Single Variable Calculus, Chapter 2, 2.3, Section 2.3, Problem 46

Given the function
$
f(x) = \left\{
\begin{array}{c}
4 - x^2 & \text{ if } & x \leq 2\\
x - 1 & \text{ if } & x > 2
\end{array}
\right.
$


a.) Find $\lim\limits_{x \to 2^-} f(x)$ and $\lim\limits_{x \to 2^+} f(x)$


For left hand limit

$
\begin{equation}
\begin{aligned}
\lim\limits_{x \to 2^-} f(x) & = \lim\limits_{x \to 2^-} (4-x^2) \\
\lim\limits_{x \to 2^-}(4-x^2) & = \lim\limits_{x \to 2^-} 4 - \lim\limits_{x \to 2^-} x^2\\
\lim\limits_{x \to 2^-} (4-x^2) & = 4 - (-2)^2
\end{aligned}
\end{equation}\\
\boxed{\lim\limits_{x \to 2^-} (4-x^2)= 0}
$


For right hand limit


$
\begin{equation}
\begin{aligned}
\lim\limits_{x \to 2^+} f(x) = & \lim\limits_{x \to 2^+} (x-1)\\
\lim\limits_{x \to 2^+} f(x) = & \lim\limits_{x \to 2^+} (x-1)\\
\lim\limits_{x \to 2^+} (x-1) =& 2 - 1\\
\end{aligned}
\end{equation}\\
\boxed{\lim\limits_{x \to 2^+}(x-1) = 1}
$

b.) Does $\lim\limits_{x \to 2} f(x)$ exists?
No, it does not exist because the left and right hand limits are different.

c.)Sketch the graph of $f$

What unusual technological devices are there in "The Veldt"?

Most of the technological innovations in "The Veldt" sound great at first. But once you realize that they're replacing human relationships and actually reducing adults to the level of helpless children, then they don't seem like such a good idea all of a sudden. A kitchen table that actually cuts your bread for you sounds like the ultimate labor-saving device, the kind of thing you'd see on an infomercial. But the problem is that, if something goes wrong with the technology—and technology always breaks down at some point—then you're in big trouble. It's no exaggeration to say that if all the appliances in their super-deluxe kitchen that do everything for them end up on the fritz, then the Hadleys could practically starve to death.
In some ways, then, the children's nursery, with its ultra-realistic lions, is simply the ultimate example of the many dangers inherent in all the various technological devices arrayed around this bright, shiny Happylife Home. Even if the Hadleys had not ended up as breakfast for the ferocious beasts of the veldt, it's likely that, sooner or later, they would've met their unfortunate ends at the robotic hands of one of their labor-saving devices. If the bread-slicing table had gone haywire, for example, the consequences simply don't bear thinking about.

Ray Bradbury's "The Veldt" is centered around technological advancement and wild inventions. The story's two most obvious unusual technological devices would be the nursery and the HappyLife Home.
The nursery can turn itself into any setting the children can imagine. The nursery can recreate the sounds, smells, and sights of whatever things the children choose to imagine. As George and Lydia note many times throughout the story, the nursery and its scenes feel very real. Unfortunately, the nursery can also be dangerous and eventually takes over as the parent of the children. 
The HappyLife Home is composed of a variety of strange - and very cool - technological devices. Inside the HappyLife Home is a table that makes food for the homeowners, a machine that ties people's shoes, machines for cleaning the house, machines for giving people baths, etc. The HappyLife Home ensures that homeowners never have to lift a finger or complete a domestic task without assistance. The HappyLife Home sounds great in theory, but in actuality it makes the homeowners feel useless and unneeded. 

How is Sodapop shown to be smart and athletic?

Very early in the book, Ponyboy introduces himself and the other Greasers to readers. When he gets to Sodapop, Ponyboy makes it clear that school is not necessarily Soda's forte. However, Ponyboy is very clear in saying that Sodapop is not an idiot. He tells readers that Sodapop understands everything.

Soda tries to understand, at least, which is more than Darry does. But then, Soda is different from anybody; he understands everything, almost.

As readers continue reading the book, we learn that Soda really understands people and relationships. While he may never score at the top of the charts on the SAT, he is incredibly intelligent when it comes to reading people. Ponyboy gives readers another example of how Soda is intelligent about a page later.

So Soda and I stay out of trouble as much as we can, and we're careful not to get caught when we can't.

Ponyboy admits that Greasers have trouble with the law every once in awhile. Then, he tells readers that Soda is careful enough (smart enough) to not get caught. This means that when Soda is going to break the law, he does so in a way that he can get away with it. That shows intelligence. Perhaps it is directed in a less than worthy direction, but it shows his intelligence.
As for athleticism, it is Darry that gets the compliments. Ponyboy is very specific that Darry is a wonderful athlete, but we are given hints that Soda is a coordinated athlete in his own right. He might not play sports on a team, but he is still a talented athlete capable of incredible feats of strength and agility.

"Yeah!" screamed Soda as he too did a flying somersault off the steps. He flipped up to walk on his hands and then did a no-hands cartwheel across the yard to beat Darry's performance.

Sodapop is Ponyboy's older brother, and he is described as being extremely handsome and funny. Even though Sodapop does not excel in the classroom (he ends up dropping out of school), there are several scenes throughout the novel that depict his intelligence. Sodapop demonstrates his intelligence through his ability to understand and relate to both Ponyboy and Darry. Sodapop acts as a mediator between his two brothers and continually resolves conflicts between Pony and Darry. Sodapop is sensitive enough to sympathize with Pony's situation but also mature enough to appreciate Darry's sacrifice. Toward the end of the novel, Sodapop attempts to repair his brothers' strained relationship by lamenting about their constant fighting and reminding them that they are siblings.
Sodapop demonstrates his athletic ability before and during the rumble against the Socs. Before the rumble, Pony describes how Sodapop flips in midair down the street and walks on his hands. Sodapop is also portrayed as a good fighter and holds his own during the rumble.

Would a Kantian think that having unpaid internships as part of students’ courses of study is morally permissible? Why or why not?

Kant argued that, in acting morally, we should always treat other people as ends in themselves, not as a means to an end. This means, among other things, that I should value other people for who and what they are, not for what they can do for me. That being the case, it seems likely that Kant would be opposed to unpaid internships, because one could argue that they treat interns simply as a resource of cheap labor, as an economic resource to be exploited for profit.
We should also bear in mind that another key element of Kant's moral philosophy is the so-called Golden Rule, which in the Christian religion is expressed as "Do unto others as you would have them do unto you." Kant argues that moral actions must be capable of being universalized—that is, applied equally to everyone—in all circumstances and under all conditions. It's difficult to see how unpaid internships could possibly fulfill such an exacting criterion. If the exploitative principle behind unpaid internships were applied universally, then the world would be in even worse shape than it currently is. In such a world, it would rapidly become the norm for people to be treated as means rather than ends in themselves. And for Kant, this would be morally unacceptable.

If we look at Kant's Critique of Practical Reason (not his Critique of Pure Reason), then yes, under certain conditions, the practice of unpaid internships would be ethical or morally permissible.
Kant's ethics are based on universal applicability. In other words, the rules must apply to all people (or all the people impacted by the ethical rule), and the moral rule has to be sensible (not universal suicide, let's say).
An unpaid internship could be easily argued as a sensible way for a student to gain real-world experience.
Unpaid internships could also be made universal: this would be an important criterion for considering them morally permissible. It would not be fair, according to Kant, if some students were able to earn money and college credit for internships and others were deprived of that opportunity. At that point, unpaid internships would become highly problematic in the Kantian system.

One could make arguments based on a Kantian deontological system of ethics either for or against internships, but the strongest are probably against internships due to their narrowly vocational nature. The main issue you will need to address for writing your paper about this is that Kant's system of ethics is rule-based rather than consequentialist. In other words, Kant would not, as this excerpt does, look at the outcomes of internships, but rather look at whether their existence was compatible with an ethical system based on rules and duty. 
Kant would begin by looking at the ultimate end of internships within the context of education. A good starting point would be to look at Kant's book on education. Kant believes that the central problem of education is how to develop moral character in a child without constraining moral freedom. Equally important is developing habits of discipline and especially self-discipline. Students, for Kant, should be educated with a view to the progress of humanity in the future, not for simply immediate or proximate goals. Rather than pandering to the desires of students and parents for narrow job training or easy and palatable edutainment, educators, according to Kant, should consider what is best for society as a whole. Thus for Kant, the sort of vocational training implied in internships (of any sort, paid or unpaid) distracts from the main purpose of education as moral and cultural formation to act as narrow vocational training; this would would be problematic. 
You could examine this in light of his notion of universalizing one's moral maxims. If everyone simply trained for immediate vocational goals, our civilization would have no people capable of complex political and ethical thought, and would lack artists, scientists, social activists, and others whose education needs to be broader and more global rather than narrower and more focused on specific jobs (which may well become obsolete in a few years due to technological change anyway). Education should be done by " enlightened experts" rather than outsourced to for-profit interests according to Kant. 

Why did the British issue the Proclamation of 1763?

When the French and Indian War came to an end, the British Empire inherited a great swath of land in the Ohio River Valley. While control over this large territory was one of the leading causes of the war, finally acquiring it did not come without its problems. There were still scattered pockets of French settlers in the region. There were also larger numbers of Native Americans there who had been allies of the French. Even though the European powers had ceased their armed struggle, several Native American groups continued the fight against the British in what was known as Pontiac's Rebellion.
The British were fearful that if large numbers of settlers crossed into the Ohio River Valley, this would prolong and expand the fighting with the Native Americans and possibly rekindle the fight with the French settlers there. The war had been costly, both financially and politically, for the English. Parliament and the king wanted their fragile peace to take root. They considered the movement of English settlers into this newly acquired land to be a possible destabilizing element that could erase the gains they had secured in the Treaty of Paris. That is why King George issued the Royal Proclamation of 1763, which was intended to stop the expansion of American settlers from moving west of the Appalachian Mountains.
Of course, history has shown that the wishes and orders of the British powers were often ignored by their colonists. Even before the Proclamation of 1763 was made, many settlers had already laid claim to lands west of the Appalachians. Many felt that these lands had been hard-won during the war and did not feel that it was right that they be denied to them. Indeed, it was impossible for the thinly-stretched British authorities to enforce the Proclamation, and it was ultimately worth little more than the parchment it was written on.
https://history.state.gov/milestones/1750-1775/proclamation-line-1763

Following the Seven Years War, known in North America as the French and Indian War, the British acquired a large amount of territory from France. Much of the newly gained land happened to be west of the Appalachian Mountains and included farmland on which American colonists hoped to settle. There was, however, a problem with this situation. The problem was the fact that the land was inhabited by Native Americans, many of whom had enjoyed relatively positive relations with the French and had assisted the French in their fight against the British.
In 1763, the British would issue the Proclamation of 1763, which forbade their colonists in the Americas from moving west of the Appalachian Mountains. The British did this because they feared that settlers moving west of the Appalachian Mountains would be encroaching upon lands of the Native Americans, which would further upset them. The British were afraid that the Native Americans would launch attacks against the settlers and thus force them into another costly conflict.
Although the British aimed to prevent conflict through the Proclamation of 1763, it upset many American colonists who had assisted the British in the French and Indian War in order to hopefully gain access to the land west of the Appalachian Mountains. Some colonists disobeyed the order and moved west of the Appalachian Mountains anyway. There were also a number of colonists who were found living west of the Appalachian Mountains and evicted. The Proclamation of 1763 ultimately hurt relations between the British and their American colonists and can be cited as a contributing factor to the growing desire for American independence.

