2022W2_MATH_101A_ALL_2022W2
.pdf
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School
University of British Columbia *
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Course
101
Subject
Mathematics
Date
Apr 3, 2024
Type
Pages
9
Uploaded by PresidentPartridgeMaster810
Gyan Edbert Zesiro
2022W2
MATH
101A
ALL
2022W2
Assignment WW11 due 04/13/2023 at 11:59pm PDT
Problem 1.
(1 point)
This assignment is not for marks. It is intended to help you
practice topics appearing at the end of the course (more Tay-
lor series and Probabilty) as you study for the final exam.
Evaluate the given limit.
lim
x
→
0
log
(
1
-
x
)+
x
+
x
2
2
4
x
3
=
Answer(s) submitted:
•
-
1
12
submitted: (correct)
recorded: (correct)
Correct Answers:
• -
0
.
0833333333333333
Problem 2.
(1 point)
Decide if the following series converges. If it does, enter the ex-
act value for its sum (not a decimal approximation); if not, enter
either ”Diverges” or ”D”.
S
=
∞
∑
n
=
0
(
-
1
)
n
2
n
+
1
1
3
2
n
+
1
Answer:
Answer(s) submitted:
•
tan
-
1
1
3
submitted: (correct)
recorded: (correct)
Correct Answers:
•
tan
-
1
1
3
Problem 3.
(1 point)
Let
F
(
x
) =
Z
x
0
e
-
2
t
4
dt
.
(a)
Find the Maclaurin series for
F
(
x
)
. Then enter the polynomial
obtained by discarding all terms involving
x
n
for
n
>
9.
Answer:
T
9
(
x
) =
(b)
Find the indicated derivative:
F
(
29
)
(
0
) =
Answer(s) submitted:
•
x
-
2
5
x
5
+
2
9
x
9
•
(
-
2
)
7
(
29!
)
7!
·
29
submitted: (correct)
recorded: (correct)
Correct Answers:
•
x
-
2
x
5
5
+
2
2
x
9
18
•
(
-
1
)
7
·
2
7
((
4
·
7
)
!
)
7!
1
Problem 4.
(1 point)
Find the infinite series representation, centred at
x
=
0, of the im-
proper integral
f
(
x
) =
Z
x
0
sin
(
5
t
)
2
t
dt
.
Enter the first five non-zero terms, in order of increasing degree.
Answer:
f
(
x
) =
+
+
+
+
+
···
What is the radius of convergence?
Answer:
R
=
Answer(s) submitted:
•
5
x
2
•
-
5
3
x
3
2
·
(
3!
)
·
3
•
5
5
x
5
2
·
(
5!
)
·
5
•
-
5
7
x
7
2
·
(
7!
)
·
7
•
5
9
x
9
2
·
(
9!
)
·
9
•
∞
submitted: (correct)
recorded: (correct)
Correct Answers:
•
1
2
·
5
x
• -
1
2
1
3
·
(
3!
)
·
5
3
x
3
•
1
2
1
5
·
(
5!
)
·
5
5
x
5
• -
1
2
1
7
·
(
7!
)
·
5
7
x
7
•
1
2
1
9
·
(
9!
)
·
5
9
x
9
•
∞
Problem 5.
(1 point)
Let
f
(
x
) =
6
x
sin
(
2
x
)+
px
2
24
-
24cos
(
5
x
)
-
300
x
2
.
(a)
Find the one and only value of the constant
p
for which
lim
x
→
0
f
(
x
)
exists.
Answer:
p
=
(b)
Using the value of
p
found in part
(a)
, evaluate the limit.
Answer:
lim
x
→
0
f
(
x
) =
Answer(s) submitted:
• -
12
•
8
625
submitted: (correct)
recorded: (correct)
Correct Answers:
• -
6
·
2
•
2
3
5
4
Problem 6.
(1 point)
Evaluate the indicated limit.
lim
x
→
0
(
1
+
4
x
+
4
x
2
)
1
/
x
=
Answer(s) submitted:
•
e
4
submitted: (correct)
recorded: (correct)
Correct Answers:
•
exp
(
1
·
4
)
Problem 7.
(1 point)
Find the value of
C
so that the function
f
(
x
) =
(
0
if
x
<
0
Ce
-
2
x
if
x
≥
0
.
is a probability density function.
