Single Variable Calculus, Chapter 3, 3.3, Section 3.3, Problem 101

How many lines through the point $(0,c)$ are normal lines to the parabola $y=x^2$ if $\displaystyle c > \frac{1}{2}$?
What if $\displaystyle \leq \frac{1}{2}$?

Recall that the slope of the normal line is equal to the negative reciprocal of the slope of the tangent line. So,


$
\begin{equation}
\begin{aligned}
m_T &= - \frac{1}{m_N}\\
m_T &= \frac{dy}{dx} = \frac{d}{dx} (x^2)\\
m_T &= 2x\\
m_N &= - \frac{1}{2x}
\end{aligned}
\end{equation}
$


We can get the equation of the normal lines by using the slopes formula at the point of tangency
at $(x,x^2)$ and at $(0,c)$ and equate it with the slope of the normal line. So...


$
\begin{equation}
\begin{aligned}
m_N &= \frac{y_2-y_1}{x_2-x_1}\\
\frac{-1}{2x} &= \frac{c-x^2}{0-x} && \text{(Applying cross multiplication)}\\
x &= 2xc - 2x^3\\
2x^3 - 2xc + x &= 0\\
x(2x^2-2c+1) &= 0
\end{aligned}
\end{equation}
$


Its either $x=0$ and $2x^2-2c+1=0$


$
\begin{equation}
\begin{aligned}
2x^2 - 2c + 1 &= 0\\
\frac{\cancel{2}x^2}{\cancel{2}} &= \frac{2c-1}{2}\\
\sqrt{x^2} &= \sqrt{c - \frac{1}{2}}\\
x &= \pm \sqrt{c-\frac{1}{2}}
\end{aligned}
\end{equation}
$


If $\displaystyle c > \frac{1}{2}$ let's say $c=2$, $\displaystyle x=\pm\sqrt{2 - \frac{1}{2}} \Longrightarrow
x = + \frac{\sqrt{6}}{2}$ and $\displaystyle x = \frac{-\sqrt{6}}{2}$

You will have 2 normal lines. However, if $\displaystyle c \leq \frac{1}{2}$, let's say $\displaystyle c=\frac{1}{2}$
and $\displaystyle c = -\frac{1}{2}$, $\displaystyle x = \sqrt{\frac{1}{2}-\frac{1}{2}} = 0$ and
$\displaystyle x = \sqrt{-\frac{1}{2}- \frac{1}{2}} = \sqrt{-\frac{1}{4}}$, there will be only 1 normal line since
square root of a negative value is undefined.

Therefore,

$
\begin{equation}
\begin{aligned}
& \text{if } c > \frac{1}{2}, \quad \text{3 normal lines} && (\text{including } x = 0 \text{ we've had here } x(2x^2-2c+1)=0 )\\
& \text{if } c \leq \frac{1}{2}, \quad \text{1 normal line}
\end{aligned}
\end{equation}
$

How does The Refugees by Viet Thanh Nguyen relate to immigration?

Immigration is a main theme in The Refugees. It is immigration that brings the refugees of the story to the United States. Most of the characters are either first- or second-generation immigrants, having left Vietnam because of the war. These characters learn that the journey to the United States is not easy, and many of them—like the narrator of “Black-Eyed Women”—arrive with little money and are forced to live in rough neighborhoods. Several characters, such as Phuong and Liem, must rely on others to help them meet the financial demands of immigrating. The immigration journey itself is often perilous or traumatic; the unnamed ghostwriter from the first story recalls how she watched pirates kill her brother just before they raped her. In addition to the difficult adjustments that come along with immigration to a new country, The Refugees also explores how immigration affects one’s concept of home at a generational level. Some of the second-generation immigrants feel more comfortable in the United States and, unlike their parents, identify primarily as Americans. By telling the stories of these refugees, Nguyen explores how the experience of immigration intersects with one’s personal identity. 

What is the difference between substances and mixtures?

Matter is anything with mass and volume. Matter is composed of particles. Particles are composed of atoms. Atoms are composed of protons, electrons, and neutrons.
There are two types of matter, pure substances and mixtures: 

Pure Substances: Pure substances are composed of identical particles. 

Mixtures: Mixtures are composed of two or more different particles.
There are two types of pure substances, elements and compounds:

Elements: The particles of elements are composed of identical atoms. Examples: magnesium (Mg), oxygen (~O_2)

Compounds: The particles of compounds are composed of two or more different atoms. Examples: calcium iodide (~CaI_2), sodium chloride (NaCl)
There are two types of mixtures, homogeneous mixtures and heterogeneous mixtures:

Homogeneous Mixtures: Homogeneous mixtures are composed of particles that are arranged evenly throughout the mixture. Examples: salt water (NaCl and ~H_2O ), air (~O_2 , ~N_2 , and ~CO_2 ) 

Heterogeneous Mixtures: Heterogeneous mixtures are composed of particles that are arranged unevenly throughout the mixture. (salads and dirt)
https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Physical_Properties_of_Matter/Solutions_and_Mixtures/Classification_of_Matter

Give a brief description of the Malay sequence in Opium Eater.

By the time of the Malay sequence, De Quincey has become hopelessly addicted to opium. Yet at the same time he's managed to reduce his dosage substantially, and proudly proclaims this year—1816—to be the happiest of his life.
One day, a Malayan sailor knocks at the door of De Quincey's cottage. He is a strange, exotic man in "dingy" pants. Not unreasonably, De Quincey wonders how and why a sailor from the other side of the world should suddenly turn up at the doorstep of a remote English country cottage. But he's unable to get any information out of his unusual visitor as the sailor doesn't speak a word of English.
Nonetheless, De Quincey proceeds to treat his unexpected guest to some unorthodox hospitality by giving him a piece of his opium stash. The sailor wolfs down the drugs in one go, leading De Quincey to fear that he might have an overdose. All seems well, however, as the Malay leaves the cottage without displaying any notable side-effects of his rash indulgence.
Whether the anecdote about the Malayan sailor is a true story, or just a drug-induced hallucination, is unclear. However, the incident has a profound effect upon De Quincey's fevered imagination, causing him to experience numerous terrifying nightmares.

How does Bradbury reveal another view of books and society through Faber, and how does this compare to Captain Beatty's view?

Upon their first meeting, Faber subtly reveals himself as a reader of books, and he tells Montag, “’I don’t talk things, sir . . . I talk the meaning of things. I sit here and I know I’m alive’” (75). He is introducing to Montag the idea that without books to present ideas for people to consider independently, society simply accepts at surface value the things pushed at them by the powers that be (aka the mass media). There is no depth of thought about anything, and people are forgetting what it means to be truly human. At their second meeting Faber explains to Montag the things we need as humans: “’Telling detail . . . Quality texture of information,’” “’leisure to digest [the information],’” and “’the right to carry out actions based on what we learn from the interaction of the first two’” (83-85). He makes it clear that books are necessary to fulfill these needs: “’The things you’re looking for, Montag, are in the world, but the only way the average chap will ever see ninety-nine percent of them is in a book’” (98).
Captain Beatty, on the other hand, has convinced himself that reading books turns one into an intellectual snob. He actually tried balking the system once, as Montag is doing, but he failed, and so he feels threatened by Montag’s fresh insight about all that books can mean to humanity. Beatty believes that by preventing anyone from becoming smarter than others through books, everyone is made equal: “’A book is a loaded gun in the house next door. Burn it. Who knows who might be the target of the well-read man?’” (60) He feels that no one has the right to make another feel inferior. Besides, the books aren’t trustworthy. He argues, “’What traitors books can be! You think they’re backing you up, and they turn on you’” (p. 107). He means that two people can often argue from opposite platforms using the same text for support. Perhaps this is what happened to Beatty when he attempted, so long ago, to free the books. But he isn’t considering what Faber already knows; when we are free to interpret the information in books independently, it should have unique meanings and insights for each reader. Our human experiences ensure it.
Oddly, you could say that Faber and Beatty agree on the basics: books can be used to gain knowledge (so the lack of them is dumbing society down), they allow us to think independently, and they are interpreted uniquely by each reader. But ultimately the differences in their views stem from the opinion of whether these things are beneficial or damaging to society.

Contrast the rationale that differed between European and U.S. imperialist actions.

When comparing US and European imperialism, it is notable that there are some similarities and some differences between the two.
When we look at European imperialism during the 1500s–1900s, it often involves the taking of a nation by military force. Countries like Britain, Spain, and France developed empires through this method. European nations utilized their military and technology superiority to conquer foreign lands. Once conquered, they would install a governor/governors to manage the territory in the name of the home country.
When we look at American imperialism, particularly in the more modern sense, it is a bit different. The US government generally doesn't just send in the military to conquer a nation and then install a governor to manage the territory. This strategy may have been more common in the past, such as the taking of Cuba and Puerto Rico during the Spanish-American War, or the taking of Hawaii, but more modern examples are less direct. The type of imperialism the US seems to engage in these days involves supporting and assisting leaders who the US favors over those that they don't. The United States will provide financial and military supports to leaders who support US policies on the global level or will provide beneficial deals for US corporations to operate inside their borders.
American imperialism may also involve the US government threatening other nations through economic means. The threat of cutting off trade with or raising tariffs on goods from other nations can be effective in getting them to cooperate with US goals.
When we look at motives for European and American imperialism, we will notice more similarities. In both cases one of the major goals of imperialism is not just the control of a territory, but access to the resources that territory holds. Gaining free or cheap access to these resources allows for extraction and usage in the home country. The added benefit of colonization is the utilization of the colony as a market for goods produced in the home country. So essentially the valuable resources are extracted, manufactured into finished products, and then sold back in the colonies for a profit. This strategy proved effective in gaining wealth for imperialist nations.
Another similarity between European and US imperialism would be the spreading of cultural values. In the age of European imperialism, one of the major goals was the spreading of Christianity. After invading and conquering a foreign nation, missionaries often arrived to spread Christian beliefs. The ethnocentric views of Europeans at this time felt that it was a duty to spread Christianity to the "uncivilized" people they had conquered. In a way they felt they were improving them by pushing European cultural and religious values on them.
This can also be seen in different ways through US imperialism. Although it may not be strictly based on spreading Christianity, the US does seem to push values of Western culture and American culture on foreign nations it aims to control. This can include the introduction of American forms of entertainment, dress, and other cultural aspects.
http://webs.bcp.org/sites/vcleary/modernworldhistorytextbook/imperialism/section_2/motives.html

How does recognizing your lenses assist in maintaining objectivity?

