This is an experiment in using the World Wide Web to illustrate the intuitive notion of the tangent plane to a graph of a function of two variables.
If you have access to a browser that supports tables, you may prefer to view the Netscape version.
If the graph of z = f(x,y) has a tangent plane at (a,b,f(a,b) )then the graph gets flatter and flatter (more "plane-like") as you get closer and closer to (a,b). This is why the earth looks flat to us as we stand on it.
(In studying several variable Calculus you encounter a definition of tangent plane which involves the derivative of the function.)
In this document you will have the opportunity to view a variety of different graphs and see how they do or don't become plane-like as you zoom into them.
The function z = 5/(1+x^2 + 6y^2) has a hill-shaped graph. Click on the image to see a QuickTime movie showing how the surface seems to flatten as you zoom closer and closer.
z = sin x * sin y looks like an egg-carton. Click on the image to zoom in and watch the bumps flatten.
f(x,y) = x^2y^2/(x^2 + y^2) isn't defined at (0,0), but if we define f(0,0)=0 we get a function that is not only continuous at (0,0), but also has a tangent plane there. Check it out visually with the QuickTime movie zoom.
You may recall from single variable calculus that f(x) = |x| lacks a tangent line at x=0. Well, its cousin f(x,y) = |x| lacks a tangent plane for any point of the form (0,y). If you click on the figure you'll get a look at z=f(x,y) from all sides. It's clear that if you zoom in on a point (0,y) the figure will look the same, and it won't flatten out a bit.
You will see that a function need only be slightly "crinkled" at a point to lack a tangent plane there.
f(x,y) = x^2y/(x^2 + y^2), f(0,0)=0 is reminiscent of Example 3, but if you click on the figure you'll see the most boring zoom of your life -- the function doesn't change; the crinkle remains and doesn't flatten at all. (This movie is an actual zoom of 100 power magnification -- I'll send you the code if you don't believe me.
if you zoom in on any point other than (0,0), then the graph of f(x,y) will flatten out and have a tangent plane. This movie, for example, shows a zoom-in on the point (0,.1). Even though it is very close to (0,0), it will flatten out.
For an explanation of all of these wonders, it is necessary to turn to calculus...