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CALCULUS III, SPRING 2005 SAMPLE MIDTERM EXAM 2 YOU MUST JUSTIFY YOUR ANSWERS. NO CALCULATORS. (1) Consider the curve r(t) = (t, t2 , t3 ). Let P = (1, 1, 1). (a) Find the equation of the normal plane to the curve at P. (b) Find the curvature κ at P. (2) Reparametrize the following curve with respect to arclength, starting at t = 0 in the direction of increasing t, r(t) = (3t + 1, 2 − 5t, 4t − 3) (3) (a) Decide if the function ( f (x, y) = x x2 +y 2 0 is continuous at (0, 0) (b) Show that the function ( 2 g(x, y) = xy x2 +y 2 0 (x, y) 6= (0, 0) (x, y) = (0, 0) (x, y) 6= (0, 0) (x, y) = (0, 0) is continuous at (0, 0). ∂g (4) (a) A function g(u, v) satisfies: g(3, 4) = 10, ∂u (3, 4) = 2, ∂g (3, 4) = −4. Use this information to estimate g(2.8, 4.1). ∂v (b) If f (x, y, z) = xyez and x = cos(t), y = sin(t), z = t2 , use . Express your answer as a the chain rule to compute df dt function of t. (5) Consider two parametrizations of the same curve: r(t) and r(s), where s = f (t). Show that if s0 = f (t0 ), then the vectors (T(t0 ), N(t0 ), B(t0)) = (T(s0 ), N(s0), B(s0 )) i.e., the TNB-frame is independent of parametrization.