Recognizing your lens refers to the perspective from which a person views events. It is important to know the perspective from which a person views events. A person’s lens often contains some predetermined viewpoints or judgments. For example, a person who lives in a rural environment will see events that occur in the world differently than a person who lives in the city. A person’s experiences also impact how a person sees things. A person who has faced discrimination or harassment will see the world differently than a person who has not experienced these things. A person’s lens usually has some bias associated with it.
A person who has been bitten by a dog will very likely view dogs differently than a person who has never had this experience. The person who has been bitten might see a dog as a threat and might be fearful of any dog he or she encounters. Another person might see a dog as a friendly, playful animal because they have not had a negative experience with dogs.
By understanding that an individual’s perspective impacts how a person sees events that occur in the world, a person might be able to understand why people from different backgrounds can see the same event very differently. Recognizing that a person’s lens contains some bias should encourage a person to try to filter out that bias in order to view an event more objectively.

How did the events ending World War I bring about World War II decades later?

Two events in particular at the end of World War I helped bring about World War II two decades later. Far and away the most important was the Versailles Treaty. The treaty forced the Germans to assume guilt for World War I. This humiliated and angered the German people, leading to the kind of discontent that allowed a demagogue like Adolph Hitler to rise to power. Second, the Germans were angered at having to pay heavy reparations for the war, mounting to more than two trillion dollars in today's money. Further, the treaty restricted the Germans to one hundred thousand army troops and forbade the Germans an air force. The armed services had been a chief way young men from humble roots could get ahead in Germany. An army career was respected and led to decent pay, so a man could marry, buy a home, and support a family if he chose that path. When this career option was cut off, many unemployed, alienated young men joined the right-wing freikorps that morphed into Hitler's brown shirts.
The formation of the Weimar Republic at the end of World War I also helped lead to the Second World War. The republic was devised hastily as a way to fend off a threatened communist takeover similar to the one that had recently occurred in Russia. However, large segments of the population did not support the idea of a republic, preferring a monarchy. The Republic also lost support as it was perceived as both decadent and weak. Many people wanted a strong absolute ruler and support for traditional values.
Both the resentment wrought by the Versailles Treaty and the weakness of the Weimar Republic gave Hitler his opening to seize power. Hitler's ascent to absolute power is key to the start of World War II: even in Germany, most people wanted to avoid another major war. Hitler, however, was determined toward conflict and conquest.

Explain the significance of Romeo referring to the Capulet's tomb as a"lighthouse."

In the play's closing scene, after Romeo has killed Count Paris outside the Capulet tomb, he reiterates his theme of honoring Juliet as the one who brought "light" to his life. Upon opening the tomb, Romeo says,

For here lies Juliet, and her beauty makesThis vault a feasting presence full of light.

It is, of course, a figurative light or brightness. Romeo is really referring to Juliet's beauty and presence as something that lights up a room. He has dwelled on this light throughout the play. When he first sees Juliet in act 1, scene 5 he says, "O, she doth teach the torches to burn bright!" and then follows up by saying she is a "snowy dove trooping with crows," emphasizing that she is bright in a world of darkness.


In act 2, scene 2, the famous balcony scene, Romeo compares Juliet to the "sun" and a "bright angel," further saying,


Two of the fairest stars in all the heaven,Having some business, do entreat her eyesTo twinkle in their spheres till they return.What if her eyes were there, they in her head?The brightness of her cheek would shame thosestarsAs daylight doth a lamp; her eye in heavenWould through the airy region stream so brightThat birds would sing and think it were not night.



Light and dark are recurring motifs throughout the play. The light is often a reference to Juliet's beauty. In an interesting juxtaposition in act 3, scene 5, however, Juliet argues that the nightingale, symbol of the night, is singing outside her bedroom window and Romeo responds by saying it is the lark, symbol of the day, which sings. In this case, the darkness is friendly to Romeo and Juliet and the light brings sadness because Romeo has been banished from Verona.

Thus, it is not surprising that Romeo should bring up the light which pervades Juliet in the final scene as he is about to commit suicide. In fact, Romeo observes that this light has not left Juliet, even in (supposed) death. He says,

Death, that hath sucked the honey of thy breath,Hath had no power yet upon thy beauty.Thou art not conquered. Beauty’s ensign yetIs crimson in thy lips and in thy cheeks,And death’s pale flag is not advancèd there.

Juliet's beauty then, is most certainly why Romeo would consider Capulet's tomb a lighthouse.

Are there any similarities between the characteristics demanded of an entrepreneur and those of a professional athlete? Would an athlete be a good prospect for entrepreneurship? Why or why not? Could teamwork be important in an entrepreneurial effort? Why or why not?

The expectations of an entrepreneur and a professional athlete are more similar than one might assume. Professional athletes are expected to put in long hours, stay committed, take care of their bodies, keep a professional reputation, and manage their personal brand. An entrepreneur must work unpredictable hours, stick with their goals, take care of themselves, network and maintain good relationships with vendors and customers, and promote their company. While their final goals may differ, the requirements along the way are similar.
Because of their dedication to themselves, their team, and their sport, athletes are excellent prospects for entrepreneurship. Working as a part of a team teaches one about being a follower and a leader, both of which are important when beginning a new business venture. An entrepreneur must be able to work with vendors, customers, employees, and a slew of other groups to successfully launch their company. Working as a part of a team gives an athlete experience working with teammates, managers, fans, and various other entities to bring together their brand.
https://www.entrepreneur.com/article/323418

There are actually some similarities between the characteristics demanded of an entrepreneur and those of a professional athlete.
Both the athlete and entrepreneur must be individuals who understand the value of perseverance, endurance, and motivation. On the field, the professional athlete must be physically fit, self-disciplined, and resilient. Similarly, in the world of commerce, the entrepreneur must exhibit these same characteristics. Competition in both sports and commerce is extremely high: both athletes and entrepreneurs must be willing to work harder than their competitors, and they must love what they do. 
Certainly, an athlete would be a good prospect for entrepreneurship because the characteristics that are necessary for success on the field are the same required in the field of commerce. Today, the most successful athletes are those who can retain their mental and emotional equilibrium in the face of setbacks and failures. Take Tom Brady, for instance. In February 2017, he guided the Patriots from a 25-point deficit to a decisive Superbowl victory. The secret to Tom's success is that he never gives up. 
The drive to succeed and the willingness to make sacrifices encapsulates the modus operandi of every successful athlete. Since these athletes also appreciate the meaning of teamwork, they are likely to make successful entrepreneurs. Below are two links that provide examples of professional athletes who went on to become famous entrepreneurs.
For example, Venus Williams (a world-renowned professional tennis player) has her own fashion line and is also the CEO of her own interior design firm. As if that is not enough, she has endorsement contracts with some of the largest companies in the world. Both Venus and her sister Serena are as successful on the tennis court as they are in the business world.
Venus has also written books about tennis and business. One of her most prominent books documenting how athletic success can be duplicated in the field of business is Come to Win: Business Leaders, Artists, Doctors, and Other Visionaries on How Sports Can Help You Top Your Profession.
Additionally, teamwork could be important in any entrepreneurial endeavor. An entrepreneur needs to be able to rely on his/her team to come up with effective solutions to problems. Also, a good team allows the entrepreneur to delegate responsibilities and projects with confidence. For more on how effective teamwork contributes to success, please refer to the link below.
https://www.businessinsider.com/professional-athletes-turned-entrepreneurs-2015-2

https://www.cnbc.com/2010/08/04/15-Pro-Athletes-Turned-Entrepreneurs.html?slide=2

Single Variable Calculus, Chapter 4, 4.5, Section 4.5, Problem 24

Use the guidelines of curve sketching to sketch the curve. $\displaystyle y = x\sqrt{2-x^2}$

The guidelines of Curve Sketching
A. Domain.
We know that $f(x)$ contains square root function that is defined only for positive values. Hence,

$
\begin{equation}
\begin{aligned}
2 - x^2 &\leq 0 \\
x^2 &\geq 2
\end{aligned}
\end{equation}
$

We have,
$-\sqrt{2} \leq x \leq \sqrt{2}$
Therefore, the domain is $[-\sqrt{2},\sqrt{2}]$


B. Intercepts.
Solving for $y$-intercept, when $x= 0 $
$ y = 0 \sqrt{2-0^2} = 0$
Solving for $x$-intercept, when $y = 0$
$0 = x\sqrt{2-x^2}$
We have, $x = 0$ and $ x = -\sqrt{2}$ and $x = \sqrt{2}$

C. Symmetry.
Since $f(-x) = -f(x)$, the function is symmetric to origin.

D. Asymptotes.
The function has no asymptotes.

E. Intervals of Increase or Decrease.
If we take the derivative of $f(x)$, by using Chain Rule and Product Rule.

$
\begin{equation}
\begin{aligned}
f'(x) &= x \left( \frac{1}{2}(2-x^2)^{\frac{-1}{2}} (-2x) \right) + 1 (2-x^2)^{\frac{1}{2}}\\
\\
f'(x) &= \frac{-x^2}{(2-x^2)^{\frac{1}{2}}} + (2-x^2)^{\frac{1}{2}} = \frac{-x^2+2-x^2}{(2-x^2)^{\frac{1}{2}}}\\
\\
f'(x) &= \frac{2-2x^2}{\sqrt{2-x^2}}
\end{aligned}
\end{equation}
$

when $f'(x) = 0$

$
\begin{equation}
\begin{aligned}
0 & = \frac{2-2x^2}{\sqrt{2-x^2}}\\
\\
0 &= 2- 2x^2\\
\\
x^2 &= 1
\end{aligned}
\end{equation}
$

The critical numbers are $x = 1 $ and $x = -1$

Hence, the intervals of increase and decrease are

$
\begin{array}{|c|c|c|}
\hline\\
\text{Interval} & f'(x) & f\\
\hline\\
-\sqrt{2} < x < -1 & - & \text{decreasing on } (-\sqrt{2}, -1)\\
\hline\\
-1 < x < 1 & + & \text{increasing on } (-1,1)\\
\hline\\
1 < x < \sqrt{2} & - & \text{decreasing on } (1,\sqrt{2})\\
\hline
\end{array}
$


F. Local Maximum and Minimum Values.
since $f'(x)$ changes from negative to positive at $x = -1$, $f(-1) = -1$ is a local minimum. On the other hand, since $f'(x)$ changes from positive to negative at $x =1,$ $f(1) = 1$ is a local maximum.

