Why were Alexander Hamilton's policies as Secretary of the Treasury important for the new republic?

Alexander Hamilton believed in paying down the national debt through the use of excise taxes and customs.  This was important as alcohol was one of the colonies' major products –– it would now be taxed in the name of paying off the national debt.  In this manner, the entire citizenry assumed some of the tax burden, but no one group felt pressured to pay for it all.  By paying down the debt, Hamilton restored the world's faith in the American economy.  Hamilton was a strong Federalist, and he tied the state debts to the national debt in order to make the states more subservient to the national Union.  Hamilton allowed the national government to sell land in order to pay off its debt, as well.  Hamilton created the First Bank of the United States in 1791 in order to allow the government a centralized banking system and he also allowed for the issuance of paper currency.  This allowed the people to have faith in the banking system and for there to be some kind of regulation in banking, though land speculation continued to be a source of boom and bust cycles in the days of the early Republic.  
https://www.treasury.gov/about/history/pages/ahamilton.aspx

What was the soldier telling the old man to do? Did the old man accept his advice?

"The Old Man at the Bridge" is set during the Spanish Civil War, a particularly bitter, savage conflict that led to death and destruction on a horrendous scale. As with all modern conflicts, the Spanish Civil War had a particularly devastating impact on civilian life. We see an illustration right at the beginning of the story, where dozens of men, women, and children are making their way to safety across a hastily-constructed pontoon bridge to escape the imminent arrival of fascist troops.
There's a real sense of urgency in the air, but an old man sitting by the side of the dusty road remains rooted to the spot, seemingly unwilling to move. A soldier engages in conversation with the old man, who reveals he left behind the animals he was caring for when forced to leave his hometown. The soldier, clearly recognizing the gravity and urgency of the situation, advises the old man to get up and try to walk. The old man gets to his feet, but soon starts swaying from side to side before sitting back down in the dust. And there he remains, mumbling to himself about taking care of the animals. There's nothing more that the soldier can do to help him.

Who do the townspeople blame for the epidemic?

The townspeople blame Hannah Tupper for the outbreak of fever in their village.
In chapter 17, a group of people gathers at Matthew Wood's house and demands that he join them in their mission to hunt down Hannah. Matthew, however, refuses to participate in the witch hunt. His refusal angers his neighbors, and one of them warns that his stubbornness will lead to his daughter's death.
The people proclaim that Hannah is a Quaker, an infidel who has been practicing witchcraft for years. However, Matthew refuses to entertain the notion of chasing down an old woman. The people eventually leave, but Kit fears that they will harm Hannah. Her fears prompt her to make her way to Hannah's house. There, she rouses the sleeping Hannah, and the two hide under some brush. 
From there, Hannah and Kit can see the people torching Hannah's house, which eventually burns down. The two then make their way to the river's edge and spend the rest of the night there. In the morning, Kit is relieved when she spots the Dolphin making its way down the river. She swims out to the ship and surprises Nat.
Later, Nat gives Hannah and her cat temporary shelter on the ship.

According to Paine, why is there a need for regulations and government in general?

Paine answers this question in a brief discussion of the difference between "society" and "government." He views society as the consequence of human interactions, both economic and emotional, each carried out with the aim of satisfying human wants and needs. Society is healthy, natural, and, he says, "in every state a blessing." Government, on the other hand, is always a "necessary evil," according to Paine. Government exists to restrain people from their worst impulses--to keep them from stealing from and killing each other. This is what Paine means when he says that government is the "badge of lost innocence." It exists, in short, to protect the rights of the people, and to guarantee their freedom and security. The best government, according to Paine, will be one which places the least burden on the people, both in terms of money and in terms of intervention in their day to day lives. Governments should be very limited in authority, and they must be representative. So the regulations established by the state should limit the freedoms of the people only to the extent necessary to protect them from each other. His larger point is that the British government has acted contrary to these principles, an argument which is intended to justify separating from it.
http://www.let.rug.nl/usa/documents/1776-1785/thomas-paine-common-sense/some-writers-have-so-confounded.php

What is the twist in the story "The Last Leaf"?

The twist, or surprise, in O. Henry's story "The Last Leaf" comes at the very end. Johnsy has been expecting to die when the last ivy leaf fall off the vine attached to the brick wall of a neighboring building. It seems likely that she actually will die when this happens because she believes in it so firmly. Johnsy is described as small and frail. 

A mite of a little woman with blood thinned by California zephyrs was hardly fair game for the red-fisted, short-breathed old duffer.

If Mr. Pneumonia can kill much stronger people by the hundreds, why should he be expected to spare little Johnsy? The reader is practically convinced that the last leaf will fall any moment and that Johnsy will really pass away when that happens. 
But somehow the brave leaf clings tenaciously to its place on the denuded vine. If we believe that Johnsy can will herself to die when the leaf falls, then we can believe that she will decide to recover, and will recover, when the leaf doesn't fall. The actual "twist" in the story does not come when the leaf doesn't fall, but when Sue reveals that it couldn't fall because it had been painted on the wall in the middle of the night.

“Mr. Behrman died of pneumonia to-day in the hospital. He was ill only two days. The janitor found him on the morning of the first day in his room downstairs helpless with pain. His shoes and clothing were wet through and icy cold....Didn't you wonder why it never fluttered or moved when the wind blew? Ah, darling, it's Behrman's masterpiece—he painted it there the night that the last leaf fell.”

It should be noted that O. Henry takes considerable pains to keep the reader from suspecting that Old Mr. Behrman might get the idea of painting a fake ivy leaf on the brick wall in order to keep Johnsy alive and inspire her to recover. The author makes Behrman an old man. He is a heavy drinker. He has given up painting and only models for other artists. And he expresses extreme skepticism about the idea of anybody dying when a leaf falls off a vine.

“Vass!” he cried. “Is dere people in de world mit der foolishness to die because leafs dey drop off from a confounded vine? I haf not heard of such a thing." 

 

What is significant about the knight’s lady love? Why does Quixote need to include her in his new reality?

Don Quixote has spent so much time reading courtly romances about knights and their ladies that it has addled the middle-aged man's mind, and, we are told, made him crazy. Whether or not he really is crazy becomes one of the open-ended questions the book never fully resolves, leaving us to draw our own conclusions.
Whatever the case, Quixote finds the everyday world too dull and lacking in opportunities for glory and valor. He therefore decides to "be the change he wants to see," to use a modern phrase, and refashions himself as a knight errant. A knight needs a lady to defend, impress, and honor, so Quixote finds one in the form of a sturdy, loud-voiced, strong, and "manly" peasant "lass" named Aldonza Lorenzo. In his imagination, Quixote decides a knight such as him can have no less than a princess as his lady, so a princess she becomes. He renames her Dulcinea, which means sweet. He decides she is a fragile lady with golden hair. Making fun of stereotypical descriptions of beautiful women, Cervantes has Quixote assign Dulcinea eyes like suns, eyebrows like rainbows, alabaster skin, pearl teeth, and coral lips—more like a plaster doll than a real person.
Dulcinea, though Quixote identifies her with Aldonza, is a make-believe character. Since she is left behind as Quixote sallies forth on his adventures, he can imagine her as he wishes. Despite her fictive nature, she inspires Quixote to many feats of courage and daring-do. She shows the importance of having ideals to live for and the way a dream can motivate us to action.

As a part of Don Quixote's delusions that he is a knight-errant living a life straight out of one of his favorite books, he believes he must have a lady love motivating his actions and to whom he can dedicate his "heroic" deeds. Quixote fixates on a woman called (at least in Quixote's head) Dulcinea del Toboso. In reality, her name is Aldonza Lorenzo and, though Quixote describes her as "a princess" and a rare beauty, she is in fact just a local woman, of common birth, who is not even particularly aware of Don Quixote's existence.
So, in short, Don Quixote needs a "lady love" to fulfill the requirements of the genre to which he is trying to shape his life. What is significant about Dulcinea, the woman he chooses, is that she is not actually anything like the highborn, stunning, elegant woman Quixote builds her up to be in his own mind.

How is industrialism and economic/social change addressed in Goblin Market? How did it affect individual life?

This is very interesting angle from which to approach this poem. The themes of "Goblin Market," on the face of it, seem fairly obvious—the pure and virginal heroine must enter into a world of dangerous men (all of whom want to sully her virtue) and can only emerge intact by resisting all temptation. Viewed from the angle of industrialization and globalization in the nineteenth-century, however, the reader is forced to focus in on precisely the kinds of men Rossetti's heroines must resist. The goblins in the poem, it cannot be forgotten, are tradesmen, and they are trading in food. In the context of Victorian London, which was increasingly becoming a melting pot of foreign vendors selling their exotic wares from all over the British Empire, this is significant.
The cries of the goblins—"Come buy, come buy"—are an echo of similar calls that would have been heard in the street markets of any town in England, but although the wares of the goblins begin with the common English "apples and quinces," they swiftly move to "swart-headed" and "wild free-born" fruits. The "swart-headed" reference is also significant, as it seems to recall other dark-headed elements introduced into London: namely, people from the Middle East and beyond. "Pine-apples" and "pomegranates," the latter a fruit which has long held a sexual connotation, continue the array of exotic wares: these "Citrons from the south" have an obvious appeal, the poem says, to innocent girls and others who cannot possibly understand the origins of the wares. The difficulty lies in deciding which of these foreign wares are safe. Western anxieties about their countries being infiltrated by foreign traders and their unknown wares are captured in the image of Laura and Lizzie struggling to resist the "goblin men" and their fruits of unknown provenance—"Who knows upon what soil they fed / Their hungry thirsty roots?"
The sisters are both convinced that the "evil gifts" of the goblin men would be damaging to them. However, this does not make it any easier for Laura to stay away. Notably, what the goblins covet from her is a part of her very person, and a part which symbolizes her opposition to them, in terms of racial symbols: " a precious golden lock." Having given this up in search of the rare delights, she becomes swiftly addicted to them, sucking "until her lips were sore."
Ultimately, Laura can only be redeemed through her sister, Lizzie, who represents a sort of judicious curator of the goblin fruits. Knowing, or fearing, that the goblin men will attack her with "gibe or curse," Lizzie presents her sister at the end of the poem with the juices to which she has become addicted, "squeez'd from goblin fruits for you." In the face of her sister's sacrifice, the goblin juice becomes "wormwood to [Laura's] tongue," as, seeing what Lizzie has endured for her sake, Laura recognizes what the fruit truly is. Engaging with the goblins, in any way at all, is best avoided, and once the initial temptation has passed, this becomes clear.
Interpreted through the lens of industrialization and globalization, then, Goblin Market becomes a caution against engaging with anything that is unknown or exotic. The two English sisters, Lizzie and Laura, can only survive by resisting temptation and clinging together, something which has an obvious sexual connotation, but also a wider applicability in terms of the growing merchant class and availability of foreign wares. The impact of this industrialization on the individual is, Rossetti suggests, simply that it offers too much choice and too little explanation: better to avoid the dizzying array of options altogether than to allow oneself to become corrupted through contact with the unknown.

What was the plan of Professor Arronax to fulfill their mission?

