3.8 Directional Derivatives and the Gradient (continued)

Vector Properties of the Gradient

Applications

Write a few sentences about why the gradient at $P$ goes in the southeast direction and not the northwest (which is also perpendicular to the thermocline/ level curve at $P$).

Draw a vector in each of the following directions starting at $P$ and state whether the directional derivative will be positive/negative/0 in each of these directions. You should write a couple of sentences for each direction vector about how the sign of the directional derivative is related to the angle between the direction and the gradient vectors.

• West - South - Northeast

Write a few sentences about why any direction moving up and left will have negative rate of change for the temperature in the direction given.

Explain what direction you would need to move in to have the greatest positive rate of change for the temperature. Write a couple of sentences to relate your answer to the gradient using Equation.

Calculate the gradient of $f$

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

Calculate $D_{\vec{u}} f (1,1)$ for each of the following vectors: -$\vec{u}=\langle -1,0\rangle$-$\vec{u}=\langle \frac{3}{5},-\frac{4}{5} \rangle$-$\vec{u}=\langle -\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}} \rangle$

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.

Let $P_1$ be your current location in the foggy park. You use your compass to find the east and north directions. At $P_1$, you find that the ground rises 1 meter per 50 meters traveled to the east and the ground rises 2.5 meters per 50 meters traveled to the north.

Use this information to find $\nabla h (P_1)$.

Use your answer to the previous part to say what direction is “uphill” at $P_1$ and state the rate of elevation increase in this direction.

You decide to walk uphill from your location $P_1$ in order to try to find the top of the mountain. After walking in the same direction for a while, you are at a point $P_2$ and notice that you are no longer walking in the steepest direction. You again locate east and north and measure the steepness of the mountain in these directions. You find that the ground rises 1.5 meters per 75 meters traveled to the east and the ground goes down 0.5 meters per 100 meters traveled to the north.

Use this new information to calculate $\nabla h (P_2)$, find the uphill direction, and find how steep the mountain is in the uphill direction at $P_2$.

Suppose you continue this method of walking in the uphill direction for a while, finding the new uphill direction, and walking in the new uphill direction. Do you think you must eventually find the top of the mountain? How you will know that you have reached the top of the mountain? Remember that you can't see very far in front of you. Write a few sentences to explain your reasoning.

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.