G. Concavity and Points of Inflection.

$
\begin{equation}
\begin{aligned}
\text{if } f'(x) &= \frac{2-2x^2}{\sqrt{2-x^2}} \quad \text{, then by using Quotient Rule and Chain Rule.}\\
\\
f''(x) &= \frac{\sqrt{2-x^2}(-4x)-(2-2x^2)\left( \frac{1}{2}(2-x^2)(-2x) \right)}{(\sqrt{2-x^2})^2}\\
\\
f''(x) &= \frac{x\cancel{(2-x^2)}\left[ -4(2-x^2)^{\frac{-1}{2}} + (2-2x^2)\right]}{\cancel{(2-x^2)}}\\
\\
f''(x) &= \frac{-4x}{\sqrt{2-x^2}} + x(2-x^2)
\end{aligned}
\end{equation}
$


when $f''(x) = 0$,
$\displaystyle 0 = \frac{-4x}{\sqrt{2-x^2}} + x(2-2x^2)$
we have $x= 0 $ as inflection point.
Thus, the concavity can be determined by dividing the interval to...

$
\begin{array}{|c|c|c|}
\hline\\
\text{Interval} & f''(x) & \text{Concavity}\\
\hline\\
-\sqrt{2} < x < 0 & - & \text{Downward}\\
\hline\\
0 < x < \sqrt{2} & + & \text{Upward}\\
\hline
\end{array}
$


H. Sketch the Graph.

Of the reforms that took place in America during the nineteenth century, which one was most important? What issues did the reform raise, and how did it affect American society?

This is a big question, and also a matter of opinion. I would argue, though, that the biggest reform of the nineteenth century was the abolition of slavery with the Thirteenth Amendment. More than anything else, the emancipation of an entire class of enslaved people did more to change American society than anything else at the time.
Ever since the founding of the United States, the country had been deeply divided over the issue of slavery. Tensions finally came to a head in 1861 when the country descended into a bloody civil war over the issue. Even with a Northern victory, emancipation was not a certainty until the passage of the Thirteenth Amendment.
About four million former slaves received free legal status. Politically, this changed the way Southern states would be represented in Congress and in the electoral college. The Constitution had previously counted a slave as three-fifths of a person for the purpose of population-based representation. As free people, they now were counted equally with others. This dramatically changed the number of representatives awarded to the South after the 1870 census. Northern Republicans were concerned that this would be damaging to their party. As a result, Republicans began to heavily court the favor of former slaves. One thing they did was set up the federal Freedman's Bureau to help former slaves adjust to a life of liberty.
The effect on society could be most deeply felt in the South. Many large plantations lost a significant amount of their labor force as emancipated slaves left for other opportunities. However, the old plantation style of farming did not end with emancipation; it merely adapted. Sharecropping became a widespread form of employment for black people in the South. In many ways, this system of using tenant farmers recreated the conditions of slavery by keeping the black tenants bound to the land.
Furthermore, racism did not end with the Thirteenth Amendment. In some ways, it only intensified. "Black Codes," laws designed to criminalize many aspects of black life, abounded in the South. These state-level laws applied exclusively to the African American population and made it easy for local law enforcement to circumvent the Thirteenth Amendment in order to persecute African Americans, imprison them, and subject them to forced labor.
So, while the reforms brought about by the emancipation of slavery had limited immediate effects, I would argue that it was still one of the most significant reforms of the era. The following two Constitutional amendments and several other federal reforms sought to empower a large population of former slaves. The hope among many reformers was that African Americans would be able to participate fully in American society. The pushback they received was massive, and more than a century would follow in which freed slaves and their descendants would struggle to achieve the full freedoms that these earlier reforms promised.
https://books.google.com.ni/books?id=VMCXjRyyKTQC&pg=PA46&dq=thirteenth+amendment+%22three+fifths%22&redir_esc=y&hl=es-419

https://www.archives.gov/historical-docs/13th-amendment

How did religion influence the development of the New England colonies?

Religion was an essential element in the foundation and development of the New England colonies. Many of the early colonists were "Dissenters," a term used to describe people in England who were not members of the Church of England. Through the mid-nineteenth century, Dissenters, including both Protestants and Roman Catholics, who "dissented" from the 39 Articles of the Church of England, were discriminated against, and could not enter Oxford and Cambridge University or hold certain political offices. Many moved to the colonies in search of religious freedom.
Many of the colonies in New England were settled by Puritans, who wanted to create devout religious communities. This led to such early excesses as the Salem Witch Trials. On the other hand, Roger Williams and the early colonies in Rhode Island advocated greater religious freedom and separation of church and state. Many of the people who moved to the United States were members of smaller or minority religious groups. This meant that there were many small, tightly knit communities in New England, leading towards what eventually became a philosophy of federalism.
The eventual evolution of the United States into a country with a "wall of separation" between church and state was due to two different impulses in these early colonies, one of tightly knot religious communities wanting strong local control to exercise their beliefs and one of people who wanted a more secular society with individual religious freedom.

Religion played the key role in the settlement of the first three communities of New England: Plymouth, Massachusetts Bay, and Rhode Island.
Although New England had far less fecund farmland than in the South, its population grew steadily. In 1700, New England's population was larger than that of Virginia and Maryland.
The first of the three important settlements in New England was that of the Pilgrims in 1620. They thought the practices of the Church of England were too Catholic, and they wanted their own religious community. After spending a decade in Holland, the Pilgrims sailed aboard the Mayflower to Plymouth.
In 1630, John Winthrop led a second group on the Arbella to New England. The Puritans sought to "purify" the Church of England rather than break away from it. However, Winthrop was determined to create a disciplined community built on religious uniformity, and religious dissenters were not tolerated.
One religious dissenter, the charismatic and intelligent Roger Williams, challenged Winthrop's harsh rules and beliefs. After he was banished to England, Williams founded a third colony, which would become Rhode Island. Williams supported religious freedom and fair treatment of Native Americans.

The Massachusetts Bay Colony was founded by Puritans, a religious group looking to purify the Church of England. To the Puritans, reading the Bible was important, and, as a result, they focused on literacy and education. They founded grammar schools and the first university in the United States, Harvard, in 1636, as a means of educating clergymen. The Puritan towns were tight-knit and formed around churches as a way of reinforcing their beliefs. Puritans mostly settled in family groups, and their towns were comprised of well-ordered, church-going populations with families that exercised strong control over their children.
The colony of Rhode Island was founded in reaction to the Puritans. The founder of Rhode Island, Roger Williams, had been banished from the Massachusetts Bay Colony. Instead of advocating the practice of Puritanism, he advocated a form of religious tolerance. In 1763, a Jewish synagogue was founded in Newport, Rhode Island.

Religion was quite important to the New England colonies. The church was the main part of a town and it also served as a meeting house. People lived in close-knit farming communities for safety but also to ensure that one could get to church on Sundays. Since the Puritans practiced a form of Christianity that believed God had chosen the "Elect" to go to heaven, and one showed that one was a member of this group by being successful, the Puritans worked hard at their occupations. This region was not dependent on slaves, though before the American Revolution some New England residents did own slaves. Children and hired hands did most of the farm labor. The sons married and created their own farming families. This led to the growth of the New England colonies and westward expansion. Religious dissent was not tolerated in the Puritan religion; one group of dissenters left the Massachusetts colony and formed Rhode Island.

What is a polygon?

A polygon is a two-dimensional (plane) shape. A polygon has straight sides, thus a circle is not a polygon because it has curved sides.
A polygon has a minimum of 3 sides. A polygon must be closed; if the sides do not fully connect it cannot be a polygon. If the shape is open, it is not a polygon.
The number of sides that a polygon has relates to its name. For example: a triangle has 3 sides, quadrilateral 4 sides, pentagon 5 sides, etc.
A polygon that has all sides and angles the same is called a regular polygon. For example, a square is a regular polygon, or regular quadrilateral, because it has 4 equal sides and 4 right angles. Some triangles are regular polygons (equilateral triangles).

A polygon is a closed figure made up of straight line segments.
A triangle is an example of a polygon. It is made up of three straight line segments that meet at three points, or vertices (the point where the sides of a polygon meet), to form the closed shape.
Other examples include quadrilaterals (a 4 sided polygon, like a rectangle), pentagons (a 5 sided polygon), and hexagons (a 6 sided polygon).
Whereas a circle, , is not a polygon. A circle is not made up of straight line segments.
A polygon can be any shape, or be made up of any number of line segments or edges, as long as the figure or shape is closed, where all the edges meet to form a points or vertices.

A polygon is a closed figure made up of straight line segments.
grad A triangle is an example of a polygon. It is made up of three straight line segments that meet at three points, or vertices (the point where the sides of a polygon meet), to form the closed shape.
Other examples include quadrilaterals (a 4 sided polygon, like a rectangle), pentagons (a 5 sided polygon), and hexagons (a 6 sided polygon).
Whereas a circle, @, is not a polygon. A circle is not made up of straight line segments.
A polygon can be any shape, or be made up of any number of line segments or edges, as long as the figure or shape is closed, where all the edges meet to form a points or vertices.

A polygon is also considered "closed". As in, the letters "M" or "K" wouldn't be considered polygons. They aren't considered closed, unlike the letters "O" and "D". "O" and "D" have a specific inside and outside to them. One might be able to specify that "M" and "K" have an inside and outside, also. But, there isn't any "drawn" border to them already. Or, in an effort to bring some reality to this, an outline of your house could be considered a polygon, as long as your front door is closed on the drawing. Then, the figure would be closed. But, if the figure has an "open" from door, it is no longer a polygon.

What is a polygon? A lost parrot.
Seriously, there is not an agreed definition of a polygon, the primary difference being whether the interior is included or not. Polygon translates as many angles.
A typical definition of a polygon is a geometric shape created by line segments (by definition these are straight, so no "curves" are allowed). The segments meet at vertices such that:
(1) No line segments intersect, except at a vertex (sides are not self-intersecting)(2) At each vertex, exactly two line segments meet (no segment ends in a bare vertex)(3) No three vertices are collinear
See the attached pictures for examples of polygons and non-polygons.
Polygons can be convex, concave, or star. We typically study convex polygons. Here are some ways to determine if a polygon is convex or not:
(1) In a convex polygon, if you extend any of the sides, no points on these lines will be in the interior of the polygon.(2) If a convex polygon, you can choose any two points in the interior of the polygon, and every point on the segment defined by these points is in the interior of the polygon.
Regular polygons are equilateral (all sides the same length) and equiangular (all angles have the same measure).
http://mathworld.wolfram.com/Polygon.html

What is a summary of Helen Keller's The Story of My Life?