Professor Aronnax is a leading marine biologist. His professional interest is piqued by tales of a deadly sea monster—thought to be a giant narwhal—that has apparently been terrorizing international shipping lanes. A number of ships have been destroyed, which has severely disrupted the flow of trade. The American government has become so concerned that it has put together an expedition to hunt down and kill the dreaded sea monster before it wreaks any further havoc.
Professor Aronnax is just the man to take part in such an expedition. With his vast knowledge and expertise, he is ideally placed to identify this strange and deadly creature. Once he has done so, he will gain the credit for discovering what he believes is possibly a giant mutation of a narwhal. Professor Aronnax will be feted all across the world for this remarkable scientific discovery; his place in history is assured. No wonder he regards the expedition as his sole purpose in life.

In the novel To Kill a Mockingbird, why is it weird for an animal to behave the way Tim Johnson did given the time of year?

In Chapter 10, Jem and Scout are outside playing with their air rifles when Jem notices Tim Johnson staggering down the middle of the road. After Calpurnia sees the dog, she immediately recognizes that it is "mad." When Calpurnia speaks to Miss Eula May on the telephone, she says,

“I know it’s February, Miss Eula May, but I know a mad dog when I see one. Please ma’am hurry!” (Lee 58).

Clearly, something is odd about an animal with rabies during the cooler months in Alabama. Apparently, rabies tends to be transmitted more often during the summer months when the weather is warmer which is why Eula May questions Calpurnia. Sheriff Tate then arrives on the scene and gives his rifle to Atticus. Atticus reluctantly takes the weapon and shoots Tim Johnson above his left eye. After witnessing his father's expert marksmanship, Jem is in awe of Atticus' abilities. Miss Maudie then explains why Atticus never bragged about the fact that he was the best shot in Maycomb. 

Precalculus, Chapter 5, 5.4, Section 5.4, Problem 50

sin(u)=-7/25
using pythegorean identity,
sin^2(u)+cos^2(u)=1
(-7/25)^2+cos^2(u)=1
cos^2(u)=1-49/625=(625-49)/625=576/625
cos(u)=sqrt(576/625)=+-24/25
Since u is in quadrant III ,
:.cos(u)=-24/25
sin^2(v)+cos^2(v)=1
sin^2(v)+(-4/5)^2=1
sin^2(v)+16/25=1
sin^2(v)=1-16/25=(25-16)/25=9/25
sin(v)=sqrt(9/25)=+-3/5
since v is in quadrant III,
:.sin(v)=-3/5
cot(v-u)=cos(v-u)/sin(v-u)
cot(v-u)=(cos(v)cos(u)+sin(v)sin(u))/(sin(v)cos(u)-cos(v)sin(u))
plug in the values of sin(v),sin(u),cos(v) and cos(u),
cot(v-u)=((-4/5*-24/25+(-3/5)*-7/25))/((-3/5*-24/25-(-4/5)*-7/25))
cot(v-u)=(96/125+21/125)/(72/125-28/125)
cot(v-u)=(117/125)/(44/125)
cot(v-u)=117/44

When Old Man Warner says, "Lottery in June, corn be heavy soon," what does that tell you about the original purpose of the lottery?

The townsfolk in "The Lottery" appear to be pagans. In many ancient pagan societies, it was believed that in order to ensure the success of the harvest, sacrifices of one kind or another had to be made to the gods. Good harvests were essential for such communities; if they failed, widespread starvation could ensue. So it was considered absolutely crucial to do whatever was necessary to appease the gods, even if it meant resorting to human sacrifice.
And that's what happens in "The Lottery." The annual lottery is held in June to make sure that once the human sacrifice has been made, a bumper harvest will inevitably follow. That's what the old man means by "Lottery in June, corn be heavy soon." What makes the annual ritual sacrifice especially disturbing is that it takes place in a modern town. Indeed, it is ironic that the townsfolk feel the need to resort to such primitive rites in order to maintain a modern standard of living. They firmly believe that if human sacrifices aren't made every year, then the crops will fail, and if the crops fail then the community will be forced to revert back to a more primitive standard of life.

Old Man Warner's comments suggest that the Lottery ritual may have started as a human sacrifice to the gods of the harvest and fertility. He seems to tie the labor of farming and cultivated crops with the better lifestyle the community has enjoyed since they stopped, as he puts it, living in caves. But he insinuates that giving up the Lottery could thrust the community back into poverty and starvation, into the days of eating "stewed chickweed and acorns." Such a diet would signify a hunter-gatherer type of society rather than an agrarian-based culture. When other people mention that some neighboring villages have discontinued the Lottery, or are contemplating discontinuing it, Old Man Warner insists that is "foolishness" and "nothing but trouble." He seems more tightly connected to the superstition that keeps the Lottery in place than some of the younger townspeople. That superstition is that bountiful harvests depend on the sacrifice of one randomly chosen human each year.

What are some comic elements in Mrs. Dalloway?

Mrs. Dalloway is primarily not a comic novel, dealing as it does with suicide, loss, compromise, failure, aging, the inability of the medical system to treat mental illness, and the repercussions of World War I. Further, Woolf has no authoritative narrator in this novel, leaving it to the reader to interpret tone and meaning.
Nevertheless, Woolf uses dark comedy to poke fun at (satirize) patriarchal illusions. For instance, she makes fun of the pomposity and self-importance of Hugh Whitbread, who attends Clarissa's party:

Hugh [Whitbread], intimating by a kind of pout or swell of his very well-covered, manly, extremely handsome, perfectly upholstered body (he was almost too well dressed always, but presumably had to be, with his little job at Court).

Using words like "manly," "extremely handsome," and "perfectly upholstered," Woolf ridicules Hugh's strutting, inflated self-image and his superficiality: he derives his sense of worth from external qualities, such as his body and his clothes. She also punctures him by referring to his job at Court, of which he is very proud and which makes him feel important, as "little."
Woolf also comically skewers the inflated, patriarchal delusions of grandeur in Peter Walsh's thoughts about himself:

He was an adventurer, reckless, he thought, swift, daring, indeed (landed as he was last night from India) a romantic buccaneer, careless of all these damned proprieties. . . . He was a buccaneer.

Peter Walsh is no buccaneer, regardless of how much he wants to harbor this fantasy (and, to his credit, he does realize this).
There is a dark humor in Peter spending half an hour following a girl in the streets. Peter rationalizes this by making up a fantasy about her, noting that "one makes up the better part of life."
Ultimately, it is these male fantasies, comic in themselves, that lead to tragedies like World War I. The inflated egos of patriarchal men, if silly and comic, take a heavy toll on society.
There is also a comic element in all the flurry and fuss over Mrs. Dalloway's party. She is an admirable person, but is a party what she should be using her energies and gifts to produce?

College Algebra, Chapter 8, 8.2, Section 8.2, Problem 18

Determine the vertices, foci and eccentricity of the ellipse $9x^2 + 4y^2 = 1$. Determine the lengths of the major and minor
axes, and sketch the graph.
If we divide both sides by $16$, then we have
$\displaystyle \frac{x^2}{\frac{1}{9}} + \frac{y^2}{\frac{1}{4}} = 1$
We'll see that the function has the form $\displaystyle \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$. Since the denominator of $y^2$ is larger, then the ellipse
has a vertical major axis. This gives $\displaystyle a^2 = \frac{1}{4}$ and $\displaystyle b^2 = \frac{1}{9}$. So,
$\displaystyle c^2 = a^2 - b^2 = \frac{1}{4} - \frac{1}{9} = \frac{5}{36}$. Thus, $\displaystyle a = \frac{1}{2}, b = \frac{1}{3}$ and
$\displaystyle c = \frac{\sqrt{5}}{6}$. Then, the following is determined as

$
\begin{equation}
\begin{aligned}
\text{Vertices}& &(0, \pm a) &\rightarrow \left(0, \pm \frac{1}{2}\right)\\
\\
\text{Foci}& &(0, \pm c) &\rightarrow \left(0, \pm \frac{\sqrt{5}}{6}\right)\\
\\
\text{Eccentricity (e)}& &\frac{c}{a} &\rightarrow \frac{\sqrt{5}}{3}\\
\\
\text{Length of major axis}& &2a &\rightarrow 1\\
\\
\text{Length of minor axis}& &2b &\rightarrow \frac{2}{3}
\end{aligned}
\end{equation}
$

What is the main argument of the poem "Sonnet 73"? With which three tools does the author prove this argument?

The main argument in William Shakespeare's "Sonnet 73" is that passion grows with age. The speaker describes this passion as a "glowing . . . fire." The speaker tell his beloved that passion increases because of the knowledge that death, which is presented as "black night," is drawing near.
Shakespeare's tools in the sonnet are meter, rhyme, and metaphors. The meter is iambic pentameter. The rhyme scheme is A-B-A-B, C-D-C-D, E-F-E-F. He uses three conceits, or extended metaphors, to establish comparisons to the speaker's age: the season, winter; the time of day, twilight; and the fire, ashes. The season or "time of year" is shown by the scarcity of leaves on the trees. The day fades into twilight "after sunset . . . in the west." The fire still glows against the background of "the ashes of . . . youth."
All three conceits connect with the finality of death, which "seals up" everything as a final repose.

The main argument of Sonnet 73 is that, in loving someone whom he knows to be in the "twilight" of his life, and who is very obviously mortal, the beloved demonstrates that his love is "more strong." It is representative of a stronger and more devoted love "to love that well which thou must leave ere long."
Shakespeare uses a number of tools to convey the message in this poem. A symbolic field of autumn pervades the poem, with an extended metaphor used to indicate that the poet's appearance reflects "that time of year" when "yellow leaves, or few, or none" can be seen "upon those boughs which shake against the cold." The poet characterizes himself as a tree in autumn, with its leaves fallen off. Next, another extended metaphor compares the poet to "the twilight of such day / as after sunset fadeth in the West." The shadow of death is alluded to when the speaker discusses "black night," "Death's second self," which takes away all who have lived. Finally, a metaphor pertaining to fire is used to describe the poet, who is now "glowing" "as on the ashes of his youth." Where once a fire burned high in the poet, his youth has now ebbed away such that there is little left to fuel the fire, leaving it only "glowing" and "ashes."
It is clear, then, the poet says, that his beloved knows the poet is getting old. He cannot avoid seeing this to be true: "this thou perceiv'st." As such, the beloved's love must be all the stronger, because he still loves the poet, despite the fact that their love cannot last much longer.