The Proclamation of 1763 was issued at the end of the French and Indian War.  There is some debate as to exactly why King George III and Britain may have issued the proclamation, but the main reason for the proclamation was that it was supposed to protect both the colonists and the Native Americans.  The proclamation itself forbade citizens or colonial governments from buying land or making agreements with the Native Americans.  This effectively shut down westward expansion of the colonies.  By shutting down the westward movement of colonists, Britain hoped to appease the Native Americans who felt threatened by rapid westward colonization.  The thinking was that the proclamation would also protect the colonists from angry Native Americans and their retaliatory raids.  The thinking worked the other way too.  If colonists were not expanding west, the colonists weren't thinking of attacking the Native Americans.  Many settlers defied the proclamation in order to secure more and better farmland, and the proclamation came to an end with American independence.  

Could you give me a detailed summary of Scott Peck's thoughts on grace (section 4) from his book The Road Less Traveled?

To understand the concept of grace as described by M. Scott Peck in his book The Road Less Traveled, it is helpful to keep in mind the ideas from which the book sets out. Its opening words are "life is difficult." Peck asserts that life represents a series of problems, and we need to summon the discipline to solve them—that is, the discipline to face the truth even when it is painful. Another word for this kind of inner strength might be "love." If we remain open to this love and use it to confront life's problems instead of avoiding them, then mental health and spiritual growth are available to us.
In the fourth and last section of The Road Less Traveled, Peck describes certain events that appear miraculous. Some people recover fully and quickly from illnesses called terminal or heal from debilitating mental disorders. Dreams and the unconscious mind sometimes deliver stunning flashes of insight, indicating that we are wiser than we think we are. Astounding coincidences and improbable, serendipitous events actually occur much more often than you would think. These types of phenomena, in Peck's view, are all related. Events that we call miracles, that foster health and growth and that take place with no discernible cause appear to be motivated by a force outside human consciousness. Such a force has never been identified by scientific research, but it has been recognized throughout history by religious and spiritual traditions. This force is called, among other names, grace—"amazing grace"—as in the Christian hymn. Peck indicates that "God's love" could be another way of referring to the same notion.
Our desire to locate the specific origins of this kind of benevolent force or explain the mechanics of its operation are a large part of the reason why we are likely to misperceive it or miss it completely. On this score, the Eastern religious traditions come closer to the mark, for they believe in the essential oneness of all phenomena, both material and spiritual. These traditions do not subscribe to a discrete notion of the self. Therefore, these forces outside the dimensions of the individual human body and the conscious mind are also in us. Grace is in us; it is a matter of welcoming its presence, rather than resisting it, with the help of the form of self-discipline Peck defines, at the outset of The Road Less Traveled, as love.

What are the meanings of the epigraphs in The Secret Life of Bees?

Each of the epigraphs offers a comment about "the secret life of bees" that also serves as a focusing lens for the chapter it precedes. By reflecting on the human implication of the epigraph concerning bees, the reader can find a deeper significance in the plot unfolding.
The first epigraph addresses "queenlessness," and Lily is clearly in need of a communal "hive" so that she can grow into her potential. As she learns from the Boatwright sisters how to care for bees, she also learns how to care for herself and others. Chapter 2 discusses "scout bees," and Lily and Rosaleen go in search of Tiburon and the Black Mary. When Lily and Zach begin to fall in love and Lily awakens to her maturing sexual self, the chapter is preceded by an epitaph about the sexual behavior of bees.
In chapter 6, we read,

The Queen must produce some substance that attracts the workers and that can be obtained from her only by direct contact. This substance evidently stimulates the normal working behavior in the hive. This chemical messenger has been called "queen substance." Experiments have shown that the bees obtain it directly from the body of the queen.

This connects to the chapter, which concerns a Daughters of Mary service in which participants feel a great release and empowerment when they touch the statue.
We can judge Lily's growth by the last epigraph. Lily has not finished her progress, and she is still trying to reconcile herself to the memory of her mother, but she seems ready to advance into adulthood secure in wisdom she has gained with the Boatwrights. Like the first one, this epigraph discusses a hive whose queen has died. The difference here is that, instead of ending on "queenlessness," this last epigraph suggest hope for a dynamic future when the new queen arrives. One senses that Lily is ready to accept the mantle of being her own queen bee.

Each epigraph in The Secret Life of Bees connects in some way to the chapter it heads. Some of those examples are simple and straightforward, but most are metaphorical. They speak to the motif of bees as an allegory for the life of this little society that we're watching from the outside—much as humans must do when observing bees in their natural habitats.
A strong example of this is the epigraph in chapter four. Very simply, the epigraph describes honeybees and their social colony. It highlights a strong central female figure who acts as figurehead for all other workers. It also mentions that male bees are more or less absent.
This model is very strongly reflected in the Boatwright home that Lily and Rosaleen are about to encounter. August Boatwright is very clearly the queen bee described in the epigraph. She is even called "Mistress of the Bees" by Lily (68). Her sisters demonstrate worker-like qualities, and males are conspicuously missing from the home—at least at the beginning.
The connection between the epigraph and the chapter it precedes is replicated throughout the text.

Observing the present political competition in our country, write about the merits and drawbacks of the democratic form of government.

The merits and drawbacks of democratic government come from the responsibility of the people themselves to choose their leaders, and thereby, sometimes directly, sometimes indirectly, choose the path the nation will follow. 
This election cycle has been particularly interesting because of the controversial nature of the candidates. Donald Trump gives voice to what has been a vocal but outnumbered wing of the Republican party. His views appeal strongly to some and horrify others. In a democracy it is possible for such a candidate to draw enough support to make a serious run at winning office because voters can be unpredictable--sometimes they are moved by momentary political pressures that might not be successful in the long run. If we lived in a less democratic society, a candidate like Trump might have a harder time making a strong run, even with the same sort of support that he has right now. 
The same could be said of Bernie Sanders' campaign. He gave front runner Hilary Clinton a much tougher battle than most expected. Again, this may have been due to temporary political pressures. The recent recession left many voters and their families worse off than they had been previously, which widened the interest in Sanders' liberal ideas of helping middle and lower class families economically. As a result of his strong run, Clinton had to move to the left a bit to weaken his base. 
Since electoral decisions rest with the voters, politicians must find ways to appeal to them. Voters are fickle, and not always very well educated, which means that politicians don't necessarily have to present sound ideas, they just have to present ideas that sound good. Can the voting public discern the difference? How well they do so determines to what extent voters are either the strength or weakness of their democracy. 

What did muckrakers do?

Muckrakers were American journalists in the Progressive Era (the 1890s–1920s) who sought to expose corruption, primarily in politics and business. The name is a reference to Paul Bunyan's Pilgrim's Progress; there is a character within the story referred to as the "The Man with the Muck Rake," who abandoned a chance at salvation in order to turn to filth. Similarly, muckrakers used their investigative skills (rakes) to dig through the polished outer appearance of politicians and big businesses (the muck), thus launching an entire movement of journalism that has yet to slow down.
The most influential people behind the muckraker movement were Lincoln Steffens, Ida M. Tarbell, and Ray Stannard Baker. Lincoln Steffens published what is now considered the first muckraking article, "Tweed Days in Minneapolis," in 1902 regarding the corruption of local politicians. These politicians were working in tandem with big businesses in order to maintain their power and influence the city's treasury.
Ida M. Tarbell's article, "The History of Standard Oil," was published one month after Lincoln Steffen's work. In the article, she revealed unfair business practices behind John Rockefeller's success.
Soon after, Ray Stannard Baker's article, "The Right to Work," exposed the violence that non-striking union members all too often faced when they disagreed with union officials who used their positions in attempts to force agreements.
These three people began the muckraking movement, a movement that is still prevalent in modern journalism. Muckraking is now referred to as "investigative journalism," but its importance in reporting unsavory practices by those in positions of power has not diminished with time.

What is the Theban Saga?

The Theban Saga is a collection of sagas, or legends, that focuses on important events in mythology around the founding of Thebes, a major city in Boeotia which is a region in mainland Greece. It also contains stories about the different kings of Thebes and their families.
The first myth of the Theban Saga tells of the founding of Thebes. Thebes was founded soon after Zeus took the form of a bull and kidnapped Europa, taking her to Crete. There, she gave birth to Minos, whose father was Zeus, and Minos eventually became the first king of Crete. Cadmus, Europa's brother, went to look for her in Greece, and he was instructed by the oracle at Delphi to found a city. In Boeotia, he founded Cadmeia, which was later known as Thebes.
The Theban Saga continues with tales about the descendants of King Cadmus of Thebes. One story talks of King Laius, who was famously murdered by his son, Oedipus, as predicted by the oracle at Delphi. Oedipus did not knowingly kill his father. However, he encountered an old man on the road, and Oedipus killed him in anger after the old man's chariot struck his chariot. Later, when Oedipus marries his mother Jocasta (also without realizing her real identity and also in accordance to the prediction of the oracle), Oedipus becomes king of Thebes. He became the king of Thebes after successfully answering one of the Sphinx's riddles.
Antigone, the daughter and sister of Oedipus, is the subject of Sophocles's tragedy of the same name. Tiresias, the blind prophet from The Odyssey, also plays a role in the Theban legends of Oedipus and Antigone.

Why is the government looking for Harrison Bergeron in Kurt Vonnegut's story "Harrison Bergeron"?

At the beginning of the story, readers learn fourteen-year-old Harrison Bergeron was taken by the Handicapper General men and placed in jail.

And it was in that clammy month that the H-G men took George and Hazel Bergeron's fourteen-year-old son, Harrison, away. 

Later on, breaking news reveals Harrison unexpectedly escaped from jail and is on the loose. Harrison was originally put in jail because he was suspected of plotting to overthrow the government. Now that Harrison has escaped, the government is searching for him because they believe him to be a threat to the government as a whole and to society's current state of "equality." 

"Harrison Bergeron, age fourteen," she said in a grackle squawk, "has just escaped from jail, where he was held on suspicion of plotting to overthrow the government. He is a genius and an athlete, is under-handicapped, and should be regarded as extremely dangerous."

Considering the handicaps Harrison is required to endure, it is clear he is incredibly strong and extremely intelligent, making him a capable candidate to overthrow the government and the Handicapper General. The government fears his escape will allow him to successfully overthrow the government once and for all. Harrison's rebellion and his attempts to bring an end to society's idea of "equality" show he has an ethical view of what it means to be truly equal.

Who are three heroes in The Hobbit? Could they be Bilbo (for being willing to give up his life for a better cause), Gandalf (for saving the group multiple times) and perhaps Bard (for slaying Smaug)?

