Question

The temperature at a point (x, y, z)

is given by T(x, y, z) = 400e⁻ˣ² - 5y² - 9z² where T is measured in degree C and x, y, z in meters.
(a) Find the rate of change of temperature at the point P (4, -1, 5) in the direction towards the point (6, -3, 6).
(b) In which direction does the temperature increase fastest at P?
(c) Find the maximum rate of increase at P.

Answer 1

To find the rate of change of temperature at point P (4, -1, 5) in the direction towards point (6, -3, 6), first calculate the gradient vector of the temperature function. Then, find the unit vector in the direction towards the second point and calculate the dot product between the gradient vector and the unit direction vector.

Step-by-step explanation:

To find the rate of change of temperature at point P (4, -1, 5) in the direction towards point (6, -3, 6), we can first find the gradient vector of the temperature function T(x, y, z) = 400e⁻ˣ² - 5y² - 9z². The gradient vector represents the direction of the steepest increase of the function.

Next, we can calculate the direction vector between the two points and normalize it to obtain the unit vector in the direction towards the second point. Finally, we can find the dot product between the gradient vector and the unit direction vector to calculate the rate of change of temperature.