Single Variable Calculus, Chapter 8, 8.1, Section 8.1, Problem 14

Show that $\displaystyle \tan h (x + y) = \frac{\tan h x + \tan hy}{1 + \tan h x \tan hy}$

Solving for the left-hand side of the equation

Using Hyperbolic Function


$
\begin{equation}
\begin{aligned}

\tan hx =& \frac{\sin hx}{\cos hx} = \frac{e^x - e^{-x}}{e^x + e^{-x}}
\\
\\
\tan h (x + y) =& \frac{e^{(x + y)} - e^{-(x + y)} }{e^{(x + y)} + e^{-(x + y)}}
\\
\\
\tan h (x + y) =& \frac{e^x e^y - e^{-x} e^{-y}}{e^x e^y + e^{-x} e^{-y}}

\end{aligned}
\end{equation}
$


Using Hyperbolic Identities


$
\begin{equation}
\begin{aligned}

& \cos hx + \sin hx = e^x \text{ and } \cos hx - \sin hx = e^{-x}
\\
\\
& \tan h(x + y) = \frac{(\cos hx + \sin hx)(\cos hy + \sin hy) - (\cos hx - \sin hx)(\cos hy - \sin hy)}{(\cos hx + \sin hx)(\cos hy + \sin hy) + (\cos hx - \sin hx)(\cos hy - \sin hy)}
\\
\\
& \tan h(x + y) = \frac{\cos hx \cos hy + \cos hx \sin hy + \sin hx \cos hy + \sin hx \sin hy - (\cos hx \cos hy - \cos hx \sin hy)(- \sin hx \cos hy + \sin hx \sin hy)}{\cos hx \cos hy + \cancel{\cos hx \sin hy} + \cancel{\sin hx \cos hy} + \sin hx + \sin hy + \cos hx \cos hy - \cancel{\cos hx \sin hy} - \cancel{\sin hx \cos hy} + \sin hx \sin hy}
\\
\\
& \tan h(x + y) = \frac{\cancel{\cos hx \cos hy} + \cos hx \sin hy + \sin hx \cos hy + \cancel{\sin hx \sin hy} - \cancel{\cos hx \cos hy} + \cos hx \sin hy + \sin hx \cos hy - \cancel{\sin hx \sin hy}}{2 \cos hx \cos hy + 2 \sin hx \sin hy}
\\
\\
& \tan h(x + y) = \frac{2 \cos hx \sin hy + 2 \sin hx \cos hy}{2 \cos hx \cos hy + 2 \sin hx \sin hy}
\\
\\
& \tan h(x + y) = \frac{\cancel{2} (\cos hx \sin hy + \sin hx \cos hy)}{\cancel{2} (\cos hx \cos hy + \sin hx \sin hy)}
\\
\\
& \tan h(x + y) = \frac{\cos hx \sin hy + \sin hx \cos hy}{\cos hx \cos hy + \sin hx \sin hy} \cdot \frac{\displaystyle \frac{1}{\cos hx \cos hy}}{\displaystyle \frac{1}{\cos hx \cos hy}}
\\
\\
& \tan h(x + y) = \frac{\displaystyle \frac{\cancel{\cos hx} \sin hy}{\cancel{\cos hx} \cos hy} + \frac{\sin hx \cancel{\cos hy}}{\cos hx \cancel{\cos hy}}}{\displaystyle \frac{\cancel{\cos hx \cos hy}}{\cancel{\cos hx \cos hy}} + \frac{\sin hx \sin hy}{\cos hx \cos hy}}
\\
\\
& \tan h(x + y) = \frac{\displaystyle \frac{\sin hy}{\cos hy} + \frac{\sin hx}{\cos hx}}{\displaystyle 1 + \frac{\sin hx}{\cos hx} \cdot \frac{\sin hy}{\cos hy}}
\\
\\
& \tan h(x + y) = \frac{\tan hy + \tan hx}{1 + \tan hx \tan hy}
\\
\\
& \text{or}
\\
\\
& \tan h(x + y) = \frac{\tan hx + \tan hy}{1 + \tan hx \tan hy}



\end{aligned}
\end{equation}
$

What comparisons between Macbeth and the story of Achilles can be made?

Achilles appears in Homer's epic The Illiad. Achilles has come to be known as one of the most well-known heroes in Greek mythology. As the story goes, Achilles's mother, Thetis, dipped him as an infant into the River Styx. She did this in order to make him invulnerable to all that he would face. Unfortunately for her, the waters of the river did not cover him completely; she left a small place on his heel vulnerable to attack. Achilles ultimately dies from an arrow being shot into his heel, where the water did not touch. This injury proved to be fatal, and Achilles dies.
Similar to Achilles, Macbeth (from William Shakespeare's Macbeth) possesses his own tragic flaw. Macbeth's ambitious nature exists as his tragic flaw. Macbeth's ambition to be king of Scotland proves to be the one thing which means more to him than anything. Because of this, Macbeth is unable to see the unfortunate events which will lead to his own demise.
Another similarity between Achilles and Macbeth lies in prophecy. When Achilles is nine years old, a prophet (seer) tells Thetis that he will die in battle against the Trojans. In order to save her son from this fate, she dresses him as a girl and sends him away. In the end, Achilles decides to join the Greek army, which puts him into war with the Trojans, and he loses his life. Although he is thought to be invulnerable, his mother's actions to save him from the prophecy fail. Achilles's free will seems to be the one thing that ensures the prophecy comes true, resulting in his death.
Macbeth similarly has a prophecy regarding his own death. Three witches tell Macbeth that his death will only come under three circumstances: Macduff will kill him, only a man not born of woman will kill him, and he will not die until Birnam Wood moves against him. Macbeth, like Achilles's mother, tries to use the prophecy's weaknesses to see that the prophecy has no possibility of coming true. Macbeth dismisses the first and second prophecies together. He states that he need not fear Macduff because Macduff is born of woman. He dismisses the final prophecy by stating that trees will not pull their roots out of the ground and walk. Therefore, he need not worry about the prophecies at all. In reality, Macbeth's own free will "allows" him to dismiss the integrity of the prophecies. In the end, Macbeth should have feared Macduff because he was not born of woman; he was born by caesarean section. While the trees themselves do not move, it does look like they do. The soldiers moving against Macbeth's castle are ordered to cut down branches and carry them before them. They do this so that it looks like their numbers are greater than they actually are.
One final, yet arguably less important, comparison between Achilles and Macbeth lies in the idea that both men were well-known and successful warriors. Achilles becomes well-known over the course of the Trojan War's battles. The war has gone on for many years, and time and time again, Achilles has proven himself a great success. Macbeth's renown in war comes as his successes in battle are relayed to the king, Duncan. Macbeth defeated Macdownald, a great foe of the king. Because of the success Macbeth has found on the battlefield, Duncan rewards him with a new title, Thane of Cawdor. The previous thane has been found guilty of treason, and Duncan needs a new thane. Given the news brought to him about Macbeth's successes for Scotland, Duncan rewards him. Essentially, both men are great warriors.

Was Shakespeare an actor or a playwright?

Both, actually, but a playwright for a living.  He played a few roles for his theatre group, Lord Chamberlain’s Men, including (possibly) the ghost in Hamlet.  But his canon of plays, especially as documented in the 1616 Folio and its opening endorsements by other theater figures in the Elizabethan and Jacobean periods, makes it abundantly clear that Shakespeare spent his entire professional life as a working playwright, well known and universally respected for his products for the stage.  In addition to his theatre work, he also wrote poems, mostly but not completely sonnets.  He has adumbrated dozens of his contemporaries, such as Christopher Marlowe, Ben Jonson, and Thomas Middleton, all very competent playwrights but virtually forgotten outside academia.  

What are Snowball's dreams for the windmill?

Snowball plans for the windmill to operate as a dynamo which will supply the farm with electrical power and significantly improve the standard of living on the farm. Snowball tells the animals that the electricity produced from the windmill will light the stalls in the evenings and allow the animals to use power tools, which will make them more efficient and dramatically reduce manual labor on the farm. Snowball even declares that the windmill will make it possible for the animals to only work three days a week. Snowball's plans for constructing the windmill and improving the standard of living on the farm are popular among the animals; however, Napoleon strongly opposes Snowball's plans and the two rival pigs engage in heated debates over the topic of the windmill.
Tragically, Napoleon usurps power, steals Snowball's blueprints, and forces the animals to work long hours building the windmill. Even after construction, the windmill never operates as a dynamo, and the animals continue to live difficult lives on the farm.

In Chapter Five of Animal Farm, Snowball surveys the farmland and develops an idea to build a windmill. When he announces this idea to the others, his dreams for this project become clear. He talks about this "fantastic" machine, for example, and says that it can do their work for them while they "graze in fields" all day. In fact, after looking over the plans, Snowball declares that the windmill can reduce the animals' working week from six days to three. This is a significant reduction which makes his plans for the windmill very popular on the farm.
In addition to its labour-saving abilities, Snowball wants to use the windmill to generate electricity. This will enable the introduction of more labour-saving devices, like the plough, as well as the ability to provide heating and lighting to individual stalls in the barn.
It is this dream, however, which brings Snowball into conflict with Napoleon and which ultimately brings about his expulsion from the farm. 

Who was the mother of Fabrizio?

Fabrizio's mother is the Marquise del Dongo. On page 10, chapter 1, the French soldier Lieutenant Robert describes her in glowing terms.

Lovely, angelically soft eyes and that pretty dark blond hair that so perfectly framed her charming face. There was a painting of Herodias by Leonardo in my room: you'd have thought it was her portrait.

In comparison, he claims her husband the Marquis del Dongo is a coward. When the French entered Italy, and later Austria, he left his wife to deal with everything while he took refuge in his castle.
She dealt with it well and became thought of as a good person. Though her husband didn't give her much money, she helped out her poor sister by secretly giving her her pearls. Equally, Lieutenant Robert describes how warmly she welcomed him and how actively and emphatically she listened to him, even though he claimed he looked a disgrace.
The author states the Marquise "adored" her son. When Fabrizio boards at a Jesuit college in Milan, she borrows money from her sister-in-law to visit him. She is upset when he decides to leave the country, but does all she can to help him.

The Marquise burst into tears on hearing of her son's strange plan... When she became convinced that nothing in the world, other than the walls of a prison, could prevent his departure, she gave him what little money she owned, then recollected that she had in her possession since the previous day, eight or ten little diamonds, worth perhaps 10,000 lire.

Marquise Del Dongo was Fabrizio's mother. Lieutenant Robert described her as a woman of much beauty, with auburn hare, eyes of "angelic sweetness" and an oval face. He also said that Leonardo Da Vinci himself could of painted her, for she was that pretty.
She is a truly warm hearted woman. She lives in a lovely home with marble in the dining room and servants galore. When married, she brought her husband a dowry of 800,000 francs. Her husband however is greedy and cold hearted, only letting her have 80 francs a month, even though she brought him so much. She is described as timid, yet she does manage to sneak diamonds from her jewelry as a way to help out his sister, who he had abandoned for going against his will. It is seen that even though his sister is starving in Paris, he refuses to help her, which is why in the end Marquise Del Dongo helps her like that.
Her character is a total contrast from her husband, as she is sweet where he is mean. She is warm where he is cold, and she is compassionate where he has no heart.

Fabrizio's mother was the Marquise del Dongo. She is described by Lieutenant Robert as a beautiful woman in full bloom with auburn hair, an oval face, and eyes of "angelic sweetness." According to Robert, she is so lovely she could have been painted by Leonardo da Vinci.
She lives in a splendid home with a marble-paneled dining room and many servants. She is a kind-hearted woman who brought her husband a dowry of 800,000 francs when they married. Unfortunately, her husband is a greedy and hard-hearted man who only allows her 80 francs a month in allowance. Although described as very timid, she manages to take a few diamonds from her jewels, which her husband keeps carefully locked up. She sends these to his sister, who married against his will and who, because she defied him, he will not help out, even though she is starving in Paris. The Marquise is depicted as a gentle woman of goodness and compassion, a sharp contrast to her sordid husband.

