What idea does Creon preach?

The play Antigone is really about the tension between the oikos (family household) and the polis (the city-state). Antigone, in wanting to honor her brother (whom Creon has labeled a traitor), speaks on behalf of family honor—the idea that the love and respect of family members should come before one's duty to the government.
Creon, however, refutes this view. Instead, everything he says reflects his celebration of the polis, even if it means putting his duty as a king before his duty as an uncle or father. In Creon's first public speech, he states,

"a man who thinks more highly of a friend than of his country, well, he means nothing to me...for I know well our country is a ship which keeps us safe, and only when it sails its proper course do we make friends."

For Creon, loyalty to one's country is most important, for without it, there would be no order or security.
It is important to note that Creon's lines are all delivered in public, so we cannot be sure what he really believes. He is often seen as a stubborn, blind, and raging fool who is cruel towards his rebellious niece; however, it's important to consider his position. He is terrified of anarchy--he witnessed Oedipus's demise, saw Eteocles and Polynices attack each other in a bloody battle, and now he feels enormous pressure as king to restore peace to Thebes. He must present a strong front, and preaching the importance of loyalty to the city is his way of establishing order. Of course, like in most Greek tragedies, his tragic flaw (hamartia) is his stubbornness. While we may see the core of his belief in the importance of citizens being loyal to their city as admirable, his refusal to consider Antigone's (or Ismene's, or Tiresias's) views leads to his downfall. 

In the play Antigone, Creon chooses to bury Eteocles with full military honors, while his brother's corpse lays decaying in an open field where carrion can feast on it. Creon decrees that anyone who touches or prays for Polyneices will be considered an enemy of the State and sentenced to death. Creon is only interested in retaining his power and maintaining social order. He refers to his enemies as "stiff-necked anarchists," and preaches the importance of following laws and decrees handed down by political officials. In Creon's opinion, individuals must respect the State regardless of their personal beliefs or grievances. In contrast, Antigone believes that citizens must follow the wishes of the gods before listening to politicians. Creon's refusal to compromise and view the situation from Antigone's perspective is a drastic miscalculation that has disastrous results. His insistence that Antigone's crime must not go unpunished results in bloodshed and hardship.

Single Variable Calculus, Chapter 4, 4.4, Section 4.4, Problem 30

Determine the $\displaystyle \lim_{x \to \infty} \sqrt{x} \sin \frac{1}{x}$



$
\begin{equation}
\begin{aligned}

\displaystyle \lim_{x \to \infty} \sqrt{x} \sin \frac{1}{x} \cdot \frac{\sqrt{x}}{\sqrt{x}} =& \lim_{x \to \infty} \frac{x}{\sqrt{x}} \sin \frac{1}{x} = \lim_{x \to \infty} \frac{1}{\sqrt{x}} x \sin \frac{1}{x}

&& \text{We can rewrite $\displaystyle \lim_{x \to \infty} x \sin \frac{1}{x}$ to $\displaystyle \lim_{x \to \infty} \frac{\displaystyle \sin \frac{1}{x}}{\displaystyle \frac{1}{x}}$}
\\
\\

=& \lim_{x \to \infty} \frac{1}{\sqrt{x}} \cdot \lim_{x \to \infty} \frac{\displaystyle \sin \frac{1}{x}}{\displaystyle \frac{1}{x}}
&&

\\
\\
=& (0)(1)
&&
\\
\\

=& 0
&&

\end{aligned}
\end{equation}
$


As $x \to \infty$, the leading term of a polynomial function will dominate the infinity, thus the other terms can be omitted.

Why did Fitzgerald begin The Great Gatsby with a quote from Thomas Parke D’Invilliers, a character from Fitzgerald’s This Side of Paradise? The quote is below: Then wear the gold hat, if that will move her; If you can bounce high, bounce for her too, Till she cry “Lover, gold-hatted, high-bouncing lover, I must have you!”

There is another reason why Fitzgerald included this quote from one of his fictional characters at the beginning of the novel. Arguably, he wanted to make it clear to the reader exactly what this story is about—that is, Gatsby's quest to win back Daisy Buchanan. Of all the events and relationships in the novel, nothing has more significance than Gatsby's experience of the American Dream, shown chiefly through his quest to win back Daisy, a quest which consumes Gatsby's every moment and every ounce of energy.
This idea is also supported by the repetition of the word "gold" in this quote. This word has connotations of high-value items, of wealth and luxury. The use of this word, therefore, reinforces the idea that this story is all about Gatsby's quest to achieve his version of the American Dream.
It does not matter that D'Invilliers is a fictional character from another of Fitzgerald's novels. What matters is that Fitzgerald is using this quote to foreshadow to the reader that this is a story about the American Dream and one man's quest to achieve it.

D'Invilliers was both a pen name for Fitzgerald and a character based on Fitzgerald's friend, the poet John Peale Bishop. The quote reflects Gatsby's conviction that if he can earn enough money and remake himself, he can win the love of Daisy Buchanan. The "gold hat" in the poem symbolizes Gatsby's fortune and the grand home and enormous parties he throws to attract her attention. As Nick points out using different language, Gatsby was one to "bounce high," dream big, and aspire to realize a vision. In the epigraph, the narrator is likewise trying to woo his lover with money (the "gold hat") and by "high-bouncing:" trying hard to get her attention. He believes that by doing so the woman will inevitably end up saying, "I must have you." It reflects a very materialistic view of love. Fitzgerald chooses it because this is how Gatsby pursues Daisy, although with less success than the gold-hatted man in this quote.

int_0^(pi/2) xsin(2x) dx Use integration tables to evaluate the definite integral.

