18.100B.MidTerm08

# 18.100B.MidTerm08 - 18.100B Midterm Exam Tuesday October 7...

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18.100B Midterm Exam Tuesday, October 7 2008, 7:30–9:00, in 4–370. Closed book, no calculators. YOUR NAME: YOUR SECTION (circle one): 18.100B MWF 12-1 18.100B TR 1-2:30 18.100C This is a 90-minute evening exam. No notes, books, or calculators are permitted. Point values are indicated for each problem. Do all the work on these pages. (Use the back sides if needed.) GRADING 1. /25 2. /20 3. /20 4. /20 5. /20 6. /25 TOTAL /130

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Problem 1. [25 points: (a) /5 (b) /10 (c) /5 (c) /5 ] (a) Define what a metric is. (b) Let d : R n × R n R be defined by d ( x , y ) = | x 1 y 1 | + 2 | x 2 y 2 | + · · · + n | x n y n | . Here x = ( x 1 , x 2 , . . . , x n ) and y = ( y 1 , y 2 , . . . , y n ) . Prove that d is a metric.
(c) Sketch the unit ball of d in R 2 : that is, the set of all x R 2 such that d ( x , 0 ) < 1 . (d) Let g : R 2 × R 2 R be defined by g ( x , y ) = | x 1 y 2 | . Is g a metric? Prove your claim.

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Problem 2. [20 points: (a) /5 (b) /5 (c) /10 ] Consider the following sequences of real numbers: x n = cos parenleftBig 3 parenrightBig y n = ( 1) n + 1 n z n = 6 n + 4 7 n 3 (a) Determine the set of subsequential limits of each of the sequences ( x n ) , ( y n ) , and ( z n ) .
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