Q 48 - Calculus

Question

In Exercises $43-52$, sketch the level curves $c=-3,-2,-1,0,1,2,3$ if they exist for the specified function.

$f(x,y)=x^2-y^2$.

Step-by-Step Solution

Answer

The required level curves $c=-3,-2,-1,0,1,2,3$ for the function $f\left(x,y\right)=x^2-y^2$ as shown as following :-

Step 1. Given Information

We have given the following function :-

$f(x,y)=x^2-y^2$.

We have to graph the level curves $c=-3,-2,-1,0,1,2,3$ for this function.

Step 2. Graph of level curves

For each value of $c$, the level curve is the graph of the equation $f\left(x,y\right)=c$.

That is we have to graph the following equations :-

$$\begin{gathered} x^2-y^2=-3, \\ {x}^2-y^2=-2, \\ {x}^2-y^2=-1, \\ {x}^2-y^2=-0, \\ {x}^2-y^2=1, \\ {x}^2-y^2=2 and \\ {x}^2-y^2=3, \end{gathered}$$

As these level curves are hyperbolas.

Then we have the following graph of these level curves :-