There are other ways in which Bilbo is a hero, too, aside from the obvious saving of the dwarves. If we look at chapter 5, "Riddles in the Dark," we see how Bilbo's encounter with Gollum adds to his heroic traits.
At the beginning, the hobbit and the creature engage in a battle of wits for Bilbo's life. Gollum has lost the ring; Bilbo has stumbled upon it. And it is the ring that ultimately allows Bilbo to make a last-ditch attempt at a riddle to keep him from being eaten. Gollum, however, doesn't see it as fair.
Arguably, it isn't. Asking about an object in a pocket is more of a guessing game than a riddle, and it angers Gollum. Bilbo ends up using the ring to escape Gollum, who is hastily chasing him. Because the ring is on Bilbo's finger, he eventually comes face to face with Gollum at the exit. Bilbo has the advantage; he has a reason to kill Gollum (for trying to eat him), but he doesn't:

He must get away, out of this horrible darkness, while he had any strength left. He must fight. He must stab the foul thing, put its eyes out, kill it. It meant to kill him. No, not a fair fight. He was invisible now. Gollum had no sword. Gollum had not actually threatened to kill him, or tried to yet. And he was miserable, alone, lost.

Instead, he chooses to leap over the creature, to spare him. That type of consideration and thought, knowing when to kill and when not to, is an important quality for a hero to have.

There are many characters in The Hobbit who act heroically. Bilbo grows to become a hero, despite his fears and anxieties. Gandalf saves the lives of the dwarves and Bilbo on multiple occasions, including when he steps in to protect them from the trolls (Bert, Tom, and William). Bard defeats the dragon with his treasured black arrow, which was passed down from his grandfather.
Another hero in the story is Beorn, the shape shifter. At the end of chapter 17, Bilbo sees the eagles, who are coming to help defeat the goblins, and then gets hit by a stone and falls unconscious. In chapter 18, after Bilbo wakes, he finds that the war is over. He learns that Beorn had arrived after the eagles and that his arrival was instrumental in defeating the enemies:

He came alone, and in bear’s shape; and he seemed to have grown almost to giantsize in his wrath. The roar of his voice was like drums and guns; and he tossed wolves and goblins from his path like straws and feathers.

Beorn arrived full of noise and anger towards the enemies; he fought with passion against the goblins:

He fell upon their rear, and broke like a clap of thunder through the ring. . . . Then Beorn stooped and lifted Thorin, who had fallen pierced with spears, and bore him out of the fray.

Beorn acts heroically when he joins the battle against the goblins. He risks his life to try to defeat wicked creatures. Additionally, he rescues Thorin after he is wounded. His bravery helps to bring the battle to its end.

All three characters you've mentioned are certainly heroes in J.R.R. Tolkien's The Hobbit. Bilbo is a hero for saving the dwarves on multiple occasions, Gandalf is a hero for initially guiding the expedition in the beginning and for averting disaster at the Battle of the Five Armies at the end, and Bard is a hero for slaying Smaug and restoring the city of Dale. In short, your instinct regarding these characters is correct.
However, I'd encourage you to consider adding Thorin Oakenshield to your list of heroes. At first glance, Thorin might seem like an odd choice: he's got a reliably quick temper, tends to doubt Bilbo, and also nearly instigates a devastating battle with the elves and men of Lake-town. That said, Thorin proves his worth in the end. He cares deeply about his kinsmen, and he comes to the rescue during the Battle of the Five Armies. Indeed, this last point is key, as Thorin fights bravely and ultimately dies of his wounds. As such, though he can be an antagonistic character at some points in the story, Thorin certainly proves his heroism by the end of the novel. 

what do pure substances and mixtures have in common

Pure substances are made up of one type of atom or molecule only (resulting in elements or compounds). For example, sodium is a pure substance and is composed of sodium atoms only. Similarly, table salt (or sodium chloride) is composed of NaCl molecules only. In comparison, a mixture is composed of two or more pure substances in a non-fixed ratio. Pure substances cannot be separated out into any other form. Mixtures, on the other hand, can be separated out into individual pure substances by physical processes.
From the above, it is clear that both pure substances and mixtures contain atoms. Furthermore, both of these contain at least one pure substance. Both pure substances and mixtures have properties like weight, color, volume, flammability, toxicity, etc. Similarly, a pure substance and a homogeneous mixture will have the same properties throughout the entire substance or mixture.
Hope this helps.

How does Chaucer use irony in the Canterbury Tales?

There are numerous examples of dramatic irony in The Canterbury Tales. Overall, Chaucer skillfully uses irony as a way of putting distance between what appears on the surface to be the case and what he thinks is really going on. Right from the outset, Chaucer adopts the literary strategy of drawing a clear distinction between Chaucer the (somewhat naive) pilgrim, the narrator of the Tales, and Chaucer the author of those very same Tales, a world-weary, cynical chronicler of human foibles, forever casting a knowing wink at the reader.
The pervading sense of irony is much in evidence in The General Prologue. Here, we are treated to a collection of character sketches of each individual pilgrim that develops Chaucer's taste for irony to its fullest extent. Chaucer, the naive pilgrim narrator has, it appears, many a good word to say about his fellow pilgrims as they gather together at The Tabard Inn. However, upon on closer inspection, we find that things are not all that they seem. The monk, for example, is lauded as "A manly man, to been an abbot able." On a superficial level, this is high praise indeed. However, Chaucer the ironist arguably wants to suggest, ever so gently, that worldliness (rather than piety) is the ideal qualification for becoming an abbot.
This is one of many examples of Chaucer using irony as a means of satirizing the Church. Here we see that there is a serious side to Chaucer's use of irony. Though a devout Christian, Chaucer nonetheless subscribed to a widely-held conviction in the Middle Ages that the Church was irredeemably worldly, hypocritical, and corrupt. Irony in The Canterbury Tales is not simply used for comic effect; it has a clear moral purpose too.
Both comic and moral elements are ably combined in the amply proportioned figure of the Prioress. Chaucer the naive pilgrim gazes admiringly at her flashy jewels, including a "brooch of ful gold shene." Yet, Chaucer the ironist cannot resist insinuating that such ostentatious displays of wealth are rather inappropriate for the head of a nunnery.
Chaucer's irony, though, is never intended to be wounding or vicious. Even the Prioress, for all her obvious faults, still proves to be an immensely stimulating companion on the long, arduous journey to Canterbury. In other words, for all its importance as a literary device, irony is never allowed by Chaucer to obscure his unforgettable, sympathetic portrait of fallen humanity in all its richness and diversity.

What argument is she making about the “objectification” of women?

Woolf argues that, historically, women have lacked the same opportunities that men have enjoyed. She is, in particular, making this case in terms of women writers and how women had been represented in literature. The notion of "room" or "a room" is literal and figurative. Women need a room, a space to write and be published. They need to be paid; thus, they need the opportunity for that employment. Room needs to be made in the profession of writing and room needs to be made in the canon of literature (in terms of allowing more female writers and in terms of more diverse representations of women in literature). Woolf adds that, historically, women in literature have been represented primarily as the objects of men's affections. Being represented mainly as these "objects" severely limits the ways women can be represented.
In the section “Shakespeare's Sister,” Woolf imagines that Shakespeare had a sister (Judith) who had the same gifts that he did. Woolf supposes that Judith would not have had Bill's (Shakespeare's) access to education. Judith would also not be permitted to act in plays (all roles at this time were played by males). There is no "room" for Judith. Woolf assumes that faced with a life of being denied the ability to develop and share her genius, Judith would be miserable and eventually commit suicide. Judith would have been treated as an object: a daughter to be married off with no consideration for what she wants out of life. Judith is an object to be exchanged. Meanwhile, her brother, Will Shakespeare, gets to thrive in his genius and make the most of his life. Woolf thinks that women (like Judith) should have the same opportunities as men. Denied the ability to be an author (of literature and of her own life), Judith is an object. If room was made for the Judiths of the world to write and have equal opportunity, they would shift from objects to authors. The author has authority over her own life. The object does not.
The same reasoning goes for how women are represented in literature. In the section on Chloe and Olivia, Woolf writes about how women in literature are mostly represented by their relation to men. (This is called the "male gaze"—that women are only seen/represented through men's perspectives.) Up until Jane Austen's time, women in literature were mainly objects of men's desires. Woolf wonders what Antony and Cleopatra would have been like if Cleopatra and Octavia might have been friends or more. Male characters in literature had always been described in great and various detail. This had not been the case with women. Woolf adds, “Suppose, for instance, that men were only represented in literature as the lovers of women, and were never the friends of men, soldiers, thinkers, dreamers . . . “ Woolf simply wants the same for women—to be more than objects of men's affections.

what characteristics does Harte give the setting

The setting of this story is Roaring Camp, and it is a settlement town in 1805. Harte definitely gives the setting a rough feel. He tells us that death is a regular occurrence in the town as is eviction from the town. We are also told that the town is just about completely filled with men. The town's only woman is a woman named “Cherokee Sal," and even she is described as "coarse" and "sinful." Roaring Camp is not a nice place to be. The men that are there are hard men, and many of them are former or even current criminals. Of the 100 men that are in the town, only a few off them are described. Readers get a bit of info about a gambler named Oakhurst, and we are told that many of the men are missing smaller body parts.

The strongest man had but three fingers on his right hand; the best shot had but one eye.

In general, Roaring Camp is not the place for the faint of heart, which is why the camp's transformation once the baby is born is so powerful.

Single Variable Calculus, Chapter 3, 3.2, Section 3.2, Problem 45

Suppose that $f(x) = 2x^2 - x^3$, find $f'(x), f''(x), f'''(x)$ and $f^4(x)$. Graph $f, f', f''$ and $f'''$ on a common screen. Are the graphs consistent with the geometric interpretations of these derivatives?

