BONUS Week 2 Homework
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School
Georgia Institute Of Technology *
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Course
6242
Subject
Mathematics
Date
Feb 20, 2024
Type
docx
Pages
6
Uploaded by BailiffNeutron10026
BONUS Week 2 Homework
Due
Jan 26 at 8:59pm
Points
8
Questions
8
Available
Jan 19 at 5am - Jan 29 at 8:59pm
Time Limit
None
Instructions
Please answer all the questions below.
This quiz was locked Jan 29 at 8:59pm.
Attempt History
Attempt
Time
Score
LATEST
Attempt 1
2 minutes
8 out of 8
Score for this quiz:
8
out of 8
Submitted Jan 25 at 2:05pm
This attempt took 2 minutes.
Question 1
1
/ 1
pts
(Lesson 2.1: Derivatives.) BONUS: If �(�)=ℓ�(2�−3), find the derivative �′
(�).
a.
2�
b.
12ℓ�(2�−3)
Correct!
c.
2/(2�−3)
This follows by the chain rule,
�′(�)=[ℓ�(2�−3)]′=(2�−3)′2�−3=22�−2
d.
�/2
(c). This follows by the chain rule,
�′(�)=[ℓ�(2�−3)]′=(2�−3)′2�−3=22�−2
Question 2
1
/ 1
pts
(Lesson 2.1: Derivatives.) BONUS: If �(�)=cos(1/�), find the derivative �′
(�).
a.
cos(1/�2)
b.
sin(1/�2)
c.
−1�2sin(1/�)
Correct!
d.
1�2sin(1/�)
By the chain rule,
[cos(1/�)]′=−sin(1/�)[1/�]′=1�2sin(1/�)
(d). By the chain rule,
[cos(1/�)]′=−sin(1/�)[1/�]′=1�2sin(1/�)
Question 3
1
/ 1
pts
(Lesson 2.2: Finding Zeroes.) BONUS: Suppose that �(�)=�4�−4�2�+4. Use any method you want to find a zero of �(�), i.e., � such that �(�)=0.
a.
�=0
b.
�=1
c.
�=ℓ�(2)=0.693
Correct!
d.
�=12ℓ�(2)=0.347
This doesn't take too much work. Namely, set
0=�(�)=�4�−4�2�+4=(�2�−2)2.
This is the same as �2�=2, or �=12ℓ�(2)=0.347
(d). This doesn't take too much work. Namely, set
0=�(�)=�4�−4�2�+4=(�2�−2)2.
This is the same as �2�=2, or �=12ℓ�(2)=0.347
Question 4
1
/ 1
pts
(Lesson 2.3: Integration.) BONUS: Find ∫01(2�+1)2��.
a. 1/2
b. 7/2
c. 7/3
Correct!
d. 13/3
We have
∫01(2�+1)2��=(2�+1)36|01=276−16=13/3
(d). We have
∫01(2�+1)2��=(2�+1)36|01=276−16=13/3
Question 5
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1
/ 1
pts
(Lesson 2.3: Integration.) BONUS: Find ∫12�2���.
a. 1
b.
�2−�
Correct!
c. 23.6
We have
∫12�2�=12�2�|12=23.60
d. 46.2
(c). We have
∫12�2�=12�2�|12=23.60
Question 6
1
/ 1
pts
(Lesson 2.3: Integration.) BONUS: Find
lim�→0sin(�)−��.
a. 1
Correct!
b. 0
If we let �(�)=sin(�)−� and �(�)=�, then lim�→0�(�)=0 and
lim�→0�(�)=0, so that
lim�→0�(�)�(�)
seems to get us into a 0/0 issue. Thus, we'll need to employ L'Hôspital's Rule:
lim�→0�(�)�(�)=lim�→0�′(�)�′(�)=lim�→0cos(�)−11=01=0.
c.
∞
d. undetermined
(b). If we let �(�)=sin(�)−� and �(�)=�, then lim�→0�(�)=0 and
lim�→0�(�)=0, so that
lim�→0�(�)�(�)
seems to get us into a 0/0 issue. Thus, we'll need to employ L'Hôspital's Rule:
lim�→0�(�)�(�)=lim�→0�′(�)�′(�)=lim�→0cos(�)−11=01=0.
Question 7
1
/ 1
pts
(Lesson 2.4: Numerical Integration.) BONUS: Find the approximate value of the integral ∫02(�−1)2�� using the lesson's form of the Riemann sum with �(�)=(�−1)2,�=0,�=2, and �=4.
a. -2
b. 1/3
Correct!
c. 3/4
We have
∫02(�−1)2��≈�−��∑�=1��(�+(�−�)��)=24∑�=14(2�4−1)2=3/4
Well, this is sort of close to the true integral of 2/3. Of course, we could've done even better if � had been bigger or if we had used the midpoint of each interval instead of the right endpoint.
d. 3
(c). We have
∫02(�−1)2��≈�−��∑�=1��(�+
(�−�)��)=24∑�=14(2�4−1)2=3/4
Well, this is sort of close to the true integral of 2/3. Of course, we could've done even better if � had been bigger or if we had used the midpoint of each interval instead of the right endpoint.
Question 8
1
/ 1
pts
(Lesson 2.6: Simulating Random Variables.) BONUS: Suppose � and � are independent Uniform(0,1) random variables. (You can simulate these using the RAND() function in Excel, for instance.) Consider the nasty-looking random variable
�=−2ℓ�(�)cos(2��),
where the cosine calculation is carried out in radians (not degrees). Go ahead and calculate �. . . don't be afraid. Now, repeat this task 1000 times (easy to do in Excel) and make a histogram of the 1000 �'s. What distribution does this look like?
Correct!
a. Normal
This is the Box-Muller method to generate normal random variables. We'll learn much more about this later on.
b. Unif(0,1)
c. Exponential
d. Weibull
(a). This is the Box-Muller method to generate normal random variables. We'll learn much more about this later on. Here's some example Matlab code
that works well. . .
%Matlab code
clear all;close all;clc;
z_vec=zeros(1000,1);
for i = 1:1000
u = rand;
v = rand;
z_vec(i)= sqrt(-2*log(u))*cos(2*pi*v);
end
nbins = 30;
bins = linspace(-5,5,nbins);
histogram(z_vec,bins)
Of course, you can do this easily in R, Excel, Python etc.
Quiz Score:
8
out of 8
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