mat125a_midterm

mat125a_midterm - MAT125A Midterm One and Solutions Feb. 1,...

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Unformatted text preview: MAT125A Midterm One and Solutions Feb. 1, 2010 1. Show that if f : R → R is bounded, then the function g ( x ) = ( x 2 f ( x ) x 6 = 0 x = 0 is differentiable at 0. What is the value of g (0)? There exists C > 0 such that | f ( x ) | ≤ C for all x since f is bounded. We have lim x → g ( x )- g (0) x- = lim x → x 2 f ( x ) x = lim x → xf ( x ) = 0 , where the last step follows from the Squeeze Theorem since ≤ | xf ( x ) | ≤ C · x → . This shows that the derivative of g exists at x = 0 and is equal to 0. 2. Let f be a uniformly continuous function on the interval [0 , 1]. Show that if { x n } is a Cauchy sequence in [0 , 1], then { f ( x n ) } is also a Cauchy sequence. There are a number of ways of doing this problem. The following might be the easiest. Since { x n } is Cauchy, it is convergent. Because f is continuous, the image { f ( x n ) } of { x n } under f is also a convergent sequence. Convergent sequences are Cauchy, so f ( x n ) is Cauchy....
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This note was uploaded on 03/18/2012 for the course MAT MAT 125A taught by Professor Bremer during the Winter '10 term at UC Davis.

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mat125a_midterm - MAT125A Midterm One and Solutions Feb. 1,...

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