In what ways did the emergence of the anti-slavery movement impact American society and politics?

Another key way in which the anti-slavery movement impacted politics is that it was the precursor to the women's suffrage movement—the first wave of feminism.
A number of white women from the North and the South spoke out against slavery, including Lucy Stone and Angelina Grimke. These women also held the view that men and women should be equal, and they learned to speak out about their own subordination through their advocacy on behalf of black people who were enslaved. Sojourner Truth also addressed repressive notions of femininity in her speech "Ain't I a Woman?" delivered at the Women's Convention in Akron, Ohio in 1851. During her speech, she held up her muscled arms, sculpted from years of farm labor, which she had been forced to handle just as any male slave, and asked if her life and her body still made her a woman or something other. The question forced white feminists to reassess notions of femininity which had been used to oppress them.
Support for the Fifteenth Amendment, which granted black men the right to vote, was mixed, and it was a point over which suffragists disagreed. Frederick Douglass had been a supporter of women's suffrage, along with Susan B. Anthony and Elizabeth Cady Stanton. However, the latter two women disagreed with him over the passage of the amendment—not because they did not want black men to vote, but because the law still excluded women.

The anti-slavery or abolition movement polarized United States society. As anti-slavery groups and individuals increasingly condemned slavery and called for its immediate cessation, slave-owners went on the offensive. Some declared that it was not good enough to characterize slavery as a necessary evil or an institution that would gradually fade away in face of modernity. Instead, they insisted it was a beneficial institution, it should expand west, and the slaves were better off on the plantations than industrial factory workers were in the Northern states.
John C. Calhoun, for example, stated that slavery had helped improve the lives of the slaves, because they were "civilized" by their contacts with European civilization. As time went on, both proponents and opponents of slavery became less willing to compromise, each side insisting that Christianity supported their argument and each side demanding that their adherents toe an increasingly hard line.
Slave rebellions tied to abolition, such as John Brown's Harper's Ferry uprising, led to harsh reprisals and even more polarization. By Lincoln's election, there was little hope that the two sides could have a meeting of the minds.

The anti-slavery movement started as something only a minority of Americans wanted.  There were always fears of what would happen if the slaves were freed, and many Americans, while not personally in favor of slavery, were afraid of a slave revolt.  The anti-slavery movement had religious undertones as well, as many claimed that if all were equal in the eyes of God, then slavery was a moral evil.  William Lloyd Garrison's newspaper The Liberator was one of the great early anti-slavery publications, and Garrison achieved fame and notoriety when he burned a copy of the Constitution because he said that he could not support a document that assisted slavery.  Slavery became such an issue in Congress that Congress passed a "gag rule," which meant that they would not talk about it.  When the United States expanded, there were worries that the balance between free and slave states would be disrupted, so Congress passed the Compromise of 1820. This created a line at the southern border of Missouri, although Missouri was a slave state, and allowed the creation of Maine as a free state.  In 1840, James Birney ran as an abolitionist candidate in response to Whigs who could not accept slavery.  Slavery was the key issue in the Compromise of 1850 as the nation sought to add new territory but not disrupt the delicate balance between free and slave states.  In 1856, the Republican Party ran on a platform of not allowing slavery into the new territories.  Slavery was also at the heart of an American bestseller; Uncle Tom's Cabin provided a dramatic portrayal of what life was like for a slave family.  The book inflamed passions in both free and slave states—more Northerners called slavery barbaric and more Southerners called the book a farce because Stowe did not study slavery closely.

In Fahrenheit 451 how is Montag going to stand up to the influence of Captain Beatty?

We could also argue that Montag uses the two-way communication device as a means of standing up to Captain Beatty's influence. Remember that this device was invented by Faber. It allows two people to communicate with each other over any distance and, more importantly, without anyone else being aware of it. All the user has to do is place the device into his ear and he is immediately connected to the person on the other end.
When Montag visits Faber's apartment, he tells Faber that he is worried about being talked around by Beatty. Faber's solution is to give him the communication device. This means that Faber can hear everything that Beatty says and craft an appropriate response for Montag.
Although the two-way communication device is discovered by Beatty, it gives Montag the confidence he needs to stand up to Beatty's pro-censorship arguments.

