A sample of ascorbic acid (Vitamin C) is synthesized in the laboratory. It contains 1.50 g of carbon and 2.00 g of oxygen. Another sample of ascorbic acid isolated from citrus fruits contains 6.55 g of carbon. How many grams of oxygen are there?

While this question may look complicated, it is surprisingly simple to complete. I will break it down word by word, and show the full process to solve.
I like to start with the last sentence of a problem to find out what I am looking for. In this problem the last sentence is "How many grams of oxygen are there?" This is straightforward, and tells us that we are looking for oxygen in the final product of something.
The second and third sentences are "It contains 1.50 g of carbon and 2.00 g of oxygen. Another sample of ascorbic acid isolated from citrus fruits contains 6.55 g of carbon." This tells us that a sample of ascorbic acid contains 1.50 g of carbon and 2.00 g of oxygen, and the carbon content of a secondary sample. Clearly, because we are looking for the oxygen content, this is the important info to know.
The first sentence, "A sample of ascorbic acid (Vitamin C) is synthesized in the laboratory," tells us nothing at all. It sets up a 'setting' for the math problem, and puts some story into it, but when you read this, you  will learn nothing that helps you solve the problem. 
In the end we are left with the quantity of components in ascorbic acid and a single component from a second sample, and are asked to find the missing quantity. As a math problem, you would write
[1.5g_(carbon)]/[2.0g_(o)]=[6.55g_(carbon)]/x
where x is the missing quantity. Solving, you get 8.7333(...) g oxygen.

How does Linda change through out Death of a Salesmen?

It's not entirely clear that Linda Loman changes all that much. All throughout the play, she acts as Willy's enabler: protecting him, his fantasies, and his delusions from the harsh intrusion of reality. At no point does she ever confront Willy about his suicidal tendencies; she seems to think that the best way for Willy's mental health to improve is through his sons striving hard for success.
Her whole world revolves around Willy, and she puts her husband's needs ahead of her own and her sons' every single time. The quintessential 50s housewife, Linda is loyal, patient, and submissive to her husband. Only she acts more like a mother than a wife, and therein lies the problem. Willy doesn't need a mother, doesn't need to be coddled; he needs someone to shake some sense into him, to get him to see that his current path can only lead to destruction for himself and sadness for those left behind. Linda is not that someone. Her inability to change rubs off on her husband, with fatal consequences.

In the novel "The Dispossessed" Anarresti culture sought to eliminate hierarchies within its society. To this end, Odo provided no formal/legal apparatus for marriages; however, Le Guin, the female author of this novel, centers her story on two individuals who are in a monogamous relationship (i.e. Shevek and Takver). Analyze this problematic tension within the story. Considering the portion of the novel we have read, how might this tension be resolved into a coherent theme? 1 or 2 paragraphs

The relationship between Takver and Shevek, while not expressly forbidden by the Annarean government, is highly controversial. While Le Guin explains that Odo originally set up a society with no legal form of marriage in order to prevent Annares from becoming a hierarchy like Urras, this policy becomes limiting to the young couple. Shevek and Takver feel a deep bond and desire to share their lives with each other. In this sense, the monogamous relationship between Takver and Shevek fits in with the larger theme of the story. A lack of formal contracts and institutions in which two people effectively belong to one another is only freeing for the Annareans who have no desire to enter into such contracts. By contrast, Takver and Shevek find Annarean society highly restrictive.
More tension is created when Shevek is sent away on a lengthy tour of labor. Takver is also sent away to continue her work on addressing the problem of starvation on Annares. Because marriage is not an officially recognized institution, the Annarean government has no obligation to keep the spouses together. The tension can be further resolved into a coherent theme when contrasted with the opposing tensions Shevek experiences on Urras. While the Urrasti government permits marriage, its social institutions create great pressure to conform in a different way. "The Dispossessed" uses these tensions to illustrate how even restrictions created from the purest motives will have a negative impact on someone.

Find a news article on the effects of imperialism today somewhere in the world.

Imperialism is when one nation exerts control over another through primarily economic or military means, usually for financial gain. Many European nations engaged in imperialist policies in the 18th and 19th centuries, continuing into the 20th century, and the United States began an imperialist period in the late 19th and early 20th centuries. Many scholars point to Theodore Roosevelt as the first imperialist U.S. president. Examples of imperialism include British control of India from 1858 to 1947; Belgian colonization of what is now the Congo, Burundi and Rwanda after King Leopold II came to power in the mid 19th century; and American occupation of the Philippines after the Spanish-American War of 1898. The impact of imperialism can have long-lasting consequences for the target nation, even after the imperialist nation has retreated from direct control of a region. This can happen because the nation has had its valuable resources extracted or reduced by the imperialist forces; it has not had the opportunity to develop its internal systems and infrastructure, and its own citizens were not put in the position to govern their own affairs and thus lack governing and organizational experience; and the country has become culturally deficient because it was forced to acclimate and live under a foreign culture. Additionally, imperialist nations often drew arbitrary geographical boundaries that ignored traditional boundaries and grouped sometimes opposing forces into one nation. It is not difficult today to see the legacy of imperialism in news across the world, in the form of civil unrest, poverty, and violence in affected nations. One example would be the current fears that the Democratic Republic of the Congo (DRC) will slide back into civil unrest after years of civil war.

Explain how Abigail's disappearance affects the resolution of the play The Crucible?

Abigail's disappearance tells you everything you need to know about this devious, slippery character. Having unleashed all this chaos upon Salem, she's nowhere to be found as the action of the play heads towards its tragic conclusion. However, instead of concluding that Abigail was a pathological liar and that the witch-trials based on those lies have no credibility, Judge Danforth still goes ahead with the proceedings despite Reverend Hale's desperate pleas. What this shows us is that, although Abigail may have started this particular fire, others have a vested interest in keeping it going. There's simply too much riding on a successful outcome of the trials (i.e., sending innocent people to their deaths) to stop them now. It's as if the whole witch-craze has taken on an unstoppable momentum all of its own, and which must be allowed to run its natural course.

Abigail's disappearance makes plain her dishonesty and deceptiveness, something the courts have been unwilling to see up until now.  She lies to her uncle, telling him that she'll be spending the night at her friend, Mercy Lewis's, house -- Mercy tells the same lie at her house -- and then Abigail robs her uncle blind.  Before boarding a ship, she breaks into Reverend Parris's strongbox, and steals his life's savings.  It begins to look as though Abigail is not the righteous, holy, instrument of God the courts initially believed her to be; instead, she appears to be a conniving and deceitful young woman, determined to serve herself and get out of Salem before the tide turns against her.
Abigail's dishonesty casts doubt on the accusations she has made thus far.  Her disappearance is just one of many reasons, however, that Reverend Parris and Mr. Hale beg Deputy Governor Danforth to postpone the hangings.  Despite the many reasons to delay, Danforth refuses, insisting that it will only cast doubt on the guilt of others who have been hanged for the same crimes already.  He seems to understand, now, that Abigail is a vicious liar, but he will not risk his credibility or authority by walking back on the convictions.

College Algebra, Chapter 4, 4.1, Section 4.1, Problem 50

A quadratic function $f(x) = 1 + x - \sqrt{2} x^2$.

a.) Find the maximum or minimum value of the quadratic function $f$, using a graphing device.







Based from the graph, the function has a maximum value of $1.2$.

b.) Find the exact maximum or minimum value of $f$.

Using the quadratic equation $ax^2 + bx + c$ with $a = - \sqrt{2}$ and $b = 1$. Thus, the maximum or minimum value occurs at

$\displaystyle x = - \frac{b}{2a} = - \frac{1}{2(- \sqrt{2})} = \frac{1}{2 \sqrt{2}}$

Since $a < 0$, the function has a maximum value

$\displaystyle f \left( \frac{1}{2 \sqrt{2}} \right) = 1 + \frac{1}{2 \sqrt{2}} - \sqrt{2} \left( \frac{1}{2 \sqrt{2}} \right)^2 = 8 + \frac{\sqrt{2}}{8} = 1.18$

How did apartheid affect people's lives, and how did they respond?

Apartheid, a system of complete racial segregation, governed nearly every aspect of life for black and other South Africans. The laws dictated where they could live and travel. Black South Africans needed pass books to travel and work, but getting these pass books was difficult and, at times, made employment impossible. Blacks could also only attend certain segregated schools, at which they received an inferior education and could only hold certain menial jobs. They were excluded from public places in white areas, unless they were passing through on the way to work.
In response, black South Africans began to resist the system of apartheid through the African National Congress (ANC) and other means. The ANC, in which Nelson Mandela was active, was a political group that at first used peaceful protests to fight for rights of blacks and colored people (people of mixed race) in South Africa. After protestors were killed at Sharpeville in 1960, however, the ANC developed a paramilitary wing. There were also student uprisings, such as the 1976 Soweto Uprising in which students protested the imposition of Afrikaans (the language of the Dutch who had settled in South Africa) as a language in schools. Women also mounted protests against pass laws, as these laws made it difficult for them and their families to survive economically. Some of these protests involved coalitions of black people, white people, and Indian people, for example. Eventually, apartheid, which was officially made legal in 1948, was brought down in the early 1990s.

Apartheid is the systematic segregation of a particular group of people by a country’s government. The term traces its origins to South Africa. In 1948, the ruling National Party (NP) instituted a system of racial segregation. They established a system of white minority rule over the black majority natives and other racial groups. The ruling party undermined the social and civil rights of the races they considered inferior.
Members of the black community had their citizenship revoked. They were evicted from their homes and forced into segregated residential areas. The segregation affected access to social amenities and institutions. Schools and hospitals, among other public services, were segregated. Black people were provided with substandard services with no political representation to voice their opposition.  
The situation led to growing dissent among the majority black population. Anti-apartheid groups were organized by the black community to protest against the situation imposed on the community. The government responded by arresting the leaders and participants. Violence erupted across the country between the government and the black community. Internal pressure by the anti-apartheid groups and international embargoes forced the government to reconsider its position and agree to end the segregation.
https://www.blackpast.org/global-african-history/apartheid-1948-1994/

https://www.history.com/topics/africa/apartheid

Apartheid established a system of white minority rule over the country of South Africa that resulted in the eviction of members of the black community from their homes. They were then forced into segregated residential areas, and interracial relationships were forbidden. The majority black population began protesting almost immediately, and violence would dominate the country for years to come.

What is a federal system of government and why did the Framers choose to design a federal system of government for the United States?

