## Problem of level curves

Sketch the level curves of $f(x,y) = xy$.
• ## Solution

Recall that
For $f(x,y) = xy$, the level curve of value $c$ is given by $$xy=c.$$
We now sketch the resulting curves for a couple values of $c$.
The $c=0$ curve consists of all points satisfying $xy=0$. That is, it has points where $x=0$ or $y=0$:
The $c=1$ and $c=2$ level curves are given by $y=1/x$ and $y=2/x$:
Similarly, the $c=-1$ and $c=-2$ level curves are given by $y = -1/x$ and $y = -2/x$. Hence our sketch of the level curves of $f$ looks like: