What did Augustus do to reform Roman politics, the military, the economy, society and religion to keep the empire together?

Augustus, the first Roman Emperor (who ruled from 27 BCE to 14 CE), wanted to restore ancient Roman religion to improve the moral behavior of the Romans. To this end, he gave himself the title pontifex maximus, or religious head of the empire, and brought back the priesthood. He also restored religious holidays such as Lustrum and started the Imperial Cult to worship the emperor as a god. This cult gained immense popularity. In addition, he restored public monuments, such as the Temple of the Gods, and built monuments that featured traditional Roman scenes, such as the Roman family, and that featured Roman gods such as Apollo and Mars. 
To restore what he considered proper moral behavior and to curtail the practice of having children out of marriage and through adulterous liaisons, he gave financial and political rewards to people who had three children, particularly sons. Men over 38 who were not married were heavily taxed, and they were not allowed to attend public games. Finally, laws required Romans to marry (a law called the ex Julia de maritandis ordinibus) and outlawed celibacy and marriage without having children. Adultery became a crime punishable by the state in the lex Julia de adultenis.
Politically, Augustus placated the Roman senators by revoking the emergency powers that the Roman leaders Julius Caesar and Mark Antony had exercised and returned these powers to the Senate and the people. However, he agreed to retain his emergency powers over rebellious provinces. After he returned power to the Senate, he took the title Augustus in 27 BCE. By taking the title Augustus, he became the first in a line of Roman emperors.
To reform the military, Augustus created the Praetorian Guard, an elite unit meant to protect the emperor. He also used the wealth collected from Egypt to pay off the troops and reduced the number of legions while sending military veterans to the provinces. He gave the veterans lands in these provinces and ensured that the provinces remained peaceful. 
He also instituted economic reforms, including creating a treasury department and standard denominations in the currency. Bronze currency, the issuing of which had been disturbed in the civil wars, was begun again, and he personally oversaw the coinage of gold and silver. Gold became part of the monetary system for the first time in Rome's history. 

Why did Hamlet request the actors to perform the play The Murder of Gonzago?

Hamlet wants the traveling actors who arrive at the castle to perform The Murder of Gonzago because he wants to the test his father's ghost's assertion that Claudius murdered him.
At this point, Hamlet has only the word of his father's ghost that Claudius crept up on him as he was sleeping and poured poison in his ear that killed him. He wonders: was that really my father's ghost, or was it an evil spirit sent by Satan to tempt me to kill an innocent man? He decides he will test whether the ghost's words are true or not by staging a play in which one person murders another by pouring poison in his ear. That play happens to be The Murder of Gonzago.
Hamlet encourages the players to be as realistic as possible as they perform the silent pantomime of the murder. He doesn't want them to ham it up because he wants to see how Claudius reacts to a scene as close as possible to the real event.
Ironically, Claudius couldn't be more pleased at Hamlet's interest in the actors. He has been worried about Hamlet's moping, depressed, angry behavior. In fact, however, the last thing Claudius should be encouraging is Hamlet's involvement with this play.
Claudius's frightened and guilty reaction to the performance shows Hamlet that he is guilty; the ghost did not lie.

Hamlet's intent is to confirm his belief that King Claudius killed Hamlet's father. By having the actors perform The Murder of Gonzago, he's having a mirror of the King's own actions presented in a theatrical setting in front of the whole court. And, as Hamlet hopes and predicts, Claudius is immediately shaken upon seeing the play, gets up, and walks out.
Hamlet has also used the play to observe other people's reactions, including that of his mother, and to provide his own running commentary expressing his general feelings about, essentially, the "phoniness" of the courtiers and of the world overall.
Hamlet's somewhat chaotic ranting is a way of his connecting the real world with the artificial representation of the world in the performance of the players. It is also his way of describing what he regards as hypocrisy: that of Claudius, his mother, and perhaps even what he views as Ophelia's falseness. His crude sexual remarks and jabs at religion ("but he must build churches, or else suffer not thinking on, with the hobby horse!") are a kind of verbal dissection of the falseness of those around him, just as The Murder of Gonzago is a dramatic dissection of Claudius's crime and of the complicity in it which Hamlet sees in his mother and perhaps Polonius as well.

Until the staging of the play within the play, Hamlet has only the assertion made by his ghostly father as proof of Claudius's guilt. So when a troupe of actors come to Elsinore, Hamlet seizes the opportunity to test Claudius by arranging the performance of a play he calls "The Mousetrap." Hamlet asks the players to perform "The Murder of Gonzago," a reference to the murder committed by Luigi Gonzaga, who poured poison in the ear of Francesco Maria I Della Rovere, Duke of Urbino, in 1538. Because the ghost has told Hamlet that Claudius murdered Hamlet's father, who was asleep in his orchard, by pouring poison into his ear, Hamlet expects to get a reaction from Claudius when he watches the scene. Claudius does not disappoint, leaving the performance obviously flustered, and Hamlet secures the proof that seems to confirm what his father's ghost has told him.

Calculus and Its Applications, Chapter 1, 1.8, Section 1.8, Problem 42

Find $y''$: For $\displaystyle y = x^{-3} + 2x^{\frac{1}{3}}$, find $f^{(5)} (x)$


$
\begin{equation}
\begin{aligned}
f'(x) &= \frac{d}{dx} \left[ x^{-3} + 2x^{\frac{1}{3}} \right] = -3x^{-3-1} + 2 \cdot \frac{1}{3} x^{\frac{1}{3}-1}
= -3x^{-4} + \frac{2}{3} x^{-\frac{2}{3}}\\
\\
f''(x) &= \frac{d}{dx} \left[ -3x^{-4} + \frac{2}{3} x^{-\frac{2}{3}} \right] = -3(-4)^{-4-1} + \frac{2}{3} \left( -\frac{2}{3}\right) x^{-\frac{2}{3}-1}
= 12x^{-5} - \frac{4}{9}x^{-\frac{5}{3}}\\
\\
f'''(x) &= \frac{d}{dx} \left[ 12x^{-5} - \frac{4}{9}x^{-\frac{5}{3}} \right] = 12(-5)x^{-5-1} - \frac{4}{9} \left( - \frac{5}{3} \right)x^{-\frac{5}{3}-1}
= -60x^{-6} + \frac{20}{27} x^{-\frac{8}{3}} \\
\\
f^{(4)}(x) &= \frac{d}{dx} \left[ -60x^{-6} + \frac{20}{27} x^{-\frac{8}{3}} \right] = -60 (-6)x^{-6-1} + \frac{20}{27}\left( - \frac{8}{3} \right)x^{-\frac{8}{3}-1}
= 360x^{-7} - \frac{160}{81} x^{-\frac{11}{3}}\\
\\
f^{(5)}(x) &= \frac{d}{dx} \left[ 360x^{-7} - \frac{160}{81} x^{-\frac{11}{3}} \right] = 360 (-7)x^{-7-1} + \frac{160}{81}\left( - \frac{11}{3} \right)x^{-\frac{11}{3}-1}
= 2520x^{-8} - \frac{1760}{243} x^{-\frac{14}{3}}
\end{aligned}
\end{equation}
$

How did J.K rowling come up with the idea to make harry potter books?

Joanne Rowling, more famously known as J.K. Rowling, made the most of a train delay while traveling in 1990. According to Bloomsbury, it was on this delayed train, while traveling from Manchester to London King Cross, that the idea for Harry Potter was born. During an interview with Scholastic, J.K. Rowling said, "Harry just sort of strolled into my head, on a train journey. He arrived very fully formed. It was as though I was meeting him for the first time." When asked if the idea came from something she saw on the train, she said, "I have no idea where the idea came from, it just fell into my head!"
Once the idea was born, J.K. Rowling spent the next several years planning out each of the seven books in the series. However, she told Scholastic that she didn't plan the specific details in advance. "I always have a basic plot outline, but I like to leave some things to be decided while I write. It's more fun," she said.
According to Newsweek, after planning out the series, J.K. Rowling completed her first manuscript for Harry Potter and the Sorcerer's Stone in 1995 -- 5 years after the idea originated.
 
Further reading:
https://harrypotter.bloomsbury.com/uk/jk-rowling-biography/
https://www.scholastic.com/teachers/articles/teaching-content/jk-rowling-interview/
http://www.newsweek.com/how-jk-rowling-created-harry-potter-510042
 

According to Bloomsbury, J.K. Rowling came up with the idea for Harry Pottery when she was delayed during travel.
In 1990, J.K. Rowling was traveling by train from Manchester to King's Cross after looking for a new place to live with the person she was dating then. The trip was delayed and during that time, she thought of the idea for Harry Potter. Rowling says:

I took the train back to London on my own and the idea for Harry Potter fell into my head. . . . Coincidentally, I didn't have a pen and was too shy to ask anyone for one on the train, which frustrated me at the time, but when I look back it was the best thing for me. It gave me the full four hours on the train to think up all the ideas for the book.

According to Newsweek, Rowling completed the first Harry Potter manuscript in 1995. It was published two years later. The last book in the main series was published in 2007.
According to Biography, Rowling's book was green-lighted by Bloomsbury because the daughter of the chairman of the publishing house read the first chapter. She liked it so much, she wanted more. They go on to say that her editor at the time told her it was "impossible to make a living writing children's books." Rowling also says parts of her life inspired the tone of the story and that the death of her mother inspired parts of Harry's character and made the series darker.
https://harrypotter.bloomsbury.com/uk/jk-rowling-biography/

https://www.biography.com/news/jk-rowling-harry-potter-facts

https://www.newsweek.com/how-jk-rowling-created-harry-potter-510042

How does Nicholas spend his time while he is in the lumber room in Saki's story "The Lumber Room"?

In “The Lumber Room” by Saki, Nicholas implements his plan to enter the forbidden room while his aunt searches for him in the gooseberry garden. Being just a little boy, Nicholas practiced using a key for days before he had the opportunity to use the real one to enter the lumber room.
While he was in the room, Nicholas was treated to a variety of items that brought him great delight and intrigue. One of the first things he did was to examine the scene depicted on a piece of tapestry. To the little boy, the scene of a hunt came to life as he imagined the sights and sounds associated with the hunter aiming the bow and arrow at a stag as dogs joined in the hunt. But, the boy could see what the hunter could not. There were wolves descending on the man and dogs, which left Nicholas wondering how the story would unfold.

First and foremost there was a piece of framed tapestry that was evidently meant to be a fire-screen. To Nicholas it was a living, breathing story; he sat down on a roll of Indian hangings, glowing in wonderful colours beneath a layer of dust, and took in all the details of the tapestry picture. 

The tapestry was not the only object of delight in the room. There were dust covered rugs, lamps, paintings, pieces of china, candlesticks, and a variety of books. In addition, there was a plain bound book, which Nicholas thought would not hold any interest for him. But when he opened it, it was filled with pictures of exotic birds, the likes of which he had never seen.

Less promising in appearance was a large square book with plain black covers; Nicholas peeped into it, and, behold, it was full of coloured pictures of birds.

Nicholas spent his time in the lumber room enraptured by its contents of curiosities.

Compare and contrast the essays "Just Walk on By" by Brent Staple and "How It Feels To Be Colored Me" by Zora Hurston.

Staples and Hurston both describe themselves as ordinary individuals who one day awoke to the realization that, to white people, they were not simply "people" but "coloured". This realization marks a "before" and "after" stage in their lives.
Staples's realization comes after an encounter with a woman on the street. It is nighttime; he is walking behind her. She is frightened of him and breaks into a run. Staples says:

I was twenty-two years old, a graduate student newly arrived at the University of Chicago. It was in the echo of that terrified woman's footfalls that I first began to know the unwieldy inheritance I'd come into—the ability to alter public space in ugly ways.

