Section 14.6: Directional Derivatives & Gradient Vector

The temperature at a point \((x, y, z)\) is given by \(T(x, y, z) = 200e^{-x^2 - 3y^2 - 9z^2}\) where T is measured in °C and x, y, z in meters. (a) Find the rate of change of temperature at the point \(P(2, -1, 2)\) in the direction toward the point \((3, -3, 3)\). (b) In which direction does the temperature increase fastest at P? (c) Find the maximum rate of increase at P.