Chapter 4

For the following exercises, evaluate each function at the indicated values. $$ W(x, y)=4 x^{2}+y^{2} . \text { Find } W(2,-1), \quad W(-3,6) $$

For the following exercises, evaluate each function at the indicated values. $$ W(x, y)=4 x^{2}+y^{2} . \text { Find } W(2+h, 3+h) $$

The volume of a right circular cylinder is calculated by a function of two variables, \(V(x, y)=\pi x^{2} y,\) where \(x\) is theradius of the right circular-cylinder and \(y\) represents the height of the cylinder. Evaluate \(V(2,5)\) and explain what this means.

An oxygen tank is constructed of a right cylinder of height \(y\) and radius \(x\) with two hemispheres of radius \(x\) mounted on the top and bottom of the cylinder. Express the volume of the cylinder as a function of two variables, \(x\) and \(y,\) find \(V(10,2),\) and explain what this means.

For the following exercises, find the domain of the function. $$ V(x, y)=4 x^{2}+y^{2} $$

For the following exercises, find the domain of the function. $$ f(x, y)=\sqrt{x^{2}+y^{2}-4} $$

For the following exercises, find the domain of the function. $$ f(x, y)=4 \ln \left(y^{2}-x\right) $$

For the following exercises, find the domain of the function. $$ g(x, y)=\sqrt{16-4 x^{2}-y^{2}} $$

For the following exercises, find the domain of the function. $$ z(x, y)=y^{2}-x^{2} $$

For the following exercises, find the domain of the function. $$ f(x, y)=\frac{y+2}{x^{2}} $$

Find the range of the functions. $$ g(x, y)=\sqrt{16-4 x^{2}-y^{2}} $$

For the following exercises, find the level curves of each function at the indicated value of \(c\) to visualize the given function. $$ z(x, y)=y^{2}-x^{2}, \quad c=1 $$

For the following exercises, find the level curves of each function at the indicated value of \(c\) to visualize the given function. $$ z(x, y)=y^{2}-x^{2}, \quad c=4 $$

For the following exercises, find the level curves of each function at the indicated value of \(c\) to visualize the given function. $$ g(x, y)=x^{2}+y^{2} ; c=4, c=9 $$

For the following exercises, find the level curves of each function at the indicated value of \(c\) to visualize the given function. $$ g(x, y)=4-x-y ; c=0,4 $$

For the following exercises, find the level curves of each function at the indicated value of \(c\) to visualize the given function. $$ f(x, y)=x y ; c=1 ; c=-1 $$

For the following exercises, find the level curves of each function at the indicated value of \(c\) to visualize the given function. $$ h(x, y)=2 x-y ; c=0,-2,2 $$

For the following exercises, find the level curves of each function at the indicated value of \(c\) to visualize the given function. $$ f(x, y)=x^{2}-y ; c=1,2 $$

For the following exercises, find the level curves of each function at the indicated value of \(c\) to visualize the given function. $$ g(x, y)=\frac{x}{x+y} ; c=-1,0,2 $$

For the following exercises, find the level curves of each function at the indicated value of \(c\) to visualize the given function. $$ g(x, y)=x^{3}-y ; c=-1,0,2 $$

For the following exercises, find the level curves of each function at the indicated value of \(c\) to visualize the given function. $$ g(x, y)=e^{x y} ; c=\frac{1}{2}, 3 $$

For the following exercises, find the level curves of each function at the indicated value of \(c\) to visualize the given function. $$ f(x, y)=x^{2} ; c=4,9 $$

For the following exercises, find the level curves of each function at the indicated value of \(c\) to visualize the given function. $$ f(x, y)=x y-x ; c=-2,0,2 $$

For the following exercises, find the level curves of each function at the indicated value of \(c\) to visualize the given function. $$ h(x, y)=\ln \left(x^{2}+y^{2}\right) ; c=-1,0,1 $$

For the following exercises, find the level curves of each function at the indicated value of \(c\) to visualize the given function. $$ g(x, y)=\ln \left(\frac{y}{x^{2}}\right) ; c=-2,0,2 $$

For the following exercises, find the level curves of each function at the indicated value of \(c\) to visualize the given function. $$ z=f(x, y)=\sqrt{x^{2}+y^{2}}, \quad c=3 $$

For the following exercises, find the level curves of each function at the indicated value of \(c\) to visualize the given function. $$f(x, y)=\frac{y+2}{x^{2}}, \quad c= any\quad constant$$

For the following exercises, find the vertical traces of the functions at the indicated values of \(x\) and \(y,\) and plot the traces. $$ z=4-x-y ; x=2 $$

For the following exercises, find the vertical traces of the functions at the indicated values of \(x\) and \(y,\) and plot the traces. $$ f(x, y)=3 x+y^{3}, x=1 $$

For the following exercises, find the vertical traces of the functions at the indicated values of \(x\) and \(y,\) and plot the traces. $$ z=\cos \sqrt{x^{2}+y^{2}} \quad x=1 $$

Find the domain of the following functions. $$ z=\sqrt{100-4 x^{2}-25 y^{2}} $$

Find the domain of the following functions. $$ z=\ln \left(x-y^{2}\right) $$

Find the domain of the following functions. $$ f(x, y, z)=\frac{1}{\sqrt{36-4 x^{2}-9 y^{2}-z^{2}}} $$

Find the domain of the following functions. $$ f(x, y, z)=\sqrt{49-x^{2}-y^{2}-z^{2}} $$

Find the domain of the following functions. $$ f(x, y, z)=\sqrt[3]{16-x^{2}-y^{2}-z^{2}} $$

Find the domain of the following functions. $$ f(x, y)=\cos \sqrt{x^{2}+y^{2}} $$

For the following exercises, plot a graph of the function. $$ z=f(x, y)=\sqrt{x^{2}+y^{2}} $$

For the following exercises, plot a graph of the function. $$ z=x^{2}+y^{2} $$

For the following exercises, plot a graph of the function. Use technology to graph \(z=x^{2} y\)

Sketch the following by finding the level curves. Verify the graph using technology. $$ f(x, y)=\sqrt{4-x^{2}-y^{2}} $$

Sketch the following by finding the level curves. Verify the graph using technology. $$ f(x, y)=2-\sqrt{x^{2}+y^{2}} $$

Sketch the following by finding the level curves. Verify the graph using technology. $$ z=1+e^{-x^{2}-y^{2}} $$

Sketch the following by finding the level curves. Verify the graph using technology. $$ z=y^{2}-x^{2} $$

Sketch the following by finding the level curves. Verify the graph using technology. Describe the contour lines for several values of \(c\) for \(z=x^{2}+y^{2}-2 x-2 y\)

Find the level surface for the functions of three variables and describe it. $$ w(x, y, z)=x-2 y+z, c=4 $$

Find the level surface for the functions of three variables and describe it. $$ w(x, y, z)=x^{2}+y^{2}+z^{2}, c=9 $$

Find the level surface for the functions of three variables and describe it. $$ w(x, y, z)=x^{2}+y^{2}-z^{2}, c=-4 $$

Find the level surface for the functions of three variables and describe it. $$ w(x, y, z)=x^{2}+y^{2}-z^{2}, c=4 $$

Find the level surface for the functions of three variables and describe it. $$ w(x, y, z)=9 x^{2}-4 y^{2}+36 z^{2}, c=0 $$

For the following exercises, find an equation of the level curve of \(f\) that contains the point \(P .\) $$ f(x, y)=1-4 x^{2}-y^{2}, P(0,1) $$