MIT18_014F10_ex3

MIT18_014F10_ex3 - EXAM 3 - NOVEMBER 19, 2010 (1) (10...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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. ...
View Full Document

Page1 / 4

MIT18_014F10_ex3 - EXAM 3 - NOVEMBER 19, 2010 (1) (10...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online