Using the definition of derivative


$
\begin{equation}
\begin{aligned}

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

\end{aligned}
\end{equation}
$


Using the 2nd derivative of the definition


$
\begin{equation}
\begin{aligned}

\qquad f''(x) =& \lim_{h \to 0} \frac{f'(x + h) = f'(x)}{h}
&&
\\
\\
\qquad f''(x) =& \lim_{h \to 0} \frac{4(x + h) - 3 (x + h)^2 - (4x - 3x^2)}{h}
&& \text{Substitute $f'(x + h)$ and $f'(x)$}
\\
\\
\qquad f''(x) =& \lim_{h \to 0} \frac{4x + 4h - 3 (x^2 + 2xh + h^2) - 4x + 3x^2}{h}
&& \text{Expand the equation}
\\
\\
\qquad f''(x) =& \lim_{h \to 0} \frac{\cancel{4x} + 4h - \cancel{3x^2} - 6xh - 3h^2 - \cancel{4x} + \cancel{3x^2}}{h}
&& \text{Combine like terms}
\\
\\
\qquad f''(x) =& \lim_{h \to 0} \frac{4h - 6xh - 3h^2}{h}
&& \text{Factor the numerator}
\\
\\
\qquad f''(x) =& \lim_{h \to 0} \frac{\cancel{h}(4 - 6x - 3h)}{\cancel{h}}
&& \text{Cancel out like terms}
\\
\\
\qquad f''(x) =& \lim_{h \to 0} (4 - 6x - 3h) = 4 - 6x - 3(0) = 4 - 6x - 0
&& \text{Evaluate the limit}
\\
\\
\qquad f''(x) =& 4 - 6x

\end{aligned}
\end{equation}
$


Using the 3rd derivative of the definition


$
\begin{equation}
\begin{aligned}

\qquad f'''(x) =& \lim_{h \to 0} \frac{f''(x + h) = f''(x)}{h}
&&
\\
\\
\qquad f'''(x) =& \lim_{h \to 0} \frac{4 - 6 (x + h) - (4 - 6x)}{h}
&& \text{Substitute $f''(x + h)$ and $f''(x)$}
\\
\\
\qquad f'''(x) =& \lim_{h \to 0} \frac{\cancel{4} - \cancel{6x} - 6h - \cancel{4} + \cancel{6x}}{h}
&& \text{Expand the equation and combine like terms}
\\
\\\qquad f'''(x) =& \lim_{h \to 0}\frac{-6\cancel{h}}{\cancel{h}}
&& \text{Cancel out like terms}
\\
\\
\qquad f'''(x) =& -6
&&

\end{aligned}
\end{equation}
$


Using the 4th derivative of the definition


$
\begin{equation}
\begin{aligned}

\qquad f^4(x) =& \lim_{h \to 0} \frac{f'''(x + h) = f'''(x)}{h}
&&
\\
\\
\text{If $f'''$ is constant, then $f'''(x + h) = f'''(x)$}
\\
\\
\qquad f^4(x) =& \lim_{h \to 0} \frac{-6 - (-6)}{h}
&& \text{Substitute $f'''(x + h)$ and $f'''(x)$}
\\
\\
\qquad f^4(x) =& \lim_{h \to 0} \frac{-6 + 6}{h} = \lim_{h \to 0} \frac{0}{h}
&& \text{Simplify the equation}
\\
\\
\qquad f^4(x) =& 0

\end{aligned}
\end{equation}
$




Graph $f, f', f''$ and $f'''$

Who are Pyramus and Thisbe in A Midsummer Night's Dream by Shakespeare?

Pyramus and Thisbe are characters in the play the craftsmen are performing.
The characters of Pyrmaus and Thisbe (or Thisby), are characters from the Roman poem Metamorphoses by Ovid.  Their story is similar to that of Romeo and Juliet.  They were two young lovers whose parents kept them apart, and they met an untimely end.  Why someone would choose this play to present at a wedding is beyond me.
The play Pyramus and Thisbe is being presented by the craftsmen, a group of men with no acting experience.   They are presenting it at the royal wedding of Theseus and Hippolyta, so they really want to get it right.  They don’t want to offend anyone. 
Quince, the leader, introduces the play as a “lamentable comedy” and passes out the parts. (Irony means nothing to Quince.)  He assigns the lead, Pyramus, to the pompous Bottom.

BOTTOM
What is Pyramus? a lover, or a tyrant?
QUINCE
A lover, that kills himself most gallant for love. (Act 1, Scene 2) 

Flute is assigned the role of the girl.  In those days, all roles were played by men, and this group is all men anyway.  Shakespeare makes a joke about the fact that young men often played the female roles when Flute says he can't play the girl because he is getting a beard. 

QUINCE
Flute, you must take Thisby on you.
FLUTE
What is Thisby? a wandering knight?
QUINCE
It is the lady that Pyramus must love.
FLUTE
Nay, faith, let me not play a woman; I have a beard coming. (Act 1, Scene 2) 

The group decides to practice in secret so no one see their production (and no one will make fun of them).  In the play, the lovers talk to each other through a wall only.  The wall is played by an actor.  The entire story is explained in the prologue, which is Shakespeare’s way of poking fun at prologues that give away the ending.  Pyramus finds a shawl that has blood on it and thinks that Thisbe is dead, when in fact the lion did not maul her.  He kills himself, and then she kills herself.

Calculus of a Single Variable, Chapter 3, 3.2, Section 3.2, Problem 22

The Rolle's theorem is applicable to the given function, only if the function is continuous and differentiable over the interval, and f(a) = f(b). Since all trigonometric functions are continuous and differentiable on R, hence, the given function is continuous and differentiable on interval. Now, you need to check if f(pi) = f(2pi).
f(pi) =sec (pi)= 1/(cos pi) = 1/(-1) = -1
f(2pi) =sec 2pi = 1/(cos 2pi) = 1/1 = 1
Since one of all the three conditions is not valid, f(pi) != f(2pi) , you cannot apply Rolle's theorem.

College Algebra, Chapter 10, 10.4, Section 10.4, Problem 22

Assume that for any given live human birth, the chances that a child is a boy or girl are equally likely.

a.) What is the probability that in a family of five children a majority are boys?

b.) What is the probability that in a family of seven children a majority are girls?

Recall that the formula for the binomial probability is given by

$C(n,r) p^r q^{n-r}$

In this case, the probability of success and failure is $p = q = 0.50$.

a.) If the family of five children has majority of boys, then the number of boys must be 3. In this case, $r=3$. So, we have

$= C(5,3)(0.50)^3 (0.50)^{5-3}$

$= C(5,3)(0.50)^3 (0.50)^2$

$= 0.3125$

b.) If the family of seven children has majority of girls, then the number of girls must be 4. In this case, $r=4$, so

$= C(7,4) (0.50)^4 (0.50)^{7-3}$

$= C(7,4) (0.50)^4 (0.50)^3 $

$= 0.2734$

sum_(n=1)^oo n(6/5)^n Use the Root Test to determine the convergence or divergence of the series.

To determine the convergence or divergence of a series sum a_n using Root test, we evaluate a limit as:
lim_(n-gtoo) root(n)(|a_n|)= L
or
lim_(n-gtoo) |a_n|^(1/n)= L
Then, we follow the conditions:
a) Llt1 then the series is absolutely convergent.
b) Lgt1 then the series is divergent.
c) L=1 or does not exist  then the test is inconclusive. The series may be divergent, conditionally convergent, or absolutely convergent.
We may apply Root test on the given series sum_(n=1)^oo n(6/5)^n when we let: a_n=n(6/5)^n .
Then, set-up the limit as:
lim_(n-gtoo) |n(6/5)^n|^(1/n) =lim_(n-gtoo) (n(6/5)^n)^(1/n)
Apply Law of  Exponents: (x*y)^n = x^n*y^n and (x^n)^m = x^(n*m).
lim_(n-gtoo) (n(6/5)^n)^(1/n)=lim_(n-gtoo) n^(1/n) ((6/5)^n)^(1/n)
                               =lim_(n-gtoo) n^(1/n) (6/5)^(n*1/n)
                                =lim_(n-gtoo) n^(1/n) (6/5)^(n/n)
                               =lim_(n-gtoo) n^(1/n) (6/5)^1
                               =lim_(n-gtoo) 6/5n^(1/n)
Evaluate the limit.
lim_(n-gtoo) 6/5n^(1/n) =6/5lim_(n-gtoo) n^(1/n)         
                    =6/5 *1  
                  =6/5 or 1.2
The limit value L =6/5 or 1.2 satisfies the condition: Lgt1 since 6/5gt1 or 1.2gt1 .
Therefore, the series sum_(n=1)^oo n(6/5)^n is divergent.

"The Lottery" was published in 1948, just after World War II. What other cultural or historical events, attitudes, institutions, or rituals might Jackson be satirizing in this story?

After The New Yorker published "The Lottery" in 1948, causing outrage throughout much of America, Shirley Jackson remained silent about the actual meaning of the story. Today, the common assumption is that the German population's acceptance of the Nazi death camps served as inspiration for her story (why would otherwise decent people commit such a heinous act?). Critics also contend that the village in the story is modeled after her hometown of North Bennington, Vermont, a place where Jackson commented on the anti-Semitism her father faced there.
However, in an article for the San Francisco Chronicle in 1948, Jackson provided a clear explanation for the story:

Explaining just what I had hoped the story to say is very difficult. I suppose, I hoped, by setting a particularly brutal ancient rite in the present and in my own village to shock the story's readers with a graphic dramatization of the pointless violence and general inhumanity in their own lives.

This idea of "pointless violence and general inhumanity" in ordinary people's lives could be seen in many historical events during this time period. In the rural South, the public often treated the lynchings of innocent black Americans as a time to gather and celebrate. During World War II, a plurality of Americans approved of the internment of Japanese Americans in camps. Jim Crow laws and segregation were entrenched in the South at this time. 
Overall, the idea that "The Lottery" refers to a specific event really limits the strength of the story. The story can be broadly interpreted as a case study of how people, when surrounded by tradition and group pressure to do something, generally follow whatever the norm may be.

This is an interesting question!
One of the major cultural institutions this story could be satirizing and commenting on is the military draft. The United States instituted the draft in 1940, and 50 million American men registered for the draft by the end of World War II. While there are major differences between the two rituals, both involve a major element of chance, social responsibility, and the likelihood of death. Because so many families had experienced this, this would have loomed large in their minds.
Of course, there are older traditions of human sacrifice. Many ancient cultures used human sacrifice, and, like the mention of "Lottery in June, corn be heavy soon," these sacrifices were often intended to please the gods, and/or as an exchange or payment for future good favor, like good crops or success in war.
http://www.bbc.co.uk/history/ancient/british_prehistory/human_sacrifice_01.shtml

https://www.nationalww2museum.org/students-teachers/student-resources/research-starters/draft-and-wwii

Calculus of a Single Variable, Chapter 3, 3.2, Section 3.2, Problem 15

The Rolle's theorem is applicable to the given function, only if the function is continuous and differentiable over the interval, and f(a) = f(b). Since all polynomial functions are continuous and differentiable on R, hence, the given function is continuous and differentiable on interval. Now, you need to check if f(-1) = f(3).
f(-1) = ((-1)^2-2*(-1)-3)/(-1+2) = (1+2-3)/1 = 0
f(3) = ((3)^2-2*(3)-3)/(3+2) = (9-6-3)/5 = 0
Since all the three conditions are valid, you may apply Rolle's theorem:
f'(c)(b-a) = 0
Replacing 3 for b and -1 for a, yields:
f'(c)(3+1) = 0
You need to evaluate f'(c), using quotient rule:
f'(c) = ((c^2-2*c-3)'*(c+2) - (c^2-2*c-3)*(c+2)')/((c+2)^2) => f'(c) = ((2c-2)(c+2) -c^2 + 2c + 3)/((c+2)^2)
f'(c) = (2c^2 + 4c - 2c - 4 - c^2 + 2c + 3)/((c+2)^2)
f'(c) = (c^2 + 4c - 1)/((c+2)^2)
Replacing the found values in equation 4f'(c) = 0
4(c^2 + 4c - 1)/((c+2)^2) = 0 => c^2 + 4c - 1 = 0
c_(1,2) = (-4+-sqrt(16+4))/2 => c_(1,2) = (-4+-2sqrt5)/2 => c_(1,2) = (-2+-sqrt5)
Since c = (-2-sqrt5) does not belong to (-1,3), only c =(-2+sqrt5) is a valid value.
Hence, in this case, the Rolle's theorem may be applied for c = (-2+sqrt5).

