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# hw6 - x-q √ x-√ x-b ´ dx converge[Putnam Exam 1995 A2...

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Putnam Exam Seminar Assignment 6 Fall 2010 Due October 18 Exercise 1. Find all real-valued continuously differentiable functions f defined on the real line such that for all x , ( f ( x )) 2 = 1990 + Z x 0 ( f ( t )) 2 + ( f 0 ( t )) 2 dt. [Putnam Exam, 1990, B1] Exercise 2. Evaluate Z a 0 Z b 0 e max { b 2 x 2 ,a 2 y 2 } dx dy where a and b are positive. [Putnam Exam, 1989, A2] Exercise 3. Evaluate Z 2 π 0 dx 1 + e sin x dx. Exercise 4. For what pairs ( a, b ) of positive real numbers does the improper integral Z b q x + a -
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Unformatted text preview: x-q √ x-√ x-b ´ dx converge? [Putnam Exam, 1995, A2] Exercise 5. Evaluate Z 1 ln( x + 1) x 2 + 1 dx. [Putnam Exam, 2005, A5]. Exercise 6. For each continuous function f : [0 , 1] → R , let I ( f ) = R 1 f ( x ) dx and J ( x ) = R 1 x ( f ( x )) 2 dx . Find the maximum value of I ( f )-J ( f ) over all such functions f . [Putnam Exam, 2006, B5]...
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