When you are hiking on a mountain or a slope you have a choice of many directions in which you can go. Starting at the same point some directions head generally upward; some directions head generally downward; and some directions are steeper than others. In this module we are interested in the directional derivative of a function
-- that is, the slope of the surface described by this function as we go in different directions starting from the same point. As an example consider the function
shown in the graph below.
Suppose that we start at the point (0, 0) and go one unit in several different directions. The graph below shows four different directions marked by curves starting at the origin and it also shows all the points we would reach if we tried all possible directions and walked one unit. When we say "we walk one unit," we mean one unit in the xy-direction. Thus, we walk from (0, 0) to the point (cos t, sin t) where t is any angle.
The TI-92 screen below shows another way of studying what happens as we walk one unit in various different directions. It shows two graphs -- one showing what happens when we start at the origin and walk one unit in the northeast direction and one showing what happens when we walk one unit in the southeast direction.
The two screens below show how these two graphs were reproduced. First we define two functions --the first function is just the function f(x, y) = x y that we have been studying. The second function describes a walk in the direction of the angle t as the time parameter x goes from 0 to 1. Notice at time x we evaluate the function f at the point (x cos t, x sin t). The notation is a bit awkward because we need to use the variable x for the time parameter as we walk along because the TI-92 expects this variable on its Y= screen.
The function Y1 shows what happens if we walk in the southeast direction; the function Y2 shows what happens if we walk in the northeast direction. Notice that northeast is the direction indicated by the angle Pi/4 and that southeast is the direction indicated by the angle -Pi/4.
Use your CAS window to investigate what happens if you walk one unit in each of the indicated directions along the given surface starting at the given point. Notice that we indicate the direction by a unit vector pointing in the desired direction.
When you're hiking on a mountain one of the most important questions is -- which way is up? or, more preciesly, in which direction should you walk to climb upward as steeply as possible?
Use your CAS window and any other tools you choose, to answer the following questions.
which direction is the steepest uphill direction?
which direction is the steepest downhill direction?
Does it make any difference where you are standing?
where a and b are constants, which direction is the steepest uphill direction?
which direction is the steepest downhill direction?
Does it make any difference where you are standing?
which direction is the steepest uphill direction?
which direction is the steepest downhill direction?
Does it make any difference where you are standing?
After you've answered the questions above, click on the button below to continue.
