4.1 Functions of several variables  (Page 5/12)

$W(x,y)=4x^2+y^2.$ Find $W(2,\mathrm{-1}),$ $W(\mathrm{-3},6).$

$W(x,y)=4x^2+y^2.$ 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, $x\text{and}y,$ find $V(10,2),$ and explain what this means.

$V(x,y)=4x^2+y^2$

$f(x,y)=\sqrt{x^2+y^2-4}$

$f(x,y)=4\text{ln}(y^2-x)$

$g(x,y)=\sqrt{16-4x^2-y^2}$

$z(x,y)=y^2-x^2$

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

$g(x,y)=\sqrt{16-4x^2-y^2}$

$V(x,y)=4x^2+y^2$

$z=y^2-x^2$

$z(x,y)=y^2-x^2,$ $c=1$

$z(x,y)=y^2-x^2,$ $c=4$

$g(x,y)=x^2+y^2;c=4,c=9$

$g(x,y)=4-x-y;c=0,4$

$f(x,y)=xy;c=1;c=\mathrm{-1}$

$h(x,y)=2x-y;c=0,\mathrm{-2},2$

$f(x,y)=x^2-y;c=1,2$

$g(x,y)=\frac{x}{x+y};c=\mathrm{-1},0,2$

$g(x,y)=x^3-y;c=\mathrm{-1},0,2$

$g(x,y)=e_{}^{xy};c=\frac{1}{2},3$

$f(x,y)=x^2;c=4,9$

$f(x,y)=xy-x;c=\mathrm{-2},0,2$

$h(x,y)=\text{ln}(x^2+y^2);c=\mathrm{-1},0,1$

$g(x,y)=\text{ln}\left(\frac{y}{x^2}\right);c=\mathrm{-2},0,2$

$z=f(x,y)=\sqrt{x^2+y^2},$ $c=3$

$f(x,y)=\frac{y+2}{x^2},$ $c=$ any constant

$z=4-x-y;x=2$

$f(x,y)=3x+y^3,x=1$

$z=\text{cos}\sqrt{x^2+y^2}$ $x=1$

$z=\sqrt{100-4x^2-25y^2}$

$z=\text{ln}\left(x-y^2\right)$

$f(x,y,z)=\frac{1}{\sqrt{36-4x^2-9y^2-z^2}}$

$f(x,y,z)=\sqrt{49-x^2-y^2-z^2}$

$f(x,y,z)=\sqrt[3]{16-x^2-y^2-z^2}$

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

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

$z=x^2+y^2$

Use technology to graph $z=x^{2}y.$

$f(x,y)=\sqrt{4-x^2-y^2}$

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

$z=1+e^{\text{−}{x}^2-y^2}$

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

$z=y^2-x^2$

Describe the contour lines for several values of $c$ for $z=x^2+y^2-2x-2y.$

$w(x,y,z)=x-2y+z,c=4$

$w(x,y,z)=x^2+y^2+z^2,c=9$

$w(x,y,z)=x^2+y^2-z^2,c=\mathrm{-4}$

$w(x,y,z)=x^2+y^2-z^2,c=4$

$w(x,y,z)=9x^2-4y^2+36z^2,c=0$

$f(x,y)=1-4x^2-y^2,P(0,1)$

$g(x,y)=y^2\text{arctan}x,P(1,2)$

$g(x,y)=e^{xy}(x^2+y^2),P(1,0)$

The strength $E$ of an electric field at point $\left(x,y,z\right)$ resulting from an infinitely long charged wire lying along the $y\text{-axis}$ is given by $E(x,y,z)=k\text{/}\sqrt{x^2+y^2},$ where $k$ is a positive constant. For simplicity, let $k=1$ and find the equations of the level surfaces for $E=10\text{and}E=100.$

A thin plate made of iron is located in the $xy\text{-plane.}$ The temperature $T$ in degrees Celsius at a point $P\left(x,y\right)$ is inversely proportional to the square of its distance from the origin. Express $T$ as a function of $x\text{and}y.$

Refer to the preceding problem. Using the temperature function found there, determine the proportionality constant if the temperature at point $P\left(1,2\right)is50\text{°}\text{C}.$ Use this constant to determine the temperature at point $Q\left(3,4\right).$

Refer to the preceding problem. Find the level curves for $T=40\text{°}\text{C and}T=100\text{°}\text{C},$ and describe what the level curves represent.