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508F10Ex2s - Exam 2 Math 508 December 9 2010 Jerry L Kazdan...

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Math 508 Exam 2 Jerry L. Kazdan December 9, 2010 9:00 – 10:20 Directions This exam has three parts, Part A asks for 3 examples (5 points each, so 15 points). Part B has 4 shorter problems (8 points each so 32 points). Part C has 4 traditional problems (15 points each so 60 points). Total is 107 points. Closed book, no calculators or computers– but you may use one 3 × 5 card with notes on both sides. Part A: Examples (3 examples, 5 points each so 15 points). Give an example having the specified property. 1. A function f C 1 ([ - 1 , 1]) but is not in C 2 ([ - 1 , 1]). 2. A bounded sequence a k in a complete metric space M where a k has no convergent subsequence. 3. A sequence of continuous functions f n ( x ) C ([0 , 1]) that converges pointwise to zero but 1 0 f n ( x ) dx 1. [A clear sketch is adequate.] Part B: Short Problems (4 problems, 8 points each so 32 points) B–1. Let f ( x ) be a smooth function with the properties: f ( - 1) = 1, f (0) = 0, and f (1) = 1. Show that f ( c ) = 2 at some c ( - 1 , 1). [Suggestion: Consider g ( x ) := f ( x ) - x 2 .] B–2. Let 2 x 0 f ( t ) dt = e cos(3 x +1) + A . Find f
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