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Unformatted text preview: a n k ) such that Σ a n k converges. 5. Let f be continuous on [0 , 1] → [0 , 1]. Show that f ( x ) = x for some x ∈ [0 , 1] (Hint: Use the Intermediate Value Theorem). 6. (a) State the Mean Value Theorem. (b) Using (a), show that if f is diﬀerentiable and f > 0 on (0 , 1), then f is strictly increasing on (0 , 1) . 1 7. Show that f ( x ) = sin 1 x for x > ,f (0) = 0, is integrable in [0 , 1]. (You may use the fact that f is continuous in (0 , 1].) 8. Let f be a continuous function on IR and deﬁne G ( x ) = Z x 2 f ( t ) dt for x ∈ IR. Show that G is diﬀerentiable on IR , and compute G . 2...
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 Spring '10
 Kim
 lim sup sn

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