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Math 10C
Spring 2007
Final Exam Solutions – Version 1
1.
[6 points] Suppose a drug is taken in 25mg doses four times a day (i.e. every six
hours). It is known that at the end of the six hours, 3% of the drug remains in the
body. Let
Q
n
represent the quantity, in milligrams, of the drug in the body right
after the
n
th
dose.
a.
[2 points] Express
Q
n
as a series.
1
21
0
25 25(0.03)
25(0.03)
... 25(0.03)
25
(0.03)
n
ni
n
i
Q
−
−
=
=+
+
+
+
=
∑
.
b.
[2 points] Explain why the series from part (a) is a geometric series.
We have a constant ratio, 0.03, between successive terms.
c.
[2 points] Assuming that the doses of the drug continue indefinitely, will
the amount of the drug in the body ever exceed 100mg? Explain.
No, since the infinite sum is equal to:
0
25
25
25
25
(0.03)
100 100
97
1 (0.03)
97
100
i
i
∞
=
==
=
⋅
<
−
∑
.
2.
[8 points] Suppose that
x
measures the time it takes for a student to complete an
exam for which the maximum time allowed is one hour. The distribution of the
completion times by the students is given by the probability density function
3
if 0
1
()
0o
t
h
e
r
w
i
s
e
cx
x
px
⎧
≤
≤
=
⎨
⎩
a.
[4 points] Determine the value of
c
that ensures that
p
(
x
) is a density
function.
1
1
4
3
0
0
1
1
1
1
4
44
xc
p x dx
cx dx
c
c
∞
−∞
⎡⎤
=⇒
=⇒ =⇒ =
⎢⎥
⎣⎦
∫∫
.
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View Full Documentb.
[4 points] What is the mean time for the students to complete the exam?
1
11
5
34
00
0
44
()
4
55
x
xp x dx
xcx dx
x dx
∞
−∞
⎡⎤
⇒⇒
⇒
=
⎢⎥
⎣⎦
∫∫
∫
hours.
3.
[ 8 points] Suppose
f
(
x
,
y
) is a linear function with the values
a.
[4 points] Write the equation of
f
in the form
f
(
x
,
y
) =
mx
+
ny
+
c
.
m
= 3,
n
= 2,
c
= 4 (where
x
and
y
are both 0), so we have that:
f
(
x
,
y
) = 3
x
+ 2
y
+ 4.
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 Spring '07
 Hohnhold
 Math

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