11.1 Problem 6

Could we go over contour plots in class? They were not in the notes and it
came up in the homework which confused me greatly.

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Sure, you only need to ask me in class.  But since the deadline on the HW is creeping up, let's address it a little bit now.  First of all, I recommend that you consult the textbook on this subject, you paid a lot of money for it and it covers this subject pretty well.    Second, we have used contour plots in class, though usually written on the whiteboard and as you point out not in the lecture notes.  Thirdly, this link takes you to a topo map of the downtown Tempe and ASU area, including Tempe Butte.  I recommend that you print the map and go climb A-Mountain and compare the features of the map to the topography of the butte.

The general notion is that each contour represents a trace of the graph of a function z=f(x,y) for some value of z, and the contour map is a whole family of contours for a variety of different values of z.  Usually the z-values are chosen to be evenly spaced, for example it looks like the contours in the topo map of Tempe are spaced every 20 feet in height.  Sometimes the contours are closer together,  this happens when the graph of the function steeper, and sometimes they are further apart, which happens when the graph is less steep.  In some regions where you don't see any contours at all the graph of the function is so flat that the height doesn't change enough to take you to the next contour. As a specific example, let's look at your contour plot b)

Notice that the contour lines are all straight lines and all parallel to the line y=x--this means that f(x,y)  depends only on the value of y-x.  The graph is pretty flat around the line y=x but on both sides the contours are getting closer and closer as you move away from y=x, hence the graph of the function is getting larger and steeper.  If you look at what function could do this, you focus in on b) (x-y)^2 which has the requisite properties.