quiz1-soln - MAT237Y – Multivariable Calculus Summer 2009...

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Unformatted text preview: MAT237Y – Multivariable Calculus Summer 2009 Solutions to the Quiz #1 Instructors: A. Hammerlindl and J. Uren 1. (a) [5 marks] State the Triangle Inequality. Solution: bardbl x + y bardbl ≤ bardbl x bardbl + bardbl y bardbl . Equivalently, bardbl x − y bardbl ≥ bardbl x bardbl−bardbl y bardbl . In terms of distances, the length of one side of a triangle is less or equal to the sum of lengths of two others. (b) [5 marks] Show that the set U = { ( x, y ) ∈ R 2 : 3 < bardbl ( x, y ) bardbl < 7 } is a neighbourhood of the point (3 , 4). Solution 1: Since bardbl (3 , 4) bardbl = √ 9 + 16 = 5 and 3 < 5 < 7 , (3 , 4) ∈ U. The function f : ( x, y ) mapsto→ bardbl ( x, y ) bardbl is continuous on R 2 . U is the preimage under f of the interval { t ∈ R : 3 < t < 7 } , which is open. Therefore, U is also open. But an open set is a neighborhood of every point it contains. Solution 2: By definition of the neighborhood, we need to show that ∃ r > 0 such that B r (3 , 4) ⊂ U. Let r = 1 and let ( x, y ) ∈ B 1 (3 , 4) . Then bardbl ( x, y ) − (3 , 4) bardbl < 1 ⇒ ( x − 3) 2 + ( y − 4) 2 < 1 ⇒ | x − 3 | < 1 , | y − 4 | < 1 ⇒ − 1 < x − 3 < 1 , − 1 < y − 4 < 1 ⇒ 2 < x < 4 , 3 < y < 5 . Therefore, bardbl ( x, y ) bardbl = radicalbig x 2 + y 2 > √ 2 2 + 3 2 = √ 13 > 3 and bardbl ( x, y ) bardbl = radicalbig x 2 + y 2 < √ 4 2 + 5 2 = √ 41 < 7 ....
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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.

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quiz1-soln - MAT237Y – Multivariable Calculus Summer 2009...

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