Exercises

1. Use the command plot3d to generate a surface plot and the command contourplot to generate a contour plot with 50 contours for the following funtion on the given domain:
   
   $$f(x,y) = \frac{x-y}{7+x^2+y^2} ~~ -4 \leq x \leq 4, ~~ -4 \leq y \leq 4$$
   
   

   a)

   What does the contour plot look like in regions where the surface plot has relative extrema?

   b)

   Rotate the 3-d graph and give an estimate of the extrema. (Extrema are the $z$ values of the highest and lowest points on the graph.)

   c)

   Click your mouse on the point on the contourplot where you think the extrema occur to get an approximate $(x,y)$ coordinate location. Evaluate the function at each of these points and compare to your estimate in part c.

2. For the given equations below, plot 2 two dimensional level curves parallel to the $xy$ plane and then plot 2 two dimensional cross sections in the $xz$ plane and again, 2 two dimensional cross sections in the $yz$ plane. Identify the type or shape of the quadric surface, ie. a sphere, cylinder, cone, elliptic cone, paraboloid, elliptic parabaloid, ellipsoid, hyperboloid of one sheet, hyperboloid of two sheets, elliptic hyperboloid of one or two sheets, or a hyperbolic parabaloid (saddle). Once you have determined the shape of the surface, supply a three dimensional plot to support your conclusion.

   a)

   $z=x^2+y^2$

   b)

   $z^2=x^2+y^2+2$

   c)

   $z=x^2-2y^2$

3. Create a contour plot for the function $$f(x,y)= \frac{\exp(x)+\exp(y)}{x^2+y^2}$$ for the $z$ values $1/2,1,2,3,4$ using two different methods; first using cross sections and then using Maple's contourplot command.

4. Consider the following function $$f(x,y) = \frac{\sin(x)}{1+y^2}$$ for $0 \leq x \leq 2 \pi$ and $-3 \leq y \leq 3$ which looks like a deep valley with a mountain opposite it. Is is possible to find a path from the point $(0,3,0)$ to $(2\pi,-3,0)$ such that the value of $z$ is always between $-0.25$ and $0.25$ ? You do not have to find a formula for your path, but you must present convincing evidence that it exists. For example, you might want to sketch your path in by hand on an appropriate countour plot.