Chapter 4

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

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

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. $$z=\cos \sqrt{x^{2}+y^{2}}$$

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

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 \(P\). $$f(x, y)=1-4 x^{2}-y^{2}, P(0,1)$$

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