Yosemite, Part 3

Exhausted after the climb in Part 2, you consider the design of a hiking trail with a gentler ascent. Hiking trails generally do not rise much more than one foot for each five feet horizontal. What value derivative does this correspond to? How does it compare with the values of the derivatives in your table?

To keep trails from following a gentle, but unacceptably long, spiral to the summit, they're usually designed to incorporate switchbacks: sharp turns in direction that nevertheless keep the trail at a constant slope. See if you can draw a hiking trail on your map, making a switchback every s = 1000 ft, that starts at (0,0) and arrives at your present position. How does the length w of your trail compare with the length of your ascent in Part 2?

Now start from the summit of Mt. Watkins. Suppose you wish to descend as quickly as possible to Tenaya Creek. Choose a sequence of points P_n, analogous to those in Part 2, marking the path on your map. Again write a general formula that gives P_n+1 in terms of P_n and grad f(x_n,y_n).

Finally, you jump in a canoe and start back toward Mirror Lake. What's the difference between following the creek and simply continuing the iterations in step 3? Why does the creek follow the path that it does?