Hurston's realization comes when she leaves her small town and goes away to school in a big city. She does not elaborate on the way in which her blackness is made known to her but implies that it is a sudden and immersive experience rather than a single incident:

I remember the very day that I became colored. Up to my thirteenth year I lived in the little Negro town of Eatonville, Florida. It is exclusively a colored town . . . During this period, white people differed from colored to me only in that they rode through town and never lived there . . . But changes came in the family when I was thirteen, and I was sent to school in Jacksonville . . . When I disembarked from the river-boat at Jacksonville . . . It seemed that I had suffered a sea change. I was not Zora of Orange County any more, I was now a little colored girl. I found it out in certain ways. In my heart as well as in the mirror, I became a fast brown—warranted not to rub nor run.

While both Staples and Hurston experience a kind of awakening to their blackness and its many implications within a predominantly white society, the awakenings are very different. Staples says:

I was surprised, embarrassed, and dismayed all at once. [The woman's] flight made me feel like an accomplice in tyranny. It also made it clear that I was indistinguishable from the muggers who occasionally seeped into the area from the surrounding ghetto.

To Staples, his blackness and his maleness suddenly combine to make him inadvertently menacing. Although he describes himself as "a softy" and "timid," this is not at all how others see him, and there is a sharp, painful disconnect between Staples's sense of himself and the way he is perceived by others. He goes on to describe other encounters—with pedestrians, with a jewelry store owner, with the manager of the building where he works—in all of which the others see Staples as a threat, or even a criminal, while Staples himself is doing nothing more than walking, or browsing, or going to work.
Staples learns that his blackness signifies aggression, danger, and ill intent. He goes to some pains to counter that impression, saying "I now take precautions to make myself less threatening." He wears business clothes, he is "calm and extremely congenial," gives "a wide berth to nervous people," and whistles snatches of classical music.

Virtually everybody seems to sense that a mugger wouldn’t be warbling bright, sunny selections from Vivaldi's Four Seasons. It is my equivalent of the cowbell that hikers wear when they know they are in bear country.

Having been identified by the wider world as a black man, Staples must work constantly to undo people's fear of black men, lest he be victimised by that fear.
Hurston, by contrast, does not feel her blackness to be problematic:

I am not tragically colored. There is no great sorrow dammed up in my soul, nor lurking behind my eyes. I do not mind at all. I do not belong to the sobbing school of Negrohood who hold that nature somehow has given them a lowdown dirty deal and whose feelings are all but about it. Even in the helter-skelter skirmish that is my life, I have seen that the world is to the strong regardless of a little pigmentation more of less.

Hurston embraces her blackness and the vibrant culture African Americans have forged for themselves in the midst of white society. She sees her heritage as a tremendous impetus to do more, do better, seize every opportunity, and revel in the astonishment of white onlookers:

Slavery is the price I paid for civilization, and the choice was not with me. It is a bully adventure and worth all that I have paid through my ancestors for it. No one on earth ever had a greater chance for glory . . . It is quite exciting to hold the center of the national stage, with the spectators not knowing whether to laugh or to weep.

Hurston states that she often ceases to be aware of her blackness, and that she "[feels] most colored when [she is] thrown against a sharp white background," but unlike Staples, these moments of racial contrast do not cause Hurston embarrassment or dismay. She is puzzled by the inability of a white friend of hers to understand the wild beauty of jazz music, but she feels a kind of pity for him, more than anything else—sad at his deafness to the riches of black culture:

He has only heard what I felt. He is far away and I see him but dimly across the ocean and the continent that have fallen between us. He is so pale with his whiteness then and I am so colored

When Hurston has experienced discrimination, she is not angry but "merely astonishe[d]" that anyone would "deny themselves the pleasure of [her] company." Her sense of self is not threatened or shaken by feeling the otherness of white people:

Among the thousand white persons, I am a dark rock surged upon, and overswept, but through it all, I remain myself.

Unlike Staples, Hurston does not feel like "a hiker . . . in bear country" needing to signify to other hikers that she is not dangerous. Consequently, she is able to truly embrace her blackness without the cognitive dissonance Staples experiences. She is not ashamed of her color and is often able to live her life as if color is not a factor.
Staples, by contrast, lives at a different intersection of race and gender, and as a black man he is unable to live any other way. In moments where he has forgotten his blackness and his maleness, the world swiftly reminds him that he is both of these things and that the combination of the two marks him as a threat. He must constantly seek to put people around him at ease, even other black people:

At dark, shadowy intersections, I could cross in front of a car stopped at a traffic light and elicit the thunk, thunk, thunk of the driver—black, white, male, or female—hammering down the door locks.

The sum of "black" and "man" is "danger," a mental arithmetic Staples must factor into any encounter he has with anyone. He does not have the luxury of forgetting his race, and he must always see himself as other people see him so that he can soothe their anxieties in advance. He seems almost to resent his blackness and the burden it has placed on him; he cannot relax into it and simply live as a person who happens to be black. To the other "hikers in bear country," Hurston may be a hiker, but Staples is a bear. Their experience of life is very different as a result.
https://www.casa-arts.org/cms/lib/PA01925203/Centricity/Domain/50/Hurston%20How%20it%20Feels%20to%20Be%20Colored%20Me.pdf

int 1/(4+4x^2+x^4) dx Find the indefinite integral

int1/(4+4x^2+x^4)dx
Let's rewrite the integrand as :
=int1/((x^2)^2+2(2)(x^2)+2^2)dx
=int1/(x^2+2)^2dx
Apply integral substitution:x=sqrt(2)tan(u),u=arctan(x/sqrt(2))
dx=sqrt(2)sec^2(u)du
=int(sqrt(2)sec^2(u))/((sqrt(2)tan(u))^2+2)^2du
=int(sqrt(2)sec^2(u))/(2tan^2(u)+2)^2du
=int(sqrt(2)sec^2(u))/(2^2(tan^2(u)+1)^2)du
Take the constant out,
=sqrt(2)/2^2int(sec^2(u))/(tan^2(u)+1)^2du
Use the identity:1+tan^2(x)=sec^2(x)
=sqrt(2)/4int(sec^2(u))/(sec^2(u))^2du
=sqrt(2)/4int1/(sec^2(u))du
=sqrt(2)/4intcos^2(u)du
Use the trigonometric identity:cos^2(x)=(1+cos(2x))/2
=sqrt(2)/4int1/2(1+cos(2u))du
Take the constant out,
=sqrt(2)/8int(1+cos(2u))du
Apply the sum rule,
=sqrt(2)/8{int1du+intcos(2u)du}
Apply the common integral:intcos(x)dx=sin(x)
=sqrt(2)/8{u+1/2sin(2u)}
Substitute back u=arctan(x/sqrt(2))
=sqrt(2)/8{arctan(x/sqrt(2))+1/2sin(2arctan(x/sqrt(2)))}
=sqrt(2)/8{arctan(x/sqrt(2))+1/2(2sin(arctan(x/sqrt(2)))cos(arctan(x/sqrt(2))))}
=sqrt(2)/8{arctan(x/sqrt(2))+sin(arctan(x/sqrt(2)))cos(arctan(x/sqrt(2)))}
Use the identities:sin(arctan(x))=x/sqrt(1+x^2),cos(arctan(x))=1/sqrt(1+x^2)
=sqrt(2)/8{arctan(x/sqrt(2))+x/sqrt(x^2+2)sqrt(2)/(sqrt(x^2+2))}
=sqrt(2)/8{arctan(x/sqrt(2))+(sqrt(2)x)/(x^2+2)}
=1/8{sqrt(2)arctan(x/sqrt(2))+2x/(x^2+2)}
Add a constant C to the solution,
=1/8(sqrt(2)arctan(x/sqrt(2))+(2x)/(x^2+2))+C
 

Single Variable Calculus, Chapter 7, 7.6, Section 7.6, Problem 72

Find the volume of the region under the curve $\displaystyle y = \frac{1}{\sqrt{x^2+4}}$ from $x = 0$ to $x = 2$ that is rotated about the $x$-axis.

By using vertical strips, notice that if you slice the curve, its cross section forms a circle with radius $\displaystyle \frac{1}{\sqrt{x^2+4}}$. Hence, its cross sectional area is as $\displaystyle A = \pi \left( \frac{1}{\sqrt{x^2 +4}} \right) = \frac{\pi}{x^2+4}$

Thus, the volume is...

$
\begin{equation}
\begin{aligned}
V &= \int^2_0 A(x) dx\\
\\
V &= \int^2_0 \frac{\pi}{x^2+4} dx\\
\\
V &= \pi \int^2_0 \frac{dx}{x^2 + 4}\\
\\
\text{We can rewrite it as } V &= \pi \int^2_0 \frac{\left( \frac{1}{4} \right)}{\frac{x^2}{4}+1} dx\\
\\
&= \pi \int^2_0 \frac{\left( \frac{1}{4} \right)}{\left(\frac{x}{2}\right)^2+1} dx
\end{aligned}
\end{equation}
$


If we let $\displaystyle u = \frac{x}{2}$, then
$\displaystyle du = \frac{dx}{2}$
Make sure that the upper and lower limits are also in terms of $u$ so...
$\displaystyle V = \pi \int^{\frac{2}{2}}_{\frac{0}{2}} \frac{2 du}{u^2 + 1} \left( \frac{1}{4} \right)$


Recall that $\displaystyle \frac{d}{dx}\left( \tan^{-1} x \right) = \frac{dx}{1 + x^2}$

$
\begin{equation}
\begin{aligned}
V &= \frac{\pi}{2} \int^1_0 \frac{du}{u^2 \pi}\\
\\
V &= \frac{\pi}{2} \left[ \tan^{-1} u \right]^1_0\\
\\
V &= \frac{\pi}{2} \left[ \tan^{-1} (1) - \tan^{-1} (0) \right] = \frac{\pi}{2} \left[ \frac{\pi}{4} -0 \right] = \frac{\pi^2}{8} \text{cubic units}
\end{aligned}
\end{equation}
$

What characteristics most help Rikki-Tikki defeat the snakes?

The basic characteristics that Rikki-tikki has that allow him to defeat snakes are the basic characteristics of a mongoose. A mongoose is a natural predator to snakes of all kinds; therefore, Rikki-tikki has been blessed with innate abilities that work in his favor. One such characteristic is his quickness. Being as fast or faster than the strike of a cobra or an Eastern brown snake is fast.  

The victory is only a matter of quickness of eye and quickness of foot—snake's blow against mongoose's jump—and as no eye can follow the motion of a snake's head when it strikes, this makes things much more wonderful than any magic herb.

Readers get a clue early on as to how quickly Rikki-tikki can move. Nagaina tries to catch Rikki-tikki by surprise from behind, and he is able to quickly jump out of the way of the strike. A few paragraphs later, Rikki-tikki defeats Karait. Readers are specifically told Karait is exceptionally quick because of his size.

If Rikki-tikki had only known, he was doing a much more dangerous thing than fighting Nag, for Karait is so small, and can turn so quickly, that unless Rikki bit him close to the back of the head, he would get the return stroke in his eye or his lip.

Another characteristic of Rikki-tikki that allows him to defeat the snakes is that he is smart. He knows when to press his advantage and when to back off.  

Rikki-tikki did not care to follow them, for he did not feel sure that he could manage two snakes at once.

Finally, Rikki-tikki is brave. Taking on dangerous snakes is brave to begin with; however, the story ends with Rikki-tikki following Nagaina into her home territory. This act is something Rikki-tikki knows is especially dangerous, but he knows he must kill her for the safety of everybody.  

