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Unformatted text preview: Practice Midterm II 1. Show that the function f ( x ) = ( x sin(- 1 /x ) x 6 = 0 x = 0 is continuous but not differentiable at 0. 2. Find all the values of x for which the series X k =1 x k k converges. 3. Suppose that f : ( a,b ) R is differentiable on ( a,b ) and f ( x ) is bounded on ( a,b ). Prove that f is uniformly continuous on ( a,b ). 4. Let f be a continuous function mapping the interval [ a,b ] into the interval [ a,b ] (in other words, f : [ a,b ] [ a,b ]). Show that there is a point z in [ a,b ] such that f ( z ) = z (that is, show that f has a fixed point). 5. Give an example of a divergent series k =1 a k whose partial sums are bounded. 1 2 Solutions 1. Since sin( x ) is bounded by 1, we have | x sin(1 /x ) | | x | 0 as x , which shows that f ( x ) is continuous at 0. Now lim x f ( x )- f (0) x = lim x sin(1 /x ) does not exist; we can show that by letting a k = 2 k and observing that sin(1 /a k ) = sin( 2 k ) = ( for even k...
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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.
- Winter '10