Exercise 8.9. Let g : R2 R be given by g(v, w) = v 2 w 2 . This exercise works out the contour plot of g via visual reasoning; later it will be an important special case for the study of what are called "saddle points" in the multivariable second derivative test. (a) Sketch the level set g(v, w) = 0. (b) The level set g(v, w) = 1 is a hyperbola (with two "branches"). Mark where it cuts either of the coordinate axes, and explain why for |v| or |w| large we have v 2 w 2 on this level set, so v w. Sketch the resulting hyperbola, with asymptotes v = w. (c) The level set g(v, w) = 1 is a hyperbola (with two "branches"). Mark where it cuts either of the coordinate axes, and explain why for |v| or |w| large we have v 2 w 2 on this level set, so v w. Sketch the resulting hyperbola, with asymptotes v = w. (d) For c > 0, check that g(v/ c, w/ c) = 1 precisely when g(v, w) = c. Using this and (b), explain why the level set g(v, w) = c is the same as the scaling up by the factor c of the level set in (b). (e) For c < 0, similarly relate the level set g(v, w) = c to what you drew in (c) using a scaling factor of p |c|. (f) Sketch a contour map for g in the vw-plane, labeling the level sets.