How do Gatsby's friends compare to the disciples of Christ?

After Christ is arrested, his disciples all flee. They aren't there for him at all. They are too busy protecting their own hides, fearful that they will get into trouble if they are found to be connected with him. Even one of Christ's closest disciples, Peter, pretends he doesn't know Christ.
Likewise, after George Wilson shoots and kills Gatsby, holding him responsible for Myrtle's death, all of Gatsby's supposed friends, except for Nick and Owl Eyes, flee. Hundreds of people attended Gatsby's lavish parties, where they were happy to eat his food and drink his alcohol, but none of them, except the two aforementioned, show up for his funeral. Even Gatsby's former business associate Meyer Wolfsheim wants nothing to do with his old friend when Nick calls him about the funeral. Daisy, who was the one driving the car when it killed Myrtle, and who might be expected to owe him some slight regard, is nowhere to be found. Like Jesus's disciples, the people around Gatsby are too busy protecting themselves to concern themselves with Gatsby.

Single Variable Calculus, Chapter 7, Review Exercises, Section Review Exercises, Problem 26

Differentiate $\displaystyle y = e^{-t} (t^2 - 2t + 2)$


$
\begin{equation}
\begin{aligned}

y =& \frac{t^2 - 2t + 2}{e^t}
\\
\\
y' =& \frac{d}{dt} \left( \frac{t^2 - 2t + 2}{e^t} \right)
\\
\\
y' =& \frac{\displaystyle e^t \frac{d}{dt} (t^2 - 2t + 2) - (t^2 - 2t + 2) \frac{d}{dt} (e^t)}{(e^t)^2}
\\
\\
y' =& \frac{(e^t) (2t -2) - (t^2 - 2t + 2)(e^t) }{e^{2t}}
\\
\\
y' =& \frac{e^t (2t - 2 - t^2 + 2t - 2)}{e^{2t}}
\\
\\
y' =& \frac{-t^2 + 4t - 4}{e^t}
\\
\\
& \text{ or }
\\
\\
y' =& e^{-t} (-t^2 + 4t - 4)


\end{aligned}
\end{equation}
$

Analyze life in the 1950s/1960s. What do you conclude the characters accept in society? What do the characters want to change about society?

Initially, there seems little to suggest that either Susan or Matthew are prepared to challenge society's norms in any way. Why should they? On the surface, they appear to have it all: a happy marriage, four lovely children, and a nice house in the suburbs. They enjoy the privilege of leading a truly blissful life, a generous reward conferred upon them by middle-class society for their outward conformity to its mores.
The Rawlings epitomise the stifling social conventions of middle-class English suburban life in the early 1960s. Matthew has a respectable job and is the sole breadwinner; Susan is a homemaker whose whole life revolves around domestic chores and taking care of the children. Nothing could be more conventional.
Yet, neither Susan nor Matthew see themselves as being in any way subject to a mindless, stultifying conformity. On the contrary, in living out their socially allotted roles, both of them genuinely believe themselves to be behaving in a perfectly rational manner. They have always prided themselves on making the right decisions in life. On a number of occasions in the story, Lessing uses the word "sensible" to describe them. We have no reason to doubt that they are, for that is precisely what they appear to be.
All that changes when the little ones are finally packed off to school. It is instructive that the seeds of tragedy are sown at this precise moment in time. As she sits there at home all alone, Susan realizes that her role as a mother has been severely circumscribed, depriving her life of whatever meaning it may once have had. This forces her to reflect upon her life to date, to question whether she and her husband have indeed made the right choices in life. Susan unconsciously seems to be reconnecting with her instinct, a feminine instinct stifled and suppressed by both her husband and society, one seemingly lost forever, cast into outer darkness on her wedding day.
She cannot truly connect. Therein lies her tragedy. Susan has been living in a bubble all these years, her perfect suburban existence untainted by pain. When pain finally does intrude unceremoniously into her little world, the pain of separation and the pain of adultery, she simply does not know how to cope effectively. Her inherited social world has been rendered utterly meaningless, yet she patently lacks the wherewithal to construct a world of her own. This way madness lies.
She cannot and will not consciously challenge society's norms; she is by no means a strident feminist yearning to throw off the yoke of patriarchal repression. However, Susan does nonetheless act as a symbol for a nascent feminist consciousness, one that would achieve full flower within a few short years after To Room Nineteen was published. For Lessing, it is a telling indictment of British society in the early 1960s that a woman can only truly achieve a sense of liberation by retreating into a world of insanity and suicide.
By doing so, Susan finally breaks down the instinct/reason dichotomy which she was forced to confront as her whole world began to collapse around her. As a "mad woman," she has rejected the constricting roles bourgeois English society has imposed upon her. She is deemed to have failed in performing her role as mother and wife. Despite this, there is still method to Susan's madness. Ironically, she has achieved a higher degree of rationality than either she or the society she has now abandoned ever previously possessed. Matthew, who as a bourgeois male has no real need to challenge society's norms, still remains trapped in a world of constructed rationality in which the old certainties retain their irresistible hold upon his limited imagination.
Between the ostensibly rational, yet ultimately meaningless realm of bourgeois respectability and the ostensibly irrational, yet ultimately hyper-rational, creative world of madness, there can be no meaningful communication. That is why Susan is forced to construct the phantom figure of a wholly fictitious lover to account to Matthew for her descent into madness. It is the only language he can understand; the sensible discourse of the society that continues even now to mould him as his wife heads towards her sad, lonely demise.

