Math 1A - Fall 1995 - Ribet - Final

# Math 1A - Fall 1995 - Ribet - Final - 1

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Unformatted text preview: 09/28/2000 THU 11:28 FAX 6434330 MOFFITT LIBRARY .001 Math HlA Fall, 1995 Professor K. A. Ribet Final Examination—December 13, 1995 This is an “0 en boo ” exam. You 111a consult our notes and textbook. Gradin is P y y g based on completeness, clarity, and accuracy Please write in complete English sentences and explain your reasoning carefully Time allotted: three hours. _ . . \$2 — 693 + 9 l (5’ paints). Evaluate iﬂm' 2a {4 points). Find f’(l) if f(:c) I {75. 2 2b {4 points). Evaluate] 932V1+r3 dag. —1 2c (4 points). Differentiate f(.’L‘) = 331/3”. 2d (4 points). Calculate diltlogﬁinﬁ». 1 3 (5 points). Working directly with the deﬁnition of “limit,” prove that lint;2 — 2 2. cs—)- 11’: 1 >2 n+2— n+1 forallinteersn>l. Itmi ht m - (V V l g — ( g help to multiply both sides by the positive quantity VT; -|— 2 -|- \/H + 1.) 4b {4 points). Use mathematical induction and the result of part (a) to show: 4a (3' points). Show that (s) 1+—2+——+---+—22(\/m_1) for TL 2 l. n+1 1 4c {2 points). Show that f (in: : 2(Vn + 1 L 1). Does this suggest a geometric 1 ﬂ interpretation of (iv)? .‘L‘ 1 1/13 1 5 ' t . h th t I :2 dt d . a (4 pom s) S ow a (2:) j; l + t2 cit +/0 1 + t2 oes not depend on 2: 5b {5’ points). Calculate [(1). 1 6 {5 points). Suppose that f(a:) is a continuous function on [0,1] for which / (f(3:))2 clan 0 vanishes. Show that f is identically 0. 7 {5 points). Suppose that l) is non—zero. Prove that m4 —|— b4 2 (a: + by!“ only when a: = {1. (If a4 + b4 : (a + b)4, apply Rolle’s theorem to ﬁst) :2 3:4 + 64 — (a: + b)4 on the interval of numbers between 0 and a.) 8 (4’ points). Let A = {misc2 3 6 and a: is rational}. Decide Whether supA exists. If supA exists, is it an element of A? ...
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