Exam I
July 6, 2011
MAC 2312 and Key
S Hudson
1) [30pt] Short calculations, 5 pts each, not much partial credit likely. Simplify terms like
sin(
π
) and
e
0
, but not like
√
121
/
44.
a)
d
dx
[
R
x
0
te
t
dt
]
b)
R
4
0
√
x
+
e
x
dx
c)
R
2
1
1
t
dt
f)
R
π
0
sin
2
x dx
e)
R
1
0
dx
2

x
f)
R
3
0

x

2

dx
=
2) [15pts] State and prove the F.T.C., Part Two, about computing the derivative of a
definite integral. You can use other definitions and theorems in your proof  but point out
when you are doing so. Partial credit is likely, even with small gaps, if you are making a
good effort to explain your reasoning.
3) [15pts] Find the exact area under
f
(
x
) =
x
2
, on [0,2] using a limit of sums (as done in
class and in Ch.5.4). Use a regular partition and the Right Endpoint Rule. This is roughly
equivalent to asking for the definition of
R
2
0
x
2
dx
and a calculation of that. To make this
long problem a bit easier, you get an outline and some formulas to choose from (but not
all are needed, or even correct).
A) The interval [0
,
2] gets split into
n
parts and
x
*
k
is the right endpoint of the
k
th
part.
Which two of these formulas are correct (choose and explain) ?
Δ
x
= 1
/n
, Δ
x
= 2
/n
,
x
*
k
=
k/n
,
x
*
k
= 2
k/n
,
x
*
k
= 2
k

1
/n
.
B) Now, we add up the areas of
n
rectangles to get an estimate of the total area,
called
E
n
. Which two of these are correct formulas for
E
n
(choose and explain) ?
∑
n
1
k/n
,
2
∑
n
1
(
x
*
k
)
2
/n
,
n
(
n
+ 1)
/
2,
n
(
n
+ 1)
/
3
n
2
, 8
n
(
n
+ 1)(2
n
+ 1)
/
6
n
3
.
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 Summer '08
 Storfer
 Calculus, Continuous function, dx, Riemann, partial credit

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