And very few mongooses, however wise and old they may be, care to follow a cobra into its hole. It was dark in the hole; and Rikki-tikki never knew when it might open out and give Nagaina room to turn and strike at him.

I need help to describe the color pink in an essay.

You do not specify whether pink is the subject of your essay or if your description constitutes just a segment of a longer essay on another topic. My attempt below considers the alternatives:
If pink is the subject of your entire essay:
Your essay should obviously commence with a thesis statement in which you divulge what the main theme of your paper is and what you wish to convey about the colour in your writing. The statement should be specific so that your audience is aware of the direction your essay will take. You could, for example write, 'We live in a world rich in color, but some colors are more significant than others. The color pink is a perfect example of such significance. It has, throughout history, retained its allure and is symbolic of the good, the bad, the mysterious, the mundane, the extraordinary, the practical and the metaphysical.'
Each following paragraph in your essay can then describe and discuss the color within the themes / main ideas you specified in your thesis statement. Your second paragraph may, for instance, provide a visual image of what pink looks like and what defines its allure. Your audience should be able to see the image in their minds since the descriptors you use convey the color as you see it. In this regard, you may depict it as 'pale red' or  'a bit of red tainted with massive dollops of white mixed in,' or 'oodles of white with just a tint of red blended in.' Whichever descriptors you use should be effective and enable the reader to see what you want them to see—your vision of pink. 
In each consecutive paragraph you can discuss the different symbolic associations and meanings pink conveys. Each paragraph would obviously deal with a different perspective. The conclusion of your essay should tie in with the introductory paragraph and convey a sense of completeness.
If the description is part of a longer essay on a different topic:
In this regard, your task should be somewhat easier. In this instance all you need to do is to use descriptors similar to the ones I've used in the second paragraph to describe it. If you need a more lengthy piece as part of an essay with a different theme, you could describe the color by mentioning its appeal to the senses. How would pink look, taste, smell, sound, or feel and what kind of emotion or mood would it convey? If someone should, for example, say, 'I feel pink today,' what kind of image would that evoke? Or imagine someone speaking of a 'pink sound.' Would it be harsh and jarring or soft and soothing? This approach can obviously also be used if pink should be the topic of your entire paper.
There are many variables you can choose from, but whatever you attempt, be creative! I have listed a number of websites you can visit for help and ideas. All the best!
https://en.wikipedia.org/wiki/Pink

https://www.bourncreative.com/meaning-of-the-color-pink/

During the Civil War, why was the North ultimately able to defeat the South?

The North was able to defeat the South due to many factors. The first factor is attrition. The North had more men serving in the armed forces than the South—at the end of the war, the North had over one million men in uniform. The North also had a greater industrial capacity to supply these men. The North had a much better infrastructure system than the South, and it could use its railroads to send men and supplies where needed. The food in the Union army was not always the best, but it was at least present under normal circumstances; many Confederate soldiers either had to forage for food or do without, especially in the latter stages of the war.
The North also had better finances. The Confederate treasury lacked the capacity to keep up with the demands of a war this size. The North, while having to implement an income tax, was never insolvent. The North also had the benefit of being recognized by major governments. The Confederacy bet their survival on being recognized by European powers who might become allies, but due to a naval blockade by the North and William Seward's war threats, Europe never came to the South's aid. Britain and France realized that they could get their cotton from North Africa and India and, thus, avoid the American situation altogether. After the Emancipation Proclamation in 1863, many Britons saw the North as being morally right. This also coincided with major Confederate defeats in the field at Gettysburg and Vicksburg, thus proving that Southern independence was not assured and that any support given would be a risk many in Europe did not wish to take.

Why do you think the Gothic element was so attractive to Romantic writers such as Irving, Poe, and Hawthorne?

Gothic literature, especially as it developed in the mid-nineteenth century, was less concerned with outright horror and the supernatural than with the individual's reactions to fear, anxiety, and uncertainty. Poe's work, for example, is mostly psychological horror. It often turns out that there is no supernatural threat to the protagonists; instead, we study the character's descent into madness, his psychological devolution.
This is true in works like "The Tell-Tale Heart," "The Black Cat," and "The Premature Burial." In the first and second works, the narrators have committed crimes and then been driven mad by their own actions, their guilt, and/or their overactive imaginations. The final story studies the protagonist's all-encompassing fear of being buried alive; his obsession leads to his heightened anxiety and borderline madness. It is ultimately the character's own psyche that destroys him in some of these tales. The same can be said of some of Hawthorne's Gothic-tinged tales, like "Young Goodman Brown," though his experience incorporates more supernatural horror than Poe's works do. Still, the protagonist is psychologically destroyed by what he learns in the woods by observing the satanic ceremony.
Washington Irving wrote slightly earlier than the other two authors, and his works are a bit more classically Gothic. I think his inspiration for his supernatural folktales is the uncertainty of "the new world" that colonists entered. There was so much they didn't know or understand about their new environment, and this presented fear and anxiety to them. The tales could function as a way to deal with the unsettling feeling colonists had in a world that seemed both promising and threatening.

Another reason the Gothic was so attractive to some Romantic writers is that it allowed them to explore the darker side of human nature and our common fears.  Instead of focusing on the sunnier side of Romanticism, with its focus on the sublime, nature, beauty, and love, the Gothic focuses on the things that scare us and the things that worry us in the middle of the night.  Poe, for example, can show us a man who claims to be perfectly sane but who murders an old man because he does not like the old man's "vulture eye"; in reality, the murderer seems to fear the eye because it reminds him of his own mortality and the inevitability of death.  Hawthorne can show us a man who gives in to temptation and meets the devil in the forest. Even though he eventually decides to return to his faith, he finds that it is too late. He has lost his faith forever.  Irving can show us a teacher who exploits his students' families and attempts to manipulate a young woman to raise his social status.  If these authors only focused on the brighter elements of our natures, these darker aspects of ourselves would remain unexplored.

Gothic horror was a sub-genre of Romanticism, as the Romantic movement sought to reverse the Enlightenment's focus on reason and emphasized sensation and emotion instead. The Gothic element, with its emphasis on suspense, horror, and the supernatural, was designed to provoke sensation and emotion in the reader and to confront the reader with elements of the supernatural that defied reason. Irving, Poe, and Hawthorne, among other Romantic writers, found this element attractive because it could provoke a strong sensation in their readers.
Irving featured Gothic elements in works such as "The Legend of Sleepy Hollow," in which the Headless Horseman haunts the mystery-infused Hudson Valley, shrouded with mist and the supernatural. The Hudson is also the setting of Irving's story "Rip Van Winkle," in which Rip, a character modeled on the simple village dweller who was a staple of folklore, falls into a twenty-year sleep after meeting Henry Hudson's men. Irving's stories are Romantic in that they feature dark, gloomy natural scenes and supernatural elements that are designed to provoke a sensation of mystery and imagination in the reader.
Poe's stories also confront the reader with situations that are contrary to reason. For example, in "The Cask of Amontillado," the narrator, Montresor, descends deeper and deeper into a wine vault until he buries his nemesis in a crypt. This type of tale is contrary to the force of reason and again presents the reader with chilling sensations and a sense of doom.
Finally, Hawthorne's tales, such as "The Birthmark," show the limits of science. In contrast to the Enlightenment, which focused on science and scientific laws, this story is about a scientist who goes awry. Aylmer, the scientist, removes his wife's one flaw--her birthmark--but kills her in the process. His search for scientific perfection is misguided, which is a Romantic idea that goes against the belief in reason presented by the Enlightenment. Hawthorne's Gothic elements provide a sense of suspense and thrill in the reader.

What did the author want you to think/feel regarding this text

Ray Bradbury’s short story “The Pedestrian” takes place in a dystopian future. It is the year 2053, and the story’s protagonist, Leonard Mead, leaves his house late at night to go for a walk. We quickly discover that, although Leonard walks nightly, his behavior is quite odd in this future: everyone else is at home, watching television. While there are other people in the world, the story is post-apocalyptic in feel. Leonard’s isolation is on full display for the reader as he sadly wonders what everyone is watching. 
Although science fiction tends to be set in the future, it often helps us see something about our own historical present. “The Pedestrian” warns us about the dehumanizing and antisocial potential of the television. Leonard is a writer, but he has not written in years. We learn that there is no longer any crime because everyone is at home watching television; similarly, there is no longer any need for fiction. This metafictional moment likely produces a sense of sadness for the reader, who is currently reading “The Pedestrian” instead of watching television. In this way, the story invites us to identify with Leonard and his sad fate. By doing so, the story warns us about the dangers of television and the culture that forms around it.

College Algebra, Chapter 4, 4.5, Section 4.5, Problem 22

Factor the polynomial $P(x) = x^3 - x^2 + x$, and find all its zeros. State the multiplicity of each zero.

To find the zeros of $P$, we set $x^3 - x^2 + x = 0$, so $x(x^2 - x + 1) = 0$ by using quadratic formula


$
\begin{equation}
\begin{aligned}

x =& \frac{-(-1) \pm \sqrt{(-1)^2 - 4 (1)(1)}}{2(1)}
\\
\\
=& \frac{1 \pm \sqrt{-3}}{2}
\\
\\
=& \frac{1 \pm \sqrt{3} i}{2}

\end{aligned}
\end{equation}
$


By factorization,


$
\begin{equation}
\begin{aligned}

P(x) =& x \left[ x - \left( \frac{1 + \sqrt{3} i}{2} \right) \right] \left[ x - \left( \frac{1 - \sqrt{3} i}{2} \right) \right]

\end{aligned}
\end{equation}
$


The zeros of $P$ are $\displaystyle 0, \frac{1 + \sqrt{3} i}{2}$ and $\displaystyle \frac{1 - \sqrt{3} i}{2}$. Each zeros has multiplicity of $1$.

How is Ichabod Crane’s failed pursuit of Katrina Van Tassel a metaphor for the inferiority of society compared to nature?

Ichabod's failed pursuit of Katrina Van Tassel works as a metaphor for the inferiority of society compared to nature, because Ichabod himself represents a man obsessed with some of the worst aspects of society.  He is a school teacher but feels no particularly strong calling to the profession.  Rather, his interest stems from the ease with which it allows him to exploit the families of his students "who happened to have pretty sisters, or good housewives for mothers, noted for the comforts of the cupboard."  He uses and abuses the system for sustenance. 
Similarly, his interest in Katrina is based not on sincere affection for her, but rather for the land that she is set to inherit via her father, Baltus Van Tassel.  Ichabod's

heart yearned after the damsel who was to inherit these domains, and his imagination expanded with the idea, how they might be readily turned into cash, and the money invested in immense tracts of wild land, and shingle palaces in the wilderness."

If Ichabod succeeds in his goal, he would turn the profit from the farm into the acquisition of additional land.  In order to achieve this end, he courts Katrina in much the same way that he learned to court the families of his pupils.  However, he fails because Katrina desires another: the much more sincere Brom Bones.  With Brom, what you see is what you get.  He doesn't pretend to be something that he is not.  He is straightforward, perhaps too much so, and he does not hide his intentions.  He is interested in Katrina for Katrina,  not the land she will inherit.  He represents the uncivilized, untamed spirit of man.  In the end he succeeds where Ichabod fails, and in turn represents the triumph of nature unbridled.

Write a short note on Achilles' anger.

The rage of Achilles is a central theme in the Iliad. Homer begins the story by asking the muse to tell him the story--not of the Trojan War--but of Achilles' rage.

