A Tangent Plane Applet

This applet is designed to build intuition about tangent planes to surfaces that are graphs of functions in two variables.
As with all major concepts in multi-variable calculus, this is done seeing how this is connected to the related concept in calculus of one variable.

• The applet has a control window and 3 graphics windows.
• There are windows for a y-z slice and an x-z slice, the two cross sections of the 3-D graph obtained by holding x=x0 and y=y0, respectively.  On the control panel there are check boxes to add the appropriate tangent line to each of these graphs.
• The third window gives a graph of the surface.  If the tangent lines are made visible on the slices, they will also be visible here.  An additional checkbox makes the tangent plane visible.  It should be noted that the tangent plane, if it exists, is the plane that contains the two tangent lines for the slice curves.

Things to notice:
• If you rotate the function graph so that you are looking at one of the tangent lines head on, the tangent plane seems to be a tangent line to the other slice curve.
• For the function we start with, the tangent plane is tangent to the surface in the sense of touching the surface at (x0, y0, f(x0, y0)), and at no other point in the neighborhood.
• The values of  (x0, y0) are controlled on the control panel.  See if you can find the point where both tangent lines are flat.  For the given function, this should be at a minimum value of the function.
• If we change (x0,y0) to (0, -1) the tangent plane cuts the surface in a path through the point.  Try to come up with a rule in terms of the x and y slices that determines if the tangent plane only touches the surface in a single point in a neighborhood of the hot spot.

A sample worksheet for this applet is available.   (Courtesy of Dr. Steve Harris.)

This applet was designed as by modifying an applet from a demo by Tom Banchoff at Brown University.  It is used with permission.   Go to the Banchoff Applet Help page.