Suppose you are standing on the side of a mountain. You are using a topographic map, which uses rectangular coordinates measured in miles. The positive x-axis represents East; the positive y-axis represents North.
Your GPS says that your current location is (1,3). Your guidbook also tells you that the height of the mountain in feet above sea level is given by
h = c + ax2-by2
where c= 9000 ft and a=b= 20 ft/mi2
(a) Starting at your present location, if you walk due west, will you be ascending or decending? At what Rate?
(b) Starting at your present location, in what map direction (2D unit vector) should you walk to travel along a level path? Also draw and label this vector on your topographical map.
(c) Starting at your present location, in what map direction (2D unit vector) do you need to go in order to climb the hill as steeply as possible? Also draw and label this vector on your topographical map.
(d) In what direction in space (3D vector) would you actually be moving if you started at your present location and walked in the map direction you found in part (c) above? Your answer should not be a unit vector.
(e) Find an equation for the tangeant plane at your present location.
(f) Find the height of (i.e., on) the tangeant plane if you move from your present location by Δx = -5/4 mi and Δy = 1/2mi