Graded Paths Lesson

These are two surfaces used in a lesson published by Margaret Reese (citation below). The lesson gives students the formulas for the surfaces of two "mountains":

*f*(*x*,*y*) = 100 - 2*x*2 - *y*2

*g*(*x*,*y*) = 100 + 3*x*2 - 2*y*2

Students are then asked to use the use the tools of mutlivariable calculus and differential equations to find (for a given starting point):

1. The steepest possible path up the mountain
2. A path which ascends the mountain at 50% of the maximum possible steepness.
3. A path which ascends the mountain at a constant grade of 50%.

When assigning this project, I had students sketch the paths they found on the model.

Project source: Reese, Margaret. Computing Project for Multivariable Calculus. *PRIMUS*. Vol. 6, Iss. 2, 1996.