Question

The elevation in meters above sea level (sea level is taken to be at zero meters) of a valley is given by E(x, y) = −x2 − 4y2, where x and y are in meters. Suppose Ralph is at a

point in the valley where (x, y) = (1, −2).
a. Sketch the level curves of the function E where it is equal to −5, −10, −15, −20, −25, and the elevation at which Ralph currently stands. (If you sketch this neatly, you can sketch your answers to the rest of this problem atop it.)
b. In what direction (as a unit vector in the xy-plane) should Ralph head from his current position to ascend from the valley as rapidly as possible? Sketch this vector based at his current location.
c. At what rate, in meters of elevation per meter of distance in the xy-plane, will Ralph ascend if he moves in the direction you found in part b.?
d. Suppose Ralph heads in the direction indicated by (1,1). Sketch this direction based at his current location. At what rate is his elevation changing? Is he headed up, down, or neither with respect to sea level, and why?
e. In what directions (as unit vectors in the xy-plane), if any, could Ralph head to keep his current elevation exactly as it is? Sketch these directions based at his current location.