What was Machiavelli's theory about how rulers should act?

There is a great deal of popular misunderstanding about Machiavelli, largely due to misunderstandings people have about The Prince.
A lot of people believe a Machiavellian is someone who wants power so he can take what he wants for his own personal self-aggrandizement--"the ends justify the means." Nothing could be further from the truth. Machiavelli actually states that such behavior is liable to cost a prince his throne. From Ch. XIX:

It makes [the Prince] hated above all things, as I have said, to be rapacious, and to be a violator of the property and women of his subjects, from both of which he must abstain. And when neither their property nor honour is touched, the majority of men live content, and he has only to contend with the ambition of a few, whom he can curb with ease in many ways.

The truth of the matter is that being a Machiavellian prince is very demanding. The first requirement for a proper Machiavellian prince is to have virtù. Without this quality--which is rendered in English as everything from "manliness" to "ability" to "skill" to "power," depending on the translation you are reading--a man cannot hope to succeed as a ruler. It is virtù that will allow the prince to survive the slings and arrows of Fortune. (All reputable scholars agree that the conflict between virtù and fortuna is central to The Prince.)
According to Machiavelli, a man of virtù embodies the qualities of the lion (strength) and the fox (cleverness):

You must know there are two ways of contesting, the one by the law, the other by force; the first method is proper to men, the second to beasts; but because the first is frequently not sufficient, it is necessary to have recourse to the second. Therefore it is necessary for a prince to understand how to avail himself of the beast and the man. [...] A prince, therefore, being compelled knowingly to adopt the beast, ought to choose the fox and the lion; because the lion cannot defend himself against snares and the fox cannot defend himself against wolves. Therefore, it is necessary to be a fox to discover the snares and a lion to terrify the wolves. Those who rely simply on the lion do not understand what they are about (Ch. XVIII).

In addition to having virtù, a proper prince must constantly study the arts of war and make sure that he has a properly functioning army that does not depend on mercenary forces (Ch. VI, Chs. XII-XIV). He must be continually wary of the nobility and careful of how he cultivates the support of the broader populace (Ch. IX, Ch. XVII). The prince must realize that the most prudent course of action is likely to make him unpopular (see many chapters, but esp. Ch. XVI). Finally, the prince must accept that, no matter what, there is always a chance Fortune may undo everything he has accomplished (Ch. XXIV-XXV).
What is underscored in Chs. XVI-XVII (and echoed to some extent in Ch. IX) is that the prince must be concerned with what is prudent, not what will please the populace in the moment. As he says in Ch. XVII:

this is to be asserted in general of men, that they are ungrateful, fickle, false, cowardly, covetous, and as long as you succeed they are yours entirely; they will offer you their blood, property, life and children, as is said above, when the need is far distant; but when it approaches they turn against you.

For example, when it comes to practicing genrosity, contrary to the standard advice given to princes (which was influenced by the philosophy of Aristotle), Machiavelli suggests restraint. Machiavelli says a prince who throws lavish festivals or builds extravagantly is foolish, because this prince may not be able to provide for his people in tough times. However, a prince who initially appears stingy because he will not host festivals will be celebrated when he is able to provide for the people during a famine or other disaster.In summary, the Machiavellian ruler is a man of virtù, who uses a combination of force/skill with arms and clear, prudent long-term vision to strengthen his principality in order that it may withstand challenges from internal factions, outside enemies, or the acts of Fortune.

What is the setting of the short story "The Birds" by Daphne Du Maurier?

The setting of Daphne du Maurier's short story, "The Birds," is the Cornish coast of England in autumn during the threat of Cold War in the 1950's and 1960's.
This time of year is appropriate to the theme of du Maurier's story because autumn is a season that presages change from the beauty and calm of the fullness of the year to a time of death with winter. At this time of the year, the birds are restless as the sea gulls run on the beaches and sandpipers scurry back and forth. Both seem to feel an "impulse to flight."