Calculus of a Single Variable, Chapter 6, 6.1, Section 6.1, Problem 46

To solve this differential equation, rewrite it as
dy = xcos(x^2)dx
Integrate both sides of the equation:
y = int xcos(x^2)dx
To take the integral on the right side of the equation, use the substitution method. Let z(x) = x^2 .
Then, dz = 2xdx and integral becomes
int xcos(x^2)dx = int(xdx) cos(x^2) = int 1/2 dz cosz = 1/2 int coszdz
This is a simple trigonometric integral: int cosz dz = sinz . Substituting the original variable, x, back into equation results in
y = 1/2 sinx^2 + C , where C is an arbitrary constant.
So, the general solution of the given differential equation is
y = 1/2sinx^2 + C .

Which heir to the Rome empire was more like the old Roman Empire? Some historians argue that the Islamic empire was more Roman, while others hold that the Carolingian empire reflected Roman culture and ideals more. what various ways in which there was religious, cultural, intellectual, and political continuity between Rome and the successor states, and which state more closely resembled the fallen Roman Empire. also identify what makes each of these empires different from Rome.

The Byzantine Empire was clearly the first successor to the Roman Empire. At one point, the Roman Empire was too large and cumbersome for one ruler to govern effectively, so the empire was divided in half between a western and eastern sphere. After the fall of the Western Empire in 476 A.D., the Eastern Empire came to be called the Byzantine Empire.
Some historians have argued that the Roman Empire did not completely collapse in 476 but continued through the Byzantine Empire. This argument is compelling because the Byzantine Empire resembled its western counterpart in many ways. The Byzantine emperor ruled with absolute authority. The Byzantine Empire also had a supreme religious leader—the Patriarch—just like the Pope in the Western Empire. The Patriarch served as the official head of the Eastern Orthodox religion.
Much like the Western Empire, the Byzantine Empire constantly tried to expand its borders and territories through military conquest. This usually brought the Byzantines into conflict with Islamic peoples to its east and south. The Byzantine Empire was never as militarily successful as its western partner and faced constant encroachment (and defeat) by nomadic tribes and Islamic armies. The Byzantine Empire eventually fell to the Islamic armies in 1453, when the Byzantine capital, Constantinople, was conquered by the Ottoman Turks.
For more information see: Herrin, Judith. Byzantium: The Surviving Life of a Medieval Empire. Princeton, NJ: Princeton University Press, 2007.

Single Variable Calculus, Chapter 6, 6.5, Section 6.5, Problem 8

Determine the average value of the function $\displaystyle f(x) = \frac{3}{(1 + x)^2}$ on the interval $[1,6]$.


$
\begin{equation}
\begin{aligned}

f_{ave} =& \frac{1}{b - a} \int^b_a f(x) dx
\\
\\
f_{ave} =& \frac{1}{6 - 1} \int^6_1 \frac{3}{(1 + x)^2} dx
\\
\\
\text{Let } u =& 1 + x
\\
\\
du =& dx

\end{aligned}
\end{equation}
$


Also, make sure that your upper and lower limits are also in terms of $u$.


$
\begin{equation}
\begin{aligned}

f_{ave} =& \frac{3}{5} \int^{1 + (6)}_{1 + (1)} \frac{1}{u^2} du
\\
\\
f_{ave} =& \frac{3}{5} \int^7_2 u^{-2} du
\\
\\
f_{ave} =& \frac{3}{5} \left[ \frac{u^{-1}}{-1} \right]^7_2
\\
\\
f_{ave} =& \frac{3}{5} \left[ \frac{-1}{7} - \left( \frac{-1}{2} \right) \right]
\\
\\
f_{ave} =& \frac{3}{14}

\end{aligned}
\end{equation}
$

When did the Ghost of Christmas Present use Scrooge's own words against him?

The Ghost of Christmas Present is trying to convey to Scrooge—as Dickens is to his audience—that words have meaning. When Scrooge brushes off the charitable requests and impulses of people asking him for help on Christmas Eve, what he says are just words to him. When he states it would be better for the poor to die and rid England of its excess population or go to prisons or workhouses, "the poor" are simply an abstraction in his mind. He is simply mouthing sentiments, hard-hearted as they are, that he has heard others express. He can't really imagine what it is to truly be in need.
But when Scrooge sees real poor people, like Cratchits, and especially Tiny Tim, his attitude changes. "The poor" are now not simply a disposable category, but real people like him with hearts and personalities and minds that are worth preserving. Having really seen the Cratchits for the first time, their struggles and sorrows become of great concern to him and can no longer be dismissed. The thought of Tiny Tim dying for no more reason than lack a little money for medical care fills Scrooge with grief. Even the poor children Ignorance and Want under the ghost's robes become objects of his compassion as is heart is awakened and stirred.
Dickens has the ghost throw Scrooge's words back at him, because he wants us to make a connection between callous words and real people. The hard-hearted way we speak of "the poor" ignores the fact that these are suffering people. Today, we might simply say "get a job" or "why don't they get a job?" or "Why should they get food stamps?" when the reality is, if we knew one or two of "these people" personally, we would probably feel much more compassion. In the end, Dickens believed, it was the role of the novelist (like the role of the Ghosts—and what is a novelist but a ghostly presence from "another world?") to put the flesh and blood on mere words so that they became attached to people and feelings.

In Stave Three, the Ghost of Christmas Present turns Scrooge's words against him on two occasions. The first of these occurs when the ghost and Scrooge are visiting the Cratchit family. On the subject of Tiny Tim, Scrooge asks the ghost if he thinks that Tim will survive. In response, the ghost says he sees a vacant seat at the dinner table next Christmas. It is at this point that the ghost uses Scrooge's words against him:

 "'What then? If he be like to die, he had better do it, and decrease the surplus population.'''

The second occasion occurs after Scrooge sees Ignorance and Want hiding underneath the ghost's robes. When Scrooge asks if there are any resources available to help these children, the ghost repeats Scrooge's words from Stave One:

"'Are there no prisons?'' said the Spirit, turning on him for the last time with his own words. 'Are there no workhouses?'"

By using Scrooge's words against him, the ghost highlights Scrooge's negative attitude towards the poor while also providing an impetus for change.

In "Lamb to the Slaughter" by Roald Dahl, how is motivation used?

I believe that this question is asking about character motivation and what motivates that particular character to act. Mary Maloney is a good character to use regarding motivation. When readers are first introduced to Mary, she exists to serve her husband. She is patiently waiting for him to get home, and once Patrick gets home, Mary flutters around the house getting him food and drink so that he can relax after his day of work.

She wasn’t really watching him, but she knew what he had done because she heard the ice cubes falling back against the bottom of the empty glass when he lowered his arm. He paused a moment, leaning forward in the chair, then he got up and went slowly over to fetch himself another.
“I’ll get it!” she cried, jumping up.

I assume that her motivation at this point is true love. Maybe complete infatuation, but the end result is the same. She is motivated to do whatever it takes to keep Patrick happy. When he announces that he is leaving her, that motivation ends. Soon after, Mary kills Patrick.
Then Mary's motivation shifts to protecting her unborn child. The text tells readers that she is completely willing to take on the punishment for her actions, but she is not willing to risk the life of her baby.

As the wife of a detective, she knew quite well what the penalty would be. That was fine. It made no difference to her. In fact, it would be a relief. On the other hand, what about the child? What were the laws about murderers with unborn children? Did they kill them both--mother and child? Or did they wait until the tenth month? What did they do?
Mary Maloney didn’t know. And she certainly wasn’t prepared to take a chance.

Protecting her child is Mary's motivation. It is what causes her to plan out an alibi for the murder and get away with her actions.

Why do auditors need accounting information?

Auditing is the review of an institution’s books of account and financial documents to determine the institution’s level of compliance with regards to legal regulations and business best practices. Auditing aims to ensure the institution presents an accurate and true position of its financial operations. Most organizations have internal auditors responsible for examining an institution’s books of accounts and other financial documents. In certain special situations, however, an independent auditing firm is brought in to evaluate the documents.
Auditors require accounting information because it forms the basis of their examinations. Auditors would have nothing to scrutinize if the necessary accounting documents are non-existent. Accounting information is important in determining the true position of the institution. Auditors will also determine whether the institution is adhering to the generally accepted protocols and standards by evaluating their financial/ accounting reports and procedures.
https://www.acfe.com/accounting-and-auditing.aspx

https://www.grfcpa.com/resource/auditor-responsibilities/

Precalculus, Chapter 7, 7.3, Section 7.3, Problem 13

EQ1: 2x+y-3z=10
EQ2: y+z=12
EQ3: z=2
In this system of equations, the value of variable z is known. So to get the values of the variables substitute z=2 to one of the equations. It is better if it is plug-in to the second equation since it composed of two variables only.
y + z=12
y+2=12
Then, solve for y.
y=12-2
y=10
Now that the values of y and z are known, solve for x. Plug-in them to the first equation.
2x+y-3z = 10
2x+10-3(2)=10
2x+10-6=10
2x+4=10
2x=10-4
2x=6
x=6/2
x=3
Therefore, the solution is (3,10,2).

Why do you think the Roman solider keeps coming around?

When he accepted Simon's offer to work in his shop, Daniel had to agree to one important condition: he must repair any items brought into the shop by Romans. Simon's rationale is that refusing to do so could lead to trouble in town; lives may be lost. Daniel reluctantly accepts Simon's condition. He hates the Romans with a passion, having fought so hard against them. He does not want the Romans to be anywhere near Judea, let alone Simon's forge, but he is sensible enough to accept Simon's point. A zealot he may be, but a complete fanatic he is not; there are potentially lives at stake here.
One day, a Roman legionary by the name of Marcus comes into the forge needing a broken bridle ring to be fixed. Mindful of Simon's instructions, Daniel grudgingly agrees to accede to the Roman's request. But nothing can change how Daniel feels about his people's oppressors. He is still dedicated to driving every last Roman from his homeland and continues to engage in clandestine resistance activity. (Though as he will later discover, Marcus is a German, not a Roman). Not surprisingly, Daniel needs to be careful that Marcus does not suspect what he is up to. He soon gets paranoid, finding it suspicious that the legionary always seems to show up on the same day that Daniel holds meetings with his revolutionary group. There is a danger that Daniel and his fellow zealots are being spied on.
However, Marcus's regular visits to the shop have nothing to do with counterrevolutionary espionage; he is falling for Daniel's sister, Leah. Although this means that Daniel has less reason to feel so nervous about being spied on, he is still not exactly thrilled that one of the hated enemy is wooing his own sister. Daniel has never accepted Romans—or the non-Romans like Marcus who fight with them—as fully human. He has always looked at them as the enemy, the other, the forces of occupation ruthlessly exploiting his sacred homeland. The very idea that his own flesh and blood could become the object of affection of one of Judea's hateful oppressors is simply too horrible to contemplate.

How do Angus and Mae make a living?

