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mat125a_midterm

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

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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 0 x = 0 is differentiable at 0. What is the value of g 0 (0)? There exists C > 0 such that | f ( x ) | ≤ C for all x since f is bounded. We have lim x 0 g ( x ) - g (0) x - 0 = lim x 0 x 2 f ( x ) x = lim x 0 xf ( x ) = 0 , where the last step follows from the Squeeze Theorem since 0 ≤ | xf ( x ) | ≤ C · x 0 . 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. Note the hypothesis that f be uniformly continuous is stronger than necessary — f is continuous would have sufficed. It was included so as to make a direct proof

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