MIT18_014F10_ex3

MIT18_014F10_ex3 - EXAM 3(1(10 points Evaluate x0 lim 1 1 x...

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Unformatted text preview: EXAM 3 - NOVEMBER 19, 2010 (1) (10 points) Evaluate x0 lim 1 1 - x log(x + 1) (2) (10 points) Evaluate x2 3x - 2 dx - 6x + 10 1 2 EXAM 3 - NOVEMBER 19, 2010 (3) (10 points) Let f be an infinitely differentiable function on R. We say f is analytic on (-1, 1) if the sequence {Tn f (x)} converges to f (x) for all x (-1, 1), where Tn f (x) is the nth Taylor polynomial of f centered at zero. Suppose there exists a constant 0 < C 1 such that (k) f (x) C k k! for every positive integer k and every real number x (-1, 1). Prove that f is analytic on (-1, 1). (4) (10 points) Let f (x) be a function defined on (0, ]. Suppose limn f (1/n) = 0 and limn f (/n) = 1. Prove that limx0+ f (x) does not exist. EXAM 3 - NOVEMBER 19, 2010 3 (5) A function f on R is compactly supported if there exists a constant B > 0 such that f (x) = 0 if |x| B. If f and g are two differentiable, compactly supported functions on R, then we define (f g)(x) = f (x - y)g(y)dy. - Note: We define - f (x)dx = limt 0 -t f (x)dx + limt t 0 f (x)dx. (10 points) Prove (f g)(x) = (g f )(x). (10 points) Prove (f g)(x) = (g f )(x). MIT OpenCourseWare http://ocw.mit.edu 18.014 Calculus with Theory Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. ...
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This note was uploaded on 01/18/2012 for the course MATH 18.014 taught by Professor Christinebreiner during the Fall '10 term at MIT.

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MIT18_014F10_ex3 - EXAM 3(1(10 points Evaluate x0 lim 1 1 x...

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