24_new - EXAM 02 gilbert (55035) This print-out should have...

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EXAM 02 – gilbert – (55035) 1 This print-out should have 14 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Determine the maximum value of f ( x, y ) = 8 x 2/3 y 1/3 subject to the constraint g ( x, y ) = 2 x + y 6 = 0 . 1. maximum = 14 2. maximum = 17 3. maximum = 15 4. maximum = 18 5. maximum = 16 correct Explanation: By the Method of Lagrange multipliers, the extreme values occur at the common solutions of ( f )( x, y ) = λ ( g )( x, y ) , g ( x, y ) = 0 . Now ∂f ∂x = 16 3 x −1/3 y 1/3 , ∂f ∂y = 16 3 x −1/3 y 1/3 . Thus ( f )( x, y ) = D 16 3 x −1/3 y 1/3 , 3 8 x 2/3 y −2/3 E . On the other hand, ( g )( x, y ) = h 2 , 1 O . But then by the condition on f and g , 16 3 x −1/3 y 1/3 = 2 λ , 8 3 x 2/3 y −2/3 = λ , which after simplification gives λ = 8 3 x −1/3 y 1/3 = 8 3 x 2/3 y −2/3 , I.E., y = x . Thus by the constraint equation, g ( x, x ) = 2 x + x 6 = 0 , I.E., x = 2. Consequently, ( 2 , 2 ) , is a critical point at which f (2 , 2) = 8(2) 2/3 (2) 1/3 = 16 . And so f has maximum value = 16 subject to the constraint g ( x, y ) = 0. keywords: 002 10.0 points The graph of g ( x, y ) = 0 is shown as a dashed line in 1 2 3 3 2 1 -0 y -1 -2 -3 x P -3 -2 -1 while the level curves f ( x, y ) = k for a func- tion z = f ( x, y ) are shown as continuous curves with values of k listed at the edges. Which one of the following properties does f have at P subject to the constraint g ( x, y ) = 0? 1. a local max at P
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EXAM 02 – gilbert – (55035) 2 2. a local min at P correct 3. neither local max nor local min at P Explanation: The principle underlying the method of La-grange Multipliers is that z = f ( x, y ) will have a local extremum at P subject to the constraint g ( x, y ) = 0 when (i) P lies on the the graph of g ( x, y ) = 0, (ii) the level curve f ( x, y ) = k through P and the graph of g ( x, y ) = 0 have a common tangent at P . For then P is a solution of the equations ( f )( x, y ) = λ ( g )( x, y ) , g ( x, y ) = 0 . This local extremum will be a local max- imum if the value of k is increasing as one approaches P on the graph of g ( x, y ) = 0, and will be a local minimum if the value of k is decreasing as one approaches P . Consequently, subject to the constraint g ( x, y ) = 0, f has a local min at P . 003 10.0 points If z = f ( x, y ) and f x (4 , 2) = 4 , f y (4 , 2) = −3 , find dz dt at t = 2 when x = g ( t ) , y = h ( t ) and g (2) = 4 , g (2) = 4 . h
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This note was uploaded on 04/04/2012 for the course M 408d taught by Professor Sadler during the Spring '07 term at University of Texas at Austin.

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24_new - EXAM 02 gilbert (55035) This print-out should have...

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