How were the American colonists able to defeat the British military?

There are several answers to this, all of which are interrelated to some degree. We can list the most important ones as follows:
1. Washington knew that as long as he kept an army alive and in the field, the rebellion could not be ended by the British.
He was defeated in battle more often than he was victorious, but each time his army survived to fight another day. The conflict became a war of attrition in which the British were worn down even in battles they won, and the cumulative effect was to exhaust them and sap them of the will to go on fighting.
This became more apparent than ever during the Southern campaign in 1780-1781, when General Nathaniel Greene, in many ways Washington's right-hand man, repeatedly engaged the British and essentially won the campaign without actually winning a single individual battle.
2. Isolated, surprise victories—especially at Trenton in December, 1776—had the effect of demoralizing the British, who were unprepared for defeat on any level.
Washington's Trenton victory also caused other European leaders to take the American cause seriously, as Frederick the Great of Prussia did in predicting at this point that the Patriots would win the war.
3. The British themselves were divided on how to manage the war and even on the overall policy pursued by the King and his administration.
The Whig Party was sympathetic to the Patriot cause, and the ambivalent situation caused the British military effort to be conflicted and unaggressive. British generals such as Howe did not wish to engage in a "total war" against a people who were perceived by many in Britain as simply trying to stand up for their rights as Englishmen. At some stages of the conflict, it may have been possible for the British to destroy Washington's army, but they did not, partly because they did not take the independence effort seriously and did not wish to make the effort to follow up their own victories with a true knock-out blow.
This was the case in the autumn of 1777 when Howe preferred, after his victory at Brandywine, merely to settle down and spend a quiet winter in Philadelphia (hosted by the wealthy Philadelphia Loyalist families) instead of pursuing Washington to his encampments first in Whitemarsh and then Valley Forge and destroying him.
4. The alliance established with the French in 1778 meant that the British were now fighting their principal European enemy and that the war had become an international conflict with naval engagements in the West Indies as well.
With the combination of Washington's and Rochambeau's armies at Yorktown and the bottling up of the outlet to the Atlantic by De Grasse's fleet, Cornwallis was forced to surrender. At that point in October, 1781, the British had been exhausted by six years of war after which they were no closer to ending the rebellion than they had been at the start of the conflict. Parliament no longer had the will to continue an effort on which they had been divided in the first place, and thus voted to begin negotiations with the aim of granting independence to the colonies—already, of course, self-declared as independent since July, 1776.

By a purely conventional military analysis, the British certainly had the advantage over the colonials. With that being stated, however, you should recognize that the Americans made use of unconventional military tactics. They knew the land better than the British did, and they made effective use of this advantage, utilizing hit and run tactics and ambushes, not always giving the British a direct confrontation.
North America in the late-eighteenth century was a very different world than North America today. There were great stretches of wilderness that the British would have had to deal with, along with very poor roads. This would have been in sharp contrast to conditions on the other side of the Atlantic, and the colonists exploited these conditions to their advantage.
Beyond this, recognize that the Americans did not fight the Revolutionary War alone. Indeed, the reason that the Battle of Saratoga was so important was that this victory convinced France to enter the war in aid of the American cause. The colonials had allies against Britain, and this support was critical in ensuring their success.

The Americans were not expected to defeat the British military in the Revolutionary War. However, the unexpected occurred, and the Americans defeated the British. There were reasons why this occurred.
The British generals made some mistakes. General Howe went to Philadelphia instead of going to Saratoga. The British planned on having three armies arrive at Saratoga, but only one army made it there. General Burgoyne also moved very slowly because he brought many unnecessary items with him.
The Americans got help from Spain and France, enemies of Great Britain, after the Americans won the Battle of Saratoga. These countries provided money and weapons. Also, some foreign soldiers came to join the American side. These soldiers also helped train the American army. The French navy helped the Americans trap the British at Yorktown. This led to the British surrender, ending the war.
The Americans had several reasons for which to fight and win. Many of the leaders of the American Revolution would likely have been killed if the Americans had lost the war. Additionally, the Americans were fighting for their freedom. These factors really motivated the Americans to fight and to win.
https://alphahistory.com/americanrevolution/why-the-americans-won-the-revolutionary-war/

https://www.history.com/topics/american-revolution/battle-of-saratoga

How can I connect "Salvation" by Langston Hughes, "Sister Flowers" by Maya Angelou, and "What Suffering Does" by David Brooks? How can I use suffering in my own life as an example?

In his essay "What Suffering Does," Brooks writes that "recovering from suffering is not like recovering from a disease. Many people don’t come out healed; they come out different." He writes that people who suffer gain a quality that he refers to as "holiness" because they see life as a "moral drama" and try to make something good emerge out of suffering. 
In considering what to write about in your own life, think about an experience in which suffering caused you to change so that you saw things in a different light. In "Sister Flowers," Maya Angelou, then a child rendered mute for several years after being raped, is introduced by Bertha Flowers, an educated woman in her town, to literature. It is literature that brings Angelou, then called Marguerite, back to speech and to life. Angelou writes:

"To be allowed, no, invited, into the private lives of strangers, and to share their joys and fears, was a chance to exchange the Southern bitter wormwood for a cup of mead with Beowulf or a hot cup of tea and milk with Oliver Twist."

In other words, by reading, Angelou can escape her bitter childhood and experience other worlds; she emerges from suffering into something better and more beautiful.
In "Salvation," Langston Hughes writes about a boy who does not experience salvation as his aunt wants him to. He pretends to experience salvation, but he experiences the suffering of guilt. This suffering moves him to a new level of awareness about himself and the pain of feeling different and thinking differently from others. Again, in this situation, pain brings the person who experiences it to a new level of awareness. In your essay, think about a way in which suffering has helped you learn about yourself or the world, and use the experiences of Angelou and Hughes and the writing of Brooks to help you understand and comment on your experience. 

1,1/2,1/6,1/24,1/120 Write an expression for the n'th term of the sequence. (There is more than one correct answer.)

Consider the ratios between adjacent terms of the given sequence:
1/2 : 1 = 1/2,  1/6 : 1/2 = 1/3,  1/24 : 1/6 = 1/4, 1/120 : 1/24 = 1/5.
We see that the n -th term of the sequence may be obtained from the (n-1)-th term by multiplying by 1/n.
Therefore the n-th term may be predicted to be equal to
a_n = 1/(n(n-1)(n-2)...) = 1/(n!)

In To Kill A Mockingbird what are the items found in the tree knothole?

The knothole in the tree is Boo Radley's sole link with the outside world. He fills it with a number of objects, which is his unique way of reaching out to Jem and Scout. When returning home from school one day, Scout sees a piece of tin-foil sticking out of the knothole. Naturally, she's curious, so she opens up the foil and looks inside to find two sticks of gum. When she gets home, she tries one of them, just to make sure it isn't poisoned. But when Jem discovers where Scout found the gum, he makes her spit it out. Boo Radley, though an endless source of fascination for kids in the neighborhood, is still very much a boogie-man figure. Who knows what he might have done to that chewing gum?
Later on, Scout and Jem find another item in the knothole. This time it's a couple of old pennies or "Indian heads" from way back in the early 1900s, buried inside a velvet jewelry box. Boo's gifts get more interesting as the story progresses. Next up is a ball of twine. Jem's still quite wary, but after a few days, when he sees the twine still there, he takes it. Slowly but surely a bond of trust is developing between Boo and the children.
Boo can still be a bit creepy, though. The little figurines of Scout and Jem carved out of soap kind of look like voodoo dolls. Scout certainly thinks so, anyway. But Boo subsequently redeems himself with more conventional gifts such as some more chewing gum (a whole pack this time), a tarnished spelling bee medal from way back when, and a broken old pocket watch.
The kids are truly fascinated by Boo's unusual gifts. They're also deeply grateful and so try to leave him a thank you note in the knothole. Unfortunately they're not able to do this because Nathan Radley, Boo's brother, has filled in the hole with cement. He claims it's because the tree is diseased and needs to be cured. But the kids aren't really convinced; they realize that Nathan's trying to keep Boo isolated from the outside world. He's hacked into his brother's little line of communication with the Finch children, ostensibly to protect him. Yet we're never really sure of his motives. He remains as much of a mystery as Boo, albeit in a different way.

Use the derivative to determine whether the function y = 8e^x - 12 is strictly monotonic on its entire domain and therefore has an inverse function.

We are asked to determine if the function  y=8e^x-12 has an inverse function by finding if the function is strictly monotonic on its entire domain using the derivative. The domain is all real numbers.
y'=8e^x and y'>0 for all real x so the function is strictly monotonic (in this case strictly increasing) on its entire domain and thus has an inverse function.
The graph of the original function, which we can see is always increasing:

Single Variable Calculus, Chapter 5, 5.4, Section 5.4, Problem 24

Find the integrals $\displaystyle \int^4_0 (2v + 5)(3v- 1) dv$

$
\begin{equation}
\begin{aligned}
\int (2v + 5)(3v- 1) dv &= \int \left( 6v^2 - 2v + 15v - 5 \right) dv\\
\\
\int (2v + 5)(3v- 1) dv &= \int \left( 6v^2 + 13v - 5 \right) dv \\
\\
\int (2v + 5)(3v- 1) dv &= 6 \int v^2 dv + 13 \int v d v - \int 5 dv\\
\\
\int (2v + 5)(3v- 1) dv &= 6 \left( \frac{v^{2+1}}{2+1} \right) + 13 \left( \frac{v^{1+1}}{1+1} \right) - 5 \left( \frac{v^{0+1}}{0+1} \right) + C\\
\\
\int (2v + 5)(3v- 1) dv &= 6 \left( \frac{v^3}{3} \right) + 13 \left( \frac{v^2}{2} \right) - 5v + C\\
\\
\int (2v + 5)(3v- 1) dv &= 2v^3 + \frac{13v^2}{2} - 5v + C\\
\\
\int^4_0 (2v + 5)(3v- 1) dv &= 2(4)^3 + \frac{13(4)^2}{2} - 5 (4) + C - \left[ 2(0)^3 + \frac{13(0)^2}{2} - 5 (0) + C \right]\\
\\
\int^4_0 (2v + 5)(3v- 1) dv &= 128 + 104 - 20 + C - 0 -0 + 0 - C\\
\\
\int^4_0 (2v + 5)(3v- 1) dv &= 212
\end{aligned}
\end{equation}
$

College Algebra, Chapter 8, Review Exercises, Section Review Exercises, Problem 40

Identify the type of curve which is represented by the equation $\displaystyle 36x^2 - 4y^2 - 36x - 8y = 31 $
Find the foci and vertices(if any), and sketch the graph

