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MIT18_100BF10_pset9

# MIT18_100BF10_pset9 - and ln x 4(a Show that sin x ≃ x is...

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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 , and ln( x ) . 4 . (a) Show that sin( x ) x

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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 ....
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