91Ó°ÊÓ

In Exercises \(1-8,\) sketch the region of integration and evaluate the double integral. $$ \int_{0}^{2} \int_{0}^{1}(3 x+4 y) d y d x $$

Find the intercepts and sketch the graph of the plane. $$ 3 x+6 y+2 z=6 $$

Plot the points on the same three dimensional coordinate system. $$ \begin{array}{l}{\text { (a) }(0,4,-5)} \\ {\text { (b) }(4,0,5)}\end{array} $$

Examine the function for relative extrema and saddle points. $$ f(x, y)=x^{2}-y^{2}+4 x-4 y-8 $$

Evaluate \(f_{x}\) and \(f_{y}\) at the point. $$ \text { Function } \quad \text { Point } $$ $$ f(x, y)=x^{2}-3 x y+y^{2} \quad(1,-1) $$

The population density (in people per square mile) for a coastal town can be modeled by \(f(x, y)=\frac{120,000}{(2+x+y)^{3}}\) where \(x\) and \(y\) are measured in miles. What is the population inside the rectangular area defined by the vertices \((0,0),\) \((2,0),(0,2),\) and \((2,2) ?\)

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. $$ \int_{0}^{1} \int_{2 y}^{2} d x d y $$

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

The population density (in people per square mile) for a coastal town on an island can be modeled by \(f(x, y)=\frac{5000 x e^{y}}{1+2 x^{2}}\) where \(x\) and \(y\) are measured in miles. What is the population inside the rectangular area defined by the vertices \((0,0),\) \((4,0),(0,-2),\) and \((4,-2) ?\)

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