$
\begin{equation}
\begin{aligned}
36 (x^2 - x + \quad) - 4(y^2 + 2y + \quad)&= 31 && \text{Group terms and factor}\\
\\
36 \left( x^2 - x + \frac{1}{4} \right) - 4 (y^2 + 2y + 1) &= 31 + 9 - 4 && \text{Complete the square; Add } \left( \frac{2}{2} \right)^2 =1
\text{ on the left and subtract. Then, add 9 on the right side and subtract 4}\\
\\
36 \left( x - \frac{1}{2} \right) - 4 (y + 1)^2 &= 36 && \text{Perfect square}\\
\\
\left( x - \frac{1}{2} \right)^2 - \frac{(y+1)^2}{9} &= 1 && \text{Divide by 36}
\end{aligned}
\end{equation}
$


The equation is hyperbola that has the form $\displaystyle \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$ with center at $(h,k)$ and horizontal transverse axis.
Since the $x^2$-term is positive. The graph of the shifted hyperbola is obtained by shifting the graph of $\displaystyle x^2 - \frac{y^2}{9} = 1$, by
$\displaystyle \frac{1}{2}$ units to the right and 1 unit downward. This gives us $a^2 = 1$ and $b^2 = 9$, so $a = 1, b =3$ and $c = \sqrt{a^2+b^2} = \sqrt{1+9} = \sqrt{10}$.
Thus, by applying transformations, we have

$
\begin{equation}
\begin{aligned}
\text{center } & (h,k) && \rightarrow && \left( \frac{1}{2}, -1 \right)\\
\\
\text{vertices } & (a,0)&& \rightarrow && (1,0) && \rightarrow && \left( 1 + \frac{1}{2}, 0 - 1 \right) && = && \left( \frac{3}{2}, -1 \right)\\
\\
& (-a,0)&& \rightarrow && (-1,0) && \rightarrow && \left( -1 + \frac{1}{2}, 0 - 1 \right) && = && \left( - \frac{1}{2}, -1 \right)\\
\\
\text{foci } & (c,0)&& \rightarrow && (\sqrt{10},0) && \rightarrow && \left( \sqrt{10} + \frac{1}{2}, 0 - 1 \right) && = && \left( \sqrt{10} + \frac{1}{2}, - 1 \right)\\
\\
& (-c,0)&& \rightarrow && (-\sqrt{10},0) && \rightarrow && \left( -\sqrt{10} + \frac{1}{2}, 0 - 1 \right) && = && \left( -\sqrt{10} + \frac{1}{2}, - 1 \right)
\end{aligned}
\end{equation}
$

Therefore, the graph is

Give examples of various kinds of irony.

In literature, there are three kinds of irony. Here are their definitions with some examples:

Verbal irony: this occurs when a character says one thing but actually means another. Often, this verbal irony takes the form of sarcasm. For example, saying that something is "great," when, in fact, it is a really terrible or silly idea. Similarly, in "The Tell-Tale Heart," the narrator says that he was kind and loving to the old man in the last week of his life. This is ironic because the narrator spent that week plotting the old man's murder. This is, therefore, the very opposite of being kind and loving.

Dramatic irony: this occurs when the reader of a story knows something ironic that the character does not. In Romeo and Juliet, for example, when Romeo comments that Juliet still looks alive when he sees her in the tomb, the audience knows she appears this way because she actually is alive.

Situational irony: this occurs when the opposite of what we expect to happen actually happens. For example, in Macbeth, Macbeth kills King Duncan and takes his throne because he believes that being king will satisfy him and his wife. In reality, being king makes him miserable and paranoid, and sends his wife mad, making it the very opposite of what he expected.

How does Edwards's tone in "Sinners in the Hands of an Angry God" influence his congregation?

In this sermon, Edwards combines a condemning tone with some pretty vivid figurative language to deliver a message that would shock his audience into turning from sin and accepting God's grace.
Edwards begins with a verse from Deuteronomy that sets the tone immediately: "Their foot shall slide in due time." This verse reflects God's anger with the Israelites who failed to follow God's laws, but Edwards quickly aligns the wickedness of his congregation with the Israelites. He uses the image of a slipping foot to show the precarious position of those living in sin; God will not tolerate it for long before he delivers a final judgement, casting those who choose sin into Hell.
Consider the imagery that Edwards uses in explaining how the members of his audience are on a path toward eternal devastation:

That world of misery, that lake of burning brimstone, is extended abroad under you. There is the dreadful pit of the glowing flames of the wrath of God; there is hell's wide gaping mouth open; and you have nothing to stand upon, nor anything between you and hell but the air; it is only the power and mere pleasure of God that holds you up.

This is a pretty terrifying image, painting a mental picture of being suspended on a thin layer of earth above a lake of brimstone—a "wide, gaping mouth" ready to capture and eternally torment each congregational member who has not chosen to truly follow God. Edwards also allows for another ending here: God currently saves each one of them from Hell and allows them time to choose a different path to a different eternity.
Edwards's tone achieved its intended effect. Reportedly, before he even finished his sermon, members of the congregation begged for salvation and fell into the floor, crying and screaming in terror. This was especially significant because this sermon was delivered in a town where there seemed to be a little pocket of "thoughtless and vain" holdouts in the Great Awakening revival which was sweeping the area. Yet the tone achieved quite an impact at the church where Edwards delivered it in Enfield and is still studied and remembered nearly 300 years later.

Edwards's tone is threatening, and he tries to convince his listeners to repent for their sins and return to God immediately by instilling fear. He uses horrible descriptions of the eternal fate that would await them if God decided to let them fall. He emphasizes that God could make this decision at any time. Edwards says, "'There is nothing that keeps wicked men at any one moment out of hell, but the mere pleasure of God." He calls God's will "arbitrary" in that God has no obligation to continue to preserve wicked people, and he could, at any moment, decide to stop.
Furthermore, Edwards argues that it is basically inevitable that we will fall without God's pleasure in restraining us. He says that we are "always exposed to destruction; as one that stands or walks in slippery places is always exposed to fall." The sinner can do nothing to prevent himself or herself from slipping right down into Hell's fire, and can only hope and pray that God will prevent it. In fact, the devil

stands ready to fall upon them, and seize them as his own, at what moment God shall permit him [....]. The devils watch them; they are ever by them at their right hand; they stand waiting for them, like greedy hungry lions that see their prey, and expect to have it, but are for the present kept back. If God should withdraw his hand, by which they are restrained, they would in one moment fly upon their poor souls.

This imagery is particularly frightening, again, and conveys Edwards's threatening tone, describing sinners as prey for terrible predator devils. Threats like this, I imagine, would convince many people to change their ways. Because the sermon has been retained, it seems clear that it made a significant impression on its audience.

Edwards uses threatening, emotionally-charged language to convince the congregation that they "hang by a slender Thread, with the Flames of divine Wrath" waiting beneath to consume them. Recorded accounts of the reception of the sermon at an Enfield, Connecticut church in July of 1741 describe people fainting, weeping, and crying out to ask how they can be saved. This reaction is understandable because the sermon details at length how formidable God's anger is and how torturous the eternal punishment will be if one lands in hell.
Near the end of the sermon, Edwards momentarily modifies his tone and alters his message. He briefly introduces the idea that salvation might still be possible when he claims it is "a Day wherein Christ has flung the Door of Mercy wide open." Edwards promptly reverts to his overall threatening tone and cautions that sinners must act quickly to avoid a terrifying fate. His last words observe that damnation is "now undoubtedly hanging over great Part of this Congregation."

In act 3, scene 1, what resolution does Hamlet come to?

In act 3, scene 1, Hamlet resolves to keep his promise of revenge. He has formulated a plan to use “players” (or actors) to reenact the murder of his father before his uncle Claudius and his mother. In watching their reactions, he hopes to discover if his father’s ghost was telling the truth about being poisoned. Hamlet wants to be absolutely certain that Claudius is guilty, because if it is true, then he feels he must kill Claudius, as he told his father's ghost he would.
Briefly, in act 3, scene 1, Hamlet considers not keeping his promise to his father’s ghost. In his most famous soliloquy, he briefly considers whether or not to commit suicide instead. Killing Claudius would be a risky feat, given that Claudius is king, and Hamlet knows it would be seen as a traitorous act. Probably, most everyone would disbelieve his “ghost” story and assume he did not want Claudius to marry his mother. Or they would assume Hamlet was jealous for the throne.
Not only does Hamlet know killing Claudius is risky in a life-threatening regard, but he also knows it will burn bridges between him and his mother for good. His mother seems to love Claudius, having married him just a month after the loss of her first husband. If he kills Claudius, his mother will grieve. Even though Hamlet is angry at his mother, some part of him longs for restoration with her. In Elizabethan times, marrying a family member was known to be a great sin, which is why Hamlet is disgusted with his mother for marrying Claudius.
Furthermore, Hamlet may feel that murdering the king would separate him from pure, godly Ophelia, whether in a practical sense (if he is caught) or in a spiritual sense (because she is a saintly sort). This may be why he decides to push her away later in this act.
Hamlet sees himself as stuck in a lose-lose situation, which is the reason for his suicide contemplation:

To be, or not to be: that is the question:Whether tis nobler in the mind to sufferThe slings and arrows of outrageous fortune,or to take arms against a sea of troubles,and by opposing end them? To die: To sleep.

Here, Hamlet contemplates if it would be “nobler” to face his future head-on, taking revenge on Claudius and suffering all the figurative “slings and arrows” that he expects to be thrown his way for doing so. He also contemplates whether suffering his fate would be better than opposing fate itself by killing himself so that he does not fulfill his evil promise.
Hamlet resolves not to kill himself, because he fears what comes after death. He reasons,

But that the dread of something after death,The undiscover’d country from whose bournNo traveller returns, puzzles the willAnd makes us rather bear those ills we haveThan fly to others that we know not of?Thus conscience does make cowards of us all . . .

Preceding this excerpt, Hamlet has just taken some time to list the various oppressions a person might avoid by being dead. But here, he exposes that he fears that the afterlife might be worse than reality. He describes death as “an undiscovered country / from whose bourn no traveler returns.” This is compounded by the fact that Elizabethans considered suicide to be a sin that would send you directly to hell in the afterlife.

Another possible anxiety in this speech is that Hamlet fears breaking his oath to his ghost father would lead to separation from him in the afterlife. Thus, Hamlet concludes that his “conscience” has made a coward of him.
The twofold resolution of Hamlet, to remain living and pursue the path of revenge, is a decision that seals his fate. The soliloquy shows just how unhealthy his state of mind is at this point in the play. Seeing his father’s ghost, realizing his uncle’s betrayal, and losing respect for his mother have taken a toll on his mental health and thrown him into a state of depression. To live, for Hamlet, is “to grunt and sweat under a weary life.” He sees every aspect of the future pessimistically, like a burden. He fails to see that suicide is never a good option. Furthermore, Hamlet is thinking with an “either/or” mentality. He does not consider any other hopeful alternatives, such as exposing Claudius, trapping him into a confession, talking to his mother or crafty Polonius (or seeking support from a myriad of other brave young soldiers faithful to him). Ultimately, his lose/lose situation is a concoction of an unhealthy mind.
One final note is that in Shakespeare’s plays, supernatural entities often appear to mess with the fates of kings. For example, the witches in Macbeth give Macbeth a prophecy with the singular purpose of causing his downfall. In Othello, his mother (a kind of witch doctor) indirectly causes Othello’s downfall by tying his happiness in love to an enchanted handkerchief she passes down to him. Many in Elizabethan times understood ghosts to be demons, evil entities that could take different forms in order to deceive and mislead the human beings they encountered. It is a possibility that the rashly made promise was not the request of Hamlet’s real father, but a demonic being intending to cast him into turmoil and bring about his destruction.

