Question

In Example 3 of: http://tutorial.math.lamar.edu/Classes/CalcIII/MultiVrbleFcns.aspx
Why are we concluding that k is only going up to k=5?

Answer 1

No one is concluding that \(k\in\{1,2,3,4,5\}\). That's just what he graphed.

"The level curves (or contour curves) for this surface are given by the equation are found by substituting \(z=k\). In the case of our example this is, \(k=\sqrt{x^2+y^2}\Longrightarrow k^2=x^2+y^2\), where \(k~\underset{\overset{\mathrm{\leftrightsquigarrow\leftrightsquigarrow\leftrightsquigarrow\leftrightsquigarrow\leftrightsquigarrow\leftrightsquigarrow\leftrightsquigarrow\leftrightsquigarrow\leftrightsquigarrow}}{}}{\mathbf{is~any~number}}\)."