Math 115 — Second Midterm
November 15, 2011
Name:
EXAM SOLUTIONS
Instructor:
Section:
1.
Do not open this exam until you are told to do so.
2.
This exam has 9 pages including this cover. There are 9 problems. Note that the problems
are not of equal difficulty, so you may want to skip over and return to a problem on which
you are stuck.
3.
Do not separate the pages of this exam. If they do become separated, write your name on
every page and point this out to your instructor when you hand in the exam.
4.
Please read the instructions for each individual problem carefully. One of the skills being
tested on this exam is your ability to interpret mathematical questions, so instructors will
not answer questions about exam problems during the exam.
5.
Show an appropriate amount of work (including appropriate explanation) for each problem,
so that graders can see not only your answer but how you obtained it. Include units in your
answer where that is appropriate.
6.
You may use any calculator except a TI92 (or other calculator with a full alphanumeric
keypad). However, you must show work for any calculation which we have learned how to
do in this course. You are also allowed two sides of a 3
′′
×
5
′′
note card.
7.
If you use graphs or tables to find an answer, be sure to include an explanation and sketch
of the graph, and to write out the entries of the table that you use.
8.
Turn off all cell phones and pagers
, and remove all headphones.
9.
You must use the methods learned in this course to solve all problems.
10.
You must clearly indicate your final answer for each problem to receive credit.
Problem
Points
Score
1
9
2
8
3
12
4
10
5
8
6
13
7
16
8
12
9
12
Total
100
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Math 115 / Exam 2 (November 15, 2011)
page 2
1
. [9 points]
Let
U
=
f
(
t
) give the number of Facebook users in millions in year
t
.
Suppose
f
(2005) = 5
.
5 and
f
′
(2005) = 4
.
9. For this problem assume that
f
(
t
) is strictly increasing.
a
. [4 points]
Find and interpret, in practical terms,
f
−
1
(5
.
5).
Solution:
The year in which there were 5.5 million Facebook users was 2005.
f
−
1
(5
.
5) =
2005
(year)
b
. [5 points]
Showing work, evaluate (
f
−
1
)
′
(5
.
5). Interpret your answer in practical terms.
Solution:
(
f
−
1
)
′
(5
.
5) = 1
/
(
f
′
(
f
−
1
(5
.
5)) = 1
/f
′
(2005) = 1
/
4
.
9
.
When the number of Facebook users was 5.5 million, in 1/4.9 (
∼
0.2) years the number
of Facebook users will have increased by approximately 1 million users.
(
f
−
1
)
′
(5
.
5) =
1
/
4
.
9
years per million users
2
. [8 points] Recall the function
T
(
x
) that took the number of followers (in millions) of a Twitter
user and returned a value from 0 to 10 called the user’s Twitter celebrity index. The derivative
of
T
(
x
) is given by the function
T
′
(
x
) =
1532
.
5
·
(0
.
6)
x
(5 + 60(0
.
6)
x
)
2
.
a
. [4 points]
If
T
(3) = 1
.
56, compute the local linearization of
T
(
x
) near
x
= 3.
Solution:
The formula for the local linearization for a function
f
(
x
) near a point
x
=
a
is
L
(
x
) =
f
(
a
) +
f
′
(
a
)(
x

a
)
.
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 Fall '08
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 Math, Derivative

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