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

sum_(n=1)^oo n/(n+1)
To verify if this infinite series diverges, apply the Divergent Test.
In the Divergence Test, 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) = DNE  
 
:. sum  a_n  diverges

So, taking the limit of a_n as n approaches infinity yields:
lim_(n->oo) a_n
=lim_(n->oo) n/(n+1)
=lim_(n->oo) n/(n(1+1/n))
=lim_(n->oo) 1/(1+1/n)
=1/(1+0)
=1
Since the result is not equal to zero, therefore, the series is divergent.

What is the external conflict in Animal Farm?

We might very well phrase the external conflict in Animal Farm in Marxist terms. Marx believed that the proletariat, or working class, was essential for the wealth enjoyed by the capitalist bourgeois class. It's clear in the speech by Old Major at the beginning of the novel that the animals are representative of the proletariat and Jones is the capitalist. The animals provide the wealth for Farmer Jones through their work and their very lives. Old Major says,

Man is the only creature that consumes without producing. He does not give milk, he does not lay eggs, he is too weak to pull the plow, he cannot run fast enough to catch rabbits. Yet he is lord to all the animals. He sets them to work, he gives the bare minimum that will prevent them from starving, and the rest he keeps for himself.

This is precisely the same argument Marx made about the social classes in Europe in the 1800s. He thought the capitalists and the bourgeoisie lived off and exploited the labor of the proletariat. Marx predicted there would be a breaking point and the proletariat would eventually rise up and stage a revolution against the upper classes. Under his hypothesis, industrialized workers would rebel, overthrowing the bourgeoisie and forming a new classless society. Unfortunately, neither Marx nor Old Major could conceive of a new ruling class (in history, the Bolsheviks; in Animal Farm, the pigs) who would continue to exploit the working class while telling them that their new society was far better than the old capitalist regime.

The main external conflict throughout Orwell's novella is Man vs. Society, which is illustrated by the conflict between animals and humans. Mr. Jones is an extension of society and neglects the animals by continually forgetting to feed them. Early in the novella, old Major speaks about the dangers of interacting with humans and vehemently declares that all humans are enemies. Following old Major's death, the pigs develop his ideas into a system of thought known as Animalism and create laws preventing animals from interacting and behaving like humans. After initially expelling Mr. Jones during the Rebellion, he returns with his men and the animals repel his attack during the Battle of the Cowshed. When Napoleon usurps power, he begins trading with humans and is taken advantage of by Mr. Frederick, who buys timber with forged bank notes from Animal Farm. The next morning, Mr. Frederick and his men attack the farm and end up destroying the windmill.
Another external conflict depicted throughout the novella is Man vs. Nature. Under Napoleon's tyrannical reign, the animals suffer from the harsh conditions during the winter. They are malnourished and the unfavorable weather prevents them from having successful harvests. The building materials needed for the windmill are also located at the bottom of a deep quarry. Animals like Boxer struggle to lift and carry the stones to the appropriate location to construct the windmill.
Man vs. Man conflict is illustrated by Napoleon and Snowball's struggle for authority and power on the farm. In an effort to destroy his political rival, Napoleon raises nine enormous dogs and trains them to attack on command. During an assembly, Napoleon instructs the dogs to attack Snowball and they end up chasing him off the farm. For the remainder of the novella, Napoleon uses Snowball as a scapegoat when anything on the farm goes wrong.

In Animal Farm, there is an external conflict between animals and humans. Orwell introduces this conflict early in the story: in Chapter One, for example, Old Major expresses this conflict when he assembles the animals in the barn to warn them about the "tyranny of human beings." He also teaches them a song from his childhood called Beasts of England which inspires them to unite against the humans. This conflict is resolved in Chapter Two when the animals successfully overthrow Mr Jones and take control of the farm for themselves.
Later in the story, this external conflict is again expressed through the animals' relationships with their human neighbours, Frederick and Pilkington. This is best shown when both men bid for a pile of seasoned timber which Napoleon wants to sell. In the end, Napoleon sells the timber to Frederick but is double-crossed when he realises that the bank notes are nothing more than forgeries (in Chapter Eight). This conflict is resolved through battle and the animals are, once again, victorious.

Beginning Algebra With Applications, Chapter 6, Cumulative Exercises, Section Cumulative Exercises, Problem 22

Solve by substitution:
$
\begin{equation}
\begin{aligned}

2x-3y =& -7
\\
x+4y =& 2

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

x+4y =& 2
&& \text{Solve equation 2 for } x
\\
x =& 2-4y
&&
\\
2x-3y =& -7
&& \text{Substitute $2-4y$ for $x$ in equation 1}
\\
2(2-4y)-3y =& -7
&& \text{Solve for } y
\\
4-8y-3y =& -7
&&
\\
-11y =& -11
&&
\\
y =& 1
&&

\end{aligned}
\end{equation}
$


Substitute the value of $y$ in equation 2


$
\begin{equation}
\begin{aligned}

x =& 2-4 (1) \qquad \text{Solve for } x
\\
x =& 2-4
\\
x =& -2

\end{aligned}
\end{equation}
$


The solution is $(-2,1)$.

Describe the Underground Railroad.

In 1831, a slave owner complained that his slave was aided in an escape by what he referred to as an "underground railroad." While this was the first known use of the term "underground railroad," this network of sympathizers, both African American and white, actually began their work in the late eighteenth century. George Washington, in 1786, attributed the escape of one of his slaves to Quakers. The goal of the Underground Railroad was to help slaves escape the South, with some traveling into free states in the North and some traveling as far north as Canada.
Terms used in railroading were also used for the Underground Railroad. For example, "conductors" were those that guided the slaves from one location to the next. Harriet Tubman was perhaps the most well-known conductor. "Stations" and "depots" were the homes or other locations where escaped slaves could find food and shelter. Those escaping slavery would typically travel at night and would be given information on how to locate the next safe place, or "station," along their route to the North.
https://www.history.com/topics/black-history/underground-railroad

In "The Open Window," why do you think Saki chose to name the visitor Framton Nuttel?

Framton Nuttel certainly is an unusual name. It helps to characterize the visitor as an odd and eccentric man. The last name of Nuttel must suggest to most readers that the visitor is a little bit nutty, just as Aunt Sappleton's last name suggests the woman is a bit of a sap. Saki may have had another reason for giving the visitor such an unusual name. Authors often have problems choosing names for their characters. If they choose a common name, such as "Bob Wilson," a number of male readers who happen to have that name (or people who know a "Bob Wilson") might think the author is writing about them! This is not important if the character is fairly innocuous. Framton Nuttel is not innocuous, though; he is depicted as an object of ridicule, so Saki probably gave extra thought to Nuttel's name and came up with a name that would keep Nuttel from being mistaken for any man in the general population.
As another example, Vladimir Nabokov created an outrageous character in his novel Lolita. Nabokov didn't want any living man to be offended, so he chose Humbet Humbert, a name that seems as weird as the protagonist and would be impossible to find duplicated in any phone book.

What does the subtitle 'childrens crusade' mean in Slaughterhouse 5?

During the early 13th century, a group of European children traveled to the Holy Land to spread the word of Christ. This became known as The Children's Crusade. Over the centuries, the precise nature of the crusade has been obscured by myth and legend. Many historians believe that the crusaders were actually poor people, rather than children. In any case, according to traditional accounts, the Children's Crusade was led by a boy, allegedly inspired by a vision of Jesus who told him to go to the Holy Land and peacefully convert Muslims to Christianity.
In relation to Slaughterhouse-Five, the subtitle has two possible meanings. The original Children's Crusade dissolved into tragic failure, with many of those involved being killed or sold into slavery. Slaughterhouse-Five is an anti-war novel, which highlights the senseless loss of innocent life that armed conflict always entails. We can see, then, close parallels between the two events.
The second possible meaning comes directly from the book itself. The soldier narrator promises the wife of one of his comrades that he'll call the book he's writing The Children's Crusade as he wants to highlight the futility and bloodshed of war. The title suggests not the heroism displayed in war, but the sacrifice of young, innocent lives, and is therefore entirely appropriate.

Calculus of a Single Variable, Chapter 6, 6.1, Section 6.1, Problem 19

Given,
y = C_1e^2x + C_2e^(-2x) + C_3sin(2x) + C_4cos(2x)
let us find
y'=(C_1e^2x + C_2e^(-2x) + C_3sin(2x) + C_4cos(2x))'
= 2 C_1e^2x +(-2) C_2e^(-2x) +2 C_3cos(2x) -2 C_4 sin(2x)
y''=(2 C_1e^2x +(-2) C_2e^(-2x) +2 C_3cos(2x) -2 C_4 sin(2x))'
=4 C_1e^2x +(4) C_2e^(-2x) -4 C_3sin(2x) -4 C_4 cos(2x)
y'''=(4 C_1e^2x +(4) C_2e^(-2x) -4 C_3sin(2x) -4 C_4 cos(2x))'
=8 C_1e^2x +(-8) C_2e^(-2x) -8 C_3cos(2x) +8 C_4 sin(2x)
y''''=(8 C_1e^2x +(-8) C_2e^(-2x) -8 C_3cos(2x) +8 C_4 sin(2x))'
=(16 C_1e^2x +(-8)(-2) C_2e^(-2x) -8(-2) C_3sin(2x) +8(2) C_4 cos(2x))
=(16C_1e^2x +16 C_2e^(-2x) +16 C_3sin(2x) + 16C_4cos(2x))
So lets check whether y'''' -16 y =0 or not
(16C_1e^2x +16 C_2e^(-2x) +16 C_3sin(2x) + 16C_4cos(2x))-16(C_1e^2x + C_2e^(-2x) + C_3sin(2x) + C_4cos(2x))
=(16C_1e^2x +16 C_2e^(-2x) +16 C_3sin(2x) + 16C_4cos(2x))-(16C_1e^2x +16 C_2e^(-2x) +16 C_3sin(2x) + 16C_4cos(2x))
=0
so,
y'''' -16 y =0

What does Masha from the play The Seagull tell us about the social, political, and cultural environment of Russia at the turn of the twentieth century?

