Chapter 13

Sketch the region of integration and evaluate the double integral. $$ \int_{0}^{2} \int_{0}^{1}(3 x+4 y) d y d x $$

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

Use Lagrange multipliers to find the given extremum. In each case, assume that \(x\) and \(y\) are positive. $$ \operatorname{Maximize} f(x, y)=x y \quad x+y=10 $$

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 \(y\). $$ z=3 x+5 y-1 $$

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

Plot the points on the same threedimensional coordinate system. (a) \((2,1,3)\) (b) \((-1,2,1)\)

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. $$ \int_{x}^{x^{2}} \frac{y}{x} d y $$

Use Lagrange multipliers to find the given extremum. In each case, assume that \(x\) and \(y\) are positive. $$ \text { Maximize } f(x, y)=x y \quad 2 x+y=4 $$

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. \(f(x, y)=4-x^{2}-4 y^{2}\) (a) \(f(0,0)\) (b) \(f(0,1)\) (c) \(f(2,3)\) (d) \(f(1, y)\) (e) \(f(x, 0)\) (f) \(f(t, 1)\)

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

Plot the points on the same threedimensional coordinate system. (a) \((3,-2,5)\) (b) \(\left(\frac{3}{2}, 4,-2\right)\)

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

Use Lagrange multipliers to find the given extremum. In each case, assume that \(x\) and \(y\) are positive. $$ \text { Minimize } f(x, y)=x^{2}+y^{2} \quad x+y-4=0 $$

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. \(f(x, y)=4-x^{2}-4 y^{2}\) (a) \(f(0,0)\) (b) \(f(0,1)\) (c) \(f(2,3)\) (d) \(f(1, y)\) (e) \(f(x, 0)\) (f) \(f(t, 1)\)

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

Plot the points on the same threedimensional coordinate system. (a) \((5,-2,2)\) (b) \((5,-2,-2)\)

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

Use Lagrange multipliers to find the given extremum. In each case, assume that \(x\) and \(y\) are positive. $$ \text { Minimize } f(x, y)=x^{2}+y^{2} \quad-2 x-4 y+5=0 $$

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. \(g(x, y)=\ln |x+y|\) (a) \(g(2,3)\) (b) \(g(5,6)\) (c) \(g(e, 0)\) (d) \(g(0,1)\) (e) \(g(2,-3)\) (f) \(g(e, e)\)

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

Plot the points on the same threedimensional coordinate system. (a) \((0,4,-5)\) (b) \((4,0,5)\)

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. $$ \int_{0}^{\sqrt{4-x^{2}}} x^{2} y d y $$

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. $$ (-2,-1),(0,0),(2,3) $$

Use Lagrange multipliers to find the given extremum. In each case, assume that \(x\) and \(y\) are positive. $$ \text { Maximize } f(x, y)=x^{2}-y^{2} \quad 2 y-x^{2}=0 $$

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

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

Find the function values. \(h(x, y, z)=\frac{x y}{z}\) (a) \(h(2,3,9)\) (b) \(h(1,0,1)\)

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. $$ \int_{0}^{2} \int_{0}^{4-x^{2}} x y^{2} d y d x $$

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. $$ (-3,0),(-1,1),(1,1),(3,2) $$

Use Lagrange multipliers to find the given extremum. In each case, assume that \(x\) and \(y\) are positive. $$ \text { Minimize } f(x, y)=x^{2}-y^{2} \quad x-2 y+6=0 $$

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

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

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

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. $$ \int_{1}^{e} \frac{y \ln x}{x} d x $$

Use Lagrange multipliers to find the given extremum. In each case, assume that \(x\) and \(y\) are positive. $$ \text { Maximize } f(x, y)=2 x+2 x y+y \quad 2 x+y=100 $$

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 \(y\). $$ f(x, y)=\sqrt{x^{2}+y^{2}} $$