Angus and Mae sell things they make to people.
Mae sews and makes quilts and other handmade items. Winnie describes seeing rugs and quilts that are half completed. Angus does woodwork. Winnie says there are wood shavings, splinters, chips and sawdust in the living room. There are dolls, wooden soldiers, a model ship, wooden bowls, and utensils.
As Winnie explores the Tuck's home, she sees these things in their loft. They're scattered about and create a mess. She feels out of place because her home is usually orderly. She thinks that maybe it doesn't bother them because they have an unlimited amount of time to clean it up. However, Mae looks at the mess and seems to approve of it.
Mae and Angus's sons go away and work odd jobs from time to time but the brothers meet up every ten years and travel to be together with their parents. They also return home at other times, when they feel like it.

Angus and Mae earn money by making and selling things such as wooden toys and handmade quilts. It seems strange to say that they make a living when you consider that they are both immortal. However, being immortal means that the whole world is your oyster; you can go anywhere and do pretty much anything. Thus, it makes sense for Angus and Mae to turn their hands to something they can do just about anywhere in the world. Immortality has its drawbacks, though, the most serious of which is the potential for boredom. Angus often seems quite depressed, and it is largely due to the sheer boredom that immortality can bring. It is not surprising, then, that he and Mae make their own things, as the worst thing in the world would be for them to be tied down to any particular line of work. It is always so much more interesting and fulfilling to put your soul into what you do.

College Algebra, Chapter 5, 5.4, Section 5.4, Problem 52

Solve the Logarithmic Equation $\log x + \log (x - 3) = 1$ for $x$.

$
\begin{equation}
\begin{aligned}
\log x + \log (x - 3) &= 1\\
\\
\log x(x -3) &= 1 && \text{Laws of Logarithm } \log_a AB = \log_a A + \log_a B\\
\\
10^{\log x(x -3)} &= 10^1 && \text{Raise 10 to each side}\\
\\
x(x - 3) &= 10 && \text{Property of log}\\
\\
x^2 - 3x &= 10 && \text{Distributive property}\\
\\
x^2 - 3x - 10 & = 0 && \text{Subtract 10 }\\
\\
(x - 5)(x + 2) &= 0 && \text{Factor}
\end{aligned}
\end{equation}
$

Solve for $x$

$
\begin{equation}
\begin{aligned}
x -5 &= 0 &&\text{and}& x + 2 &= 0 \\
\\
x &= 5 &&& x &= -2
\end{aligned}
\end{equation}
$

The only solution in the given equation is $x = 5$, since $x = -2$ will give a negative value.

analysis chapter 18, 25 and 29

Chapter 18: "We're The Only Colored People Here"
In this vignette, Maud Martha and her husband, Paul, go to the World Playhouse (a theater). There, they find that they are the only African Americans in attendance. Maud worries about the impression they exude. Paul is also uncomfortable, but the two begin to revel in their experience once the movie begins.
The couple's experience exposes the prejudices that permeate their society. For example, Maud Martha attracts curious glances from white women at the World Playhouse. They assume that Maud Martha's long hair is what attracted the light-skinned Paul to her. Beauty, and by extension, the scope of attraction, is defined by entrenched preconceptions.
The name of the theater is also ironic: while it suggests inclusivity, the reality is far different. At the end of the movie, Maud nurses regret that she can never indulge in casual conversations with the white patrons. Inclusiveness, as the theater's name and its movie offerings suggest, is nothing more than an illusion.
Chapter 25: "The Self Solace"
In this vignette, Maud Martha is at the beautician. During her time there, the owner (Sonia Johnson) purchases a set of lipsticks from a white saleswoman (Miss Ingram). The name of the lipstick is Black Beauty. It is geared toward women with darker complexions. The name and shade of the lipstick stereotype African American women, failing to account for the wide diversity in skin tone within the population itself.
During her exchange with Sonia, Miss Ingram uses a racial epithet. Maud Martha is initially upset, but since Sonia does not challenge Miss Ingram, Maud decides that she must have misheard. After Miss Ingram leaves, Sonia confirms that Maud Martha's ears did not deceive her. However, Sonia makes excuses for Miss Ingram's thoughtless language, perhaps as a way to rationalize her fear of confrontation.
The exchange between Sonia and Miss Ingram highlights the explosive implications of a racial epithet. Sonia is obviously flustered but fearful of appearing contentious. Her rationalizations are a form of self-solace for the pain Miss Ingram has inflicted on her. Racism was so entrenched in Brooks' 1930s era that few Caucasians questioned the implications of their speech.
Chapter 29: "Millinery"
In this chapter, Maud Martha is at the hat shop. She indicates interest in a hat, only to discover that it costs more than she is willing to pay. The white saleswoman treats Maud with barely concealed disdain. Through indirect characterization (Brooks makes us privy to the saleswoman's thoughts), the former's cruel and vindictive nature is exposed.
The saleswoman laments that her own daughter is dating a Greek man ("revolting" enough, but at least fashionable for their time, she thinks). When it comes to Maud Martha's "kinky" hair, the saleswoman exceeds her previous callousness. She dislikes the fact that "concoctions of smelly (hair) grease" leave a lingering smell after African-American customers try on hats. Here, beauty is shown to be confined to a narrow, exclusive standard. "Kinky" hair is portrayed as unnatural, ugly, and worthy of contempt. Maud Martha's insecurity is thus a direct result of such widely-held misconceptions.

int sqrt(x)/(x-4) dx Use substitution and partial fractions to find the indefinite integral

Indefinite integral are written in the form of int f(x) dx = F(x) +C
 where: f(x) as the integrand
           F(x) as the anti-derivative function of f(x)
           C  as the arbitrary constant known as constant of integration
To evaluate the integral problem: int sqrt(x)/(x-4)dx , we may apply u-substitution by letting:
u=sqrt(x) then u^2 =x and 2u du = dx
Plug-in the values, we get:
int sqrt(x)/(x-4)dx=int u/(u^2-4)* 2udu
                  = int (2u^2)/(u^2-4)du
 To simplify, we may apply long division:(2u^2)/(u^2-4) =2 +8/(u^2-4)
To expand 8/(u^2-4) , we may apply partial fraction decomposition.
The pattern on setting up partial fractions will depend on the factors  of the  denominator. The factored form for the difference of perfect squares: (u^2-4)= (u-2)(u+2) .
 
For the linear factor (u-2) , we will have partial fraction: A/(u-2) .
For the linear factor (u+2) , we will have partial fraction: B/(u+2) .
The rational expression becomes:
8/(u^2-4) =A/(u-2) +B/(u+2)
Multiply both side by the LCD =(u-2)(u+2) .
(8/(u^2-4)) *(u-2)(u+2)=(A/(u-2) +B/(u+2)) *(u-2)(u+2)
8=A(u+2) +B(u-2)
We apply zero-factor property on (u-2)(u+2)  to solve for values we can assign on u.
u-2=0 then u=2
u+2 =0 then u =-2
To solve for A , we plug-in u=2 :
8=A(2+2) +B(2-2)
8 =4A+0
8=4A
8/(4) = (4A)/4
A = 2
To solve for B , we plug-in u=-2 :
8=A(-2+2) +B(-2-2)
8 =0 -4B
8=-4B
8/(-4) = (-4B)/(-4)
B = -2
Plug-in A = 2 and B =-2 , we get the partial fraction decomposition:
8/(u^2-4)=2/(u-2) -2/(u+2)
 Then the integral becomes:
int (2u^2)/(u^2-4)du= int [2+8/(u^2-4)]du
                    =int [2 +2/(u-2) -2/(u+2)]du
Apply the basic integration property: int (u+-v+-w) dx = int (u) dx +- int (v) dx+- int (w) dx .
int [2 +2/(u-2) -2/(u+2)]du =int 2du +int 2/(u-2) du int -2/(u+2)du
For the first integral, we may apply basic integration property: int a dx = ax+C.
int 2 du = 2u
For the second and third integral, we may apply  integration formula for logarithm: int 1/u du = ln|u|+C .
int 2/(u-2) du =2ln|u-2|
int 2/(u+2) du =2ln|u+2|
Combining the results, we get:
int (2u^2)/(u^2-4)du = 2u +2ln|u-2| -2ln|u+2| +C
 Apply logarithm property: n*ln|x| = ln|x^n| and ln|x| - ln|y| = ln|x/y|
int (2u^2)/(u^2-4)du = 2u + ln|(u-2)^2| - ln|(u+2)^2| +C
                    = 2u + ln|(u-2)^2/(u+2)^2| +C
Plug-in u =sqrt(x) on 2u + ln|(u-2)^2/(u+2)^2| +C , we get the indefinite integral as:
int sqrt(x)/(x-4)dx =2sqrt(x) +ln|(sqrt(x)-2)^2/(sqrt(x)+2)^2| +C
              OR 2sqrt(x) +ln|(x-4sqrt(x)+4)/(x+4sqrt(x)+4)| +C

Why were the Japanese Americans moved off Terminal Island?

In December 1941, the Japanese attacked Pearl Harbor, bringing the United States into the Second World War. Among other things, the Japanese attack created a great deal of animosity and prejudice towards Japanese-Americans. Even though the vast majority were hard-working, law-abiding, and patriotic, (like Jeanne's father in the story) the Japanese-American community came under intense suspicion. Many people falsely believed that they were potentially disloyal, traitors who would somehow assist the Japanese war effort against the United States.
In a climate of growing suspicion, paranoia, and prejudice, the Roosevelt Administration initiated a policy of forced internment of Japanese and Japanese-American citizens. Whole families were uprooted and moved to settlements where their movements could be more effectively monitored by the authorities. Some people were moved to ghettos; others to what can only be described as concentration camps. In Farewell to Manzanar Jeanne's family, along with other Japanese and Japanese-American citizens, is moved from Terminal Island to the immigrant ghetto of Boyle Heights in Los Angeles.
The rationale of this program of state-sponsored forced relocation was that it was strategically necessary to protect the Long Beach naval station. In reality, however, there was no imminent threat to national security from people like Jeanne's family. They were largely the victims of racist mass hysteria.

What are three instances where characters stand up to authority? Provide quotes.

Montag and Faber both stand up to authority several times throughout the novel. At the beginning of Part III, Beatty tells Montag that he will trace the green bullet back to its owner and Montag switches the safety off of the flamethrower. Beatty then dares Montag to pull the trigger. Bradbury writes,

"And then he was a shrieking blaze, a jumping, sprawling gibbering manikin, no longer human or known, all writhing flame on the lawn as Montag shot one continuous pulse of liquid fire on him" (113).

While Montag is on the run from the authorities, he decides to hide books in a fireman's home as a way to sabotage the system. Bradbury writes,

"He hid the books in the kitchen and moved from the house again to the alley and looked back and the house was still dark and quiet, sleeping" (123).

When Montag arrives at Faber's home to discuss how to escape the city, Faber tells him to follow the river and join a group of traveling intellectuals. Before Montag leaves, Faber mentions how he will stand up to authority by saying,

"You might hole up with them for a time and get in touch with me in St. Louis. I'm leaving on the five a.m. bus this morning, to see a retired printer there, I'm getting out in the open myself, at last" (126).

What does the trapper do during the buffalo stampede in James Fenimore Cooper's The Prairie to stay safe?