The Story of my Life by Helen Keller is an autobiography that recounts Helen’s experiences as she adjusts to the world as a blind and deaf person. Helen begins the story by describing her earliest memories of sights and sounds and her memory of contracting the illness that resulted in her deafness and blindness. Helen learned sign language after her illness, but she describes the isolation she felt from the world around her and the frustration she felt while trying to learn.
At the age of six, Helen’s life changes drastically when she is referred to a teacher who has had tremendous success educating blind and deaf children. Helen devotes the rest of the book to describing her experiences learning to read, write, and speak under the tutelage of her teacher, Anne Sullivan. She describes the sensory experiences Miss Sullivan encouraged that helped her first learn words, and then learn the meaning of words, and then gain a fuller understanding of their meaning in the world around her. Helen describes moments of insight that came over the course of her learning as she was able to connect her learning activities to her childhood memories of sights and sounds. By the end of the book, the author’s descriptions of past and present come together to give the story continuity and meaning.

The Story of My Life is Hellen Keller's autobiography. Stricken with illness early in life that left her blind and deaf, Keller claims to tell her story as part of an attempt to validate her experiences with her disability, and to make the disabled subject more intelligible to the public.
The book is split into three parts. The first part recalls Keller's initial despair, as she tried to navigate life and obtain an education without readily available tools to make blind people's everyday experience easier. She finds solace in reading and writing, learning to use language as her primary tool.
Part two is composed of various letters written to Keller's friends and family. In them, she reveals insecurities about her proficiency at verbal expression, preferring the written form after finding it difficult to get feedback with which she could measure her progress with spoken language. The section is organized such that in each successive letter, Keller's written proficiency is improved. It is as much an autobiographical work as a demonstration of the fruits of her industriousness.
The third and final part of the book reconstructs Keller's narrative using the observations of Keller's teacher, Anne Sullivan, who stayed alongside her for almost half a century. Sullivan gives an account of Anne's prodigious intellect and drive, tracing her improvement from the early days, in which she picked up on new vocabulary at an exponential rate, through her later adoption of the Braille system and acceptance into Radcliffe College at Harvard.

The Story of My Life is the autobiography of Helen Keller, written in 1903 while she was a student at Radcliffe College. She describes her childhood memories prior to losing her hearing and vision, focusing on her memories of speech and early love for language. Later in her childhood, Keller became increasingly frustrated with her difficulties with communication and lashed out when she felt her sign language was insufficient. Much of the book focuses on her experiences with her teacher, Anne Sullivan, who Keller often refers to as Teacher. Sullivan was a teacher at the Perkins Institute for the Blind and started teaching a seven-year-old Keller to communicate by spelling out words and how to read Braille. By the age of ten, Keller could communicate with Sullivan and read Braille fluently, and had even learned how to speak. The book details her attempts to continue developing language skills with her teacher, and describes her later success as an honors student at Radcliffe College.

Whose family won possession of the disputed land in the lawsuit?

At the beginning of the short story, Saki writes that Ulrich von Gradwitz's grandfather had won possession of the disputed territory and "wrestled it" from the illegal possession of Georg Znaeym's family. The disputed territory is described as being a narrow strip of land with little game. Despite the uselessness of the territory, the two families have been feuding for three generations, and the land dispute has become personal for Ulrich von Gradwitz. Ulrich detests Georg Znaeym, who illegally poaches in the disputed territory and refuses to acknowledge the Court's past decision. At the beginning of the story, Ulrich and his group of men are traveling through the narrow strip of land looking for Georg in hopes of murdering him. Ulrich ends up running into Georg in the middle of the forest and is about to murder him when a severe storm causes a beech tree to fall on both of them.

How would you describe Mary Maloney's behavior before and after the murder in "Lamb to the Slaughter"?

Before committing the murder, Mary behaves in the same way that she would on any other evening. She prepares the home for her husband's arrival by lighting the lamps and then preparing the glasses for their drinks. When Patrick arrives at the house, she removes his coat and fixes them both a drink.
After the murder, Mary washes her hands, "fixes" her make-up and puts the lamb in the oven before going to the grocery store. There, she buys some vegetables before coming home and reporting her husband's death to the police. When the police arrive, she answers some questions while they look around for the murder weapon. In the middle of the search, she feeds them the leg of lamb she used to murder Patrick.
What is significant about Mary's behavior before and after the murder is that she never deviates from her domestic role. Before the murder, she focuses on the well-being of her husband. After murdering him, she continues as normal, buying items from the store and then feeding the police officers at her home. It is, perhaps, this continuity in her behavior that allows her to get away with her crime.

Pick 3 voter behaviors and theorize (speculate) why voters of those groups have traditionally voted for Democrats or Republicans. What values do you share that would move you to pick one party over the other? Explain.

There are many different types of voter behaviors and patterns. For example, African Americans have traditionally voted Democrat, as have members of unions (though this was not necessarily true of unions in the last election). These groups traditionally find Democrats more sympathetic to their cause and think that the government can help their interests. Veterans have tended to vote Republican, as they see Republicans as more supportive of the military and of veterans. Voters who are religious tend to vote Republican, in part because they feel that Republicans are pro-life, while Democrats are pro-choice. Men tend to vote Republican, while women tend to vote Democrat. Finally, voters who are more highly educated tend to vote Democrat.
You can think about how your identity would sway you to vote for one party or the other. For example, if you have a college degree (or are in the process of earning one), would this make you more likely to vote for a Democrat? If you identify as a man, would you be more likely to vote Republican? If you identify as a woman, would that make you more likely to vote Democrat? People who are transgender also tend to vote Democrat.

Why does Helen Keller say that the happy days of her life did not last for long?

Helen says that the happy days did not last because she became very ill and almost died, and was left blind and deaf.
The happy days that Keller speaks of are the early days of her childhood. She lived with her mother and father in a little house on her family land. As a toddler, little Helen was thrilled by the simplest things. Life was great. Then she became very ill. 

Then, in the dreary month of February, came the illness which closed my eyes and ears and plunged me into the unconsciousness of a new-born baby. They called it acute congestion of the stomach and brain. The doctor thought I could not live. (Ch. 1) 

The doctors did not know exactly what she had, so they did not know if she would survive it or what it would do to her.  She ended up both blind and deaf. This was a big blow at such a young age. Because she was barely walking or talking, she did not really have much communication and was plunged into a scary world of darkness.

I especially remember the tenderness with which my mother tried to soothe me in my waking hours of fret and pain, and the agony and bewilderment with which I awoke after a tossing half sleep, and turned my eyes, so dry and hot, to the wall, away from the once-loved light, which came to me dim and yet more dim each day. (Ch. 1) 

Since she was so young, losing her sight and hearing was very frightening. There was little her mother could do to help make her feel better or explain what was happening, because she was so sick. Poor Helen was in a great deal of pain in addition to losing more and more of her sight and hearing each day.

In "The Lottery," what allusions, symbols, or irony does Jackson show from the names Delacroix, Graves, Summer, and Bentham?

Dickie Delacroix is mentioned quite early in the story. He and two other schoolboys have "made a great pile of stones in one corner of the square and guarded it against the raids of the other boys." He is, clearly, very excited about stoning an innocent person to death later, and it even feels like a game to him and the other boys, just like so many people who had no compunctions about crucifying Christ and attended the crucifixion as though it were some kind of spectator sport.
Mr. Graves, the postmaster, is quite intimately connected with the lottery's proceedings. He carries the stool on which Mr. Summers places the box. The pair of them, together, had "made up the slips of paper and put them in the box," and, between lotteries, the box is sometimes kept in Mr. Graves's barn or even the post office where he works. He is one of the people Mr. Summers talks to "interminably" just prior to the lottery. Mr. Graves even helps with the lottery once it is discovered that Bill Hutchinson has drawn the marked slip of paper. In other words, Mr. Summers and Mr. Graves are the two people most involved in running the lottery, and their names seem to symbolize how the lottery might seem to readers versus how the lottery really is to the characters, respectively. Mr. Summers's name seems so light, just like calling this tradition "a lottery" as though it is something one would want to win, and holding it in the actual summer when the kids are happy to be out of school and formality seems to break down due to the heat. Certainly, his name seems ironic; we don't expect a summer season "lottery" to involve a person's violent death. Mr. Graves's name, on the other hand, calls attention to the deadliness of the tradition, even how serious and sober—or "grave"—a tradition it is. The people are somewhat grave on that day because they know how serious it is.
Bentham is only mentioned once, and very briefly as Mr. Summers conducts the lottery, calling out names one by one. The name probably refers to Jeremy Bentham, the philosopher who founded the concept of utilitarianism; he believed that an action is utilitarian when it benefits more people than it harms. By this definition, then, the lottery could be seen as utilitarian because it brings pleasure to all the people who are not chosen and displeasure to a few. However, it certainly seems cruel to the one chosen in the lottery, and this revelation, perhaps, ought to encourage us to consider whether such a trade is really ethical.

Delacroix in French literally means "of the cross." It's also the name of a famous French Romantic painter. In the story, there is a family called Delacroix, and by using this name, Jackson is possibly referring to the distorted nature of the village's ritual sacrifice. Jesus Christ, God's only begotten son, willingly sacrificed himself on the cross to save humankind. Yet the sacrifice of Tessie Hutchinson, in which Mrs. Delacroix plays such an enthusiastic part, is completely unwilling.
Mr. Graves, as postmaster, wields enormous power over the village, controlling its communications with the outside world. As such, he is the keeper of the flame, as it were, the man who ensures the survival of this twisted, diabolical custom. His name is significant because a grave is precisely where the "winner" of the lottery will end up.
Mr. Summers's name is somewhat ironic as there's nothing remotely summery about the grotesque ritual over which he presides. Nevertheless, the annual sacrifice does take place each summer, and so his name does reflect how closely involved he is with the lottery.
The name Bentham could possibly refer to Jeremy Bentham, an English philosopher of the late eighteenth and early nineteenth centuries. He devised a moral and philosophical system called utilitarianism, which held that moral actions should be judged not in themselves or by their motivations but by whether they produce the greatest good for the greatest number of people. The lottery in the story could be seen as an example of this. The stoning to death of Tessie Hutchinson in itself is a wicked act, but the villagers involved in her ritual murder genuinely believe that it is all for the greater good.

