Exercise 24

Let $E(x,y) = \frac{100}{1+(x-5)^2 + 4(y-2.5)^2}$ represent the elevation on a land mass at location $(x,y)$. Suppose that $E$, $x$, and $y$ are all measured in meters.

1. Find $E_x(x,y)$ and $E_y(x,y)$.

2. Let $\vu$ be a unit vector in the direction of $\langle -4,3 \rangle$. Determine $D_{\vu} E(3,4)$. What is the practical meaning of $D_{\vu} E(3,4)$ and what are its units?

3. Find the direction of greatest increase in $E$ at the point $(3,4)$.

4. Find the instantaneous rate of change of $E$ in the direction of greatest decrease at the point $(3,4)$. Include units on your answer.

5. At the point $(3,4)$, find a direction $\vw$ in which the instantaneous rate of change of $E$ is 0.