23s08Exam2-OLD-Answers

23s08Exam2-OLD-Answers - x 2 + y 2 25 } . Then the volume...

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OLD MATHEMATICS 23 EXAMINATION 2 SPRING 2008 Reminder: The actual examination may be very different. (1) let z = f ( x,y ) where f is differentiable, x = g ( t ), y = h ( t ), g (3) = 2, g 0 (3) = 5, h (3) = 7, h 0 (3) = - 4, f x (2 , 7) = 5, f y (2 , 7) = - 8. Find dz dt when t = 3. Justify your answer. Answer : 57. ± (2) Find the absolute maximum and minimum values of f ( x,y ) = 2 x 3 + y 4 on the closed unit disk D = { ( x,y ) | x 2 + y 2 1 } . Answer : The absolute maximum value is 2 and the absolute minimum value is - 2. ± (3) Use Lagrange multipliers to find the maximum value of f ( x,y,z ) = 2 x + 6 y + 10 z subject to x 2 + y 2 + z 2 = 35. Answer : 70. ± (4) Set up a double integral giving the volume of the solid inside the sphere x 2 + y 2 + z 2 = 25 and outside the cylinder x 2 + y 2 = 16. Answer : Let D = { ( x,y ) | 16
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Unformatted text preview: x 2 + y 2 25 } . Then the volume is given by V = ZZ D 2 p 25-x 2-y 2 dA. (5) Evaluate the above integral. Answer : 36 . (6) Find the area of the part of the surface z = xy lying within the cylinder x 2 + y 2 = 1. Answer : 2 3 (2 2-1). (7) Evaluate the integral Z a-a Z a 2-y 2 ( x 2 + y 2 ) 3 2 dxdy . Answer : a 5 5 . (8) A lamina occupies the part of the disk x 2 + y 2 4 which lies in the upper half plane. If the density is a positive constant, nd the center of mass of the lamina. Answer : ( x, y ) = ( , 8 3 ) . 1...
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This note was uploaded on 10/06/2008 for the course MATH 22 taught by Professor Dodson during the Spring '05 term at Lehigh University .

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