In Salinger's The Catcher in the Rye, what hard choice does Holden Caulfield have to make, and how does it affect him and others?

The hard choice that Holden Caulfield is faced with in the novel concerns his decision to run away from home in order to avoid entering the competitive world of adults or remain with his family to receive much-needed psychiatric help. Shortly after flunking out of his third prep school, Holden decides that he will sneak back home, visit his younger sister, Phoebe, and run away to somewhere in the country, where he can be by himself and avoid "phonies." Holden is depicted as a traumatized, neurotic teenager, who is struggling with the transformation from adolescence to adulthood. Holden desperately wishes to avoid entering the world of competitive, phony adults and wants to remain young. During an enlightening conversation with Phoebe, Holden mentions that he would like to become a catcher in the rye to prevent children from growing up and becoming adults.
If Holden decides to run away, his family would be devastated by the loss of their second son and more than likely inform the authorities, which would further complicate Holden's relationship with them. Phoebe and D. B. would also be extremely concerned about their brother if he were to run away. As a naive teenager, Holden would endanger his life and well-being by completely abandoning his support base in favor of living in the wilderness alone. Eventually, Holden is influenced by his sister and decides to not run away from home. He is then taken to a mental hospital or sanatorium in California, where he is psychologically evaluated. It is uncertain if Holden ever recovers from his mental instability, but there is hope that he will receive the treatment he needs in California.

In Salinger's The Catcher in the Rye, the hard choice that Holden has to make is whether to grow up and face adulthood or not. This also includes the choice to face his parents after being dismissed from his third private high school or to run away. If Holden stays to face the consequences of his actions, then his parents may be able to get him the help he needs to face adulthood. If, however, Holden chooses to run away, he will only perpetuate his fear of adulthood and possibly keep running for the rest of his life. In addition, Holden and his family members will all experience the consequences of whatever he decides. For example, his parents and siblings will mourn the loss of another brother if Holden disappears from their lives, especially after experiencing Allie's death a few years previously.
The one who might suffer the most by Holden running away is his little sister, Phoebe. In fact, she packs a suitcase with plans to run away with him! Once Holden realizes that Phoebe won't go home without him, he changes his mind about running away and says the following:

"'I'm not going away anywhere. I changed my mind. So stop crying and shut up,' I said. The funny part was, she wasn't even crying when I said that. I said it anyway, though. 'C'mon, now. I'll walk you back to school. C'mon, now. You'll be late'" (207).

The above passage shows Holden actually making the decision to stay for Phoebe's sake. He makes a very adult-like decision by placing his own desires and fears aside for his little sister. This is the first step that Holden takes towards adulthood. As a result, Phoebe stops crying and Holden takes her to the zoo and for a carousel ride. Eventually, Holden receives the help he needs by checking into a hospital near his older brother in California, and the family doesn't have to suffer a life of estrangement from Holden.

College Algebra, Chapter 4, 4.5, Section 4.5, Problem 52

Determine all the zeros of the polynomial $P(x) = 2x^3 - 8x^2 + 9x - 9$.
Based from the theorem, the possible rational are the factors of $9$ divided by the factors of the leading coefficient 2 which are $\displaystyle \pm \frac{1}{1}, \pm \frac{3}{1}, \pm \frac{9}{1}, \pm \frac{1}{2}, \pm \frac{3}{2} \text{ and } \pm \frac{9}{2}$. Thus, by using synthetic division and trial and error,


Hence,

$
\begin{equation}
\begin{aligned}
x &= \frac{-(-2) \pm \sqrt{(-2)^2-4(2)(3)}}{2(2)}\\
\\
&= \frac{2\pm\sqrt{-20}}{4}\\
\\
&= \frac{2\pm 2\sqrt{-5}}{4}\\
\\
&= \frac{1 \pm \sqrt{5}i}{2}
\end{aligned}
\end{equation}
$


To find the complex roots, we use quadratic formula.

$
\begin{equation}
\begin{aligned}
x &= \frac{-2 \pm \sqrt{2^2 - 4(1)(3)}}{2(1)}\\
\\
&= \frac{-2 \pm \sqrt{-8}}{2} = \frac{-2\pm2\sqrt{-2}}{2} = -1\pm\sqrt{2}i
\end{aligned}
\end{equation}
$

Thus, the zeros of $P$ are $\displaystyle 3, \frac{1 + \sqrt{5}i}{2} \text{ and } \frac{1 - \sqrt{5}i}{2}$

What does holism, as an ethnographic practice, permit the anthropologist to achieve; what are the weaknesses and strengths of such an approach?

Holism, first developed by Jan Christiaan Smuts (who was also the second Prime Minister of South Africa), is the idea that all systems should be viewed together rather than as separate parts. Just as physical systems are connected in the universe, so do human systems need to be examined together. 
As applied to anthropology or ethnography, holism means that cultures can best be understood by looking at the whole culture, not just its parts. Holism in anthropology usually involves taking a "four-field" approach that includes linguistics, physical anthropology, archeology, and cultural anthropology or social anthropology. All approaches should be used to understand a person or people.
The advantages of holism is that an anthropologist can get a broader view of a people or social structure in a more complex way that integrates different types of approaches. Cultures are complex, and this approach appreciates their complexity. The disadvantages are that such an approach may not be as rigorously scientific as other approaches because holism involves an element of subjectivity or judgment. In addition, holism results in observations that are at times uncertain and not conclusive, as holism produces theories. 

dy/dx = xcos(x^2) Use integration to find a general solution to this differential equation.

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

When was Orwell's 1984 written?

George Orwell published 1984 in 1949, meaning that, though the year 1984 is now in the past for contemporary readers, it represented the not-so-distant future for Orwell. As such, the year becomes the perfect period in which to set a dystopia with eerie similarities to totalitarian regimes of the twentieth century. For instance, many similarities could be drawn between the party of Big Brother and the recently defeated Nazis. Likewise, Orwell's vision can in many ways be seen as a commentary on established totalitarian regimes of his day, such as Stalin's Russia. Thus, Orwell is imagining a near-future in which the totalitarian regimes of the middle twentieth century have grown into all-powerful entities capable of totally dominating human life across the globe. It's a chilling vision, especially since its relatively contemporary setting makes it impossible to avoid drawing parallels between Orwell's dystopia and our own society. 

What are three things Puritans wanted to reform in the Church of England?

The list of things the Puritans wanted to reform in the Church of England is long, but here are three important ones:
Christian iconography in the church itself—This means that Puritans objected to stained glass windows, ornaments, religious paintings, and sculptures. Puritans held their services in simple, plain wooden meetinghouses with simple wooden pews for worshipers.
Puritans did not like musical accompaniment in sacred music. Instead of using organs or harpsichords, Puritans preferred to sing psalms and hymns in unison.
The Puritans did not want their important rituals to resemble those practiced in Catholicism, many of which the Anglicans retained. Puritans did not exchange wedding rings in marriage ceremonies or kneel for Communion. A Puritan baptism did not include the minister making a sign of the cross.

John F. Kennedy: Evaluate John F. Kennedy’s presidency. Be sure to discuss his successes and failures as president in both domestic and foreign affairs, as well as his legacy for American history.

Perhaps John F. Kennedy's greatest legacy is the optimism he brought to the Oval Office, epitomized in his famous 1961 inaugural address in which he stated, "Ask not what your country can do for you; ask what you can do for your country." His wife, Jacqueline Kennedy (later Onassis), also added a great deal of glamour and culture to the life of the White House. 
You should sort through the information about his presidency to decide what you think are his successes and failures, but, while the topic is still debated, some historians believe that Kennedy enflamed the Cold War. He increased the number of American advisors sent to Vietnam, which would become a full-fledged war under his successor, Lyndon B. Johnson. He also brought the world to the brink of a nuclear showdown during the Cuban Missile Crisis in 1962 and mounted a failed operation to topple Fidel Castro during the Bay of Pigs operation in 1961.
Domestically, he supported a progressive set of policies he called the New Frontier. Though long on idealism, few of his policies became a reality. However, he did start the Peace Corps, and he supported the Civil Rights movement and rights for people, like his sister Rosemary, who had intellectual disabilities. Many of his policies and programs would have to wait until after his death to become realities.
Kennedy's assassination in 1963 made him a martyr, and Americans are still obsessed with the death of such a young and vibrant leader. Many people felt that his death marked the end of an era of optimism, as the concluding years of the 1960s ushered in the Vietnam War and a great deal of domestic and international conflict.
https://www.jfklibrary.org/learn/about-jfk/jfk-in-history

How did the Russians eventually come in contact with the Byzantine Empire?

In the ninth century CE, Muslim travelers met Scandinavian Rus merchants in the middle Volga region. According to their testimonies, the Khazars, who controlled the steppe in the lower Volga region, were fighting the Rus vikings in an effort to prevent them from reaching the Caspian Sea. In the second half of the ninth century, these Scandinavian Viking warriors and merchants succeeded in subjugating, taxing, and partially incorporating the Eastern European Slavic population into their new state of Kievan Rus in the upper Dnieper basin, with its center and capital in Kiev. Kievan Rus also governed the cities of Ladoga and Novgorod in the north and therefore controlled the famous riverine pathway known in Russian historical tradition as “the Path from the Varangians to the Greeks.” “Varangians” meant Vikings, or Scandinavians, and “Greeks” meant Byzantines.
Prevented by the Khazars from expanding towards the Caspian Sea and completely taking over the thriving trade between eastern/central Europe and the Muslim Middle East, the Rus Vikings instead expanded further into the areas of modern-day Ukraine and the Black Sea region; this expansion brought them into direct contact with the Byzantine Empire, whose territory in Asia Minor included the southern coast of the Black Sea; the Byzantine Empire also had outposts in Crimea. In 860 CE, the Rus Vikings unsuccessfully attacked Constantinople, the capital of the Byzantine Empire. In the tenth century, the Byzantines signed two treaties with the Rus Vikings; the ancient Russian chronicle “L’etopis’ Vremennykh L’et,” preserves the text of these treaties.
In the middle of the tenth century, the Rus ruler Sviatoslav defeated the Khazars; he then attempted to create a new power base in the Balkans, but the Byzantine army forced him to withdraw. His mother, Olga, was baptized in Constantinople and his son Grand Prince Vladimir adopted Eastern Orthodox Christianity from the Byzantine Empire (988) and became an ally of the Byzantine emperors. In this way, Kievan Rus joined Georgia, Armenia, Serbia, and Bulgaria in becoming an integral part of the Byzantine cultural world, which historians call the Byzantine commonwealth. At the same time, the Slavic and Viking populations of Rus were merging; they would eventually become Russians, Belorussians, and Ukrainians.

