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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.
 Fall '09
 RomauldStanczak
 Calculus, Multivariable Calculus

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