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(Solved): The temperature at a point \( (x, y, z) \) is given by \[ T(x, y, z)=100 e^{-x^{2}-5 y^{2}-7 z^{2} ...
![The temperature at a point \( (x, y, z) \) is given by
\[
T(x, y, z)=100 e^{-x^{2}-5 y^{2}-7 z^{2}}
\]
where \( T \) is measu](https://media.cheggcdn.com/media/ed3/ed3b35e7-3d5b-4a23-8321-993f8c791958/phpj3mpld)
The temperature at a point \( (x, y, z) \) is given by \[ T(x, y, z)=100 e^{-x^{2}-5 y^{2}-7 z^{2}} \] where \( T \) is measured in \( { }^{\circ} \mathrm{C} \) and \( x, y, z \) in meters. (a) Find the rate of change of temperature (in \( \left.{ }^{\circ} \mathrm{C} / \mathrm{m}\right) \) at the point \( P(4,-1,2) \) in the direction toward the point \( (6,-2,3) \). \[ { }^{\circ} \mathrm{C} / \mathrm{m} \] (b) In which direction does the temperature increase fastest at \( P \) ? (c) Find the maximum rate of increase at \( P \).
Expert Answer
(a). Given , The temperature at point (x,y,z) is : T(x,y,z)=100e?x2?5y2?7z2 Gradient vector of T is : ?T(x,y,z)= or , ?T= ?200e?x2?5y