Spring_2024_Calculus_II_HW 2 (1)
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Date
Feb 20, 2024
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Uploaded by ColonelBadgerMaster1065
Calculus II
Professor Ross Flek - Spring 24
Written Assignment 2
Due: Friday, February 23rd, 23:59 via Gradescope
Instructions
•
This homework should be submitted via Gradescope by 23:59 on the date listed above.
You can find
instructions on how to submit to Gradescope on our Campuswire channel.
•
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∗
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your work with your phone and convert them into a single pdf file. Gradescope will only allow
pdf files to be uploaded.
•
You must show all work
. You may receive zero or reduced marks for insufficient work.
Your work
must be neatly organised and written
. You may receive zero or reduced marks for incoherent work.
•
If you are writing your answers on anything other than this sheet, you should only have
one question
per page
. You can have parts a), b) and c) on the page for example, but problems 1) and 2) should be
on separate pages.
•
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•
These problems are designed to be done without a calculator. Whilst there is nothing stopping you using
a calculator when working through this assignment, be aware of the fact that you are not permitted to
use calculators on exams so you might want to practice without one.
•
The maximum score for this assignment is
50
.
The problems on this assignment will be
graded on
correctness and completeness
.
MATH-UA.0122-004/006 - Written Assignment 2
Problem 1:
(20 points)
Evaluate the following integrals.
Clearly indicate the technique(s) you are
using in each one.
(a)
(5 pts)
Z
1
√
x
+ 1 +
√
x
dx
(b)
(5 pts)
Z
1
0
ln(
x
)
√
x
dx
2
MATH-UA.0122-004/006 - Written Assignment 2
(c)
(5 pts)
Z
sin (
x
) cos (
x
)
sin
4
(
x
) + cos
4
(
x
)
dx
(Hint: let
u
= sin
2
(
x
); once you have rewritten in terms of
u
, you
will have to complete the square in the denominator of your result and use another substitution)
(d)
(5 pts)
Z
(
2
x
2
+ 1
)
e
x
2
dx
3
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MATH-UA.0122-004/006 - Written Assignment 2
Problem 2:
(10 points)
Calculate the following integrals using partial fraction decomposition.
(a)
(5 pts)
Z
∞
2
4
x
2
x
4
−
1
dx
4
MATH-UA.0122-004/006 - Written Assignment 2
(b)
(5 pts)
Z
x
3
+ 6
x
2
+ 3
x
+ 6
x
3
+ 2
x
2
dx
5
MATH-UA.0122-004/006 - Written Assignment 2
Problem 3:
(5 points)
The widths, in metres, of a kidney-shaped swimming pool were measured at
2-metre intervals as indicated in the diagram below. Use Simpson’s Rule to estimate the area of the pool.
6
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MATH-UA.0122-004/006 - Written Assignment 2
Problem 4:
(15 points)
The probability density function for the
standard normal distribution
, the
familiar bell-shaped curve, is given by the function
f
(
x
) =
1
√
2
π
e
−
x
2
/
2
.
(a)
(5 pts) Show that
Z
∞
−∞
f
(
x
)
dx
is convergent. (Hint: first show that
−
x
2
2
≤
1
2
−
x
. It will then follow
that
e
−
x
2
2
≤
e
1
2
−
x
and both functions are positive. Now you can construct a function whose integral
you can compare with the original integral)
7
MATH-UA.0122-004/006 - Written Assignment 2
(b)
(5 pts) The area under
f
(
x
) from
x
=
−
1 to
x
= 1 represents the probability that a normal random
variable is within one standard deviation of the mean. Use Simpson’s rule with
n
= 4 to approximate
this probability.
Show a sufficient amount of work to demonstrate you can do the problem. You may use a calculator
on your last step and give your answer to 6 decimal places.
(c)
(5 pts) How accurate is the probability you calculated in part (b)? That is, find the maximum error.
(Use the fact that
f
(4)
(
x
) =
1
√
2
π
(
x
4
−
6
x
2
+ 3
)
e
−
x
2
/
2
)
8