How did the Nile shape Ancient Egypt?

As other contributors have already pointed out, the Nile River was critical to Egyptian civilization (so much so that it's difficult to imagine Egyptian civilization existing as we know it without the presence of the River). Indeed, if you look back towards the Neolithic Era and the early stages of Ancient History, you tend to find that many of the earliest centers of human habitation and civilization were in River Valleys, and the Nile is no exception.
Additionally, when discussing the importance of the Nile, it's worth noting the degree to which Egyptian civilization was able to maintain, for so much of its history, a centralized political structure, where all authority emanating downwards from the Pharaoh. I'd suggest that the Nile was a key component to Egyptian unification and state-building, which allowed for rapid communications and transport of goods and people.

The river Nile played a critical role in ancient Egyptian civilization. The river flooded annually and deposited nutrient-rich silt on its banks that created an environment that supported agriculture. The Egyptians planted crops such as wheat, beans, and cotton, as well as built canals that sustained agriculture all year round. Agriculture was the bedrock of Egyptian civilization.
The annual occurrence of the Nile floods led to the belief in gods and establishment of a social structure. The Egyptians believed that as long as the gods were happy, the Nile would continue flooding and bringing along the abundant blessings. At the apex of the social structure were gods, followed by pharaohs, scribes, merchants, artisans, farmers, and slaves. Individuals in each social stratum had a specific responsibility to perform, although education made it possible for people from lower social strata to progress to higher ones.
http://www.bbc.co.uk/history/ancient/egyptians/nile_01.shtml

https://pages.vassar.edu/realarchaeology/2017/04/09/impact-of-the-nile-river-on-ancient-egypt/

Without the Nile, Egyptian civilization would not have been possible. With the exception of a few oases, most of Egypt, other than the areas bordering the Nile, is inhospitable desert. Everything that demands water—drinking, cooking, animal husbandry, creating pottery—depended on the Nile bringing water from the southern mountains to lower Egypt. The Egyptians also relied on the Nile for transportation.
It has also been argued that much of the character of early Egyptian civilization depended on the need to maintain an extensive network of irrigation channels, which worked extremely quickly during the annual floods to ensure that water was distributed to the fields. This required complex cooperative action and a strong legal system ensuring that people did not divert water or fail to maintain channels running through the lands they cultivated. This may have been one reason Egypt developed a complex and sophisticated civilization quite early in its history.

The Nile River shaped ancient Egypt in that it provided a waterway for trading, a means of defending against enemies, and a water source for agriculture.  Each year, the river flooded the land around it.  This caused silt deposits to develop and made the land fertile for agriculture.  Wheat and papyrus were important crops in ancient Egypt.  Agricultural life in Egypt was dependent on the Nile River because much of Egypt was desert.
Egyptian merchants both exported and imported goods.  Sailing vessels traveled along the Nile River for trading purposes.  Boats from Egypt could travel north to the Mediterranean Sea.  From there, they could get to Asia and Europe to trade.
In the southern part of the Egyptian Kingdom, the cataracts provided protection along the Nile.  Fortresses were built along the Nile for defense.  Military forces from Egypt also sailed up and down the river.

Single Variable Calculus, Chapter 2, 2.4, Section 2.4, Problem 19

Show that the statement $\displaystyle\lim\limits_{x \to 3} \frac{x}{5} = \frac{3}{5}$ is correct using the $\varepsilon$, $\delta$ definition of limit.

Based from the defintion,


$
\begin{equation}
\begin{aligned}

\phantom{x} \text{if } & 0 < |x - a| < \delta
\qquad \text{ then } \qquad
|f(x) - L| < \varepsilon\\

\phantom{x} \text{if } & 0 < |x-3| < \delta
\qquad \text{ then } \qquad
\left|\frac{x}{5} - \frac{3}{5}\right| < \varepsilon\\

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}
& \text{But, } \\
& \phantom{x} & \left|\frac{x}{5} - \frac{3}{5}\right| = \left|\frac{1}{5} (x-3)\right| = \frac{1}{5}|x-3| \\
& \text{So, we want}\\
& \phantom{x} & \text{ if } 0 < |x-3| < \delta \qquad \text{ then } \qquad \frac{1}{5}|x-3| < \varepsilon\\
& \text{That is,} \\
& \phantom{x} & \text{ if } 0 < |x-3| < \delta \qquad \text{ then } \qquad |x-3| < 5\varepsilon\\


\end{aligned}
\end{equation}
$


The statement suggests that we should choose $\displaystyle \delta = 5 \varepsilon$

By proving that the assumed value of $\delta$ will fit the definition...



$
\begin{equation}
\begin{aligned}
\text{if } 0 < |x-3| < \delta \text{ then, }\\
\left|\frac{x}{5} - \frac{3}{5}\right| & = \left| \frac{1}{5} (x-3) \right| = \frac{1}{5}|x-3| < \frac{\delta}{5} = \frac{5 \varepsilon}{5 } = \varepsilon
\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

& \text{Thus, }\\
& \phantom{x} \quad\text{if } 0 < |x-3| < \delta \qquad \text{ then } \qquad \left|\frac{x}{5} - \frac{3}{5}\right| < \varepsilon\\
& \text{Therefore, by the definition of a limit}\\
& \phantom{x} \qquad \lim\limits_{x \to 3}\frac{x}{5} = \frac{3}{5}


\end{aligned}
\end{equation}
$

What message does The Help send, in terms of injustice and the nature of humans in terms of injustice?

One can take this message in different potential directions. Overall, one might see the message in The Help involving the necessity of courage in fighting injustice.
Among the white families depicted in this novel, a code of enforced unity prevails. Individually, few of the characters have much courage to impose injustice on their own, but with the backing of the larger community, they perpetuate all kinds of obvious indignities and injustices. The generational, or systemic, privilege that supports this community amplifies their power over others, encouraging them to maintain power and privilege by denying it to others.
Similarly, the maids are isolated in the homes in which they work, economically desperate for the income this belittling work provides, and disempowered by the larger fabric of Jackson, Mississippi. In Constanine's case, even their children are taken from them.
Characters in the beginning of the novel who feel isolated or powerless in Jackson (e.g., the maids, Skeeter, Celia) eventually form a community whose solidarity in helping Skeeter write her book gives them not only an emotional but also a social power over the elitist white community.
One claim the novel may make about the nature of humans is that few people have the courage to stand up to injustice on their own, but when the burden of fighting evil is shared, even the most simple and humble of people can move mountains.

The message that The Help sends in terms of the nature of humans who face injustice is that people will fight against injustice any way they can. The characters in the book who are African-American domestics in Jackson, Mississippi, in 1962 are in a position of very little power in the white society, but they often find ways to fight injustice. They aren't allowed to speak back to their employers, but they find other ways to express themselves. For example, Minny, an African-American maid, gives Miss Hilly, a white woman who uses gossip as a weapon against the maids, a cake laced with feces. More importantly, the maids work with Skeeter, a sympathetic white woman, to construct a narrative of what their lives are like, providing insight into the lives of African-American domestics to an audience outside the south. 
Skeeter and the other women also work against injustice against women. Though Skeeter is pressured to get married, she decides instead to work as a writer and refuses to get married just because it's conventional. There are many layers of injustice in the society portrayed in the book that involve race, gender, and their intersection, and though the characters are affected by these societal power dynamics, they also find ways to work together against them. 

Calculus of a Single Variable, Chapter 9, 9.6, Section 9.6, Problem 29

Recall the Root test determines the limit as:
lim_(n-gtoo) root(n)(|a_n|)= L
or
lim_(n-gtoo) |a_n|^(1/n)= L
Then, we follow the conditions:
a) Llt1 then the series is absolutely convergent.
b) Lgt1 then the series is divergent.
c) L=1 or does not exist then the test is inconclusive. The series may be divergent, conditionally convergent, or absolutely convergent.
We may apply the Root Test to determine the convergence or divergence of the series sum_(n=0)^oo 6^n/(n+1)^n .
For the given series sum_(n=0)^oo 6^n/(n+1)^n , we have a_n =6^n/(n+1)^n
Applying the Root test, we set-up the limit as:
lim_(n-gtoo) |6^n/(n+1)^n|^(1/n)=lim_(n-gtoo) (6^n/(n+1)^n)^(1/n)
Apply the Law of Exponents: (x^n/y^n) = (x/y)^n and (x^n)^(m)= x^(n*m) .
lim_(n-gtoo) (6^n/(n+1)^n)^(1/n) =lim_(n-gtoo) ((6/(n+1))^n)^(1/n)
=lim_(n-gtoo) (6/(n+1))^(n*(1/n))
=lim_(n-gtoo) (6/(n+1))^(n/n)
=lim_(n-gtoo) (6/(n+1))^1
=lim_(n-gtoo) (6/(n+1))
Evaluate the limit.
lim_(n-gtoo) (6/(n+1)) =6lim_(n-gtoo) 1/(n+1)
=6 *(lim_(n-gtoo) 1)/(lim_(n-gtoo)(n+1))
=6* (1/(oo+1))
=6* (1/oo)
=6*0
=0
The limit value L =0 satisfies the condition: Llt1 .
Thus, the series sum_(n=0)^oo 6^n/(n+1)^n is absolutely convergent.

Does a theory of civil disobedience necessarily depend on there being a Natural Law?

Some of the most famous practitioners of civil disobedience, most notably Martin Luther King, Jr., have based their actions on their belief in a higher law, a moral code that superseded the laws of men. They claimed, as King wrote (quoting St. Augustine), that "an unjust law is no law at all." People were thus bound to "natural law," and could only obey actual laws to the extent that they conformed to it. Natural laws, of course, were understood to mean those established by God. While King, Gandhi, and others based their actions on natural law, a belief in this concept is not essential to civil disobedience. A person could refuse to obey a law because they judged that it did more harm to people than good. This would be something like a utilitarian perspective on civil disobedience. A person could also exercise civil disobedience in order to draw attention to a law they saw as ineffective or discriminatory. A belief in natural law is not essential to civil disobedience.
https://plato.stanford.edu/entries/natural-law-ethics/

The main difference between European colonies and protectorates in Africa had to do with their ...

