This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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....
View
Full
Document
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.
 Winter '10
 Bremer

Click to edit the document details