In Anton Chekhov’s play The Seagull, Masha is just one of many characters who feels unrequited love for another person.
The daughter of the managers of Sorin’s estate, Masha is a depressed young woman who uses alcohol and tobacco (snuff) to escape mentally from her boring, unfulfilling life. Part of the reason Masha is despondent is her unreturned affection for Treplev, who is in love with Nina. Meanwhile, a relatively impoverished teacher named Medvedenko is hopelessly infatuated with Masha. In his conversation with Masha at the beginning of act 1, Medvedenko asserts that she would love him if he were richer. Masha vehemently disagrees, arguing that poverty is tolerable as long as she can have the lover she desires. This shows that women during this time period were beginning to see marriage as a romantic pursuit rather than a means of economic stability or social climbing. This also shows that men largely thought women were only interested in men because of their wealth.
However, Masha eventually does marry Medvedenko despite her lack of love for him. This suggests that women like Masha were also willing to settle for an unhappy marriage if their quest for love was unsuccessful. This means that conforming to traditional gender roles was still seen as the only path for women. Masha likely felt societal and internal pressure to marry and have children when she realized that she could never have Treplev.
In act 2, the other characters in the play belittle Masha in various ways. Dorn insinuates that she is unattractive when he says that the twice-as-old Arkadina appears more youthful, and no one acknowledges her request of Nina to recite lines from Treplev’s play. This suggests that Masha is an outsider in this group. Unlike the majority of them, she is of lower social class and is not an artistic type. Their dismissal of her might reflect how class struggles at the time were beginning to cause political unrest—which eventually led to the Bolshevik Revolution in 1917.
Each of these things regarding Marsha’s character arc reveal something about Russian society at the time.

int sec^3 (pix) dx Find the indefinite integral

Indefinite integrals are written in the form of int f(x) dx = F(x) +C
 where: f(x) as the integrand
           F(x) as the anti-derivative function 
           C  as the arbitrary constant known as constant of integration
For the given  integral problem int sec^3(pix) dx , we may evaluate this using u-substitution.
Let: u = pix then du = pi dx or  (du)/pi =dx .
The integral becomes:
int sec^3(pix) dx =int sec^3(u) * (du)/pi
 Apply the basic properties of integration: int c*f(x) dx= c int f(x) dx .
int sec^3(u) * (du)/pi =1/piint sec^3(u) du
Apply integration formula for secant function:
int sec^n(x) dx = (sec^(n-1)(x)sin(x))/(n-1) + (n-2)/(n-1) int sec^(n-2)(x) dx
We get:
1/piint sec^3(u) du =1/pi [(sec^(3-1)(u)sin(u))/(3-1) + (3-2)/(3-1) int sec^(3-2)(u) du]
        =1/pi [(sec^2(u)sin(u))/(2) + (1)/(2) int sec^(1)(u) du]
For the integral of  int sec^(1)(u) du or  int sec^(u) du , we may apply int sec(theta) d theta = ln(sec(theta)+tan(theta))+C .
Then,int sec^(u) du =ln(sec(u)+tan(u))+C
The complete indefinite integral will be:
1/piint sec^3(u) du =1/pi [(sec^2(u)sin(u))/(3-1) + (1)/(2) int sec^(1)(u) du]
           =1/pi [(sec^2(u)sin(u))/(2) + (1)/(2)[ln(sec(u)+tan(u))]]+C
           =1/pi [(sec^2(u)sin(u))/2+ln(sec(u)+tan(u))/2]+C
           =(sec^2(u)sin(u))/(2pi)+ln(sec(u)+tan(u))/(2pi)+C
          or  tan(u)/(2picos(u)) + 1/(2pi)ln((1+sin(u))/cos(u))+C
Plug-in u = pix , we get the final indefinite integral as:
int sec^3(pix) dx=(sec^2(pix)sin(pix))/(2pi)+ln(sec(pix)+tan(pix))/(2pi)+C
                        or tan(pix)/(2picos(pix)) + 1/(2pi)ln((1+sin(pix))/cos(pix))+C

Please identify the basic duties of employers to their employees, and discuss which of these duties the employer failed to do, based on the documentary “A Dangerous Business: The McWane Story.” http://www.pbs.org/wgbh/pages/frontline/mcwane/

Under Occupational Safety and Health Administration (OSHA) regulations, employers are mandated to provide a safe workplace for employees. Included among key employer responsibilities is to have a workplace free from serious recognized hazards, to regularly examine workplace conditions for OSHA compliance, to keep records of work-related injuries and illnesses, and to not discriminate against employees who exercise their rights, such as whistleblowers.
McWane Inc., one of the largest iron pipe foundry companies in North America, was fined $8 million for dozens of safety and environmental crimes at several of the company’s plants. In 2003, investigations by Frontline, The New York Times, and the Canadian Broadcasting Corporation revealed horrific safety and environmental violations that had led to thousands of injuries and nine deaths at McWane plants. The privately held company and eight of its executives and managers were convicted of 125 environmental, health, and safety crimes. The company had violated the Clean Water Act and Clean Air Act and been involved in lying to government officials and conspiracy to cover up violations. While courts have imposed almost $20 million in criminal fines, and government regulatory agencies have imposed millions more in fines, only one McWane official has served any time in jail.
David Uhlmann, as section head of the Environmental Crime Section of the Department of Justice, guided the nationwide investigation of McWane violations in partnership with the Environmental Protection Agency and OSHA. Uhlmann concluded that the Environmental Crime Section had never seen such criminal violations in its 20 year history, citing the company’s “culture of lawlessness.”

What is Rainsford's position on hunting before arriving on the island?

In the opening of "The Most Dangerous Game," Rainsford and Whitney are having a conversation about hunting. Whitney remarks, "great sport, hunting," and Rainsford agrees, saying, "best sport in the world." As the conversation goes on, Rainsford states that animals have no feelings, or if they do, no one cares about them. Rainsford is a big game hunter of international renown and tells Whitney that there are really only two kinds of creatures in the world: the hunters and the hunted.
Later, when Rainsford meets General Zaroff, the Cossack immediately recognizes Rainsford as "the celebrated hunter" and tells him that he enjoyed the book he wrote about hunting snow leopards in Tibet. When the two men sit together at dinner, Rainsford admires the mounted heads that decorate Zaroff's home, particularly a sizable Cape buffalo. As Zaroff confides to Rainsford that he has stocked his island with the most dangerous game of all, Rainsford is fascinated—until he learns that it is humans that Zaroff hunts.

According to what Kipling wrote in "The White Man's Burden," what would happen to the imperialists?

"The White Man's Burden" expresses the idea that imperialists are engaged in a noble but thankless task. The native peoples they colonize will, Kipling argues, benefit. But they will hate the imperialists. Only other imperialists (in this case, the British) will understand and respect those who take on the burden.
In essence, the argument is that native people aren't competent to rule themselves. The imperialist is acting selflessly, taking over in order to deliver the benefits of Western civilization. But, like unreasonable children, the natives will resent being controlled.
To understand this point of view -- which is today recognized as racist -- it helps to consider the historical context. Kipling wrote this poem for the express purpose of persuading the United States to act as an imperial power in the Philippines. Many citizens, like Mark Twain, opposed this. They believed that the Philippines should be left to rule itself.
But Kipling argued that the United States had a moral obligation to take on the duties of a colonial power. As he wrote in a letter to Theodore Roosevelt, the United States had created this obligation when it ended centuries of Spanish control over the islands. The Philippines were in disarray, and Americans bore responsibility for improving things:  

"America has gone and stuck a pick-axe into the foundations of a rotten house, and she is morally bound to build the house over, again, from the foundations, or have it fall about her ears."

What exactly were Americans supposed to build? Kipling envisioned a mix of tasks, some of which we'd recognize today as humanitarian. His poem makes reference to ending hunger and providing people with medical care ("Fill full the mouth of Famine / And bid the sickness cease"). He also alludes to ending warfare among different factions in Filipino society, and creating the infrastructure of civilization, like better ports and roads:

"Take up the White Man's burden, No tawdry rule of kings,But toil of serf and sweeper, The tale of common things. The ports ye shall not enter, The roads ye shall not tread,Go make them with your living, And mark them with your dead."

But again and again, Kipling makes it clear that the native people -- whom Kipling describes in patronizing, racist terms ("half-devil and half-child") -- will resist and resent imperial efforts. He writes sarcastically of how the imperialist will be "rewarded" by "blame of those ye better, The hate of those ye guard."
Kipling's words suggest the analogy of a child being ruled by a parent. The parent toils for the child's own good, but the child doesn't understand that and is ungrateful. But the analogy doesn’t only apply to the relationship between the United States and the Philippines. In the final stanza, Kipling also suggests that the United States isn't quite grown up yet: Americans need to become a colonial power and assume imperial responsibilities. The British might not thank them for doing so, but they will respect them as peers who have taken on the noble burden.
Kipling, Rudyard (1990) The Letters of Rudyard Kipling, Thomas Pinney, Editor. London, Macmillan, Vol II, p. 350.

Why does Kira appeal to the women's sense of fear and self-interest?

In chapter 2, Kira returns to the village after attending her mother's dead body in the Field of Leaving, only to find that the women of the village intend to kill her so that they can have the area on which her family's cott had stood. They want to build a pen for chickens and tykes there. Vandara is the leader of this group of women, and the women look toward her for instructions. Some of the women have rocks in their hands. Once one woman threw a rock, Kira knew, the others would follow, and she could be quickly stoned to death. She knows it is useless to appeal to their compassion. The women in the community don't value thinking of others and seem to have never learned to be kind or empathetic. Therefore, Kira knows she must appeal to their fear. She begins reciting the law that the women are subject to. Conflicts must be taken to the guardians for resolution. If a dispute results in a death without going to the guardians, then the one who causes the death will be executed. This knowledge, this fear of capital punishment, saves Kira from stoning. Vandara becomes her accuser, and the Council of Guardians allows Kira to live.

How does Hamlet respond to Ophelia at the beginning of Act 3?

In Act III, scene i of Hamlet, Hamlet appears to Ophelia after Ophelia, the king and queen, Rosencrantz and Guildenstern, and Polonius discuss Hamlet's seeming descent into madness. Gertrude tells Ophelia that she hopes that Ophelia and Hamlet will come back together as they were before, which makes Hamlet's response to Ophelia moments later even more difficult for Ophelia to understand.
Hamlet's response to Ophelia when she asks him, "How does your honor for this many a day?" is full of strangeness—all the more strange for her simple and polite question. Ophelia mentions some things that Hamlet has given her, and he responds by contradicting her and saying that he never gave her anything. He then asks a series of odd questions about Ophelia's beauty and goodness—rhetorical questions that he doesn't expect Ophelia to answer—and soon he tells Ophelia "Get thee to a nunnery" after criticizing her for unreliable qualities she does not actually exhibit.
These responses all combine to confuse and alarm Ophelia. She feels sure that Hamlet has gone insane, and when he exits, she says, "Oh, what a noble mind is here o'erthrown!"

What was Silas Marner's vocation?

Silas Marner is a linen weaver. He works out of his own cottage, as weavers did in the early years of the nineteenth century before textile weaving became mechanized and the work was done in big factories under arduous conditions. Silas has to spin flax fiber into thread and then weave the threads into linen. The fact that Silas Marner works at home enables him to adopt the little golden-haired girl named Eppie who wanders into his home one night after her mother dies in the snow. The identity of the child's father is unknown. Marner is able to look after the little girl personally even though she is very young when she first comes into his life. They develop a loving relationship, and she becomes a substitute for him in place of the hoard of gold he accumulated with years of work and had stolen from him on night by a drunken wastrel named Dunstan.

Who is John O’Neill? Why is he relevant to the narrative of the book?

As the FBI agent in charge of counterterrorism, John O’Neill is Osama bin Laden’s primary adversary in The Looming Tower by Lawrence Wright. He is vital to the narrative because he creates a conflict for the terrorists. However, John O’Neill's worthiness as an adversary becomes debatable when his flaws are revealed. We find out that he is a philanderer with a bad temper and large amounts of debt. He is then forced to retire from the FBI after leaving a briefcase containing confidential documents in a public place. It is the ultimate irony when John O’Neill becomes the head of security at the Word Trade Center and dies in the 9/11 bombings. So, he not only loses his life, but he also loses in his conflict with Osama bin Laden.

