02 Multivariable Functions Free Online FlashCards

What is a function of several variables?

A function of several variables assigns exactly one output to each input in a specified domain.

In $z=f(x,y)$, what do $x$, $y$, and $z$ represent?

For a function of two variables, $x$ and $y$ are inputs and $z=f(x,y)$ is the output.

Evaluate $f(1,2)$ for $f(x,y)=x^2+3xy-y^2$.

Substitution gives $f(1,2)=1^2+3(1)(2)-2^2=3$.

What is the domain of $f(x,y)=\sqrt{9-x^2-y^2}$?

The domain is $x^2+y^2\le 9$, the closed disk of radius $3$ centered at the origin.

What restriction defines the domain of $f(x,y)=\frac{1}{x-y}$?

The domain excludes points where the denominator is zero: $x-y\ne0$. Geometrically, it is the plane excluding the line $y=x$.

What is the graph of $z=f(x,y)$?

The graph is the set $\{(x,y,z)\in\mathbb{R}^3:z=f(x,y)\}$, generally a surface in three-dimensional space.

What is a vertical trace?

A vertical trace fixes one input variable, such as $x=a$ or $y=b$, reducing the surface to a two-dimensional curve.

What are the level curves of $f(x,y)=x^2+y^2$?

For $f(x,y)=x^2+y^2$, the level curve $f(x,y)=c$ is $x^2+y^2=c$, a circle of radius $\sqrt{c}$ when $c>0$.

What shape do the level curves of $f(x,y)=2x-y$ have?

For $f(x,y)=2x-y$, the level curves are $y=2x-c$: parallel lines with slope $2$.

What is a level surface?

A level surface for $w=f(x,y,z)$ is the set of points satisfying $f(x,y,z)=c$.

What must be true for a two-variable limit to exist?

A multivariable limit exists only if the function approaches the same value along every path toward the point.

Why does $\lim_{(x,y)\to(0,0)}\frac{x^2-y^2}{x^2+y^2}$ not exist?

The limit does not exist: along $y=0$ the value is $1$, while along $x=0$ it is $-1$.