In the novel, Natty Bumppo is the trapper, or "the old man." To stay safe during the buffalo stampede, he first shoots a buffalo at the forefront of the stampede and then runs towards the approaching herd with his arms raised. Then, standing firm, he divides the stampede into two columns. Captain Middleton and Paul Hover (the bee hunter) join him in this protected area, and this is how the trapper and his friends stay safe.
Accordingly, the buffalo stampede occurs in Chapter 19. Prior to the stampede, the trapper (Natty) points out a flight of vultures traveling across the sky. He tells Captain Duncan Middleton that the appearance of vultures usually precedes a buffalo stampede.
In short order, the first lines of buffalo make their way through the prairie; eventually, the plains are filled with herds of bison. Natty's companions are entranced by the spectacle, but he has seen such a scene many times in his life. He advises his friends to prepare for the buffalo onslaught, warning them that hiding will prove fruitless.
First, Inez and Ellen, the two women, are placed at the edge of the thicket, farthest from the approaching herd. Asinus, Doctor Battius' donkey, is placed in the center. Meanwhile, Natty and his three male companions "divided themselves in such a manner as they thought would enable them to turn the head of the rushing column, should it chance to approach too nigh their position." Eventually, however, Natty's companions panic and call on Natty to save them.
This is when Natty aims at the buffalo at the forefront of the stampede and shoots. The buffalo falls dead, and this alerts the rest of the herd to the presence of men on the plain. At this point, Natty rushes toward the approaching herd with his arms raised. Intimidated by Natty's sudden appearance, the herd abruptly comes to a stop. Eventually, threatening bellows from the rear cause the stampede to begin anew. This time, however, the herd divides into two columns around Natty, Middleton, and Hover.
The three are able to hold off the beasts from breaking their ranks for a time. Eventually, however, the buffalo push closer and closer to the men, and the situation becomes dire. At this point, Asinus saves the day. With a loud bray, he scatters the frightened beasts. The bison change course (away from the thicket they were approaching), and the stampeding columns eventually reunite at the opposite end of the thicket.

All their efforts would have proved fruitless, however, against the living torrent, had not Asinus, whose domains had just been so rudely entered, lifted his voice, in the midst of the uproar. The most sturdy and furious of the bulls trembled at the alarming and unknown cry, and then each individual brute was seen madly pressing from that very thicket, which, the moment before, he had endeavored to reach, with the eagerness with which the murderer seeks the sanctuary. As the stream divided, the place became clear; the two dark columns moving obliquely from the copse, to unite again at the distance of a mile, on its opposite side.

Why does Crystal enjoy church so much?

The church provides Crystal with a spiritual home, a vital support network at a time when she seems to have hit rock bottom. And things are pretty grim for Crystal; having been evicted, she's out on the streets with no money, selling herself to survive. Crystal has spent her whole life moving from place to place, first in the care system, and then as an adult with numerous mental health issues. The church is about the only place that can provide some kind of stability in her life, some measure of permanence. Crystal's faith offers her much-needed hope, albeit in a somewhat unorthodox fashion. Even when she has little money left—which is virtually the whole time—she donates most of what she has to the church. Crystal believes that by doing this, God will eventually bless her with wealth. This hope, as with everything else associated with the church, gives Crystal something to live for.

Calculus: Early Transcendentals, Chapter 4, Review, Section Review, Problem 26

y=sqrt(1-x)+sqrt(1+x)
Domain of function: -1<=x<=1
a) Asymptotes
a) Asymptotes
The function has no undefined points, so there are no vertical asymptotes.
For horizontal asymptotes , check if at x->+-oo , function behaves as a line y=mx+b,
However +-oo is not in the domain , so there are horizontal asymptotes.
b) Maxima/Minima
y'=(1/2)(1-x)^(-1/2)(-1)+(1/2)(1+x)^(-1/2)
y'=1/2(1/sqrt(1+x)-1/sqrt(1-x))
Let's find critical numbers by solving for x at y'=0,
1/2(1/sqrt(1+x)-1/sqrt(1-x))=0
1/sqrt(1+x)=1/sqrt(1-x) rArrsqrt(1+x)=sqrt(1-x)
squaring both the sides yields,
1+x=1-xrArr2x=0
x=0

Now let's find Maxima/Minima ; which shall be at either critical numbers or at the end points of the function.
y(0)=sqrt(1-0)+sqrt(1+0)=2
y(-1)=sqrt(1-(-1))+sqrt(1-1)=sqrt(2)
y(1)=sqrt(1-1)+sqrt(1+1)=sqrt(2)
Global Maximum=2 at x=0
Global Minimum = sqrt(2) at x=-1 and x=1
c) Inflection Points
Let's find the second derivative
y''=1/2((-1/2)(1+x)^(-3/2)-(-1/2)(1-x)^(-3/2)(-1))
y''=-1/4(1/(1+x)^(3/2)+1/(1-x)^(3/2))
For inflection points, solve for x at y''=0,
(1/(1+x)^(3/2)+1/(1-x)^(3/2))=0
1/(1+x)^(3/2)=-1/(1-x)^(3/2)
squaring both the sides and refining,
(1+x)^3=(1-x)^3
1+x^3+3x(1+x)=1-x^3-3x(1-x)
2x^3+3x(1+x+1-x)=0
x(x^2+3)=0
x=0 , ignore complex zeros
However if we plug in x=0 in y'' it is not true.
so there is no solution of x, hence there are no inflection points.

After Romeo kills Tybalt, he states, “I am fortune’s fool!” Explain how this is shown in the play.

In saying that he is "fortune's fool," Romeo refers to his terrible luck. If you think about it, he has had pretty bad luck throughout. First, Rosaline doesn't return his feelings, and she rebuffs him. Next, he falls in love at first sight with a girl he meets at a party, but she turns out to be the daughter of his family's sworn enemy. Then, she falls in love with him and marries him anyway (yay!), but her vengeful cousin, Tybalt, comes after him. Though Romeo tries to refuse to fight, his best friend, Mercutio, gets involved and is killed when Romeo comes between them to break up the fight. Next, his own anger gets the best of him, despite his resolve not to fight Tybalt, and he kills his wife's beloved cousin. I cannot really think of a single instance of Romeo benefiting from good luck, with the possible exception of Juliet's falling in love with him (though this, too, actually leads to more bad luck down the road).
Even after Romeo says this, the bad luck continues. He doesn't get the message from Friar Lawrence that Juliet isn't really dead. Next, he is able to find a corrupt apothecary who is willing to sell him deadly poison. Finally, he drinks the poison just moments before Juliet wakes up from her death-like sleep.

Using pages 3–22 in Elie Wiesel's Night, give 10 pieces of textual evidence of Jewish optimism.

From your question, you're looking for examples of Jewish optimism. Here are some that may interest you.
The first example of Jewish optimism can be found on page 4. Elie tells us that his father tried to drive the idea of studying the Kabbalah from his mind. Despite his father's efforts, Elie remained optimistic. He sought out Moishe the Beadle in order to learn as much as he could about the Jewish mystic arts. Elie tells us that he and Moishe eventually spent hours trying to decipher Kabbalist revelations from the Zohar.
Before we continue, I would like to point out that Jewish optimism is more than just surface positivity. Instead, it is the conviction that all of us have control over our lives and that we do not have to let life's challenges limit our success. Thus, Jewish optimism is about overcoming opposition, stumbling blocks, and drawbacks. It is about pushing forward, even when success seems unattainable. In this vein, Elie's refusal to give up on his dream of studying Kabbalah constitutes Jewish optimism.
Another example of Jewish optimism centers on Moishe's return. After he miraculously survives being shot, he returns to Sighet. Despite the difficulties of the journey and his own suffering, he makes his way back. This in itself is Jewish optimism: the propensity to press forward despite the obstacles.
Upon his return, Moishe stands outside the synagogue and tries to tell everyone the truth about his experiences. And, despite being ignored and insulted, Moishe continues to warn his neighbors about what is coming. This is, of course, Jewish optimism at play. Moishe is more focused on saving others than himself.
A fourth example of Jewish optimism lies in how Elie's Jewish neighbors regard the war. They never doubt that the Russians will defeat the Germans, and they scoff at the idea that one man (Hitler) could exterminate millions of people. Even after hearing that the Fascists have overtaken Budapest, Elie's neighbors still refuse to worry about their fate. They optimistically reason that the Fascists will never encroach on territory outside Budapest. Within three days, however, German tanks make their way into Sighet.
A fifth example of Jewish optimism lies in how the Jews of Sighet respond to the billeting of German soldiers in Sighet. Because the soldiers are respectful and behave courteously, the Jews believe that legendary tales of German cruelty are false. They allow themselves to relax. On the seventh day of Passover, the Germans arrest every Jewish community leader.
A sixth example of Jewish optimism lies in Elie's father's response to new developments. The text tells us that the Germans eventually forbid Jews to travel anywhere after six o'clock in the evening. Additionally, all Jews are to wear a yellow star on their persons. Elie's father chooses to retain his optimism:

My father's view was that it was not all bleak, or perhaps he just did not want to discourage the others, to throw salt on their wounds: "The yellow star? So what? It's not lethal. . . ."

A seventh example of Jewish optimism centers on how the Jews respond when two ghettoes are set up in Sighet. Despite the barbed wire surrounding them, Elie's Jewish neighbors retain their optimistic attitude. They reason that the wire makes their enclosed surroundings a sort of Jewish republic. Most of them believe that life will continue as normal in the ghettoes until the Russians come to restore the old order of things.
An eighth example of Jewish optimism lies in how the Jews respond to the German order to transport everyone to the "factories." Despite their private fears, Elie's Jewish neighbors try to believe the best about the situation. The text tells us that Elie's father keeps checking with the Jewish Council, in case the "order had been rescinded. To the last moment, people clung to hope."
A ninth example of Jewish optimism lies in how Elie's family reacts as they await their turn to be transported out of the ghetto. Despite the loss of friends and neighbors, everyone remains hopeful. The text tells us that Maria, a former maid, begs Elie and his family to go with her to a safe shelter in her village. Elie's father refuses, and therefore, none of the family leaves with Maria.
Finally, in a most touching demonstration of Jewish optimism, Elie's neighbors try to downplay the enormity of the situation. They tell each other that their departure might be for their own good. They also remind each other that the war will conclude soon. Others joke that the Germans merely want to steal their valuables and that is why the Jews have to leave.
http://awonderingjew.com/2014/06/dont-worry-be-happy-jewish-views-on-optimism/

Why was Holling very impressed with Danny's party?

Danny Hupfer is Holling's Jewish friend. He's reached the age where he will be required to perform his bar mitzvah, the formal rite of passage into manhood for Jewish boys. Danny takes his forthcoming bar mitzvah very seriously, and prepares thoroughly for the occasion. Holling attends the ceremony, and is genuinely impressed with how Danny conducts himself under the weight of so much expectation. Danny sings, and reads from the Torah. Holling immediately understands the significance of what's happening: Danny's taking his place among a historical and religious community going back thousands of years. Naturally, Danny understands the significance of this too, which earns the admiration of Holling. Right before his very eyes, Danny turns from a boy into a man.
But Holling's father doesn't really see the point. He doesn't believe that you can become a man simply by "chanting a few prayers." He thinks that the way to do that is by getting a good job and providing for your family. But Holling admirably stands his ground, telling his father that being a man is more than about just your job; it's about having the freedom to choose for yourself.

Why is nucleus called the director of the cell?