Montag stands up to Captain Beatty by choosing to follow his heart, consulting Faber regarding the truth about literature, quitting his job, and shooting Beatty using a flamethrower. Captain Beatty is a proponent of government censorship and believes that literature and the pursuit of knowledge are harmful to society. Beatty attempts to persuade Montag into believing that a book is similar to a loaded gun and tries to convince him that firefighters are providing a necessary service. Despite Beatty's arguments, Montag searches for answers by reading illegal novels and discussing the significance of literature with Faber. When Captain Beatty attempts to arrest Montag, Montag demonstrates his rebellious, independent personality by shooting Beatty with a flamethrower. Montag willingly becomes an enemy of the state and flees the city in order to live as a hobo intellectual. Montag's decision to leave his old life behind in pursuit of knowledge instead of listening to Captian Beatty illustrates how he stands up to the captain.

sum_(n=1)^oo 1/(n+3) Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series.

sum_(n=1)^oo1/(n+3)
The integral test is applicable if f is positive, continuous and decreasing function on infinite interval [k,oo) where k>=1 and a_n=f(x) . Then the series sum_(n=1)^ooa_n converges or diverges if and only if the improper integral int_1^oof(x)dx converges or diverges.
For the given series a_n=1/(n+3)
Consider f(x)=1/(x+3)
Refer to the attached graph of the function. From the graph,we observe that the function is positive, continuous and decreasing for x>=1
Let's determine whether function is decreasing by finding the derivative f'(x),
f'(x)=-1(x+3)^(-2)
f'(x)=-1/(x+3)^2
f'(x)<0  which implies that the function is decreasing.
We can apply the integral test, since the function satisfies the conditions for the integral test.
Now let's determine whether the corresponding improper integral converges or diverges,
int_1^oo1/(x+3)dx=lim_(b->oo)int_1^b1/(x+3)dx 
=lim_(b->oo)[ln|x+3|]_1^b
=lim_(b->oo)ln|b+3|-ln|1+3|
=lim_(b->oo)ln|b+3|-ln4
lim_(b->oo)(b+3)=oo
Apply the common limit:lim_(u->oo)(ln(u))=oo
=oo-ln4
=oo
Since the integral int_1^oo1/(x+3)dx diverges, we can conclude from the integral test that the series diverges.
 

College Algebra, Chapter 4, 4.4, Section 4.4, Problem 30

Determine all rational zeros of the polynomial $P(x) = 2x^4 - x^3 - 19x^2 + 9x + 9$, and write the polynomial in factored form.

The leading coefficient of $P$ is $2$ and the possible zeros are $\pm 1, \pm 2$. They are divisors of the constant term $9$. We get $\pm 1, \pm 3, \pm 9 $.

Using the rational zeros theorem

possible rational zero of $P = \displaystyle \frac{\text{factor of 9}}{\text{factor of 2}}$

The possible rational zeros of $P$ are

$\displaystyle \pm \frac{1}{1}, \pm \frac{1}{2}, \pm \frac{3}{1}, \pm \frac{3}{2}, \pm \frac{9}{1}, \pm \frac{9}{2}$

Simplifying the fractions, we get

$\displaystyle \pm 1, \pm 3, \pm 9, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{9}{2}$

Using Synthetic Division







We find that $1$ is a zero and that $P$ factors as

$2x^4 - x^3 - 19x^2 + 9x + 9 = (x - 1)(2x^3 + x^2 - 18x - 9)$

We now factor the quotient $2x^3 + x^2 - 18x - 9$. The factors of $9$ are $\pm 1, \pm 3, \pm 9$ and the factors of $2$ are $\pm 1, \pm 2$. The possible rational zeros are

$\displaystyle \pm 1, \pm 3, \pm 9, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{9}{2}$

Using Synthetic Division







We find that $1$ is not a zero and that $3$ is a zero and that factors as

$2x^4 - x^3 - 19x^2 + 9x + 9 = (x - 1)(x - 3)(2x^2 + 7x + 3)$

We now factor $2x^2 + 7x + 3$ using trial and error, so we get

$2x^4 - x^3 - 19x^2 + 9x + 9 = (x - 1) (x - 3) (2x + 1) (x + 3)$

The zeros of $P$ are $\displaystyle 1, 3, \frac{-1}{2}$ and $-3$.

