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University of Toronto at Scarborough
MAT A37H
Summer 2011
Assignment # 9
You are expected to work on this assignment prior to your tutorial during the week of
July 18th. You may ask questions about this assignment in that tutorial.
At the beginning of your TUTORIAL during the week of July 25th you need to submit
the following homework problems.
STUDY:
Chapter 11, Chapter 13.
HOMEWORK PROBLEMS:
1. Let
n
∈
N
be arbitrary. Use MVT to prove Bernoulli’s inequality for
x >
0:
(1 +
x
)
n
≥
1 +
nx
.
2. Does such a function exist?
•
f
is continuous and diﬀerentiable for all
x
∈
R
, and
•
f
(0) =

1 ,
f
(2) = 4, and
•
f
0
(
x
)
≤
2 for all
x
∈
R
.
3. Spivak #13.20(a)(b)(c)
4. Let
f
(
x
) =
0
,
if 0
≤
x
≤
1
2
1
,
if
1
2
< x
≤
1
1
.
5
,
if 1
< x
≤
2
Show that
R
2
0
f
exists by applying the integrability reformulation (Theorem 2 of Chap
ter 13).
EXERCISES:
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 Summer '11
 Katherine
 Math

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