Let z = f(x, y) = x^2 + 4y^2, and consider the z-value 16. Find the level set for the z-value 16, and describe it geometrically. (Recall that the level set such that z = 16 is a set of points in the domain.) Find a parametrization ~r(t) for the level set in the xy-plane. Hint: Use sine and cosine. Find the time t such that the parametrization gives the point ~r(t) = (0, 2), and find the derivative of ~r at that time. Find the gradient of z at the point (0, 2), and show that the gradient of z at (0, 2) is perpendicular to ~r 0 (t) at (0, 2).

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