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A midterm exam for math 251 at simon fraser university, held on march 7, 2006. The exam consists of 6 questions, worth a total of 50 points, covering topics such as limits, critical points, directional derivatives, differentials, and lagrange multipliers.
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Midterm 2 7 March 2006, 5:30โ6:20pm
Instructor: Ralf Wittenberg
Instructions
(a) [4 points]
lim (x,y)โ(0,0)
3 xy x^2 + 2y^2
(b) [4 points]
lim (x,y)โ(0,0)
3 xy^2 x^2 + 2y^2
F (x, y, z) = x^2 z โ
y z^2
(z > 0).
(a) [5 points] Find the directional derivative of F at the point P (2, โ 3 , 1) in the direction of the vector v = i + 5j โ 2 k.
(b) [2 points] Find the maximum rate of change of F at the point P (2, โ 3 , 1). In the direction of which unit vector u is the directional derivative a maximum?
(c) [3 points] Find an equation to the tangent plane to the surface
F (x, y, z) = x^2 z โ
y z^2
at the point P (2, โ 3 , 1).
x^2 y^3 z^4
(a) [4 points] Find the differential dw.
(b) [4 points] Suppose x increases by 1%, y increases by 2% and z increases by 3%. Using your answer from (a): By approximately what percentage will the value of w increase or decrease?