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exam 2 winter 18 solutions.pdfMath 115 — Second M...
exam_2_winter_18_solutions.pdfMath 115 — Second Midterm —
exam 2 winter 18 solutions.pdfMath...
exam_2_winter_18_solutions.pdfMath 115 — Second Midterm —
Page 1
Math 115 — Second Midterm — March 20, 2018.
EXAM SOLUTIONS
1.
Do not open this exam until you are told to do so.
2.
Do not write your name anywhere on this exam.
3.
This exam has 11 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.
4.
Do not separate the pages of this exam. If they do become separated, write your UMID (not
name) on every page and point this out to your instructor when you hand in the exam.
5.
Note that the back of every page of the exam is blank, and, if needed, you may use this space
for scratchwork. Clearly identify any of this work that you would like to have graded.
6.
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.
7.
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.
8.
The use of any networked device while working on this exam is not
permitted.
9.
You may use any one calculator that does not have an internet or data connection except a
TI92 (or other calculator with a “qwerty” 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 single 3
00
×
5
00
notecard.
10.
For any graph or table that you use to ﬁnd an answer, be sure to sketch the graph or write
out the entries of the table. In either case, include an explanation of how you used the graph
or table to ﬁnd the answer.
11.
Include units in your answer where that is appropriate.
12.
Problems may ask for answers in
exact form
. Recall that
x
=
√
2 is a solution in exact form
to the equation
x
2
= 2, but
x
=1
.
41421356237 is not
.
13.
Turn oﬀ all cell phones, smartphones, and other electronic devices
, and remove all
headphones, earbuds, and smartwatches. Put all of these items away.
14.
You must use the methods learned in this course to solve all problems.
Problem
Points
Score
1
13
2
12
3
12
4
12
5
15
6
5
Problem
Points
Score
7
10
8
14
9
7
Total
100
Page 2
Math 115 / Exam 2 (March 20, 2018.)
page 2
1
. [13 points]
Some values of the twice diﬀerentiable function
f
(
x
) and of its ﬁrst and second
derivative are given by the following table
x
0
1
2
4
5
6
7
f
(
x
)
1
4
4
.
3
5
f
0
(
x
)
8
0
.
25
0
.
6
2
f
00
(
x
)
4
0
.
1
0
.
2
Suppose the function
f
(
x
) is deﬁned and invertible for
∞
<x<
∞
.
In the following
questions, you will ﬁnd some
of the missing values using the information given. If there is not
enough information given to answer the question, write “
NEI
”. Show your work.
a
. [4 points] The function
a
(
x
) = ln(1 +
f
(
x
)) satisﬁes
a
0
(2) = 2. Find
f
(2).
Solution:
a
0
(
x
)=
1
1+
f
(
x
)
f
0
(
x
)
2=
8
1+
f
(2)
8=2+2
f
(2)
f
(2) = 3
Answer:
f
(2) = 3.
b
. [3 points] Let
b
(
x
)=
f
(
x
)
f
0
(
x
) and
b
0
(0) = 4. Find
f
0
(0).
Solution:
b
0
(
x
)=(
f
0
(
x
))
2
+
f
(
x
)
f
00
(
x
)
4=(
f
0
(0))
2
+
f
(0)
f
00
(0)
4=(
f
0
(0))
2
+4
f
0
(0) = 0
.
Answers:
f
0
(0) = 0.
c
. [3 points] The quadratic approximation
Q
(
x
) of the function
f
(
x
) at
x
= 1 is
Q
(
x
)=
1
2
x
+
3
2
. Find
f
(1),
f
0
(1), and
f
00
(1).
Solution:
Answers:
f
(1) = 2 ,
f
0
(1) =
1
2
,
f
00
(1) = 0
d
. [3 points] Let
h
(
x
)=
f

1
(
x
). Find the value of
h
0
(5).
Solution:
h
0
(
x
)=
1
f
0
(
f

1
(
x
))
, then
h
0
(5) =
1
f
0
(
f

1
(5))
=
1
f
0
(6)
=
1
0
.
6
=
5
3
.
Answer:
h
0
(5) =
5
3
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