Chapter 7

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

In Exercises \(1-10,\) evaluate the partial integral. $$ \int_{0}^{x}(2 x-y) d y $$

Find any critical points and relative extrema of the function. $$ f(x, y)=x^{2}-y^{2}+4 x-8 y-11 $$

Find the first partial derivatives with respect to \(x\) and with respect to \(y .\) $$ z=3 x+5 y-1 $$

In Exercises \(1-12,\) use Lagrange multipliers to find the given extremum. In each case, assume that \(x\) and \(y\) are positive. $$ \begin{array}{ll}{\text { Objective Function }} & {\text { Constraint }} \\\ {\text { Maximize } f(x, y)=x y} & {x+y=10}\end{array} $$

Find the intercepts and sketch the graph of the plane. $$ 4 x+2 y+6 z=12 $$

In Exercises \(1-4,\) plot the points on the same three dimensional coordinate system. $$ \begin{array}{l}{\text { (a) }(2,1,3)} \\ {\text { (b) }(-1,2,1)}\end{array} $$

Sketch the region of integration and evaluate the double integral. $$ \int_{0}^{3} \int_{0}^{1}(2 x+6 y) d y d x $$

Evaluate the partial integral. $$ \int_{x}^{x^{2}} \frac{y}{x} d y $$

Find any critical points and relative extrema of the function. $$ f(x, y)=x^{2}+y^{2}+2 x-6 y+6 $$

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

Find the function values. $$ f(x, y)=4-x^{2}-4 y^{2} $$ $$ \begin{array}{llll}{\text { (a) } f(0,0)} & {\text { (b) } f(0,1)} & {\text { (c) } f(2,3)} & {} \\ {\text { (d) } f(1, y)} & {\text { (e) } f(x, 0)} & {\text { (f) } f(t, 1)}\end{array} $$

Use Lagrange multipliers to find the given extremum. In each case, assume that \(x\) and \(y\) are positive. $$ \begin{array}{ll}{\text { Objective Function }} & {\text { Constraint }} \\\ {\text { Maximize } f(x, y)=x y} & {2 x+y=4}\end{array} $$

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. $$ \begin{array}{l}{\text { (a) }(3,-2,5)} \\ {\text { (b) }\left(\frac{3}{2}, 4,-2\right)}\end{array} $$

Sketch the region of integration and evaluate the double integral. $$ \int_{0}^{1} \int_{y}^{\sqrt{y}} x^{2} y^{2} d x d y $$

Evaluate the partial integral. $$ \int_{1}^{2 y} \frac{y}{x} d x $$

Find any critical points and relative extrema of the function. $$ f(x, y)=\sqrt{x^{2}+y^{2}+1} $$

Find the first partial derivatives with respect to \(x\) and with respect to \(y .\) $$ f(x, y)=3 x-6 y^{2} $$

Find the function values. $$ f(x, y)=x e^{y} $$ $$ \begin{array}{llll}{\text { (a) } f(5,0)} & {\text { (b) } f(3,2)} & {\text { (c) } f(2,-1)} \\ {\text { (d) } f(5, y)} & {\text { (e) } f(x, 2)} & {\text { (f) } f(t, t)}\end{array} $$

Use Lagrange multipliers to find the given extremum. In each case, assume that \(x\) and \(y\) are positive. $$ \begin{array}{ll}{\text { Objective Function }} & {\text { Constraint }} \\\ {\text { Maximize } f(x, y)=x^{2}+y^{2}} & {x+y-4=0}\end{array} $$

Find the intercepts and sketch the graph of the plane. $$ 3 x+3 y+5 z=15 $$

Plot the points on the same three dimensional coordinate system. $$ \begin{array}{l}{\text { (a) }(5,-2,2)} \\ {\text { (b) }(5,-2,-2)}\end{array} $$

Sketch the region of integration and evaluate the double integral. $$ \int_{0}^{6} \int_{y / 2}^{3}(x+y) d x d y $$

Find any critical points and relative extrema of the function. $$ f(x, y)=\sqrt{25-(x-2)^{2}-y^{2}} $$

