sum_(n=1)^oo n^k*e^(-n) Use the Integral Test to determine the convergence or divergence of the series, where k is a positive integer.

The Integral test says that this sum will converge if and only if this integral also converges:int_{1}^{infty} x^k e^{-x} dx When integrating this, we would use integration by parts, and we would need to use it k times. The first set of parts is u = x^k, dv = e^{-x} dx, du = k x^{k-1} dx, v = -e^{-x}
 
int u dv = u v - int v du = - x^k e^{-x}|_1^infty + int_{1}^{infty} k x^{k-1} e^{-x} dx
Then we repeat for u_1 = x^{k-1} , and so on until we have only the e^{-x} term left.
But the important thing is that the last term would only be in terms of a constant times int e^-x dx , which clearly converges; and then all the other terms would look like this, for some integer 1 leq p leq k and some constant C:
C x^p e^{-x} |_{1}^{infty} The value of this term at x = 1 we can simply calculate; no problem there, it will be some finite number. The limit as x goes to infinity we can also determine by the fact that e^x always increases faster than any polynomial as x gets very large, and thus for any value of p, this limit must be zero.Thus, we have k-1 terms that are finite (zero minus a finite value), plus one final term that is a convergent integral. Therefore the whole integral converges; therefore the sum converges.

What themes does Rupi Kaur explore in milk and honey?

In her debut poetry collection milk and honey, Rupi Kaur explores themes of abuse and trauma, survival, healing, sex, love, heartbreak, and gender dynamics. All of these themes can be found throughout milk and honey, but the four sections into which the book is divided each focuses mainly on one or two themes in particular. The first section, “The Hurting,” focuses on the female speaker’s history as a survivor of childhood sexual abuse. In the second section, “The Loving,” Kaur weaves in themes of love and sex as the speaker recounts her first major romantic relationship as an adult. The third and longest section, “The Breaking,” involves the theme of heartbreak, as the speaker describes how her relationship with the man introduced in “The Loving” comes to an end. Later in this section Kaur focuses more on the theme of gender, as the narrator begins to speak out against sexism and to decide for herself what being a woman means rather than to accept the gender roles that have been defined for her by men. The theme of healing is perhaps the most significant in the collection and forms the focus of the last section, appropriately titled “The Healing.” The poems in this final chapter reflect the speaker’s healing process, newfound empowerment, and ability to unlearn the harmful lessons she learned about love, sex, and femininity in the past.

Chapter-2.pmd The year is 1916. You are a general in the Tsarís army on the eastern front. You are writing a report for the government in Moscow. In your report suggest what you think the government should do to improve the situation.

First, generals were not invited to criticize the Imperial Russian government--in many cases, these positions were not granted based on merit but rather on who one knew inside the imperial court.  Also, the seat of the imperial Russian government was St. Petersburg--Moscow would not become popular with Russian leadership until after the Revolution.  
The Russian Army had a lot of problems on the Eastern Front, but the most glaring problem was that of logistics.  The average Russian soldier went into battle poorly equipped and poorly trained.  While the Western Front was a war of attrition, the Eastern Front became a war of movement as Russia lost land to both the Austro-Hungarian Empire and Germany.  If I was a general who had the capacity to criticize the government and cause change, I would ask for better rail lines--much of Russia's supply problems were caused by not using uniform rail gauges--this simple change would have given the troops more food and these better conditions would have improved non-combatant casualties and improved desertion rates.  I would also encourage better cooperation among generals, as key battles early in the war such as Tannenburg were lost due to generals working in competition instead of cooperation with each other.  

Why does Hamlet delay? Why does he eventually act?

Hamlet has many human qualities that Shakespeare uses to such marvelous effect. It is why so many people can identify with him. It is what makes him so timeless. The reason Hamlet delays in the play is because delay, procrastination, "thinking too much" however one may characterize it, it is a particularly human quality. The ability to reason contrasted with bestial oblivion is a dichotomy that Shakespeare peppers throughout the play. So is thinking too much a problem? or is it a blessing? Shakespeare doesn't really provide the answers but he does pose the question.
But, look further. If delay is a human condition does it appear in other characters? Oddly, it does. One of the most obvious is Pyrrhus when he is about to kill Priam. He stops himself for however briefly it takes the First Player to ponder the weather of all things. Brilliant writing. Claudius delays sending Hamlet to England. Laertes delays in killing Claudius and then Hamlet. Notice that Laertes postured that he would cut Hamlet's throat in the churchyard and yet he got his fingers on Hamlet's throat in the graveyard and did... nothing! But, Laertes had reasons, I hear someone say. Of course! There are always reasons. My favorite delay is in the Murder of Gonzago. See where Hamlet announces the entrance of Lucianus, nephew to the king. Then Hamlet engages in petty banter with Ophelia leaving Lucianus to wait and make faces of evil intent until he can regain the stage from Hamlet's rantings. But, Hamlet draws attention to Lucianus' delay in poisoning the Player King as if it wasn't Hamlet's fault but Lucianus' fault. "Begin Murderer" Hamlet yells. "Pox, Leave thy damnable faces and begin." Then to cover his outburst Hamlet quickly narrates, "Come, the croaking raven doth below for revenge." Only then can Lucianus reclaim the stage and his murderous intentions.
Again, brilliant writing. But, the real subtlety is how well Shakespeare delays Hamlet's ultimate revenge on Claudius. In Romeo and Juliet the Prologue tells us the star crossed lovers are "now the two hours' traffic of our stage."  Polonius in an off the cuff remark in act 2 scene 2 tells us that Hamlet, "he walks four hours together here in the lobby." Indeed when produced in its entirety, the play takes 4 hours. Shakespeare then is the real reason Hamlet delays?

These questions have been debated for centuries. They are called "The Hamlet Problem." I think the best answer was given by Samuel Taylor Coleridge, who said Hamlet procrastinates because he thinks too much. When he acts on impulse, he can act very courageously. A good example is the way he is the first to board the pirate ship which kidnaps him. When Hamlet finally kills Claudius, it is impulsively, in the heat of emotion. He doesn't have time to think about it. He holds the poisoned foil right in his hand.
When Hamlet is on his way to England, he discovers his escorts Rosencrantz and Guildenstern are carrying a letter asking the English king to have him executed immediately upon arrival. Hamlet forges a substitute letter in which he has Claudius asking the English king to execute Rosencrantz and Guildenstern. This is before Hamlet is kidnapped by the pirates and then ransomed in Denmark by Horatio. At that point, Hamlet knows his life is in extreme danger because Claudius will soon receive word from England that Rosencrantz and Guildenstern have been beheaded. Claudius will know Hamlet forged a new letter and will certainly have Hamlet executed as a punishment and for the king's own self-protection. The following dialogue shows this.

HORATIOIt must be shortly known to him from EnglandWhat is the issue of the business there.
HAMLETIt will be short. The interim is mine.

Hamlet knows he must kill Claudius without further deliberation or Claudius will have him killed. He does not know Claudius has already made plans to have him killed in the forthcoming fencing match with Laertes.
Hamlet had other reasons for his previous delays. One was that he sincerely believed the ghost he met in Act I, Scenes 4 and 5 might be a devil instead of his father. In Act II, Scene 2 he says to himself:

The spirit that I have seenMay be a devil; and the devil hath powerT' assume a pleasing shape; yea, and perhapsOut of my weakness and my melancholy,As he is very potent with such spirits,Abuses me to damn me. I'll have groundsMore relative than this. The play's the thingWherein I'll catch the conscience of the King.

Hamlet had to stage the play-within-a-play to prove to himself that Claudius was actually guilty, as the ghost had told him. After that, he felt free to act against the king, but he was forestalled by being sent to England as an ambassador. 
One other reason Hamlet might have been slow in carrying out his obligation to kill Claudius is that Hamlet is aware he is supposed to inherit the throne when his uncle dies. All he has to do is wait. If he kills Claudius, it might be impossible for Hamlet to inherit the throne, especially if everyone believes he is mad. They would think Hamlet was motivated purely by ambition and would not believe Claudius murdered Hamlet's father in order to usurp the throne and marry King Hamlet's wife Gertrude. As a result, Hamlet is torn between acting and waiting for the crown to drop into his lap.

Compare and Contrast William Wordsworth and Samuel Coleridge's views on the imagination and nature in regards to the poems "Tintern Abbey" and "Frost at Midnight".

There are many similarities between "Tintern Abbey" and "Frost At Midnight." This is not particularly surprising, as Wordsworth and Coleridge were close friends and collaborators who shared a common aesthetic vision. The first thing to note is that there are similarities at the structural level, with both poems written in blank verse. But it is in terms of tone and content that the similarities really begin to emerge.
The one main difference between the two poems lies in the language used. Wordsworth, in keeping with his and Coleridge's statement of intent in "Lyrical Ballads," is more direct in his language, much simpler, whereas Coleridge noticeably departs from his earlier intentions, writing in a more abstract vein. He uses a number of metaphors whose meanings aren't at all clear at first glance. For instance,

The Frost performs its secret ministry.

Nature inspires in both men a deep, philosophical reflection. But in the case of Coleridge it's a much more abstract style of philosophizing ("abstruser musings"), heavily influenced by his in-depth study of German Idealism.
To return to the similarities, both men are engaged in reminiscence. Wordsworth recalls the time five years ago when he last visited the area he is about to see once more. Coleridge recalls aspects of his childhood. In both cases, nature is the catalyst for reflection. For arch-romantics such as Wordsworth and Coleridge, the natural world is not just a source of the pretty and the picturesque; it is a creative force with a life of its own. It also has an important didactic function: it teaches us. But what it teaches us can change over time. When Wordsworth first visits Tintern Abbey, he's overwhelmed by the beauty of the natural world around him. And the memory of that stunning vision has sustained him ever since.
But now things have changed. Now, Wordsworth sees in nature the fundamental connection of all things, an underlying unity that transcends the ostensible gap between human beings and nature:

"And I have felt 
A presence that disturbs me with the joy 
Of elevated thoughts; a sense sublime 
Of something far more deeply interfused, 
Whose dwelling is the light of setting suns, 
And the round ocean and the living air, 
And the blue sky, and in the mind of man: 
A motion and a spirit, that impels 
All thinking things, all objects of all thought, 
And rolls through all things."

Wordsworth is still "a lover of meadows and woods," but now he doesn't just appreciate nature, he understands it.
 
Coleridge's reminiscence of childhood is also triggered by nature, in his case the appearance of frost on the window panes at midnight and the owlet's plaintive cry. He is alone with his thoughts and begins to reflect on years past. Wordsworth's memories of his previous visit to Tintern Abbey are happy; they kept him going as he struggled against the harshness of city life. Coleridge, however, recalls the boredom of schooldays, his memories brought on by a fluttering film of soot, a "stranger" on the fire-place grate.
 
For Coleridge, no less than Wordsworth, nature is a great teacher. The little piece of flapping ember doesn't just take him back to his schooldays, it allows him to connect past with present and present with future. Like Wordsworth, he sees the underlying unity of things. In the figure of his sleeping baby, he has presentiments of a better childhood for his son than the one he endured himself. For his child will be brought up in a natural environment, where he will soon come to understand the world as it really is by unifying it with his imagination.
 
