# 1704 - Math 20300 Spring 2006 Final Exam Answers: 1a) 4( x...

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Math 20300 Spring 2006 Final Exam Answers: 1a) b) 4( 1) 7( 5 0 xy z !" # 1 1 47 5 x y z ! ! # # c) 90 2 2a) 23 2 3, 2 , z z x yx ye y e " b) 3 7 c) 3( 3( 0 z ! "! " # 3a) Absolutely convergent – use the limit comparison test with 3 2 1 n \$ . b) Conditionally convergent. Use the integral test to show that the series is not absolutely convergent and the alternating series test to then show it is conditionally convergent. c) Divergent – the terms of the series do not have a limit. ( 2 1 2 41 n n % " as , so in the limit the terms of the alternating series oscillate between n %& 1 2 and 1 2 ! .) 4. The volume is 8 times the volume of the portion in the first octant. This gives 2 3 22 4 2 00 0 81 r V r dz dr d 6 ( ! ## )) ) (The upper z -limit is found by expressing the boundary condition 31 44 x y z #! ! in cylindrical coordinates.) 5. The series is absolutely convergent in the interval 11 13 , * + , - . / , i.e. 1 3 4 x ! 0 . 6. The critical points are (0 , , and ,0) (3,6) (3, 6) ! . Local maximum: (0 Saddle points: and (3 . (3,6) , 6) ! 7. 2 1 1 2 1 2 8 x M dy dx " )) , 2 1 1 1 2 ln 2 4 x y Mx dy dx " , 2ln 2 x # .

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## This note was uploaded on 10/29/2011 for the course MATH 190 taught by Professor Ocken during the Spring '11 term at CUNY Hunter.

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1704 - Math 20300 Spring 2006 Final Exam Answers: 1a) 4( x...

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