Question

Draw a Contour Map of the function showing several level Curves.

\(f\left( {x,y} \right) = \sqrt {{y^2} - {x^2}} \)

Short answer

The contour map is:

Step-by-step solution

Assume a Constant

Let us remember that the level curves of a function of several variables shows where the graph of function has height k, where k is the z-value of level curve.

The level curves of given function are

\(\begin{aligned}{l}f(x,y) &= k\\\sqrt {{y^2} - {x^2}} &= k\\{y^2} - {x^2} &= {k^2}\\{y^2} &= {k^2} + {x^2}\\y &= \pm \sqrt {{k^2} + {x^2}} \end{aligned}\)

Which are upward and downward parabolas instead of\(k = 0\)when they are linear functions \(y = \pm x\).

Construct Contour Map

The level curve for \(k = 0,1,2,3\)are shown on the following graph: