Section 1
Functions Of Several Variables
Find the domain of the function. $f(x, y)=\sqrt{x}+\sqrt{y}$
Find the domain of the function. $f(x, y)=\sqrt{x+y}$
Find the domain of the function. $f(x, y)=\frac{y}{x}-\frac{x}{y}$
Find the domain of the function. $f(x, y)=\sin \frac{1}{x y}$
Find the domain of the function. $g(x, y)=\sqrt{x^{2}+y^{2}-25}$
Find the domain of the function. $g(x, y)=\sqrt{25-x^{2}-y^{2}}$
Find the domain of the function. $f(x, y)=\frac{1}{x+y}$
Find the domain of the function. $f(u, v)=\ln \frac{u^{2}+v^{2}}{\left(u^{2}-v^{2}\right)^{2}}$
Find the domain of the function. $f(x, y, z)=\sqrt{1-x^{2}-y^{2}-z^{2}}$
Find the domain of the function. $f(x, y, z)=\frac{1}{x y z}$
Find the domain of the function. $g(x, y, z)=\frac{x}{y}-\frac{y}{z}+\frac{z}{x}$
Find the domain of the function. $f(x, y, z)=\frac{x y z}{(x+y)^{3}-(x+z)^{3}}$
Sketch the level curve $f(x, y)=c$. $f(x, y)=3 x-y ; c=2,3$
Sketch the level curve $f(x, y)=c$. $f(x, y)=6 x^{2} ; c=6,24$
Sketch the level curve $f(x, y)=c$. $f(x, y)=x^{2}+4 y^{2} ; c=1,4$
Sketch the level curve $f(x, y)=c$. $f(x, y)=x^{2}-y ; c=-2,2$
Sketch the level curve $f(x, y)=c$. $f(x, y)=x^{2}-y^{2} ; c=-1,0,1$
Sketch the level curve $f(x, y)=c$. $f(x, y)=2 y-\cos x ; c=0,1,2$
Sketch the graph of $f$. $f(x, y)=x+2 y$
Sketch the graph of $f$. $f(x, y)=2 x-3 y+4$
Sketch the graph of $f$. $f(x, y)=\sqrt{4-x^{2}-y^{2}}$
Sketch the graph of $f$. $f(x, y)=\sqrt{4 x^{2}+9 y^{2}}$
Sketch the graph of $f$. $f(x, y)=x^{1 / 3}$
Sketch the graph of the equation. $z=2$
Sketch the graph of the equation. $x=-3$
Sketch the graph of the equation. $z=y^{2}$
Sketch the graph of the equation. $z=x^{3}+1$
Sketch the graph of the equation. $z=y^{3}-1$
Sketch the graph of the equation. $x=\sqrt{1-y^{2}}$
Sketch the graph of the equation. $x=\sqrt{4-y^{2}-z^{2}}$
Sketch the graph of the equation. $x=\sqrt{y^{2}+4 z^{2}}$
Sketch the graph of the equation. $y=\sqrt{1-x^{2}-z^{2}}$
Sketch the level surface $f(x, y, z)=c$. $f(x, y, z)=2 x-4 y+z ; c=-1$
Sketch the level surface $f(x, y, z)=c$. $f(x, y, z)=x^{2}+y^{2}+z^{2} ; c=2$
Sketch the level surface $f(x, y, z)=c$. $f(x, y, z)=4 x^{2}+4 y^{2}+z^{2} ; c=1$
Sketch the level surface $f(x, y, z)=c$. $f(x, y, z)=x^{2}+y^{2}-z^{2} ; c=0$
Sketch the level surface $f(x, y, z)=c$. $f(x, y, z)=z-1-x^{2}-y^{2} ; c=2$
Sketch the quadric surface. $\frac{x^{2}}{4}+y^{2}+\frac{z^{2}}{9}=1$
Sketch the quadric surface. $x^{2}+2 y^{2}+3 z^{2}=6$
Sketch the quadric surface. $x^{2}+z^{2}=4$
Sketch the quadric surface. $y^{2}+z^{2}=9$
Sketch the quadric surface. $z=x^{2}+\frac{y^{2}}{9}$
Sketch the quadric surface. $x=y^{2}+\frac{z^{2}}{4}$
Sketch the quadric surface. $z^{2}=x^{2}+4 y^{2}$
Sketch the quadric surface. $x^{2}=9 y^{2}+4 z^{2}$
Sketch the quadric surface. $y=1-x^{2}$
Sketch the quadric surface. $x=z^{2}+3$
Sketch the quadric surface. $z=y^{2}-4 x^{2}$
