Preview Activity 11.9.1.
In this activity, we will use a function corresponding to elevation (as a function of 2D location) to explore what properties a point corresponding to a local maximum or local minimum (of a function of two variables) must have. Similar to Activity 11.7.5, 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.
We also brought a level that we can place on the ground at our feet to measure how steep the change in elevation will be in a particular direction. In other words, the level can measure the directional derivative at our current location. Conceptually, we want to understand how the measurements we can take with our level will allow us to identify when we have reached the top of the mountain.
(a)
In Activity 11.7.5, we saw how we can move toward the top of the mountain by taking steps in the “uphill” direction (the direction of greatest rate of increase for elevation). We want to state some conditions for our measurement using the level that will determine if we have reached the top of the mountain.
Suppose you are actually standing at the top of the mountain (which will correspond to a local maximum for the elevation) and you set the level at your feet in the East direction. Will the level measure that the elevation is increasing, decreasing, or constant elevation? Write a couple of sentences to justify your answer.
(b)
If you are at the top of the mountain and set the level at your feed in the South direction, will the level measure that the elevation is increasing, decreasing, or constant elevation? Write a couple of sentences to justify your answer.
(c)
Write a few sentences to explain why your level MUST show a zero rate of change in every direction at the top of the mountain.
(d)
If you took a step to the West from the top of the mountain, should your level measurement in the West direction at your new location be increasing, decreasing, or constant elevation? Explain your reasoning.
(e)
If you took at step in any direction from the top of the mountain, should your level measurement in the direction you took the step at your new location be increasing, decreasing, or constant elevation? Explain your reasoning.
(f)
If you were at the lowest point in the foggy park, write a few sentences to explain why your level MUST show a zero rate of change in every direction.
(g)
If you took at step in any direction from the lowest point in the park, should your level measurement in the forward direction (the direction in which you took the step) at your new location be increasing, decreasing, or constant elevation? Explain your reasoning.
(h)
Is it possible on your hike in the fog that you find a point where your level shows constant elevation in every direction but you are NOT at the top of the mountain or lowest point on the mountain? Explain what the the terrain would look like at this kind of location.
