Match the function to its contour diagram

Zen level — Class exercise, Section 12.3, Contour Diagrams

These are the nine functions from the zen-level graph exercise, now as contour diagrams. Match each function (a)–(i) with its diagram (I)–(IX). Useful questions: Is the diagram symmetric about the origin, the axes, or both? Are the values positive, negative, or both? What happens far from the origin?

(a) \(z = xy\,e^{-(x^2+y^2)}\)
(b) \(z = \cos\!\left(\sqrt{x^2+y^2}\right)\)
(c) \(z = \sin y\)
(d) \(z = -\dfrac{1}{x^2+y^2}\)
(e) \(z = \cos^2 x\,\cos^2 y\)
(f) \(z = \dfrac{\sin(x^2+y^2)}{x^2+y^2}\)
(g) \(z = \cos(xy)\)
(h) \(z = |x|\,|y|\)
(i) \(z = (2x^2+y^2)\,e^{\,1-x^2-y^2}\)

Answer key

(a) \(z = xy\,e^{-(x^2+y^2)}\) → (IV). Four separate families of closed curves, one in each quadrant, positive in the first and third and negative in the second and fourth: the four lobes. The axes are the zero contour and the function dies out far away, so there are no contours near the edges.
(b) \(z = \cos\!\left(\sqrt{x^2+y^2}\right)\) → (IX). Concentric circles whose values cycle between −1 and 1 forever; the rings stay the same width because cos r keeps its amplitude. Compare (f).
(c) \(z = \sin y\) → (VI). No x, so the contours are horizontal lines, with values going up and down periodically as you move along the y-axis.
(d) \(z = -\dfrac{1}{x^2+y^2}\) → (II). Concentric circles with negative values that plunge toward −∞ at the origin: the circles bunch up tightly near the center and spread out far away.
(e) \(z = \cos^2 x\,\cos^2 y\) → (I). A grid of closed curves (bumps) centered at the points (mπ, nπ), values between 0 and 1, with the zero level along the lines x = ±π/2, ±3π/2, … and y = ±π/2, …. Compare (g).
(f) \(z = \dfrac{\sin(x^2+y^2)}{x^2+y^2}\) → (VIII). Concentric circles whose values shrink outward (the ripples die out) and whose spacing shrinks too, because sin(r²) oscillates faster and faster. The center is the peak, value near 1.
(g) \(z = \cos(xy)\) → (VII). The contours are the hyperbolas xy = const, bunching up toward the corners; the value is 1 all along both axes.
(h) \(z = |x|\,|y|\) → (III). Hyperbolas in all four quadrants, all with positive values (unlike xy), symmetric across both axes.
(i) \(z = (2x^2+y^2)\,e^{\,1-x^2-y^2}\) → (V). Closed curves around a rim: the value is 0 at the origin, rises to a crest, then falls off. The crest is higher on the x-axis (value 2 at (±1, 0)) than on the y-axis (value 1 at (0, ±1)), which is why the contours above 1 are two separate ovals on the x-axis while the low ones circle the whole crater. (At the points (0, ±1) the function has saddle points with value exactly 1; that level curve would cross itself there, so it is not drawn.)