Wordsworth, too, looks to a close relative as he surveys the future beyond. His sister still retains the childlike sense of wonder at nature that he himself enjoyed five years ago. But he is certain that one day she, too, will come to recognize the interconnectedness of all things.

What is the central idea

The central idea is this book is that Junior must choose to be a "part-time Indian" because he wants to live a better life than the Indians on the reservation do. When he lives on the reservation, he is close to his family and friends, but he can't get a good education. He sees that people like his parents and sister, Mary, have experienced the death of their dreams by not getting off the reservation. He therefore chooses to attend school in the white town of Reardan, where he feels that he is misunderstood and ridiculed for being an Indian. In Junior's life, he simply can't entirely win—as he must choose between a limited life on the reservation and a life with misunderstanding and hostility (on the part of both whites and Indians on the reservation who feel he has shunned them) when he chooses to go off the reservation for school. In his quest for identity, he is always caught between two worlds.

The central idea in Alexie’s novel is evinced in Junior’s struggle for identity. Torn between his desire to maintain his cultural heritage and his aspirations for a better life through education, he becomes an object of derision and bullying at home on the Spokane reservation, as well as at his new school in Reardan. This either-or situation created by Alexie for Junior—either he can be an Indian or he can assimilate into White culture, but he cannot do both—fuels the novel’s main conflict. Junior does not feel as if he truly belongs in either culture and his efforts to fit in, both at home and at school, often fail. Ultimately, Junior realizes that he does not have to adhere to a single set of expectations and accepts that his differences, although they might always make him a bit of an outsider, are what make him unique and valuable as a person.

Discuss any parallelism between Oates’s “Where Are You Going, Where Have You Been?” and any of the versions of the fairy tale “Little Red Riding Hood.”

"Where Are You Going, Where Have You Been?" is a classic short story written by Joyce Carol Oates. The story explores themes of trust, sexuality, and religion.
In the story, Connie, a young and beautiful teenager, is obsessed with her looks. She does not get along well with her family members, especially her older sister, who is somewhat mousy and still lives at home with the family.
Connie is interested in boys, and things change for her when she meets a stranger named Arnold Friend. He is older and charismatic. As the story progresses, Connie realizes that Arnold Friend might actually be way older than she is and possibly dangerous.
This story can probably be related to many traditional fairy tales. "Little Red Riding Hood," which has versions written by Perrault and the Brothers Grimm, is one that stands out as a definite parallel. In this story, a young girl, Little Red Riding Riding Hood, ventures into the forest to visit her grandmother. She is naive about the element of fear that lurks in the forest and almost ends up getting eaten by a wolf who is wearing her grandmother's clothing.
Both Connie and Red are naive characters. They are both too trusting of strangers. Connie willingly gets into a car with Arnold Friend, believing that he is only eighteen and a boy that she might like. Red notices her grandmother's changed physical features, stating, "What big ears you have" and so on, but she doesn't take into consideration that it might not really be her grandmother.
It is assumed that Arnold Friend is a predator and possibly even the Devil. The ending to "Where Are You Going" is ambiguous, but many assume that Connie gets hurt or even dies. Although Red doesn't die, she is nearly preyed upon. The wolf in general gets his reputation as a predator. When portrayed in the musical Into the Woods, his character is more of a sexual predator, singing the song "Hello, Little Girl."
With Red, the forest is the unknown and dangerous area to be, whereas staying at home is supposed to be safe. The same goes for Connie. She is safe inside her home, until she feels compelled to leave with the threatening Arnold Friend at the end of the story.
Both stories can be seen as classic cautionary tales.

What are the universal truths regarding human conditions in The Crucible and what approach does Miller use to develop that theme

Arthur Miller develops two related universal truths, or themes, in this play through the morally ambiguous character of John Proctor. First, through Proctor's development, we see that redemption is always possible. Early in the play, we learn that Proctor has broken one of the Ten Commandments and committed adultery with Abigail Williams. His sin has dramatically affected his self-esteem, and he now lives "with [a] deep hatred of himself." Proctor thinks of his wife as good and pure while he is soiled and sinful. However, by the end of the play, after he's begun and then stopped himself from confessing a lie to save his life, Proctor develops a more positive self-image. During the confession, he says that he is "not worth the dust on the feet of them that hang," but once he rescinds his confession, he tells his judges, "You have made your magic now, for now I do think I see some shred of goodness in John Proctor." In refusing to add a new sin on top of the other sins he's committed, Proctor is able to redeem himself, and he can think of himself as good once again.
Second, through Proctor's character, we see that one's self-respect is more important than anything else. For a time, in Act Four, it seems that Proctor is going to confess a lie. However, when he is asked to name the names of other witches, he refuses. When he is told he must sign the confession, which is to be nailed to the church door, he thinks twice about continuing with the lie. He cries out, "I have given you my soul; leave me my name!" While Proctor signs himself to lies, he cannot feel any self-respect; when he finally tears the confession up, he regains that self-respect despite the fact that it means his life is forfeit. Ultimately, he knows that his self-respect is worth more than his reputation in the town, the judgment of the court, and even his life.
By developing the moral ambiguity of John Proctor, Miller illuminates these two related themes. 

How would you story map Charlie and the Chocolate Factory by Roald Dahl?

Exposition: Willy Wonka, a candy factory owner, had closed his factory's doors to the public for a long time. One day, it is revealed that he plans to open the doors once again, but only to a select few people who find Golden Tickets hidden in chocolate bar wrappers.
While this is going on, we are introduced to Charlie, a kind young boy who lives with his parents and both sets of grandparents. His family lives in poverty, but is very close and optimistic.
Rising Action: Charlie finds a Golden Ticket and visits the factory with the other winners (all spoiled, bratty children and their parents). Willy Wonka takes the children through the factory, and each child is picked off one by one due to their greed.  
Climax: Charlie is the last child remaining, and Willy Wonka reveals he plans to give Charlie the factory.
Falling Action: Willy Wonka explains his plan to Charlie and we see the other children leave the factory.
Resolution: Willy Wonka takes Charlie on a ride in his glass elevator, and they go to Charlie's house to tell the rest of his family the news.

Precalculus, Chapter 4, 4.2, Section 4.2, Problem 53

True
tan(a) = tan(a - 6pi)
proof
tan(a - 6pi) = sin(a - 6pi)/cos(a - 6pi)
Now,
sin(a - 6pi) = ?
this is of the form
sin(a-b) = sin(a)cos(b) - cos(a)sin(b)
so ,
sin(a - 6pi) =sin(a)cos(6pi) - cos(a)sin(6pi)
= sin(a) (1) -0 = sin(a)

cos(a - 6pi) = ?
this is of the form
cos(a-b)= cos(a)cos(b) +sin(a)sin(b)
so,
cos(a - 6pi) = cos(a)cos(6pi) +sin(a)sin(6pi)
= cos(a) (1) + 0 = cos(a)
so , now
tan(a - 6pi) = sin(a - 6pi)/cos(a - 6pi) = sin(a)/cos(a) = tan(a)

What are some themes of Things Fall Apart?

One theme of Achebe's Things Fall Apart is the importance of customs and traditions.  Okonkwo, through much of the novel, is attempting to balance Igbo traditions of leadership and masculinity with the new influences that are making their way into his culture.  The reader is shown a great many traditions and customs during this book.  For example, the use and meaning of sharing kola nut between men is a favorite passage of mine about tradition and custom.  
Another theme is masculinity.  The novel does a nice job of presenting the faults of hypermasculinity.  For example, Okonkwo wants to be a better man than his father, who Okonkwo saw as weak.  In order for Okonkwo to not be weak and show weakness, he believes that he must be aggressive and violent to a fault.  He beats his wives and even volunteers to violently kill his surrogate son in order to not appear weak.  Okonkwo looks down on men that he considers have effeminate characteristics because he sees it as weakness.  
 

Give two lies squealer uses to prove that snowball was a traitor from the very beginning

After Snowball is driven from the farm he is used as a scapegoat for every conceivable ill. Napoleon's dictatorship becomes ever more repressive and there are growing food shortages thanks to his greed and incompetence. Rumors swirl around the farm that Snowball keeps creeping back at night and causing trouble. After all, if Napoleon is the wise, benevolent leader he claims to be, then Snowball's sabotage is the only rational explanation for the farm's mounting problems.
But not everyone is convinced. So up pops Squealer, Napoleon's chief propagandist, to give everyone the party line. He claims Snowball was a traitor. Squealer says Snowball has been taken by Mr. Frederick and is now actively plotting with him to overthrow the farm's Animalist regime. And not only that, he was also in cahoots with Jones right from the very start, trying to undermine Animalism from within. At the Battle of the Cowshed it was Snowball whose treachery almost led to a catastrophic defeat.
And at that crucial battle for the future of the Animalist revolution, Squealer says it was the heroic Napoleon who cried "Death to humanity!" as he bit Mr. Jones in the leg. Like every other word that comes out of Squealer's porky mouth, this is a total lie. But Boxer, for one, appears convinced because

If Comrade Napoleon says it, it must be right.

As subsequent events show, Boxer comes to pay dearly for his gullibility and fanatical belief in the ideology of Animalism.

How does hunger plague Richard in chapter 5 of Black Boy?

Hunger plagues Richard incessantly in chapter 5.
The reason Richard is always hungry is because he lives with his grandmother, who is a strict Seventh-Day Adventist. Many people who profess the religion do not eat meat. So, Richard must subsist on a diet of starches, greens, and lard. It is ironic, however, that Richard must rely on lard for sustenance, as it is an animal byproduct.
In chapter 5, Richard finds that hunger is his constant companion. When he discovers that his classmates work mornings, evenings, and Saturdays to earn extra money, he decides to do the same. However, when Richard asks his grandmother for permission to work on Saturday, his request is denied. Richard's grandmother observes Saturday as the Sabbath and believes that it would be a sin for her grandson to work on that day. Richard ends up having to forego lunch every day.
Instead, he must watch in agony while his classmates purchase delicious sandwiches at lunchtime. To deflect uncomfortable questions about his home life, Richard lies about being so hungry at noon. Essentially, Richard lives close to actual starvation. His suffering leads him to a decision. He will be so successful one day that he will never be hungry again. To broaden his horizons, he decides not to head home immediately after school each day. Instead, he visits different venues and learns about the world around him.
During this time, Richard endures extreme hunger. His decision not to go home means that he can only eat two meals a day (mush at eight in the morning and greens later in the evening). For Richard, it is only the feeling of freedom he gets from his explorations that helps him forget his hunger pangs.

What is a summary of Written on the Body?

