A spaceship is at location $(1,1,1)$ and the temperature of the ship's hull when at location $(x,y,z$ will be $$ T(x,y,z) = 200 +e^{-x^2-2y^2-3z^2} $$ where x,y,z are in meters.
a) In what direction should the ship proceed in order to decrease temperature most rapidly?
b) If the ship travels at $e^8$ m/sec, how fast will the temperature decrease if it proceeds in this direction
Since $\nabla T(x,y,z) = (-2xe^{-x^2-2y^2-3z^2},-4ye^{-x^2-2y^2-3z^2},-6ze^{-x^2-2y^2-3z^2}) $ we have that T has continuous partials so T is differentiable and so we know that the largest rate of change occurs in the direction $\nabla T(1,1,1) = (-2e^{-6},-4e^{-6},-6e^{-6})$ Would this be the direction the ship should proceed in order to decrease temperature most rapidly? How do we know temperature will decrease?
