Tangent Planes


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 your WWW browser doesn't seem to work with this version (or loading is too slow), try the Non-Tabled 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.


Example 1

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. 693K

Example 2

z = (sin x)(sin y) looks like an egg carton. Click on the image to zoom in and watch the bumps flatten.

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Example 3

f(x,y) = (x^2)(y^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.


But Wait...
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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. 741k

Note:

You will see that a function need only be slightly "crinkled" at a point to lack a tangent plane there.

Example 4

f(x,y) = x^2 y/(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.)

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However, 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 (0,0.1). Even though it is very close to (0,0), it will flatten out. 368K
For an explanation of all of these wonders, it is necessary to turn to calculus ...

E. Klotz and E. Magness
The QuickTime movies were made using software by Jeremy Dilatush.
14 September 1995