ertet - f at P in the direction of the vector 3 i 4 j(b...

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MATH 241H - EXAM 2 - October 11, 2004 No Calculators - Show your work. [20] 1. Let f ( x, y ) = ± x 2 - 2 y 2 x 2 + y 2 if ( x, y ) 6 = (0 , 0) 1 if ( x, y ) = (0 , 0). (a) Find lim ( x,y ) (2 , 1) f ( x, y ) or explain why it doesn’t exist. (b) Find lim ( x,y ) (0 , 0) f ( x, y ) or explain why it doesn’t exist. (c) Find f x (0 , 0) or explain why it doesn’t exist. (d) Find f y (0 , 0) or explain why it doesn’t exist. [15] 2. A rectangular block of ice is melting so that the length is decreasing at a rate of 3 inches per hour, the width is decreasing at a rate of 2 inches per hour, and the height is decreasing at a rate of 4 inches per hour. How fast is the volume decreasing when the length is 10 inches, the width is 5 inches, and the height is 8 inches? [25] 3. Let f ( x, y ) = 3 x 2 + 4 y 2 and let P = ( - 2 , 1). (a) Find the directional derivative of
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Unformatted text preview: f at P in the direction of the vector 3 i + 4 j . (b) Find the maximal directional derivative of f at P . (Not the direction in which f increases most rapidly, but the maximal possible value of D u f (-2 , 1) as u varies.) (c) Find an equation of the plane tangent to the graph z = f ( x, y ) at the point (-2 , 1 , 16). [20] 4. Find all critical points of the function f ( x, y ) = x 3 + y 2-6 xy and determine whether each is a local maximum, local minimum, or saddlepoint. [20] 5. Use the method of Lagrange multipliers to find the absolute maximum and minimum values of f and the points at which they occur when f ( x, y ) = x 2-4 y is restricted to the circle x 2 + y 2 = 25....
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This note was uploaded on 04/04/2010 for the course MATH 113 taught by Professor Staff during the Spring '08 term at Maryland.

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