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prelim2_sp05

# prelim2_sp05 - Fall 05 Ignore Questions 6-7-8 which are on...

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Math 192, Prelim 2 April 14, 2005 1) a) (7 points) Give an example of a function f ( x, y ) that has a local maximum at (0 , 0). Show that your example is valid. b) (7 points) Give an example of a function f ( x, y ) that has a saddle point at (0 , 0). Show that your example is valid. 2) (11 points) Determine all the maxima and minima of the function f ( x, y ) = x 2 + xy on the region of points ( x, y ) satisfying x 2 + xy + y 2 1. 3) (11 points) Find the average distance from a point in the interior (or on the boundary) of a circle of radius R to the center of the circle. 4) (11 points) Let C be a circle of radius 2 + 2 2 centered at the origin. Find the area of the region outside C but inside the region bound by r = 2 + cos θ . 5) The questions below are true/false. You need not give reasons. You will be penalized for wrong answers, just like the SAT’s. So do not guess!! a) ( ± 3 points) If f ( x, y ) is given and ±u is a unit vector in two dimensions then the directional derivative of f in the direction of ±u at ( x 0 , y 0 ) is f x ( x 0 , y 0 ) ± i + f y ( x 0 , y 0 ) ± j . b) ( ± 3 points) If f ( x, y, z ) is a scalar function then the expression grad(div(
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