Jeanette Winterson's Written on the Body takes the reader through the ungendered, unnamed narrator's past and present love life. The narrator has boyfriends and girlfriends, although it appears that relationships with women leave more significant emotional scars. S/he is focused on Louise, a lover s/he met while living with another partner, Jacqueline. Louise hails from Australia and has red hair, a beautiful face, and a husband, Elgin. The narrator falls quickly in love with Louise and ends the relationship with Jacqueline. She reacts by ransacking the narrator's flat, causing a fight between the two. The narrator quickly invites Louise to move in.
Elgin is heartbroken by the betrayal and tries to convince the narrator to give up his relationship with Louise. Elgin informs the narrator that Louise has cancer, which he is willing to help her fight if the narrator will break up with Louise so that Louise and Elgin can reunite. The narrator agrees and writes Louise a tearful goodbye letter before packing up and leaving London.
The narrator moves into an old house in the countryside of Yorkshire and takes a job at a wine bar managed by Gail. S/he ultimately takes Gail home, although s/he never sleeps with Gail. S/he continues to mourn Louise, both worrying about her health and wishing she was still in the narrator's life. S/he shares the story about Louise with Gail, who shares her maternal side, telling the narrator it was cowardly to leave Louise behind. The narrator decides to go to London to find Louise but cannot locate her despite checking her flat, Elgin's home, and her mother's house. The narrator does find Elgin, and the two have a physical fight resulting in injuries for Elgin.
The narrator spends the next six weeks in London, reminiscing and visiting familiar places. Ultimately, s/he returns to the country, where she finds Louise, looking very ill. The reader is left unsure as to whether Louise and the narrator find happiness.

Before revealing her haircut to Jim, Della says, “Please God, make him think I am still pretty.” How does Jim feel about Della after he discovers her new haircut?

While the story is written in 3rd person, the perspective is limited mostly to Della's experiences and thoughts. We are not given direct insight to Jim's thinking. As we look at his reactions from the outside, though, we can at least accurately guess that he still finds his wife beautiful.
Jim's initial response is one of confusion, as he seems to look around the house for Della's missing hair. He explains that this reaction has nothing to do with his wife's new look, though. "'I want you to understand me, Dell,' he said. 'Nothing like a haircut could make me love you any less. But if you'll open that, you may know what I felt when I came in.'" Jim has sold his prize possession to help Della adorn hers. Della quickly comforts him, "My hair grows so fast, Jim!"
After Della reveals her gift to Jim, "Jim sat down and smiled." At this point, we can judge that Jim is not displeased, irritated, or dismayed by Della's actions. Jim realizes that they both had the same thought - to sell their possessions to give a better gift to the other. The narrator never comments on Della's physical beauty because the love she has for Jim makes her more attractive than any hair cut could.

Jim frightens Della because she can't understand his expression as he sees her short hair. We are told,

His eyes were fixed upon Della, and there was an expression in them that she could not read, and it terrified her. It was not anger, nor surprise, nor disapproval, nor horror, nor any of the sentiments that she had been prepared for.

Jim at first is stunned and dismayed because—he has just sold his watch to buy her a Christmas gift of beautiful tortoiseshell combs for her to wear in her long hair. He can't, initially, absorb what she has done, so he simply stares and tries to process the new reality. Jim obviously never expected her to do this.
Once Jim realizes, however, that Della sold her beloved hair to buy him a watch chain, he feels deeply moved by her sacrifice and enfolds her in his arms. Jim realizes she has sold her hair because she loves him so deeply that she was willing to give up what was most precious to her for his sake. This makes his love for her grow more than any physical gift she could give him.

Jim's initial reaction is one that Della simply doesn't understand. He's so incredibly quiet; his eyes are full of something, but she doesn't quite know what. It isn't anger, it isn't surprise, but whatever it is, it fills Della with fear. That's a perfectly understandable reaction. Della is scared stiff that Jim will no longer find her beautiful when he sees that her lovely locks have been cut off. At this point, Jim seems really perplexed, even more so when he discovers that Della has sold her hair to buy him a Christmas gift. But confusion quickly dissolves into a warm, loving embrace. Jim understands that Della's selfless act was an expression of love. He can empathize with her because he's done exactly the same thing: selling something precious to buy a Christmas gift for someone he loves.

y=-3/(x-4)-1 Graph the function. State the domain and range.

The given function y = -3/(x-4)-1 is the same as:
y = -3/(x-4)-1*(x-4)/(x-4)
y = -3/(x-4)-(x-4)/(x-4)
y=(-3-(x-4))/(x-4)
y=(-3-x+4)/(x-4)
y = (-x+1)/(x-4)
To be able to graph the rational function  y = (-x+1)/(x-4) , we solve for possible asymptotes.
Vertical asymptote exists at x=a that will satisfy D(x)=0 on a rational function f(x)= (N(x))/(D(x)) . To solve for the vertical asymptote, we equate the expression at denominator side to 0 and solve for x .
In y = (-x+1)/(x-4), the D(x)=x-4.
Then, D(x) =0  will be:
x-4=0
x-4+4=0+4
x=4
The vertical asymptote exists at  x=4 .
To determine the horizontal asymptote for a given function: f(x) = (ax^n+...)/(bx^m+...) , we follow the conditions:
when n lt m     horizontal asymptote: y=0
        n=m        horizontal asymptote: y =a/b
        ngtm       horizontal asymptote: NONE
In y = (-x+1)/(x-4) , the leading terms are ax^n=-x or -1x^1 and bx^m=x or 1x^1 . The values n =1 and m=1 satisfy the condition: n=m. Then, horizontal asymptote  exists at y=(-1)/1 or y =-1 .
To solve for possible y-intercept, we plug-in x=0 and solve for 
y =(-0+1)/(0-4)
y =1/(-4)
y = -1/4 or -0.25
Then, y-intercept is located at a point (0, -0.25).
To solve for possible x-intercept, we plug-in y=0 and solve for x .
0 =(-x+1)/(x-4)
0*(x-4)= (-x+1)/(x-4)*(x-4)
0 =-x+1
x=1
Then, x-intercept is located at a point (1,0)
Solve for additional points as needed to sketch the graph.
When x=3 , the y =(-3+1)/(3-4)=-2/(-1)=2 . point: (3,2)
When x=5 , the y = (-5+1)/(5-4)=(-4)/1=-4 . point: (5,-4)
When x=7 , the y =(-7+1)/(7-4)=(-6)/3=-2 . point: (7,-2)
When x=-2 , the y =(-(-2)+1)/(-2-4)=3/(-6)=-0.5 . point: (-2,-0.5)
Applying the listed properties of the function, we plot the graph as:

You may check the attached file to verify the plot of asymptotes and points.
As shown on the graph, the domain: (-oo, 4)uu(4,oo)
and range: (-oo,-1)uu(-1,oo) . 
The domain of the function is based on the possible values of x. The x=4 excluded due to the vertical asymptote.
The range of the function is based on the possible values of y . The y=-1 is excluded due to the horizontal asymptote. 

Compare the main characters of books 5 and 10 of The Odyssey. What are the similarities? What are the differences?

Books 5 and 10 center around Odysseus, unlike books 1–4, which focus on his son, Telemachus. In this respect, books 5 and 10 actually follow the same main character. Circe and Calypso, however, are central characters in Books 10 and 5 respectively, not only because the two are central personalities in those two chapters, but more importantly because they they do seem to serve as mirrors to one another, fulfilling comparable roles within the poem's narrative structure.
Circe and Calypso are similar in a number of respects. They are both minor goddesses, they both reside on an island far removed from civilization, and they both create lengthy delays in Odysseus's voyage back to Ithaca before helping and preparing him for the next stage in his voyage. Even in their descriptors (The Odyssey, having originally been passed down orally, is filled with various mnemonic devices to aid memorization and recitation) the two overlap. Calypso is occasionally referred to as “the nymph with lovely braids” (see Fagles, Book 5, lines 34 and 63), a description she shares with Circe: “We reached the Aeaean islands next, the home of Circe the nymph with lovely braids” (Fagles, translation, Book 10, lines 148–149). Both are associated with animals—Circe more famously, given her powers of enchantment, but Calypso's cave is explicitly described as being a gathering place of all manner of birds and plants and flowers (Book 5, lines 71–81). It's a recurring theme in Greek mythology and heroic myth—the conflict between civilization and nature (and the conquest of the one over the other). With both Calypso and Circe, we see personalities who seem to be suspended somewhere between the two—they both have servants, and they both have the trappings of civilization (Circe, for example, is described as residing in a palace), but they're also apart from it, isolated as they are on their islands far removed from human interaction. So the similarities are striking—to a degree, they do seem like mirror images of one another.
That being said, there are differences between them too. For all that Circe arrives later in the narrative, her encounter with Odysseus is actually set earlier in the chronology (it should be remembered, much of The Odyssey is Odysseus himself telling his story to the Phaeacians). From that perspective, Calypso is actually one of the last encounters he has on his way back to Ithaca. When he encounters Circe, he still has many of his followers behind him, followers he is ultimately responsible for, whereas by the time Calypso enters the picture, Odysseus is the only survivor that remains, found by Calypso adrift at sea. This is probably the most obvious difference between the two nymphs: Circe is introduced as a malicious force that Odysseus needs to conquer. She uses magic to transform humans into animals, and she turns that sorcery against Odysseus's own people. With the aid of Hermes (who will also arrive later to release him from Calypso's keeping—yet another parallel between the two nymphs), Odysseus is able to overcome Circe's magic, but it is Odysseus himself who must forcibly overcome her before the nymph will grant him aid. Calypso, it must be said, is herself antagonistic (let's not forget, she does entrap him unwilling on her island, intending to make him her husband), but the nature of that antagonism is changed—Circe is an opponent he must overcome and conquer, a malicious force converted into an ally, whereas Calypso is an opponent he can only endure.
This response was written with reference to the Penguin Edition of The Odyssey, as translated by Robert Fagles. See: Homer, The Odyssey. Translated by Robert Fagles. New York: Viking Penguin, 1996.

What is an example of irony in Ray Bradbury's "The Pedestrian"?

Of course, the central irony of the story is that a simple activiity—walking—is considered abnormal in the dystopian society of Bradbury’s story. This “normal” habit of walking the deserted streets is “regressive,” suggesting some previous time when people routinely walked for pleasure.
Another source of irony is the fact that there is not much difference between the darkened houses Mead passes in 2053 and the darkened houses one would pass walking at night through a suburban neighborhood in 1951 (when the story was written) or 2018 (when this is being written). The image of people disconnected from each other and the outside world, their sense of reality mediated by television, is hardly one that requires science fiction.
But I suspect that the real irony of the story centers around the idea of writing itself. Mead identifies himself to the police car as “a writer,” which the police car decides means “no profession.” Mead silently agrees: “He hadn't written in years. Magazines and books didn't sell any more. Everything went on in the tomblike houses at night now, he thought, continuing his fancy. The tombs, ill-lit by television light, where the people sat like the dead, the gray or multicolored lights touching their faces, but never really touching them.” In a way, Bradbury’s writing of the story is a kind of refutation of Mead’s contention that “magazines don’t sell” (i.e., Bradbury himself is a writer, and the very text in which Mead appears would seem to refute his claim that writing isn’t possible or lucrative). But there is also a sense in which Mead’s nightly walks are an attempt to gather material for writing, material which is not forthcoming or accessible. It’s ironic that Mead is picked up for this activity, since it is truly harmless (Mead likely will never write anything). It’s ironic that the stories on television, which so absorb the townspeople, are the things that inhibit the human interactions that make stories (and writers) possible.