Why is The Crucible a modern tragedy?

The Crucible is a modern tragedy because it was written in 1953 in the modern era. Although it was set in the 1600s during the Salem witch trials, it was really a commentary on a current event: the McCarthy era "witch hunts" that were going on the early 1950s.
At the end of World War II, the Soviet Union and the United States (countries that were allies during the war) broke with each other. What emerged was called a "cold war" because it was fought through propaganda, proxy wars in smaller countries, and military build ups rather than an actual "hot" war between the two countries.
In this country, certain segments of society became very fearful of communist infiltration and takeover. This led to a "Red scare," red being the color associated with communism. Under an executive order signed by President Truman allowing Congress to screen people to test their loyalty to the U.S. and to root our communists, fascists, and other subversives, Senator Joseph McCarthy started conducting wide-ranging investigations. This led to many innocent people having their lives or careers destroyed by being unfairly labelled communists. Miller's play is a commentary on this tragic modern situation. Unlike a Greek tragedy, which focuses on a tragic flaw in a person of high stature, such as a king, Miller's play shows what can happen to ordinary people when lies and hysteria are allowed to take over a society.

Arthur Miller's The Crucible is a modern tragedy as opposed to a classical tragedy primarily because of when it was written, published, and performed. The play was first published in 1953, and while it does retain some elements of the classical tragedy of Ancient Greece, it also is inspired by contemporary current events (in the 1950s) and complicates the standard expectations of tragedy, catharsis, and tragic hero.
The classical Greek tragedy, originating in the ancient world, follows the downfall of a tragic hero, who is generally an upper-class/powerful figure who has a tragic flaw that will lead to his demise. The hero is relatable to the audience on some level, so he cannot be perfect. We watch his flaw lead him to make poor decisions or take actions that will cause his death or fall. Through watching the tragic reversal of fate, we feel catharsis, or a purging of emotions.
The modern tragedy, on the other hand, does not necessarily follow the strict structure of the classical tragedy. Modern tragedies can have tragic heroes who are more ordinary, everyday people, and they are often placed in more realistic or average circumstances. The Crucible is a tragedy set during the Salem Witch Trials in Colonial Massachusetts. The characters are average Puritans living in Salem who get caught up in an extraordinary hysteria. The character who is closest to a tragic hero is John Proctor, a respected man in the community but not a rich man nor a powerful leader in the town. His flaw of lust leads him to betray his wife by having an affair with Abigail. This sets much of the witch accusations in motion, since Abigail goes on to accuse Mrs. Proctor of being a witch. The girls at the center of the hysteria are depicted as trying to deflect blame from themselves for their own transgressions (participating in a voodoo ceremony with Tituba). Eventually, John is faced with execution, and he must either confess to being a wizard or be put to death. He can be considered heroic because he chooses to die rather than lie or ruin his name. The play does end with his downfall (his death). The characters, including John, are common people, and though he does fall quite a bit in the play, his devolution is not as extreme as, say, Oedipus in Sophocles's classical tragedies. The cathartic effect on the audience is also questionable. The purpose of the play wasn't necessarily to have the audience purge emotions as it was to draw parallels between the witch trials and the McCarthy Era in U.S. politics.
The Crucible is a modern tragedy because it was written in the 1950s, is based on real events, and depicts the lives of ordinary characters who are simply caught up in an extraordinary situation.

What literary device is used in "The Cask of Amontillado" how does it develop the message of the story?

Though his dark subject matter is not universally appealing, Edgar Allan Poe was a superb technician and his skill in crafting this revenge tale through the use of symbolism is undeniable.
A single example among the many uses of symbolism is found in the names of the principal characters, Montresor ( French for "my treasure") and Fortunato (ltalian for "fortunate one").  Montresor is pathologically angry with Fortunato for a series of slights that are described only as "a thousand injuries". Exacting revenge through Fortunato's destruction is what is most precious to Montresor.
Fortunato, symbolically costumed as a fool for Carnival, is merrily oblivious to Montresor's anger and readily follows him underground to sample a rare sherry. When Montresor overpowers and entombs Fortunato, he literally buries his "treasure." Fortunato's name, then, can only be deemed ironic.
 Poe, Edgar Allan.  "The Cask of Amontillado"
http://xroads.virginia.edu/~hyper/poe/cask.html

In a hydrogen atom the electron is separated from the proton by an average distance of approximately 5.3*10^-11 m. Calculate the magnitude of the electrostatic force of attraction exerted by the proton on the electron.

Use Coulomb's law to determine the magnitude of the electrostatic force of attraction exerted by the proton on the electron. The magnitude of both the electron and proton charge is equal to the elementary charge e .
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elefor.html

Is it ethical to imprison non-dangerous white-collar criminals to set an example, or should they merely suffer financial penalties?

The description of "non-dangerous white-collar criminals" itself is somewhat problematic. Many such criminals have indeed been dangerous to their victims. The victims of such criminals can be ruined financially. One of Madoff's victims committed suicide. Victims of identity theft, revenge porn, and other technologically enabled crimes can have their lives ruined. White-collar crime can include corruption, which weakens democratic institutions and leads to oligarchy and increasing inequality in which the very wealthy feel increasingly entitled to act with impunity, as though they are above the law. White-collar criminals also have relatively high rates of recidivism.
Merely assessing financial penalties for white collar crimes sends the message that a poor person who steals food should be imprisoned while a rich person who steals millions of dollars should just pay a fine. This is fundamentally unjust. The message it sends to society is that it is a crime to be poor and hungry and steal food but being rich and greedy and stealing additional money one does not actually need is merely a minor miscalculation.
In terms of imprisonment, for all criminals the issue is the same, namely whether sending them to prison is likely to protect society from them and prevent future crime. While no one should be sent to prison as "an example" as that implies imprisonment for some reason other than justice being applied to the individual case, the two other criteria of protection and prevention apply just as much to people who steal millions from bank accounts as to pickpockets who lift wallets. "White-collar" criminals present clear dangers to our society and letting them off with minor fines does not protect society from their actions nor deter them from re-offending.
https://www.nytimes.com/2016/11/08/business/dealbook/determining-a-punishment-that-fits-the-crime.html

To what extent does Paine's characterization of America in Rights of Man still hold true today? How can I write an essay on this?

The best way to approach this essay is to define the key components of Paine's characterization of America. How does he describe the United States in Rights of Man, published in 1791?
The next part is going to depend on your own viewpoint about the state of our country in the present day. You'll have to ask yourself about each of the components that Paine lays out, then decide for yourself whether or not you think that parts of his description still hold true today.
Let's pin down a few facts right off the bat. Paine was born in England, but he supported American independence from England. He was also a supporter of the French Revolution. In Rights of Man, he discusses both of these topics. Keep in mind that the United States was a brand-new country at the time, and it was founded, to some extent, on ideals. 
Could we describe Paine's depiction of the US, then, as idealistic? Indeed, some scholars would go even further, describing his vision of America as utopian. Consider the following passage from Rights of Man:

If there is a country in the world where concord, according to common calculation, would be least expected, it is America. Made up as it is of people from different nations, accustomed to different forms and habits of government, speaking different languages, and more different in their modes of worship, it would appear that the union of such a people was impracticable; but by the simple operation of constructing government on the principles of society and the rights of man, every difficulty retires, and all the parts are brought into cordial unison. There the poor are not oppressed, the rich are not privileged. Industry is not mortified by the splendid extravagance of a court rioting at its expense. Their taxes are few, because their government is just: and as there is nothing to render them wretched, there is nothing to engender riots and tumults.

Let's go through a few of these points as prompts for your essay. Is America still made up of people from many nations who practice different religions and speak different languages? The answer is yes, of course. 
What about the poor not being oppressed, or the rich not being privileged? I doubt you'd agree with that. I'd suggest scanning recent news articles for examples of policies or decisions that seem to favor the rich over the poor.
Which brings us to the next point. I don't know what your personal politics are: this essay is yours, and you'll have to respond according to what you believe. But the notion that nothing could "engender riots and tumults" seems like the exact opposite of the America we're living in today, and I encourage you to use this as a basis for arguing that Paine's characterization of the US is idealistic at best.

How do the witch trials in The Crucible empower some individuals who were previously powerless? Is it possible that the situation that happened in The Crucible is capable of taking place today? How would it be different/similar?

Throughout the Puritan town of Salem, young women were expected to be obedient, quiet, and God-fearing. Living in a society where males were culturally and economically dominant, young females were essentially powerless. Abigail Williams is the niece of Reverend Parris and is treated as an afterthought before the witch trials begin. She is dismissed from working at John Proctor's home and lives in the shadow of her uncle. Mary Warren is another young girl who goes unnoticed throughout the community before the witch trials begin. However, both characters find prestige and influence by accusing others of witchcraft. Abigail is the ringleader of the group of girls that are accusing citizens of witchcraft. She is respected by the judges and enjoys the power of manipulating Salem's community. Mary Warren also enjoys her position as an official of the court. She even brags to John Proctor at the beginning of Act Two about the importance of her position. Abigail, who was somewhat of a social outcast, and Mary Warren, a timid young girl, both are empowered by their positions in the Salem court.
It is not out of the realm of possibility that a situation similar to the events that transpired in Salem would happen in today's society. Arthur Miller's play was inspired by the second Red Scare, which was when Senator Joseph McCarthy falsely accused many citizens of having communist ties. Miller explores the concept of hysteria in Salem, Massachusetts, which parallels McCarthy's 1950s communist "witch-hunt." Although citizens would probably not be accused of witchcraft in today's society, it is possible that hysteria could result in false confessions from innocent individuals. The government could utilize propaganda to stir American citizens into a state of hysteria and create a situation where innocent individuals are forced to falsely accuse their neighbors of various crimes.

What are the pros and cons of the structure of the US legal system as they pertain to the government's ability to enact large-scale health legislation?

