Java Applet Worksheet
(By Dr. Steve Harris)
link: Tangent Planes
button: Tangent Lines and Planes
Spread the graphs out where you can see them all, as well as the control panel.
Use the Tangent Planes applet to examine the question, does the tangent plane at (a, b) intersect the surface only at the point (a, b), or also at other points nearby? In other words: Is the tangent plane on only one side of the surface (near to (a, b), anyway), or is it on both sides?
Turn on the Tangent plane button in the control panel; you might also want to turn on the x-tan line and y-tan line buttons.
1) For the function the program starts with, is the tangent plane on only one side of the surface, or on both sides? You can manipulate the point of tangency with the Òhot point" on the Domain window, and you can rotate the picture in the Function Graph window.
2) Change to the surface z = x2 + y2 (don't forget to hit "return" in the f(x, y) panel), and answer the same question.
Change to the surface z = x2 + y3. For which
values of the point of tangency is the tangent plane on only one side
surface (at least, nearby the point of tangency), and for which values
point of tangency is the tangent plane on both sides of the graph (even
the point of tangency)? You can read off the coordinates of the point
tangency in the C1 and C2 panels.