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Unformatted text preview: Fall 2006 ARE211 Midterm Exam Due date: Mon Oct 23, ’06, noon. (Howard’s mailbox in Giannini) There are 6 questions Rules, regulations and advice a) You must work on this exam by yourself. You are not allowed to talk to anybody about it or exchange notes (not even old class notes) with anybody during the exam period. You can use your own notes, the lecture notes, the answer keys to the problem sets, as well as books such as Simon & Blume. b) Please write up the answers neatly; otherwise points may be subtracted because your answer is not legible. c) Conciseness is a virtue of proofs. We will definitely take points off for excessively long proofs. It is just not fair to give somebody the same points for writing a proof covering three pages that other people did perfectly in four lines. ( None of the questions requires a proof that is long, so if your answer is getting really long and messy, you must be missing something ) d) Some questions, or parts of questions might require a little trick which might be difficult to see. So, please work on the easy questions, and the easy parts of questions that have hard parts, first and save the difficult questions for the end. Each question has a “theme,” so if you are having trouble with figuring out what the question is getting at, try a different part to see if you can pick up the theme. e) If you believe a question needs clarification, or even worse, is incorrect (I hope this doesn’t happen) send me an email. If I consider it appropriate to reply, I will reply to the whole class so everybody gets the same response. You might want to check your email periodically, just to see if there’s any correspondence. Please note well: i) last year, more than half the email questions I received were undecipherable, and had to be returned for further clarification. This was a major pain. Hence please follow the following simple instructions for submitting questions. ii) always specify the precise part of the question to which you are referring. iii) your questions will (should) always relate to very fine details of the question. So if you think I’ve mis-worded a question (which, by the way, is unlikely, though possible) specify in a full sentence the re-wording you think is correct. Write mathematical symbols in words, i.e., if you want to write x ∈ B ( x,δ ) ⊂ X , write ”x in B(x,delta) subset X” and I’ll know what you mean. 1 Problem 1 : Metrics and continuity . (10 points) Let X be an arbitrary universe, let d be an arbitrary metric. For arbitrary y ∈ X , consider the function f y : X ⇒ R n defined by f y ( · ) = d ( · ,y )....
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- Fall '07
- Howard, Convex function, Giannini, Extrema., Blume.