The structure of the US legislative system responsible for enacting large-scale health legislation has pros and cons that are currently actively debated ("legal system" refers to the system of court procedures and processes overseen by an independent judge and responsible for interpreting civil law and common law [Cornell Law School]). One significant pro for the structure of our legislative system's ability to enact large-scale health legislation is that the individual's right to the pursuit of happiness, which is generally seen as including prosperity, is protected. This means that medical professionals can practice medicine how they want and at what price for services they want. One oft discussed con against the structure of our legislative system's ability to enact large-scale health legislation is that since the medical industry is protected, certain individuals are left without adequate recourse to health care, sometimes at great harm to their physical well being and quality of life. Another pro for our system's ability to enact large-scale health legislation is that, in our federalist republic, individual states are more free to enact legislation according to their own assessment of and perspective on need. Medical marijuana offers a perfect example of this as some states, such as California, Montana and Texas, have legalized medical marijuana while others, like Idaho, Kansas and Nebraska, haven't (as of July 2018, NCSL). Another con against our system's ability to enact large-scale health legislation is that our health care delivery system "has not produced superlative health outcomes for its people" (ncbi.nlm.nih.gov). A corollary to this is the con that the US is one of less than a half-dozen highly developed countries that haven't enacted large-scale health legislation to provide universal health care coverage (world health care map, The Atlantic).
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=8&cad=rja&uact=8&ved=2ahUKEwikje-ZzfLdAhXkY98KHT3ICnoQFjAHegQIABAC&url=http%3A%2F%2Fwww.jblearning.com%2Fsamples%2F0763763802%2F63800_CH01_Final.pdf&usg=AOvVaw3mPyyfXhSbXqHWYfx4qTav

Although many of the obstacles to health care reform are grounded in political will, entrenched interests, and legislative gridlock, there are several ways in which the legal system impacts the ability to reform healthcare.
Perhaps the most overwhelming obstacle to healthcare reform on a national level is federalism and the issue of states' rights. The greatest cost efficiencies in health care reform can be found in national systems that can develop economies of scale in everything from data processing to purchasing. If the US federal government negotiated, for example, pricing for prescription medication, it could reduce their costs significantly while still allowing pharmaceutical companies adequate margins to support research and development. Having funding and benefits vary from state to state with a complex mixture of different systems such as the public Medicare, Medicaid, Veteran's Administration, and private insurance with various public subsidies is needlessly complex and expensive in terms of overhead costs. On the other hand, many "blue" states have been able to experiment with new models of health care funding and access and have been significantly more agile than the federal government.
The next obstacle to reform involves regulation and licensing. For example, nurse practitioners can offer basic family health care considerably less expensively than physicians. They have the ability to fill in gaps in health care availability for underserved populations in poor or rural areas. Scope of Practice laws, however, limit their ability to offer care and often make it more expensive. Ongoing reforms to these laws can help reduce costs and increase access to health care.
Laws passed by conservative Republicans to restrict abortion and attacks on Planned Parenthood limit the ability to expand affordable family planning and women's health care. Laws designed to reduce services for and deport illegal immigrants also have a negative impact on health care. This is because public health depends on preventing or treating communicable diseases in entire populations. Offering free and accessible health care to everyone on US soil, no matter whether they are immigrants, visitors, or citizens, prevents the spread of diseases. Laws which insist that students are vaccinated as a condition of school attendance have had a strong positive effect on public health.
Finally, malpractice laws encourage doctors to practice defensively rather than effectively, often ordering tests that are not needed or overmedicating. On the other hand, public disclosure laws have a strong positive effect on physician performance.
https://www.amnhealthcare.com/latest-healthcare-news/limits-nurse-practitioner-practice-current-future-trends/

https://www.ncbi.nlm.nih.gov/books/NBK236545/

To what degree does Mama display a consistent persona in narrating the story?

The character of Mama is a mainstay in Alice Walker's short fiction story. Throughout the changes over time, including the fads which Dee would embrace, Mama does not change. She wears serviceable clothing, which may be unfashionable and masculine, but is appropriate for the work she must perform. Mama does not embrace the African heritage, yet she treasures family quilts that have been passed on for generations for their practical use, as well as their symbolic value of maternal love and family connections. Dee's capricious nature is a foil to Mama's steadfast one.
It is Mama's voice as the narrator, which endures, just as the quilts endure due to their hardy construction. It is she who holds on to given names and understands their value as opposed to the adoption of newer names. She has no time or energy to devote to trends, unlike her daughter Dee. She stands and sweeps her yard of dirt, creating an oasis of calm that opposes the stormy nature of Dee's personality.

In Alice Walker's "Everyday Use," Mama's consistent persona is the antithesis of Dee's ever-evolving one. As she and Maggie wait for Dee to arrive, Mama thinks about dreams she has had such as appearing on Johnny Carson or some such show and surprising Dee while Dee, who "has made it," graciously thanks her now-glamorous mother for being the reason she has made it. However, Mama's self-awareness that she can never be or look like what Dee wants adds to the stability that she has been able to provide for herself and Maggie throughout their lives. She has no grand illusions about her femininity—she thinks about how she has "always been better at a man's job"—or how society sees her. Because of this consistent, honest view of herself, Mama is able to remain largely unaffected by Dee's constantly changing versions of herself—from her name change, to unique hairstyles, to greetings in Arabic. Mama knows who she is and makes no apology for it even when her daughter seems to expect one from her.
It is this tenacious true-to-yourself part of Mama's nature which so frustrates Dee but which in the end draws a smile of appreciation from Maggie.

When he was mayor of New York City, Michael Bloomberg proposed a ban on any "sweet drinks" more than 16 ounces. Although it was ruled illegal by the courts, other cities (notably Mexico City) have started to introduce this kind of legislation. The new mayor of New York, Bill Di Blasio, is thinking about reintroducing this proposal. Is this an intrusion into personal freedoms? How should suppliers of these products react?

Clearly, a ban on these drinks would intrude on personal freedom.  The issue is not whether it intrudes on personal freedom but whether it does so excessively.
Essentially every government law infringes on personal freedom in some way.  The speed limit intrudes on our freedom to drive whatever speed we want.  The ban on cocaine and heroin infringes on our freedom to do whatever we think will make us feel good.  The ban on sugary drinks would infringe on our right to eat and drink what we want.  However, while laws always infringe on our freedoms, we accept them because we think they do not intrude too much in comparison to the bad things that they prevent.
It is in this context that we have to think about the proposed NYC ban.  Does it do enough good to overcome the way in which it infringes on our freedom?  A ban would presumably help at least somewhat to reduce the problems associated with obesity in the city.  This would make people healthier and reduce the government’s need to spend on health care.  The question you have to answer for yourself, then, is whether that is enough of a benefit to justify telling people what they can and cannot drink.  One thing to think about as you consider this question: where do we draw the line?  Could the government ban us from eating large steaks (or all meat) because it thinks fat is unhealthy?  This is a matter of personal opinion.  What do you think?
As for what companies should do about this, I would say they should do two things.  First, they should advertise, telling people to fight for their right to eat and drink what they want.  Second, they should work to develop new drinks that would have less sugar. If they could do that, they would be covered in the event that such a ban does eventually end up happening.
https://www.huffpost.com/life/topic/soda

When Holden gets on the elevator, whom does he tell the operator he is going visit?

At the beginning of chapter twenty-one, Holden catches a break when he attempts to sneak into his family's apartment to speak with Phoebe. When Holden gets into the elevator, he notices that the regular operator, Pete, isn't working and there is a new person in his position. Holden also mentions that the new operator seems somewhat on the "stupid side" and he casually tells the operator to take him to the Dicksteins. The elevator operator responds by telling Holden that the Dicksteins are not at home and are currently attending a party on the fourteenth floor. Holden then tells the man that he is the Dicksteins's nephew. The operator responds by suggesting that Holden wait in the lobby but Holden feigns a leg injury and says that he should wait in a chair outside their door. When Holden gets off the elevator, he limps towards the Dicksteins and waits until the elevator door shuts before entering his family's apartment.

In chapter twenty-one, Holden catches a very lucky break when he takes the risk to sneak into his family's apartment to visit Phoebe at night. Pete, the attendant who usually staffs the elevator at night, is not on duty. In his place is "some new guy" that Holden has never seen. Moreover, Holden deems him to be "on the stupid side." When Holden enters the elevator, he asks to be taken to the floor his family's apartment shares with the Dicksteins. When the elevator attendant tells Holden that the Dicksteins are on the fourteenth floor at a party, Holden quickly improvises a leg injury and says he must sit outside their door and elevate his leg. To his relief, the elevator attendant doesn't question him or insist that he wait downstairs. Within minutes, Holden is on his way into his own apartment undetected.

Find the arc length from (0,3) clockwise to (2,sqrt(5)) along the circle x^2+y^2 = 9

Use the arc length formula,
L=intds
ds=sqrt(1+(dy/dx)^2)dx , if y=f(x), a<= x<= b
Given x^2+y^2=9
=>y^2=9-x^2
y=(9-x^2)^(1/2)
dy/dx=1/2(9-x^2)^(1/2-1)*(-2x)
=-x/sqrt(9-x^2)
Plug in the above in ds,
ds=sqrt(1+(-x/sqrt(9-x^2)))^2dx
ds=sqrt(1+x^2/(9-x^2))dx
ds=sqrt((9-x^2+x^2)/(9-x^2))dx
ds=3/sqrt(9-x^2)dx
The limits are x=0 and x=2,
L=int_0^2 3/sqrt(9-x^2)dx
=3int_0^2 1/sqrt(9-x^2)dx
Now let's first evaluate the indefinite integral by using integral substitution,
Let x=3sin(u)
dx=3cos(u)du
int1/sqrt(9-x^2)dx=int1/sqrt(9-(3sin(u))^2)3cos(u)du
=int(3cos(u))/sqrt(9-9sin^2(u))du
=int(3cos(u))/(sqrt(9)sqrt(1-sin^2(u)))du
=int(3cos(u))/(3cos(u))du
=int1du
=u
substitute back u and add a constant C to the solution,
=arcsin(x/3)+C
L=3[arcsin(x/3)]_0^2
L=3[arcsin(2/3)-arcsin(0)]
L=3arcsin(2/3)
~~2.19

What stylistic strategies does Coelho use to make The Alchemist come across as a mythic, universally applicable story?

The Alchemist, written by Paulo Coelho, comes across as mythic and universally applicable. This is a timeless story because it teaches the lesson of realizing "personal legends," or following one's dreams, and pursuing these dreams by making decisions. This is something humans will always aspire to do and will wish to learn about.
This story is universally applicable and timeless because Coelho does not use specifics in his descriptions of places or people. Although there are specific people and places mentioned, such as Santiago (the main character) and the fact that the story begins in Spain, most of the places and characters are left vague.
When Santiago describes the people he is meeting, he often refers to them as the things they represent in his life or the jobs they do for a living, such as "the merchant," "the fortune teller," "the Englishman," and "the Alchemist." Even the girl Santiago falls in love with, Fatima, is not explicitly described. By not describing these characters and keeping their identities vague, Coelho makes them more relatable to readers so that they could stand in for any person.
When Santiago describes places, he calls them by basic names like "the Pyramids," "the oasis," and "the crystal merchant's shop." These names could indicate any place, which makes them more universal to readers.
Coelho does not ever mention the time frame of this story, either. Because readers do not know the exact year or time of year the story takes place, it is easier for readers to picture themselves in the story.
All of these reasons make this story universally applicable. At least in part because of the story's ambiguity, it became an international bestseller and was published in many languages and countries.

The mythic takes place outside of history. The main stylistic strategy Coehlo uses to give his work a mythic feel is to place it apart from the historical and the specific.
The tale does start off in a place, for we are told early on that:

The girl was typical of the region of Andalusia, with flowing black hair, and eyes that vaguely recalled the Moorish conquerors.

So we know that Santiago, the main character, starts off in Spain. We learn that the story is set after the time of the "Moorish conquerors," but we are given no specific dates to know when this might be and only a general sense of place. We learn Santiago is heading to the pyramids, but we do not know much beyond that. We also know the story is set in the past, because the characters do not have automobiles, phones, planes, and other modern technologies, but it is an unspecified past—hence a mythic past. 
Also, many of the characters do not have names, but are, in fact, representative types, such as a merchant or an old woman. Again, this lack of specificity as to proper names lends a mythic, universal quality to the novel. Santiago himself is often referred to as the "boy," making him seem a mythic representative of all boys who aspire to follow their dreams.