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

Find the function values. $$ g(x, y)=\ln |x+y| $$ $$ \begin{array}{llll}{\text { (a) } g(2,3)} & {\text { (b) } g(5,6)} & {\text { (c) } g(e, 0)} \\ {\text { (d) } g(0,1)} & {\text { (e) } g(2,-3)} & {\text { (f) } g(e, e)}\end{array} $$

Use Lagrange multipliers to find the given extremum. In each case, assume that \(x\) and \(y\) are positive. $$ \begin{array}{ll}{\text { Objective Function }} & {\text { Constraint }} \\\ {\text { Maximize } f(x, y)=x^{2}+y^{2}} & {-2 x-4 y+5=0}\end{array} $$

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

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

Sketch the region of integration and evaluate the double integral. $$ \int_{0}^{1} \int_{0}^{\sqrt{1-x^{2}}} y d y d x $$

Evaluate the partial integral. $$ \int_{0}^{\sqrt{4-x^{2}}} x^{2} y d y $$

Examine the function for relative extrema and saddle points. $$ f(x, y)=(x-1)^{2}+(y-3)^{2} $$

Find the least squares regression line for the points. Use the regression capabilities of a graphing utility or a spreadsheet to verify your results. Then plot the points and graph the regression line. $$ (-2,-1),(0,0),(2,3) $$

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

Find the function values. $$ h(x, y, z)=\frac{x y}{z} $$ $$ \begin{array}{ll}{\text { (a) } h(2,3,9)} & {\text { (b) } h(1,0,1)}\end{array} $$

Use Lagrange multipliers to find the given extremum. In each case, assume that \(x\) and \(y\) are positive. $$ \begin{array}{ll}{\text { Objective Function }} & {\text { Constraint }} \\\ {\text { Maximize } f(x, y)=x^{2}-y^{2}} & {2 y-x^{2}=0}\end{array} $$

Find the intercepts and sketch the graph of the plane. $$ 2 x-y+3 z=4 $$

Sketch the region of integration and evaluate the double integral. $$ \int_{0}^{2} \int_{0}^{4-x^{2}} x y^{2} d y d x $$

Examine the function for relative extrema and saddle points. $$ f(x, y)=9-(x-3)^{2}-(y+2)^{2} $$

Find the least squares regression line for the points. Use the regression capabilities of a graphing utility or a spreadsheet to verify your results. Then plot the points and graph the regression line. $$ (-3,0),(-1,1),(1,1),(3,2) $$

Find the first partial derivatives with respect to \(x\) and with respect to \(y .\) $$ z=x \sqrt{y} $$

Find the function values. $$ f(x, y, z)=\sqrt{x+y+z} $$ $$ \text { (a) } f(0,5,4) \quad \text { (b) } f(6,8,-3) $$

Use Lagrange multipliers to find the given extremum. In each case, assume that \(x\) and \(y\) are positive. $$ \begin{array}{ll}{\text { Objective Function }} & {\text { Constraint }} \\\ {\text { Maximize } f(x, y)=x^{2}-y^{2}} & {x-2 y+6=0}\end{array} $$

Find the intercepts and sketch the graph of the plane. $$ 2 x-y+z=4 $$

Sketch the region of integration and evaluate the double integral. $$ \int_{-a}^{a} \int_{-\sqrt{a^{2}-x^{2}}}^{\sqrt{a^{2}-x^{2}}} d y d x $$

Evaluate the partial integral. $$ \int_{1}^{e^{y}} \frac{y \ln x}{x} d x $$

Examine the function for relative extrema and saddle points. $$ f(x, y)=2 x^{2}+2 x y+y^{2}+2 x-3 $$

Find the first partial derivatives with respect to \(x\) and with respect to \(y .\) $$ f(x, y)=\sqrt{x^{2}+y^{2}} $$

Find the function values. $$ V(r, h)=\pi r^{2} h $$ $$ \begin{array}{llll}{\text { (a) } V(3,10)} & {\text { (b) } V(5,2)}\end{array} $$