livinia the housefly finds herself caught in the oven at the point (0,0,1). the temperature at the points in the oven is given by the function

T(x,y,z) = 10 (xe^(-y2) + ze^(-x2))

where the units are in degrees celsius.

(i.)so if livinia begins to move towards the point (2,3,1) at what rate in deg/cm does she find the temp. changing?
(ii.)in what direction should she move in order to cool off as rapidly as possible?
(iii.)suppose that livinia can fly at a speed of the square root of 2 (cm/sec.) if she moves in the direction of part (ii) at what rate will she find the temp. to be changing?

if you can help that would be awesome
thanks
danielle

Hi Danielle,

I can help get you started.

Let i = (1, 0, 0), j = (0, 1, 0) and k = (0, 0, 1) then the gradient of T(x,y,z) is given by

The rate of change of the temperature T in the direction given by the unit vector u is then

For part (i.) livinia is at (0, 0, 1) and moves toward (2, 3, 1) so u is the unit vector in the direction

(2, 3, 1) - (0, 0, 1) = (2, 3, 0).

Thus find the gradient at (0, 0, 1) and take the dot product of this gradient with u.

For part (ii.) the function T(x,y,z) decreases most rapidly in the direction of .

Penny