An example of irony in Ray Bradbury’s “The Pedestrian” would be that perhaps the most sane character in the story is taken to an asylum because he enjoys walking. Leonard Mead, the main character, takes walks at night while the other citizens are in their homes watching their “viewing screens.” In our society, we would consider walking to be an example of perfectly normal behavior. However, in this dystopian society, Leonard Mead is not the norm.
Mr. Mead is aware that his chosen activity is somewhat abnormal as he notices that, “lights might click on and faces appear and an entire street be startled by the passing of a lone figure.” A police car, the only one that now patrols, finds Mr. Mead walking one night. Mr. Mead again considers how his behavior although normal to him, is seen as peculiar when he refers to how the light of the police car, “held him fixed, like a museum specimen, needle thrust through chest.” Just as one studies a museum specimen and its characteristics, Mr. Mead must be studied in an attempt to find out why he is unusual and "regressive." The car itself questions Mr. Mead and determines that he must be taken to the “Psychiatric Center for Research on Regressive Tendencies." His individuality makes him a threat in a society where most citizens can be controlled through watching their viewing screens.

It's a little tricky to identify irony in Ray Bradbury's "The Pedestrian," but there are a few interpretive claims you might make if you need to identify that particular literary device.
Situational irony occurs when the opposite of what we'd expect happens in a particularly interesting or critical way. For example, it would be situationally ironic if the firehouse (the building belonging to the people whose job it is to put out fires) burned down.
You might argue that it's ironic that Leonard Mead continually thinks of the urban landscape he wanders in natural terms:

The street was silent and long and empty, with only his shadow moving like the shadow of a hawk in mid-country. If he closed his eyes and stood very still, frozen, he could imagine himself upon the center of a plain, a wintry, windless Arizona desert with no house in a thousand miles, and only dry river beds, the streets, for company.

Mead's thoughts paint a picture of a city being reclaimed by nature, with sidewalks "vanishing under flowers and grass." He uses this to emphasize the lack of a human presence in the city at night; people sit at home and watch television instead of going out or visiting friends. This is ironic because in Bradbury's vision, the industrialized, technological future actually leads humans to abandon their environment and make less of an impact on the landscape—the opposite of what we might expect. In short, our expectations about what progress should look like are reversed; this vision of progress looks like decline. By the same token, humanity's unnatural technological advances actually cause nature outside to flourish (the opposite of what we would predict).
It’s also ironic that the man we as readers identify as the most normal character in the story—the one who doesn’t want to sit at home and watch TV every night—is regarded by his society as deviant to the point of criminality: at the end of the story, he is taken away to the “Psychiatric Center for Research on Regressive Tendencies.” Though we regard Leonard Mead as natural (both in his behavior and in his affinity with the natural world), his society sees him as unnatural—the exact opposite of what we might expect.

College Algebra, Chapter 7, 7.4, Section 7.4, Problem 36

Solve the system $\left\{\begin{equation}
\begin{aligned}

\frac{1}{2} x + \frac{1}{3} y =& 1
\\
\\
\frac{1}{4} x - \frac{1}{6}y =& \frac{-3}{2}

\end{aligned}
\end{equation} \right.$ using Cramer's Rule.

For this system we have


$
\begin{equation}
\begin{aligned}

|D| =& \left| \begin{array}{cc}
\displaystyle \frac{1}{2} & \displaystyle \frac{1}{3} \\
\displaystyle \frac{1}{4} & \displaystyle \frac{-1}{6}
\end{array} \right| = \frac{1}{2} \cdot \left( \frac{-1}{6} \right) - \frac{1}{3} \cdot \frac{1}{4} = \frac{-1}{6}
\\
\\
|D_{x}| =& \left| \begin{array}{cc}
1 & \displaystyle \frac{1}{3} \\
\displaystyle \frac{-3}{2} & \displaystyle \frac{-1}{6}
\end{array} \right| = 1 \cdot \left( \frac{-1}{6} \right) - \frac{1}{3} \cdot \left( \frac{-3}{2}\right) = \frac{1}{3}
\\
\\
|D_{y}| =& \left| \begin{array}{cc}
\displaystyle \frac{1}{2} & 1 \\
\displaystyle \frac{1}{4} & \displaystyle \frac{-3}{2}
\end{array} \right| = \frac{1}{2} \cdot \left( \frac{-3}{2} \right) - 1 \cdot \frac{1}{4} = -1

\end{aligned}
\end{equation}
$


The solution is


$
\begin{equation}
\begin{aligned}

x =& \frac{|D_x|}{|D|} = \frac{\displaystyle \frac{1}{3}}{\displaystyle \frac{-1}{6}} = 2
\\
\\
y =& \frac{|D_y|}{|D|} = \frac{-1}{\displaystyle \frac{-1}{6}} = 6

\end{aligned}
\end{equation}
$

How does the Epic of Gilgamesh represent society?

The Epic of Gilgamesh has many archetypal overtones, and it represents human society in numerous ways. Aside from clear themes such as religion, the search for meaning and immortality or legacy, and the need for community, the story of Enkindu within the overall narrative is a particularly specific representation of humanity as a whole.
Enkindu is the man-beast who becomes Gilgamesh's best friend. Found in the wild, he represents mankind's origins as uncivilized beasts. He eventually comes into society and learns to be truly human by gaining community—particularly with Gilgamesh. The two end up working together to find the secret to immortality, which is the existential goal of humanity. During all of this, Gilgamesh's city becomes a renowned civilization of which he is the king, showing the ascent of humanity through its pursuits.

Society is presented as an interesting force in The Epic of Gilgamesh. It is a rather traditional society, with men and women handling different roles. Gilgamesh himself rules over the ancient Sumerian city Uruk, which is the main social and cultural setting of the story.
However, society is not represented as perfect. Gilgamesh is initially a corrupt figure who bullies his subjects into doing whatever he wants them to, whether it's fighting him or sleeping with him. Though society has rules, those in power are free to trespass them.
Society is contrasted with nature, represented by the wild man Enkidu. He initially shuns society, living with the animals, until he is "civilized" by an extended sexual encounter with a woman. Then he becomes a social being and eventually Gilgamesh's dearest friend. He challenges Gilgamesh's bullying and ultimately helps him grow into a better person.
So, society is a double-edged sword in Gilgamesh: it can corrupt as well as enlighten and ennoble.

The question hinges on what you mean by “society.” I’ve always thought that the figure of Gilgamesh represents a kind of transition in the way ancient societies are understood. That is, Gilgamesh is a figure who clearly has mythological origins—he is part god, a heritage that makes him incredibly strong and wise. His rule over Uruk is more than by “divine right”—he actually is divine! There is a sense in which his conflict with Ishtar supersedes his role as king, or, rather, that his killing the bull and defeating Enkidu are somehow evidence of his kingly capabilities. What makes Gilgamesh so compelling, however, is his humanity, his friendship with Enkidu, and his quest for immortality after Enkidu’s death. His decision to take Utnapishtim’s advice and return to rule Uruk as a mortal suggests that society is not intertwined with the gods, but is, in fact, a fully human development, and made all the stronger for that.

Although many elements of the Epic of Gilgamesh are mythical or fictional, the protagonist Gilgamesh was an historical king of Uruk whose rulership is attested by ancient king lists. Many aspects of the epic are consistent with information obtained from archaeological and other non-literary sources. 
The epic represents the society of Uruk as one in which the monarch has absolute power. Even when subjects disapprove of his actions or he acts in a manner that violates social norms, there is no mechanism by which ordinary people can prevent him from doing as he pleases. 
Religion is an important element in Gilgamesh's society, which is ordered as a theocracy, with there being no clear distinction between secular and religious authority. The portrait of the constant intervention of the gods in human affairs reflects the actual importance of temples in the administration of Uruk.
The society is represented as having very distinct gender roles and strong social stratification. The society is extremely hierarchical. As well as class distinctions, the society is highly urbanized with a strong division between city and countryside. The society is also strongly militaristic, and in a constant state of potential conflict with neighboring kingdoms.

How many divisions of Christianity are there, and which is the right one to go with?

To answer the first part of the question, some history might help. Jesus of Nazareth was Jewish; he lived and died as such, and during his ministry, he was very clear about who he was. “Do not think that I have come to abolish the Law of the Prophet; I have not come to abolish them but to fulfill them” (Matthew 5:17). In other words, he came to fulfill the prophecy (that God would send a savior) and deliver the message that the time had come to move on from the traditions, routines, and laws that focused on ritual and less on faith and forgiveness. It is important to keep in mind that during his life, he preached to people who were living in terrible poverty and under the brutal domination of Roman occupation, so this was a message that people were dying to hear.
When he died, his followers spread his message of hope throughout the Middle East; again, this was a message that people needed. People gathered where they could (mostly in small areas in their houses) and studied together. The letters of Paul in the New Testament are his reflections and advice to the people he encountered in various areas (Thessalonians, Galatians, Philippians, Corinthians, and Romans). The traditional Jewish beliefs began to clash with the “new religion” almost immediately, and followers of Christ branched off into another religion (Christianity), but because they were being persecuted, they went underground and remained fairly quiet. This persecution continued with the Romans until around the fourth century.
In 312 AD, Emperor Constantine the Great had a vision that God would lead him to military victory, and upon winning the Battle of the Milvian Bridge, he converted to Christianity. Constantine enacted the Edict of Milan, which allowed the worship of all religions: “we have also conceded to other religions the right of open and free observance of their worship for the sake of the peace of our times, that each one may have the free opportunity to worship as he pleases; this regulation is made so that we may not seem to detract from any dignity or any religion” (Constantine, Edict of Milan). Then, in 325, Constantine called a meeting of three hundred major leaders in Christianity to meet and unify their beliefs so that people would be clear about what Christ actually taught. Although the group did not agree on many points, the origins of the Bible were born at the Council of Nicaea; they decided which books and writings would be included as codified canon and which would not.
With Christians given the right to worship openly and a clarified set of beliefs, Christianity spread quickly. Catholicism is considered by many to be the first official church, but it split into two distinct factions, the Roman Catholic Church in the West and the Eastern Orthodox Church in Eastern European countries and Greece. Due to widespread corruption, people started to demand a reform in the Roman Catholic Church, and in 1517, Martin Luther kick-started the Protestant Reformation by nailing his Ninety-five Theses to the doors of All Saints’ Church in Wittenberg, Germany. So, another split (this time in the West) was created.
Over time, these divisions splintered into other factions, and today, there are many different variations on a theme. According to the World Christian Encyclopedia (Bar­rett, Kurian, and John­son; Oxford Uni­ver­sity Press), there are roughly 33,000 denominations of Christianity in over 238 countries. Groups can also be divided into larger cultural blocks as well as smaller traditional affiliations within a denomination. The varying branches make sense, since Christ’s teachings revolved around faith rather than ritual and tradition; people are able to take the basic beliefs of Christianity and tailor the traditions around what makes sense to them. That is not to say, though, that Christians are not unified in certain ideas. These core beliefs revolve around the teachings of Jesus Christ, set forth in the New Testament.
Which is the “right” denomination? That is a question for the individual to figure out. However, researching the history, the literature, the language, and the core beliefs and actions of the founder (Jesus and the disciples) and weighing them against specific denominations and/or churches and the people who run them might help.