An Act Three, Scene One of Shakespeare's Hamlet, Hamlet delivers the iconic "To be or not to be" soliloquy. In this passage, he ponders whether he should continue to live when life is merely a matter of great suffering. Hamlet suggests that the only thing that is stopping us from shuffling off the mortal coil is a fear of the unknown and of what lays in wait for us in the afterlife, suggesting that it is preferable to "bear those ills we have" rather than "fly to others that we know not of." The resolution that Hamlet ends up making in this scene is to live rather than commit suicide.
Hamlet also makes a decision around his relationship with Ophelia when she comes to him in the midst of his thoughts. Hamlet firmly rejects her, telling her to "get thee to a nunnery."
Having spied on Hamlet and watched his erratic behavior, Claudius ends the scene by stating, "Madness in great ones must not go unwatch'd." 

why was jimmy cross embarrassed by the violent ending of the movie bonnie and clyde?

Jimmy takes Martha out on a date to the movies. The fact that it becomes their one and only date gives you some idea as to how successful it was. The film they go to see is Bonnie and Clyde, a landmark in cinema that pushed the boundaries of how violence is presented on screen. Generally regarded as a cinematic milestone, it's not exactly the most obvious date movie, although it's still a kind of love story.
During the final scene of the movie, when Bonnie and Clyde are shot and killed in an ambush, Jimmy places his hand on Martha's knee. She turns and looks at him in a way that suggests that he really ought not to have done that. Not surprisingly, Jimmy is hugely embarrassed by Martha's rebuff. For the rest of his life he'll always remember the feel of her tweed skirt and the knee beneath it as the sound of gunfire in the movie dragged on, slow and oppressive.

What is the conflict between the mother and narrator in "Two Kinds" by Amy Tan?

The conflict between Jing Mei and her mother stems from the latter's need to turn her daughter into a prodigy of sorts.
As the text tells us, Jing Mei's mother is focused on her mission to make Jing Mei a piano prodigy. This becomes a point of conflict between the two. While Jing Mei's mother thinks making her daughter into a star (on par with Shirley Temple) is the right thing to do, Jing Mei isn't so sure. She feels pressured by her mother and resents the need to put on an obedient front before relatives and the larger Chinese-American community.
Jing Mei just wants to be herself and be allowed to choose her own path in life. Thus, her mother's simultaneous need to keep up the family's reputation and to compel unquestioning obedience is seen as oppressing to her. In the end, Jing Mei rebels by playing horribly in a piano recital; in the aftermath of the disastrous performance, Jing Mei engages in an emotionally charged argument with her mother. She accuses her mother of not accepting her and expecting her to be something she's not.
Jing Mei's final words during the argument decide the result of the conflict, but it proves to be an unsatisfying victory. By cruelly referencing her mother's dead babies from her first marriage, Jing Mei manages to hurt her mother as well as to dispirit her. She reports the piano lessons stopped soon after, and she stopped playing the piano entirely. It is years before Jing Mei realizes the faith her mother had in her in that conflict-ridden period of her youth.

Is Macbeth a static or dynamic character?

Macbeth is a dynamic character because he does fundamentally change over the course of the play.  Initially, he is described as "brave" (1.2.18) and "valiant" (1.2.26), and he is awarded a new title as a result of his courageous and loyal service to the crown: he becomes the new Thane of Cawdor when the old one is executed for disloyalty.  Even his wife, the person who (potentially) knows him the best, thinks that he is "too full o' th' milk of human kindness" to ever do anything to betray his king—a man who is also his kinsman and friend (1.5.17).
By the end of the play, however, Macbeth has not only murdered Duncan (a crime for which he, admittedly, does feel a great deal of guilt), he also arranged for the murders of his best friend, Banquo, as well as Banquo's young son, Fleance.  (Fleance, unfortunately for Macbeth, succeeds in escaping with his life.)  In fact, at this point in the play he tells his wife, "We are yet but young in deed," meaning that they will have to continue killing in order to secure their positions as king and queen (3.4.176).  Additionally, Macbeth arranges to have the innocent wife and children of a rival, Macduff, killed in their home; he does so with no evidence of guilt or remorse.  By the end of the play, he has become a callous killer, much changed from the Macbeth we met in the beginning.

What are three examples of dramatic irony in "The Cask of Amontillado"?

Dramatic irony occurs when characters in a story are unaware of things known by the reader, thereby creating suspense or humor. In the case of Edgar Allan Poe's "The Cask of Amontillado," we know from the very first paragraph that Montresor, the narrator, seeks vengeance for the unspecified "thousand injuries" inflicted upon him by Fortunato. However, Fortunato remains blissfully oblivious to Montresor's malice until the end of the tale, meaning that many of the two characters' statements take on different meanings when viewed from the perspective of Fortunato or from the perspective of readers. Here are three examples:
When Montresor first brings up the cask, he tells Fortunato, "My friend, no; I will not impose upon your good nature." Fortunato accepts this compliment blindly, but readers know that Montresor is lying through his teeth.
Shortly thereafter, the two have this conversation:

"Enough," he said; "the cough's a mere nothing; it will not kill me. I shall not die of a cough."
"True—true," I replied . . .

Here, Montresor's agreement that the cough won't be fatal sounds to Fortunato like simple acknowledgement. Yet readers understand that Montresor knows the cough won't kill Fortunato, because Montresor plans to kill Fortunato long before the disease has time to turn deadly!
A third example occurs when Montresor describes his coat of arms, "A huge human foot d'or, in a field azure; the foot crushes a serpent rampant whose fangs are imbedded in the heel." Fortunato takes the statement at face-value as a factual description of the Montresor family crest. But readers know that Montresor believes that he represents the foot stomping upon the snake that is Fortunato, an inference supported by a Latin motto that translates to "no one attacks me with impunity." Moreover, in a second layer of irony, readers know that Montresor is obsessed and murderous. In all likelihood, they see Montresor, not Fortunato, as the treacherous serpent biting at the heel.
https://literarydevices.net/dramatic-irony/

What was the Red Scare's effect on the government, society, labor, and industry?

I'll address the Red Scare of the 1920s, since it was the one that involved the most change.  After Russia fell to the Bolsheviks in 1917-1918, America was afraid that Russian agents would seek to use labor unrest in order to topple the government.  A. Mitchell Palmer, Postmaster General under Woodrow Wilson and Attorney General under the Harding administration, began to round up anyone whom he thought was a Communist sympathizer.  This was a main reason that Progressive issues left the national political landscape--people were worried about anything that looked like too much government involvement in the private sector.  Palmer deported many suspected radicals whose only offense was having Eastern European or Jewish origins.  Politically, the U.S. put quotas on European immigration, limiting Eastern European immigration to a trickle at a time when millions of deserving refugees wanted to flee the Soviet Union.  Socially, it became popular to criticize leftist ideas as being "un-American."  Even though due process was not followed through with the deportations of suspected radicals, Americans began to look at recent immigrants from Eastern Europe suspiciously.  Labor union membership also dropped, and business began to actively ask government to get back to the laissez-faire practices of the late 1800s, which it soon adopted.  

In the book Old Yeller what is the first 34 pages about

Depending on the edition you have, the exact amount of plot in 34 pages may differ slightly. The following summary of the first 34 pages was taken from my edition, so be aware that there might be a bit more/less than whichever version you're referring to!
Old Yeller takes place shortly after the Civil War in Texas; the first-person narrator Travis Coates is left to take care of the farm and his family when his father heads off for a cattle drive. The morning after his father leaves, Travis meets Old Yeller, an ugly little dog who's gnawing at a piece of scrap meat. Arliss, Travis's younger brother, demands that the dog be taken in. Travis initially attempts to reject this idea, for he finds the dog to be dirty and ugly.
The next day, Travis goes hunting and kills a deer to feed his brother and mother. When he returns, he finds Arliss and Old Yeller dancing in the family's supply of drinking water. As a result, Travis swears a vendetta on the poor little dog!

What evidence shows that karana took pride in her appearance even though she was alone on the island?

Even though she lives alone on the island, Karana takes great pride in her appearance. In chapter 18, we are told that Karana takes time to make a yucca skirt for herself even as she spends time taming her pet birds, Tainor and Lurai.
Accordingly, Karana takes time to soften yucca fibers in water before she braids them into twine. She designs her skirt to have lengthwise folds. The beautiful skirt is open on both sides and hangs gracefully to her knees. Karana also makes a sealskin belt to go with the skirt. She completes her look with sealskin sandals, which she proudly wears on walks over the dunes.
Often, Karana makes a wreath of flowers to wear on these walks. More often than not, she also wears her beautiful new skirt and sandals during her lonely excursions. We can see that Karana takes great pride in her appearance from the way she fashions her clothes: they are both beautiful and durable.
In chapter 20, we have further evidence that Karana takes great pride in her appearance. We are told that she kills and skins cormorants from Tall Rock so that she could one day fashion a skirt for herself from cormorant feathers. Later, when Karana moves to the cave in the ravine, she makes sure to take along her prized skirt, beads, earrings, and the cormorant feathers. Despite her loneliness, she always makes sure that she looks presentable, a clear sign that she takes great pride in her appearance.

How does George seem to feel about his handicap?

George seems resigned to his handicap. He tells Hazel that he doesn't mind it anymore, as he's gotten used to wearing it. His handicap has become a part of who he is.
However, the forty-seven pounds of bird-shot padlocked round his neck is a great burden; Hazel notices that George is often exhausted from carrying the canvas bag of bird-shot round his neck. Although Hazel wishes that they could make a hole in the bottom of the bag in order to take out a few lead balls, George isn't thrilled with the idea.
He tells Hazel that, if he's caught taking out any of the lead balls, he'll get two years in prison, and he'll be fined two thousand dollars for every ball he takes out. He also argues that he will set a bad example for others. Before long, George maintains that they'll be back to the "dark ages" where everybody competes against everybody else. He contends that society will suffer the minute everyone gets away with disobeying the laws.
George doesn't realize the irony of his statements. He and Hazel are living under a tyrannical system of laws, but he seems fine with preserving the status quo. George has been brainwashed into believing that the laws in his country are meant to preserve freedom and justice, but in reality, they do just the opposite.

How did the government of Great Britain justify the Crimean War and the suppression of the Indian Mutiny?

