31L Test #3
(spring, 2009, Tomberg)
Name______________________________
I reaffirm the Duke Community Standard.
You must clearly show all your work on each problem to receive full credit.
work may
Unsupported
receive
credit
no
.
All
integrals should be worked out by hand unless you are instructed otherwise
. If you
have read the previous three sentences place an X in the following box.
œ
1.
Let be a continuous function on the interval [a,b]and let
,
,
,
,
be a partition
of
0
B &+BB ±B &,
0
12
8
[a,b] . Complete the following:
We define
to be______________________________________,
(
a
b
0—B².B
where
= ____________________.
?
B
2.
Environmentalists, monitoring the flow of waste water from a chemical plant, recorded the data in the
table below.
time (days
)
02468
1
0
12
waste water (hundred of gallons / day
)
4.0
0
3.5
0
1.7
5
0.6
5
0.2
4
0.1
5
0.15
a)
Give upper and lower estimates for the amount of waste water that flowed from the plant during the
12 day period. Your answers should contain units.
b) How often would the environmentalists have had to record the flow rate of the waste water in order
for the error of their estimate to less than 10 gallons? Your answers should contain units.
3.
Solve the initial value problem
cos(5 ) ,
3.
.C
.>
&
C
>
C—!²&
4.
Solve the initial value problem
2
.>
#B
& B
C
/ ³ C—!²&
#
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Below you see the graphs of
and
If the area under
between
and
is approximately
0
B
0—B&
±
0
BB
ww
3
3²"
ab
23 (the shaded region),
what corresponding quantity on the graph of is 23? Illustrate your answer on the graph
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 Spring '08
 Staff
 Calculus, Derivative, Integrals

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