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# examsoln_2000f - MATH 247 FALL 2000 FINAL EXAM BRIEF...

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MATH 247 — FALL 2000 — FINAL EXAM — BRIEF SOLUTIONS NAME: Total: 200 points. Do 8 out of 12 questions. You MUST indicate which 8 questions are to be graded; otherwise, just the first 8 problems will be graded. EXPLAIN every answer. No books, notes, calculators or computers allowed on this exam. 1 (25 points) . (a) [8 points] A function f ( x ) on [ a, b ] is called bounded if there exists M R such that | f ( x ) | ≤ M for all x [ a, b ]. Negate this, so obtaining the definition of an unbounded function. Solution. For all M R there exists x [ a, b ] such that | f ( x ) | > M . (b) [8 points] Define what it means to say that “ a n converges to L ”. Solution. For all ε > 0 there exists N N such that for all n N we have | a n - L | < ε . (c) [9 points] Negate your answer in part (b), thus obtaining a definition of “it is false that a n converges to L ”. Solution. There exists ε > 0 such that for each N N there exists n N such that | a n - L | ≥ ε . 1

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2 (25 points) . Consider a function f : Z R such that f (1) = 2, f ( m ) > 0 for all m Z , and f ( j - k ) = f ( j ) f ( k ) for all j, k Z . Using these properties, find a formula for f ( m ) , m Z . (Hint: play around to guess a formula, and then use induction ideas to give a proper proof.) Solution. The formula f ( j - k ) = f ( j ) /f ( k ) looks like the law of exponents, and so we guess f ( m ) = a m for some a . Then since f (1) = 2 we guess a = 2.
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examsoln_2000f - MATH 247 FALL 2000 FINAL EXAM BRIEF...

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