A federal system of government is one in which power is distributed between a central authority and smaller political units such as states. Advocates of federalism claim that this method of governance is to be preferred as it brings power closer to the people at the local level. In countries with large, diverse populations, federalism can be a better way of giving people a voice in how things are run. A federal system can also act as a check upon the central government overreaching itself and expanding its powers beyond what the voters think acceptable.
The framers of the Constitution designed a federal system because they thought it would prevent central government from having too much power. The American colonists had just fought a long, bloody war against what they perceived to be the tyranny of the British. The United States had been ruled from a centralized government far away in London, and the result had been a systematic assault on American liberty. So the framers set about constructing a system that would protect Americans' hard-won freedoms; a system in which a considerable degree of sovereignty would still reside with the individual states.
However, the question of the precise division of power between the Federal government and the states was a bone of contention during the Constitutional Convention. Some argued that the proposed new Constitution actually gave too much power to the Federal government, whatever the framers' intentions, potentially turning it into an instrument of tyranny greater than anything devised by the British. Others argued that leaving ultimate sovereignty in the hands of the states, as had been the case under the Articles of Confederation, would lead to the United States being unable to speak with one voice on matters of concern such as foreign affairs and the economy.
Given such radical differences of opinion, the federalist system that subsequently emerged from the Constitutional Convention was, inevitably, something of a compromise. But the precise terms of the distribution of power between the Federal government and the states would remain highly contentious for many years to come, leading indirectly to the Civil War.

What are some quotations from Hunger of Memory by Richard Rodríguez?

This memoir containing different essays about Rodriguez's own experience growing up in a system without bilingual education has many important quotations about being educated within the United States.  I have chosen four quotations that connect to the important themes of the memoir.  

Memory teaches me what I know of these matters. The boy reminds the adult. I was a bilingual child, but of a certain kind: "socially disadvantaged," the son of working class parents, both Mexican immigrants.

I want to include this particular quotation first because it connects to the title:  Hunger of Memory.  It also introduces the reader to the author's upbringing, which is the main focus of the memoir.  It is important to realize that it is Rodriguez's memory that is both the focus and the teacher here.  It is also important to realize that Rodriquez grew up with a public language (English) and a private language (Spanish).  Even though he admits to being "socially disadvantaged," it is this distinguishing factor between public and private language that Rodriguez believes leads directly to his success.

The boy who first entered a classroom barely able to speak English, twenty years later concluded his studies in the stately quiet of the reading room in the British Museum. Thus with one sentence I can summarize my academic career. It will be harder to summarize what sort of life connects the boy to the man.

Here Rodriguez admits that summarizing a life is harder than summarizing success in education.  However, he immediately connects this idea to the importance of a public language vs. a private language.  Previously, Rodriquez admits that having Spanish as a private language did make him "socially disadvantaged," but his academic success speaks for itself.  

To many persons around him, he appears too much the academic. ... He has used education to remake himself. They expect—they want—a student less changed by his schooling. If the scholarship boy, from a past so distant from the classroom, could remain in some basic way unchanged, he would be able to prove that it is possible for anyone to become educated without basically changing from the person one was.

In case there is any doubt, here is a quotation that shows that Rodriguez is against bilingual education.  When he uses the word "they," he is referring to people who believe the fairly new idea of teaching children in their native language is preferable.  Rodriguez believes these people to be wrong.  They want to see an "unchanged" person to be academic.  Rodriguez's point is that change is imperative.  The most important change, says Rodriquez, is distinguishing between a public language and a private language (and never combining the two within the educational system).

Of all the institutions in their lives, only the Catholic Church has seemed aware of the fact that my mother and father are thinkers—persons aware of the experience of their lives. Other institutions—the nation’s political parties, the industries of mass entertainment and communications, the companies that employed them—have all treated my parents with condescension.

Along with Rodriguez's anit-affirmative action stance, he also believes that the changes in the Roman Catholic Church have not been good for the community.  The traditional Catholic Church empowered Rodriguez's parents and gave them a firm foundation.  Changes in the Church seem to water it down and negate the mystery found within it.  This is a good comparison to what Rodriguez says about language.  Any time English as a public language is "watered down" to include Spanish in an effort to pander to the minority, the result is a detriment to the privately Spanish-speaking population.  Similarly, any time the Roman Catholic Church is "watered down" to include changes in order to reflect the masses, the result is a detriment to the traditional Roman Catholic Population.

How do I make a good research question regarding homeschool education in the 21 century?

Start with some general research to find out which aspects of your topic interest you the most, and see what kind of information is already available that you can use to support your own arguments. Respond to others’ claims with your own questions, and think about the type of research you’d have to do to find answers. Consider how much time you have to finish the paper, and how long it has to be. Has your question already been answered completely, or do you have a new perspective on the topic that you want to explore? Is there enough information out there to provide answers without being overwhelming?
A good research question addresses a specific issue, without being too broad (“How do parents in the United States home school their children?” would have to cover so much information that your research would turn into a book) or too narrow (“How many children are home schooled?” can be answered with just a few numbers, so that doesn’t provide enough material for a thorough research paper).

What was Afro-Eurasia like prior to the arrival of Islam?

Afro-Eurasia is a vast landmass, including everything from the southernmost tip of Africa to the far northern areas of Scandinavia and Russia and from China in the east to France and Britain in the west. This area was home to many different cultures in the late sixth and early seventh century as Islam arose. This answer will cover several of the major cultures of the period and region. 
Perhaps the most notable event in western Europe in this period was the collapse of the Roman Empire. The area from Italy north to Hadrian's wall in Britain and much of North Africa had been under the sway of the western, Latin-speaking part of the Roman Empire as a result of conquests originating in the Punic Wars. With the fall of the city of Roman, and barbarian conquests of former Roman territories, this area fell into what are sometimes called the "dark ages", a period where much of Europe was split into various smaller kingdoms, marked by a decline in culture and prosperity. 
In the Greek east, the Roman Empire survived in the form of the Byzantine Empire, Greek speaking and Christian but still following many Roman traditions and laws. Although the Eastern Schism occurs substantially later, in this period we can see the beginnings of the divisions of Christianity into Orthodox and Roman, as the Greek and Latin traditions began to diverge. 
In Persia, the Sassanid Empire continued earlier cultural traditions, following the Zoroastrian religion. It was the main rival of the Byzantine Empire. Its eventual defeat by the Arabs led to the rise of Islam in Iran, replacing Zoroastrianism. 
China was flourishing under the Tang dynasty in this period; it was a powerful and stable dynasty renowned for its art. The Gupta Empire was flourishing in India, where Buddhism and Hinduism were the major religions. 
 
 

Be prepared to answer EVERY PART of the question using VERY SPECIFIC references to historical events and circumstances. 1. Discuss the differences between the worldviews of the European settlers and Native Americans. Choose one European group and one specific Indian tribe to illustrate your discussion. Make sure you discuss relationships between people and the natural world as well as the role of women. Use specific historical events and information to illustrate your explanation. (About 200 words)

You can answer this question in a number of ways and use different historical sources. You might want to look at chapter 1 of Howard Zinn's People's History of the United States. This chapter describes the interaction between the Arawak tribe in the Bahamas and Columbus's crew from Spain (though Columbus was from Italy). 
In his writings, Columbus noted right away that the Arawaks did not have the same sense of private property as Europeans did. He wrote:

"They. . . brought us parrots and balls of cotton and spears and many other things, which they exchanged for the glass beads and hawks' bells. They willingly traded everything they owned."

Zinn notes that the Europeans found the Arawaks' generosity astounding, as it was so dissimilar to what the Europeans practiced. The Europeans were far more interested in making money from commerce and in controlling the Arawaks and their resources.
Zinn also quotes Las Casas, a Spanish priest who observed the interaction between Columbus and the Arawaks. Las Casas wrote of the Arawaks' interaction with the natural world:

They prize bird feathers of various colors, beads made of fishbones, and green and white stones with which they adorn their ears and lips, but they put no value on gold and other precious things. They lack all manner of commerce, neither buying nor selling, and rely exclusively on their natural environment for maintenance.

The Arawaks were able to interact with their environment to get what they needed but did not value hoarding natural resources as the Europeans did.
Las Casas also noted of the women among the Arawaks:

"Marriage laws are nonexistent and men and women alike choose their mates and leave them as they please, without offense, jealousy or anger. They multiply in great abundance; pregnant women work to the last minute and give birth almost painlessly; up the next day, they bathe in the river and are as clean and healthy as before giving birth. If they tire of their men, they give themselves abortions with herbs that force stillbirths, covering their shameful parts with leaves or cotton cloth; although on the whole, Indian men and women look upon total nakedness with as much casualness as we look upon a man's head or at his hands.

Women in the Arawak tribe were treated far more fairly than women in European society at the time. The women's sexual freedom surprised European observers, who were used to women being subject to strict laws about childbirth and marriage and the patriarchal society of Europe.

Single Variable Calculus, Chapter 8, 8.2, Section 8.2, Problem 16

Determine the integral $\displaystyle \int \cos \theta \cos^5 (\sin \theta) d \theta$

Let $u = \sin \theta$, then $du = \cos \theta d \theta$. Thus,


$
\begin{equation}
\begin{aligned}

\int \cos \theta \cos^5 (\sin \theta) d \theta =& \int \cos^5 u du
\\
\\
\int \cos^5 u du =& \int \cos^4 u \cos u du
\\\\
\int \cos^5 u du =& \int (\cos^2 u)^2 \cos u du \qquad \text{Apply Pythagorean Idendity } \cos^2 u + \sin^2 u = 1
\\
\\
\int \cos^5 u du =& \int (1 - \sin^2 u)^2 \cos u du

\end{aligned}
\end{equation}
$


Let $v = \sin u$, then $dv = \cos u du$. Thus,


$
\begin{equation}
\begin{aligned}

\int (1 - \sin^2 u)^2 \cos u du =& \int (1 - v^2)^2 dv &&
\\
\\
\int (1 - \sin^2 u)^2 \cos u du =& \int (1 - 2v^2 + v^4) dv &&
\\
\\
\int (1 - \sin^2 u)^2 \cos u du =& v - \frac{2v^{2 + 1}}{2 + 1} + \frac{v^{4 + 1}}{4 + 1} + c &&
\\
\\
\int (1 - \sin^2 u)^2 \cos u du =& v - \frac{2v^3}{3} + \frac{v^5}{5} + c
&& \text{Substitute value of } v
\\
\\
\int (1 - \sin^2 u)^2 \cos u du =& \sin u - \frac{2 (\sin )^3}{3} + \frac{(\sin u)^5}{5} + c
\\
\\
\int (1 - \sin^2 u)^2 \cos u du =& \sin u - \frac{2 \sin^3 u}{3} + \frac{\sin^5 u}{u} + c
&& \text{Substitute value of } u
\\
\\
\int (1 - \sin^2 u)^2 \cos u du =& \sin (\sin \theta) - \frac{2 \sin^3 (\sin \theta)}{3} + \frac{\sin^5 (\sin \theta)}{5} + c