In both cases, Great Britain was motivated by the need to protect its empire, which at that time stretched across the globe. The Russian Empire was perceived as a clear threat to the British, not least because of its vast size. As the Ottoman Empire was thought to be a check on Russian expansionism, the British joined the Crimean War on the Ottoman side in the hope of putting paid to Russian territorial ambitions.
Had the Russians prevailed, it is likely that at some point they would have turned their attentions further East, where vast swathes of its empire bordered territories occupied by the British. It was thought necessary, then, by both the British government and by public opinion, that the Russians should be put in their place, lest they get any ideas about encroaching on British interests in Asia.
As far as the Indian Mutiny was concerned, the British government regarded its suppression as essential to the maintenance of good order in the Empire. As with the Russians in the Crimean War, had the mutineers prevailed, it would've sent out a clear message that the British Empire could be undermined with impunity. It would've given Britain's rival colonial powers encouragement that the British Empire was built on shaky foundations and was vulnerable to sudden outbreaks of violence. That being the case, it was deemed essential for the colonial authorities to put down the rebellion by all available means.

The Crimean War was a response to the Russian occupation of the Danubian principalities (modern-day Romania) in 1853. The Turks declared war against Russia, and the French and British declared war on Russia after the Russians destroyed a Turkish fleet on the Black Sea. The British justification for the war was that they had to protect Turkish sovereignty against the Russians. However, the real motivations of the British were to check Russian power and reduce the Russian imperial interest in the Middle East. In the end, Russia asked for peace terms in 1856, and the Black Sea became neutralized.
The Indian Mutiny, also called the Sepoy Mutiny of 1857, began when the Sepoys, soldiers in the Bengali Army, revolted against the use of British cartridges for their rifles because they contained pork and beef fat. The beef fat was offensive to Hindus, while the pork fat offended Muslims, and a large-scale revolt ensued that was brutally crushed by the British. Atrocities were committed on both sides.
The British justified their suppression of the mutiny, which was in part a reaction against British rule in India, by characterizing the Sepoys as hideous animals and by vilifying them. In addition, the British press portrayed both Hindus and Muslims as belonging to violent and demonic religions. The Indian massacres of British civilians were used as evidence of the Indians' demonic and brutal natures. You can read accounts of the British coverage of the revolt—coverage that never questioned the rightness of British imperialism or the supposedly inferior nature of the Indians.

What does the Wife of Bath mean by "engender"?

The Wife of Bath uses this word in line 465, when she says,

For al so siker as cold engendreth haylA likerous mouth moste han a likerous tayl.

In modern English, this means something like: "For just as surely as cold engenders hail, a gluttonous or greedy mouth must have a lecherous tail." She is using the word in the same sense that it would be used in modern English—effectively, to engender something means to bring forth or give birth to it. So, here, she is saying that cold engenders hail in the sense that cold gives rise to hail, or leads to this kind of weather; hail does not exist when it isn't cold, and when it is cold, there is a high likelihood that this cold will produce hail or snow. She is using this as an example of something which is "siker," or sure; there is a direct causality between the two things. Something which is as likely as cold giving rise to hail is something that is very likely indeed.

Why did Milo recieve an unusual package?

At the beginning of The Phantom Tollbooth, Milo is describing his today. The reader learns that Milo is bored throughout his entire day. When he is at school, he is thinking about being at home and when he is at home, he is thinking about being at school. No matter what Milo is doing, it is does not seem to occupy him.
On this particular day, Milo gets home and finds a package that has a bright blue envelope next to it. The letter in the envelope reads, "FOR MILO, WHO HAS PLENTY OF TIME." When he opens the package, Milo discovers the phantom tollbooth and begins to assemble it. He starts his toy car and drives through the tollbooth, for he has nothing better to do.
At this point we don't know who sent the package and we don't know why it was sent but we can infer that Milo is going to learn something about time from the letter that was attached to the package. As Milo adventures through the Land's Beyond, he meets a variety of characters who teach him various lessons about the importance of using your time instead of wasting it.

In The Phantom Tollbooth, the main character is a young boy by the name of Milo. Milo's perennially bored and doesn't find anything in life remotely interesting. He especially hates learning; he thinks that seeking knowledge is the biggest waste of time there is.
All that changes, however, when Milo receives a mysterious package one day. Next to the package is a bright blue envelope, which reads "FOR MILO, WHO HAS PLENTY OF TIME." Inside the package, of course, is the phantom tollbooth of the book's title. Once he's assembled the tollbooth, Milo's going to rev up his toy car and drive straight through it down the highway to Dictionopolis. Milo's received the package—we don't know exactly who sent it—because someone in a strange, far away land has decided that he needs to be shown the way to the Kingdom of Wisdom. And we might readily agree that someone who finds seeking knowledge a waste of time is desperately in need of whatever wisdom this distant land can bestow.

what is the antiderivative of (-2x+5)^6

We are asked to find an antiderivative of (-2x+5)^6 :
This is essentially equivalent to evaluating the indefinite integral int (-2x+5)^6
Now int u^n du=1/(n+1) u^(n+1) with u a differentiable function of x. Here we can use u-substitution with u=-2x+5 and du=-2dx. Multiplying the integrand by -2 and (-1/2) (thus multiplying by 1) we get:
int (-1/2)(-2)(-2x+5)^6 dx or -1/2 int (-2x+5)^6(-2dx)
Integrating we get (-1/2)(1/7)(-2x+5)^7+C where C is some real constant.
Thus an antiderivative for (-2x+5)^6 is -1/14 (-2x+5)^7 .
** We can check by taking the derivative, using the chain rule to get:
d/(dx)[ -1/14(-2x+5)^7]=(7)(-1/14)(-2x+5)^6(-2)=(-2x+5)^6 as required.
** Note that we find an antiderivative, as a whole family of functions exist whose derivative is the given function. We can add any constant to the given antiderivative to get another antiderivative.
** An alternative to integrating is to use guess and revise. We might guess that the function we seek is (-2x+5)^7 . Upon checking by taking the derivative, we note that we are off by a factor of -14 (since the derivative is 7(-2x+5)^6(-2)=-14(-2x+5)^6 ), so we introduce the factor -1/14 to compensate.
http://mathworld.wolfram.com/IndefiniteIntegral.html

How does Ralph treat Piggy? Why do you think he treats him like that?

In order to answer this question, it helps to know when in the story the question is referring to. Ralph's attitude toward and treatment of Piggy changes over the course of the story. When Ralph and Piggy first interact with each other, Ralph is not especially kind. He's outright antagonistic in fact, and he acts aloof and maintains an "I'm better than you" type of attitude. This treatment starts right away. Piggy asks Ralph his name, but Ralph does not reciprocate the question. It's as if Piggy and his name aren't worth Ralph's time because he has already judged Piggy not his equal.

"What's your name?"
"Ralph."
The fat boy waited to be asked his name in turn but this proffer of acquaintance was not made; the fair boy called Ralph smiled vaguely, stood up, and began to make his way once more toward the lagoon.

Ralph is even insulting toward Piggy's family:

"Sucks to your auntie!"

Ralph then rolls around on the ground laughing hysterically when he finds out that people called Piggy "Piggy" back in school.

He dived in the sand at Piggy's feet and lay there laughing.
"Piggy!"

Ralph probably treats Piggy that way because that is how he probably always treats people that he doesn't consider his equal. Ralph is described as a fairly good-looking and capable kid. He's used to getting his way and having people look up to him. His initial treatment of Piggy is his teenage way of letting Piggy know that there is a dominance hierarchy on this island that is based on looks and fitness. Fortunately for Ralph, Piggy is a fairly forgiving and loving individual. As Ralph's ruling power begins to dwindle in the face of Jack's power grab, it is Piggy that stands by Ralph, and Ralph is grateful for that. Consequently, Ralph treats Piggy much more cordially as the book continues.

Ralph is initially rude and dismissive towards Piggy at the beginning of the novel. Ralph is relatively unsympathetic towards Piggy and calls him by his embarrassing nickname in front of the boys. Ralph also splashes water in Piggy's face, does not reciprocate his friendly advances, and does not allow Piggy to come along while they explore the island. One could assume that the reason Ralph is rude to Piggy is because he realizes that Piggy is an outsider. Unlike the other boys, Piggy is physically weak and overweight, has poor vision, and is relatively annoying. Ralph more than likely avoided unpopular, irritating boys like Piggy in school, which is probably why he initially treats him poorly. As the novel progresses, Jack and his hunters gradually descend into savagery and Piggy is one of the few boys Ralph can trust. Piggy and Ralph's relationship grows, and Piggy becomes Ralph's closest ally.

What metaphors are used in "The Passionate Shepherd to His Love" by Christopher Marlowe?

In this poem, the titular "passionate shepherd" is describing to his love how he will deck her in glorious array made only from nature. The shepherd is not a wealthy man, and, as such, the riches he offers to his love are drawn from the beauty of the nature which surrounds him.
The "melodious birds" in his natural world "sing Madrigals" to serenade the "falls"—but, as a madrigal is a very complex polyphonic type of choral music, we can understand that this is a metaphor indicating that the birds sing beautifully. They are not really singing madrigals, although the image of them serenading the falls is an example of personification. Likewise, the "kirtle" he offers will not really be "embroidered" with "leaves of myrtle"—the embroidery is metaphorical; the suggestion is that myrtle leaves will form some kind of decoration for the kirtle.
This poem is heavily laden with imagery, but I'm not sure whether there are many more outright metaphors in it. When the shepherd says that he will make his love a "belt of straw and Ivy buds," he may be making the belt out of unconventional materials, but will it still be a real, rather than a metaphorical, belt? Arguably, yes, as there is no requirement that a belt be made of leather or cloth. It is perfectly possible to make a belt out of these materials which would span the lover's waist, just as a crown of thorns is still a crown, albeit not one made of gold. It isn't metaphorical, like saying that someone is "crowned in glory." The same could also be argued of "beds of Roses." In this instance, the speaker is saying he will make a bed for his love out of roses; it will not literally be a bed as we might understand it, with mattress and pillows, but will serve the same function. So, isn't it really, rather than metaphorically, a bed? In large part, it seems that the shepherd's attempts to lure his love are not based in offering metaphorical delights, but in explaining how comfort can be provided to her through the bounty of nature; not, perhaps, in the ways she has been used to, but in ways which will serve just as well.

Christopher Marlowe's "The Passionate Shepherd to His Love" is full of figurative language, especially the metaphor. In contrast to a simile, which uses "like" or "as" to compare two different things, the metaphor uses an implicit (implied but not explicitly stated) analogy to equate two different things. Marlowe's "Shepherd" abounds with examples of metaphor. For example, at the end of the second stanza, Marlowe's shepherd describes a scene in which "Melodious birds sing Madrigals" (8). The birds aren't actually singing madrigals (a style of song sung by humans); rather, Marlowe's using a metaphor to compare the musical nature of the birds' songs to a madrigal. By doing so, he idealizes the birds' singing and paints a vivid image of natural beauty for his audience. Likewise, in the next line, the shepherd says, "I will make thee beds of Roses" (9), and here he's using a metaphor to idealize the comfort with which he and his love will be sleeping in his idyllic, pastoral paradise. There are other examples of metaphor in the poem, and they're pretty easy to spot once you get the hang of it, so I'd encourage you to search Marlowe's verse for more examples. 
https://www.poetryfoundation.org/poems/44675/the-passionate-shepherd-to-his-love

Does the musing of the banker come true?

