Question

The temperature at a point (x, y, z) is given by

T(x, y, z) = 400e⁻x² ⁻ 5y² ⁻ 7z²
where T is measured in °C and x, y, z in meters.
(a) Find the rate of change of temperature at the point P(4, −1, 2) in the direction towards the point (6, −5, 4).
(b) In which direction does the temperature increase fastest at P?
(c) Find the maximum rate of increase at P

Answer 1

The rate of change of temperature at a point can be found using the gradient and a given direction. The direction of fastest temperature increase is indicated by the gradient, and the maximum rate of increase is the magnitude of the gradient vector at that point.

Step-by-step explanation:

The rate of change of temperature at a point in a given direction is calculated using the gradient of the temperature function and the direction vector. To find this at point P(4, −1, 2) towards the point (6, −5, 4), we compute the gradient of T at P, which is the vector of partial derivatives, and then project it onto the direction from P to (6, −5, 4) after normalizing that direction vector.

The direction in which the temperature increases fastest at P is given by the gradient of T at P. This is the direction of the maximum rate of increase of the temperature. The maximum rate of increase at P is simply the magnitude of the gradient vector at P.