Is Beowulf the ideal epic hero? If not, what is he lacking?

In many ways, Beowulf is the archetypal epic hero, embodying much that was held to be good about a warrior in the Anglo-Saxon context. He is brave, has superhuman strength, and respects the kingly position of Heorot, taking on the king's greatest enemy alone as a means of honoring the king. In this, he is an excellent vassal, and, later, he grows to be an excellent king, dedicated to leading his people and improving their lives. The key example of Beowulf's bravery is, of course, the section in which he fights and defeats Grendel, which puts him in extreme personal danger, as a service to Heorot.
Beowulf's military strength, however, is inherently tied to his key weakness: his "ofermod," or over-confidence. We can detect the suggestion of this early in the poem, when he boasts of his previous exploits: heroic boasting is a key part of Anglo-Saxon warrior culture, but there is some indication that Beowulf exaggerates in order to build up his personal reputation. He may have superhuman strength, but is it really possible that he could hold his breath underwater for as long as he claims? Where a truly ideal hero would perform his feats selflessly, the use of the word "ofermod" in connection to Beowulf's decision to fight the dragon alone is telling: it suggests that the poet feels Beowulf is not pure in his motives. That is, Beowulf does want to save his people from the dragon, but he is also driven by the desire to increase his own reputation, and this need for personal reputation ultimately leads to his death.

In what chapter can I find quotes about Scout showing compassion?

In chapter 30, Scout demonstrates her compassion by caring for Boo Radley while Atticus and Sheriff Tate discuss who is responsible for Bob Ewell's death. When Atticus and Sheriff Tate leave Jem's room to discuss the events on the porch, Scout demonstrates compassion by taking Boo's hand and leading him to the front porch. Scout says,

"Come along, Mr. Arthur . . . you don’t know the house real well. I’ll just take you to the porch, sir." (Lee, 276)

When Scout takes Boo onto the porch, she once again illustrates her compassion by considering how the lights from the living room affect Boo. Scout says,

Feeling slightly unreal, I led him to the chair farthest from Atticus and Mr. Tate. It was in deep shadow. Boo would feel more comfortable in the dark. (Lee, 276)

Scout understands that Boo may be uncomfortable in the unfamiliar environment and eases his mind by sitting down next to him while Sheriff Tate and Atticus talk. After Sheriff Tate explains why he refuses to tell the community about Boo's heroics, Scout demonstrates her compassion by leading Boo to Jem's room to say goodbye before she walks Boo home.

How does Steve's environment lead to his incarceration? What is the author saying about living environments?

In Monster, Steve's environment directly leads to his incarceration. Considering Steve is accused of murder merely by association, it can be gathered that those involved with the murder, Bobo Evans and James King, are the larger cause of why Steve is on trial. Realistically, these two men, whose power Steve tends to respect, utilize their massive amount of influence to perk Steve's interest in the crime. Furthermore, Steve's neighborhood is laregely populated by minorities (notabaly African Americans), who, unfortuantely are subject to prejudice.
In terms of actual environment, Steve's neighborhood is far from priveledged. Many of Steve's friends and acquaintences have resorted to crime out of desparation. The lack of economical opportunities force the citizens of the neighborhood to act out. The author, Walter Dean Myers, seems to suggest to the reader that being born into an underpriveledged area partially dooms children and teenagers; their peers are likely to turn to crime, and even those who don't are likely to get lumped with the others. The author seems to call to the readers to understand the significant negative effects that segregation, poverty, and racism have on many levels.
It's important to note that while writing his screenplay, Steve aims to paint his neighborhood in a positive light; his home, as lousy as it might be, is still his home, and thus he sees his neighborhood with a different set of eyes than an outsider would. Yet, if Steve were to grow up in a more positive and priveledged environment, the likelihood of getting pinned for the robbery/murder is far less probable.