Portillo, Tom – Exam 1 – Due: Oct 10 2006, 11:00 pm – Inst: David Benzvi
1
This
printout
should
have
17
questions.
Multiplechoice questions may continue on
the next column or page – find all choices
before answering.
The due time is Central
time.
001
(part 1 of 1) 10 points
Estimate the area,
A
, under the graph of
f
(
x
) =
3
x
on [1
,
5] by dividing [1
,
5] into four equal
subintervals and using right endpoints.
1.
A
≈
15
4
2.
A
≈
77
20
correct
3.
A
≈
39
10
4.
A
≈
79
20
5.
A
≈
19
5
Explanation:
With four equal subintervals and right end
points as sample points,
A
≈
n
f
(2) +
f
(3) +
f
(4) +
f
(5)
o
1
since
x
i
=
x
*
i
=
i
+ 1. Consequently,
A
≈
3
2
+ 1 +
3
4
+
3
5
=
77
20
.
keywords: Stewart5e, area, rational function,
Riemann sum
002
(part 1 of 1) 10 points
Find an expression for the area of the region
under the graph of
f
(
x
) =
x
2
on the interval [1
,
6].
1.
area =
lim
n
→ ∞
n
X
i
= 1
‡
1 +
8
i
n
·
2
6
n
2.
area =
lim
n
→ ∞
n
X
i
= 1
‡
1 +
6
i
n
·
2
5
n
3.
area =
lim
n
→ ∞
n
X
i
= 1
‡
1 +
5
i
n
·
2
5
n
correct
4.
area =
lim
n
→ ∞
n
X
i
= 1
‡
1 +
5
i
n
·
2
6
n
5.
area =
lim
n
→ ∞
n
X
i
= 1
‡
1 +
6
i
n
·
2
6
n
6.
area =
lim
n
→ ∞
n
X
i
= 1
‡
1 +
8
i
n
·
2
5
n
Explanation:
The area of the region under the graph of
f
on an interval [
a, b
] is given by the limit
A
=
lim
n
→ ∞
n
X
i
= 1
f
(
x
*
i
) Δ
x
when [
a, b
] is partitioned into
n
equal subin
tervals
[
a, x
1
]
,
[
x
1
, x
2
]
, . . . ,
[
x
n

1
, b
]
each of length Δ
x
= (
b

a
)
/n
and
x
*
i
is an
arbitrary sample point in [
x
i

1
, x
i
].
Consequently, when
f
(
x
) =
x
2
,
[
a, b
] = [1
,
6]
,
and
x
*
i
=
x
i
, we see that
area =
lim
n
→ ∞
n
X
i
= 1
‡
1 +
5
i
n
·
2
5
n
.
keywords:
area, limit Riemann sum, cubic
function
003
(part 1 of 1) 10 points
A function
h
has graph