Math 116 — First Midterm
February 8, 2010
Name:
EXAM SOLUTIONS
Instructor:
Section:
1.
Do not open this exam until you are told to do so.
2. This exam has 10 pages including this cover. There are 10 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
00
×
5
00
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.
Problem
Points
Score
1
11
2
13
3
8
4
12
5
10
6
12
7
7
8
6
9
6
10
15
Total
100
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View Full DocumentMath 116 / Exam 1 (February 8, 2010)
page 2
1
. [11 points] There is a classic result in mathematics, which states that the number of prime
numbers less than any number
x
≥
2 is approximated by the function li(
x
) =
R
x
2
dt
ln
t
.
a
. [3 points] Is li(
x
) increasing, decreasing, or neither for
x
≥
2? Provide justiﬁcation for
your answer.
Solution:
This function is increasing. To see that it is increasing, we take its derivative
d
dx
li(
x
) =
d
dx
Z
x
2
dt
ln
t
=
1
ln
x
>
0
.
OR
Since ln
x
is positive for
x
≥
2, we know
f
(
x
) =
1
ln
x
>
0. Since
f
(
x
) is the derivative of
li(
x
), we know li(
x
) is increasing.
b
. [3 points] Is li(
x
) concave up, concave down, or neither for
x
≥
2? Provide justiﬁcation
for your answer.
Solution:
This function is concave down. To see it is concave down, take the second
derivative:
d
2
dx
2
li(
x
) =
d
dx
1
ln
x
=

1
x
ln
2
x
<
0
.
OR
Since ln
x
is an increasing function, we know
f
(
x
) =
1
ln
x
is decreasing, which means
f
0
(
x
)
<
0. Since
f
0
(
x
) is the second derivative of li(
x
), we know li(
x
) is concave down.
c
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 Fall '07
 Irena
 Math, Calculus, Derivative, dx, Convex function

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