Directional Derivatives and the Gradient

We will look algebraically calculating and explaining the vector properties of the gradient using the function $f(x,y)=xy-2y^2+\frac{x^2}{2}+2y-x$ at the input $(1,1)$.

1. Calculate the gradient of $f$

2. Calculate $\nabla f (1,1)$.

3. Calculate $D_{\vec{u}} f (1,1)$ for each of the following vectors:

   1. $\vec{u}=\langle -1,0\rangle$
   2. $\vec{u}=\langle \frac{3}{5},-\frac{4}{5} \rangle$
   3. $\vec{u}=\langle -\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}} \rangle$

4. What direction will correspond to the greatest rate of increase for $f$ at the input $(1,1)$? Find the value of the directional derivative in this direction.

This activity considers directional derivatives and gradients for a function that measures elevation. Suppose you are hiking in a foggy park and can only see a few feet in front of you. There is nothing blocking you from walking in any particular direction, but because of the fog you cannot see where the highest point on the mountain is. You want to try to find the top of the mountain, but you don…

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 $(3,4)$. I…

Find all directions in which the directional derivative of $f(x,y) = ye^{-xy}$ is 1 at the point $(0,2)$.

Find, if possible, a function $f$ such that . If not possible, explain why.