(1) (10pt) Suppose f : R 2 R is a differentiable function, and : I R 2 is an arclength parametrized curve, with coordinate expression (s) = (x(s), y(s)). Show that if f((s)) = 0 for all s I, then (0.1) n(s) = 1 p (fx) 2 (fy) 2 (fx, fy), t(s) = 1 p (fx) 2 (fy) 2 (fy, fx). Here fx and fy are partial derivatives of f with respect to x and y respectively. (2) (15pt) Explain why (0.2) fxx(x (s))2 2fxyx (s)y (s) fyy(y (s))2 fxx (s) fyy (s) = 0. Hint: differentiate f(x(s), y(s)) = 0 twice and use chain rule