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18.01 Single Variable Calculus
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Fall
200
6
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�
�
18.01
Practice
Questions
for
Exam
3
–
Fall
2006
�
1
�
�/
2
x
dx
3
1.
Evaluate
a
)
b
)
cos
x
sin
2
x
dx
2
0
�
1 + 3
x
�/
3
1
2.
Evaluate
x
dx
directly
from
its
definition
as
the
limit
of
a
sum.
0
n
1
Use
upper
sums
(circumscribed
rectangles).
You
can
use
the
formula
�
i
=
n
(
n
+
1).
2
1
3.
A
bank
gives
interest
at
the
rate
r
,
compounded
continuously,
so
that
an
amount
A
0
deposited
grows
after
t
years
to
an
amount
A
(
t
) =
A
0
e
rt
.
You
make
a
daily
deposit
at
the
constant
annual
rate
k
;
in
other
words,
over
the
time
period
�
t
you
deposit
k
�
t
dollars.
Set
up
a
definite
integral
(give
reasoning)
which
tells
how
much
is
in
your
account
at
the
end
of
one
year.
(Do
not
evaluate
the
integral.)
4.
Consider
the
function
defined
by
F
(
x
) =
�
x
�
3
+
sin
t
dt
.
Without
attempting
to
find
0
an
explicit
formula
for
F
(
x
),
a)
(5)
show
that
F
(1)
�
2;
b)
(5)
determine
whether
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 Math, Calculus, Derivative, one year

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