Answer(s) submitted:
•
2
submitted: (correct)
recorded: (correct)
Correct Answers:
•
2
2
Problem 8.
(1 point)
The probability density function for a certain random variable
X
is given by
p
(
x
) =
2
153
x
if 0
≤
x
≤
9
2
8
-
2
136
x
if 9
≤
x
≤
17
0
otherwise.
Find the probability that
X
is between 8 and 10.
Pr
(
8
≤
X
≤
10
) =
Answer(s) submitted:
•
1
9
+
1
4
-
19
136
submitted: (correct)
recorded: (correct)
Correct Answers:
•
0
.
22140522875817
Problem 9.
(1 point)
The probability density function for a certain random variable
X
is given by
p
(
x
) =
2
253
x
if 0
≤
x
≤
11
2
12
-
2
276
x
if 11
≤
x
≤
23
0
otherwise.
Find the expected value of
X
.
E
(
X
) =
Answer(s) submitted:
•
2
253
·
3
·
11
3
+
1
12
23
2
-
11
2
-
2
276
·
3
23
3
-
11
3
submitted: (correct)
recorded: (correct)
Correct Answers:
•
11
.
3333333333333
Problem 10.
(1 point)
Suppose
that,
after
measuring
the
duration
of
many
tele-
phone calls, a telephone company found their data was well-
approximated by the density function
p
(
x
) =
0
.
8
e
-
0
.
8
x
, where
x
is the duration of a call, in minutes.
(a)
What percentage of calls last between 2 and 3 minutes?
Percent =
percent
(b)
What percentage of calls last 2 minutes or less?
Percent =
percent
(c)
What percentage of calls last 4 minutes or more?
Percent =
percent
Solution:
SOLUTION
(a)
The fraction of calls lasting from 1 to 2 minutes is given by the
integral
Z
3
2
p
(
x
)
dx
=
Z
3
2
0
.
8
e
-
0
.
8
x
dx
=
e
-
1
.
6
-
e
-
2
.
4
≈
0
.
11118
,
or about 11.118 percent.
(b)
A similar calculation (changing the limits of integration) gives
the percentage of calls lasting 2 minutes or less as
Z
2
0
p
(
x
)
dx
=
Z
2
0
0
.
8
e
-
0
.
8
x
dx
=
1
-
e
-
1
.
6
≈
0
.
7981
,
or about 79.81 percent.
(c)
The percentage of calls lasting 4 minutes or more is given by
the improper integral
Z
∞
4
p
(
x
)
dx
=
lim
b
→
∞
Z
b
4
0
.
8
e
-
0
.
8
x
dx
=
lim
b
→
∞
(
e
-
3
.
2
-
e
-
0
.
8
b
) =
e
-
3
.
2
≈
0
.
04076
or about 4.076 percent.
Answer(s) submitted:
•
e
-
1
.
6
-
e
-
2
.
4
·
100
•
e
-
0
-
e
-
2
·
0
.
8
·
100
•
e
-
0
.
8
·
4
·
100
submitted: (correct)
recorded: (correct)
Correct Answers:
•
100
e
-
0
.
8
·
2
-
e
-
0
.
8
·
3
•
100 1
-
e
-
0
.
8
·
2
•
100
e
-
0
.
8
·
4
3
Problem 11.
(1 point)
The probability density function for the duration of riders’
screams on a roller coaster is given by
f
(
x
) =
(
1
10
π
(
1
-
cos
(
8
x
))
if 0
≤
x
≤
10
π
0
otherwise.
Find the mean duration of riders’ screams over the course of the
ride.
The mean duration of riders’ screams is
seconds.
Answer(s) submitted:
•
5
π
submitted: (correct)
recorded: (correct)
Correct Answers:
•
15
.
707963267949
Problem 12.
(1 point)
The random variables
A
,
B
, and
C
have the probability density
functions (PDFs) shown below.
A=
B=
C=
(Click on graph to enlarge)
(a)
Which random variable has the largest expected value?
Enter A, B, or C:
(b)
Select the correct inequality:
•
A. Var
(
B
)
<
Var
(
C
)
•
B. Var
(
B
) =
Var
(
C
)
•
C. Var
(
B
)
>
Var
(
C
)
Answer(s) submitted:
•
B
•
A
submitted: (correct)
recorded: (correct)
Correct Answers:
•
B
•
A
4
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