The temperature at the point (x, y) on a metal plate is given by T(x, y) = x2 - y + 10. Let P = (1, 1).
a. Find T(P), and draw the level curve through P.
b. Find ∇T, and plot (∇T)(P), with tail located at P.
c. At P, along which direction (i.e. unit vector) does the temperature increase most rapidly? At what rate?
d. Find the rate of change of temperature at P in the direction of the origin.
e. Find the rate of change of temperature at P in the direction of i.
f. A bug at P moves with unit velocity at an angle of 60o with the direction of most rapid decrease of T. At what rate is T changing for the bug?