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18.100A
Introduction
to
Analysis
–
Practice
Final
3
hours
Directions.
You
can
use
the
textbook,
but no other
material.
To justify the
arguments,
quote
theorems
, by name or
number,
verifying their hypotheses
where
necessary.
Work problems
totalling 100.
You
can
include just parts
of
some
problems.
Problems
will
be
graded
in
their
numerical
order,
stopping at 100 or
as
close
above
it as
possible,
the
total
scaled
back to 100 if
it goes
over.
∞
1
1.
(15:
5,
10)
Let
f
(
x
) =
.
x
2
+
n
2
1
a)
Prove
that
f
(
x
) is defined
and
continuous
on (
−∞
,
∞
).
(Do not use part (b).)
b) Prove
f
(
x
) is differentiable
on (
−∞
,
∞
).
(Hint:
Use intervals
of the form [
−
a, a
].)
2.
(15:
10,5)
a)
Let
S
be
a compact subset of
the open
first quadrant
Q
=
{
(
x, y
) :
x >
0
, y
>
0
}
.
Prove
there
is
a point
x
in
S
such
that the ray connecting it to the origin
has
maximum
slope (i.e.,
no other point in
S
gives
a ray with
bigger
slope).
b)
Does (a) continue to be true if
S
is
only assumed
to be closed,
not compact? If
yes,
show how to modify the proof in (a) to prove it;
if
no, give a counterexample.
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 Fall '12
 ArthurMattuck
 Topology, Metric space, Compact space, Cauchy sequence, R. Prove

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