Unformatted text preview: 09/22/2000 FRI 11:37 FAX 6434330 MOFFITT LIBRARY .001 Math H1A Fall, 1995
Professor K. A. Ribet
Midterm Exam—November 2, 1995 This is an “open book” exam. You may consult your notes and textbook.
Grading is based on completeness, claritjg and accuracy. Please write in
complete English sentences and explain your reasoning carefully. 5691 _ 2691 . t ‘ . .
1 (4 porn 3) Evaluate Ill—’11; ———b _ 2 expressing the limit as a derivative. by using rules of differentiation, ﬁrst 2 {6 points) Let c be a real number. Use Rolle’s Theorem to show that the
equation 3:5 — 6:1: + c = 0 has at most one solution in the interval [—1, 1]. 3 ( 6 points). Using techniques of differentiation, ﬁnd the equation of the
line tangent to y :; \3/ 932 —— 1 at the point (—3,2). 4 f 6 points). Use l’Hopital’s Rule to evaluate gin}; ﬁgJ—l. 5 ( 8 points ) Let S be a non—empty bounded set of real numbers. Suppose
that neither a := inf 3 nor b := sup 5' is in S. (i) For each a: E (05,17), Show that there are numbers 73 and u in S which
satisfya<t<mandzv<u<b. (ii) Assume now that S has the following property: if S contains two num—
bers t and u, then S contains all numbers between t and u. Prove that 5' = (a, b). ...
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