Rage - Goddess, sing the rage of Peleus' son Achilles,murderous, doomed, that cost the Achaeans countless losses,hurling down to the House of Death so many sturdy souls,great fighters' souls, but made their bodies carrion,feasts for the dogs and birds,and the will of Zeus was moving toward its end.

Homer blames Achilles for the loss of countless Greek lives, and this blame is justified. Achilles rages at Agamemnon for stealing his concubine, and in his rage he refuses to fight against the Trojans. Achilles was the most powerful Greek warrior, so his absence from the battlefield (and the compassion of the goddesses who sympathized with him) resulted in the deaths of numerous Greek soldiers.
Later on, Achilles does fight, but not until his dear friend, Patroclus, dies in combat at the hands of Hector. Once again, Achilles rages, but this time he directs his rage toward Hector. He kills Hector, but he does not behave honorably afterwards. You see, revenge never satisfies one the way he believes it will. Achilles was still angry, so he desecrated Hector's dead body, debasing himself in the process.

Where did Brian shown confidence in the book Hatchet?

Brian shows confidence in coming up with innovative solutions for his survival. 
For someone who is young and does not have a ton of survival experience, Brian is actually fairly confident.  Many boys in his situation would give up or get discouraged much more easily than he does.  Brian has his ups and downs, but generally speaking he is intelligent and innovative, and continues to work hard.  He doesn’t let mishaps get him down. 
An example of this is his desire to get a better food source.  Brian wants to develop a fish spear.  He makes a plan and carries it out, confident that he will succeed. It is not successful at first. 

He had been so sure, so absolutely certain that it would work the night before. Sitting by the fire he had taken the willow and carefully peeled the bark until he had a straight staff about six feet long and just under an inch thick at the base, the thickest end. (Ch. 12) 

Brian has to “invent” a bow and arrow.  Again, he is confident that this will work.  He doesn’t give up because he is hungry and can’t just go to McDonald’s and order a burger and fries.  When one thing doesn’t work, he knows that he will be able to make something else work.  He believes in himself. 
When Brian misses a search plane, he gets frustrated and feels as if he wants to die.  However, he regains his confidence again.  He realizes that he is clever, and has come this far.  He is confident that he can survive on his own. 

He was new. Of course he had made a lot of mistakes. He smiled now, walking up the lake shore after the wolves were gone, thinking of the early mistakes; the mistakes that came before he realized that he had to find new ways to be what he had become. (Ch. 13) 

Brian fishes out the survival pack, and its contents are almost superfluous because he has been so successful on his own.  However, it has a radio that the search planes use to find Brian.  If Brian had not maintained his calm and confidence, mistakes and all, he never would have been found.

When and why do individuals and groups stop believing in a political process and turn to terror and violence to achieve what they see as justice?

Individuals and groups may resort to violence when they realize that the government or the administration will not make any concessions towards their demands. Noticeable breakdowns in the political process lead to loss of confidence among the people. For instance, political processes are seen to be breaking down when the governing institution imposes itself on the governed through force, coercion, and intimidation. In such governments, the leadership would want to increase administrative powers or extend leadership terms against the will of the people. In such a situation, public protests would ensue to fight against the undemocratic practices of the leadership. The protests might start off peacefully but turn violent, forcing the government to respond through police action. A meeting of the protesters and the police on the streets can increase the chance of violence between the two groups.
A real-life example of such a situation occurred in Ukraine in 2013. President Viktor Yanukovich failed to pay attention to the demands of the people through the opposition. The opposition proposed a reduction of presidential powers, constitutional reforms, and changes in the governing system.  Yanukovich rejected the proposal and held on to the excessive presidential powers. He stifled parliamentary processes to pass amendments by coercing and intimidating the members of parliament. The stalemate led to the spread of violence from Kiev to the rest of Ukraine. The situation led to his self-imposed exile to Russia and ouster as president of Ukraine.
http://www.lse.ac.uk/international-development/conflict-and-civil-society/past-programmes/csrc-publications-resources

https://www.aljazeera.com/indepth/opinion/2014/02/violence-death-failed-politica-2014220153327383148.html

https://www.rt.com/news/ukraine-kiev-death-toll-955/

Calculus of a Single Variable, Chapter 5, 5.8, Section 5.8, Problem 89

To solve the differential equation of (dy)/(dx)=(x^3-2x)/(5+4x-x^2) .we may express it in a form of variable separable differential equation:N(y) dy = M(x) dx
dy=(x^3-2x)/(5+4x-x^2) dx
Then apply direct integration on both sides:
int dy= int (x^3-2x)/(5+4x-x^2) dx
For the left side, we apply the basis integration property: int dy = y
For the right side, we may apply long division to expand:
int (x^3-2x)/(5+4x-x^2)dx= int [-x-4+(20)/(5+4x-x^2)]dx
Then apply basic integration property: int (u+-v) dx= int(u) dx +- int (v) dx
where we can integrate each term separately.
int [-x-4+(20)/(5+4x-x^2)]dx=int (-x) dx -int 4dx +int (20)/(5+4x-x^2)]dx

For the integration of int (-x) dx , we may apply Power Rule integration: int u^n du= u^(n+1)/(n+1)+C
int (-x) dx = - int x dx
=-x^(1+1)/(1+1) = -x^2/2
For the integration of -int 4 dx , we may apply basic integration property:int c*f(x)dx= c int f(x) dx
-int 4dx = -4 int dx
= -4x

For the integration of int 20/(5+4x-x^2)dx , we apply partial fractions:
(20)/(5+4x-x^2) = 20/(6(x+1)) -20/(6(x-5))
Then, int (20)/(5+4x-x^2)dx=int [20/(6(x+1)) -20/(6(x-5))]dx
Apply basic integration property: int (u+-v) dx= int(u) dx +- int (v) dx
int [20/(6(x+1)) -20/(6(x-5))]dx =int 20/(6(x+1))dx-int 20/(6(x-5))dx
Apply apply the basic integration property: int c*f(x)dx= c int f(x) dx and basic integration formula for logarithm: int (du)/u = ln|u|+C .
int 20/(6(x+1))dx-int 20/(6(x-5))dx =(20/6)int 1/(x+1)dx-(20/6)int 1/(x-5)dx
=(20/6)ln|x+1|- (20/6)ln|x-5|
For the right side, we get:
int [-x-4+(20)/(5+4x-x^2)]dx= -x^2/2-4x+(20/6)ln|x+1|- (20/6)ln|x-5|+C
Note: Just include the constant of integration "C" on one side as the arbitrary constant of a differential equation.
Combining the results from both sides, we get the general solution of the differential equation:
y =-x^2/2-4x+(20/6)ln|x+1|- (20/6)ln|x-5|+C
y =-x^2/2-4x+(10/3)ln|x+1|- (10/3)ln|x-5|+C

Calculus of a Single Variable, Chapter 8, 8.6, Section 8.6, Problem 41

To evaluate the integral problem: int_1^2 x^4ln(x) dx , we follow the formula from basic integration table. For the integrals with logarithm, the problem resembles the formula:
int x^n ln(x) dx = x^((n+1)) ( ln(x)/(n+1)- 1/(n+1)^2), n!= -1 .
By comparison of x^n with x^4 , we let n=4 which satisfy that condition n!=-1 to be able to use the aforementioned integral formula.
Then the integral problem is evaluated as:
int_1^2 x^4ln(x) dx= [x^((4+1)) ( ln(x)/(4+1)- 1/(4+1)^2)]|_1^2
= [x^(5) ( ln(x)/5- 1/5^2)]|_1^2
= [x^(5) ( ln(x)/5- 1/25)]|_1^2
= [(x^(5) ln(x))/5- x^5/25]|_1^2
Apply definite integral formula: F(x)|_a^b = F(b) - F(a) .
[(x^(5) ln(x))/5- x^5/25]|_1^2=[(2^(5) ln(2))/5- 2^5/25]-[(1^(5) ln(1))/5- 1^5/25]
=[(32 ln(2))/5- 32/25]-[(1ln(1))/5- 1/25]
=(32 ln(2))/5- 32/25 -(1ln(1))/5+ 1/25
= (32 ln(2))/5 -0/5+ (1-32)/25

=( 32ln(2))/5 -31/25 or 3.196 (approximated value).

What are the differences in beauty and culture between Indian and Western women and how do they influence Indian women?

Increasing globalization and market dominance of the Western world as a huge impact on traditional culture, including ideals of beauty and aesthetics. In India, traditional values of beauty vary from region to region. Generally, classical beauty in India values women with full lips, expressive eyes, full physique, and thick, black hair. The 16th century Ananga-Ranga has an entire passage dedicated to describing what makes a woman beautiful.  Mentioned are traits like clean teeth, small ears, wide hips but a trim waist, and a sweet voice. In contrast, the same passage describes the following as ugly: a woman with facial hair, yellowed eyes, being notably tall or short, and having a temper. 
In the same time period, Western women were subject to similar valuation of their personality traits, and in both cultures women with mild tempers and sweet voices were considered preferable. Both cultures (pre-colonial India and Renaissance Europe) also had a similar taste for women who were of a certain voluptuous form, as long as it did not lead to excess of the body. One major difference between Indian and European beauty ideals of the time is that in Europe, the complexion was expected to be much lighter. Women with alabaster skin and golden blonde or reddish hair were considered to be most beautiful.
Since the colonial period, Indian women have been increasingly subject to the beauty standards of the Western world. In the context of colonization (and even post-colonial society) the colonizer has more power in determining what is beautiful or valuable in society. As such, a woman might do well to appeal to the colonizer's ideals of beauty. In particular, skin tone and complexion have become major indicators of both class and beauty. From the colonial period onward, fair skin indicated being of higher class because it either meant a person was of Anglo descent or could afford a lifestyle which allowed them to keep their skin a light color through using skin creams or avoiding sunlight. This ideal persists in India today, and many creams which lighten the skin are readily available and encouraged to young women. Indian women are also influenced by the Western ideal of women with slim bodies, as portrayed in media. Though the ideal of wide hips and a full bust persists in Indian beauty, women are increasingly dieting or purchasing garments which alter the shape of the body in order to appear thinner in a particular body part. 
In order to be considered attractive and succeed in a highly competitive world, Indian women are increasingly pressured to appeal to Western beauty ideals, though some traditional values do remain.
https://digitalcollections.sit.edu/cgi/viewcontent.cgi?referer=&httpsredir=1&article=2143&context=isp_collection

https://kaffeinerush.wordpress.com/2012/04/10/classic-indian-beauty/

https://www.ncbi.nlm.nih.gov/pubmed/17166111

How do you believe former President Hoover would respond to FDR's inauguration speech?

President Hoover would have responded that President Roosevelt had inaccurate ideas of what caused the Great Depression and what was needed to deal with the impact of the Great Depression. President Hoover was a strong supporter of big business. He would have stated that the bankers and the stockbrokers, whom Roosevelt called money changers, had tried to do what they felt was best for the economy. These people, Hoover would have responded, were not motivated by greed and by personal profit at the expense of the American people.
President Hoover also would have disagreed that the government should be actively involved in trying to deal with ending the Great Depression. Hoover believed in a laissez-faire philosophy, in which the government would play a limited role in economic matters. Hoover would have pointed out that some of the programs that he initiated, such as the Reconstruction Finance Corporation and the Emergency Relief and Construction Act, did little to ease the impact of the Great Depression. He would have stated that more government involvement, as Roosevelt was proposing, wouldn’t have helped to ease the impact of the Great Depression.
https://www.presidency.ucsb.edu/documents/inaugural-address-8

How did Defoe develop a deep understanding of the outcasts of society?