The nucleus is called the "director" of the cell because it controls the cell's activities. The nucleus is the is organelle that directs all of the other functions of the cell. DNA, the genetic material of an organism, is located in the nucleus. DNA is made of amino acids, which combine to form proteins. Proteins then act to direct specific functions including acting as enzymes to drive chemical reactions. This entire process begins in the nucleus of the cell, making it the cell's director.

The nucleus is sometimes referred to as the "director" of a cell because it controls the functions of the cell. Just as a director of a play or movie controls what happens in the show, the nucleus "directs" the cell by regulating the genes the cell contains.
The nucleus is a membrane-protected organelle found in eukaryotic cells (those found in plants and animals). It contains many important things like DNA and RNA and helps the cell to do important activities like move, absorb nutrients, create proteins, and create new cells. The nucleus is made up of multiple different parts that all work together to help it function. It holds the instructions for the rest of the cell's parts to do their jobs correctly. Because it is in charge of helping the cell function in this way, the nucleus is called the "director" of the cell.

The nucleus can be thought of as the "director" of the cell because it directs all the activities of the cell. It is sometimes also called the brain of the cell.
The nucleus is a membrane-enclosed organelle of a cell and is found in eukaryotic cells (for example, plant cell, animal cell, etc.). It stores the DNA of the cell which can be thought of as the code for building proteins, which in turn regulates all the activities of the cell. Through regulation of the protein synthesis, the nucleus is able to direct cell activities, such as metabolism, growth, reproduction, etc.
This ability of the nucleus to control the activities of the cell is the reason it is known as the director or brain of the cell.
Hope this helps.

What are the differences and similarities between "The Tell-Tale Heart" and "Miss Brill"?

One similarity between these two short stories is that both deal with an elderly, isolated person. In "The Tell-Tale Heart" the old man believes the narrator to be his friend, and the narrator tells the reader that he loves the old man. Yet one thing about the man drives him into a state of homicidal madness—his eye.
In Katherine Mansfield's story Miss Brill has bonded, in her own mind, with the people in a park she frequents, where they listen to a military band. She's otherwise an isolated person like the victim in Poe's tale—and in some sense like Poe's narrator as well. She even begins to imagine herself and the others in the park as a kind of acting company, each member playing his or her role, until a young couple are heard making nasty comments about her and the fur she wears. The fur could possibly be seen as Miss Brill's "defect" as the old man's "eye" is in Poe. The old man is killed by the crazed narrator because of the eye, and Miss Brill's illusions about fellowship with the others frequenting the park are destroyed, at least partly because of the fur.
Other than this one parallel that can be drawn, these two stories have little in common. Poe's is a narrative of the bizarre and the macabre. As he and his early nineteenth-century contemporaries typically do, he explores the irrational in human nature, depicts psychosis, and portrays events where the reader cannot be sure how much is "real" and how much is feigned or hallucinated by the narrator. Mansfield's story, by comparison, is realistic, gentle, and poignant. Yet it, as well, deals with a person who, like Poe's narrator, seems to be locked in a dream world of his or her own making.

Why does Godfrey take a secret interest in the upbringing of Eppie, and how does he go about doing this in Silas Marner?

Godfrey takes a clandestine interest in the upbringing of Eppie because she is his real daughter, born of Molly, a woman whom he married secretly. Godfrey provides for Eppie by sending things and having things done in and around the cottage.
When Silas Marner appears at the Squire's in Chapter XIII, he has Godfrey's child in his arms. Marner seeks Dr. Kimble, who parts the guests and leaves to examine the woman who has collapsed in the snow. Before he departs, the doctor tells someone to find Dolly Winthrop to help with the child. Godfrey rides out to find Dolly, and to go to Marner's cottage to see for himself if Molly is alive or dead.
When he returns, having learned of Molly's death, Godfrey considers that now is his opportunity to woo Nancy Lammeter. But, he will not forget the child: 

As for the child, he would see that it was cared for — he would never forsake it; he would do everything but own it.

In Chapter XVI Godfrey "did very kindly by the weaver." Among other deeds, he sends the weaver an oaken table and nice beds, and he offers Silas and Eppie some good soil for a garden.One day, after he has had the area near Silas drained, his brother Dunstan's skeleton is found at the bottom of the well that went dry from the draining. He had fallen in after robbing Marner of his gold. After returning this gold, Godfrey goes to Silas Marner's cottage in Chapter XIX and tells Silas Marner how sorry he is that his gold was stolen. Years later, he asks if he may provide for Eppie:

"You'd like to see her taken care of by those who can leave her well off, and make a lady of her; she's more fit for it than for a rough life...." 

When Marner says that he does not understand, Godfrey asks to adopt Eppie. Silas tells Eppie to speak, saying "I won't stand in your way." But, Eppie thanks Mr. and Mrs. Cass and tells them she cannot leave her father: "I couldn't give up the folks I've been used to."
When he realizes that he no longer can take a secret interest, Godfrey tells Silas and Eppie the truth. But, this has no positive effect for him. Eppie insists upon living with Silas, who has been a real father to her. 

How do I get a Pokestop in my house?

Pokémon Go players now have the opportunity to nominate special areas in or near their hometowns as PokéStops. This new nominating feature is currently available in beta mode and has been released in a number of different countries thus far. In order to nominate new PokéStops, a player must be at Trainer Level 40 in Pokémon Go and a resident of one of the following countries: Argentina, Belize, Bolivia, Brazil, Chile, Colombia, Costa Rica, Denmark, Ecuador, El Salvador, Germany, Guatemala, Guyana, Honduras, India, Indonesia, México, Nicaragua, Panamá, Paraguay, Perú, Philippines, Portugal, Singapore, South Korea, Suriname, Taiwán, Thailand, Uruguay, Venezuela (Bolivarian Republic), or Vietnam.
Unfortunately, nominations for PokéStops cannot be accepted for any private residences, so it is not possible to have a PokéStop at your home. Examples of locations that are eligible for PokéStop nominations include historical sites, famous buildings, public parks, trail signs, schools, churches, and transit stations.
Once you have entered, you will receive a congratulatory email if your nomination is accepted. Niantic Games makes it clear, however, that even if your nomination is accepted, it may still not appear live in Pokémon Go. Nianitic reserves the right to use the nomination in other games.
https://niantic.helpshift.com/a/pokemon-go/?ticket_form_id=319928&p=web&s=pokestops&f=submitting-a-pokestop-nomination&l=en

Niantic Labs, the developer of Pokemon Go, used to be tied to Google. Before Pokemon Go, Niantic developed a geolocation game called Ingress, in which players must visit specific location portals. These portals were often based on historic and cultural sites, such as monuments. Some were chosen based on popular sites found through geo-tagged photos on Google. Others were submitted by users and accepted by Niantic. The popular Ingress portals became the locations for gyms and Pokestops.
In 2019, Pokemon Go began to test a new beta feature that allows level 40 players to nominate locations for future Pokestops. However, this feature is only allowed in select countries, such as Brazil, South Korea, and Peru.
Veteran Ingress players have reported that in an old version of Ingress, high-level players are able to submit new portals for review. If those portals are accepted in Ingress, they could become Pokestops in Pokemon Go. This is a long process, and it is only available to high-level players using the Redacted version of the game. Additionally, the locations are supposed to have a cultural or historical significance or be visually unique. Players also must be able to safely access the location, so a private home is unlikely to be approved.
https://www.dexerto.com/pokemon/pokemon-go-pokestop-gym-submissions-284622

https://www.gamesradar.com/why-your-local-church-and-water-towers-are-pokemon-go-gyms-and-pokestops/

Although Niantic is not currently accepting requests for Pokemon Go stops, many hope they will reopen the form at some point so that you can ask them to place one in your area.
Niantic originally used stops from another of their games, Ingress, as their Pokestops. This means that they directly imported that information into Pokemon Go. Because of this, many rural areas don't have Pokestops—which can be a real problem for people who want to play the game but don't live near any cities.
For a short period of time after releasing the game, Niantic accepted requests for Pokestops. You might want to check back periodically to see if they have reinstated the request box. The form is found on the Pokemon Go support page. You have to click on the "Submit a request" button and then select that you wish to add a Pokestop or Gym. The form then asked for additional information, such as name suggestions and an address.
While it might seem fun to have a Pokestop outside of your house, remember that you would not be the only person to use it. Do you really want people camped out in front of your house all day and night? Probably not. Niantic will not add Pokestops or Gyms to homes (unless there is a special needs case), so make sure to request a public location.
If, however, you are a special needs person or have a disability, rumors say that Niantic will consider adding a Pokestop or Gym within reach of your location.

Single Variable Calculus, Chapter 4, 4.3, Section 4.3, Problem 50

Suppose that $f(t)$ is the temperature at time $t$ where you live and when at $t = 3$ you feel uncomfortably hot. What will you feel on the following conditions?

$
\begin{equation}
\begin{aligned}
\text{a.) } f'(3) &= 2, &&& f''(3) &= 4\\
\text{b.) } f'(3) &= 2, &&& f''(3) &= -4\\
\text{c.) } f'(3) &= -2, &&& f''(3) &= 4\\
\text{d.) } f'(3) &= -2, &&& f''(3) &= -4\\
\end{aligned}
\end{equation}
$


a.) Since $f'(3) > 0$, it means that the temperature is rising and if $f''(3) > 0$, it means that the temperature is not leveling off at any time. If I already feel hot, I might die on a heat stroke at some time.

b.) Since $f'(3) > 0$, it means that the temperature is rising. However, $f''(3) < 0$ it means that the temperature is about to level off. So I will feel hot for only a short period of time.

c.) Since $f'(3) < 0$, it means that the temperature is dropping and if $f''(3) > 0$, it means that the temperature is going to increase. So it looks like I will feel uncomfortable only for a short period of time.

d.) Since $f'(3) < 0$ and $f''(3) < 0$, the temperature is always decreasing. So I might suffer from Hypothermia at some time.

Calculus of a Single Variable, Chapter 10, 10.3, Section 10.3, Problem 32

Parametric curve (x(t),y(t)) has a horizontal tangent when its slope dy/dx is zero, i.e. dy/dt=0 and dx/dt!=0 .
Curve has a vertical tangent if its slope approaches infinity i.e. dx/dt=0 and dy/dt!=0
Given equations of the parametric curve are:
x=t^2-t+2
y=t^3-3t
dx/dt=2t-1
dy/dt=3t^2-3
For horizontal tangents:
dy/dt=0
3t^2-3=0
=>3t^2=3
=>t^2=1
=>t=+-1
Corresponding points on the curve can be found by plugging the values of t in the parametric equation,
For t=1,
x_1=1^2-1+2=2
y_1=1^3-3(1)=-2
For t=-1,
x_2=2^2-2+2=4
y_2=2^3-3(2)=2
Horizontal tangents are at the points (2,-2) and (4,2)
For vertical tangents,
dx/dt=0
2t-1=0
=>t=1/2
Corresponding points on the curve for t=1/2 are,
x=(1/2)^2-1/2+2
x=1/4-1/2+2
x=(1-2+8)/4
x=7/4
y=(1/2)^3-3(1/2)
y=1/8-3/2
y=(1-12)/8
y=-11/8
Vertical tangent is at the point (7/4,-11/8)

Why was the tactic of nonviolence so successful in achieving change?

