Maple #4 is due Thursday 10 Oct 2002

For FULL credit STAPLE your sheets together.
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Actually unstapled assignments will not be accepted.

You are to plot the five functions below. 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. 

The MATLAB function is given below, it has missing *'s and e's don't work
in Maple

matlab:= 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).

1. Have Maple find the both partials of f = matlab above, f_x and f_y
[Read f_x as f sub x.] and all four second partials of f, f_xx, f_xy,
f_yx and f_yy.] [Hint let f:=be the expression for the function, and
just do diff(f,...)] Show f_xy = f_yx, by making maple compute f_xy-f_yx.
[no plot]

2. Use Maple's dotprod and grad [first do with(linalg)] commands to
find the directional derivative of x^2y^3+sin(x)cos(y) at (pi, pi/2) 
in the direction of <5, 12>.  Give both a symbolic and numerical answer.
[Hint directions are unit vectors, ans:=eval(dotprod(....),{x=Pi,y=Pi/2})
and evalf(ans).] [no plot]

3. Find the quadratic Taylor polynomial (call it T) about (1,1) to
f=1/(3+x^2-2x+y^2-2y) and plot the two graphs together in a fashion
you can tell which is which. Make sure your title also makes it clear
which is which (for example, it could say which is on the bottom).
Use the range 0 <= x, y <= 2 (both x & y 0..2). [Hint: mtaylor] [one plot]

4 & 5 Use the function f = 8y^3+12x^2-24xy. First use maple
to solve({f_x=0,f_y=0} to find the two critical points, then for
each critical point x=a,y=b repeat a plot like #3 but using the
range x=a-1..a+1, y=b-1..b+1. (This funtions is a example in the
text.) [two plots].