In this activity, you will make contour plots by hand and look at how the spacing of contours in your plot should give an idea about how different surface shapes can be distinguished.
Let \(f(x,y) = x^2+y^2\text{.}\) Draw the level curves \(f(x,y) = k\) for \(k=1\text{,}\)\(k=2\text{,}\)\(k=3\text{,}\) and \(k=4\) on the axes below. Be sure to label the scale of the axes. Explain what the surface defined by \(f\) looks like.
Let \(g(x,y) = \sqrt{x^2+y^2}\text{.}\) Draw the level curves \(g(x,y) = k\) for \(k=1\text{,}\)\(k=2\text{,}\)\(k=3\text{,}\) and \(k=4\) on the axes below. Use the same scale on these axes as in the previous part. Explain what the surface defined by \(g\) looks like.
Compare and contrast the graphs of \(f\) and \(g\text{.}\) How are they alike? How are they different? Use traces for each function to help answer these questions.