There are many divisions of Christianity such as Baptist, Protestant, Episcopalian, Pentacostilism etc. , the choice in which one to choose is completely up to you by your own convictions. Please refer to James 1:26-27, which lets us know that only the pure religion is that which tends to the widows & orphans. Note that doesn't mean that just because a denomination as what is stated above decides to do as James 1:26-27 says is the "right one" to go with. No, it means the pure religion such as dealing with the core or heart of a religion "blessed are the pure in heart for they shall see God" 

Like with most religions, Christianity can be divided into a large number of divisions or denominations. There are five major groups of denominations that include the Church of the East, Oriental Orthodoxy, Eastern Orthodoxy, Roman Catholicism, and Protestantism—which itself is made up of smaller denominations such as Adventist, Baptist, Calvinist, Lutheran, and Methodist.
While Catholicism has the highest number of followers, each division of Christianity has its own practices and specific beliefs. Since Christianity is a religion that exists throughout the entire world, there are many different forms one can find it in, even amongst any specific division.
Each division of Christianity differs due to a long history of the religion; many contrasting and conflicting ideas have been discussed passionately, and many important religious figures have existed. While Methodism may identify with the evangelist priest John Wesley, Lutheranism identifies with the teachings of Martin Luther. Catholicism, on the other hand, principally operates under the belief that Catholicism itself is the only true church founded by Jesus Christ.
In terms of deciding which division is the right fit for an individual, there is no objective answer. When considering a religion or denomination to follow, it's important to weigh one's personal beliefs to find what they fit best. Talking to religious leaders to discuss the ideologies and practices of each religion/denomination is also helpful in gathering an understanding of which would be most appropriate.

In "Rikki-Tikki-Tavi," how does the family eating breakfast on the veranda create suspense?

The scene in "Rikki-Tikki-Tavi" where the family is eating breakfast on the veranda is full of suspense. First, Kipling foreshadows the life-and-death stakes of the scene twice before the scene actually occurs. When Nagaina is talking to Darzee's wife, Nagaina predicts that "before night the boy in the house will lie very still." Darzee's wife lets Rikki know of the cobra's murderous intent by telling him, "She means killing!" 
The way Kipling describes the scene is full of tension. Nagaina is set to strike at Teddy, and the humans are very fearful. We can picture them: They aren't eating, they aren't moving a muscle, and their faces have gone pale. The father whispers to the little boy to remain perfectly still. Nagaina is coiled near enough to Teddy's chair that she can reach him in one strike. Her head is swaying menacingly as she stares unblinkingly at the humans. 
The dialogue further adds to the suspense. The father whispers a warning, and Nagaina speaks threateningly to Rikki without turning around to face him. She says if he comes a step closer, she will strike. 
The foreshadowing, description, and dialogue all work together to create a highly suspenseful moment in the story. 

Briefly provide the context of the quote (what is happening at the time these words appear and who is speaking these words) and explore the significance of the quote in terms of the story as a whole—in relation to its theme, plot and major characters/relationships. The quote is: “As Martin watched the tranquil slumber of his wife the ghost of the old anger vanished. All thoughts of blame or blemish were distant from him now. Martin put out the bathroom light and raised the window. Careful not to awaken Emily he slid in to the bed. By moonlight he watched his wife for the last time. His hand sought the adjacent flesh and sorrow paralleled desire in the immense complexity of love.” (From Carson McCullers’s “A Domestic Dilemma.”)

Carson McCullers's "A Domestic Dilemma" was first published in Ballad of the Sad Cafe and Other Stories. This short story introduces the reader to Martin Meadows, his wife Emily, and their children, baby Marianne and six-year-old Andy. It is revealed that Emily is "sick;" in fact, she appears to be an alcoholic. Emily lies to Martin about her drinking, but by the time he comes home from work each day she is already tipsy enough for the problem to be evident. At one point Emily drops Marianne, drawing blood, and vows never to drink again. In fact, she doesn't stop, but Martin hires a housekeeper to keep an eye on the children, and Emily cuts back enough on her drinking that incidents are rare.
When Martin arrives home on this Thursday, he finds Emily inebriated, having served the children toast with cayenne pepper (she mistook it for cinnamon) for dinner. Martin argues with his wife, who is attempting to hide the sherry from him, and leaves her in their room while he makes the children a proper dinner. Emily wanders downstairs while the children are eating and causes a scene. She is afraid Martin is speaking ill of her and defends herself against him. Andy is upset by the interaction, and Martin sends him and Marianne to bed.
Emily comes to her senses and realizes she's scared her child. Martin brings her to their room and gets her into bed. He gives the children their baths and puts them to sleep. He reflects on how much he loves his children and resents Emily's implication that he prefers his daughter to his son. Martin makes himself dinner and reads a book but has a hard time concentrating. He obsessively worries that his children might be hurt by their mother. His anger grows, and he describes the feelings as hate for his wife.
Martin heads to bed and finds his wife asleep. Noticing her peaceful countenance, the narrator describes the scene:

As Martin watched the tranquil slumber of his wife the ghost of the old anger vanished. All thoughts of blame or blemish were distant from him now. Martin put out the bathroom light and raised the window. Careful not to awaken Emily he slid into the bed. By moonlight he watched his wife for the last time. His hand sought the adjacent flesh and sorrow paralleled desire in the immense complexity of love.

While Martin spent the majority of the story resenting his wife and loving his children, when she is asleep, he can relax and remind himself of the woman he fell in love with. Throughout the story, the reader gets the sense that Martin might leave his wife or have her committed. The story highlights the very fine line that can exist between love and hate.

Beginning Algebra With Applications, Chapter 6, 6.1, Section 6.1, Problem 20

Determine whether the system of equations is independent, inconsistent or dependent.







Based from the graphs, the system of equations is inconsistent. It has no solution because the two graphs doesn't intersect.

Calculus: Early Transcendentals, Chapter 4, 4.9, Section 4.9, Problem 58

Graph of f can be seen in the picture below (blue line).
From this we can sketch graph of its antiderivative. We start drawing from left to right. As long as f is positive (graph above x-axis) antiderivative is increasing and it reaches maximum when f(x)=0. It continues to decrease until f reaches zero again (minimum of antiderivative). After that (right of that) graph of f lies above x-axis so the antiderivative is increasing.
Graph of antiderivative can be seen in the picture below (red line).

How is the theme of social/political change on individual life portrayed in "The Waste Land" by T.S. Eliot?

T.S. Eliot's "The Waste Land" was published in 1922 and reflects the unease and disillusionment in society after the horrors of the First World War. The effect of this war in particular upon the individual is marked in the poem: the war occasioned enormous social and political change, but also made doing the simplest things difficult for those who had gone to war and then been forgotten by the government they served. In this poem, Eliot moves between classical allusions and narrow focus on everyday scenes, a technique which emphasizes the sense that the world is now somehow ill-fitting—or rather, that those who once belonged to it now fit in it poorly.
The poem begins with allusions to Russia and France, but settles on its main social focus, London:

A crowd flowed over London Bridge, so many,
I had not thought death had undone so many.


But the deaths of others have certainly undone many in London. Eliot shows the fragmented conversations of men who, we understand, have returned from the war with shell-shock:


“My nerves are bad tonight. Yes, bad. Stay with me."
. . .
I think we are in rats' alley
Where the dead men lost their bones.

The conversation is clipped, erratic, and betrays the deep desire of these people for a connection which now seems out of reach, and a certainty which has been lost:


“What are you thinking of? What thinking? What?
“I never know what you are thinking. Think.”

By contrast to these anxiety-ridden, clipped words from the returning soldiers, the women in the poem speak in a stream-of-consciousness flow which mimics real life and yet is so far removed from the men's speech that it marks the disjunction between the two. Those who did not experience the war can no longer connect with those who did; society has been riven by this event.

When Lil’s husband got demobbed, I said—
I didn’t mince my words, I said to her myself,
HURRY UP PLEASE ITS TIME
Now Albert’s coming back, make yourself a bit smart.
He’ll want to know what you done with that money he gave you
To get yourself some teeth.


The barman's refrain, "HURRY UP PLEASE ITS TIME," creates a sense of social realism and would have been relatable to the contemporary reader; moreover, however, it indicates that these people have stayed in the pub all evening, right up to closing time (midnight). They are seemingly unable to determine what to do with themselves.

There is a sense in this poem that every speaker is always on the edge of madness. Bursts of realism, such as the public house scene, are interspersed with dreamy reveries, echoes of death, which descend into fragmented allusions and nonsense:


White bodies naked on the low damp ground
And bones cast in a little low dry garret,
Rattled by the rat’s foot only, year to year.
But at my back from time to time I hear
The sound of horns and motors, which shall bring
Sweeney to Mrs. Porter in the spring.
O the moon shone bright on Mrs. Porter
And on her daughter
They wash their feet in soda water
Et O ces voix d’enfants, chantant dans la coupole!

It is as if the speaker cannot focus on one thing for any length of time, or perhaps seizes upon any possible distraction. What is certain is that he does not wish to dwell upon the rats to which the poem frequently refers, a motif representative of the decay in society. This society seems unlivable, impossible; and yet Eliot's classical allusions, at the same time, suggest that this is simply a cycle we all go through. An immortal, perhaps, would have experienced this kind of social disfunction many times:


(And I Tiresias have foresuffered all
Enacted on this same divan or bed;
I who have sat by Thebes below the wall
And walked among the lowest of the dead.)


"The Waste Land" is a long and hugely significant poem which you could spend many hours unpacking. For the purposes of your question, it also has a lot to yield about social and political change and the general discomfort in society at this time. You might consider also the question of "the third who walks always beside you"—who does this represent, and why?—and the way the poem breaks down to end in fragments in a foreign language.

In what ways is Creon a foil to Oedipus?

In order to understand the nature of your question about Sophocles's Oedipus Rex, let us first look at the definition of a foil: a "character that contrasts with another character, usually the protagonist, to emphasize the other character's traits" (berkeley.edu). In this case, Creon, the foil, contrasts with Oedipus, the protagonist, to emphasize the latter's character traits. The most striking example of this is the way each character reacts to negative accusations.
In the play, Oedipus is the King of Thebes. He is shown to be reckless and bold with his actions and words. When news comes from the oracle about the cause and potential cure of the Theban plague, Creon asks if he would like to hear the news in private. Oedipus, wanting to show his people that he is a prideful and valiant ruler, tells him to "[s]peak forth to all." He publicly vows to find Laius's murderer in order to end his people's suffering. At this point in the play, he seems honorable: he is quick to action when his people are counting on him.
However, his rashness soon becomes a negative trait when he consults Teiresias. Initially, he praises the blind prophet:

Even if you are blind, you know what plague / Clings to the state, and, master, you alone / We find as her protector and her saviour.

