91Ó°ÊÓ

For the following exercises, find the derivative of the function.\(f(x, y)=e^{x y}\) at point \((6,7)\) in the direction the function increases most rapidly

The temperature \(T\) in a metal sphere is inversely proportional to the distance from the center of the sphere (the origin: \((0,0,0))\). The temperature at point \((1,2,2)\) is \(120^{\circ} \mathrm{C}\). a. Find the rate of change of the temperature at point \((1,2,2)\) in the direction toward point \((2,1,3)\). b. Show that, at any point in the sphere, the direction of greatest increase in temperature is given by a vector that points toward the origin.

For the following exercises, find an equation of the level curve of \(f\) that contains the point \(P\). $$ g(x, y)=e^{x y}\left(x^{2}+y^{2}\right), P(1,0) $$

Find the maximal directional derivative magnitude and direction for the function \(f(x, y)=x^{3}+2 x y-\cos (\pi y)\) at point \((3,0)\).

A thin plate made of iron is located in the \(x y\) -plane. The temperature \(T\) in degrees Celsius at a point \(P(x, y)\) is inversely proportional to the square of its distance from the origin. Express \(T\) as a function of \(x\) and \(y\).