\end{aligned}
\end{equation}
$


@ 2nd term


$
\begin{equation}
\begin{aligned}

\frac{1}{8} \int^{\pi}_0 \cos 2t dt =& \frac{1}{8} \int^{2 \pi}_0 \cos u \cdot \frac{du}{2}
\\
\\
\frac{1}{8} \int^{\pi}_0 \cos 2t dt =& \frac{1}{16} \int^{2 \pi}_0 \cos u du
\\
\\
\frac{1}{8} \int^{\pi}_0 \cos 2t dt =& \frac{1}{16} \left[ \sin u \right]^{2 \pi}_0
\\
\\
\frac{1}{8} \int^{\pi}_0 \cos 2t dt =& \frac{1}{16} (\sin 2 \pi - \sin 0)
\\
\\
\frac{1}{8} \int^{\pi}_0 \cos 2t dt =& \frac{1}{16} (0)
\\
\\
\frac{1}{8} \int^{\pi}_0 \cos 2t dt =& 0

\end{aligned}
\end{equation}
$


@ 3rd term


$
\begin{equation}
\begin{aligned}

\frac{1}{8} \int^{\pi}_0 \cos^2 2t dt =& \frac{1}{8} \int^{2 \pi}_0 \cos^2 u \cdot \frac{du}{2}
\\
\\
\frac{1}{8} \int^{\pi}_0 \cos^2 2t dt =& \frac{1}{16} \int^{2 \pi}_0 \cos^2 u du
\qquad \text{Apply half-angle formula } \cos 2 u = 2 \cos^2 u - 1
\\
\\
\frac{1}{8} \int^{\pi}_0 \cos^2 2t dt =& \frac{1}{16} \int^{2 \pi}_0 \left(\frac{\cos 2 u + 1}{2} \right) du
\\
\\
\frac{1}{8} \int^{\pi}_0 \cos^2 2t dt =& \frac{1}{32} \int^{2 \pi}_0 (\cos 2u + 1) du

\end{aligned}
\end{equation}
$


Let $v = 2u$, then $dv = 2 du$, so $\displaystyle du = \frac{dv}{2}$. When $u = 0, v = 0$ and when $u = 2 \pi, v = 4 \pi$


$
\begin{equation}
\begin{aligned}

\frac{1}{32} \int^{32}_0 (\cos 2u + 1) du =& \frac{1}{32} \int^{4 \pi}_0 (\cos v + 1) \cdot \frac{dv}{2}
\\
\\
\frac{1}{32} \int^{32}_0 (\cos 2u + 1) du =& \frac{1}{64} \int^{4 \pi}_0 (\cos v + 1) dv
\\
\\
\frac{1}{32} \int^{32}_0 (\cos 2u + 1) du =& \frac{1 }{64} \left[ \sin v + v \right]^{4 \pi}_0
\\
\\
\frac{1}{32} \int^{32}_0 (\cos 2u + 1) du =& \frac{1}{64} (\sin 4 \pi + 4 \pi - \sin 0 - 0)
\\
\\
\frac{1}{32} \int^{32}_0 (\cos 2u + 1) du =& \frac{1}{64} (0 + 4 \pi - 0 - 0)
\\
\\
\frac{1}{32} \int^{32}_0 (\cos 2u + 1) du =& \frac{4 \pi}{64}
\\
\\
\frac{1}{32} \int^{32}_0 (\cos 2u + 1) du =& \frac{\pi}{16}

\end{aligned}
\end{equation}
$


@ 4th term


$
\begin{equation}
\begin{aligned}

\frac{1}{8} \int^{\pi}_0 \cos^3 2t dt =& \frac{1}{8} \int^{2 \pi}_0 \cos^3 u \cdot \frac{du}{2}
\\
\\
\frac{1}{8} \int^{\pi}_0 \cos^3 2t dt =& \frac{1}{16} \int^{2 \pi}_0 \cos^3 u du
\\
\\
\frac{1}{8} \int^{\pi}_0 \cos^3 2t dt =& \frac{1}{16} \int^{2 \pi}_0 (\cos^2 u)(\cos u) du
\qquad \text{Apply Trigonometric Identities } \cos^2 u + \sin^2 u = 1
\\
\\
\frac{1}{8} \int^{\pi}_0 \cos^3 2t dt =& \frac{1}{16} \int^{2 \pi}_0 (1 - \sin^2 u)(\cos u) du

\end{aligned}
\end{equation}
$


Let $v = \sin u$, then $dv = \cos u du$. When $u = 0, v = 0$ and when $u = 2 \pi, v = 0$. Therefore,


$
\begin{equation}
\begin{aligned}

\frac{1}{16} \int^{2 \pi}_0 (1 - \sin^2 u)(\cos u du) =& \frac{1}{16} \int^0_0 (1 - v^2) dv
\\
\\
\frac{1}{16} \int^{2 \pi}_0 (1 - \sin^2 u)(\cos u du) =& \frac{1}{16} \left[ v - \frac{v^3}{3} \right]^0_0
\\
\\
\frac{1}{16} \int^{2 \pi}_0 (1 - \sin^2 u)(\cos u du) =& \frac{1}{16} (0)
\\
\\
\frac{1}{16} \int^{2 \pi}_0 (1 - \sin^2 u)(\cos u du) =& 0

\end{aligned}
\end{equation}
$


Combine the results of integration term by term


$
\begin{equation}
\begin{aligned}

\int^{\pi}_0 \sin^2 t \cos^4 t dt =& \frac{\pi}{8} + 0 - \frac{\pi}{16} - 0
\\
\\
\int^{\pi}_0 \sin^2 t \cos^4 t dt =& \frac{2 \pi + 0 - \pi - 0}{16}
\\
\\
\int^{\pi}_0 \sin^2 t \cos^4 t dt =& \frac{\pi}{16}
\end{aligned}
\end{equation}
$

College Algebra, Chapter 1, 1.5, Section 1.5, Problem 28

Find all real solutions of the equation $\displaystyle \sqrt{4 - 6x} = 2x$


$
\begin{equation}
\begin{aligned}

\sqrt{4 - 6x} =& 2x
&& \text{Given}
\\
\\
(\sqrt{4 - 6x})^2 =& (2x)^2
&& \text{Square both sides}
\\
\\
4 - 6x =& 4x^2
&& \text{Divide both sides by } 4
\\
\\
1 - \frac{3}{2} x =& x^2
&& \text{Add } \frac{3}{2} x \text{ and subtract } 1
\\
\\
x^2 + \frac{3}{2} x - 1 =& 0
&& \text{Factor out}
\\
\\
(x + 2)\left( x - \frac{1}{2} \right) =& 0
&& \text{Zero Product Property}
\\
\\
x + 2 =& 0 \text{ and } x - \frac{1}{2} = 0
&& \text{Solve for } x
\\
\\
x =& -2 \text{ and } x = \frac{1}{2}
&&
\\
\\
x =& \frac{1}{2}
&& \text{The only solution that satisfy the equation } \sqrt{4 - 6x} = 2x

\end{aligned}
\end{equation}
$

What is the setting of the Watson's home in The Watsons Go to Birmingham--1963 by Christopher Paul Curtis?

The Watsons live in Flint, Michigan. As the story opens, as Kenny says, "it was about a zillion degrees below zero" (page 1). The family turns up their thermostat, but the furnace is making noises and seems like it's going to "blow up." To keep warm, the family is huddled together on the couch under a blanket. They are wearing scarves and other winter clothing. Outside is their car, a 1948 Plymouth (which is 15 years old), which they have named the Brown Bomber. The family is clearly close, and the father helps the family keep their minds off the cold by making jokes. Later, the family travels to Birmingham, Alabama, to bring their oldest son, Byron, to the house of Momma's mother, Grandma Sands, in an attempt to teach Byron how to show respect and manners. 

What lesson does Huck learn in The Adventures of Huckleberry Finn by Mark Twain?

Huck learns many lessons throughout his journey on the Mississippi with Jim. The most important of these involve caring and moral responsibility. 

Huck is a child who has never had a chance to develop a real bond with another human being. His Pap is a drunk who doesn't care if Huck lives or dies. Huck only knows how to use and manipulate another—the lesson he learned from Pap—or how to evade responsibility, the lesson he taught himself when trying to get away from the Widow Douglas and Miss Watson. Once Huck begins to spend every night with Jim, he falls back on his own habits at first and plays merciless tricks on Jim. When Jim scolds Huck for playing a trick on him, Huck begins to feel bad, realizing he has hurt another human being, even if Huck still sees the human being as "only" a slave. 
Jim gives Huck the nurturing and care he has never had, and the two become inseparable companions. Through the time they spend together, Huck learns what it's like to have companionship and even love. 
His twisted understanding of society's mores still tell him, however, that he is doing "wrong" to keep a slave from capture. It is only when Huck tears up the letter he has written to Miss Watson and emphatically says, "All right, I'll go to hell" that we see he has learned to defy the mores of a corrupt society, even if he naively believes this rebellion will send him to hell. He learns to listen to his true conscience, not the perverted mores of an unjust society. 

Single Variable Calculus, Chapter 5, 5.2, Section 5.2, Problem 4

a.) Suppose that $\displaystyle f(x) = \sin x, 0 \leq x \leq 3 \frac{\pi}{2}$, find the Riemann sum with $n = 6$, taking the sample points to be right end points. What does Riemann sum represents? Illustrate with a diagram.

With $n = 6$, we divide the interval $\displaystyle (0, 3 \frac{\pi}{2})$ into 6 rectangles with widths

$\displaystyle \Delta x = \frac{\displaystyle \frac{3 \pi}{2} - 0}{6} = \frac{\pi}{4}$ at $\displaystyle x = 0, x = \frac{\pi}{4}, x = \frac{\pi}{2}, x = \frac{3 \pi}{4}, x = \pi, x = \frac{5 \pi}{4}$ and $\displaystyle x = \frac{3 \pi}{2}$.

Evaluating $f(x)$ on the right end points (starting from $\displaystyle x = \frac{\pi}{4}$)


$
\begin{array}{|c|c|}
\hline\\
x & f(x) = x^2 - 2x \\
\hline\\
\displaystyle \frac{\pi}{4} & \displaystyle \frac{\sqrt{2}}{2} \\
\hline\\
\displaystyle \frac{\pi}{2} & 1 \\
\hline\\
\displaystyle \frac{3 \pi}{4} & \displaystyle \frac{\sqrt{2}}{2} \\
\hline\\
\pi & 0 \\
\hline\\
\displaystyle \frac{5 \pi}{4} & \displaystyle - \frac{\sqrt{2}}{2} \\
\hline\\
\displaystyle \frac{3 \pi}{2} & -1\\
\hline
\end{array}
$


Now, the total area of the rectangle is..

