2. (a) The temperature on the surface of an object is given by the function:

T(x, y, z) = x²y+3y²z+4

where T is measured in degrees Celsius and x, y, and z are measured in

metres. Determine the temperature along the contour passing through the point

(3,1,2) and calculate the temperature gradient at point (3,1,2) in the direction

a =î-2j+2k.

[6 marks]

(b) A cupboard door is fitted with a 'soft close' device, which controls the angle

8(t) of the door (in degrees) according to the equation

d²0

dt²

de

+7+100 = 0.

dt

If the door is opened to an angle of 60° and released from rest at time t = 0,

find 8(t). Assuming that time t is measured in seconds, estimate the time taken

for the door to close.

[7 marks]

(c) Given that x(0) = 4 and x(0) = 0, use the method of undetermined

coefficients to solve the following differential equation:

x + 4x + 4x = 8t².