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Unformatted text preview: Quiz 8 Name: Math 240  Calculus III April 7, 2009 Note: In order to receive full credit, you must show work that justifies your answer. 1. (6 points) Define functions f and g such that f x y = e xy p x 2 + y 2 and x y = g u v = 3 u 3 u 2 5 v . If u v = 5 5 and the rate of change of the outputs is ˙ f 1 ˙ f 2 = 60 12 , find ˙ u ˙ v , the rate of change of the inputs. Solution : By the chain rule, ˙ f 1 ˙ f 2 = D ( f ◦ g ) u v ˙ u ˙ v = Df g u v Dg u v ˙ u ˙ v . Computing the determinant matrices: ˙ f 1 ˙ f 2 = " ye xy xe xy x √ x 2 + y 2 y √ x 2 + y 2 # 3 2 u 5 ˙ u ˙ v . When u = v = 5, x = 12 and y = 0, so we have 60 12 = 0 12 1 3 10 5 ˙ u ˙ v = 120 60 3 ˙ u ˙ v which we solve to find ˙ u = 4 and ˙ v = 7. 2. (4 points) Find the directional derivative of f ( x, y ) = e x cos y at (0 , π 4 ) in the direction of v = h 1 , 1 i . Solution : The gradient of f is ∇ f ( x, y ) = h e x cos y, e x sin y i , so ∇ f ( , π 4 ) = D √ 2 2 , √ 2 2 E . The directional derivative is then....
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This note was uploaded on 09/09/2009 for the course MATH 240 taught by Professor Cremins during the Spring '07 term at Maryland.
 Spring '07
 Cremins
 Calculus, Linear Algebra, Algebra

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