The main difference between European colonies and protectorates in Africa had to do with their governments (Answer B). Colonies and protectorates are similar because they are both territories of a larger state (country). However, the main difference lies in the way each is ruled. Colonies are dependent states that have rulers who are accountable to the larger (mother) country, and colonies are considered part of the mother country. All government, trade, and activities of a colony are under the control of the mother country and enforced by the mother country's officials and laws. Protectorates, on the other hand, are smaller nations that retain autonomy and sovereignty as they enter into an agreement with a larger state for protection. A larger state might offer protection to a smaller one to gain access to ports, natural resources, or national borders, but protectorates are not directly governed by the larger country. Protectorates act independently in government and matters of rule from the larger country.

A colony has limited or in a number of cases, no autonomy, and is ultimately controlled in its domestic and foreign affairs by a "mother country." A protectorate, on the other hand, is a separate state or kingdom with substantial domestic autonomy but that is nonetheless protected by a more powerful sovereign state, which usually controls the protectorate's foreign relations.
Colonies are essentially territories outside a sovereign nation that that nation controls. For example, the British Empire consisted of several colonies worldwide, some of which enjoyed varying degrees of local rule, but all of which were ultimately controlled by the British government.
Protectorates enjoy considerably more self government, particular in the area of domestic affairs. For example, France oversaw the Kingdom of Cambodia for a time as a protectorate and was responsible for protecting the kingdom from foreign invasion, while the Khmer king and his officials oversaw such domestic affairs as taxation. Colonies and protectorates share some similarities -- citizens in colonies and protectorates often serve in the armies of the more powerful nation to which they are tied, and often have robust trade relationships with that nation as well.

The answer to this question should be B: governments. A European colony was entirely under the jurisdiction of the mother country. The difference between a protectorate and a colony in any instance is a matter of government: a colony is ruled by a government designed and implemented by another country and has no sovereignty of its own, whereas a protectorate, as the term suggests, receives protection and/or some kind of mentorship from a larger or more powerful nation and would usually offer something in return. So, a protectorate would have its own government and its own sovereignty and laws but would be protected by another country and would usually be expected to support that country in times of war and allow the protector country to utilize the raw materials, and so on, of the protectorate.

Why is chapter 10 called "The Arctic Tern"?

Doug has a rough life.  His brothers are criminals in the making, and they abuse him almost as much as his father does.  Doug's father is not afraid to verbally and physically abuse Doug, and alcohol only makes it worse.  Doug has also been moved to a new city and new school, which means his close friend Holling Hoodhood can't be there for him.  All of these things pile up on Doug and give him a bit of a rough edge, which his teachers don't respond too well to at first either.  About the only thing in Doug's life that offers him comfort and peace is looking at the "Arctic Tern" plate painting by John James Audubon in the public library.  
Doug's closest friend, Lil, is also sick with cancer.  The doctors give her a one in four chance of surviving.  She's not doing well.  In chapter ten, Doug goes to visit her several times, and in the final moments of the chapter Doug is by her side again.  It's clear that Lil is scared.  She's shaking and crying, and Doug reaches out to hold her hand.  He tells her to think of "a whole lot" of Arctic terns flying around her.  He tells Lil to imagine them showing her the next great thing that is going to come into her life.  The sequence is incredibly sad, yet it is a hopeful scene as well.  Lil is comforted by imaging the terns, and throughout the novel, the image of the Arctic tern has offered comfort to both Doug and Lil.  It's appropriate to name this chapter after such a hope giving image.  

How was the Constitution different from the Articles of Confederation?

The Constitution gave extensive powers to the Federal government in Washington. Under the Articles of Confederation there was no central government to speak of. This is because most Americans were fearful of investing a centralized authority with too much power. The Americans had just fought to get rid of what they considered the tyranny of British colonial rule and so the last thing they wanted was to see it reestablished, only this time by Americans.
In practical terms, however, the Articles had proved an inadequate instrument of government for the new nation. For one thing, it proved impossible to develop a coherent foreign policy with ultimate political authority residing with the states. Without a centralized authority, the United States was also unable to pay back the enormous debts it had accrued during the Revolutionary War.
The Constitution sought to put an end to these problems by granting extensive powers to the executive branch in the form of the President and to the United States Congress. The new proposals meant that, from now on, the United States would be able to speak with one voice on the international stage in relation to both economic and foreign policy. On the domestic front, it would also mean that the US government could deal more effectively with threats to internal stability instead of relying on state authorities.

The Articles of Confederation and Perpetual Union was, in effect, the first Constitution of the United States. Under its terms, the thirteen colonies were unified under a weak central government, but the individual colonies functioned as independent countries in matters of taxation as well as interstate and foreign commerce. The sole governing body was the Congress of the Confederation. At this point, there were no executive or judicial branches of government. Congress had the ostensible power to wage war, conduct foreign affairs, and regulate currency. In practice, however, Congress could not exercise these powers, as it had no authority to force the states to supply money or personnel to serve as troops. Although Congress could mediate disputes between the states, the states were under no obligation to accept the results.
The US Constitution was adopted because, in order to survive as an independent nation, the United States needed the stability of a stronger government. The Constitution added the executive and judicial branches to the government and provided for a system of checks and balances in order to prevent any one branch of government from becoming too powerful. It delineated the specific responsibilities of each branch. Additionally, in order to ensure that each state received fair representation, the Constitution divided Congress into a dual legislative branch composed of the House of Representatives and the Senate. A few years later, the ten amendments that made up the of the Bill of Rights were introduced into the Constitution. These amendments guaranteed that individual citizens received fundamental protections, which included freedom of speech, the right to assemble peacefully, the right to bear arms, and the right to a fair trial by jury.
https://www.archives.gov/founding-docs/more-perfect-union

There were several ways that the Constitution was different from the Articles of Confederation. The Articles of Confederation was created to have a weak federal government. For example, the federal government could not tax, could not control trade, and could not require people to join the army. There was one house in the legislative branch. There would be between two and seven members from a state in Congress. Each state, however, had only one vote in Congress, regardless of the size of the state’s population. A three-person committee ran the executive branch. The people were very afraid that one person would have too much power and begin to act like a king. In order to pass a law, nine of the thirteen states had to agree to the law. All thirteen states had to agree to change the Articles of Confederation. There also were no federal courts in the Articles of Confederation.
The Constitution had several differences. The federal government had more power. The federal government had the ability to tax the people. The federal government could also control interstate and foreign trade. There were two parts of the legislative branch. In one house of Congress, called the Senate, states would have equal representation. In the other house, called the House of Representatives, representation was based on the size of the state and therefore was unequal. The President led the executive branch. If the President acted improperly, the President could be impeached. The Constitution also did not require all the states to approve a change to the Constitution. There were federal courts created by the Constitution so states would have a place where they could resolve disputes.
The Constitution was written to try to resolve some of the issues that were created by the weak federal government that was established by the Articles of Confederation.
https://www.usconstitution.net/constconart.html

Consider a wheel of mass M and radius R up against a ledge of height h , where hltR . What horizontal force F do you have to apply to the axle to roll the wheel up over the ledge?

The wheel will roll up if the torque due to the horizontal force (tau_F) exceeds the torque due to the force of gravity (tau_g) .
tau_g=Mgsin(alpha)=Mg[sqrt(R^2-(R-h)^2)/R]
tau_F=Fcos(alpha)=F(R-h)/R
tau_Fgttau_g
F(R-h)/RgtMg[sqrt(R^2-(R-h)^2)/R]
FgtMg*sqrt(R^2-(R-h)^2)/(R-h)
This is the solution. Notice if h->R Then F ->oo . If h->0 , F->0 . This is intuitively reasonable.
http://hyperphysics.phy-astr.gsu.edu/hbase/torq.html

How are the witches presented in Macbeth?

In Macbeth, the witches are presented with a few distinct characteristics. Banquo describes them as looking "not like th' inhabitants o' the' Earth" (1.3.42). He also suggests that they have beards. In act 1, scene 3, Macbeth and Banquo happen upon the witches suddenly, which also suggests that the three weird sisters have the ability to appear as if from nowhere, but even before Macbeth's first interaction with them, Shakespeare establishes their demeanor in a way that reflects their strange appearance. At the start of this scene, the three witches are reconvening the meeting that was established at the very start of the play, and this exchange takes place:

First Witch: Where hast thou been, sister?Second Witch: Killing swine.Third Witch: Sister, where thou?First Witch: A sailor's wife had chestnuts in her lap, And munch'd, and munch'd, and munch'd— 'Give me,' quoth I: 'Aroint thee, witch!' the rump-fed ronyon cries. Her husband's to Aleppo gone, master o' the Tiger: But in a sieve I'll thither sail, And, like a rat without a tail, I'll do, I'll do, and I'll do.Second Witch: I'll give thee a wind.First Witch: Thou'rt kind.Third Witch: And I another.First Witch: I myself have all the other, And the very ports they blow, All the quarters that they know I' the shipman's card. I will drain him dry as hay: Sleep shall neither night nor day Hang upon his pent-house lid; He shall live a man forbid: Weary se'nnights nine times nine Shall he dwindle, peak and pine: Though his bark cannot be lost, Yet it shall be tempest-tost. Look what I have.Second Witch: Show me, show me.First Witch: Here I have a pilot's thumb, Wreck'd as homeward he did come.

This scene goes a long way in establishing the character of the witches. One of them just happened to be killing pigs before they met up. There is no mention of her being a butcher, so it seems like she was just doing this for fun. Further, the First Witch relates a story of a "great wrong" she has suffered. She saw a woman eating chestnuts and rudely demanded to have some. The woman told her to back off, and now because of that simple, brief affront, the witch is going to seek out the husband of the woman and torture him until he begs for death. As if that wasn't intense enough, she then shows off a finger of a dead sailor. Through this scene alone, we see the witches as more than just "weird." They are manipulative, they are evil, and perhaps scariest of all, they are petty. They behave terribly and then get mad at others for not responding as they want them to. Their entire manipulation of Macbeth also seems to be happening for no reason other than their own desire to watch his world collapse around him. They have nothing to gain from it. They simply want to watch the world burn.

Does Blanche want Mitch? Why or why not?

Blanche wants Mitch to save her and help her change her current situation. There is a great reason for her to want this, she has lost it all:
the family home of Belle Reve
her husband, to suicide
her sister, to Kowalski
both her parents, told old age and disease
her reputation
her job
her money
her friends
any prospects for a job
her stability, in every way and form
It comes to no surprise that Blanche would beg for someone, anyone, to help her get out of the massive rut that she is stuck in. Mitch is easy to manipulate and seems to be as needy as Blanche. He is ideal for her, because he is likely to do anything she asks him to do. He would have done it, too, had it not been due to Stanley Kowalski's meddling to separate the potential couple.
Mitch was, therefore, Blanche's last card to play before everything was really over for her.

