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practicefinal - a n k such that Σ a n k converges 5 Let f...

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MATH 131A section 2: Practice Final. Please write clearly, and show your reasoning with mathematical rigor. You may use any correct rule about the algebra or order structure of R from Section 3 without proving it. 1. By using product rule and induction, show that the derivative of f ( x ) = x m with m IN equals mx m - 1 . 2. (a) Show that 2 is irrational. (b) Show that r + 2 with any rational number r is an irrational number. (c) Show that irrational numbers are dense in IR . You can use (b) and the fact that rational numbers are dense in IR . 3. Let ( s n ) be a bounded sequence in IR . Show that there exists a (not necessarily monotone) subsequence whose limit is lim sup s n . 4. Let ( a n ) be a sequence such that lim inf | a n | = 0. Prove that there is a subsequence (
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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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