculating the partial derivatives, $\frac{\partial T}{\partial x} = -200x e^{-\frac{x^2 + y}{2}}$ and $\frac{\partial T}{\partial y} = -100 e^{-\frac{x^2 + y}{2}}$. At $(4,15)$, the gradient is $\nabla T = - 4 \mathbf{i} - \frac{1}{2} \mathbf{j}$.

The temperature at the point (x,y) on a metal plate is modeled by \text { The temperature at the point } ( x , y ) \text { on a metal plate is modeled by } The temperature at the point (x,y) on a metal plate is modeled by

$\text { The temperature at the point } ( x , y ) \text { on a metal plate is modeled by }$