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MULTIVARIABLE CALCULUS
EXAM 2
FALL 2012
Name: Honor Code Statement:
Directions: Complete all problems. Justify all answers/solutions. Definitions are worth 4 points, true/false questions are worth 2 points each, and any other problem is worth 10 points. Calculators are not permitted. Best of luck.
(1) Calculate the directional derivative of f (x, y, z) = xyz at the point a = (− 1 , 0 , 2) in the direction parallel to the vector u = 2 √k− 5 i.
(2) Define: curvature κ of a path x in R^3.
Date: November 15, 2012. 1
(3) Calculate the plane tangent to the surface whose equation is x^2 − 2 y^2 +5xz = 7 at the point (− 1 , 0 , − 56 ) by first computing the gradient (and then using this computation).
(4) Compute the length of the path x : [0, 2 π] → R^2 , x(t) = (3 cos(t), 3 sin(t)).
(7) Identify and determine the nature of the critical points of the function f (x, y) = x^2 − y^3 − x^2 y + y.
(8) Set up (but do not solve) the system of equations (via the method of Lagrange multipliers) for finding the critical points of f (x, y, z) = x + y + z subject to the constraints x + 2z = 1, y^2 − x^2 = 1.
