Each group should professionally write up the answers to
this lab and hand it in.
The purpose of the lab is
- to use technology to visualize surfaces in space;
- to understand how surfaces are built by one-parameter families
of one-dimensional curves;
- to compute tangent vectors to a surface in the
direction of a coordinate axis, and understand the geometric meaning
of these vectors;
- to discover the ``tangent plane'' to a surface as the span of
linearly-independent tangent vectors.
Log in to your account and launch Maple. The first command
you should type into Maple is
which will enable you to use a host of Maple plotting commands.
In this lab, we will exclusively study the
graph of the function
To make your life easier, define this expression in Maple
F := -x^4 + 2*x^2 - y^4 + 2*y^2 + x*y - 1;
and visualize the graph of this function near the origin by
plot3d( F, x=-1.6..1.6, y=-1.6..1.6);
Use the menu-bar on the plot window to add ``Boxed'' axes to your
graph. Also, color-code the function by its height and display the
surface as a ``Patch and Contour'' plot.
- If we intersect the graph of F with the vertical plane
y=1, we will get a function of one variable. You can think of this
function as g(x) = F(x,1).
- Explicitly write out this single-variable function.
- Plot or sketch the graph of g.
How does the graph of this one-variable function relate to the graph of F?
- What are the critical points of g? How do they
relate to geometric features on the graph of F?
- Parametrize the tangent line to g at the point x=1/2. Hint: The line
passes through the point (1/2, g(1/2)). [Note: it is tempting to
represent the tangent line as the graph of a function, but we want the
tangent line to be parametrized.]
- Test your computation by using Maple's plot command to
represent the tangent line and the graph of g on
the same plot. (Hint: recall that you can think of the graph of
g as a parametrized image by considering . Look on
your Maple summary sheet to see how to display two parametrized curves on
the same plot.) Print out this Maple plot and include it in your lab report.
- How does the derivative of g
relate to a partial derivative of F? Using this connection,
find a tangent vector to the graph of F in the
x-direction at the point (1/2,1,F(1/2,1)).
- Define h(y)=F(1/2,y) to be the single-variable function
obtained by restricting the domain of F to the line x=1/2?
Answer the previous questions for h. In particular
- Determine the relationship between the graph of h and the
graph of F.
- Parametrize the tangent line to h at the point y=1.
- Plot the graph of h and its tangent line at (1,h(1)).
- Find a tangent vector to the graph of F in the
y-direction at the point (1/2,1,F(1/2,1)).
- We now have a point, (1/2,1,F(1/2,1)), on the surface and two
vectors that are tangent to the surface. If the tangent vectors are linearly
independent, then they span a plane. We will call such a plane a
- Verify that one tangent vectors is not a multiple of the other,
and therefore conclude that the tangent vectors are linearly independent.
- The tangent plane is therefore the span of these two
vectors. If and are the two tangent vectors, then the
tangent plane is parametrized by
Write out each component of the function that parametrizes the
tangent plane and verify your answer by using Maple's
plot3d command to simultaneously plot the tangent plane and the
graph of F. (Hint: parametrize the graph of F as
Submit this plot with your lab writeup.
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Copyright: 1996 by the Regents of the University of Minnesota.
Department of Mathematics. All rights reserved.
Last modified: Oct 23 1996
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