Preview Activity 11.9.1.
In this activity, we will use a function that takes a two-dimensional location as its input and outputs the elevation of a surface at that point 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 11.7.4, imagine that you are hiking in a foggy park and cannot see anything more than 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’t have a map to consult or trail to follow or even a line of sight to other landmarks. You do have a compass that works in the fog, which you can use to determine directions. You also have a level that can be placed on the ground at your feet to measure how steep the change in elevation is in a particular direction. In other words, the level can measure the directional derivative at a location. Conceptually, our goal in this preview activity is to understand how the measurements you can take with your level allows you to identify when you have reached the top of the mountain.
(a)
In Activity 11.7.4, you saw how you can move toward the top of the mountain by taking steps in the “uphill” direction, which is the direction of greatest rate of increase for elevation. Suppose you are standing at the top of the mountain, which corresponds to a local maximum for elevation, and you set the level at your feet while facing east. Will the level measure that the elevation is increasing, decreasing, or constant? 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 while facing south, will the level measure that the elevation is increasing, decreasing, or constant? 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 small step to the west from the top of the mountain, should your level measurement to the west at your new location indicate that the elevation is increasing, decreasing, or constant? Explain your reasoning.
(e)
If you took a small 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 indicate that the elevation is increasing, decreasing, or constant? Explain your reasoning.
(f)
Now imagine that you are at the lowest point in the foggy park. Write a couple of 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 at your new location in the the direction in which you took the step indicate that the elevation is increasing, decreasing, or constant? 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 type of location.