When Teiresias initially faults at giving him the information he seeks before telling him that he is the cause of the plague, Oedipus quickly changes his tune, insulting the prophet and mocking his blindness. As you can see, his hastiness has gone from being a desirable trait to a reckless one. He does not respond well to being accused of wrongdoing.
Creon, on the other hand, is very prudent and logical in both his actions and his words. Where Oedipus acts rashly, he appears to think things through first. The first example of this can be seen in his second line in the play:

If you wish to hear in front of this crowd / I am ready to speak, or we can go within.

Instead of blurting out the news, he gives Oedipus the option to retire to a more secluded area first. This may seem like a small detail, but it is mirrored exponentially when he is accused by the King of being a traitor:

CREON: And was the idea voiced that my advicePersuaded the prophet to give false accounts?[. . .]Was this accusation laid against meFrom straightforward eyes and straightforward mind?[. . .]I agree with you. What you have said is just.Tell me what you say you have suffered from me.

As can be seen in these lines, Creon reacts to accusations in a much calmer, logical manner than does Oedipus. The latter, when told that he is the cause of the Theban plague, is quick to turn to insults and threats. Creon, when branded a traitor by the King, does not respond rashly, opting instead to debate his innocence. It is important to keep these qualities in mind when looking for further examples of how Creon is a foil to Oedipus.
https://www.ocf.berkeley.edu/~daradib/chsntech/review/english/lit/review-terms.pdf

While Oedipus is rash, Creon is measured. Oedipus angers easily when Teiresias, the blind prophet, will not answer his questions as he investigates who killed Laius. He jumps to the conclusion that Teiresias has been working with his brother-in-law (and uncle), Creon, in order to secure the crown for Creon himself. When Oedipus confronts Creon with his suspicions, Creon presents a very calm and reasonable argument as to why he wouldn't want to be king in the first place. In his current position, brother to the queen and brother-in-law to the king, he enjoys lots of power and authority, but he doesn't have to bear the responsibilities of being king. He enjoys the privileges without having to make unpopular decisions or feel responsible for the welfare of the entire kingdom. Oedipus yells and storms, while Creon speaks rationally and coolly. In this way, Creon can be seen as even-tempered and cool making Oedipus appear all the more hot-headed and rash.

What reason did the Founding Fathers give for their decision to write out a declaration?

In the Preamble section of the Declaration of Independence, the Founding Fathers stated why they were issuing the Declaration of Independence. The Founding Fathers believed that when a group of people is going to break free from another government and declare its independence, they should tell the world why they are doing this.
The Founding Fathers stated in the document that all people have certain rights that cannot be taken away by anybody or by any group. These rights include the rights of life, liberty, and the pursuit of happiness. When a government begins to abuse its powers by taking away the rights of the people, the people have no choice but to get rid of the government. In the Declaration of Independence, the Founding Fathers listed all of the grievances that the people had against the King of England. This section is the longest section of the document.
Thus, the Founding Fathers wanted the world to know why the colonists were declaring their independence from Great Britain.

How did Hamilton’s financial plan enable the nation to expand economically?

Hamilton's plan was to place the nation on sound financial footing by paying off its debts. Hamilton created a national bank which would control the entire banking industry in the United States. Hamilton also created a system where the debts of the individual states were rolled into the national debt, thus further ensuring that debts incurred at home and abroad during the war were paid. Hamilton sought to fund this system through land sales to the public; this allowed for American expansion and also put the federal government in charge of the land acquired in the Treaty of Paris settlement. Hamilton also sought to increase revenues through tariffs and excise taxes. By making the United States a responsible debtor with a central banking system to pay off its loans, Hamilton created a system in which the world was encouraged to invest in the United States. Foreign investment further drove the American economy and made the United States strong financially. While the central banking system would be controversial in the South and West as settlers did not trust the Eastern banks, one cannot overestimate the importance of Hamilton's regulatory system for the commerce of early America.

What are real life examples (countries or states) of the different models of democracy? a) Protective democracy b) Classic democracy (based on Athens) c) Developmental democracy

In political theory, a protective democracy is one that is tasked primarily with maintaining and protecting the rights of its citizens. This ideal is fundamental to late eighteenth and early nineteenth-century liberalism, where it is associated with the writings of John Stuart Mill and Jeremy Bentham, among others. Under this model, the government would only intervene in the lives of its citizens, and especially in the operations of the free market, in a very limited way. The United States, whose people and leaders are very reluctant to embrace social welfare programs and government control of key industries, has historically been a pretty good example of a protective democracy. Basically, protective democracy assumes that the only reason government should exist is to protect its citizens from each other. People would only, then, get involved in civic life in order to protect their own interests. 
A developmental democracy has a somewhat more benign view of human nature, and how it is reflected in people's participation in government. According to this theory, people participate in government out of a sense of civic-mindedness, what some political philosophers used to call virtue. According to this idea, our representatives must be truly representative, in that their actions in government should be consistent with the public will and the public good. This is, perhaps, how Americans understand their government, and it might be best used to describe local government, where individuals are elected to represent wards, precincts, or districts according to what their constituents want. 
The first two categories were used by the political scientist William E. Hudson in his book Democracy in Peril, published in 2006. Classical democracy was not one of his categories, but it is usually used to describe the democratic system created in ancient Athens. In this system, citizens directly participated in government. Government officials were chosen by drawing lots, and all citizens were expected to contribute their voice directly to city politics. Of course, only a very small percentage of the Athenian people were actually citizens, so this ideal falls short of even our modern conception of democracy. But we can see echoes of this model in towns that are still run by town meetings (there aren't very many of them left) or even in local grassroots organizations that make decisions for neighborhoods.
 
http://www.kropfpolisci.com/democracy.capitalism.hudson.pdf

In general a protective democracy is based on the idea that there are some inherent inequalities that can and must be corrected by governmental interference. The United States, although imperfect in many ways, is a real world example of a protective democracy in that is a built upon a system of checks and balances for each branch of government. The executive, legislative, and judicial systems all require some level of vetting or checking by the other and none of these branches can act without potential consequences from one or both of the others. For example, the legislative branch can pass a law but the executive branch can veto this law, or the judicial branch can deem this law contrary to the Constitution and the law becomes invalid.
A classic democracy or Athenian type democracy relies on direct participation of citizens. The mark of direct democracy most important to note is that there is no interference in the interaction between citizens and voting.  Local county governments in the United States are an example of this in terms of ballot measures.  A direct question is asked of the public, such as "should tax money be spent to improve a specific highway", and citizens then vote yes or no.  The measure is passed based on the number of votes it receives that indicate the choice of the public.
A developmental democracy focuses on the direct moral improvement of the citizens and the inclusion and protection of the rights of individual in the democratic process. A modern day example would be the town halls that many United States cities and towns have begun to reintegrate into their political processes.  This involves the thorough explanation of a topic to be voted on, a question and answer period where citizens can ask direct questions of their legislators and then voting upon those referendums that were presented.  

y=x^2, y=x^3 Find the x and y moments of inertia and center of mass for the laminas of uniform density p bounded by the graphs of the equations.

For an irregularly shaped planar lamina of uniform density (rho) bounded by graphs y=f(x),y=g(x) and a<=x<=b , the mass (m) of this region is given by,
m=rhoint_a^b[f(x)-g(x)]dx
m=rhoA  , where A is the area of the region
The moments about the x- and y-axes are,
M_x=rhoint_a^b 1/2([f(x)]^2-[g(x)]^2)dx
M_y=rhoint_a^bx(f(x)-g(x))dx
The center of mass (barx,bary) is given by,
barx=M_y/m
bary=M_x/m
Now we are given y=x^2,y=x^3
Refer the attached image. Plot in red color is of y=x^2 and blue color is of y=x^3
Curves intersect at (1,1)
Now let's evaluate the area of the region,
A=int_0^1(x^2-x^3)dx
A=[x^3/3-x^4/4]_0^1
A=[1^3/3-1^4/4]
A=(1/3-1/4)=(4-3)/12
A=1/12
Now let's evaluate the moments about the x- and y-axes,
M_x=rhoint_0^1 1/2[(x^2)^2-(x^3)^2]dx
M_x=rho/2int_0^1(x^4-x^6)dx
M_x=rho/2[x^5/5-x^7/7]_0^1
M_x=rho/2[1^5/5-1^7/7]
M_x=rho/2(1/5-1/7)
M_x=rho/2(7-5)/(35)
M_x=rho/35
M_y=rhoint_0^1x(x^2-x^3)dx
M_y=rhoint_0^1(x^3-x^4)dx
M_y=rho[x^4/4-x^5/5]_0^1
M_y=rho[1^4/4-1^5/5]
M_y=rho[5-4)/(20)
M_y=rho/20
barx=M_y/m=M_y/(rhoA)
Plug in the values of M_y and A ,
barx=(rho/20)/(rho1/12)
barx=12/20
barx=3/5
bary=M_x/m=M_x/(rhoA)
bary=(rho/35)/(rho1/12)
bary=12/35
The coordinates of the center of mass are (3/5,12/35)
 

sum_(n=1)^oo n/sqrt(n^2+1) Verify that the infinite series diverges

sum_(n=1)^oo n/sqrt(n^2+1)
To verify if the series diverges, apply the nth-Term Test for Divergence.
It states that if the limit of  a_n is not zero, or does not exist, then the sum diverges.

lim_(n->oo) a_n!=0     or    lim_(n->oo) a_n =DNE
:. sum a_n  diverges

Applying this, the limit of the term of the series as n approaches infinity is:
lim_(n->oo) a_n
=lim_(n->oo) n/sqrt(n^2+1)
=lim_(n->oo) n/sqrt(n^2(1+1/n^2))
=lim_(n->oo) n/(nsqrt(1+1/n^2))
=lim_(n->oo) 1/sqrt(1+1/n^2)
=(lim_(n->oo)1)/(lim_(n->oo)sqrt(1+1/n^2))
=1/sqrt(0+1)
=1
The limit of the series is not zero. Therefore, by the nth-Term Test forDivergence, the series diverges.

Why is Tokchae a prisoner? What reasons does he give for not leaving his father?

The first sentence of the story gives readers important setting details:

The northern village at the border of the Thirty-eighth Parallel was snugly settled under the high, bright autumn sky.

The 38th Parallel is an important location because it means that the story is taking place at the dividing line between North Korea and South Korea. During the war, villages located on or near this line changed hands several times as each side pushed their front line deeper into enemy territory. Tok-chae and Song-sam grew up as boyhood friends in this particular village, but the war caused them to end up on opposite sides of the conflict. Tok-chae has been captured because he's the "enemy," and Song-sam is the officer in charge of taking his childhood friend to the prisoner camp.
As Song-sam is escorting his friend, Song-sam has a flash of anger because Tok-chae wound up fighting for the other side:

Song-sam felt a sudden surge of anger in spite of himself and shouted, “So how many have you killed?”

Song-sam then asks why Tok-chae stayed behind to fight instead of fleeing like he did. Tok-chae says that he wanted to, but his father wouldn't leave. His father saw no point in running away and didn't know what he would do in another place. Tok-chae simply couldn't bring himself to leave his father. Tok-chae says,

I wanted to be with him in his last moments so that I could close his eyes with my own hand.

