MIT18_100BF10_pset9

MIT18_100BF10_pset9 - , and ln( x ) . 4 . (a) Show that...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
18.100B : Fall 2010 : Section R2 Homework 9 Due Tuesday, November 9, 1pm Reading: Tue Nov. 2 : differentiability, mean value theorem, Rudin 5.1-11 Thu Nov.4 : l’Hospital’s rule, Taylor’s theorem, Rudin 5.12-19 1 . Assume that f : R −→ R and that for some C > 0 and α > 0 we have for any x, y R | f ( x ) f ( y ) | ≤ C | x y | α . (a) Prove that if α > 1 then f is constant. [ Hint: What is the derivative of a constant function?] (b) If α 1 , is f necessarily differentiable? 2 . Problem # 2 page 114 in Rudin . 3 . (a) Problem # 7 page 114 in Rudin . (b) Show that for any polynomial P ( x ) P ( x ) ln( x ) lim = 0 and lim = 0 . x →∞ e x x →∞ P ( x ) For the second limit (of course) assume that P ( x ) is not constant. You may also use your calculus knowledge of derivatives of polynomials, e x
Background image of page 1

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

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: , and ln( x ) . 4 . (a) Show that sin( x ) ≃ x is a good approximation for small x by using Taylor’s theorem to obtain | sin( x ) − x | ≤ 6 1 | x | 3 ∀ x ∈ R . (b) Use (a) to calculate the limit for different values of a ∈ R and c > of the function x a sin( | x | − c ) (from Rudin pg.115 #13) as x → ∞ . 5 . (a) Assume f : (0 , 1] −→ R is differentiable and | f ′ ( x ) | ≤ M for all x ∈ (0 , 1] . Define the sequence a n = f (1 /n ) and prove that a n converges. (b) Problem # 26 page 119 in Rudin . MIT OpenCourseWare http://ocw.mit.edu 18.100B Analysis I Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
View Full Document

This document was uploaded on 10/26/2011 for the course MATH 18.100B at MIT.

Page1 / 2

MIT18_100BF10_pset9 - , and ln( x ) . 4 . (a) Show that...

This preview shows document pages 1 - 2. Sign up to view the full document.

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