Exercises

1. Generate a surface plot and contour plot for the following function on the given domain:
   
   $$f(x,y) = (x^2-y^2)e^{(-x^2-y^2)}, ~~ -3\leq x \leq 3, ~~ -3\leq y\leq 3$$
   
   

   a)

   What does the contour plot look like in the regions where the surface plot has a steep incline? What does it look like where the surface plot is almost flat?

   b)

   What can you say about the surface plot in a region where the contour plot looks like a series of nested circles?

2. Consider the following function from last week's lab
   
   $$r(x,y) = \frac{2x+y-1}{1+x^2+2y^2} ~~ -3 \leq x \leq 5, ~~ -3 \leq y \leq 5$$
   
   
   which represents the deviation, in inches, of last year's rainfall from the average annual rainfall in a certain area. Use a contour plot and shade in the region on your printout of this domain in which the deviation was between $-0.4$ inches and $-1.1$ inches.

3. Consider the following function $$f(x,y) = e^{(-x^2)} \cos(y)$$ for $-2 \leq x \leq 2$ and $\frac{\pi}{2} \leq y \leq \frac{5\pi}{2}$ which looks like a deep valley with a mountain opposite it. Is it possible to find a path from the point $$(1.5,\frac{\pi}{2},0)$$ to $$(-1.5,\frac{5\pi}{2},0)$$ 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.