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MAT 371
Advanced Calculus
/160
May 8, 2005
Final Exam
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1.
Give precise deﬁnitions of ﬁve technical terms (these may be nouns, adjectives, verbs, or
surnames) from our class that start with the letter “c”. You may choose the context (e.g.,
subsets, sequences in metric spaces, or realvalued functions – but you must be precise).
2. a.
State the
supremum
(or
“least upper bound”
)
axiom
for the real numbers.
Deﬁne the technical terms that you use in the axiom.
b.
State the
mean value theorem
of diﬀerential calculus.
c.
State a major theorem that involves both
“compact”
and
“continuous”
.
3. a.
Suppose that (
a
n
)
∞
n
=1
is a sequence in a metric space (
X,d
).
Prove that if the sequence (
a
n
)
∞
n
=1
has a limit, then this limit is unique.
b.
Give an example that shows that the set
{
a
n
:
n
∈
Z
+
}
of values of a sequence (
a
n
)
∞
n
=1
in a metric space (
X,d
) can have more than one limit point. Justify your assertions.
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This note was uploaded on 04/02/2009 for the course MAT 371 taught by Professor Thieme during the Spring '07 term at ASU.
 Spring '07
 thieme
 Calculus

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