The tactic of nonviolence has been used to achieve change around the world. Activists employing the tactic of nonviolence often use numerous means to reach their ends. They are known to organize mass noncooperation, economic boycotts, and civil disobedience, among other tools of non-violent action. Such actions have been shown to be successful when the activists engage long enough. Mahatma Gandhi and the nonviolence movement in India demonstrated that it was possible to succeed over a violent oppressor without the use of violence. The same was also demonstrated by Rev. Martin Luther King, as well as with other nonviolent activists during the American Civil Rights struggle.
Success for nonviolence is based on the fact that the tactic draws a clear line between the oppressor and the oppressed. The tactic identifies the victim by presenting their issues and forces the general population to side with the oppressed against their oppressors. The oppressors will characteristically continue pursuing a violent path, which will earn the oppressed support and sympathy. In the end, the oppressors are overpowered by the sheer number of the oppressed and their sympathizers.
http://www.bbc.co.uk/ethics/war/against/nonviolence.shtml

http://www.mshistorynow.mdah.ms.gov/articles/62/the-civil-rights-movement-in-mississippi-on-violence-and-nonviolence

Compare how Mr. Jones and Napoleon ruled the animals.

In the beginning, it would seem that Farmer Jones and Napoleon have a very different approach to the treatment of the animals. While a farmer would typically care for the animals on his farm because they are necessary to keep the farm going and to make money, Farmer Jones is very careless with his animals. In his alcoholic state, he forgets to feed and care for his animals, leaving them in poor conditions. Napoleon, in contrast, seems to have the best interests of the animals in mind, and proposes to take care of and ensure the well being of all animals on the farm.
However, as the reader moves through the story, it is at first foreshadowed through the taking of the milk and the puppies, and then blatantly seen, that not only does Napoleon not care for the animals, he has a brutal, dictator mentality that forces the animals to work harder for worse care than they had with Farmer Jones. Napoleon seizes the opportunity to manipulate the animals into submission and sets himself up as the ruler.
Being a farmer, one would expect Mr. Jones to treat his animals as animals, expecting them to work and controlling them. In return, the animals should expect to be fed and cared for. As an animal, Napoleon should have had a respect for the work of and care for the animals. Because of Napoleon's approach, it would seem that his treatment of the animals was far worse than that of Farmer Jones.

There are close similarities between the respective regimes of Mr. Jones and Napoleon. For one thing, they share a profound contempt for the animals on the farm, ruthlessly exploiting them for their own selfish gain. The main difference, however, lies in the fact that Napoleon is a good deal more ruthless and efficient at exploiting the animals than the alcoholic Mr. Jones. And yet Napoleon is an animal himself!
Mr. Jones treats the animals badly because that's just how it's always been. As far as he's concerned farm animals exist purely as a food resource for humans. There's no ideology involved here; Mr. Jones is a farmer, and the animals are there to be used for food and profit. That's how the farm has always operated. Humans are in charge, and the animals do as they're told. If not, then Mr. Jones can always rely on his trusty whip to keep them in line.
Napoleon, on the other hand, looks upon the animals as a source of political power. Having driven Mr. Jones from the farm, he sees a great opportunity to establish himself as dictator. Napoleon's desire for total control is based on a deeply cynical attitude towards his fellow animals. As an animal himself, he knows what they're capable of if given a chance to run their own affairs. This is something that Mr. Jones would never be able to fathom in a million years. Yet instead of joining with the other animals to build a genuinely Animalist community in which everyone works together for the common good, he chooses to dominate, control, and exploit to satisfy his own lust for power. In other words, Napoleon, unlike Mr. Jones, really should know better.

In 1491 by Charles C. Mann, what is the main thesis?

Charles Mann's work 1491 is an archaeological exploration of pre-Columbian American societies. He examines the historical dynamics between Native American tribes and tries to extrapolate as much about their culture as possible from the little we have on record of their lives prior to the arrival of Europeans.
The primary thesis in his text is that Native American society was much more advanced and civilized than we imagine or was understood by Europeans. The idea of Native Americans as savages is a very pejorative one which discounts their many accomplishments. Mann argues in this text that the Native Americans did not live in some sort of untamed wilderness but had a mutually beneficial relationship with nature and were able to live comfortably and efficiently while impacting the environment as little as possible.
Mann also argues that the society was much more advanced, both intellectually and technology, than they are given credit for. Native American people were skilled hunters and had their own forms of technology that served their purposes incredibly effectively, including easily made canoes and portable housing. The main reason for their treatment as "savages" comes from European ignorance and ethnocentrism which led them to believe that the lack of large cities or immobile communities may them less advanced because they were unlike Europeans.

In 1491, Mann makes two main arguments. The first is that the Native Americans did not live in an untamed wilderness. In fact, he argues, they were far more agriculturally sophisticated than the Europeans who conquered them. As evidence, he notes the large number of foods the Europeans brought back from the New World, such as potatoes (which became a European staple), tomatoes, and corn. He argues that the Native Americans had a different approach to land management than the Europeans. Rather than divide land up into plots to be given to individuals and fenced in, Native Americans treated the ecosystem in a large scale, collectivist way, burning large swathes of forest, building up soil en masse in different places, and preserving other spots as hunting grounds. Though this could look to untrained eyes like a wilderness, in fact it represented an advanced form of cultivation from which we can still learn. Mann even contends that the rain forest in South America shows evidence of deliberate cultivation by the natives to amend and enrich the soil and create a lush agricultural environment.
Second, Mann contends that the pre-1492 native population was larger than has been estimated. He believes the devastation from diseases brought by the Europeans was more extensive and destructive than previously understood. We don't realize how advanced the native cultures once were in the Americas were because by the time we started exploring, say, the interior of what is now the United States, the civilizations that had flourished there were gone. What we saw an untamed wilderness was in fact a post-holocaust landscape in which native civilizations had been destroyed by disease.

Charles Mann writes mainly to support the idea that the half of the world that was probably unknown to Europeans before 1492 was the site of many diverse civilizations. He selected the year 1491 specifically because Columbus arrived in 1492, and he acknowledges the limits of accurately documenting specific dates one year earlier. In supporting the idea of Native American civilization, Mann also writes against two dominant, interrelated ideas that have persisted since early European exploration: that the indigenous people were uncivilized (further elaborated in the Enlightenment as the “noble savage”) and that the territory was largely empty (what William Denevan called the “pristine myth”).
His thesis depends in part on the kind of information he uses, and that he reminds us has only recently become available, which is based on research in anthropology, archaeology, geography, history, and other disciplines. But he also reviews earlier accounts going back to the fifteenth century and points to the religious and political programs of the people who described the inhabitants of what became known as the Americas—continents for which those people had different, now largely unknown names. Throughout those territories, people labored ceaselessly to alter the landscape and grow crops, engaged in trade, and built massive cities; because of these and other factors, their total population far exceeded Europe's at the time.

1491 is the title of a 2005 book by Charles C. Mann in which Mann looks at the archaeological record to demonstrate that the people of the Americas have lived there longer than previously thought and that their civilizations were every bit as vibrant and advanced as the civilizations of Europe and Africa of the same time period.  Mann states that the old theory that the native Americans arrived via the Bering Strait is largely incorrect and that Pre-Columbian societies enjoyed trade networks which spanned that entire continent.  Mann also devotes a chapter to the Columbian Exchange in which he examines the role of earthworms and wheat in the development of the Americas for Europeans.  
Mann's work is unique in that he adds to the historiography surrounding Pre-Columbian civilizations.  Mann does not use the words "native" or "indigenous" as the Pre-Columbian people were not native to the area.  He also uses science to prove that the Pre-Columbian societies did not live peacefully with nature; rather, they adjusted their environment to suit their own needs in ways similar to European and African societies.  

College Algebra, Chapter 4, 4.1, Section 4.1, Problem 70

Suppose at a certain vineyard it is found that each grape vine produces about 10 pounds of grapes in a season when about 700 vines are planted per acre. For each additional vine that is planted, the production of the each vine decreases by about 1 percent. So the number of pounds of grapes produced per acre is modeled by

$A(n) = (700 + n)(10 - 0.01n)$

where $n$ is the number of additional vines planted. Find the number of vines that should be planted to maximize grape production.

We rewrite the function as


$
\begin{equation}
\begin{aligned}

A(n) =& (700 + n) (10 - 0.01 n)
\\
\\
A(n) =& 7000 - 7n+ 10n - 0.01n^2
\\
\\
A(n) =& 7000 + 3n - 0.01n^2

\end{aligned}
\end{equation}
$


The function $A$ is a quadratic function with $a = -0.01$ and $b = 3$. Thus, its maximum value occurs when

$\displaystyle n = - \frac{b}{2a} = - \frac{3}{2(-0.01)} = 150 $ vines planted

The maximum production is $A(150) = 7000 + 3(150) - 0.01(150)^2 = 7225 $.

It shows that $150$ vines should be planned in order to have a maximum production of $7225$ grapes.

What kind of place was crusoe looking for to take shelter

Having found himself washed up on a desert island, Robinson Crusoe urgently needs to find shelter. So he hastily constructs a makeshift tent using some sails from the wrecked ship and a couple of poles he's cut for the purpose. Although the tent is strong enough to withstand strong winds during the night, Crusoe knows that he's going to need a more secure dwelling place. In particular, he needs to find somewhere that will protect him from "savages" and wild animals. The tent is inadequate as it rests upon a low, moorish ground near the sea. As well as being vulnerable to attack from natives, animals, and elements alike, it's also nowhere near any supply of fresh water.
After thinking over the matter, Crusoe establishes four essential criteria for any future shelter:
First, it must be in a healthy place and have access to fresh water;
Second, it must provide shelter from the heat of the sun;
Third, it needs to offer protection from both man and beast;
And fourth, it must afford him a view of the sea so that he can see any ships that might appear on the horizon.

Was Columbus responsible for the behavior of his men?

We should hold Columbus to account for the abuses carried out during his expeditions. He led these expeditions, and on those grounds alone, as a leader, he holds a share of responsibility for the conduct of those under him, but his culpability goes farther than that. Columbus was not ignorant of his men's actions. They were often working under his orders or with his approval. For example, consider the enslavement of Native Americans: this was Columbus's own policy, which he enacted out of belief in its profitability. The same also applies to much of the violence carried out by his men, as well as the tribute systems he set up in the Americas. Even in his own lifetime, this got him into trouble. In 1500, he was brought back to Spain to be put on trial for cruelty.

Upon reaching the New World, Christopher Columbus' men committed many atrocities against the indigenous peoples. As their leader, he should be held responsible for the human rights violations his men committed, even if he was not physically involved in the acts. Leaders set the tone for how a mission will be carried out. Had Columbus had concern for the treatment of the indigenous peoples, he would have set very clear expectations for how his men would treat them and punishments for violating those expectations. Instead, he gave his men license to behave badly and did not punish them when they did so (although in many other respects he was a very strict leader). Columbus receives credit for the achievements which would not have been possible without his men (most notably orchestrating the first permanent European presence in the New World), so it is only fair that he should receive blame for the failure of his men to treat the indigenous Americans with dignity.