Early in "The Bet," the banker muses that the lawyer will lose the bet because he will not cope more than "three or four years" in solitary confinement. He believes this because he is certain that "voluntary" confinement is much harder to bear than "compulsory" confinement, like a court-imposed prison sentence.
Not all of these musings, however, come true. The banker is right in his prediction that the lawyer will lose the bet but he loses by choice, not as a result of being unable to cope with the conditions of his confinement. While the lawyer did suffer from "loneliness" and "depression" in the early years of his confinement, he coped relatively well with being locked away. In fact, the lawyer only walks out five hours before the deadline and only forfeits his right to the money because he no longer values material wealth.

How does cost of operations relate to business strategy?

You can understand how the cost of operations relates to business strategy through focusing on three questions: (1) what is the business strategy, (2) what operations will be needed to accomplish the business strategy, and (3) what are the costs of the operations?
(1) Companies must develop a business strategy; usually a business strategy is done through strategic planning. Companies usually develop three-year or five-year strategic plans that will help the Board of Directors and executives make decisions to accomplish the goals they have for the company. (2) To accomplish the business strategy, the company must operate well, which means various departments such as finance, human resources, legal, customer service, and so on (however a business defines “operations”) need to function so that the departments contribute to accomplishing the business strategy. (3) Companies must decide how much money they will spend on operations to accomplish the business strategy. However, companies must be logical and realistic. For example, if a business strategy requires a company to open five new stores, the company must budget out how much it would cost to operate five new stores. If the company does not have that much money, the company must either raise the extra money it needs or decrease the number of stores it wants to open.
Without the adequate amount of money to cover the operating costs, the business strategy cannot be accomplished. As a result, it is important for companies to understand the operations costs that relate to their business strategies so they know if they have enough money to accomplish the business strategies, if they have to raise more money, or if they should modify their business strategies.

int (2x-5) / (x^2+2x + 2) dx Find or evaluate the integral by completing the square

We have to evaluate the integral: \int \frac{2x-5}{x^2+2x+2}dx
We can write the integral as:
\int \frac{2x-5}{x^2+2x+2}dx=\int\frac{2x-5}{(x+1)^2+1}dx
Let x+1=t
So, dx=dt
Now we can write the integral as:
\int \frac{2x-5}{(x+1)^2+1}dx=\int \frac{2(t-1)-5}{t^2+1}dt
                       =\int \frac{2t-7}{t^2+1}dt
                        =\int \frac{2t}{t^2+1}dt-\int\frac{7}{t^2+1}dt         --------------->(1)
 
Now we will first evaluate the integral \int \frac{2t}{t^2+1}dt
        
Let t^2+1=u
So, 2tdt=du
Hence we can write,
\int \frac{2tdt}{t^2+1}=\int \frac{du}{u}
            =ln(u)
             =ln(t^2+1)             
 
Now we will evaluate the second integral : \int \frac{7}{t^2+1}dt
\int \frac{7}{t^2+1}dt=7\int \frac{1}{t^2+1}dt
               =7tan^{-1}(t)
 
Substituting both these integral results in (1) we get,
\int \frac{2x-5}{x^2+2x+2}dx=ln(t^2+1)-7tan^{-1}(t)+C   where C is a constant
                      =ln((x+1)^2+1)-7tan^{-1}(x+1)+C
                       =ln(x^2+2x+2)-7tan^{-1}(x+1)+C

How does Ishiguro represent Kathy in Never Let Me Go?

This is a big question, one that drives a lot of the novel's tension. First of all, if you haven't already done so, you should definitely read Ishiguro's Never Let Me Go because it is a beautiful, powerful, and strange story. 
Kathy is the first-person narrator of Never Let Me Go, so right away Ishiguro gives us an intimate connection to Kathy, as well as seemingly complete access to her thoughts. Kathy controls how we view the entire story and its characters, including her own character.
Kathy's POV, however, can be frustrating at times. Ishiguro, through Kathy's narration, depicts Kathy as rather hesitant and unsure. Many times throughout the story, Kathy doubts the quality of her own memory, telling us that she can't be sure the way she recalls something is how it actually happened. Kathy's actions and life events, however, starkly contrast with her hesitant storytelling—Kathy is one of the oldest carers, and therefore one of the story's survivors (again, if you haven't read the story, I really don't want to spoil it for you by revealing more!). Kathy also describes to us the decisive way she moves forward in her life, telling us,"...I made my decision, and once I'd made it, I never wavered. I just got up one morning and told Keffers I wanted to start my training to become a carer." Thus, Ishiguro brilliantly places Kathy's narration at odds with her actual actions, disorienting the reader and revealing Kathy's unassuming and self-effacing nature.
Though Kathy seems to lack confidence in her storytelling at times, the events of the story and the ways in which Ishiguro contrasts her with the other characters, also reveal how stoic and strong she is. Because of her stoicism and strength (and because of the nature of the world in which the book takes place) Kathy tends to downplay the horrific details of some of the story's events. This tends to throw us readers off, and as a result, it takes us a bit longer to realize how terrible the story's situation really is.

Calculus and Its Applications, Chapter 1, 1.7, Section 1.7, Problem 42

Differentiate $\displaystyle f(x) = \frac{(5x - 4)^7}{(6x + 1)^3}$.



$
\begin{equation}
\begin{aligned}

f'(x) =& \frac{\displaystyle (6x + 1)^3 \cdot \frac{d}{dx} (5x-4)^7 - (5x-4)^7 \cdot \frac{d}{dx} (6x + 1)^3 }{[(6x + 1)^3]^2}
\\
\\
f'(x) =& \frac{\displaystyle (6x + 1)^3 (7) (5x-4)^6 \cdot \frac{d}{dx} (5x-4) - (5x-4)^7 (3) (6x + 1)^2 \cdot \frac{d}{dx} (6x+1) }{(6x+1)^6}
\\
\\
f'(x) =& \frac{(6x+1)^3 (7) (5x-4)^6 (5) - (5x-4)^7 (3) (6x+1)^2 (6) }{(6x+1)^6}
\\
\\
f'(x) =& \frac{35 (6x+1)^3 (5x - 4)^6 - 18 (5x - 4)^7 (6x+1)^2}{(6x+1)^6}
\\
\\
f'(x) =& \frac{(6x+1)^2 (5x - 4)^6 [35 (6x+1) - 18 (5x - 4)]}{(6x+1)^6}
\\
\\
f'(x) =& \frac{(5x - 4)^6 (210x + 35 - 90x + 72)}{(6x+1)^4}
\\
\\
f'(x) =& \frac{(5x-4)^6 (120x + 107)}{(6x+1)^4}

\end{aligned}
\end{equation}
$

What is an example of Man vs. Society in The Old Man and the Sea?

Hemingway's short novel The Old Man and the Sea is primarily a tale of man vs. nature, as the old man referred to in the title, Santiago, contends first with the ocean and then with the massive marlin he catches all alone on his small fishing boat off the coast of Cuba. This man vs. nature struggle takes up the bulk of the story. It is only in the opening of the novel that one can find any evidence of man vs. society. At the start of the story, Santiago has gone eighty-four days since his last catch, and his inability to bring home any fish has lead the other people in his village to label him "salao," or unlucky. The parents of his young apprentice take the label so seriously that they refuse to let their son continue to work for Santiago. This social stigma can be read as part of the reason that Santiago is so stubborn about going out to catch fish and bringing the marlin home with him. This can be considered man vs. society because of the way Santiago struggles against the town's perception of him.

In ‘‘Rikki- Tikki-Tavi" by Rudyard Kipling, how does Rikki get to the bungalow?

The best place to start looking for the answer to this question is the third paragraph of the story. This paragraph tells readers that a high summer flood washes Rikki-tikki out of his burrow and carries him down a roadside ditch. Rikki-tikki clings on to anything he can find in an effort to prevent himself from drowning, and his efforts exhaust him to the point where he passes out. When Rikki-tikki finally regains consciousness, he is in the middle of a garden path in bright sun. This is when Teddy shouts out,

Here's a dead mongoose. Let's have a funeral.

Fortunately for Rikki-tikki, Teddy's mother thinks that the mongoose might not be dead. She suggests that they dry him off and give him a closer look. Teddy, his mother, and his father take Rikki-tikki into the house, wrap him in warm blankets, and set him by the fire. Rikki-tikki eventually wakes up and becomes the curious and fierce mongoose that decades of readers have come to know and love:

"There are more things to find out about in this house," he said to himself, "than all my family could find out in all their lives. I shall certainly stay and find out."

Rikki-Tikki-Tavi, the mongoose in Rudyard Kipling's story, originally lived in a burrow with his father and mother. One day a summer storm washes him away from his home. The rushing water carries him into a roadside ditch. At first he is conscious during this flood, but at some point he loses consciousness. That's when the boy, Teddy, finds him. Teddy thinks the animal is dead, and he plans to conduct a funeral for him. But his mother thinks he might be alive and suggests they take him back to their bungalow to dry him off. So they bring him in from the garden path onto which he has washed up and dry him off in the big house. The man determines he isn't dead, merely "half-choked." They wrap him in a cotton cloth and get him warm, and he sneezes himself awake. 

Calculus of a Single Variable, Chapter 3, 3.2, Section 3.2, Problem 24

You need to notice that the given function is continuous on [0,1] and differentiable on (0,1), since it is a polynomial function.
You need to verify if f(0)=f(1), hence, you need to evaluate the values of function at x = 0 and x = 1.
f(0) = 0 - 0^(1/3) = 0
f(1) = 1 - 1^(1/3) = 1-1 = 0
Since f(0)=f(1) = 0 and the function is continuous and differentiable on the given interval, the Rolle's theorem may be applied, hence, there is a point c in (0,1), such that:
f'(c)(1 - 0) = 0
You need to find the derivative of the function:
f'(c) = (c - c^(1/3))' => f'(c) = 1 - (1/3)c^(1/3-1)
f'(c) = 1 - 1/(2*root(3)(c^2))
Replacing the found values in equation f'(c)(1 - 0) = 0 yields:
1 - 1/(2*root(3)(c^2)) = 0 => 1/(2*root(3)(c^2)) = 1 => (2*root(3)(c^2)) = 1 => root(3)(c^2) = 1/2
Raise to 3rd power both sides:
c^2 = (1/2)^3 => c^2 = 1/8 => c_1 = sqrt 1/(2sqrt2); c_2 = -1/(2sqrt2)
Notice that c_2 = -1/(2sqrt2) does not belong to (0,1).
Hence, applying Rolle's theorem to the given function yields that there is c = sqrt 1/(2sqrt2) in (0,1) , such that f'(c) = 0.