$\displaystyle \frac{\pi}{4} \left[ \frac{\sqrt{2}}{2} + 1 + \frac{\sqrt{2}}{2} + 0 - \frac{\sqrt{2}}{2} - 1 \right] = 0.5554 \text{ units}^2$

The Riemann sum represents ab estimate of the area between the curve and the $x$-axis. Although in some cases, some areas result to a negative value because some rectangles are located below the $x$-axis. With this, you have to take the absolute values of such areas to get the actual area.

b.) Repeat part (a) with midpoints as sample points.

By using midpoints,

Evaluating $f(x)$ at midpoints


$
\begin{array}{|c|c|}
\hline\\
x & f(x) = \sin x \\
\hline\\
\displaystyle \frac{\displaystyle 0 + \frac{\pi}{4}}{2} = \frac{\pi}{8}
& 0.38327 \\ \hline\\
\displaystyle \frac{\displaystyle \frac{\pi }{4} + \frac{\pi}{2} }{2} = \frac{3 \pi}{8}
& 0.9239 \\
\hline\\
\displaystyle \frac{\displaystyle \frac{\pi}{2} + \frac{3 \pi}{4} }{2} = \frac{5 \pi }{8}
& 0.9239 \\
\hline\\
\displaystyle \frac{\displaystyle \frac{3 \pi}{4} + \pi}{2} = \frac{7 \pi}{8}
& 0.3827 \\
\hline\\
\displaystyle \frac{\displaystyle \pi + \frac{5 \pi}{4}}{2} = \frac{9 \pi}{8}
& -0.3827 \\
\hline\\
\displaystyle \frac{\displaystyle \frac{5 \pi}{4} + \frac{3 \pi}{2}}{2} = \frac{11 \pi}{8}
& -0.9239\\
\hline
\end{array}
$


Now, the total area of the rectangle is ..

$\displaystyle \frac{\pi}{4} [0.3827 + 0.9239 + 0.9239 + 0.3827 - 0.3827 - 0.9239] = 1.0262 \text{ units}^2$

Beginning Algebra With Applications, Chapter 5, 5.4, Section 5.4, Problem 28

Using the point slope formula, find the equation of the line that passes through the point whose coordinates are $(-5,0)$ and has slope of $\displaystyle \frac{-1}{5}$


$
\begin{equation}
\begin{aligned}

(x_1, y_1) =& (-5,0)
&& \text{Let $(x_1, y_1)$ be the given point}
\\
m =& \frac{-1}{5}
&& \text{$m$ is the given slope}
\\
y - y_1 =& m(x- x_1)
&& \text{Point slope formula}
\\
y -0 =& \frac{-1}{5} [x-(-5)]
&& \text{Substitute $-5$ for $x$, $$ for $y$ and } \frac{-1}{5} \text{ for $m$}
\\
y =& \frac{-1}{5} ( x + 5)
&& \text{Rewrite the equation in the form } y = mx+b
\\
y =& \frac{-1}{5} x - 1
&&

\end{aligned}
\end{equation}
$

Which ancient principles can be found in American government today?

The government of the United States borrows a number of important principles from ancient civilizations, namely classical Greece and Rome. While much of the structure of today's government is more of a direct outgrowth of the Enlightenment of the seventeenth century, many ancient ideas are present in it as well.
The eighteenth-century French philosopher Baron de Montesquieu gets a lot of credit for the idea of a government made up of separate branches with different powers. However, the idea goes much further back. The Greek philosopher Plato first promoted the idea in the fourth century BCE. He wrote in The Laws that a successful government should combine the functions of a democracy, an oligarchy, and a tyranny so that the people would benefit from the positives of each without being subjected to all their negative aspects. The idea of three separate branches with different powers similar to today's Executive, Legislative, and Judicial branches was later formulated by Aristotle, Plato's student. This idea of a mixed form of government would inspire later philosophers and the framers of the United States Constitution.
The form of our legislature also borrows many ideas from the Roman Republic. Rome was too large to have a direct democracy like Classical Athens. Instead, eligible citizens elected representatives to make laws on their behalf in the Senate. The legislature of the United States is likewise composed of elected representatives.

In October Sky by Homer Hickam, what does "Auk" stand for?

As Homer tries to build his prototype rocket, he gives it the nickname Auk 1. This is a humorous nickname in the context of the story, namely because of the boys’ eventual success. He gets Mr. Bykovski to held him solder together the pieces of the rocket to make a homemade version of Sputnik. When the rocket is finished, Homer dubs it the Auk.
The Auk is a flightless, extinct bird, indicating Homer’s doubt that the rocket would be successful and fly. However, he was very pleased with the end result. Ironically, the Auk has some limited success and the boys’ passion is truly ignited, leading them to have much success and eventually launching them to stardom and careers in science and technology in the future.

In Homer Hickman's October Sky, a group of young boys in a small mining town have decided that they are determined to build a rocket. Luckily, this small mining town has a resident, Mr. Bykovski, who is a genius that can build rockets in his home.
One day, one of the boys, Homer, who is a part of the rocket club, Bear Creek Missile Agency, goes Mr. Bykovski's house to ask him to help him build a miniature rocket. Mr. Bykovski agrees to help solder the pieces of the tiny rocket together. The result of this first rocket is the first of many attempts from the Bear Creek Missile Agency to launch their rockets.
Homer ironically names the rocket "Auk 1". An Auk is a now-extinct flightless bird. Naming the rocket Auk certainly reflects the boys' hesitation that the rocket will actually launch and also reflects their ironic sense of humor.

In chapter 6, Homer goes round to Mr. Bykovski's house to see if he can help him build a rocket. Homer wants to create his own mini-Sputnik by welding a washer to a metal tube. Mr. Bykovski knows better, however, and suggests soldering the two together instead of welding them. He agrees to do the soldering work himself and tells Homer that he'll have his rocket ready for him the next day.
When Homer returns the following night, Mr. Bykovski presents him, as promised, with his rocket. Homer's pretty pleased at Mr. Bykovski's handiwork: a long, soldered tube with a wooden cone for a head. He decides to christen his new rocket "Auk 1." An auk was a flightless bird that became extinct—rather like the dodo—precisely because it couldn't fly.
Calling his new rocket by that name indicates that, although he's pleased with it, Homer doesn't have particularly high hopes that it will get off the ground. At the same time, the name of the rocket's also designed to give the impression that building rockets isn't a complete waste of time, and it is actually making Homer much smarter. After all, it's fair to say that not too many people in Coalwood actually know what an auk is, and so perhaps Homer thinks that the local townsfolk will be impressed by his unusual knowledge.

October Sky (originally titled Rocket Boys) a story about a boy who, despite growing up in a small mining town, develops an interest in rocketry. Sonny, the main character, founds the Bear Creek Missile Agency (BCMA), a club in which he and some friends attempt to build rockets using various fuels. Their first rocket was titled Auk 1, and it was powered by black powder; all of their subsequent rockets also bore the name "Auk." This name was ironic because an Auk was a seabird that was unable to fly. Moreover, Auks have been extinct since the 19th century. Much like the bird, Auk 1 was not particularly successful at flying; it barely flew six feet before crashing. This did not deter the boys from their quest, however; the "Rocket Boys" ultimately launched 35 rockets.
https://www.smithsonianmag.com/smithsonian-institution/with-crush-fisherman-boot-the-last-great-auks-died-180951982/?no-ist

I need to compose a short parable about what God is like using something in my life and world that is familiar to me.

Obviously, as an educator who is not personally acquainted with you, I cannot know what are important and familiar elements of your personal life; therefore I will focus on the parable as a genre and provide some tips concerning how you might approach the task of writing one.
The parable is an important literary genre in many religious traditions, and is often quite similar to moral fables such as those of Aesop. It is usually relatively short narrative concluding with a moral point or parallel. Often used to convey religious or moral conceptions to children or the illiterate, parables are characterized by simple, direct language and use of elements from everyday life. The characterizations tend to be simple and generic. Characters rarely have names and are not individuated (they are often just described as "a laborer" or "a rich man").
For a modern parable, you might take an event or experience shared by your peers such as driving, attending school, shopping for groceries, or cooking. For example, you could contrast buying fast food with cooking a healthy dinner and use that to explain that God often requires us to do things in ways that are difficult but which end up being good for us. 

Single Variable Calculus, Chapter 2, 2.4, Section 2.4, Problem 5

We need to use a graph to find a number $\delta$ such that if $\displaystyle |x - \frac{\pi}{4} | < \delta $ then $|\tan x - 1| < 0.2$









First, we will get the values of $x$ that intersect at the given curve to their corresponding $y$ values. Let $x_L$ and $x_R$
are the values of $x$ from the left and right of $\displaystyle \frac{\pi}{4}$ respectively.

$
\begin{equation}
\begin{aligned}
y & = \tan x_L &
y & = \tan x_R\\

0.8 & = \tan x_L &
1.2 & = \tan x_R\\

x_L & = \tan^{-1}[0.8] &
x_R & = \tan^{-1}[1.2]\\

x_L & = 0.6741 &
x_R & = 0.8761
\end{aligned}
\end{equation}
$


Now, we can determine the value of $\delta$ by checking the values of $x$ that would give a smaller distance to $\displaystyle \frac{\pi}{4}$.


$
\begin{equation}
\begin{aligned}
\frac{\pi}{4} - x_L = \frac{\pi}{4} - 0.6747 & = 0.1107\\
\frac{\pi}{4} - x_R =0.8761 - \frac{\pi}{4} & = 0.0907
\end{aligned}
\end{equation}
$


Hence,
$\quad \delta \leq 0.0907$

This means that by keeping $x$ within $0.0907$ of $\displaystyle \frac{\pi}{4}$, we are able to keep $f(x)$ within $0.2$ of $1$.

Although we chose $\delta = 0.0907$, any smaller positive value of $\delta$ would also have work.

The school cafeteria serves pizza every sixth day and applesauce every eighth day. If pizza and applesauce are both on today's menu, in how many days will they be together on the menu again?

You have to find the least common multiple. Pizza will be served on day 6, day 12, day 18, and day 24. Apple Sauce will be served on the 8th day, 16th day, and 24th day. This means apple sauce and pizza will both be served together on day 24.

We need to find the least common multiple of 6 and 8.
First factor both numbers as a product of primes:
6=2*3; 8=2*2*2=2^3
The least common multiple can be found by taking the product of each prime factor that occurs to the highest power that it occurs in either factorization.
We have a factor of 2 which occurs to the 3rd power, and a factor of 3.
Thus LCM(6,8)=2^3*3=24
The next day the two occur together is 24 days later.
http://mathworld.wolfram.com/LeastCommonMultiple.html

To solve, let's apply the least common multiple (LCM) of a set of numbers.
Pizza is served every sixth day. To get the days in which it will be served, take the multiples of 6.
The multiples of 6 are:
6, 12, 18, 24, 30, ... 6n
So pizza will be served on day 6, day 12, day 18, day 24, day 30 until day 6n.
Applesauce is served every 8th day. To get the days in which it will be served, take the multiples of 8.
8, 16, 24, 32, 40, ... 8n
So applesauce will be served on day 8, day 16, day 24, day 32, day 40 until day 8n.
Base on the list above, the LCM of 6 and 8 is 24.
Therefore, the pizza and apple sauce are served together every 24th day.

