508F10Ex2solns

508F10Ex2solns - Math 508 Exam 2 Jerry L. Kazdan December...

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Unformatted text preview: 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 00 5 00 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]). Solution : f ( x ) := x | x | 2. A bounded sequence a k in a complete metric space M where a k has no convergent subsequence. Solution : In ` 2 the unit vectors e 1 := (1 , , ,... ), e 2 := (0 , 1 , ,... ), .... Another example: In C ([0 , 1]) the functions f k ( x ) := x k , k = 0 , 1 , 2 , ... [since every subse- quence converges pointwise to the discontinuous function f ( x ) := 0 , x [0 . 1) but f (1) = 1. Another example: In L 2 ((- , )), the functions sin kx , k = 1 , 2 ,... [since they are orthonor- mal]. 3. A sequence of continuous functions f n ( x ) C ([0 , 1]) that converges pointwise to zero but R 1 f n ( x ) dx 1. [A clear sketch is adequate.] Solution : Let f n ( x ) C ([0 , 1]) be the tent function whose graph is straight lines from (0 , 0) to (1 /n, n ) to (2 /n, 0) to (1 , 0). Part B: Short Problems (4 problems, 8 points each so 32 points) B1. Let f ( x ) be a smooth function with the properties: f (- 1) = 1, f (0) = 0, and f (1) = 1. Show that f 00 ( c ) = 2 at some c (- 1 , 1). [Suggestion: Consider g ( x ) := f ( x )- x 2 .] Solution : Let g ( x ) := f ( x )- x 2 . Then g (- 1) = g (0) = g (1) = 0. By the mean value theorem there is at least one point point c 1 (- 1 ,...
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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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508F10Ex2solns - Math 508 Exam 2 Jerry L. Kazdan December...

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