Maple #6 is due Thursday 17 Feb 2000 

For FULL credit STAPLE your sheets together.
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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 questions
in 1 and 2.

1. Find the quadratic Taylor polynomial (call it T(x,y)) about (0,0) to
f(x,y)=cos(x)*cos(y) and plot the two graphs together for -1 <= x, y <= 1.
Which graph is on the "bottom"?
2. 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"?
3. Use Maple to find the critical points of f(x,y)=x^3+3xy^2-3x^2-3y^2+4
and to identify local mins, local maxs and saddle points.
4. Use Maple to find the critical points of f(x,y)=y^3+x^2-6xy+3x+6y-7
and to identify local mins, local maxs and saddle points.
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.