Consider the function f(x, y) = x^2 (y^2/9). (a) Sketch a contour plot with (at least) levels k = 0, 1, 2, 3.

Consider the function f(x, y) = x^2 − (y^2/9). (a) Sketch a contour plot with (at least) levels k = 0, ±1, ±2, ±3. Plot a point at (2, 6) and roughly sketch a "path of greatest descent" starting from this point. That is, a path along which the directional derivative is as low a number as possible. (b) Find a formula for your curve in part a. That is, find a curve r(t) = hx(t), y(t)i such that −∇f = r 0 (t) and r(0) = <2, 6>. (This is technically an ODE, but one you can solve.) (c) In your curve from part b, solve for y in terms of x and plot the function. Is it close to your guess from part a? (d) Compare, qualitatively, what happens to the limit lim t→∞ r(t) in the three cases: (i) r(0) = <10, .01> (ii) r(0) = <10, 0> (iii) r(0) = <10, −.01>