College Algebra, Chapter 4, 4.1, Section 4.1, Problem 36

Determine the maximum or minimum value of the function $f(t) = 10t^2 + 40t + 113$.

Using the quadratic equation $ax^2 + bx + c$, where $a = 10$ and $b = 40$. Thus, the maximum or minimum value occurs at

$\displaystyle x = -\frac{b}{2a} = - \frac{40}{2(10)} = -2$

Since $a > 0$, the function has the minimum value

$\displaystyle f (-2) = 10(-2)^2 + 40(-2) + 113 = 73$

What is the impact of the bird's song on the poet in "Ode To Nightingale?"

The poet admires the nightingale's song because the bird sings with "full-throated ease." The poet recognizes a freedom of creativity and art in the song. In the third stanza, the poet notes that the nightingale does not have the concerns that he, a human being, has:

Fade far away, dissolve, and quite forgetWhat thou among the leaves hast never known,The weariness, the fever, and the fretHere, where men sit and hear each other groan;

The nightingale's song is beautiful and the poet recognizes the creativity of that song as something comparable to his art of poetry. So, he sees a common connection there. But the poet is particularly fascinated because the song comes from a creature who is not burdened by the realities of aging, sorrow, and death. To the poet, the nightingale sings without those concerns. He, on the other hand, writes poetry with those concerns always in mind. Keats was always too aware of his own mortality. And as an artist trying to create poetry that could be timelessly celebrated, he is thinking of immortality. He sees/hears that immortality in the nightingale's song because the song comes from a place in which mortality is not a concern:

Thou wast not born for death, immortal Bird!No hungry generations tread thee down;

The poet wonders what it's like to sing/create without those concerns. He also longs for his own poetry to achieve the kind of immortality he hears in the nightingale's song.
 
https://www.owleyes.org/text/ode-nightingale/read/ode-nightingale

Why is Mr. Birling impressed with Gerald?

Mr. Birling's an inveterate social climber. Although he's amassed fantastic wealth, he still feels a slight twinge of inferiority on account of his humble origins. More than anything else, he craves social respectability in the eyes of the aristocracy. And this is where Gerald fits in. Gerald is a polished young man from an elite, old money background. Mr. Birling is excited at the prospect of his daughter's impending marriage to the son of Lord and Lady Croft. This will mean the uniting of two great families, old money and new. The forthcoming marriage represents a good deal for both families: Gerald Croft will marry into money, whereas Sheila Birling will burnish the good name of her family with the luster of noble lineage. No wonder her father's so impressed with his prospective son-in-law.

Why does George Orwell use animals instead of humans?

The idea of using animals in a satiric allegory has a long history. Probably the most famous example, prior to Animal Farm, is in Book IV of Swift's Gulliver's Travels, in which Gulliver encounters the realm of the Houyhnhnms, a race of intelligent horses who embody virtues that human beings, in Swift's view, lack. In John Dryden's satiric religious allegory The Hind and the Panther, the two title animals represent, respectively, the Roman Catholic Church and the Church of England. But your question as to why Orwell uses animals in his own satirical work can best be answered if you look first at the actual events of human history portrayed by Orwell in allegorical terms.
If we see Animal Farm in its most specific meaning as a parable about the Russian Revolution, the animals symbolize the Russian people apart from the ruling class of the Czar and his administration, the aristocracy, and the wealthy people generally. In less specific terms, the animals can be seen to stand for working-class people or, more broadly, non-wealthy people in any place and time throughout history. The wealthy, the rulers, are symbolized by the human Mr. Jones, the owner of the Manor Farm before the animals chase him out. Orwell is making the sharpest contrast possible in his portrayal of the ruling class, on the one hand, and the rest of the population—especially the working class—on the other. By making it a human vs. animal distinction, any doubts should be erased about how severely working people, in Orwell's view, have been oppressed throughout history.
But as in other satires, the use of animals also gives an irreverent, humorous tone to the allegory, in spite of Orwell's sympathies obviously being so strongly with the ordinary working people of the world. This is in keeping with the rather cynical message of his fable, in which it turns out that even among the oppressed ones who carry out the Revolution there are those who are primarily interested in seizing power for themselves, rather than in helping the other "animals."
This, of course, is exactly what happened in the actual history of the Soviet Union. What presumably was meant to be a workers' state with equality for all ended up a brutal dictatorship under Stalin, who made Russia even less free than it had been under the czars and killed millions of his own people.

How do I write a comparison/contrast essay for "The Jilting of Granny Weatherall" and "The Yellow Wallpaper"? What will be a theme to compare/contrast?

One of the first things a person needs to do when constructing a compare and contrast essay is decide upon what formatting he or she is going to use. There are two formats for a compare and contrast essay: point-by-point and block.
In point-by-point, the essayist alternates points made between the two things he or she is discussing. In block, the essayist discusses the entirety of one subject and ends with the other.
One theme from both Katherine Ann Porter's "The Jilting of Granny Weatherall" and Charlotte Perkins Gilman's "The Yellow Wallpaper" is mental illness or mental deterioration. Both protagonists in both short stories deal with mental illness and coming to terms (or not) with their own mental instability.
In order to construct a compare and contrast essay regarding these two stories, one must create a thesis statement which speaks to the purpose of the essay. One example thesis statement is as follows: Both Katherine Ann Porter's "The Jilting of Granny Weatherall" and Charlotte Perkins Gilmore's "The Yellow Wallpaper" possess a strong message on mental illness using each text's theme.
Next, one will need to find points to bring up regarding each text's use of the theme of mental illness to support the thesis. One could focus upon the setting (the bedroom), other characters (John and the Doctor), and how the protagonist seems to feel about her mental deterioration.
In setting up a point-by-point, the outline would look something like this:
Introduction
Body Paragraph One (Point one for "Jilting")
Body Paragraph Two (Similar point one for "Yellow")
Body Paragraph Three (Point two for "Jilting")
Body Paragraph Four (Similar point for "Yellow")
Conclusion
The number of body paragraphs is determined by identifying the number of points one is bringing up to support his or her thesis. For example, if one has 3 points to make, one must have six body paragraphs.
When using the block format, one discusses all points (for this example) regarding one text's support and then the other text's.
Formatting would look like this:
Introduction
"Jilting" Point One
"Jilting" Point Two
"Yellow" Point One
"Yellow" Point Two
Conclusion
One must make sure that the alternating points for the texts are comparative or contrasting in nature. For example, neither Granny's doctor nor John (the husband of the protagonist in "The Yellow Wallpaper," and her doctor) seems to be doing what is best for either of their patients. This point would be made in both body paragraph one and two (in point-by-point) or in body paragraphs one and three (using the sample formatting described above).
On a side note, many times, students are misled by the name "Compare and Contrast" essay. One can do three things in this type of essay: compare only, contrast only, or compare and contrast.

Single Variable Calculus, Chapter 8, 8.2, Section 8.2, Problem 18

Determine the integral $\displaystyle \int \cot^5 \theta \sin^4 \theta d \theta$


$
\begin{equation}
\begin{aligned}

\int \cot^5 \theta \sin^4 \theta d \theta =& \int \frac{\cos^5 \theta}{\sin^5 \theta} \sin^4 \theta d \theta
\\
\\
\int \cot^5 \theta \sin^4 \theta d \theta =& \int \frac{\cos^5 \theta}{\sin \theta} d \theta
\\
\\
\int \cot^5 \theta \sin^4 \theta d \theta =& \int \frac{\cos^4 \theta}{\sin \theta} d \theta
\\
\\
\int \cot^5 \theta \sin^4 \theta d \theta =& \int \frac{(\cos^2 \theta)^2}{\sin \theta} \cos \theta d \theta
\qquad \text{Apply Pythagorean Identities } \cos^2 \theta + \sin^2 \theta = 1
\\
\\
\int \cot^5 \theta \sin^4 \theta d \theta =& \int \frac{(1 - \sin^2 \theta)^2}{\sin \theta} \cos \theta d \theta

\end{aligned}
\end{equation}
$


Let $u = \sin \theta$, then $du = \cos \theta d \theta$. Thus,


$
\begin{equation}
\begin{aligned}

\int \frac{(1 - \sin^2 \theta)^2}{\sin \theta} \cos \theta d \theta =& \int \frac{(1 - u^2)^2}{u} du
&&
\\
\\
\int \frac{(1 - \sin^2 \theta)^2}{\sin \theta} \cos \theta d \theta =& \int \left( \frac{1 - 2u^2 + u^4}{u} \right) du
&&
\\
\\
\int \frac{(1 - \sin^2 \theta)^2}{\sin \theta} \cos \theta d \theta =& \int \left( \frac{1}{u} - \frac{2u^2}{u} + \frac{u^4}{u} \right) du
&&
\\
\\
\int \frac{(1 - \sin^2 \theta)^2}{\sin \theta} \cos \theta d \theta =& \int \left(
\frac{1}{u} - 2u + u^3 \right) du
&&
\\
\\
\int \frac{(1 - \sin^2 \theta)^2}{\sin \theta} \cos \theta d \theta =& \ln u - \frac{2u^{1 + 1}}{1 + 1} + \frac{u^{3 + 1}}{3 + 1} + c
&&
\\
\\
\int \frac{(1 - \sin^2 \theta)^2}{\sin \theta} \cos \theta d \theta =& \ln U - \frac{\cancel{2}u^2}{\cancel{2}} + \frac{u^4}{4} + c
&&
\\
\\
\int \frac{(1 - \sin^2 \theta)^2}{\sin \theta} \cos \theta d \theta =& \ln u - u^2 + \frac{u^4}{4} + c
&& \text{Substitute value of } u
\\
\\
\int \frac{(1 - \sin^2 \theta)^2}{\sin \theta} \cos \theta d \theta =& \ln (\sin \theta) - (\sin \theta)^2 + \frac{(\sin \theta)^4}{4} + c
&&
\\
\\
\int \frac{(1 - \sin^2 \theta)^2}{\sin \theta} \cos \theta d \theta =& \ln (\sin \theta) - \sin^2 \theta + \frac{\sin^4 \theta}{4} + c



\end{aligned}
\end{equation}
$

Find the area of the region bounded by the graphs of the equations f(x)=-x^(2)+4x and y=0

We are asked to find the area of the region bounded by f(x)=-x^2+4x and y=0 .
We can factor the quadratic to y=-x(x-4); the x-intercepts are 0 and 4.
The equivalent question then is to find the definite integral int_0^4 (-x^2+4x)dx .
int_0^4 (-x^2+4x)dx=-x^3/3+2x^2 |_0^4
=(-64/3+32)-(0+0)=32/3
So the area bounded by the given curve and the x-axis is 32/3.
An alternative approach is to use a Riemann sum:
The area can be found by lim_{n->oo} sum_{i=1}^{n} f(c_{i})Delta x_{i}
If we use a regular partition and choose the right-hand endpoint in each subinterval we get:
A=lim_{n->oo} sum_{i=1}^{n} (-((4i)/n)^2+4((4i)/n))(4/n)
=lim_{n->oo}[(-64)/(n^2)sum_{i=1}^n(i^2/n-i)]
=lim_{n->oo}[(-64)/n^2(1/n((n(n+1)(2n+1))/6)-(n(n+1))/2)]
=lim_{n->oo}[(-64)/n^2(1/3n^2+1/2n+1/3-n^2/2-n/2)]
=lim_{n->oo}[(-64)/3-(32)/n-(64)/n^3+32+(32)/n]
=(-64)/3+32=32/3=10.bar(6)
The graph:

Why do you believe Jefferson chose to include the list of grievances in the document rather than summing them up with a single blanket statement?

