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mtpII125

# mtpII125 - Practice Midterm II 1 Show that the function f(x...

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Practice Midterm II 1. Show that the function f ( x ) = ( x sin( - 1 /x ) x 6 = 0 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 0 ( 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

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2 Solutions 1. Since sin( x ) is bounded by 1, we have | x sin(1 /x ) | ≤ | x | → 0 as x 0 , which shows that f ( x ) is continuous at 0. Now lim x 0 f ( x ) - f (0) x = lim x 0 sin(1 /x ) does not exist; we can show that by letting a k = 2 πk and observing that sin(1 /a k ) = sin( π 2 k ) = ( 0 for even k ± 1 for odd k . It follows, of course, that f ( x ) is not differentiable at 0.
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mtpII125 - Practice Midterm II 1 Show that the function f(x...

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