I told you guys I would post an explanation of how to solve problem 7 in Section 14.3. I recommend pulling out your book, because I'm not going to re-draw the diagrams for the same reason I didn't draw them in class. If you'll recall, the problem asked you to identify the function f and its partial derivatives fx and fy based on the graphs.
The general strategy is to look at cross-sections by looking at the intersections of the surfaces with planes parallel to the xz-plane (planes with the form y=k)and planes parallel to the yz-plane (those with the form x=k).
First, let's fix x=3 and look in the y direction. Specifically if we fix x=3 that corresponds to the "front edge" of the surface as drawn. Notice there are two types of intersections: the graph in (a) looks one way, whereas (b) and (c) look the same.
Look at these two graphs and try to figure out which is the derivative of the other. You should figure out that the graph on the left is the derivative of the graph on the right. Specifically, since we've fixed x, this is the partial derivative with respect to y, fy. Thus (a) represents fy. Now we have to figure out which of (b) and (c) represents f and which represents fx.
Consider the cross section of (b) and (c) at y=-1.5. They look something like this:
Again, which looks like the derivative of the other? You should see that (b) is the derivative of (c). Thus since we're fixing y and looking in the x direction, this is the partial derivative with respect to x, fx. Hence we've figured out that (b) is fx, and (c) is f itself.