Setting down a litany of woes like this is so much more effective than just putting all the colonists' grievances into a couple of sentences. Listing all the grievances separately adds a suitably condemnatory tone to the Declaration of Independence; it's as if a serial criminal is being confronted with the damaging effects of his lengthy rap sheet.
The Declaration was a rhetorical document more than anything else; it was designed to convince those still unsure about the wisdom of breaking free from the British that this was the right thing to do. So Jefferson, in listing the grievances separately, rams home the point more effectively that the British have consistently ignored the legitimate concerns of the Americans and that independence is now the only way forward. Jefferson was also doubtless aware that adopting this rhetorical strategy would make it easier for each individual grievance to form the basis of anti-British propaganda in the ensuing struggle.

In the Declaration of Independence, Jefferson was trying to make the rational argument for American self-rule as well as convince those who were undecided on the issue that the time for independence was now. By listing the grievances individually, Jefferson hopes to incite feelings of anger in those colonists who may not have made up their minds. In 1776, not all colonists felt as though separation was the proper way to deal with Parliament. Many feared American self-government—the new country would be vulnerable to attack and no one was sure how it would be governed. When Jefferson listed the problems faced by the colonists, such as trials without juries and unreasonable tax collection methods, he wished to make the colonists feel as though separation would be better than just staying under British rule.

Are Nora’s duties to herself really more important than the other roles (wife, mother, and friend) she plays?

Well, it depends. Do you believe that one's duties to oneself or one's duties to others are more important? For Nora, she realizes that she has been a pawn manipulated by the men in her life, and that she's been commanded by them since her youth. In the end, it occurs to her that she was her father's "doll" who then became her husband's doll, she's never really been happy, and she's only been valued because she's been so obedient and compliant. She's been who society, embodied by her father and husband, wanted her to be rather than figuring out who she really is and then being that person. Because she's been denied this opportunity as a result of her femininity, she does seem to deserve the same chance as a man to achieve self-actualization.
However, she does have children to consider, even if we discount her husband. While Nora might seem cold for abandoning her children, we must consider that the world has been cold to her. Her opportunities for living have been circumscribed, and she rebels against the limitations society imposes on her. This seems to me to be very brave, and it is only the existence of her children that problematizes her decision (in my view).

Please give a detailed summary of the poem The Phenomenal Woman By Maya Angelou.

Maya Angelou, who was born April 4th, 1928 and died May 28th, 2014, was an American writer and poet. Her poem "Phenomenal Woman" appeared in a collection of four poems in 1995.
 
The poem is four stanzas long and uses poetic elements of repetition, rhyme scheme, and metaphor. Angelou repeats the line "I'm a woman, phenomenally. Phenomenal woman, that's me" for emphasis and effect. The rhyme scheme is not regular, but at least two lines rhyme in each stanza.
 
Maya Angelou celebrates her womanhood in this poem. In the first stanza, she explains that she is not the stereotypical physical beauty, and yet, she is an object of desire for men. She tries to explain that her appeal is not in her dress size or her chiseled, perfect features. It's contained in elements like the span of her arms and hips, and in the way she walks. She is confident, even joyful in her womanhood. Her mystique radiates from the inside out.
 
In stanza two, Angelou talks about how men react to her when she walks into a room. Using the metaphor of bees, she compares the men swarming her to the way they swarm around a hive. She is the queen bee, and they are all working for her. She talks about where this mystique comes from again, saying it's in the fire of her eyes, the flash of her smile, and the joy in her feet. The eyes as the windows to her soul, her smile as an outward expression of joy, and the lightness of her walk are all manifestations of her inward confidence and radiance.
 
In stanza three, Angelou discusses the fact that men have often wondered what they see in her because it isn't typical beauty. It's not even something they can describe. She says they try to touch her inner mystery, but they can't. This is something she says she's tried to show them, but they still don't understand it. It's deeper than superficiality, a treasure she possesses that can't be taken away. Still, she tries to explain it in physical terms, pointing to the arch of her back and the sun of her smile. 
 
In the fourth stanza, she explains that this inward treasure that she possesses, that manifests in certain unexpected physical attributes, is why she is proud. She doesn't have to fight for attention or be loud. She exudes grace and power and says to the reader this fact ought to make them proud. She states that she is a woman, "phenomenally." The full text of the poem can be found below.

Pretty women wonder where my secret lies. 
I’m not cute or built to suit a fashion model’s size   
But when I start to tell them, 
They think I’m telling lies. 
I say, 
It’s in the reach of my arms, 
The span of my hips,   
The stride of my step,   
The curl of my lips.   
I’m a woman 
Phenomenally. 
Phenomenal woman,   
That’s me. 


I walk into a room 
Just as cool as you please,   
And to a man, 
The fellows stand or 
Fall down on their knees.   
Then they swarm around me, 
A hive of honey bees.   
I say, 
It’s the fire in my eyes,   
And the flash of my teeth,   
The swing in my waist,   
And the joy in my feet.   
I’m a woman 
Phenomenally. 


Phenomenal woman, 
That’s me. 


Men themselves have wondered   
What they see in me. 
They try so much 
But they can’t touch 
My inner mystery. 
When I try to show them,   
They say they still can’t see.   
I say, 
It’s in the arch of my back,   
The sun of my smile, 
The ride of my breasts, 
The grace of my style. 
I’m a woman 
Phenomenally. 
Phenomenal woman, 
That’s me. 


Now you understand 
Just why my head’s not bowed.   
I don’t shout or jump about 
Or have to talk real loud.   
When you see me passing, 
It ought to make you proud. 
I say, 
It’s in the click of my heels,   
The bend of my hair,   
the palm of my hand,   
The need for my care.   
’Cause I’m a woman 
Phenomenally. 
Phenomenal woman, 
That’s me.
https://www.poetryfoundation.org/poems/48985/phenomenal-woman

What is the conflict in Lois Lowry's The Giver, and how is it resolved?

The major conflict of Lois Lowry's The Giver has to do with Jonas figuring out who or what is his antagonist. At first, it seems as though the ones who run the government are to blame for limiting the freedom of its citizens by controlling what they see, hear, and do with their lives. However, as Jonas trains with the Giver, he becomes aware of what really controls their lives--Sameness. Sameness is the antagonist; therefore, the conflict lies within it.
Sameness is the controlling factor in his life and his community. During the first memory that he receives in chapter 11, Jonas experiences sledding down a snowy hill. Afterward, he asks the Giver why they don't have snow in their lives. The Giver explains the following:

"Climate Control. Snow made growing food difficult, limited the agricultural periods. And unpredictable weather made transportation almost impossible at times. It wasn't a practical thing, so it became obsolete when we went to Sameness" (83-84).

Jonas soon learns that over past generations, the people of the community had made similar choices depending on what they were willing to sacrifice to achieve specific goals. For example, the ability for citizens to see colors was sacrificed long ago in order to take away personal preferences. If people don't prefer one person or object over another, then life without prejudices can be achieved. The consequence of a life without colors, though, is that people's abilities to choose are sacrificed. For instance, when Jonas sees and understands colors for the first time, he complains as follows:

"It isn't fair that nothing has color! . . . If everything's the same, then there aren't any choices! I want to wake up in the morning and decide things! A blue tunic, or a red one?" (97).

If limited freedom to choose isn't enough to cause Jonas to revolt against Sameness, the fact that "release" means to kill someone does. Jonas learns that Sameness also requires that the rate of births is controlled in order to avoid hunger and starvation. As a result, Nurturers like Jonas's father have an obligation to kill the smaller of a pair of twins to keep population numbers down. In chapter 19, the Giver shows Jonas a recording of his father injecting a lethal drug into a twin's head. Jonas is devastated. It is after this that he and the Giver decide to overthrow Sameness.
Because Sameness is maintained by a Receiver who stores all of the memories of the world inside of him, they decide to release these memories into the community. Once that happens, everyone in the community will be aware of the sacrifices that have been made to limit their freedoms and Sameness will end. Therefore, the conflict between Sameness and Jonas is resolved when he leaves the community, releases the memories into the community, and ends the controlling grip that Sameness has over the people.

Two cannabinoid receptors have been identified. What are they called, and where is each one generally located?

The two cannabinoid receptors are abbreviated CB1 and CB2. They are similar to each other in that they share roughly 50% of their DNA identity, but this greatly changes their tissue distribution and signaling mechanisms.
CB1 receptors are generally located on nerve cells in the brain and spinal cord. Specifically, the CB1 receptors can be found in the cerebellum, the basal ganglia, and the hippocampus. CB1 receptors are also present in other organs and tissues like the spleen, white blood cells, and intestinal tract.
CB2 receptors are found in white blood cells, in the tonsils, and in the spleen. CB1 receptors are there as well. However, the CB1 receptors are in much smaller quantities. An important function of the CB2 receptors is that they help regulate cytokine release.

How does Orwell describe his feelings about the British Empire and his role in it, both at the time he took part in the incident described and of writing the essay, after having had the opportunity to reflect upon these experiences?

Orwell certainly hates British imperialism at the time of the incident he describes in his essay. He writes,

For at that time I had already made up my mind that imperialism was an evil thing and the sooner I chucked up my job and got out of it the better.