Daniel Defoe presents the case of a man with a very intriguing biography, one that certainly allowed him to understand the role of the outcast. As a very prolific author, he wrote a number of controversial political pamphlets early in his career. Through these pamphlets he was found guilty of sedition and libel and he was pilloried and spent some time in jail.
Of course, Defoe's most famous work is Robinson Crusoe (1719), one of the most widely read and translated novels in the world. It tells the story of a shipwrecked castaway who initially tries to save himself from the desert island he's stranded on but eventually makes peace with his lot in life and his separation from society. There is some suggestion that this work may be non-fiction and reflect the experiences of a family friend.
Whatever the case, Defoe was always a bit of an eccentric and an outsider, as most writers are. But Defoe was particular in this respect. Raised and educated by dissenting Protestants, he experienced all sorts of strange occurrences, not the least of which included surviving the plague and the Great Fire of London when he was young.

Why was the outcome of the French and Indian War significant to the colonists?

The defeat of the French in North America removed what had previously been a long-standing threat in American life. While the French possessions were very sparsely settled compared to British North America, they posed a continual threat in times of war, especially because of the indigenous allies the French had. The defeat of the French meant that, for one of the very first times in the American colonies, the possibility of attacks by the French was effectively eliminated. This had the effect of removing one of the ties to Britain, because the American colonies were suddenly less dependent on Britain for defense, even as British troops remained stationed in North America.
More seriously, the war left Great Britain massively in debt. Part of its war effort in North America had depended on generous subsidies to colonial legislatures, particularly the New England colonies, to raise militias and keep them in the field year after year. That had enabled the eventual defeat of the French, but by 1763, Great Britain's national debt had gone from 75 million pounds to 133 million pounds. Part of that had been a consequence of the larger war in Europe (remember, the French & Indian Wars were just one theater of the Seven Years War in Europe), but many British politicians felt that the American colonists needed to help pay back the sizable debts that had been incurred.
This combined with the need to pay for the troops who were still stationed in North America led to the passage of laws, such as the Stamp Act, that were a direct tax to raise revenue for Britain. This entirely circumvented colonial legislatures, who now complained that they were being taxed without representation. While the Stamp Act was repealed after substantial protest, it stirred colonial American grievances over British tax policy and fostered a sense of unity among the aggrieved colonists.

Calculus and Its Applications, Chapter 1, 1.8, Section 1.8, Problem 2

If the function is $y = x^4 - 7$, determine $\displaystyle \frac{d^2y}{dx^2}$

$
\begin{equation}
\begin{aligned}
\frac{dy}{dx} &= \frac{d}{dx} (x^4 - 7)\\
\\
&= \frac{d}{dx} (x^4) - \frac{d}{dx} (7) \\
\\
&= 4x^{4- 1}\\
\\
&= 4x^3
\end{aligned}
\end{equation}
$


Then,

$
\begin{equation}
\begin{aligned}
\frac{d^2y}{dx^2} &= \frac{d}{dx} (4x^3) \\
\\
&= 4 \cdot x^{3-1}\\
\\
&= 12x^2
\end{aligned}
\end{equation}
$

How can children play a role in environmental conservation?

When it comes to environmental conservation, teaching children to have respect for nature and make sustainable, environmentally friendly decisions sets them on a lifelong path of responsible action. While some aspects of environmental conservation may be difficult for children to understand fully, they can certainly learn that some actions can be harmful to the world around us. Encouraging children to recycle materials appropriately is an easy task that helps them make a big impact. Real-world examples of the importance of conservation, like nature walks and picking up litter (with the necessary safety precautions), helps children gain a sense of ownership over their actions and make connections between the idea of conservation and their lived experiences. 
There are many conservationist activities that can be done as a family to develop a child's sense of responsibility in caring for the environment. Planting a tree, tending a home garden, and taking public transport on family outings are all great activities.
As with all things, it's okay to start small! Many adults find it difficult to act in a way that is totally environmentally friendly, but by encouraging children to make just a few steps towards conservation, they may develop a life-long habit of caring for the environment.
http://footprintseducation.org/kidz-zone/how-can-i-help.php

http://www.dec.ny.gov/education/40248.html

https://learn.eartheasy.com/articles/environmental-websites-for-kids/

Mary would like to save $10,000 at the end of 5 years for a future down payment on a car. How much should she deposit at the end of each week in a savings account that pays 1.2%, compounded monthly, to meet her goal?

This type of a savings account is known as a sinking fund. The monthly payment M required to produce the desired amount of money (Future Value, or FV), when the period interest rate is R, is determined by the formula
M = FV*R/((1+R)^N - 1) , where N is the number of periods.
In this problem, the future value is FV = $10,000 minus the amount of money Mary would like to save. The interest is compounded monthly, and the annual interest rate is 1.2%, so the period (monthly) interest rate R is
(1.2%)/12 = 0.1% = 0.001 .
The number of periods (months) in 5 years is N = 12(5) = 60.
Plugging these values in the formula above, we get
M=$10,000*0.001/((1+0.001)^60 - 1) = $161.80 , rounded to the nearest cent.
The monthly payment Mary would have to make is $161.80. If she will deposit money once a week, she should deposit the amount of
($161.80)/4 = $40.45 .
Mary should deposit $40.45 each week in the given savings account in order to save $10,000 at the end of 5 years.

Single Variable Calculus, Chapter 8, 8.1, Section 8.1, Problem 34

Evaluate $\displaystyle \int t^3 e^{-t^2} dt$ by making a substitution first, then by using Integration by parts.
If we use $ z = -t^2$, $t^2 = -z$, then $dz = -2t dt$

so,

$
\begin{equation}
\begin{aligned}
\int t^3 e^{-t^2} dt = \int t^2 \cdot t e^{-t^2} dt &= \int - z \cdot e^z \cdot \left( \frac{dz}{-z} \right)\\
\\
&= \int \frac{1}{2} ze^z dz\\
\\
&= \frac{1}{2} \int z e^z dz
\end{aligned}
\end{equation}
$


By using integration by parts, if we let $u =z $ and $dv = e^z dz$, then
$du = dz$ and $v = e^z$

$
\begin{equation}
\begin{aligned}
\frac{1}{2} \int ze^z dz = uv - v\int du &= \frac{1}{2} \left[ ze^z - \int e^z dz \right]\\
\\
&= \frac{1}{2} \left[ ze^z - e^z \right]\\
\\
&= \frac{e^z}{z} [z -1]
\end{aligned}
\end{equation}
$

but, $z = -t^2$
So, $\displaystyle \frac{e^z}{z} [z-1] = \frac{e^{-t^2}}{2} \left[ -t^2 -1 \right] + c$

Based on the story "Sketches from the 'Cattle Shed'" by Ding Ling, how would you suggest I write a document (from Ding Ling's perspective) to submit to the government which criticizes the Cultural Revolution. Suppose you are a member of a group which is a "committee to criticize the Cultural Revolution" in China.

When writing this thought exercise, there are many implied and explicit critiques of the Cultural Revolution within the story you could use to create your arguments. Consider discussing such products of the Revolution as the inhumanity of internment, economic damage, social isolation and alienation, and censure.
Ding Ling's Internment and Censure
As a Chinese woman who defied the expectations and limitations of her culture, Ding Ling had a unique perspective on the effects of the Cultural Revolution. She experienced the pitfalls of communism on a personal level throughout her internment. Ding Ling was married to a man who was arrested for his leftist political ideals, and she was later censured for the political sentiments in her works. She was expelled from her own party in 1957 for advocating for women's rights and she was imprisoned for a five-year period during the Cultural Revolution. In "Sketches from the 'Cattle Shed,'" Ding Ling discusses her feelings of deep isolation and the horrors of her own internment. She was kept in solitary confinement, an experience that gave her a unique understanding of the Marxist concept of alienation. If she were to write an essay to the government criticizing the Cultural Revolution, the practical and emotional devastation of internment would likely be discussed in light of her own experiences.
Women's Rights
Each of these experiences can be found in the subject matter of "Sketches from the 'Cattle Shed,'" so it is highly likely that a letter from Ding Ling would cover each of them to some degree. She was also well-known for her passionate advocacy for women's rights, so you could also focus a portion of the document on the ways in which the Cultural Revolution leads to the oppression of women. The concept of filial piety was central to the Cultural Revolution, and one of the three foundational concepts within the philosophy of filial piety is that women are required to obey the men in their family. This view shaped the expectations the Cultural Revolution imposed on Chinese women like Ding Ling, so your essay could also discuss the various ways in which this ideology limited their careers and other personal choices and freedoms.
https://vc.bridgew.edu/cgi/viewcontent.cgi?referer=&httpsredir=1&article=1626&context=jiws

https://www.britannica.com/biography/Ding-Ling

https://www.nytimes.com/1988/06/19/books/in-short-fiction-961488.html

What is a quote from Moneyball that describes Billy Beane as a smart or great general manager?

One quote that describes Oakland Athletics' Billy Beane as a smart general manager relates to his resourcefulness.
Moneyball highlights baseball's financial landscape. Teams that play in large media and consumer markets like the Boston Red Sox or the New York Yankees are part of baseball's upper class because of their financial position. At the same time, small-market teams like the Oakland Athletics are at a competitive disadvantage because they lack financial resources. In order for Oakland's general manager Billy Beane to put forth a quality team, he had to be resourceful. He had to harvest data and look for value where he could find it.
This was not the only part of his daunting task to make Oakland a competitive baseball team. Beane also had to communicate his vision to athletes and executives who were operating under outdated paradigms. This required a specific skill set that makes him a great general manager:  

It was hard to know which of Billy's qualities was most important to his team's success: his energy, his resourcefulness, his intelligence, or his ability to scare the living s**t out of even very large professional baseball players.

In Moneyball, Billy Beane's greatness as a general manager is not only his ability to judge talent through sabermetric analysis. He is resourceful in the way he conveys this methodology and approach to athletes and baseball scouts who are mired in a philosophy that was not going to work under Oakland's financial reality. Sometimes, he has to lead by example. Other times, he has to use direct force and intimidation to make his point.
Beane's resourcefulness comes from being a former athlete and now seeing reality as an executive. Straddling both worlds enhances his resourcefulness. This unique skill set is a significant reason why he is seen as a great general manager.

Calculus of a Single Variable, Chapter 8, 8.8, Section 8.8, Problem 12

Any integral with infinite bounds is an improper integral therefore this is an improper integral.
int_-infty^0 e^(3x) dx=
Substitute u=3x => du=3dx, u_l=3cdot(-infty)=-infty, u_u=3cdot0=0.
1/3int_-infty^0 e^udu=1/3 e^u|_-infty^0=1/3(e^0-lim_(u to -infty)e^u)=
1/3(1-0)=1/3
As we can see, the integral converges and its value is equal to 1/3.
The image below shows the graph of the function and area under it corresponding to the integral. We can see that as x goes to minus infinity the function converges to zero and it does so "very fast" (exponentially to be more specific). Therefore, it should be no surprise that the above integral is a convergent one.

How is "The Tiger in the Tunnel" by Ruskin Bond a story of the extraordinary courage of an ordinary watchman?

The Tiger in the Tunnel is a story of extraordinary courage because it depicts the strength of human resolve in the face of extreme odds.
In the story, Baldeo is the night watchman at the signal station; his job is to light the signal lamp and to make sure that there are no obstructions in the tunnel to endanger the safe passage of the night train. Baldeo courageously performs his tasks without fear, despite the presence of dangerous predators in the surrounding countryside.
Fearlessly, he ignores the rumors about a famous tunnel tiger, confident that he can rely on his trusty axe to defend himself. When he is eventually faced with the forbidding animal, Baldeo does not shrink from the task at hand. The author makes clear that the tiger is swifter on foot and far stronger; he depicts Baldeo as "puny" compared to the mighty animal. Yet, the author also describes Baldeo's determination to fight the tiger to the death, despite the incredible odds against him. In the end, despite Baldeo's brave stand, the tiger kills him. It is this indomitable courage in the face of terrible odds that makes The Tiger in the Tunnel a story of the extraordinary courage of an ordinary watchman.

