Maple #6 is due Thursday 28 Feb 2002 

For FULL credit STAPLE your sheets together.
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Actually unstapled assignments will not be accepted.
	Draft mode of web maple is not acceptable, use PDF mode.

Each plot must be rotated to a "nice" position and "look good (smooth)".
Each plot must have include axes and the title must include your name
and what was plotted. Note that the expressions below are not
necessarily in a form that Maple likes. Be sure to answer the question
in 1.

1. Find the quadratic Taylor polynomial (call it T(x,y)) about (1,1) to
f(x,y)=1/(3+x^2-2x+y^2-2y) and plot the two graphs together
for 0 <= x, y <= 2. Which graph is on the "bottom"?
2&3 Plot the following spacecurves (from test 1 fall 96)
  2a. <cos t, sin t, cos 4t>
  2b. <cos 4t, sin 4t, 4t>
  3a. <t cos t, t sin t, t>
  3b. <4t, sin 4t, cos 4t>
4. The curve <cos 3t, sin 5t, cos 7t>, 0 <= t <= 2pi is particular hard
to figure out from a spacecurve. One can use "tubeplot" with a "nice"
radius and get a better picture of the curve. Do so. (careful, tubeplot
can be very slow, decrease the radius and increase the numpoints.)
5. Use Maple's dotprod and grad commands to find the directional
derivative of the matlab demo function
f(x,y)= 3(1-x)^2 e^(-x^2-(y+1)^2)
    - 10(x/5-x^3-y^5)e^(-x^2-y^2) -(1/3)e^(-(x+1)^2-y^2).
at (1/2, 0) in the direction of
<3, 4> [Hint subs(x=1/2,y=0, dotprod(....));.] (Give both a
symbolic and numerical answer [Hint evalf].)
[Remember, either define e:=exp(1) or replace e^x with exp(x). A common
error is to replace e^x with exp^x, and unfortunately Maple doesn't
notice the error, and does something dumb.] Be sure to look carefully
at your function f, at least one person has typed incorrectly on each
assignment.]