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Handout 1

Problem 1
Use Maple for this Problem! Look at the graphs of the function for -2<x <2, -2<y< 2 and

  1. a=1, b=0, c= 1,
  2. a=1, b=2, c= 1,
  3. a=1, b=4, c= 1,
  4. a=1, b=6, c= 1
  5. a=1, b=0 ,c= 4
  6. a=1, b=0, c = 6,
and at MORE values of your own choice. Do you have any conjectures on how the graph depends on the parameters a,b, and c? Try to cover some special cases: e.g. suppose a >0 and c >0; what happens as b varies ?

Explain!

Problem 2
Find functions whose graphs fit the descriptions below. Use Maple to print graphs verifying your solution. Include axes in the graphs.

  1. A bowl which opens upward and has its vertex at 5 on the z axis.
  2. A parabolic cylinder opening upward from along the line y=x in the xy plane. (A parabolic cylinder is the shape formed by extruding a parabola perpendicular to the plane in which it lies.)
  3. A cone with circular cross section having a vertex in the plane z=5 and passing through the origin.

Problem 3* (Extra Credit)
Explain why level curves of

for function values either

a)
small (e.g. c = .1), or
b)
large (e.g. c = 10)

in absolute value are generally predictable. Sketch what these should look like qualitatively.

HINT: implicitplot(, color=red);