508F06Ex2s(1)

508F06Ex2s(1) - Signature Printed Name Exam 2 Math 508...

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Signature Printed Name Math 508 Exam 2 Jerry L. Kazdan December 8, 2006 12:00 – 1:20 Directions This exam has two parts, Part A has 3 shorter problems (8 points each, so 24 points), Part B has 5 traditional problems (15 points each, so 75 points). Closed book, no calculators – but you may use one 3 00 × 5 00 card with notes. Part A: Short Problems (3 problems, 8 points each). A–1. A continuous function f : R R has the property that Z x 0 f ( t ) dt = cos( x ) e - x + C, where C is some constant. Find both f ( x ) and the constant C . A–2. A function h : R R with two continuous derivatives has the property that h (0) = 2, h (1) = 0, and h(3)=1. Prove there is at least one point c in the interval 0 < x < 3 where h 00 ( c ) > 0 by ±nding some explicit m > 0 (such as m = 2 / 3) with h 00 ( c ) m . A–3. Say a smooth function u ( x ) satis±es u 00 - c ( x ) u = 0 for 0 x 1 (here c ( x ) is some given contunuous function). If c ( x ) > 0 everywhere, show that there is no point where u ( x ) is both positive and has a local maximum. If we also knew that u (0) = 0 and u (1) = 0, why can we conclude that u ( x ) = 0 for all 0 x 1? Part B: Traditional Problems (5 problems, 16 points each) B–1. Given that two functions f : R R and
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This note was uploaded on 03/06/2012 for the course MATH 508 taught by Professor Staff during the Fall '10 term at UPenn.

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508F06Ex2s(1) - Signature Printed Name Exam 2 Math 508...

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