Sketch the quadric surface. $x=4 z^{2}-y^{2}$
Sketch the quadric surface. $y^{2}-x^{2}=4$
Sketch the quadric surface. $z^{2}-y^{2}=9$
Sketch the quadric surface. $z^{2}+4 y^{2}-2 x^{2}=1$
Sketch the quadric surface. $4 x^{2}+y^{2}-z^{2}=16$
Sketch the quadric surface. $z^{2}-4 y^{2}-x^{2}=1$
Sketch the quadric surface. $x^{2}-9 y^{2}-4 z^{2}=36$
In each of the following, determine a function $f$ of two variables (different from $F$ ) and a function $g$ of one variable such that $F=g \circ f$.a. $F(x, y)=\sqrt{4-x^{2}-y^{2}}$b. $F(x, y)=e^{x \sqrt{y}}$
Match the given function with one of the given surfaces and one of the given sets of level curves. $f(x, y)=|x|+|y|$
Match the given function with one of the given surfaces and one of the given sets of level curves. $g(x, y)=x^{2}+y$
Match the given function with one of the given surfaces and one of the given sets of level curves. $h(x, y)=x e^{-\left(x^{2}+y^{2}\right)}$
Match the given function with one of the given surfaces and one of the given sets of level curves. $k(x, y)=\sin x \sin y$
Express the height $h$ of a right circular cylinder as a function of the volume $V$ and radius $r$.
Express the radius $r$ of the base of a right circular cone as a function of the volume $V$ and height $h$.
Express the surface area $S$ of a rectangular box with no top as a function of the dimensions $x, y$, and $z$.
Express the amount $A$ of metal required to make a storage box in the shape of a rectangular parallelepiped as a function of the length $x$, width $y$, and height $z$ if the box is to have 12 compartments in 2 rows of 6 each and no top (Figure 13.19).
Express the cost $C$ of painting a rectangular wall as a function of the dimensions $x$ and $y$ (in meters) if the cost per square meter is $\$ 0.30$.
Express the cost $C$ of painting a rectangular wall as a function of the dimensions $x$ and $y$ (in meters) if the cost per square meter is $\$ 0.30$ and the wall contains a window1 square meter in area.
The strength of the electric field at $(x, y, z)$ due to an infinitely long charged wire lying along the $z$ axis is given by$$E(x, y, z)=\frac{c}{\sqrt{x^{2}+y^{2}}}$$where $c$ is a positive constant. Describe the level surfaces of $E$
The magnitude of the gravitational force exerted on a unit mass at $(x, y, z)$ by a point mass located at the origin is given by$$F(x, y, z)=\frac{c}{x^{2}+y^{2}+z^{2}}$$where $c$ is a positive constant. Describe the level surfaces of $F$
Suppose a thin metal plate occupies the first quadrant of the $x y$ plane and the temperature at $(x, y)$ is given by $$T(x, y)=x y$$Describe the isothermal curves, that is, the level curves of $T$.
Let $f(x, y)=(x+1)(y+2)$ for $x \geq 0$ and $y \geq 0 .$ Sketch the level curves $f(x, y)=3$ and $f(x, y)=4$. (If $f$ represents a utility function for two competing goods such as beer and wine, then the level curves are called indifferencecurves.)