quiz5 - Problem 4. A not uncommon calculus mistake is to...

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Putnam Exam Seminar Quiz 5 Fall 2010 October 18 Problem 1. Find the unique function u ( t ) so that u 0 ( t ) = u ( t ) + Z 1 0 u ( s ) ds and u (0) = 1. [Putnam Exam, 1958, 3] Problem 2. If u = 1 + x 3 3! + x 6 6! + · · · , v = x + x 4 4! + x 7 7! + · · · , w = x 2 2! + x 5 5! + x 8 8! + · · · , then prove that u 3 + v 3 + w 3 - 3 uvw = 1 . [Putnam Exam, 1939, 14] Problem 3. For all real x , the real-valued function y = f ( x ) satisfies y 00 - 2 y 0 + y = 2 e x . (a) If f ( x ) > 0 for all real x , must f 0 ( x ) > 0 for all real x ? Explain. (b) If f 0 ( x ) > 0 for all real x , must f ( x ) > 0 for all real x ? Explain. [Putnam Exam, 1987, A-3]
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Unformatted text preview: Problem 4. A not uncommon calculus mistake is to believe that the product rule for derivatives says that ( fg ) = f g . If f ( x ) = e x 2 determine, with proof, whether there exists an open interval ( a,b ) and a nonzero function g dened on ( a,b ) such that this wrong product rule is true for x in ( a,b ). [Putnam Exam, 1988, A-2]...
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