In Rabindranath Tagore's "The Lost Jewels," what is the character sketch of Mani based on the statement "Mani did not understand Bhusan, it is true..."?

In the story, Mani is the pampered wife of a wealthy man, Bhusan. Although Mani lives a privileged life, she is vaguely dissatisfied with her marriage. For his part, Bhusan's life of ease seems to have robbed him of an intrinsic and vital part of his manhood.
Mani comes to take Bhusan for granted. She accepts caresses and priceless jewels from her husband with undisguised contempt and little gratitude. Mani does not understand Bhusan; she finds it difficult to relate to his atypically tame nature. There is little sexual attraction between the two.
According to the narrator, Mani's female nature yearns for the novelty of a masculine temperament untouched by modern civilization. He cites Bhusan's relinquishment of his "barbaric nature" as the main reason for Mani's apathy.

"The wife of a man who is, of his own accord, submissive is altogether out of employment. All those weapons which she has inherited from her grandmothers of untold centuries are useless in her hands: the force of her tears, the fire of her anger, and the snare of her glances lie idle."

Mani essentially rejects her husband and creates a separate, dynamic existence for herself within her marriage. She spurns her social obligations and neglects her religious duties:

"Bhusan's wife did not talk very much, nor did she mix much with her neighbors. To feed Brahmans in obedience to a sacred vow, or to give a few pices to a religious mendicant, was not her way."

(Note that the pice is an obsolete Indian denomination from the days of colonial India).
Instead, Mani puts all of her energy into building wealth; since she is "always working and saving," she is never "sick nor sorry." Mani's industry sustains her and preserves her enthusiasm for life. Essentially, she invests in her own happiness and ceases to rely on Bhusan for emotional fulfillment. When Bhusan begins to experience business difficulties, Mani acts to protect her own wealth. She retains Modhu's counsel.
The text tells us that, while Mani may not understand Bhusan, she is familiar with the sort of principles that actuate a man like Modhu. She knows that Modhu is compelled only by his own self-interest. With Modhu's help, Mani makes preparations to transfer all of her jewels to her father's house. Now, here's an interesting quote that explains why the lack of rapport between Mani and Bhusan eventually leads Mani to betray her husband.

"When he ought to have been angry, Bhusan was only distressed. Man is the rod of God's justice, to him has been entrusted the thunderbolt of the divine wrath, and if at wrong done to himself or another it does not at once break out into fury, then it is a shame. God has so arranged it that man, for the most trifling reason, will burst forth in anger like a forest fire, and woman will burst into tears like a rain-cloud for no reason at all. But the cycle seems to have changed, and this appears no longer to hold good."

Basically, Mani finds it difficult to "understand" Bhusan because she is operating from a different set of standards regarding gender roles:

"Bhusan, who ought to have been born five or six centuries hence, when the world will be moved by psychic forces, was unfortunate enough not only to be born in the nineteenth century, but also to marry a woman who belonged to that primitive age which persists through all time."

Mani's feminine nature yearns to unite with a primeval masculine nature. She can only understand what she innately believes about gender roles. Because Bhusan's mild nature doesn't fit into her personal narrative about masculinity, Mani rejects her husband. In the story, the narrator tells us that Mani is synonymous with Nitya Kali, a goddess who represents endless time. Basically, the author hints that Mani's lack of understanding stems from her refusal to accept anything other than a primal interpretation of gender roles.

"Each looked for an easier triumph, and a result less fundamental and astounding. Both read the same Bible and pray to the same God, and each invokes His aid against the other. It may seem strange that any men should dare to ask a just God's assistance in wringing their bread from the sweat of other men's faces, but let us judge not, that we be not judged. The prayers of both could not be answered. That of neither has been answered fully. The Almighty has His own purposes." What theme is presented in this passage from President Lincoln’s second inaugural address?

In his second inaugural address Lincoln is referring to the soon to be concluded Civil War and its combatants. Both sides have prayed to the same God; both sides have prayed for an easy victory. Yet neither side has had its prayers answered fully. As Lincoln sees it, this is only right and proper, for God moves in mysterious ways to perform his wonders. Although it was perfectly natural that people on both sides of the conflict desperately wanted a speedy conclusion to the war, ultimately its continuation was God's will. Lincoln wants to remind his audience of how everything is ultimately part of a divine plan, much bigger than any single one of us. Having had our hopes and prayers dashed for a swift cessation of hostilities, all that we can now do is to recognize that God has his purposes and that we must humbly yield to them.
https://www.theatlantic.com/past/docs/issues/99sep/9909lincaddress.htm

What were the problems faced by Helen Keller?

When Helen Keller was young, she was struck with an illness that made her blind and deaf. As a result, she found it difficult to communicate anything other than her most basic wants. To solve this problem, Keller learned from her teacher, Anne Sullivan, how to express herself by writing signs on another's palm. This process, including learning the idioms and expressions involved in everyday conversation, took many years and a great deal of practice. She also learned to speak; she had spoken before her illness as a young child but had to learn again to use her voice.
She also had to prepare to enter Radcliffe, the women's school associated with Harvard. The problem was that Anne Sullivan could not sign all the words in the books Helen Keller had to read, and the books had to be embossed in Braille, which took some time. Her studies presented obstacles, as she had to use a Braille writer at times, and she had to draw mathematical figures on a cushion with wires because she couldn't see the figures on the board. It was also difficult for her to understand the Braille symbols for numbers and figures. She eventually overcame these obstacles and was admitted to Radcliffe.

Helen Keller faced many problems, which she recounts in her autobiography, The Story of My Life. All these problems center around being blind and deaf, which occurred when Keller was a very young child.
Initially, Helen had no language with which she could communicate to her family. She learned to talk a little before going blind and deaf due to illness, but she could no longer hear her family. This meant her behavior was often challenging. She threw temper tantrums, and her family members gave into her desires in order to avoid these. 
Annie Sullivan, her teacher, tamed the temper tantrums by not giving into them, and Sullivan worked diligently with Helen on developing language through finger spelling. Ultimately, Helen made that connection at the well, with water running over her hands. After her acquisition of language, Helen's behavior was significantly better because she was able to communicate. 
Even though Helen could communicate, she did have other challenges in her life. There was the challenge of learning about the world when she was missing two senses. Annie Sullivan had her experience as many things as possible. 
There were educational challenges. For example, Helen wanted to learn to speak. She received help from a speech specialist and did learn to use her voice. 
Helen ultimately went to Radcliffe, the sister college of Harvard. Annie Sullivan was one of the people who helped Helen "read" her text books by spelling the words into her hand. After multiple operations, though, Annie's eyes were weak, and she struggled to help Helen.
Helen Keller spent her entire life overcoming challenges and also reaching out to people and inspiring them. 

Are Dylan and Mingus the same sex? Did Dylan have romantic drives for Mingus? Migus is black, correct?

Dylan and Mingus are indeed both the same sex in this young adult novel. Both are boys. Dylan, a white boy living in a largely black neighborhood, and Mingus, a black boy, keep their close friendship relatively quiet. Racial tensions run high in the area, and the book connects these themes, along with the secrecy enforced upon Dylan and Mingus by the racial situation, to the thread of homoeroticism that runs through the novel.
Dylan is fascinated by Mingus, and, in many ways, it is aspects of his blackness, unusual in Dylan's eyes, which capture Dylan's attention. For example, he wants to touch Mingus's hair, because he is intrigued by how it might feel. At the end of the novel, Dylan refers to Mingus as his "lover" as well as his "best friend." However, it is not clear in the story whether he truly has "romantic" inclinations toward Mingus—at least, not in the modern sense of the word. Rather, the novel explores the idea of an old-fashioned "romantic friendship" in a modern context, centering on the ways in which close friendships between teenagers can often blur into something more intense and ambiguous.

What type of characters are George Wilson and Mr. Gatz in the novel The Great Gatsby? (Stock, Dynamic..etc)

In The Great Gatsby, George Wilson is a minor character who acts as a foil--a character who has character traits that are the opposite of another character. Mr. Gatz is a flat character, a personage who has only one or two personality traits and does not change throughout the narrative.
George Wilson is a product of the wasteland of the Valley of Ashes, just as Tom Buchanan is a product of the rich world of East Egg and New York City. George is a man that Nick describes as

...a blonde, spiritless man, anemic, and faintly handsome. When he saw us … hope sprang into his light blue eyes” (Ch.2)

and he describes Tom Buchanan in a different manner:

...a sturdy, straw-haired man of thirty with a rather hard mouth and a supercilious manner. Two shining, arrogant eyes had...dominance over his face....His speaking voice, a gruff husky tenor...[has] a touch of paternal contempt in it. (Ch.1)

George loves his wife Myrtle deeply; when she is killed, he is deeply changed emotionally--"deranged by grief." In fact, George grows deeply depressed. Reacting differently, Tom is angry that someone could run Myrtle down in such a cruel manner. But, while George tries to avenge Myrtle's death, Tom easily discards the memory of Myrtle and, instead, helps Daisy escape her culpability for her crime by giving Wilson Gatsby's name so he thinks that Jay Gatsby ran over his wife. Certainly, Tom and Daisy are rightfully called "careless people" by Nick. 
As a foil to Tom Buchanan, George Wilson represents the lower classes trapped in their poverty, while Tom represents the amoral upper class, concerned only with their own pleasure.______________________________________________________
Mr. Gatz, Jay Gatsby's father, is a flat character, a personage who remains the same throughout the narrative, but he does serve a purpose. He arrives at Gatsby's house in order to bury his son. Mr. Gatz is extremely proud of his son, Jimmy Gatz, as he calls Gatsby. He walks proudly up and down the hallways, impressed with the size and opulence of the house. He pulls out pictures, proudly displaying them, saying,

"He knew he had a big future in front of him. And ever since he made a success he was very generous with me."

Mr. Gatz also displays schedules that Jimmy wrote out years ago as a demonstration of how his son was destined to succeed. Both he and Nick attend Gatsby's funeral. There Nick looks anxiously for other cars, as does Mr. Gatz, whose appearance in the narrative reveals the falseness of Gatsby's life.

College Algebra, Chapter 2, 2.1, Section 2.1, Problem 50

Find the lengths of the medians of the triangle with vertices $A(1,0), B(3,6)$ and $C(8,2)$
Recall that a median is a line segment from a vertext to the midpoint of the opposite side, so.



First, we must get the midpoint on every sides of the triangle.