In what ways does Conrad reflect the British attitude of Orientalism toward Africa in "Heart of Darkness," and in what ways does he challenge such attitudes? Explore through Edward Said's framework of Orientalism.

Orientalism refers to the tendency of the West to romanticize Islamic or Middle Eastern cultures. According to Edward W. Said in his book "Orientalism," Western powers tend to fetishize Middle Eastern cultures as backward, exotic, and hedonistic. Orientalism characterizes the Occidentals (Westerners) as civilized peoples and the Orientals (Easterners) as uncivilized barbarians.
Edward Said maintains that the Western tendency to infantilize Middle Eastern peoples is just one way to justify Western imperialist and hegemonic ambitions. Additionally, the literary portrayal of the West as masculine/strong and the East as feminine/weak validates the Darwinian view of Easterners as culturally and biologically inferior. The fictionalized and simplistic characterization of Easterners as degenerate heathens in need of saving reinforces only the Western conviction at the expense of Eastern reality. Having said this, I would argue that the text and narrative of The Heart of Darkness betrays Conrad's typical British prejudice as well as his sympathy towards the native peoples.
Much like his peers, Conrad portrays the natives as savages and cannibals incapable of civilized conduct. In the novel, we see Conrad's stance reproduced in Marlow's voice. When Marlow comes across a chain gang at one of his company's upriver stations, he notes the "deathlike indifference of [the] unhappy savages" who march past him. To Marlow, the scene is surreal but expected. After all, the native prisoners are "criminals" who must reap the consequences of their aberrant behavior. A little later, however, he comes across some native laborers who are obviously in the throes of death. The text tells us that he is utterly horrified by the physical suffering the dying laborers endure. Marlow realizes that he does not view the dying natives as "criminals" or "enemies." Their palpable suffering and hopeless plight inspires something akin to sympathy in him.
Later, Marlow sails upriver to Kurtz's location with the manager of the station and some "pilgrims." The twenty crewmen on Marlow's repaired steamer are called "cannibals." Here, Conrad's Orientalism betrays itself quite clearly. The natives are portrayed as servile creatures under the direction of intellectually capable westerners. Marlow indulgently calls these cannibals "fine fellows." 

They were men one could work with, and I am grateful to them. And, after all, they did not eat each other before my face: they had brought along a provision of hippo-meat which went rotten, and made the mystery of the wilderness stink in my nostrils.

As the small group of travelers sail upriver, Marlow maintains that he can sometimes hear the faint sound of drums beating in the distant forest. The sound of the drums is equated with the primitivism of the native peoples. It is this primivitism that conjures up images of violent savagery in the Western mind. Marlow's apprehension is echoed in Kurtz's call to "exterminate the brutes."

...we whites, from the point of development we had arrived at, ‘must necessarily appear to them [savages] in the nature of supernatural beings—we approach them with the might of a deity,’ and so on, and so on. ‘By the simple exercise of our will we can exert a power for good practically unbounded,’ etc., etc

Here, Conrad demonstrates the typical Orientalism of his English peers: the natives are a threat to civilization; therefore, they must be dominated, educated, and initiated into good society. Despite his biased stance, Conrad also challenged the prevailing attitudes of his time. A very good example can be seen in the passage below:

It was unearthly, and the men were—No, they were not inhuman. Well, you know, that was the worst of it—this suspicion of their not being inhuman. It would come slowly to one. They howled, and leaped, and spun, and made horrid faces; but what thrilled you was just the thought of their humanity—like yours—the thought of your remote kinship with this wild and passionate uproar. Ugly. Yes, it was ugly enough; but if you were man enough you would admit to your self that there was in you just the faintest trace of a response to the terrible frankness of that noise, a dim suspicion of there being a meaning in it which you—you so remote from the night of first ages—could comprehend."

Here, Marlow/Conrad proclaims that all humans have the capacity for primitive or raw rage. He argues that it takes courage to even admit this. Essentially, Conrad is challenging the notion of the Other (as Easterners are characterized) as intrinsically inferior or foreign to the Western nature. 
 

Name three flash backs in the story Killings by Andre Dubus. For each flashback scene you analyze, think about how the event affects other literary elements, such as characterization and theme. Think about how the flashback scenes affect the pacing of the story. Perhaps they have a cumulative effect of creating suspense. For each flashback you plan to discuss, consider the writer’s choices in including the scene. You could reflect on how the story would be different if the writer had not used the flashback technique. What would be the effect if the events were told in a traditional, linear, chronological way? Additionally, you may wish to evaluate the effectiveness of the flashback scenes as a narrative technique.

Andre Dubus begins his short story Killings at the cemetery where his main protagonist, Matt Fowler, Matt’s wife, Ruth, and his oldest son Steve, have just buried Frank, the youngest of the Fowler’s three children—the baby of the family. As the story proceeds, Matt discusses an as-yet-to-be-introduced character who one can quickly surmise is responsible for Frank’s death. Matt visits with an obviously close friend, Willis, and the two engage in a conversation that clearly revolves around this unidentified figure, whose presence among the town’s citizenry obviously greatly upsets Matt and Ruth. The interaction between Matt and Willis begins with the former commenting, “He walks around the Goddamn streets.” In this small New England town, sightings of the man Matt and Ruth hold responsible for their youngest child’s death are frequent and inevitable. Additionally, Matt and Willis lament the deterioration of the town, with crime apparently rising: “. . .we got junkies here now too.” In short, Dubus has quickly depicted the decline of a town and of one particular family suffering the loss of their son and the continued freedom of the man they hold responsible. As the opening passages come to an end, it is revealed that Matt is contemplating taking the law into his own hands to avenge Frank’s death. It is now that Dubus employs his first flashback.
The second part of Killings describes the life of Richard Strout, a once-promising athlete who was failing in school and whose prospects were, once the possibility of professional sports was eliminated, limited at best. This biographical detail leads directly to the flashback. Strout is depicted as a thug, a ne'er-do-well with tendencies towards violence: 

“One night he beat Frank. Frank was living at home and waiting for September, for graduate school in economics, and working as a lifeguard at Salisbury Beach, where he met Mary Ann Strout, in her first month of separation.”

As Frank explains his wounds later that evening to his parent, it is clear that Dubus is using this flashback to provide the necessary background for the reader to understand the opening scene at the cemetery. The flashback sets the tone for the history of tension between Frank and Richard Strout. Absent this background information, the anguish tearing at the Fowlers and Matt and Ruth’s desire for revenge would be missing. The tone of the story would be confused, with the reader left to surmise the context in which everything that preceded this flashback took place. We now know that Frank is involved with the young, divorced mother whose ex-husband is the violent, oft-drunk Richard Strout, and that Richard has physically beaten Frank in at least one instance prior to the latter’s eventual death. And Mary Ann is four years older than Frank, contributing to Matt's and particularly Ruth’s disdain for the relationship. Finally, both Richard and Mary Ann’s history of extramarital liaisons further complicates the picture.
 
Dubus continues to use flashbacks to fill in the gaps, to illuminate the extent to which Matt and Ruth contended with what they believed could lead to negative consequences. Matt is the more forgiving, a man understanding his grown son’s biological and emotional demands. Ruth, the protective mother, is far more wary of Frank and Mary Ann’s relationship. Another flashback describes Matt and Frank going to a baseball game at Boston’s Fenway Park, the long drive an opportunity for Matt to talk to his son about the difficulties of such a young person becoming engaged in such a serious relationship, his plans for the future potentially impeded by such an entanglement. All of this builds the foundation for the act of revenge to follow. Dubus is meticulously establishing the inherent goodness of the Fowlers and the innate evil lurking in the heart of the story’s antagonist, Richard Strout. These flashback sequences are immediately followed by the scene of Richard shooting Frank, another flashback sequence, and one that illuminates the depravity of Richard’s nature. The passage begins, “Richard Strout shot Frank in front of the boys. They were sitting on the living room floor watching television. . .Strout came in the front door and shot Frank twice in the chest and once in the face with a 9 mm automatic.”
 
By providing these flashbacks, Dubus sets a very ominous tone and compels the reader to be silent and acquiescent witnesses to the act of vengeance that follows. Certainly, the reader could sympathize with Matt and Ruth’s situation without the flashback scenes, but the story would be empty without them. We are presented with human beings whose lives are tragically torn apart, and without the details provided in the flashback sequences, the reader would be bereft of the emotional baggage that places him or her squarely in the Fowlers' camp. Questions of revenge and of vigilantism provide the grist for many a contentious debate. By depicting Frank in almost angelic terms while portraying his killer as violent and seemingly beyond redemption, Dubus allows for the complicity of the reader in the vigilantism that provides the story’s climactic sequence.

In the account of the treatment of peasants on the farm captured by foreign soldiers during the Thirty Years' War, as recounted in the novel, what is revealed about the perspectives of the peasants?

From what I can see, you would like an account of the capture and ransacking of Simplicissimus's family farm in chapter 4 of book 1.
In this chapter, Simplicissimus relates that the first thing the soldiers did (upon capturing the farm) was to stable their horses. Then, some began butchering (presumably) farm animals, boiling water, and roasting meat, as if they were preparing for a banquet. Meanwhile, others immediately fell to works of "ruin and destruction."
The soldiers ransacked the farmhouse from top to bottom. Some tied together bundles of cloth, apparel, and household goods to carry away, as if they were going to have a large sale. Others shook the stuffing out of the beds and replaced them with slabs of ham, bacon, and other dried meat, as if this made the beds more comfortable to sleep on.
The soldiers destroyed what they could not carry away. They burned beds, tables, chairs, and benches. Kitchenware, stoves, and windows were demolished. Others ran their swords through the hay and straw, cutting it up. Meanwhile, other soldiers flattened copper goods and tinware so that they could easily carry these away. 
Simplicissimus relates that the women of the house (including the maids) were raped and the men tortured mercilessly. He tells of one torture called the "Swedish draught." Accordingly, the soldiers laid one of the peasants on the ground, thrust a gag into his mouth, and poured foul-smelling liquid down his throat. During the Thirty Years' War, the "draught" was said to contain excrement, urine, or even manure. The soldiers tortured the peasant until he agreed to reveal where the rest of the household were.
Simplicissimus reveals that this is how his father, mother, and sister Ursula were captured. Meanwhile, other soldiers made thumbscrews out of their pistols in order to torture the peasants. The thumbscrew was a medieval torture instrument, used to crush victims' toes, fingers, or thumbs.
Each of the soldiers devised their own methods of torture to torment their peasant victims. One peasant was burned in the baking oven, while another had his head crushed with a cord and wooden implement. Simplicissimus tells us that his father was luckier than most, perhaps because of his position as the head of the household. The soldiers tied up Simplicissimus' father, set him before a fire, and smeared his feet with damp salt. Then, they let the farm goat lick off the salt, which led to gales of laughter from Simplicissimus's father.
In this way, Simplicissimus's father was led to tell the soldiers about the "secret treasure, which proved far richer in pearls, gold, and trinkets than any would have looked for among peasants." 
As for the women of the household, Simplicissimus relates that all were raped, even his mother and sister. He remembers their pitiful screams while the soldiers had their way with them. Simplicissimus later chanced upon the household maid in the stables, and she warned him that none could save him from the plight the adults had already suffered. She advised him to run away. 
The narrator reveals that the peasants were resigned to their fate. From the peasants' perspective, their lives were expendable. They understood that they were of little consequence to the soldiers who mistreated them. These peasants had long accepted that they had little control over their destinies. This is why the maidservant told Simplicissimus to run away: with the farm destroyed and his family soon to die, he would have to fend for himself. Thus, running away would be his only option to retain a measure of control over his life.