He detests fulfilling the role of an imperial policeman, and he is disturbed and feels guilty about the way the locals jeer at Europeans and about the way prisoners are treated in Burma.
While he hates the empire, he is somewhat muddled and is perplexed about the whole picture of colonialism. He writes, "But I could get nothing into perspective." It is only after he looks back at the incident of shooting the elephant that he understands the rationale for crumbling empires (like the British in Burma) acting as they do. He realizes that he shoots the elephant only to avoid looking like a fool, and this corrupt motivation fuels the empire's feeble grasp on power. While he has always hated imperialism, he only comes to understand it after the incident he describes in the story takes place.

At the time Orwell (or, to be precise, the narrator) took part in the events described in "Shooting an Elephant," he describes himself as conflicted about the British Empire. On the one hand, he views the empire as fundamentally corrupt and morally wrong. On the other, he has little but contempt for the Burmese people themselves:

With one part of my mind I thought of the British Raj as an unbreakable tyranny, as something clamped down...upon the will of prostrate peoples; with another part I thought that the greatest joy in the world would be to drive a bayonet into a Buddhist priest's guts. 

Orwell goes on to say that this internal conflict is fairly typical of those in the imperial service, and this is underscored by his response to the incident with the elephant. Orwell does not really want to shoot the beast, which has calmed down after its rampage through the bazaar. But he feels that he has to, because the Burmese people expect it from him. He is expected to act with authority and with violence, and he realizes that, in his role as a representative of the Empire, he must live up to their expectations. So by the end of the story, he observes that he killed the elephant "solely to avoid looking a fool" in the eyes of the subjects of the British Empire. In retrospect, he recognizes that the essence of empire is violence.
http://orwell.ru/library/articles/elephant/english/e_eleph

At the time of the incident, Orwell describes his feelings about the British Empire in very negative terms. He talks about imperialism being an "evil thing," for example, and how running the empire is "dirty" work. He also refers to the British as "oppressors." To illustrate these points, he gives the example of Burmese prisoners, locked up in "wretched cages" and scarred from beatings.
Regarding his role in the British Empire, Orwell feels an "intolerable sense of guilt." Moreover, although he hates his job, he cannot simply give it up, which further adds to his sense of guilt and feelings of hatred towards the British Empire.
After reflecting on the incident, there is some change in Orwell's attitude. He is "glad" to have been on the side of the oppressors, if only because it gave him legal justification and pretext for killing the elephant. Moreover, from his tone in the final line, he seems amused as he wonders if anyone realized his true motive for killing the elephant: that he didn't want to look like a fool.

How has James Reeves used aural imagery in his poem "In the Sea"?

I think you mean the poem "The Sea," by James Reeves. Aural imagery is imagery which appeals to our sense of hearing, rather than our sense of sight—so, the use of figurative language to create a vivid sense of place in terms of how something sounds. In this poem, Reeves uses aural imagery frequently so that the reader can almost hear the scene being described. Onomatopoeia, such as "rumbling, tumbling," creates sound pictures for us; Reeves also imagines the words the "dog" of the sea might be saying, such as "Bones, bones, bones!" Obviously, this is a continuation of the central metaphor: the sea is not really a dog, nor is it really speaking, but we understand that its "moans" bear some resemblance to these human words. They help us to create a mental picture of how the sea sounds. Likewise, the "hollos" of the sea on blustery days are accompanied by the onomatopoeiac "snuffs" and "sniffs" of the dog to create a rich tapestry of sound.
Reeves uses aural imagery, too, to describe quietude. When the sea is silent, the contrast of the dog who "scarcely snores" is drawn starkly—the absence of sound is almost as illustrative as the existence of it.

Imagery is description that uses the five senses of sight, sound, taste, smell, and touch. An aural image is a sound image.
In his poem "In the Sea," Reeves compares the sea to a dog. The great gray dog of the sea is sometimes hungry, sometimes restless, and sometimes calm and at peace.
Reeves uses sound imagery to help describe all three states of the doglike sea. When it is hungry, it makes sounds such as "clashing" its teeth, which would be a way to describe the crashing of waves. The sea/dog also "moans."
In the second stanza, the sea at night is compared to a restless dog. The winds that "roar" over the sea at night are likened to a dog that "howls."
Finally, the "reedy" sound of the calm sea is compared to the sound of a dog at rest, "quiet, so quiet," hardly snoring.
Whether these sound images work successfully is up to the reader to decide, but it is interesting to think of the sea as like a dog—frisky, howling, and quiet by turns.

What was Reconciliation in Australia?

Reconciliation was (and continues to be) formal efforts made by the Australian government to affirm unity and respect between Native Australians- Aboriginal and Torres Strait Islanders- and the non-Indigenous Australians descended from colonizers or more recent waves of immigration. 
In the 1950s and 1960s, the roots of Reconciliation were planted as people (both Indigenous and non-Indigenous) began campaigning to remove clauses from the Australian Constitution which either deprived Indigenous people of their civil rights or used discriminatory language. Since then, Reconciliation has included justice for people who have been barred from entering public establishments or receiving service on the basis of their Indigenous ethnicity, granting pensions to elderly and infirm Indigenous Australians, and returning or maintaining land rights to people who had historical associations with parts of Australian territory. 
Reconciliation continues today as the Australian government seeks to remedy the institutions which have prevented Indigenous Australians from attaining an equal quality of life to those of non-Indigenous descent. Ensuring access to healthcare, nutrition, and education are some of these ongoing efforts. 

What different forms of power are conveyed in Death and the Maiden? How might Dorfman's message apply to other contexts (times and places)?

Thank you and could you please delete your answer thanks
 

Death and the Maiden examines the unstable power dynamics that are created during the shift from one political agenda and infrastructure to another. 
While Dorfman's play does not name its geographic setting, it is clearly influenced by Dorfman's experience as an exile from Chile after the coup led by General Augusto Pinochet, who assumed the role of a militaristic dictator of the country. The eventual replacement of Pinochet with a democratic government and the introduction of the Rettig Commission (which investigated crimes committed under Pinochet, but did not issue judgment or publicly name those who committed such crimes) failed to deal justly with the trauma of those who suffered through human rights violations under Pinochet's reign. This is because while those in support of the newly elected President Patricio Aylwin wanted to see those who committed atrocities under Pinochet held accountable, great political pressure still existed to deal softly with these perpetrators--many who were still firmly planted in high positions within the government and society.
Thus, the varying forms of power that are addressed within Dorfman's play emerged: the power of a victim in search of personal rather than public justice (Paulina), the power of criminals who have enacted injustice and managed to get away with it (Paulina's "possible" rapist, Roberto Miranda), and those who value the idealogical concept of "peace" over the aforementioned kind of reparative justice (Paulina's husband Gerardo).
As we watch Paulina seek to exact her vigilante justice on Miranda throughout the play while dealing with the denial and dissent of Gerardo, it is clear that these three power dynamics are unable to wholly coincide within the shifting landscape of the "new" country. These rivaling powers--which seem to leave no room for compromise--result in a horrific cycle of violence which continues the perpetration of crime and the violation of human life. 
Ultimately, it is clear that the themes that Dorfman is exploring here regarding unspeakable violence, forgiveness, and the conflict between past, present, and future can be applied to our modern world. Even today, we are faced with unresolved atrocities which seem to have left a wound that cannot be closed for many people; one example might be the Turkish government's refusal--even a century later--to acknowledge the genocide of 1.5 million Armenian people that it perpetrated beginning in 1915. In 2017, countless North Koreans are facing unfathomable human rights violations in prison camps under the regime of Kim Jung-Un. Contextually speaking, we are very much still immersed in a time and place in which victims go unrepresented by their governments out of sheer political maneuvering. The United States, for example, has failed to acknowledge the Armenian genocide due to a fear of losing the Republic of Turkey as a key regional ally in the conflict in the Middle East. The US has also maintained a "hands off" approach with North Korea as a matter of political diplomacy, fear of nuclear repercussions, etc. 
https://www.armenian-genocide.org/genocide.html

https://www.hrw.org/asia/north-korea

How does Odysseus's quote, “I am a man no more no less,” show how he has developed over the course of the hero’s journey?

After the years of suffering and trials he has gone through, Odysseus is reflecting on his own triumphs and failures and on the generally imperfect life that he, or any man, inevitably leads. In some sense, Odysseus blames himself for the length of time it has taken him—ten years—to return to Ithaca. This also needs to be seen in the context of not having wanted to leave Penelope and his infant son, Telemachus, in the first place to go to war. He spent years living with another woman, namely, the goddess Calypso, and he became the lover of the sorceress Circe all while still being in love with Penelope and wanting to return to her and his son. He must also feel that some of his decisions during the return journey backfired, such as the blinding of the cyclops Polyphemus and, as a result, making an enemy of the cyclops's father, Poseidon.
After twenty years of both the war and the return journey, Odysseus has experienced literally everything that the gods and general misfortune could inflict upon him. He has learned more about himself, the world of men and women, and the gods than even the other participants in the war, both the Greeks and Trojans, have been able to learn. These experiences have developed his inner psyche to the point that the statement about being a man reflects the turmoil, contradictions, and imperfections of the human condition in the intensified form in which he has seen them.

What are quotes in Lord of the Flies that show Simon represents morality and Ralph represents civilized behavior?

Ralph is the elected chief of the boys and is a proponent of civilization. He desperately attempts to establish a civil society on the island and creates a list of priorities that will increase the boys' chances of rescue. In chapter 5, Ralph holds an assembly to discuss the lack of civility on the island and encourage the boys to follow through with the directives and tasks they are given. Ralph attempts to establish order at the beginning of the assembly by saying,

We have lots of assemblies. Everybody enjoys speaking and being together. We decide things. But they don’t get done (Golding, 112).

When the boys proceed to interrupt during the assembly and speak out of order, Ralph is quick to chastise them by saying,

There’s too much talking out of turn . . . because we can’t have proper assemblies if you don’t stick to the rules (Golding, 128).

In chapter 11, Ralph, Piggy, and Samneric travel to the opposite side of the island in order to retrieve Piggy's glasses. After Piggy admonishes Jack and his tribe of savages for their behavior, Ralph once again demonstrates his civilized behavior by asking,

Which is better, law and rescue, or hunting and breaking things up? (Golding, 259).

Simon is depicted as a Christ figure who represents morality and supreme knowledge. There are numerous examples of Simon's benevolence and morality throughout the novel. Simon demonstrates his morality by helping the littluns retrieve fruit. Golding writes,

Then, amid the roar of bees in the afternoon sunlight, Simon found for them the fruit they could not reach, pulled off the choicest from up in the foliage, passed them back down to the endless, outstretched hands (78).

When Jack refuses to give Piggy a share of the meat, Simon once again reveals his morality by giving him a piece. Golding writes,

Simon, sitting between the twins and Piggy, wiped his mouth and shoved his piece of meat over the rocks to Piggy, who grabbed it (104).

