Fall2007
ARE211
Final Exam  Answer key
This is the final exam for ARE211. As announced earlier, this is an openbook exam. However,
use of computers, calculators, Palm Pilots, cell phones, Blackberries and other nonhuman aids is
forbidden.
Read all questions carefully before starting the test.
Allocate your 180 minutes in this exam wisely. The exam has 180 points, so aim for an
average
of
1 minute per point. However, some questions & parts are distinctly easier than others. Make sure
that you first do all the easy parts, before you move onto the hard parts. Always bear in mind
that if you leave a partquestion completely blank, you cannot conceivably get any marks for that
part. The questions are designed so that, to some extent, even if you cannot answer some parts,
you will still be able to answer later parts. Even if you are unable to show a result, you are allowed
to use the result in subsequent parts of the question. So don’t hesitate to leave a part out. You
don’t have to answer questions and parts of questions in the order that they appear on the exam,
provided that you clearly indicate the question/partquestion you are answering.
Finally,
always
keep in mind the famous maxim KISS (keep it simple, stupid).
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2
Problem 1 [20 points]
Fix some metric
d
that applies to all parts of the following question.
A function
f
is
uniformly
continuous
on
X
if for all
epsilon1 >
0,
∃
δ >
0 such that for all
x, x
prime
∈
X
,
d
(
x, x
prime
)
< δ
implies
d
(
f
(
x
)
, f
(
x
prime
))
< epsilon1
.
Now let
{
x
n
}
be a Cauchy sequence in
X
⊂
R
, let
f
:
X
→
R
be some
function, and consider the sequence
{
y
n
}
defined by, for all
n
,
y
n
=
f
(
x
n
).
A) [
10 points
]
If
f
is continuous on
X
, is
y
n
Cauchy? If so, prove it. If not provide a counter
example. (Your counterexample must specify the function
f
, its domain
X
and a Cauchy
sequence in
X
.)
No. Counter example is
f
(
x
) = 1
/x
, defined on
X
=
R
++
; Cauchy sequence is
x
n
= 1
/n
. In
this case,
f
(
x
n
) =
f
(1
/n
) =
n
, which is clearly not Cauchy.
B) [
10 points
]
If
f
is uniformly continuous on
X
, is
y
n
Cauchy? If so, prove it. If not provide
a counter example. (Your counterexample must specify the function
f
, its domain
X
and a
Cauchy sequence in
X
.)
Yes.
Fix
epsilon1 >
0
.
We need to show that there exists
N
∈
N
, such that
m, n > N
implies
d
(
y
n
, y
m
)
< epsilon1
.
Since
f
is uniformly continuous, there exists
δ >
0
such that
d
(
x, x
prime
)
< δ
implies
d
(
f
(
x
)
, f
(
y
))
< epsilon1
. Since
{
x
n
}
is Cauchy, there exists
N
such that
m, n > N
implies
d
(
x
n
, x
m
)
< δ
which in turn implies
d
(
f
(
x
n
)
, f
(
x
m
))
< epsilon1
.
Problem 2 [40 points]
Fix a vector
v
0
∈
R
n
, two natural numbers
J >
1 and
K >
1, and a nonempty set
Q
⊂ {
1
, ...JK
}
.
Now let
M
denote the set of all
n
×
JK
matrices
M
such that
M
= [
x
1
, ...,
x
JK
] where
x
m
=
braceleftBigg
v
0
if
m
∈
Q
v
k
if
m
∈
/ Q
and
m
=
k
+
jK
, for
j
∈ {
1
, ..., J
}
and
k
∈ {
1
, ..., K
}
for some
n
×
K
matrix
V
= [
v
1
, ...,
v
K
].
(Note that each distinct element of
M
is defined by a
different
vector
V
.)
A) [
7 points
]
Think of an example of an element of
M
, for
K
= 3,
J
= 4 a set
Q
that has at
least 3 elements and a
n
×
K
matrix
V
. Remember KISS! Write down
Q
,
n
,
v
0
and
V
Then
write down your matrix
M
.
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 Fall '07
 Simon
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