To evaluate the integral problem: int_0^(pi/2) xsin(2x) dx ,we may first  solve for its indefinite integral. Indefinite integral are written in the form of int f(x) dx = F(x) +C
 where: f(x) as the integrand
           F(x) as the anti-derivative function 
           C  as the arbitrary constant known as constant of integration
We follow a formula from basic integration table to determine the indefinite integral function F(x) . For the integrals with logarithm, the problem resembles the formula:
int x sin(ax) dx= -(xcos(ax))/a+sin(ax)/a^2 +C .
By comparing x sin(ax) with xsin(2x) , we determine that  a= 2 .
 Plug-in a=2 to the integral formula, we get:
int_0^(pi/2) xsin(2x) dx=-(xcos((2)x))/(2)+sin((2)x)/(2)^2|_0^(pi/2)
=-(xcos(2x))/2+sin(2x)/4|_0^(pi/2)
After solving the indefinite integral from, we may apply definite integral formula: F(x)|_a^b = F(b) - F(a) .
-(xcos(2x))/2+sin(2x)/4|_0^(pi/2) =[-((pi/2) *cos(2*(pi/2)))/2+sin(2*(pi/2))/4]-[-(0*cos(2*0))/2+sin(2*0)/4 ]
=[-((pi/2) *cos(pi ))/2+sin(pi) /4]-[-(0*cos(0))/2+sin(0)/4 ]
=[-(pi*(-1))/4+0 /4]-[-(0*1)/2+(0)/4 ]
=[pi/4+0]-[0+0]
= [pi/4] - [0]
=pi/4
 

int_0^2 dx/(x^2-2x+2) Find or evaluate the integral by completing the square

To be able to evaluate the given integral: int_0^2 (dx)/(x^2-2x+2) , we
complete the square of the expression:x^2-2x+2 .
To complete the square, we add and subtract (-b/(2a))^2 .
 The x^2-2x+2 resembles the ax^2+bx+c where:
 a=1 , b =-2 and c=2 .
Then,
(-b/(2a))^2 =(-(-2)/(2(1)))^2
              =(2/2)^2
              = (1)^2
              =1
Add and subtract 1 :
x^2-2x+2 +1-1
Rearrange as: (x^2-2x +1) +2-1 = (x-1)^2+1
Plug-in x^2-2x+2 = (x-1)^2+1 in the given integral:int_0^2 (dx)/(x^2-2x+2) .
int_0^2 (dx)/(x^2-2x+2) =int_0^2 (dx)/((x-1)^2+1)
This resembles the basic integral formula for inverse tangent function:
int (du)/(u^2+a^2) =1/a *arctan(u/a)+C
Then  indefinite integral F(x)+C,
int (dx)/((x-1)^2+1^2) =1/1 *arctan((x-1)/1)+C
                                =arctan(x-1)+C 
For the definite integral, we apply: F(x)|_a^b= F(b)-F(a) .
arctan(x-1)|_0^2 =arctan(2-1) -arctan(0-1)
                    =arctan(1) -arctan(-1)
                    =pi/4 -(-pi/4)
                    =pi/4 +pi/4
                    =(2pi)/4
                    =pi/2 

In "Once Upon a Time," what is the relationship between the characters in the story and community in which they live?

The characters in the story live in a terribly divided community, and so this analysis must take place in two parts: one must examine both the family's relationship with their neighbors and their relationship to the people of color who live outside their suburb.
The suburb in which the family lives is understood, particularly in the context of apartheid South Africa, to be both white and afraid of those who are not. Race is a recurring theme throughout the story, no less present for the family's denials of racism:

He was masked; it could not be said if he was black or white, and therefore proved the property owner was no racist.

The family is guided by their fear. They insure against disaster, and in cases in which no such insurance is available, they actively attempt to prevent damage by installing security measures such as alarms, fences, wire coils, and so on. The community of the suburb is homogenous; the actions taken by the family are by no means unique. Gordimer states that: 

The alarm was often answered — it seemed — by other burglar alarms, in other houses, that had been triggered by pet cats or nibbling mice.

Guided by a shared fear and persistent racism, the family is shown to be typical of the community of the suburb.
The community outside the suburb is greatly feared by the family. They are described as rioters, burglars, loafers, hooligans, loiterers; that the story opens with the narrator's fear of being murdered implies that the family is afraid of being killed as well. They are "people of another color," not only living in another part of the city but "quartered" there. The trustworthy ones were made housemaids and gardeners: servants. 
Despite the family's persistent fear of the people from outside the suburb, their housemaid and gardener are very loyal: when their son is caught in the coil, they are described as "the hysterical trusted housemaid and the weeping gardener" while the parents of the child are simply described as "the man" and "the wife." Faced with the death of a child, the maid and gardener, whose own community is somewhat dehumanized over the course of the story, are shown to be more human than the parents of the child. 
 

Describe the memory that was painful to Brian in Hatchet.

At the beginning of the novel, Brian mentions "The Secret," which continually haunts his memories and is the cause of his parents' divorce. Brian mentions that his mother was the one who initiated the divorce and says that his father did not understand why she wanted to leave. Brian then says that "The Secret" was the reason his mother wanted to end her marriage.
The painful memory that Brian recalls is explained towards the beginnings of chapters 4 and 7. Brian remembers riding his bike with his friend Terry past the Amber Mall and briefly seeing his mother in a strange station wagon. Brian then recalls seeing his mother sitting next to a short man with blond hair. In chapter 7, Brian also mentions that he remembers seeing his mother lean over the seat and kiss the blond man on the lips. Essentially, Brian witnesses his mother kissing a stranger and does not tell either of his parents that he knows the real reason behind their divorce, which is infidelity on his mother's part.