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MA 227, Fall, 2011CALCULUS III
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TEST II
10 questions, 10 points each.SHOW ALL YOUR WORK! CIRCLE YOUR ANSWER!
Question 1 Find the gradient of the function f (x, y) = x^2 e−xy^ at the point (1, 0).
Question 2 Find the directional derivative of the functionvector ~v = ~i + 2~j − 2 ~k at the point (2, 1 , −1). f (x, y, z) = xz^2 + yz in the direction of the
Question 3 Find local maximum, minimum and saddle points (if any) of the function f (x, y) = x^2 − 6 xy + 4y^2 − 5 y + 1.
Question 6 Let f (x, y) = xy + x y^2 and x = st, y = s^2 + t. Find partial derivatives ∂f∂s and ∂f∂t.
Question 7 Let f (x, y) = y sin(x) + x^2 y. Find all second partial derivatives: f (^) xx′′, f (^) xy′′, f (^) yy′′.
Question 8 Find equation of the tangent plane to the surface x^3 − y^3 + z^2 = −8 at the point (− 1 , 2 , 1).
Question 9 Find the maximum rate of change ofdirection does it occur? f (x, y) = y√x − xy at the point (1, −1). In which