What aspects of Greek and Etruscan cultures can be seen in ancient Rome?

Because the Etruscans were influenced by the Greeks, it can be difficult to separate out the Greek and Etruscan influences on ancient Rome. However, there are certain distinct areas in which Etruscan influence can be seen. First, although we tend to think of gladiatorial displays as Roman, they actually were originally an Etruscan practice. Next, much of the engineering and technological prowess we associate with the Roman Empire had its origins in Etruscan practices. The Etruscans pioneered the development of road and bridge networks, use of vaults and arches in architecture, and underground sewer systems. 
The Greek influence on Roman culture was equally pervasive. This was famously expressed by Horace in the line:

Conquered Greece took captive her savage conqueror
and brought her arts into rustic Latium. (II.i: 156-157)

The Roman gods were, to a large degree, assimilated to the Greek pantheon and mythology. Roman students studied the Greek language and learned Greek literature, rhetoric, and philosophy in schools. Roman theater originally consisted mainly of adaptations and translations of Greek plays and similarly Roman art, especially sculpture, imitated Greek models. 

Why did Della sell her hair?

The Gift of the Magi is set before Christmas, with Della wishing to buy a gift for her husband. However, she only has $1.87, far too little to buy a worthwhile gift. As a result, she sells her hair for $20, and with the money she is able to buy a chain to attach to his watch. This ties together the two most prized possessions of their household: for Della, it is her hair she values above all other possessions, and for her husband, it is his watch.
This brings us to the twist ending of O. Henry's tale, because just as Della sold her hair to be able to buy a present for Jim, so too did Jim sell his watch in order to gain the funds to buy his wife a gift in turn (in this case, combs for her hair). Yet, one should not forget that O. Henry closes the story in a note of approval and admiration for these two characters:

"And here I have lamely related to you the uneventful chronicle of two foolish children in a flat who most unwisely sacrificed for each other the greatest treasures of their house. But in a last word to the wise of these days let it be said that of all who give gifts these two were the wisest. Of all who give and receive gifts, such as they are wisest. Everywhere they are wisest. They are the magi."

Ultimately, then, this story isn't one about human foolishness or poor decision-making. It's about a husband and wife both subordinating their own interests in favor of the other's. From this perspective, these gifts actually hold tremendous value due to the thought, consideration, and sacrifice that went into them.

Della in "The Gift of the Magi" is renowned for having beautiful hair. She may be poor—she doesn't have much in life—but she does have a lovely set of locks. Christmas is fast approaching, and Della so desperately wants to buy her husband Jim a special gift. But because she doesn't have enough money she has no choice but to sell the only precious item she possesses: her hair. So she cuts off some of her lovely locks and sells them to a fancy hairdresser, who will presumably use her hair to make wigs. With the money she earns from selling her hair, Della buys Jim a chain for his watch.
Unfortunately for Della, she's wasted her time. The gift she bought for Jim is absolutely useless as Jim sold his watch to buy Della a gift. The gift he bought is also useless: some brand new combs for Della's lovely hair.

How can we see evidence of Toffler's three wave theory in contemporary society?

Alvin Toffler's theory is based on his book, The Third Wave, which he published in 1980 with the intention of making it part of a trilogy.
He posits that the First Wave was the agrarian society that prevailed after hunter-gatherer cultures. This society flourished until the Industrial Revolution in the mid-eighteenth century. At that time, the Second Wave began: the Industrial Age. According to Toffler, key aspects of this Second Wave were the creation of the nuclear family, corporate structure, and a factory-type educational system. The Third Wave, according to Toffler, is the post-industrial age. This wave is characterized by a new emphasis on information technology. When Toffler published the book, computers were coming into practical, everyday use.
Historically, Toffler's ideas are sensible. The Neolithic Revolution, which occurred around 3,000 BCE (or BC, depending on your preferred mode of demarcation), began in the Fertile Crescent -- the region that is now Iraq, Syria, Jordan, Egypt, and several other Middle Eastern nations, but then was mostly comprised of Mesopotamia. Technologies in irrigation began here, allowing for crops to flourish. Hunter-gatherer groups had been nomadic. However, the construction of farms allowed for people to settle and form villages and, ultimately, societies.
Within these societies, nuclear families were constructed. In this regard, I would disagree with Toffler's claim that the nuclear family developed in the Industrial Age. More prosperous families developed an interest in maintaining property among those who shared bloodlines. Marriage facilitated proper inheritance. One also began to see the development of more fixed gender roles a bit later in Akkadian societies.
Toffler is certainly right in stating that corporate culture began in the Industrial Age. Arguably, so did globalism. In 1793, the cotton gin was invented by Eli Whitney. A decade later, the United States made the Louisiana Purchase, annexing territory that was ideal for the cultivation of cotton. Demand for textiles was very high in both New England and Great Britain. Textile mills in both places depended on Southern cotton. 
The Industrial Age, which we began to call "manufacturing" after World War II, flourished until the mid-twentieth century. However, by the 1980s, many manufacturing jobs had disappeared from both the United States and Great Britain. Though it is important to note that manufacturing remains strong in some Western nations, such as Germany, Toffler, an American writer, probably had the United States foremost in his mind in his analysis of Western societies.
Our contemporary age is still transitioning from the Second Wave to the Third Wave, just as those who lived during the Industrial Revolution were struggling to transition from the First Wave to the Second Wave.
The concern, generally, is over jobs and the fear of being replaced by machinery and automated systems. Though capitalist systems thrive on innovation and progress, these features also inevitably leave behind those who cannot keep up with progress. Certain regions that once relied on manufacturing, and became wealthy as a result of it -- think of Detroit or Flint, Michigan during the postwar period -- have since fallen into depression, while cities that were built or transformed within the Information Age (e.g., Silicon Valley, Seattle, San Francisco) are very wealthy. 
Toffler also predicted other technological advances in the Third Wave that are now commonplace, including cable television, cloning, and mobile communication. 

Why was the prince called a happy prince?

Like Siddhartha Gautama, the founder of Buddhism, the prince in this story was sheltered from all knowledge of suffering. The prince explains to the swallow that was he happy during his lifetime because he lived in a palace surrounded by a high wall that blocked out anything unpleasant. He played with companions in his garden and led dances at night. He never explored what was outside the palace walls. Therefore, because he was untroubled, he was called the Happy Prince. He says he was happy because at that time he equated pleasure with happiness.
Following his death, however, the Prince (like the Buddha after he grew up) can see the suffering and poverty all around him. This has made him very sad, especially because, as a statue, he is immobilized. Nevertheless, when he meets the swallow, who can fly around, he now has a way to help other people. He becomes happy in a new way, through insisting that the costly gold and jewels of his statue be given to help the poor. The swallow distributes the wealth to those most in need. This is a deeper and more profound form of happiness than what the prince experienced in life, and he is rewarded at the end with a place for his heart in heaven.

During his life on earth, the prince was called the Happy Prince because he was always happy.
Accordingly, the prince never knew unhappiness when he was alive, as he lived in splendor and ease. He lived in a palace where sorrow was never allowed to enter; a high wall separated his palace grounds from the outside world. Because his existence was so sheltered, the prince never knew of the misfortunes and miseries his subjects experienced in their own lives. The prince tells the swallow that his own courtiers used to call him the Happy Prince, only because he was always found to be cheerful and untroubled.
The prince remembers living a life that consisted of largely unvarying daily pleasures; during the day, he cavorted with his companions in the garden, and in the evenings, he led dances in the Great Hall of the palace. It was only when he died that he became sad, and this was because they placed his statue so high up on the city grounds that he could see the surrounding miseries and degradation in the city.

Comment on the title of the play The Duchess of Malfi.

The play Duchess of Malfi is named after the character and real life historical tragic figure of Duchess of Malfi who was the regent of the southern Italian town of Amalfi between 1498 and 1510. She was a popular regent, as voiced at the beginning of the play by her lover and, later, husband Antonio:

(Her) days are practised in such noble virtue
That her night, nay more, her very sleep
Are more in heaven than other ladies' shrifts'

While it is debatable whether of not she's the main protagonist—she and her children are killed as early as act 4—it is her decision to marry someone whom her family sees as beneath her that starts the play's tragic chain of events, eventually leading to the deaths of all the play's principle characters.
As her brother Ferdinand states in the final scene, just after the Bosola and the Cardinal have suffered their fatal stab wounds,

My sister, O my sister! there's the cause on't

She is the cause, but through no fault of her own (Webster represents her as a virtuous figure through out). The fault lies with her brothers, the Cardinal, and Ferdinand. The Cardinal wants her to remain an unmarried window so the land of Amalfi remains in the family, and Ferdinand wants her to remain an unmarried widow so he can keep her to himself.
From this perspective, the play's title contains the two things most central to the play's antagonist's atrocious behavior: the land of Amalfi and the Duchess herself.

The play’s full title, The Tragedy of the Duchess of Malfi, presents an interesting conundrum as to what type of tragedy befalls and to whom. Though based upon an actual historical figure and events, some debate whether the play’s tragic focus is truly the Duchess (as the title so easily leads one to believe), when it also could be Bosola’s or something much larger in scope: a depiction of the misfortunes brought about by a corrupt system or world, for instance. Then, one also must consider the nature of the tragedy or revenge play to determine which aspects—personal, social, or political—are weightiest. The themes that Webster incorporated into his drama also provide clues to the title’s scope of influence. Among these are morality, duty, class conflict, deception, marriage, and family.

Is it probable that cyanobacteria were the first organisms to photosynthesize?

That depends on what mean by "cyanobacteria," and what kind of photosynthesis you have in mind. 
Photosynthesis is the conversion of light energy into chemical energy that an organism can use as fuel. There are two different kinds of photosynthesis -- oxygenic and anoxygenic. Oxygenic photosynthesis releases oxygen as a byproduct; anoxygenic photosynthesis does not.
There is widespread agreement among biologists that cyanobacteria were the first organisms to engage in oxygenic photosynthesis. However, the first photosynthesizing organisms ("phototrophs") may have been the ancestors of cyanobacteria.
Although biologists are still attempting to reconstruct this evolutionary history, the emerging consensus is that anoxygenic photosynthesis evolved first. In 2006, Armen Mulkidjanian and his colleagues conducted a genetic study of extant photosynthesizing bacteria species in order to reconstruct the likely evolution of photosynthesis. Their results led them to argue that the first photosynthesizing organisms were anoxygenic ancestors of cyanobacteria (see pnas.org link below). Later, these ancestral, photosynthesizing organisms gave rise to oxygenic cyanobacteria.
These events are thought to have taken place between about 3.5 and 2.5 billion years ago. After the emergence of oxygen-producing cyanobacteria, huge volumes of oxygen were pumped into the earth's atmosphere for the first time.
https://www.livescience.com/51720-photosynthesis.html

https://www.newscientist.com/article/mg19125654.200-revealing-the-dawn-of-photosynthesis/

https://www.pnas.org/content/103/35/13126.full

Who wrote the United States Constitution? What role did James Madison play?