In chapter 7, Simon also portrays his morality by volunteering to travel back through the dark forest at night in order to inform Piggy and the littluns that everyone is alive. Simon volunteers himself by saying,

I’ll go if you like. I don’t mind, honestly (Golding, 178).

What were the significant differences between Persian empire rule and Assyrian empire rule?

The Assyrian Empire did not tolerate any form of dissent from the people and mostly resolved such issues violently. On the other hand, the Persians ruled with tolerance and employed diplomacy in dealing with the populace. For instance, the Assyrians were known for burning and looting cities and exiling the inhabitants, whereas the Persian king, Cyrus the Great, instituted measures to stop his military from engaging in such wanton destruction.
The Assyrians did not extend self-rule to conquered people. They imposed their culture and religion on the society and made no attempts to honor local traditions. The Persians extended some level of autonomy to the conquered groups, and the people were known to enjoy religious and social freedoms.
Diplomatic strategies employed by the Persians enabled the empire to grow and maintain stability, while the Assyrian Empire crumbled due to its rigid and warlike nature, which created many enemies internally and externally.
http://brfencing.org/hist643/Assyria%20&%20Persia.html

In chapter 8 of Nectar in a Sieve, how does the tannery change the lives of the four women who met so long ago?

The tannery has indeed changed the lives of the four women. Although three of the women lament the changes, Kunthi is positively inclined towards the tannery. She is appreciative of the economic benefits it brings to the village. As an example, Kunthi tells Rukmani and Janaki that her two eldest sons already bring home more than a man's wages from working at the tannery.
Kunthi is also pleased that the tannery has apparently changed their dreary village into a growing town. She is simultaneously ecstatic that this change is accompanied by the appearance of droves of young men to the village. Kunthi delights in going into town, "where, with her good looks and provocative body, she could be sure of admiration, and more, from the young men."
Meanwhile, Janaki is less than enthused with Kunthi's attitude. She frets that her husband's shop is struggling to compete against the newer, bigger shops in town. Eventually, Janaki and her family have to leave the village in order to search for new opportunities elsewhere. Although Rukmani is sad about this new development, she is forced (by circumstances) to refrain from dwelling on her feelings.
Rukmani and her husband are mainly small-scale farmers, and they must continue to survive even as the economic landscape in their village changes. Both Rukmani and Kali envy the Muslim wives of the men who work as tannery officials. These wives have servants, and they don't appear to have to work for their sustenance. Rukmani rejects such a life; despite the challenges inherent in farming, she would rather revel in the "open fields and the sky and the unfettered sight of the sun" than submit herself to a life of lonely seclusion.
From the above, we can see how the tannery has changed the lives of the four women. While Kunthi embraces the benefits the tannery brings, Janaki's life has been destroyed by the tannery. Her family has had to relocate and invent new opportunities in response to the changes. Meanwhile, Kali and Rukmani choose to stay, but their lives are made more difficult as a result of their decision.

In the following passage from Lord of The Flies, how does structure contribute to the atmosphere and theme? Ralph shuddered. The lagoon had protected them from the Pacific and for some reason only Jack had gone right down to the water on the other side. Now he saw the landsman's view of the swell and it seemed like the breathing of some stupendous creature. Slowly the waters sank among the rocks, revealing pink tables of granite, strange growths of coral, polyp, and weed. Down, down the waters went, whispering like the wind among the heads of the forest. There was one flat rock there, spread like a table, and the waters sucking down on the four weedy sides made them seem like cliffs. Then the sleeping leviathan breathed out—the waters rose, the weed streamed, and the water boiled over the table rock with a roar. There was no sense of the passage of waves, only this minute long fall and rise and fall.

This quotation is taken from the sixth chapter of William Golding's Lord of the Flies. Although the excerpt is narrated in the third person, we are looking at the scene through the eyes of Ralph. Because Ralph is among the most mature and civilized of the boys, he serves as a normative viewpoint, one whose perceptions are reliable and with whom the audience is likely to empathize.
At this point in the story, Samneric have been terrified by their sighting of "the beast," which is actually the corpse of a dead paratrooper. The boys are searching for the beast on the only part of the island they have not yet explored, a sort of peninsula connected to the rest of the island by a narrow natural bridge.     
At this point in the story, the boys are facing three obstacles. The first is nature itself, from which they need shelter. The second is the external "beast." The third is the "beast" within themselves. The water for Ralph seems to embody or foreshadow all three of these thematic antagonists. Structurally, the image serves a function of transition and foreshadowing; the "slumbering leviathan" suggests that the menaces that Ralph perceives have not yet fully awoken but, like the tide, are about to unleash their forces on the boys.
The table-like rock suggests something stable and civilized that will be drowned by the primal and powerful force of the ocean. The force of the tide and phrases such as "strange growths" also suggest something primeval and terrifying, contributing to the atmosphere of horror that is developing in the novel. Thematically, this also emphasizes that nature, both physical and human, is seen in the story as potentially immensely powerful and destructive when not tamed by civilization. 

How does Ray Bradbury keep Fahrenheit 451 from being pessimistic?

Ray Bradbury prevents his narrative from becoming too pessimistic because there is a subtle mood of hope generated through characters and incidents in the plot.
In other dystopian works, such as George Orwell's 1984 and Aldous Huxley's Brave New World, the protagonists feel a terrible sense of aloneness and alienation. But, throughout the narrative of Fahrenheit 451, Montag can communicate with such people as Clarisse, professor Faber, and later Granger and the others who memorize books. Because there are those with whom he can share ideas and communicate, Montag does not suffer as badly from alienation as do other characters in dystopian worlds. He also escapes from his society, whereas the main characters of the other works do not. One critic writes, "Montag’s conversion to reading is significant in that he suddenly finds himself in light rather than darkness."
Montag enters no terrible Room 101 as does Orwell's Winston, nor is he sent to desolate geographic areas as is Huxley's Bernard. Montag happily joins a community with people who are like him so he will find meaning in his life. These people are the light of the future, the "sea of faith" of which Bernard merely hopes to be a part.

How did the Russian Revolution and the German victory of 1917 influence the development of the Entente's grand strategy during World War 1?

The Russian Revolution made a profound change in the Entente's grand strategy. The Russian Revolution came at a time when French soldiers were revolting against brutal treatment in the trenches, and American soldiers were slow in coming to the help of the Allies. Russia did not immediately become a communist nation; rather, it went through a period under the provisional government where the Duma tried to maintain power. When the government finally passed into the hands of the liberal Alexander Kerensky, the Allies put pressure on Russia to keep its armies in the field by promising financial aid to the new government. The army under the Kerensky government actually enjoyed renewed patriotism and enjoyed some successes against the German and Austro-Hungarian armies. However, these gains were overturned when the Russian army overextended, and the German army regrouped. Russian soldiers began to mutiny in the field, and Russian sailors turned against their commanders; this served to undermine the Kerensky government and ultimately led to its downfall. After the Brest-Litovsk treaty with Germany in March 1918 when Russia signed over some of its most valuable territory in its western domains, the Allies quit making loans to the nation because they felt betrayed. The government under Lenin eventually repudiated the loans that were currently outstanding. Instead of launching a joint offensive in 1918 in order to hopefully put enough pressure on Germany to end the war, the Allied Powers on the Western front took a defensive posture against the German Spring Offensive in 1918.

Intermediate Algebra, Chapter 5, 5.1, Section 5.1, Problem 52

Write the expression $6^{-1}-4^{-1}$ with only positive exponents. Then, simplify the expression.
Remove the negative exponent by rewriting $6^{-1}$ as $\dfrac{1}{6}$. A negative exponent follows the rule of $a^{-n} = \dfrac{1}{a^n}$
$\dfrac{1}{6}-4^{-1}$
Remove the negative exponent by rewriting $4^{-1}$ as $\dfrac{1}{4}$. A negative exponent follows the rule of $a^{-n} = \dfrac{1}{a^n}$
$\dfrac{1}{6}-\dfrac{1}{4}$

To subtract fractions, the denominators must be equal. The denominators can be made equal by finding the least common denominator (LCD). In this case, the LCD is $12$. Next, multiply each fraction by a factor of $1$ that will create the LCD in each of the fractions.
$\dfrac{1}{6}\cdot\dfrac{2}{2}-\dfrac{1}{4}\cdot\dfrac{3}{3}$
Complete the multiplication to produce a denominator of $12$ in each expression.
$\dfrac{2}{12}-\dfrac{3}{12}$
Combine the numerators of all fractions that have common denominators.
$\dfrac{1}{12}(2-3)$
Subtract $2$ to $3$ to get $-1$
$\dfrac{-1}{12}$

Calculus: Early Transcendentals, Chapter 7, 7.1, Section 7.1, Problem 29

You need to use the substitution -2y = u , such that:
-2y= u => -2dy = du => dy= -(du)/(2)
Replacing the variable, yields:
int y*e^(-2y) dy = (1/4)int u*e^u du
You need to use the integration by parts such that:
int fdg = fg - int gdf
f = u => df = du
dg = e^u=> g = e^u
(1/4)int u*e^u du =(1/4)(u*e^u - int e^u du)
(1/4)int u*e^u du = (1/4)u*e^u - (1/4)e^u + c
Replacing back the variable, yields:
int y*e^(-2y) dy = (1/4)((-2y)*e^(-2y) - e^(-2y)) + c
Hence, evaluating the integral, using substitution, then integration by parts, yields int y*e^(-2y) dy = ((e^(-2y))/4)(-2y - 1) + c

Intermediate Algebra, Chapter 2, 2.7, Section 2.7, Problem 78

Solve the inequality $|0.5x - 3.5| + 0.2 \geq 0.6$.


$
\begin{equation}
\begin{aligned}

|0.5x - 3.5| + 0.2 \geq & 0.6 \\
|0.5x - 3.5| \geq & 0.4

\end{aligned}
\end{equation}
$


The absolute value inequality is rewritten as

$0.5x - 3.5 \geq 0.4$ or $0.5x - 3.5 \leq -0.4$,

because $0.5x - 3.5$ must represent a number that is more than $0.4$ units from on either side of the number line. We can solve the compound inequality.


$
\begin{equation}
\begin{aligned}

0.5x - 3.5 \geq & 0.4 && \text{or} &&& 0.5x - 3.5 \leq & -0.4
&&
\\
5x - 35 \geq & 4 && \text{or} &&& 5x - 35 \leq & -4
&& \text{Multiply each side by } 10
\\
5x \geq & 39 && \text{or} &&& 5x \leq & 31
&& \text{Add each side by } 35
\\
x \geq & \frac{39}{5} && \text{or} &&& x \leq & \frac{31}{5}
&&

\end{aligned}
\end{equation}
$


The solution set is $\displaystyle \left( \frac{31}{5}, \frac{39}{5} \right)$.