Java Applet
Worksheet

Tangent
Planes

(By Dr. Steve Harris)

## To access Professor May's applets (based on work of Tom Banchoff):

http://www.slu.edu/classes/maymk/MathApplets-SLU.html

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 = x**^{2} + y^{2} (don't forget to hit
"return"
in the **f(x, y)** panel), and answer the
same question.

3)
Change to the surface **z = x**^{2} + y^{3}. For which
values of the point of tangency is the tangent plane on only one side
of the
surface (at least, nearby the point of tangency), and for which values
of the
point of tangency is the tangent plane on both sides of the graph (even
nearby
the point of tangency)? You can read off the coordinates of the point
of
tangency in the C_{1} and C_{2} panels.

Return to the Tangent Planes Applet page.

Last updated By Mike May, S.J.,
October 14, 2004.