Blanche explicitly says that she does indeed want Mitch. With his courtliness, immaculate dress sense, and his old-fashioned Southern manners, he's a breath of fresh air. Certainly, he's a welcome change from most of the men she comes across, the "apes" as she calls them—particularly Stanley:

“I want to rest! I want to breathe quietly again! Yes—I want Mitch…very badly!”

Blanche desperately needs to be loved, needs to be protected from a harsh, unforgiving world which has treated her so badly. And Mitch appears just the man for the job. Blanche also comes to look upon Mitch as some kind of savior figure, someone who can save her from a disreputable past. But this is pure fantasy on Blanche's part.
In fact, Mitch's desire for Blanche is also a fantasy. He perceives her to be a fine, upstanding, respectable Southern belle. However, when he finds out the true nature of her sordid past, the scales fall from his eyes immediately. She's not the woman he thought she was; she's the complete antithesis of the ideal fantasy figure he's constructed to fill a gaping emotional void. And when Mitch finds out about the real Blanche, she gets to see the real Mitch—a violent, would-be rapist who's just like every man she's ever met.
Ultimately, they both realize that it's not what they want, but what society will allow, that really matters in such a time and such a place.

What, in your view, is the role and dramatic significance of Aunt Lizzy within the play Philadelphia, Here I Come!?

Aunt Lizzy is the spark that sets Gar on his path. In many ways, she's the catalyst that allows the story to take place.
When Aunt Lizzy and her husband visit Gar, she suggests that he move to Philadelphia to live with them. She wants to be closer to the man she views as a stand-in for her own child, which she and her husband were unable to have.
Without the suggestion of the trip and the affection that Lizzy gives Gar, he might not have considered leaving. But Lizzy offers the view of something different. This change is especially appealing because his first love is being married to someone else. Gar wants a better life, but he also struggles with not knowing what to do.
Lizzy also has significance because she embodies Gar's internal struggle. He wants love and affection, but he is somewhat put off by her manner and appearance. Like with everything in his life, he can't choose what's right for him or what he needs.

Aunt Lizzy is probably the most important person in Gar's life and a huge part of the reason he decides to move to the United States. Gar's mother has passed away and his Aunt Lizzy (his mother's sister) has always been a big part of his life. Although he hasn't seen her for a while, Gar has a whole fantasy constructed around his arrival in the United States that involves Aunt Lizzy. When Gar was a young boy, she would come to visit the family in Ireland and practically beg Gar to come home with her to Philadelphia so she could spoil him:

"She cries “My son, Gar, Gar, Gar . . .” as she throws her arms around him."

As he is reminiscing, Gar hears about about Kate Doogan's wedding the previous weekend; Kate was the love of his life and they were once engaged. Then the story goes into flashback mode and we see a time when Gar was on his way to talk to Kate's father to ask his permission to ask her hand in marriage, but Brian begins to feel like he came from a socially inferior family and he loses his nerve and Kate goes on to marry someone else. Brian mourns the loss of that relationship and regrets his timidity.
Brian wants to forget about Kate and move forward to a new life as represented by Aunt Lizzy.
https://www.bbc.co.uk/bitesize/guides/zsj7cj6/revision/1

To a large extent, Aunt Lizzy acts as the catalyst for Gar's decision to emigrate to the United States. In that sense, her role has great dramatic significance. Gar doesn't have a mother, and Aunt Lizzy, his late mother's sister, acts as a kind of mother substitute—though if she is a mother, there's something cloying and overpowering about the mother love she so liberally lavishes upon Gar, making him feel deeply uncomfortable. Nevertheless, he still responds to Lizzy's invitation to leave Ireland for the United States.
Although Aunt Lizzy is over-affectionate toward her nephew, too much affection's better than none, and there's precious little affection around in his hometown. Ultimately, Lizzy comes to represent for Gar the prospect of a new life in the United States, with all the many good things and bad things that that will entail, which Gar will experience for himself when he finally sets foot on American soil. Gar needs a new mother away from his mother country, and Aunt Lizzy is a personification of that need.

In Reading Lolita in Tehran, why does Nafisi choose to study Lolita? What are the reasons behind this choice?

In the chapter on Lolita, Nafisi focuses on the theme of oppression in literature. This is her way to understand and cope with the oppression she experiences as a woman in Iran. She understands that her situation in totalitarian, fundamentalist Iran is similar to that described in Nabokov's Invitation to a Beheading and Lolita, and she takes strength from the idea that the victims of totalitarianism don't have to end up broken and destroyed.
Nafisi also examines A Thousand and One Nights as the story of a woman who resists severe oppression. Finally, in Lolita she finds a young girl resisting the oppression of her pedophile captor—who tries to maintain total control of her— in all the subtle ways the little girl can. She reads Lolita as an example of a man exerting disproportionate power over an innocent victim, just as the fundamentalist regime in Iran was doing to women in general.
Overall in her book, Nafisi sees literature as a way to fight back against totalitarianism. Literature, she writes

can be called moral when it shakes us out of our stupor and makes us confront the absolutes we believe in.

Totalitarianism, in contrast, believes only in absolutes and doesn't offer people the plurality of viewpoints a novel does. Reading novels like Lolita, which condemn destroying one person at the whim of another, is Nafisi's way of fighting back against her own oppression.

In Reading Lolita in Tehran, narrator and author Azar Nafisi chooses to study the controversial novel Lolita in her women's study group because the main character's life mirrors the oppression that she and the other women feel. As occupants of post-revolutionary Iran, Nafisi and her former students face the oppression of a restrictive Islamic regime. They are forced to cover themselves and threatened if they do not conform to the regime's ideals of what a woman should be and do. Similarly, Lolita is a young girl who is defined by her relationship to a much older man. At one point, Nafisi says that "not only her life but also her life story is taken from her," in reference to Lolita. She sees this as a parallel to the freedom that was taken from Iranian women, and the act of joining together to read forbidden books is a way to avoid falling victim to oppression a second time. The imbalance of power in the dynamic between Humbert and Lolita mirrors the dynamic between the women in the group and the men around them. If this dynamic is not present in their own marriages and families, it is present in greater society.
Lolita is only one of the many books that were forbidden in the new Iranian regime. Despite the risk, Nafisi and the others feel that it is important to carve out their own spaces of freedom where they can. In this sense, reading Lolita is an act of rebellion. Most of the women have a copied version from the book because it has been banned by the government. Nasifi notes that the women in her study group also have "fragility and courage" in common, both of which are qualities that can be seen peeking through Lolita's character even through Humbert's biased lens.

What are specific examples of courtly love between Romeo and Juliet in Shakespeare's play?

Romeo's relationship with Rosaline is more characteristic of courtly love than is the relationship between Romeo and Juliet. For example, courtly love involves no physical contact but a great deal of mooning on the part of the man, who is distracted and sleepless because he is obsessed with his beloved and willing to go to all extents to defend her. The woman, on the other hand, is haughty and pure. At the beginning of the play, when Romeo imagines he is in love with Rosaline, Benvolio tells Lady Montague that he has seen Romeo "underneath the grove of sycamore / That westward rooteth from this city side, / So early walking did I see your son" (I.2.111-113). Benvolio sees Romeo taking an early morning walk under sycamore trees, but Romeo is so distracted while thinking about his love that he does not even see his friend. Romeo's parents wonder what is wrong with him, but, as is characteristic of a courtly lover, Romeo keeps his love a secret from them and prefers to think about his love in private.
At the beginning of Romeo's relationship with Juliet, he also acts like a courtly lover, admiring her beauty from afar when he sees her at Capulet's ball. He is full of praise for her, saying of Juliet, "It seems she hangs upon the cheek of night / Like a rich jewel in an Ethiope’s ear" (I.5.43-44). He compares the brightness of her beauty to a jeweled earring hanging from the ear of an African person. Juliet, like a courtly lover, tells Romeo when he first takes her hand, "Good pilgrim, you do wrong your hand too much, / Which mannerly devotion shows in this" (I.5.97-98). She compares his hands to those of a pilgrim, who touches saints. She is pretending to only see pure intentions in Romeo's touch. 
However, once Juliet and Romeo meet, they quickly discard most of the conventions of courtly love and kiss. Juliet is not the traditional coy mistress. For example, after Romeo hears her declaring her love for him, she says, "Fain would I dwell on form. Fain, fain deny / What I have spoke. But farewell compliment!" (II.1.88-89). This means that she wants to follow conventions and deny what she has just said, but she is bidding farewell to what is considered conventional. She quickly discards the conventions of courtly love in her relationship with Romeo. 

What are all the themese in "Rappaccinis Daughter"?

One major theme of this story is the idea that human beings are a mixture of good and evil and that we cannot exist without one or the other. We are, by our natures, both.
Beatrice, for example, seems so good and sweet and kind, but poison is a part of her makeup, and when Giovanni attempts to remove this "evil" and destructive part of her, she dies because a human being cannot be all good—it is simply not within our nature to be so. Giovanni, likewise, seems good when he loves Beatrice, but he turns cold and dissatisfied with her. He cannot love her as she is, good and bad, and so he attempts to change her, to perfect her. In doing so, he strips her of her nature and brings about her death.

How did Rosa Parks influence change?

Rosa Parks played a major role in the civil rights movement but perhaps not in the exact way that students are often taught in school. A great deal of people, myself included, grew up thinking that Rosa Parks was just a random woman who got on a bus in 1955 and refused to give up her seat for a white passenger. While this is part of what happened, it isn't the entire story. Rosa Parks's action that day, December 1, 1955, was a result of not only other recent events but also Parks's lifelong attempt to garner equal rights for African Americans. She was active in the Montgomery NAACP, even serving as a secretary and getting heavily involved in investigations of other atrocities against African Americans more than a decade before her actions on the bus. Further, a number of other people had performed acts of civil disobedience in the past similar to that of Parks. But Parks became a civil rights icon because her act occurred at a specific time—just days after the acquittal of the brutal murderers of Emmett Till—and because the NAACP, of which Parks had been a member since 1943, strongly backed her during the court case. She was quiet, gainfully employed, married, and well-respected. She was an excellent example of what a person should be, and she spent much of her life helping to enact change.