fa01prelim3

fa01prelim3 - V be calculated from measurements of r and h...

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Prelim 3 November 27, 2001 Show all your work. Your reasoning is important. No calculators are allowed. You are allowed to use a 5 by 8 index card. The number of points each problem is worth is indicated in parentheses. Keep in mind there may be simpler methods than direct applications of calculus. Good luck. 1. (16 points) Find the directional derivative of f ( x,y ) = 2 x 2 + 3 xy - y 2 at the point (2,3) in the same direction as v = i + 3 j . 2. (16 points) Find the centroid (i.e. center of mass with uniform density) of the region bounded by y = 1 - x 2 and the x -axis. 3. Let g ( x,y ) = x 2 + 2 xy + 4 y 2 , where x = 2cos θ and y = sin θ . a. (8 points) Find dg . b. (8 points) Find the maximum and minimum values of g as a function of θ . 4. (16 points) The volume of a right circular cone of diameter D and height h is given by V = πD 2 h 12 . About how accurately may
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Unformatted text preview: V be calculated from measurements of r and h that are in error by 2% and 4%, respectively? 5. Let f ( x,y ) = x 2 y and g ( x,y ) = 2 y 2 + x 2 . a. (6 points) Sketch some level curves for f and g on the same axis. b. (14 points) Show that at any point ( x,y ) with y 6 = 0, the level curves intersect at right angles. 6. (16 points) Find the values of x and y > x such that F ( x,y ) = Z y x (6-t 2 ) dt has its maximum value. Give an explanation why your solution is a maximum. Extra Credit: (10 points) In the plane, find s and t that minimize the sum of the distance from A = ( t, 0) to B = ( s, 1-s ) plus the distance from B = ( s, 1-s ) to C = (1-t 2 , 1), where s and t are any real numbers....
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This note was uploaded on 09/29/2009 for the course 1920 1920 taught by Professor Connelly during the Fall '10 term at Cornell.

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