$
\begin{equation}
\begin{aligned}
M_{AC} &= \frac{1+8}{2} \text{ and } \frac{0+2}{2} = \left( \frac{9}{2},1 \right)\\
\\
M_{AB} &= \frac{1+3}{2} \text{ and } \frac{0+6}{2} = (2,3)\\
\\
M_{BC} &= \frac{3+8}{2} \text{ and } \frac{6+2}{2} = \left( \frac{11}{2},4 \right)
\end{aligned}
\end{equation}
$


Then, the medians can be computed by using the distance formula from a vertex to the midpoint of the opposite side. So median $b$ can be determined by setting the distance of vertex $B$ to midpoint $AC$

$
\begin{equation}
\begin{aligned}
b &= \sqrt{(6-1)^2 + \left( 3 - \frac{9}{2} \right)^2}\\
\\
&= \sqrt{5^2 + \left( - \frac{3}{2} \right)^2}\\
\\
&= \sqrt{25+ \frac{9}{4}}\\
\\
&= \sqrt{\frac{109}{4}}\\
\\
&= \sqrt{\frac{109}{2}} \text{ units}
\end{aligned}
\end{equation}
$

College Algebra, Chapter 4, 4.6, Section 4.6, Problem 46

Find the intercepts and asymptotes of the rational function $\displaystyle r(x) = \frac{x - 2}{(x + 1)^2}$ and then sketch its graph.

The $x$-intercepts are the zeros of the numerator $x = 2$.

To find the $y$-intercept, we set $x = 0$ then

$\displaystyle r(0) = \frac{0 - 2}{(0 + 1)^2} = \frac{-2}{1} = -2$

the $y$-intercept is $2$.

The vertical asymptotes occur where the denominator is , that is, where the function is undefined. Hence the line $x = -1$ is the vertical asymptote.

We need to know whether $y \to \infty$ or $y \to - \infty$ on each side of each vertical asymptote. We use test values to determine the sign of $y$ for $x$- values near the vertical asymptotes. For instance, as $x \to -1^+$, we use a test value close to and to the right of $-1$ (say $x = -0.9$) to check whether $y$ is positive or negative to the right of $x = -1$.

$\displaystyle y = \frac{(-0.9) - 2}{(-0.9 + 1)^2}$ whose sign is $\displaystyle \frac{(-)}{(+)}$ (negative)

So $y \to - \infty$ as $x \to -1^+$. On the other hand, as $x \to -1^-$, we use a test value close to and to left of $-1$ (say $x = -1.1$), to obtain

$\displaystyle y = \frac{(-1.1) -2}{(-1.1 + 1)^2}$ whose sign is $\displaystyle \frac{(-)}{(+)}$ (negative)

So $y \to - \infty$ as $x \to -1^-$.

Horizontal Asymptote. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is the line $y = 0$.

Graph. Use the information we have found together with some additional values in the table to sketch the graph of the function.

$\begin{array}{|c|c|}
\hline\\
x & r(x) \\
\hline\\
-6 & -0.32 \\
-4 & -0.666 \\
-2 & -4 \\
2 & 0 \\
4 & 0.08 \\
6 & 0.0816\\
\hline
\end{array} $

Are there campaign finance rules that could avoid the hydraulic "money-shift" tendencies that go along with campaign finance regulation?Can independent expenditures that support or criticize a candidate be limited without violating free-speech rights?Should campaign finance regulations be considered useless and ended altogether, in favor of full disclosure of who is giving money to whom, together with unlimited campaign spending?Are activities of PACs and 527 committees healthy or destructive to democratic processes?

After the Bipartisan Campaign Reform Act of 2002 (also known as McCain-Feingold) was passed, which banned soft money donations to political parties, many critics felt that it only succeeded in shifting money through what is often figuratively called a hydraulic process to other cracks. According to Raymond J. La Raja and Brian F. Schaffner, authors of Campaign Finance and Political Polarization: When Purists Prevail (please see the link below), there is no way to stop this flow of money. The authors state,


When well-intentioned reformers pass laws that limit contributions, the amount of money in politics does not change significantly, but its flow migrates. In other words, the rules do not necessarily create effective 'dams' that block money from entering politics, but instead expand or carve new 'canals' that channel its flow in other directions."

Therefore, experts often feel there is no way to stop the hydraulic "money-shift" tendencies that accompany campaign finance rules. 
As of now, individual contributions to a candidate can be limited under Federal Election Commission (FEC) rules. For example, an individual can only give $2,700 to a federal candidate per election. Several cases have put the question of whether individual campaign contributions are protected by the First Amendment before the Supreme Court. In 2010, the court ruled in Citizens United v. FEC that nonprofit corporations should not be subject to individual campaign contribution limits. These same rights also apply to for-profit corporations, unions, and other types of associations. 
As it is difficult to stop the flow of money into campaigns, many people believe campaign finance rules should be dropped entirely. In addition, many people believe that it is a First Amendment right to be able to donate as much money as one wants to a campaign or issue. I believe full disclosure of who is giving how much to which campaign would end a lot of corruption in campaigns. For example, the blog Open Secrets, in partnership with the Boston Globe's Spotlight Team, recently found out that Boston lawyers gave thousands of dollars to a Montana Senator and then received "bonuses" for these amounts from their law firm (see the story in the link below). Cases like this show rules that make it mandatory to reveal who is contributing to what (and where the money really comes from) would make the political process more transparent. 
PACs and 527s have also been criticized for being destructive to the democratic process. PACs, or political action committees, can channel money to political candidates. While corporations can't contribute directly to PACs, they can sponsor a PAC and ask for employee donations. 527s can receive unlimited donations from individuals and corporations and can run political ads. They are also tax exempt. An example of a 527 was Swift Boat Veterans for Truth, which was against Kerry's Presidential campaign in 2004. These entities still allow a lot of corporate and individual money to be channeled into political campaigns, and it's not always clear where the money is really coming from.
https://www.fec.gov/introduction-campaign-finance/understanding-ways-support-federal-candidates/

https://www.opensecrets.org/news/2016/10/millions-in-political-donations-fueled-by-matching-bonuses-at-boston-law-firm/

please explain in details

Precalculus, Chapter 9, 9.4, Section 9.4, Problem 28

You need to use mathematical induction to prove the inequality, hence, you need to perform the following two steps, such that:
Step 1: Basis: Prove that the statement holds for n = 1
If xStep 2: Inductive step: Show that if P(k) holds, then also P(k + 1) holds.
P(k): (x/y)^(k+1) <= (x/y)^k holds
P(k+1): (x/y)^(k+2) <= (x/y)^(k+1)
(x/y)^(k+1)*(x/y) <= (x/y)^(k+1) => x/y <= 1 => x <= y true
Hence, since both the basis and the inductive step hold, the statement P(n): (x/y)^(n+1) <= (x/y)^n holds for all indicated values of n.

How do historians know that the Harappans grew wheat and barley?

Clues to the type of food grown or caught in any ancient society are usually found through a combination of historical study, logical estimation, and archeological evidence.
If a civilization has permanent structures or advanced housing and city buildings, a historian will begin with the assumption that this culture was agrarian or lived off crops that could be grown and animals that could be raised (as opposed to nomadic hunter-gatherer societies that would need to be continually on the move in order to find their food source and hunt it down). Permanent buildings signify a permanent food source that can be controlled (e.g. crops).
Next, by identifying dwelling places, granaries, and other store houses, or even the local dumping grounds, archaeologists can identify what was part of daily life and local diet. They may look for the bones of animals or fruit pits—anything that will last over the centuries. In the case of the Harappan people or the civilizations of the Indus valley, archaeologists found wheat remnants in the store houses used to process the grain for further nutritional use (such as flour). For further information, archaeologists will also examine the bones and teeth of any human remains to understand more about the diet of the local people.
Here you’ll find some pictures of Mahenjo Daro, an excavation site responsible for much of the information we know about the Harappan Civilization.
https://www.bbc.co.uk/bitesize/topics/zxn3r82/articles/zghy34j

https://www.harappa.com/

Calculus of a Single Variable, Chapter 3, 3.3, Section 3.3, Problem 25

Given: f(x)=(x^5-5x)/5=(1/5)x^5-x
Find the critical x values by setting the derivative equal to zero and solving for the x value(s).
f'(x)=x^4-1=0
(x^2+1)(x^2-1)=0
x=1, x=-1
The critical values for x are x=1 and x=-1.
If f'(x)>0 the function is increasing over an interval.
If f'(x)<0 the function is increasing over an interval.
Choose an x value less than -1.
f'(-2)=15 Since f'(-2)>0 the function is increasing in the interval (-oo, -1).
Choose an x value between -1 and 1.
f'(0)=-1 Since f'(0)<0 the function is decreasing in the interval (-1, 1).
Choose an x value greater than 1.
f'(2)=15 Since f'(2)>0 the function is increasing in the interval (1, oo).
Because the function changed direction from increasing to decreasing a relative maximum will exist at x=-1. The relative maximum occurs at the point
(-1, 4/5).
Because the function changed direction from decreasing to increasing a relative minimum will exist at x=1. The relative minimum occurs at the point (1, -4/5).

Single Variable Calculus, Chapter 5, 5.5, Section 5.5, Problem 42

Find the definite integral $\displaystyle \int^{\frac{\pi}{2}}_{\frac{\pi}{2}} \frac{x^2 \sin x}{1 + x^6} dx$

Let the function $\displaystyle f(x) = \frac{x^2 \sin x}{1 + x^6}$. We have that $\displaystyle f(-x) = \frac{(-x)^2 \sin (-x)}{1 + (-x)^6} = \frac{x^2 (- \sin x)}{1 + x^6} = - \frac{x^2 \sin x}{1 + x^6} = -f(x)$

Hence $f$ is an odd function. Thus, it is symmetric to the origin, we can say that the difference of the area below the curve from $\displaystyle \left[ \frac{- \pi}{2}, 0 \right]$ and area below the curve from $\displaystyle \left[ 0, \frac{\pi}{2} \right]$ is zero. Therefore,

$\displaystyle \int^{\frac{\pi}{2}}_{\frac{- \pi}{2}} \frac{x^2 \sin x}{1 + x^6} dx = 0$

Write a short note about James Joyce.

I'm not quite sure what kind of note about Joyce you're specifically looking for but, since your question is filed under his short story "Eveline," I'll introduce Joyce as a writer within the context of that story.
An Irishman born in Dublin in 1882, Joyce was the oldest son of an initially financially stable family that gradually descended into poverty. His experience living in the squalor of nineteenth and early twentieth-century Dublin was to inform all of Joyce's work, and though he spent most of his life abroad in Paris, Trieste, and Zurich, Joyce based all of his writing in Dublin and is famous for depicting the struggles of "paralyzed" Dubliners (Joyce famously stated that one of the leading themes of Dubliners was paralysis, as it mirrored the frustrated stagnation of the people living in Ireland's major city). The themes of paralysis and poverty are of central importance in "Eveline," as Eveline is trapped in a stagnant job and a unsupportive family life (her father routinely takes her wages and spends them on alcohol). Forced to care for her family in the wake of her mother's death, Eveline is thus trapped or paralyzed, even though she is still young and ostensibly has her whole life ahead of her. It's especially interesting that, given the opportunity to leave Ireland and her dysfunctional family behind her, Eveline decides to stay. Her reasons for doing so are ambiguous but involve a promise of looking after home that she made to her late mother. In the end, Eveline is condemned to her life of stasis, and her potential for growth and renewal seems to be snuffed out even though she is young. A note of melancholy therefore pervades this short story, a melancholy born from an inability to act or pursue one's desires.