quiz8 - Quiz 8 Name: Math 240 - Calculus III April 7, 2009...

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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 , 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 ( , 4 ) = D- 2 2 ,- 2 2 E . The directional derivative is then....
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quiz8 - Quiz 8 Name: Math 240 - Calculus III April 7, 2009...

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