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Unformatted text preview: MAC 2313 Dec 7, 2010 Final Exam and Key Prof. S. Hudson 1) Find the position and velocity vectors of the particle, given that a ( t ) =- cos( t ) i- sin( t ) j , v (0) = i and r (0) = j . 2) Find the directional derivative of f ( x,y ) = 4 x 3 y 2 at P (2 , 1) in the direction of a = 4 i- 3 j . 3) Show that the following is independent of path and compute it: R (3 , 2) (0 , 0) 2 xe y dx + x 2 e y dy . 4) True-False: If D is an open set in R 2 then every point of D is an interior point o f D . If u is a fixed unit vector and D u f ( x,y ) = 0 for all points ( x,y ) then f is a constant function. If f is a differentiable function on the closed unit disk x 2 + y 2 1 in R 2 , then f has a critical point somewhere on this disk. If the line y = 2 is a contour of f ( x,y ) through (4 , 2) then f x (4 , 2) = 0. If the graph of z = f ( x,y ) is a plane in 3-space, then both f x and f y are constant functions....
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