The Constitutional Convention occurred in 1787 with George Washington presiding.  James Madison created the foundational document for the U.S. Constitution.  He is considered the Father of the Constitution because of this.  Other Founding Fathers were also involved in the creation of the Constitution.  
James Madison's original document was called the Virginia Plan.  It contained a total of fifteen resolutions.  It discussed the separation of powers, which was a new idea compared to previous documents.  The document also suggested representatives based on population.
The ideas in the document which became the U.S. Constitution were also influenced by other great thinkers.  Thomas Paine and Thomas Jefferson were among those who wrote about liberty and freedom.  These ideas were important to the U.S. Constitution.  It is believed that a man named Gouverneur Morris put many of the ideas of the Constitution into words.  There were almost a dozen other men who also contributed to the writing and composition of the document.
https://www.archives.gov/founding-docs

Who is a round character in the story "Lily Daw and the Three Ladies," and give three examples of how Welty developed the character you name.

Typically, when we talk about a "round character," we mean a character in a story that is fully developed. They feel like they have some reality to them, we learn a lot about them, and they are usually the ones who encounter an obstacle and are changed in some way because of it. The term "round character" is in contrast to someone we would term a "flat character." A "flat character" is someone that typically remains the same through the text and does not have many different facets or layers to uncover.
The characters in "Lily Daw and the Three Ladies" remain largely the same, but the characters that would make the best case for a round character are Aimee and Lily.
While Aimee does not appear to change very much throughout the text, she is the one who makes the discovery that Lily's marriage is real and rushes around trying to help Lily and her prospective lover. The reader gets more emotion from Aimee than from the other women, and we are told that "Aimee cried sorrowfully, as she thought how far away it was" (Welty). Aimee is also the one who faces the conflict most directly, as she discovers the identity of the man that Lily plans on marrying and has to try to stop the train. At the end, it is Aimee again who realizes that Lily has left her hope chest on the train, and she feels sadness for her.
Lily also does not appear to undergo significant change, and she does not speak very much for herself. However, she gets the reader's real sympathy because of the way she is pushed around. When the other women come to tell Lily she is going to Ellisville, she is pathetically packing her tiny hope chest. When the women discover that the man is real and try to get Lily off the train again, Lily begins to cry and is confused at the change that the women have now pushed her into. Finally, when she does reunite with the man she spoke about, he "kisse[s] Lily with a smack, after which she hung her head" (Welty). The reader gets the sense that Lily is pushed into one situation after another without really understanding or enjoying her circumstances, and she is the most sympathetic figure in the story.
https://archive.nytimes.com/www.nytimes.com/books/first/w/welty-stories.html?_r=1

Why would a country have a low GDP per person? Why is Madagascar so poor?

That's the trillion-dollar question, isn't it? If we knew precisely what determined the GDP of a country, development economics would be a solved problem.But I can tell you some things that most economists believe are important to the development (or lack thereof) of a country's GDP.The first is institutions. In order to achieve and maintain prosperity, a country must have institutions that support economic development, which generally means some form of democracy and free market economy. The precise form can vary (Switzerland is more democratic than the United Kingdom, while the United States has more free markets than France, yet all are about equally prosperous), and there are some exceptions, especially in the short run (Saudi Arabia has horrible totalitarian institutions, but will remain wealthy as long as the oil continues to flow). But in general, you need to have a free government and a free market if you really want to achieve a high level of economic development.Does Madagascar have these things? Not really. Madagascar has a moderately free government with a high level of corruption, and in the 1970s implemented a number of socialist economic policies that undermined investment and stunted economic growth. Starting in the 1980s many of these policies were reformed, and Madagascar transitioned to a more liberal export-oriented economy. But with corruption high, investments intended to support growth have been skimmed off to line the pockets of individual government officials.Which brings me to the next two key features: infrastructure and education. Successful economies always have a large infrastructure base of roads, rails, electric power, and clean running water; most regions of Madagascar have none of these things. Prosperous countries have educated populations where almost everyone is literate and a large fraction go to college; Madagascar has hardly anyone with college degrees and a large fraction of the population who are outright illiterate.Trade is another important source of wealth, which Madagascar has had trouble maintaining due to their geographic isolation. Their large sea coast could help with trade (it is certainly better than being landlocked), but they are very far from most of the major trade hubs like Tokyo, New York, London or Singapore. Notice what I haven't mentioned: Land or natural resources. Madagascar does have many natural resources, but these actually turn out not to be all that relevant to economic development. They don't hurt necessarily (at one time economists actually thought they did hurt, an idea called the "Resource Curse", but it turns out that the problem isn't the natural resources per se but the extractive institutions that tend to build up around them if they are the only source of wealth. The US and Norway are quite rich in natural resources, and they do just fine because they have sound institutions.)To make matters worse, Madagascar has a long history of colonialism and extractive institutions, starting centuries ago with French colonization. Even after independence they were never fully able to shed these institutions, and still have an economic system that works better at enriching a small few than it does at achieving overall prosperity.Reforming Madagascar to achieve economic growth will be no small task, but we must not give up hope. Other countries such as Korea and Japan have lifted themselves out of similar levels of poverty in the past. Once we figure out exactly how, perhaps we can achieve it elsewhere as well.
https://www.heritage.org/index/country/madagascar

https://www.infoplease.com/world/countries/madagascar

https://www.wildmadagascar.org/overview/FAQs/why_is_Madagascar_poor.html

When Tom gets his paper back, does he finish his work?

Tom never finishes his work. When he breaks his own window and manages to get back inside his apartment, he decides not to stay there and rest as he has planned. Instead he gets his hat and topcoat and prepares to go to the movie-theater to find his wife. He takes the crumpled yellow sheet out of his pocket and lays it on his desk, weighting it down with only a pencil. But when he opens the door to leave his apartment, the warm air from the hallway rushes in creating a draft. The draft is much stronger now than it had been the first time it caused his precious yellow sheet to be blown out the narrow opening in the window. The difference is that now the whole window pane is gone. He broke it to get inside and had then crawled through to safety after

...picking away the protruding slivers and big wedges of glass from the window frame, tossing them in onto the rug.

Ironically, the draft picks up the yellow sheet again. This time Tom isn't even concerned when he sees his worksheet perversely following the identical route it had taken before. The story ends with these words:

He turned to pull the door closed and the warm air from the hall rushed through the narrow opening again. As he saw the yellow paper, the pencil flying, scooped off the desk and, unimpeded by the glassless window, sail out into the night and out of his life, Tom Benecke burst into laughter and then closed the door behind him.

 
 

What are the burdens that Holden carries? I know the death of his brother is one but need to know what else.

In addition to Allie's tragic death, Holden also witnesses the suicide of one his classmates while attending Elkton Hills. Holden elaborates on this traumatic experience in chapter 22 during a conversation with Phoebe. When Phoebe asks Holden to name one thing he likes, Holden can only think of James Castle. Holden recalls the time when James Castle was being bullied by his classmates and refused to take back something he said about Phil Stabile. According to Holden, Phil Stabile and six of his friends locked James in his room and proceeded to beat him severely.
Holden then remembers hearing the sound of James Castle's body hitting the ground. Holden also recalls running outside and witnessing James Castle's lifeless body mangled on the pavement. James Castle was even wearing Holden's turtleneck sweater that he let him borrow.
Holden also alludes to adults sexually molesting him or making advances towards him at the end of chapter 24. After Mr. Antolini wakes Holden in the middle of the night by rubbing his head and admiring him, Holden immediately leaves the apartment. Holden then says,

When something perverty like that happens, I start sweating like a bastard. That kind of stuff's happened to me about twenty times since I was a kid. I can't stand it (Salinger, 104).

While it is hard to determine whether or not Holden was actually molested by an adult, one cannot dismiss such a comment.

In interpersonal communication use the three-part management strategy (state my feelings, describe the other person's behavior that I object to, and state a cooperative response that we both can live with) for: 1. My friend helped me get a job at his company. He reminds me every time he needs something and tells me if it wasn't for him I wouldn't have a job here. 2.Someone who is always making statements like, "You can't be serious!" and "You've got to be kidding" whenever I talk about a promotion or making plans.

Interpersonal communication strategies are useful in making it possible to talk to somebody about a conflict. In your question you listed three parts of a strategy designed to accomplish this without letting emotions or defensiveness get in the way. Here's how I would deal with each of the two scenarios using this method.
Scenario 1
State your feelings:
It is important to acknowledge the other person's feelings right away. People are naturally defensive, so your first job is to soften that a little, even before you state your own feelings. I'd say something like: “I really appreciate you helping me get this job, but right now I'm responsible for doing xyz. I'll help you if I can once I've done what I am required to do.”
Describe the other person's behavior:
“When you say that if it wasn't for you I wouldn't have this job, you make it sound like you don't think I'm doing the job well enough to deserve it. That's not what you mean, is it?”
State a cooperative response that you both could live with:
Try to offer them something they want if you can, but still be firm about doing what you have to do.
“I'll be glad to help you whenever I can, but I have to make sure I complete my job properly first. “
Scenario 2
State your feelings:
“I thought I was doing a good enough job to deserve a promotion.”
Describe the other person's behavior:
This time, since they've been making negative comments, give them a chance to offer some input.
“Your comments make it sound like I'm doing something wrong. Do you think I need to improve in some area before I should seek a promotion.”
State a cooperative response that you both could live with:
Show a willingness to cooperate.
“I'll be glad to think about how to improve in whatever areas you mention.”
Notice that in both of these scenarios, we are acknowledging the other person's feelings and views respectfully, while giving them a chance to respond. It isn't just a matter of saying something like “Hey, listen here buddy. You're going to stop doing this and that's all there is to it.”
Also, we can't always expect these types of conflicts to be solved immediately. It may seem at first like you have been unsuccessful, but sometimes, after people have had a chance to mull things in their mind for awhile, they will change their behavior, if you stay calm and courteous.  

How does Bud fulfill the meaning of his name throughout the course of the novel?

At the beginning of the novel, Bud recalls a conversation he had with his mother about his name. She would remind him that his name was Bud and that he should never let anyone call him Buddy. She told him that Buddy was a dog's name and that when people call him Buddy they are being "false-friendly." She then told him that a bud is a "flower-to-be" that is waiting for the right love and warmth to unfold. Throughout the novel, Bud refuses to allow people to call him Buddy and develops friendships with various caring individuals. Eventually, Bud travels to Grand Rapids, where he meets his grandfather, Herman Calloway. The Dusky Devastators of the Depression take Bud into their group, and he becomes a part of their family. Bud fulfills the meaning of his name by finding a home full of caring, sympathetic individuals. Bud is happy to be surrounded by people who support and love him. Myers implies that Bud will enjoy a life full of happiness where he will blossom into a talented, morally upright young man. 

