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Unformatted text preview: Mathematics 131A - Sample Questions for the Final Examination Instructor : D. E. Weisbart June 8, 2009 NAME (please print legibly): Your University ID Number: Signature: This is not a sample exam! The final will be much shorterabout 8 questions at most. Calculators, notes and books may not be used in this examination. Full credit may not be given if insufficient work is shown. Note that you should be able to prove Fermats Theorem, Rolles Theorem and the Mean Value Theorem. You should be able to prove the Fundamental Theorem of Calculus I and II. You should also be able to prove the Bolzano-Weierstrass Theorem and the Heine Borel Theorem. You should also know how to use the ratio and root test. You should know how to sum a geometric series. Below are a bunch of sample questions that have showed up on exams for previous courses I have taught. 1 Problem 1. a. Prove that 1 + 3 + 5 + + (2 n- 1) = n 2 . b. Suppose a subset S of the natural numbers has the property that n S implies that n + 1 S . Is S necessarily the natural numbers? If so, explain why. If not, give a counter example....
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- Spring '08