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Activity 11.1.6.

In this activity, we will be making contour plots by hand and looking at how the spacing of contours in your plot should give an idea about how different surface shapes can be distinguished.

Figure 11.1.25. Left: Level curves for \(f(x,y) = x^2+y^2\text{.}\) Right: Level curves for \(g(x,y) = \sqrt{x^2+y^2}\text{.}\)
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Figure 11.1.25. Left: Level curves for \(f(x,y) = x^2+y^2\text{.}\) Right: Level curves for \(g(x,y) = \sqrt{x^2+y^2}\text{.}\)
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(a)

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 left set of axes given in Figure 11.1.25. (You decide on the scale of the axes.) Explain what the surface defined by \(f\) looks like.

(b)

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 right set of axes given in Figure 11.1.25. You should use the same scale on these axes as in the previous task. Explain what the surface defined by \(g\) looks like

(c)

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.