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

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 $ ^\circ 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, -1, 3) $. (b) In which direction does the temperature increase fastest at $ P $? (c) Find the maximum rate of increase at $ P $.

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 $ ^\circ 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, -1, 3) $.
(b) In which direction does the temperature increase fastest at $ P $?
(c) Find the maximum rate of increase at $ P $.

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Key Concepts

Gradient

The gradient of a multivariable function is a vector formed by its partial derivatives. It indicates the direction of the steepest ascent of the function and plays a crucial role in understanding how the function changes in space.

Directional Derivative

The directional derivative quantifies the rate of change of a function in any specified direction. It is found by taking the dot product of the gradient vector and a unit vector pointing in that direction, thus revealing how the function varies along that line.

Unit Vector

A unit vector is a vector of magnitude one that defines a direction in space. When computing directional derivatives, it is essential to use a unit vector so that the rate of change reflects only the influence of direction and not the length of the vector.

Maximum Rate of Increase

The maximum rate at which a function increases at a given point is equal to the magnitude of the gradient at that point. This maximum is achieved in the direction of the gradient vector, making it a critical concept in gradient-based optimization and analysis.

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