In The Crucible, who called Elizabeth out and why?

According to Cheever, Abigail Williams made the claim that Elizabeth's spirit stuck needles into her (Abigail's) stomach. Cheever is then summoned to the Proctor house to question Elizabeth and look for evidence of this type of witchcraft. He finds a poppet (doll) with needles stuck in it. This is what Abigail claimed. She concluded that Elizabeth stuck needles in the doll and used her (voodoo) powers of witchcraft to inflict that violence on Abigail herself.
Of course, this is untrue. Abigail is pretending, and she had a role in using Mary Warren to plant the doll (poppet) in Elizabeth's home. As a result of Abigail's accusation and finding the poppet in her home, Cheever is forced to arrest Elizabeth.
Abigail accuses Elizabeth out of jealousy. She and John Proctor, Elizabeth's husband, had an affair, but John ended it. Abigail, still in love with John, did not want their relationship to end. When he abandons her, she takes her revenge upon Elizabeth. At this point, John Proctor's only hope is to convince Mary Warren to admit it was her poppet and the accusations of witchcraft are all false. John is outraged that his wife has been charged, but he must know his affair with Abigail indirectly contributed to his wife's arrest.

Calculus of a Single Variable, Chapter 9, 9.6, Section 9.6, Problem 21

When using Root test on a series sum a_n , we determine a limit as:
lim_(n-gtoo) root(n)(|a_n|)= L
or
lim_(n-gtoo) |a_n|^(1/n)= L
Then, we follow the conditions:
a) Llt1 then the series is absolutely convergent.
b) Lgt1 then the series is divergent.
c) L=1 or does not exist then the test is inconclusive. The series may be divergent, conditionally convergent, or absolutely convergent.
We may apply the Root Test to determine the convergence or divergence of the series sum_(n=1)^oo n^3/3^n . when we let: a_n =n^3/3^n .
Applying the Root test, we set-up the limit as:
lim_(n-gtoo) |n^3/3^n|^(1/n) =lim_(n-gtoo) (n^3/3^n)^(1/n)
Apply Law of Exponents: (x/y)^n = x^n/y^n and (x^n)^m= x^(n*m) .
lim_(n-gtoo) (n^3/3^n)^(1/n) =lim_(n-gtoo) (n^3)^(1/n)/(3^n)^(1/n)
=lim_(n-gtoo) n^(3*1/n)/3^(n*1/n)
=lim_(n-gtoo) n^(3/n)/3^(n/n)
=lim_(n-gtoo) n^(3/n)/3^1
=lim_(n-gtoo) n^(3/n)/3
Evaluate the limit.
lim_(n-gtoo)n^(3/n)/ 3 =1/3 lim_(n-gtoo) n^(3/n)
=1/3 *1
=1/3
The limit value L =1/3 satisfies the condition: Llt1 since 1/3lt1 .
Therefore, the series sum_(n=1)^oo n^3/3^n is absolutely convergent.

In 2081 there had been 213 amendments added to the US Constitution. What does this show us about the government in the story?

Perhaps the addition of around 190 amendments to the Constitution since 1961, the publishing date of "Harrison Bergeron," suggests the increasing authoritarianism of the US government. For the most part, it is very difficult to pass an amendment to the Constitution, considering only 27 have been passed in nearly 240 years. For so many amendments to pass would suggest that strongmen and strongwomen, as is the case in this story, have pushed through these amendments with threats of violence.
Just thinking about "Harrison Bergeron," it's very clear that the government has become authoritarian. The removal of handicaps is punished with Draconian laws ("Two years in prison and two thousand dollars fine for every ball I took out . . ."). Those, like Harrison himself, who defy societal expectations are removed from the public and put in prisons. Then, as the end of the story shows, people who defy the government, as Harrison does, do not get a trial, but are instead killed on the spot.

The fact that there have been 213 amendments added to the United States Constitution in the short story indicates that America's government is continually making changes in order to appease the entire population. Given the fact that today the Constitution has only been amended twenty-seven times since its inception in 1787, Vonnegut's future American society has taken extreme steps to change the outlook and landscape of the nation.
In the short story, the last three amendments added to the Constitution specifically concern issues of equality. After the 213th amendment was added to the Constitution, the entire population of America was completely equal in every sense of the word. With the perseverance and vigilance of the agents of the United States Handicapper General, citizens became equal in intelligence, athleticism, and appearance. Overall, the high number of amendments indicates that the future United States government is continually changing in an attempt to please seemingly everyone in society.

Discuss Keat's "Hyperion" as an example of romantic poetry.

Romanticism is a great period of literature. Different Romantic authors vary the formula characteristics a bit from piece to piece; however, there are some key elements of Romanticism that are quite consistent throughout the period. If I had to pick three characteristics that are almost never missing, it would be the emphasis on nature, imagination, and emotion over reason.
This particular Keats poem immediately checks off the nature box within the first stanza. Readers are taken to a quiet and lonely "vale" that is far from any kind of hustle and bustle of cities and large populations.

Deep in the shady sadness of a vale
Far sunken from the healthy breath of morn,
Far from the fiery noon, and eve's one star.

Often, a Romantic author's imagination will focus on the supernatural. That doesn't necessarily means ghouls and goblins. It means anything beyond the natural world, and Keats does this quite early in the poem by personifying and describing Saturn.


