quiz1summer2009 - (a) [5 marks] lim ( x,y,z ) → (0 , , 0)...

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MAT237Y – Multivariable Calculus Summer 2009 Quiz #1 Tuesday, May 26 from 6:10pm to 7:00pm. Instructors: A. Hammerlindl and J. Uren 1. (a) [5 marks] State the Triangle Inequality. (b) [5 marks] Show that the set U = { ( x, y ) R 2 : 3 < b ( x, y ) b < 7 } is a neighbourhood of the point (3 , 4). 2. [10 marks] Consider the set S = { ( x, y ) R 2 : 2 x 2 5 y 2 = 3 } . Prove whether or not S satisFes the following properties. (No marks for guessing.) (a) Is S open? (b) Is S closed? (c) Is S compact? (d) Is S connected? 3. Determine whether the following limits exist. Justify your answer.
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Unformatted text preview: (a) [5 marks] lim ( x,y,z ) → (0 , , 0) xyz 2 x 4 + y 4 + z 4 (b) [5 marks] lim ( x,y ) → (0 , 0) 3 x 4 − 2 y 3 x 2 + y 2 4. [10 marks] Suppose { x n } ∞ n =1 is a sequence of positive real numbers such that lim n →∞ x n = 0 and suppose y n ∈ R m are such that b y n b ≤ x n for n ∈ N . Prove that the sequence { y n } ∞ n =1 converges....
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This note was uploaded on 10/21/2010 for the course MATHEMATIC MAT237Y1 taught by Professor Romauldstanczak during the Fall '09 term at University of Toronto- Toronto.

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