91Ó°ÊÓ

Find the average value of \(f(x, y)\) over the region \(R\). $$ \begin{aligned} &f(x, y)=x\\\ &R \text { : rectangle with vertices }(0,0),(4,0),(4,2),(0,2) \end{aligned} $$

Find the first partial derivatives with respect to \(x, y\), and \(z\). $$ w=x^{2}-3 x y+4 y z+z^{3} $$

Find the critical points and test for relative extrema. List the critical points for which the Second-Partials Test fails. $$ f(x, y)=\left(x^{2}+y^{2}\right)^{2 / 3} $$

Sketch the region \(R\) whose area is given by the double integral. Then change the order of integration and show that both orders yield the same area. $$ \int_{0}^{4} \int_{\sqrt{x}}^{2} d y d x $$

Find the average value of \(f(x, y)\) over the region \(R\). $$ \begin{aligned} &f(x, y)=x y\\\ &R: \text { rectangle with vertices }(0,0),(4,0),(4,2),(0,2) \end{aligned} $$

Find the minimum distance from the curve or surface to the given point. (Hint: Start by minimizing the square of the distance.) Circle: \((x-4)^{2}+y^{2}=4,(0,10)\) Minimize \(d^{2}=x^{2}+(y-10)^{2}\)

Find the minimum distance from the curve or surface to the given point. (Hint: Start by minimizing the square of the distance.) Plane: \(x+y+z=1,(2,1,1)\) Minimize \(d^{2}=(x-2)^{2}+(y-1)^{2}+(z-1)^{2}\)

Find the critical points of the function and, from the form of the function, determine whether a relative maximum or a relative minimum occurs at each point. $$ f(x, y, z)=(x-1)^{2}+(y+3)^{2}+z^{2} $$

Find the average value of \(f(x, y)\) over the region \(R\). $$ \begin{aligned} &f(x, y)=x^{2}+y^{2}\\\ &R: \text { square with vertices }(0,0),(2,0),(2,2),(0,2) \end{aligned} $$

Find the first partial derivatives with respect to \(x, y\), and \(z\). $$ w=\frac{2 z}{x+y} $$