Sat gray-hair'd Saturn, quiet as a stone,
Still as the silence round about his lair;
Forest on forest hung about his head
Like cloud on cloud.



Regarding emotion, a Romantic author is allowed to explore various emotions; however, they tend to lean more toward being sad and melancholy than any other emotion. This particular emotional vibe is again set fairly early on for the reader when the narrator describes Thea's face.


But oh! how unlike marble was that face:
How beautiful, if sorrow had not made
Sorrow more beautiful than Beauty's self.


About 35 lines later, Thea finds Saturn. The sad and melancholy emotions are again emphasized here when she tries to wake him.


Saturn, sleep on:—O thoughtless, why did I
Thus violate thy slumbrous solitude?
Why should I ope thy melancholy eyes?
Saturn, sleep on! while at thy feet I weep.

How did the ideas of the French Revolution spread outside France?

The ideas of the French Revolution (1789--1799) spread in various ways. The French Revolution was a momentous event and its effects could not be limited to France.
One way ideas spread was through the thousands of French emigres. These were the nobles who fled France to save their lives. They were, of course, very hostile to the French Revolution.
A second way these ideas spread was through France's position as the intellectual powerhouse of Europe. France was the home of the philosophes, so the ideas of the French Revolution influenced many people in far-flung places. In one example, they inspired the Haitian war for independence.
Another way the ideas were spread was through military conquest. Napoleon conquered much of Europe and put relatives on thrones throughout Europe. After Napoleon's defeat in 1815, the conservative Great Powers of Europe tried to reverse the effects of the French Revolution.

Based on ideals of liberty and equality, the French Revolution destroyed the monarchy, curbed the power of the Catholic Church, created fairer taxation for citizens, and established democracy as the new form of government. Destruction of the feudal model of governance, when combined with new industrial opportunities, helped foster lasting economic growth.
The ideals perpetuated by the French Revolution had a ripple effect throughout the world, fueling similar revolutions in other countries: Haiti, Italy, Switzerland, Prussia, and South America for instance. Those who emigrated to other countries in an effort to escape the war also helped promote the spread of French culture and revolutionary ideals.
Finally, the French Revolution had an enormous impact on art—and literature in particular—by spawning the greatest literary movement of all time: Romanticism, which inspired the works of authors such as Keats, Byron, Coleridge, and Shelley.
https://scholar.harvard.edu/files/jrobinson/files/jr_consequeces_frenchrev.pdf

Beginning Algebra With Applications, Chapter 5, 5.2, Section 5.2, Problem 106

Graph $3x+4y =12$ by using $x$- and $y$-intercepts

$x$-intercept:


$
\begin{equation}
\begin{aligned}

3x+4y =& 12
&& \text{Given equation}
\\
3x + 4(0) =& 12
&& \text{To find the $x$-intercept, let } y = 0
\\
3x =& 12
&& \text{Divide by } 3
\\
x =& 4
&&

\end{aligned}
\end{equation}
$


The $x$-intercept is $(4,0)$

$y$-intercept:


$
\begin{equation}
\begin{aligned}

3x+4y =& 12
&& \text{Given equation}
\\
3(0) + 4y =& 12
&& \text{To find the $y$-intercept, let } x=0
\\
4y =& 12
&& \text{Divide by } 4
\\
y =& 3
&&

\end{aligned}
\end{equation}
$



The $y$-intercept is $(0,3)$

Graph the ordered pairs $(4,0)$ and $(0,3)$. Draw a straight line through the points.

Calculus: Early Transcendentals, Chapter 7, 7.1, Section 7.1, Problem 30

int_1^sqrt(3)arctan(1/x)dx
If f(x) and g(x) are differentiable functions, then
intf(x)g'(x)dx=f(x)g(x)-intf'(x)g(x)dx
If we write f(x)=u and g'(x)=v, then
intuvdx=uintvdx-int(u'intvdx)dx
Using the above method of integration by parts,
intarctan(1/x)dx=arctan(1/x)*int1dx-int(d/dx(arctan(1/x)int1dx)dx
=arctan(1/x)*x-int(1/(1+(1/x)^2)*d/dx(1/x)int1dx)dx
=xarctan(1/x)-int(x^2/(x^2+1)*(-1x^-2)*x)dx
=xarctan(1/x)+intx/(x^2+1)dx
Now let's evaluate intx/(x^2+1)dx
substitute t=x^2+1,=>dt=2xdx
intx/(x^2+1)dx=intdt/(2t)
=1/2ln|t|
substitute back t=x^2+1
=1/2ln|x^2+1|
intarctan(1/x)dx=xarctan(1/x)+1/2ln|x^2+1|+C
C is a constant
Now evaluate the definite integral,
int_1^sqrt(3)arctan(1/x)dx=[xarctan(1/x)+1/2ln|x^2+1|]_1^sqrt(3)
=[sqrt(3)arctan(1/sqrt(3))+1/2ln(3+1)]-[1arctan(1/1)+1/2ln(1+1)]
=[sqrt(3)pi/6+1/2ln(4)]-[pi/4+1/2ln2]
=[sqrt(3)pi/6+1/2ln(2^2)]-[pi/4+1/2ln(2)]
=(sqrt(3)pi/6+ln(2)-pi/4-1/2ln(2))
=(sqrt(3)pi/6-pi/4+1/2ln(2))
=(2sqrt(3)-3)pi/12+1/2ln(2)