The restless urge of autumn, unsatisfying, sad, had put a spell upon them and they must flock, and wheel, and cry; they must spill themselves of motion before winter came.

For some reason, however, the birds are more restless than ever. Perhaps, then, there is a certain pathetic fallacy with the stirrings of the Cold War. That is, as the fear of communism spreads and increases suspicion among people, perhaps nature senses this trepidation and paranoia, sending currents of anxiety that affect the birds. Not only are they present and restless, but the farmer Nat Hocken notices that there are more birds than is customary for this town on the coast.
It is the resilient nature of Nat Hocken that keeps him in the fight; however, even he eventually succumbs to the power of Nature that is disturbed by unnatural occurrences in this horrifying tale. 
 

Single Variable Calculus, Chapter 3, 3.1, Section 3.1, Problem 25

If $f(x) = 3-2x + 4x^2$, find $f'(a)$.

Using the definition of the derivative


$
\begin{equation}
\begin{aligned}

\qquad f'(a) &= \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}
&& \\
\\
\qquad f'(a) &= \lim_{h \to 0} \frac{3 - 2 (a + h) + 4 (a + h)^2 - (3 - 2a + 4a^2)}{h}
&& \text{Substitute $f(a + h)$ and $f(a)$}\\
\\
\qquad f'(a) &= \lim_{h \to 0} \frac{3 - 2a - 2h + 4(a^2 + 2ah + h^2) - 3 + 2a - 4a^2}{h}
&& \text{Expand the equation}\\
\\
\qquad f'(a) &= \lim_{h \to 0} \frac{\cancel{3} - \cancel{2a} - 2h + \cancel{4a^2} + 8ah + 4h^2 -\cancel{3} + \cancel{2a} - \cancel{4a^2}}{h}
&& \text{Combine like terms}\\
\\
\qquad f'(a) &= \lim_{h \to 0} \frac{4h^2 + 8ah -2h}{h}
&& \text{Factor the numerator}\\
\\
\qquad f'(a) &= \lim_{h \to 0} \frac{\cancel{h}(4h + 8a - 2)}{\cancel{h}}
&& \text{Cancel out like terms}\\
\\
\qquad f'(a) &= \lim_{h \to 0} (4h + 8a - 2) = 4(0) + 8a - 2
&& \text{Evaluate the limit}

\end{aligned}
\end{equation}
$


$\qquad\fbox{$f'(a) =8 a - 2$} $

Who did the Confederates turn to for help during the Civil War?

The Confederacy turned to both France and the British Empire for assistance in the early years of the American Civil War. Their primary leverage was cotton: the Confederate states dominated the global cotton market, and the British and French textile industries depended on Southern cotton to function. It was the Confederates' hope that access to cotton could be used as a bargaining chip to secure alliances with these foreign powers and gain official recognition as a sovereign government from them.
Britain was the primary aim of the Confederates' diplomatic efforts; they had tried to court France, but the French were reluctant to commit themselves to the Confederacy unless Britain took the lead. Britain's textile industry formed a major part of its economy in the 1860s, so the threat of a "cotton famine" was a serious one, and the British government did consider intervention in the American war—even going so far as to supply the Confederacy with warships. In 1861, Queen Victoria issued a proclamation of Britain's neutrality with regard to American affairs, but there was much support for the Confederacy among wealthy British citizens, who sent money and supplies through the Union naval blockade to the Confederate government.
As the Confederate army advanced into Northern territory and repeatedly emerged victorious from battles with the Union, there was real concern in Europe that the United States would not survive the war. Tentative arrangements were put into place for a Confederate victory, and for a period the Confederate government entertained serious hopes of gaining official assistance from the British.
Those hopes were dashed in 1863 when Lincoln issued the Emancipation Proclamation, reframing the Civil War as a conflict over the ethics of slavery rather than “states’ rights." British Empire had abolished slavery in all of its territories in 1833, after decades of protracted and often ugly debate, and public sentiment was strongly anti-slavery. The British government could not, therefore, be seen to take the side of the Confederacy, either explicitly through treaties (as the Confederates had hoped) or implicitly through supplying money and materiel. The British dropped all further involvement with the Confederacy and in fact publicly condemned it for its continued use of slavery. The French, who were only tangentially interested in intervening in the war, followed Britain's lead and withdrew all support from the Confederacy.
You may find the following resources useful:
British Involvement in the American Civil WarThe Response from Overseas to the American Civil War