t5 - 2 f x 2 =-6 x , 2 f y 2 =-6 y , 2 f xy = 15. (3) (20...

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Methods of Calculus, MAC3233 Test 5 December 14, 1990 (1) (30 points) Find the indicated quantities. (a) f ( x,y ) = 4 x 3 + 3 x 4 y 5 + 5 y 7 , ∂f ∂y = (b) q ( x,y,z ) = yz y + x , ∂q ∂y (1 , 3 , 5) = (c) g ( x,y ) = (8 x 4 y + 6 x ) 26 , ∂g ∂x = (d) f ( A,B ) = ( AB ) 8 , ∂f ∂B = (e) h ( x,y,z ) = ln( x + z 3 ) e yz , ∂h ∂z = (f) r ( x,y ) = x 4 y + x 2 y 4 , 2 r ∂y 2 = (2) (30 points) Find ALL points ( x,y ) where f ( x,y ) has a possible relative maximum or minimum. Then use the second derivative test to state the nature of f ( x,y ) at each point. (a) f ( x,y ) = 8 x 2 - 4 xy + y 2 - 3 y (b) f ( x,y ) = 15 xy - x 3 - y 3 Note: ∂f ∂x = 15 y - 3 x 2 , ∂f ∂y = 15 x - 3 y 2 ,
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Unformatted text preview: 2 f x 2 =-6 x , 2 f y 2 =-6 y , 2 f xy = 15. (3) (20 pts.) I want to use 5 rectangular pieces of glass to build an aquarium of volume 4000 cubic inches (the top will be open). Find the dimensions which minimize the amount of glass which I must use. (4) (20 points) Use the method of Lagrange multipliers to minimize 7 x 2-2 x + y 2 subject to the constraint 2 x-5 y =-102. 1...
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