MIT18_014F10_pr_ex3

# MIT18_014F10_pr_ex3 - a contraction ±or any x ∈ R prove the sequence f n x is Cauchy where f n x = f f f x(the n times composition oF f with

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PRACTICE EXAM 3 3 (1) Evaluate t + t 1+ t 2 dt (2) Evaluate 5 x 3 x 2 9 dx 3 (3) Suppose that lim x a + g ( x ) = B = 0 where B is fnite and lim x a + h ( x ) = 0, but h ( x ) ± ± ± ± ( ) g x ± ± ± ± = 0 in a neighborhood oF a . lim h ( x ) Prove that = . x a + (4) Let f ( x ) : [0 , ) R + be a positive continuous Function such that lim x →∞ f ( x ) = 0. Prove there exists M R + such that max x [0 , ) f ( x ) = M . (5) A sequence is called Cauchy iF For all > 0 there exists N Z + such that For all m,n > N , | a m a n | < . Prove that iF { a n } is a convergent sequence, then it is Cauchy. (The converse is also true.) A Function f : R R is called a contraction iF there exists 0 α < 1 such that | f ( x ) f ( y ) | ≤ α | x y | . Let f be

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Unformatted text preview: a contraction. ±or any x ∈ R , prove the sequence { f n ( x ) } is Cauchy, where f n ( x ) = f f f ( x ) (the n times composition oF f with itselF). ◦ ◦···◦ 1 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_pr_ex3 - a contraction ±or any x ∈ R prove the sequence f n x is Cauchy where f n x = f f f x(the n times composition oF f with

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