How does Vonnegut's story, "Harrison Bergeron," compare to Plato's Allegory of the Cave?

The short answer is that the characters in Vonnegut's story find themselves in a situation very similar to the one described in Plato's scenario. They are prisoners in a carefully restricted, artificial reality. Ordinarily, their freedom is restricted so much that they can't even perceive clues that another, greater reality exists. When they are confronted with clear evidence from a prisoner who has escaped and discovered the truth, the prisoners remain unconvinced. Pain and the misleading nature of their perceptions prevent the prisoners from attempting escape.
To appreciate these correspondences, let's first take a look at key elements in the Allegory of the Cave. Then we'll consider the specific ways in which "Harrison Bergeron" matches up.
In Plato's allegory, people are held prisoner and allowed to look only at the wall opposite them. They have lived there since they were children, chained and shackled, and cannot move around or look behind them. The only action they see is the movement of shadows across the wall. These are produced by people and objects moving behind them, illuminated by a fire. The prisoners have no way of knowing what causes the shadows they see, so they take them at face value. They don't suspect they are the signs of something else — the existence of a larger world.
What, asks Plato, might happen if one of the prisoners were allowed to turn around and look at the fire? Because his eyes were accustomed to darkness, the light would hurt his eyes and he would look away. He would likely choose to resume his former ways (staring at the wall with the shadows), and he would resist the idea that the fire represented true reality. It is visually easier to look at the shadows, and they are familiar, so he accepts them more readily as reality.
Similarly, if a prisoner left the cave, he'd be overwhelmed by the painful, bright light. If he remained outdoors, his eyes would eventually adjust, and he would be able to see clearly what the real world looks like. Over time, he would figure out how shadows work, and realize how the subjective experiences of the prisoners have misled them about reality — a wider world exists beyond the shadows.
If the escaped prisoner returned to the cave to liberate the others, though, he'd immediately be blinded in the darkness. Unable to see anything (because his eyes haven't yet readjusted), he would appear to the other prisoners to have been harmed by his experiences. Therefore, they wouldn't be receptive to his message about the advantages of quitting the cave. They would fear leaving, and even fight to stay.
That's the Allegory of the Cave. How is "Harrison Bergeron" similar to it?
1. The people in the story inhabit a world where the government carefully controls what they can see, and they are compelled to wear devices and shackles that prevent them from perceiving the larger reality.
In Vonnegut's story, people are prisoners of a police state. The United States Handicapper General monitors each individual to make sure that he or she has no competitive advantage over anyone else. If a person has more than his or her share of a valued trait, this advantage is neutralized by devices, weights, and shackles.
For instance, George, whose intelligence is "way above normal," has a "handicap radio in his ear" that blasts painful, disoriented noises every 20 seconds. This prevents him from maintaining a coherent train of thought, and appears to interfere with his memory processes. Two of the eight ballerinas wear the same device, and Harrison wore earphones that function like George's radio. Apparently, then, a sizable fraction of the population has their thoughts controlled in this way. In addition, people wear onerous weights — like the "forty-seven pounds of birdshot in a canvas bag" hung around George's neck — that interfere with their ability to work, move with grace, investigate the world, or otherwise resist the government's control. Kept under control by these means, people allow themselves to be convinced that removing handicaps would destroy society.
2. Harrison Bergeron is analogous to the prisoner who has escaped Plato's cave, learned the truth, and returned to free the other prisoners.
When Harrison escapes from jail, he doesn't attempt to leave his society (the allegory of the cave). Instead, he goes to the heart of this society's mass communication system — the television studio — and removes all his physical handicaps on camera. Much of his motivation might be selfish. He announces that he is "a greater ruler than any man who ever lived" and makes it clear he intends to use his advantages to be an autocrat:

"I am the Emperor!" cried Harrison. "Do you hear? I am the Emperor! Everybody must do what I say at once!" He stamped his foot and the studio shook.

Harrison wants to enjoy the beauty and talents that the government has hidden with shackles, masks, and weights. He strips the ballerina and musicians of their handicaps — freeing these prisoners from the conditions imposed by the cave — and he does so in front of all the television viewers. So his actions present people with the reality that had been hidden from them.
3. Just as Plato's prisoners reject the truth after encountering the returning prisoner, so do the people in "Harrison Bergeron."
Diana Moon Glampers kills Harrison and the ballerina, and orders the musicians to put their handicaps back on. The television burns out, and George is hit with an especially painful, forceful blast ("a doozy") that wipes out his memory. Hazel, who has been crying, can't remember what caused it, except that it was "something real sad on television," and both she and George agree that she should "forget sad things." Like the prisoners in the cave, they associate hints of escape with something painful, so they are content to remain in their situation, ignorant of the truth.
http://classics.mit.edu/Plato/republic.8.vii.html

Intermediate Algebra, Chapter 5, 5.1, Section 5.1, Problem 56

Evaluate the expression $\dfrac{3^{-3}}{2^{-2}}$.
Remove the negative exponent by rewriting $3^{-3}$ as $\dfrac{1}{3^3}$. A negative exponent follows the rule of $a^{-n} = \dfrac{1}{a^n}$
$\dfrac{1}{3^3}\cdot\dfrac{1}{2^{-2}}$
Cubing a number is the same as multiplying the number by itself $3$ times. In this case, $3$ cube is $27$.
$\dfrac{1}{27}\cdot\dfrac{1}{2^{-2}}$
Remove the negative exponent by rewriting $2^{-2}$ as $\dfrac{1}{2^2}$. A negative exponent follows the rule of $a^{-n} = \dfrac{1}{a^n}$
$\dfrac{1}{27}\cdot\dfrac{1}{\dfrac{1}{2^{2}}}$
Remove the parentheses from the denominator.
$\dfrac{1}{27}\cdot\dfrac{1}{\dfrac{1}{4}}$
Simplify the multi-level fraction by multiplying by the reciprocal of the denominator.
$\dfrac{1}{27}\cdot 4$
Multiply $\dfrac{1}{27}$ by $4$ to get $\dfrac{4}{27}$ .
$\dfrac{4}{27}$

Explain the relationship between the limits to presidential power and the importance of the president's informal powers.

Presidential Powers
Before we discuss limits on president power and how that connects to informal powers, it is helpful to understand what the president can do, according to the Constitution:
The President can execute (carry out) the laws made by Congress
The President is commander in chief of the Army and Navy of the United States
The President has the power to grant pardons for offenses, except in cases of impeachment
The President can make treaties with 2/3rds consent of the Senate
The President can nominate ambassadors, Supreme Court justices, cabinet officers and other executive officers with consent of the Senate
Limits to Presidential Power
The President's powers are limited by the system of checks and balances present in our government.  This system is also written into the Constitution.  The Legislative branch can overrule a Presidential veto with a 2/3rds vote, stop budgeting, and impeach the President.  The Judicial branch cannot be tampered with; once appointed, judges sit in their position for life.  The Courts can also use judicial review, which examines actions by the Legislature or Executive branch for constitutionality.  If the Courts find an action unconstitutional, they can reverse it.
Connections to Informal Powers
The President's informal powers are:
Setting the agenda for Congress
Issuing executive orders
Sending out troops without a declaration of war
Conducting initiatives and actions concerning foreign policy
How are these informal powers connected to the limits and restrictions on formal Presidential power?  
Since the Legislative branch has the power to override and veto Presidential action, the President can use his informal power of setting the agenda wisely- if the agenda does not contain any hot button issues, or if the President waits until he has a Congress that shares his partisan views, he can assume that his actions will be accepted or passed.  It requires a 2/3rds vote in both the House and Senate to override Presidential action.  
The President has the power to issue executive orders, a rule or law issued DIRECTLY from the President without going through Congress.  This bypasses the lengthy process of turning a bill into law.  However, it is important to note that is the executive order is unconstitutional, then the Supreme Court can overrule it.  
The President has the ability to send troops abroad without waiting for the Senate to formally declare war.  This has been used several times in the recent past with the understanding that immediate action needed to take place before the Senate could vote on declaring war, like when George W. Bush sent troops to Afghanistan in the wake of the 9/11 terror attacks.
The President's information powers are a way to bypass some of the checks on Executive power, but they do not give the President free reign as head of the country.
 
http://law2.umkc.edu/faculty/projects/ftrials/conlaw/prespowers.html

Single Variable Calculus, Chapter 5, 5.4, Section 5.4, Problem 28

Find the integrals $\displaystyle \int^2_1 \frac{y+5y^2}{y^3} dy$

$
\begin{equation}
\begin{aligned}
\int \frac{y+5y^2}{y^3} dy &= \int \left( \frac{y}{y^3} + \frac{5y^7}{y^3} \right) dy\\
\\
\int \frac{y+5y^2}{y^3} dy &= \int \left( \frac{1}{y^2} + 5y^4 \right) dy\\
\\
\int \frac{y+5y^2}{y^3} dy &= \int y^{-2} dy + 5 \int y^4 dy\\
\\
\int \frac{y+5y^2}{y^3} dy &= \left( \frac{y^{2+1}}{2+1} \right) + 5 \left( \frac{y^{4+1}}{4+1} \right) + C\\
\\
\int \frac{y+5y^2}{y^3} dy &= \frac{y^{-1}}{-1} + \cancel{5} \left( \frac{y^5}{\cancel{5}} \right) + C\\
\\
\int \frac{y+5y^2}{y^3} dy &= -y^{-1} + y^5 + C\\
\\
\int \frac{y+5y^2}{y^3} dy &= \frac{-1}{y} + y^5 + C\\
\\
\int^2_1 \frac{y+5y^2}{y^3} dy &= \frac{-1}{2} + (2)^5 + C - \left[ \frac{-1}{1} + (1)^5 + C \right]\\
\\
\int^2_1 \frac{y+5y^2}{y^3} dy &= \frac{-1}{2} + 32 + C + 1 - 1 - C\\
\\
\int^2_1 \frac{y+5y^2}{y^3} dy &= \frac{-1+64}{2}\\
\\
\int^2_1 \frac{y+5y^2}{y^3} dy &= \frac{63}{2}
\end{aligned}
\end{equation}
$

The Supreme Court justices have often been active in the selection of new justices. Do you think this is appropriate?

Assuming you mean the Supreme Court's calls in the past year to fill the vacancy left by the death of Justice Scalia, then yes, it is appropriate because the role of the Supreme Court is to uphold the US Constitution and the laws of the United States against all who violate them, including the legislature.
Article III of the Constitution created the Supreme Court of the United States. Since then, laws passed by Congress have set the number of justices that are to sit on the Court.
However, when Justice Scalia died and President Obama (of the Democratic Party) nominated his replacement, Merrick Garland, the Republican-controlled Congress refused to partake in the nomination process. Instead, they let the clock tick out until President Obama left office—a period of 293 days—so a Republican could become president and nominate a different Supreme Court justice.
During this time, several sitting Supreme Court justices voiced displeasure with the political maneuver to pack the Supreme Court with votes that were meant to be friendly to one political party. These sitting justices were active in criticizing Congress and urged them to uphold the law and select a new member of the Supreme Court.

