6 Captain Ralph is in trouble near the sunny side of Mercury.
The temperature of the ship’s hull when he is at location (x, y, z) will be given by
T = e
−x
2−2y
2−3z
2
, where x, y, and z are measured in meters. He is currently at
(1, 1, 1).
(a) In what direction should he proceed in order to decrease the temperature most
rapidly?
(b) If the ship travels at e
8 meters per second, how fast will be the temperature
decrease if he proceeds in that direction?
(c) Unfortunately, the metal of the hull will crack if cooled at a rate grater than
√
14e
2 degrees per second. Describe the set of possible directions in which he may
proceed to bring the temperature down at no more than that rate.