Activity 11.7.5.
In this activity, we will make sense of the directional derivatives and gradients in terms of a function that measures the elevation. We are hiking in a foggy park and cannot see anything more than a few feet in front of us. There is nothing blocking us from walking in any particular direction but because of the fog we cannot see where the highest point on the mountain is. We want to try to find the top of the mountain, but we don’t have a map or trail or any line of sight to other landmarks. Our compass still works in the fog, so we can tell what direction North/South/East/West are.
In order to use some of our calculations tools from multivariable calculus, we will think of the elevation at different locations in the park given by a function \(h(x,y)\) where \(x\) is your location in the East (positive \(x\))/West (negative \(x\)) direction and \(y\) is your location in the North (positive \(y\))/South (negative \(y\)) direction.
(a)
Let \(P_1\) be your current location in the foggy park. You use your compass to find the East and North directions. At \(P_1\text{,}\) you find that the ground rises 1 meter per 50 meters traveled in the East direction and the ground rises 2.5 meters per 50 meters traveled in the North direction.
Use this information to give \(\nabla h (P_1)\text{.}\)
(b)
Use your answer to the previous task to say what direction is “uphill” at \(P_1\) and state how steep the mountain is in this direction.
(c)
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 notice that you are no longer walking in the steepest direction. So at your new location, which we will call \(P_2\text{,}\) you find the East and North directions and measure the steepness of the mountain in these directions. You find that the ground rises 1.5 meters per 75 meters traveled in the East direction and the ground goes down 0.5 meters per 100 meters traveled in the North direction.
Use this new information to calculate \(\nabla h (P_2)\text{,}\) find the uphill direction, and give state how steep the mountain is in the uphill direction at \(P_2\text{.}\)
(d)
If we use this method of walking in the uphill direction for a ways and then finding the new uphill direction, do you think we will have to find the top of the mountain? You should write a few sentences to justify your ideas and be sure to state how you will know you are at the top of the mountain. Remember that you can’t see very far in front of you.
