quadric_think.mws as present
in the folder Maple 7/Math 222 /Lecture 1 on lab machines.
Surfaces 1
An introduction to three dimensional graphs. Lots of examples of
contours. Includes some cases where computer pictures are
misleading. Hyperboloids of one and two sheets with an
opportunity to look at sections and what they tell us about
graphs.
Experiments3.mws as present
in the folder Maple 7/Math 222 /Lecture 3 on lab machines.
Linear_Approximation
Some examples of tangent lines and planes graphically. Efficient
generation in Maple of the linear approximation of a function
at a point. Use of the transform plots package to
geometrically compare the mapping properties of a function
with those of its linear approximation.
This worksheet looks geometrically at three different kinds of
singularities. A high point is the sketch of an argument at
the end about why the cube root of (x^3-3xy^2) fails to be
differentiable at the origin. A good opportunity to talk about
the idea of what it means for an approximation to be linear.
PDE_Change.mws as present
in the folder Maple 7/Math 222 /Lecture 6 on lab machines.
Quadrics and Eigenvalues
This worksheet explores the relationship between symmetric
matrices and quadratic forms. It shows how the eigenvalues of such a
matrix relate to the geometric character of the graph of the quadratic
form. It also discusses in the context of an example how the
eigenvectors of the symmetric matrix determine a rotation of
coordinates making the quadratic form diagonal.
A worksheet intended for people to use as a reinforcement on the correlation between eigenvalue signs and the graphs of quadratic forms. Helping with max-min as well.
Taylorexp.mws as present
in the folder Maple 7/Math 222 /Lecture 6 on lab machines.
Taylor Series
Explores multivariate Taylor series. Compares graphically
various Taylor polynomials to the original function. Investigates how
errors depend upon the size of the region on which the approximation
is being used. Uses estimates on the size of partial derivatives to
bound errors in Taylor polynomial approximation. (These latter
are obtained from Maple graphs of the partial derivatives.)
LagMultExp as present
in the folder Maple 7/Math 222 /Lecture 7 on lab machines.
Lagrange Multipliers
The geometry of Lagrange multipliers is explored in the context of the optimization problem for y e^x on an ellipse. Solutions are also obtained numerically using fsolve.
plot3dr_exp.mws as present
in the folder Maple 7/Math 222 /Lecture on lab machines.
Parameterized Surfaces
A variety of parameterization and reparameterization examples are presented. Many are quadric surfaces. Singularities of parameterizations are discussed. The worksheet also presents the natural frame (T_u, T_v, N) associated with a parameterization.