Beginning Algebra With Applications, Chapter 8, 8.1, Section 8.1, Problem 130

Write an expression in factored form for the shaded portion in the diagram.

a.







The area of the rectangle is equal to $A_{\text{rectangle}} = LW$, where $L = 4r$ and $W = 2r$. So,

$A_{\text{rectangle}} = 4r(2r) = 8r^2$

And the area of the two circles is


$
\begin{equation}
\begin{aligned}

A_{\text{circle}} =& 2 (\pi r^2)
\\
=& 2 \pi r^2

\end{aligned}
\end{equation}
$


Then, by subtracting the area of the rectangle to the area of circle, we get the area of the shaded portion as


$
\begin{equation}
\begin{aligned}

A_{\text{rectangle}} - A_{\text{circle}} =& 8r^2 - 2 \pi r^2
\\
=& 2r^2 (4 - \pi)

\end{aligned}
\end{equation}
$


b.







Based from the figure, the area of the square is $A_{\text{rectangle}} = (2r)^2 = 4r^2$ and the area of the circle is $A_{\text{circle}} = \pi r^2$
Then, by subtracting the area of the square to the area of the circle, we obtain the area of the shaded portion as



$
\begin{equation}
\begin{aligned}

A_{\text{rectangle}} - A_{\text{circle}} =& 4r^2 - \pi r^2
\\
=& r^2(4 - \pi)

\end{aligned}
\end{equation}
$

In Jerome's Three Men in a Boat, what important decisions do the three friends make as a part of their preparations for their pleasure trip and why?

The three men prepare for the river trip over the course of the first four chapters of the book. They have to debate every decision, and this is why it takes them a long time. In the first chapter, they decide to go on a boat trip along the River Thames, instead of on an ocean or sea voyage, when they could get seasick. In the second chapter, they debate the advantages and disadvantages of camping out every night. They decide to camp out in good weather and to stay in pubs or inns whenever it rains, since camping out in wet weather gets too messy. In the third chapter, they make a list of supplies for the trip. They decide to use a boat with a canvas cover, instead of taking an extra tent to set up, which would require extra work and extra baggage. They decide what clothing they will take, and what would be appropriate attire for boating. In the fourth chapter, they decide what food and what kind of stove they will take. In all of these instances, they have to pare down their original lists to only what they will need and what is likely to work best on a boat. Then they begin to pack. J., the narrator, packs the equipment and the clothing. George and Harris pack the food hamper. Montmorency gets in the way while they do. By the beginning of the fifth chapter, they’re finally ready to get going.

How does Harper Lee present prejudice throughout the novel?

Prejudice is a common theme throughout To Kill a Mockingbird. The story is set in the Deep South during the 1930s, a time and a place where racial prejudice was deeply ingrained in society. Harper Lee portrays prejudice in all its ugliness through the words that people speak and the actions they carry out. Examples of the former would include copious uses of the n-word. The near-universal use of this notorious epithet is an indication of just how deeply the roots of prejudice reach in this society.
In Maycomb, prejudiced actions are also disturbingly commonplace. The trial of Tom Robinson—the most blatant example of prejudice in the whole book—is a prime illustration of this. There's no evidence whatsoever to suggest that he's guilty of the rape and assault of Mayella Ewell, but in a society as prejudiced as this, that's considered to be of minor importance. Tom Robinson is "guilty" of nothing more than being a black man accused of raping a white woman. The very fact that he's been accused is sufficient to condemn him in the eyes of most people in Maycomb. If that isn't prejudice, then nothing is.
Harper Lee is also keen to highlight the effect that prejudice has on people—not just its victims, but also those with a prejudiced outlook. In particular, she shows us how prejudice prevents such people from realizing how ridiculous they are. We see this in the scene involving Aunt Alexandra's missionary circle, where the assembled ladies express concern over the spiritual and material welfare of a remote African tribe while at the same time using racial slurs against Maycomb's African American community.
A similar lack of self-awareness is evident when Miss Gates condemns the Nazis for their persecution of the Jews. This is the very same woman who expresses her approval at the wrongful conviction of Tom Robinson, which she believes will serve to keep African Americans "in their place."

Harper Lee presents prejudice throughout the novel by depicting the thoughts and actions of the racist community members of Maycomb. Lee also depicts prejudice by illustrating how negative rumors adversely affect Boo Radley's life. Lee uses specific characters such as Mrs. Dubose, Mrs. Merriweather, and Bob Ewell to depict the prejudice against African Americans in Maycomb. Lee also uses characters like Miss Stephanie Crawford to portray the prejudiced feelings towards Boo Radley. The community member's racist comments and negative attitude towards Atticus also depict their prejudiced beliefs. Lee also portrays how the community's prejudice harms specific individuals. Tom Robinson becomes a victim of racial injustice and the rumors surrounding Boo Radley give him a negative reputation throughout Maycomb. Despite the overwhelming prejudice throughout the community, Atticus remains tolerant and morally upright. He teaches his children to treat others equally which aids in Scout and Jem's moral development.

At a hammer-throwing event, the speed that an athlete spins is such that they complete one revolution in 0.50s. The hammer weighs 7.0 kg. The athlete stretches their arms out such that they are holding the end of a chain attached to the hammer (ignore the weight of the chain). The hammer is at a distance of 0.60m away from the axis of rotation of the athlete. Assume that the hammer has no vertical motion and travels in a circle parallel to the ground. What is the velocity of the hammer? What is the acceleration of the hammer? What is the centripetal force?

As the hammer spins, it's acceleration is produced by its changing direction, even if the velocity remains constant. This is centripetal acceleration and force. To calculate these values, we need to know a little bit about circular motion.
To calculate the speed, you need to know the distance traveled in a period of time. We know the time required is 0.5 seconds, and the distance can be calculated by using the circumference of a circle. The circumference is 2*pi*r or 2*(3.14)*(.6). Therefore the circumference is 3.77 meters, and the velocity is 7.54 meters/second.
To find the acceleration, use the formula for centripetal acceleration, which is v^2/r. (7.54^2)/(0.6) = 95.75 meters per second squared.
Finally, to find the force, you multiply mass times acceleration - 95.75*7 kg, equaling 670.25 Newtons.

In the situation described in this problem, the hammer can be modeled as an object undergoing uniform circular motion. In this kind of motion, the speed of the object is constant, but since the direction of motion is constantly changing (as the object moves around in circle), there is acceleration directed towards the center of the circle, known as centripetal acceleration.
The speed of the hammer can be calculated as v = C/T, where C is the circumference of the circle and T is the time it takes to complete one revolution, 0.50 s. The circumference formula is C = 2*pi*r, where r is the radius of the circle.
Then, v = (2*pi*r)/T = (2*pi*0.60)/(0.50) = 7.5 m/s (rounded to the two significant digits, as are the rest of the given values).
The acceleration of the hammer can be found from the formula for centripetal acceleration:
a_c = v^2/r = (7.5)^2/0.60 = 95 m/s^2.
The centripetal force—that is, the force responsible for the centripetal acceleration—is produced by the tension in the chain attached to the hammer. It equals the product of the hammer's mass and the centripetal acceleration:
F = ma_c = 7*95 N = 670 N (again, rounded to the two significant digits).
Note that this value is much (ten times) greater than the hammer's weight, which is mg = 7*9.8 N = 67 N.

Centripetal force is the force required to keep an object moving in a circle. Objects in motion tend to stay in motion, so, to prevent spinning objects from flying off in the straight line that their momentum would normally carry them, another force, perpendicular to their straight momentum, must accelerate them towards the center of their rotation. When talking about planetary orbits, this force is gravity; when talking about race cars, this force is the friction of tires on asphalt; when talking about the hammer toss, this force is the tension of the string.
To calculate the hammer's centripetal force and acceleration, we need the circular velocity of the hammer. Since we have the rotation time, we can calculate the hammer's speed based on the the circumference of its circuit. This is equal to 2πr = (2)(3.14)(0.6) = ~3.77 meters. The hammer travels that time in half a second, so its velocity is ~7.54 m/s
The Centripetal Acceleration Formula is simple: a = v^2 / r. Plugging in our the velocity from part a and the radius from the prompt, we get (7.54)^2 / 0.6 = ~95.75. The tension on the hammer string is equivalent to an acceleration of 95.75 meters per second squared.
Finally, the force affecting the hammer is simply that acceleration times the mass or 7 * 95.75 = 670.25 Newtons of force.
http://hyperphysics.phy-astr.gsu.edu/hbase/cf.html

http://www.softschools.com/formulas/physics/centripetal_acceleration_formula/71/

How do the Christ figures (think about which characters could represent Christ) in Uncle Tom's Cabin underscore the novel’s basic Christian messages?

There are two characters who resemble Jesus in Uncle Tom's Cabin, Tom and Eva. Like Jesus in Christian scriptures, Tom and Eva make the ultimate sacrifice, giving their lives to save others. While Jesus is said to have died for humanity's sins, Tom's death helps Emmeline and Cassy escape and leads to all of the slaves on Shelby Farm in Kentucky being freed. Both Tom and Jesus are beaten and endure tremendous pain and suffering in the process.
Eva is a rich woman who gives money to the poor, reflecting Jesus's teaching about loving thy neighbor as thyself. She literally fulfills the third verse of the Sermon on the Mount (and the beatitude which states "Blessed are the meek, for they shall inherit the earth") by providing for their material needs.
Like Jesus, Tom has doubt about his faith. Tom and Jesus express their doubt when they are being either beaten or crucified, respectively, and in their dying moments.

In addition to Tom and Eva, the Quakers are also Christlike figures in Uncle Tom's Cabin. They uphold the values of pacifism and compassion, and they are vehemently opposed to slavery as an institution because it gives human beings absolute power over others and allows them free rein to abuse those they "own." As a result, the Quakers view slavery as incompatible with the Christian messages of brotherhood and selfless love.
The Quaker couple Rachel and Simeon Halliday agree to help out Eliza when she runs away from her owners, even though it is not legal in the eyes of the state to do so. Their civil disobedience puts their necks on the block as well, and this willingness to risk punishment for the sake of other people makes them Christlike.

Uncle Tom and Eva are the two main Christ-like figures in the novel. Through them, Stowe shows that true Christians sacrifice themselves for the good of others. Slavery, in contrast, sacrifices the good of others (the slaves) for the convenience of the slave owners. Slavery, therefore, is an evil, un-Christian institution.
Uncle Tom underscores a Christian ethic by continually sacrificing himself for other people. He doesn't try to run away when he is about to be sold at the beginning of the novel, even though he is deeply grieved by the news. Being sold will separate him from his wife and children. It will leave him open to being purchased by a cruel master. Nevertheless, he knows Mr. Shelby is selling him because of the high price he will bring. This will save Mr. Shelby from having to sell many of his ––or even break up the plantation––to settle his debts. Tom sacrifices himself to protect the other slaves. He knows that Shelbys are kind masters and that the slaves there are in a good situation. Likewise, at the end of the book, Tom allows himself to be beaten to death rather than violate his conscience and turn on his fellow slaves. He dies to protect them, the ultimate Christ-like sacrifice.
Like Uncle Tom, the angelic little Eva has a moral compass different from that of the society around her. She is not old enough to free the slaves or do anything as bold and courageous as Tom does in defying Simon Legree, but she puts the needs of others ahead of herself. She treats the slaves with great kindness, dignity, and love. The slaves all love her in return. When she is dying, she extorts everyone in the room, slave and free, to be good Christians and love and serve one other.
If more people behaved like the true Christians Tom and Eva, Stowe believed, putting the needs of others first, there would be no slavery.