To access the cross sections applet (based
on work of Tom Banchoff) follow the link:
http://www.slu.edu/classes/maymk/banchoff/CrossSection.html
Spread the four graphs out
where you can see them all, as well as the control panel.
We want to start with a simple function whose
graph we understand and use it to see how cross sections can be used to
build the surface which is the graph of a function in two
variables. The applet opens with the graph of the function f(x,y)
= x^2-y^2, a simple saddle where cross sections with y fixed are
parabolas opening up and cross sections with x fixed are parabolas
opening down.
1) Use the opening example, f(x,y) = x^2-y^2, to
see how a graph can be
constructed.
First check the x-slice box and see how the intersection of the surface
with the plane gives a curve.
Use the y0 control buttons to compare this curve with curves of
parallel slices.
Check the x-wireframe box to show the collection of slice curves.
Repeat the same procedure with the role of x and y reversed.
Hide the surface and note how the wireframe gives a good approximation
of the graph.
2) Now move to a less familiar example and repeat
the process.
Hide the surface and replace the function with f(x, y) =x^2 + y^2 -
x^2*y^2.
Look at the collection of x-slices. What values of y0 give
particularly interesting x-slices?
What happens to the equation of the x-slices for those y0 values?
Based on the x-slices, try to describe the surface.
Repeat the process with the role of x and y
reversed.
Show the surface and how good a description of the surface you made
from the cross sections.
The process above is helped if we can describe the
curves produced by
fixing x or y.
3) From your knowledge of graphs of functions of a
single variable describe the slice curves of f(x,y) = x^2/2 + y^3/3 -
x/2 - y/3 + 1.
Graph the surface to check your work.
4) From your knowledge of graphs of functions of a
single variable
describe the slice curves of f(x,y) = x^2/2 + y^2/2 + x*y^2.
Graph the surface to check your work.
Return to the CrossSection Applet page.
Last updated By Mike May, S.J.,
January 17, 2006.