What was the final decision in the Plessy v. Ferguson case?

In Plessy v. Ferguson (1896), the Supreme Court ruled that the doctrine of "separate but equal" was constitutional. That meant that racial segregation was legal in public places, including trains and other modes of transportation, schools, drinking fountains, and other public spaces. This doctrine meant that the United States, particularly in the south, maintained separate public accommodations and schools for African-American and white people. Though the schools that African-American children attended were supposed to be equal, they were in fact not equal. Instead, African-American children in the south attended schools with far fewer resources than white schools. The doctrine of separate but equal was not overturned until the 1954 Supreme Court case Brown v. Board of Education, in which the court ruled that separate but equal schools were inherently unequal (meaning that even if these schools had the same resources, the fact that there were some schools for white people and some for African-Americans created inequality). 

What do Tess's words and actions say about the village and the nature of the ritual in "The Lottery"?

Most of the words that come out of Tessie Hutchinson's mouth are basically complaints about the fairness of the process of the lottery. That, in itself, shows the asinine nature of the practice, and the equally ignorant nature of the followers of such practice. It is clear the villagers cannot be bright and independent thinkers, otherwise this ridiculous tradition would have never become a reality in the first place. Also, if the villagers were rational beings, the lottery would have been abolished by the common sense of the people. Instead, we see the opposite.
The villagers are entirely bound to the macabre tradition of the lottery. Their day depends on it. There are roles to be followed in the process, and there are no questions raised prior to the moment when it is the turn of the "winner" to die. Still, it is admitted in the story that the villagers do not quite know why they are really doing the lottery.

Although the villagers had forgotten the ritual and lost the original black box, they still remembered to use stones. The pile of stones the boys had made earlier was ready; there were stones on the ground with the blowing scraps of paper that had come out of the box Delacroix selected a stone so large she had to pick it up with both hands and turned to Mrs. Dunbar. "Come on," she said. "Hurry up.

Tess's words of "It's not fair," reflect the blindness of following a practice for the sake of following. It also reflects the villagers' selfishness: Notice how Tess does not say anything about the fairness, or lack of fairness, of the lottery until the name of her family comes up, and until it is her turn to be stoned to death. Perhaps it has been the same with all the other villagers who also thought, like Tess, that the lottery "was not fair" when it was all too late.

Give an analysis of the purpose and language in Act 1 scene 5.

Act 1 Scene V is an absolutely crucial part of Hamlet. It establishes the overriding theme of the play, provides a catalyst for the protagonist's actions, and begins the process of Hamlet's development as a character. In this scene, the ghost of Hamlet's father appears to him on the battlements of Elsinore and tells him that he was murdered by his brother Claudius, who also seduced his wife Gertrude. Here, Shakespeare is establishing the play's dominant conflicts: the external conflicts between Hamlet, Claudius, and Gertrude; and the internal conflict that Hamlet has with himself—how to reconcile his nature as a thoughtful Christian prince with his desire for terrible revenge. And from now on, the entire play will hinge upon this central theme.
Hamlet is an intelligent young man, highly educated and prone to introspection. He also takes very seriously his duty as a Christian prince, which among other things, involves following Christ's injunction to turn the other cheek. The Ghost, however, forces Hamlet to turn his back on his Christian principles as embodied in the New Testament and instead embrace the Law of Talon—an eye for an eye, a tooth for a tooth—as found in the Old Testament. This irruption of a wholly unfamiliar world, a world of a dark and distant past, into Hamlet's life completely upends all that he'd previously held holy and just. Now he realizes the full measure of Claudius's evil, he must reject everything he's ever believed in to avenge his father's brutal murder.
It's notable that the disruption of Hamlet's life by a voice from the past was accompanied by an archaic form of language used by the Ghost to tell the tragic tale of his demise. He uses expressions such as "murder most foul," and "harrow up thy soul," which even at the time would've been considered somewhat old-fashioned. But such language is entirely appropriate to a voice from the spirit world, a voice that hearkens back to the Catholic belief in Purgatory, something that would've been rather exotic to Shakespeare's overwhelmingly Protestant audience.
Religious language and imagery are also used by the Ghost in recounting the sordid details of his murder. He was asleep in the garden when Claudius crept up on him and poured poison down his ear. The parallels with the Garden of Eden are not hard to detect. Hamlet's father was in Paradise before Claudius came along, like the snake who tempted Adam and Eve:

The serpent that did sting thy father’s life Now wears his crown.

By using the imagery of the Garden of Eden and the corruption of humankind by Satan disguised as a serpent, the Ghost is shaking Hamlet out of his complacent worldview, one characterised by a combination of New Testament Christianity and Renaissance humanism. That said, although Hamlet immediately vows to take revenge, he will do so in such a way that he will remain true to his ideals, or as true to them as he can be. That is why he's going to pretend to be mad, or to "put his antic disposition on;" it also explains why he restages The Murder of Gonzago to try and expose Claudius's villainy; and it's the main reason why Hamlet appears to vacillate so much in carrying out his act of vengeance. He will carry out revenge, alright, but in a method more becoming of his values.
There's a certain lurid quality to the Ghost's language, especially in relation to his murder. He refers to a "vile and loathsome crust" upon his body. He also pulls no punches in relation to his betrayal at the hands of Gertrude and Claudius. Theirs is a sordid, corrupt relationship in which the royal bed of Denmark is a "couch for luxury and damned incest." This is very far from the kind of sophisiticated, intelligent discourse that the student prince is used to and in which he usually engages. Yet the words of his father's ghost have their effect, and it's noticeable that, as the play progresses, Hamlet's own language will become progressively more vulgar and profane. One only has to think of his vicious tongue-lashing of Ophelia in Act III Scene I: "Get thee to a nunnery." And even more extreme is the intemperate langauge used by Hamlet towards Getrude in Act III Scene IV:

Nay, but to live In the rank sweat of an enseamèd bed, Stewed in corruption, honeying and making love Over the nasty sty—

There is a close parallel here between Hamlet's words and those of the Ghost in relation to Gertrude and Claudius's "incestuous" marriage. This is not the language of a son towards his mother, much less of a highly intelligent Christian prince; it is the language of hatred and resentment, a holdover from an atavistic pre-Christian world in which brutality and revenge were the norm. The Ghost, and the language that he uses, act as a reminder to Hamlet and to the audience, that for all our outward displays of civilized learning and religion, in the very depths of our souls, we are never too far away from a more primitive past.

Could someone share some personal impressions, to help me understand the key points of the Every Student Succeeds Act (ESSA)? And, how does it relate to the four groups in education known as English Language Learners (ELL), Learning Disabled Learners (LDL), Advanced Learners (AL), and Economically Disadvantaged Learners (EDL)?

A key feature to the Every Student Succeeds Act is that it gives individual states new flexibility on how to exactly improve education in various districts.  For example, states can pick which long term and short term goals to work on; however, the goals need to address test proficiency, English language proficiency, and graduation rates.  The focus is to require that "all students in America be taught to high academic standards that will prepare them to succeed in college and careers." 
Another key point of the Every Student Succeeds Act is that it increases focus on lower performing schools.  States are required to "identify and intervene" in a high school that has a graduation rate of 67% or lower.  What's good about the focus is that the ESSA allows individual school districts to figure out and implement a plan to help the struggling schools.  States will monitor the progress of the school, and if no improvements are made, the state can take over the school to implement its own plan. 
Standards based education is still a priority with the ESSA in that states are required to adopt "challenging" academic standards; however, individual states are free to pick their own set of standards.  In fact, the ESSA expressly prohibits the Secretary of Education from even encouraging a state to pick one set of standards over another. For example, no state would be required to adopt the Common Core State Standards.  
The ESSA has an increased focus on English Language Learners.  This is evidenced by the fact that ELLs' accountability has been moved from Title III to Title I. By a student's third year in the country, his/her test scores are treated no differently than any other student's test scores. 
As for students with learning disabilities, the ESSA maintains its focus on high standards.  Only 1% of students (overall) are going to be allowed to take alternative tests.  A consequence of this 1% cap is that the testing needs of many students with learning disabilities could be ignored. 
Regarding Advanced Learners and Economically Disadvantaged Learners, a provision in the ESSA allows states to use up to 3% of their Title I money to create new programs.  The aim is for money to go to schools and districts in lower income areas, which should help those students reach higher levels of achievement.  Some of those students are likely Advanced Learners; therefore, the ESSA is attempting to spread funding around so that more students can be Advanced Learners.  Michael Magee, the CEO of Chiefs for Change, says the following about the ESSA.

 ". . . it will help a great many schools create new tutoring programs or Advanced Placement classes, or devise new ways to use technology."
https://educationpost.org/this-little-provision-of-essa-could-make-a-big-difference-for-low-income-kids/

Single Variable Calculus, Chapter 7, 7.4-1, Section 7.4-1, Problem 44

Find the derivative of the function $\displaystyle y = \sqrt[4]{\frac{x^2 + 1}{x^2 -1}}$, using log differentiation

$
\begin{equation}
\begin{aligned}
\ln y &= \ln \sqrt[4]{\frac{x^2+1}{x^2-1}}\\
\\
\ln y &= \ln \frac{\sqrt[4]{x^2+1}}{\sqrt[4]{x^2 - 1}}\\
\\
\ln y &= \ln \sqrt[4]{x^2 + 1} - \ln \sqrt[4]{x^2 - 1}\\
\\
\ln y &= \ln (x^2 + 1)^{\frac{1}{4}} - \ln (x^2 - 1)^{\frac{1}{4}}\\
\\
\ln y &= \frac{1}{4} \ln (x^2 + 1) - \frac{1}{4} \ln (x^2 -1)\\
\\
\frac{d}{dx} \ln y &= \frac{1}{4} \frac{d}{dx} \ln (x^2 +1 ) - \frac{1}{4} \frac{d}{dx} \ln (x^2 -1)\\
\\
\frac{1}{y} \frac{dy}{dx} &= \frac{1}{4} \cdot \frac{1}{x^2 +1} \frac{d}{dx} (x^2 + 1) - \frac{1}{4} \cdot \frac{1}{x^2 -1} \frac{d}{dx} (x^2 - 1)\\
\\
\frac{1}{y} y' &= \frac{1}{4(x^2+1)} \cdot 2x - \frac{1}{4(x^2 -1)} \cdot 2x\\
\\
\frac{y'}{y} &= \frac{x}{2(x^2+1)} - \frac{x}{2(x^2-1)}\\
\\
y' &= y \left[ \frac{x}{2(x^2+1)} - \frac{x}{2(x^2-1)} \right]\\
\\
y' &= \sqrt[4]{\frac{x^2+1}{x^2 -1 }} \left[ \frac{x}{2(x^2+1)} - \frac{x}{2(x^2-1)} \right]
\end{aligned}
\end{equation}
$