How does the setting of The Crucible set the atmosphere and tone of the story?

The setting of The Crucible is the town of Salem, Massachusetts. It's a crucial setting for what follows, as it's a very small town where everyone knows each other's business. This creates a stifling, restrictive atmosphere in which it is virtually impossible to lead your life free from prying eyes and wagging tongues. And such an atmosphere provides the ideal conditions for a witch-craze to develop.
Salem's not just a small town, but a Puritan town, a Puritan theocracy governed by a narrow interpretation of the Bible. As die-hard Calvinists, the townsfolk already believe that everyone is utterly depraved, mired in sin. So it doesn't take much for them to suspect that people they see every day and with whom they go to church might actually be practicing witchcraft in secret. All the ingredients of a potential witch-craze are already in place long before Abigail Williams and the other girls start messing with the forces of darkness in the forest one night.

The Crucible takes place in Salem, Massachusetts, in 1692. Salem, a city near Boston, was part of the Massachusetts Bay Colony, which was founded by Puritans. The Puritans ruled the colony as a theocracy, a government in which religious leaders were in control and religion was the law of the land. Puritans saw the world as a battle between good versus evil, and they saw the woods and lands outside their towns as the devil's domain. This establishes the paranoid tone of The Crucible, in which the Puritans believe witchcraft is at work, and establishes an atmosphere in which people still believe in witches' power to control them and wreak havoc in the town. In 1692, a witch hunt actually occurred in Salem, and 20 people were executed. Miller bases his play on some of these real events.

y^2=28x Graph the equation. Identify the focus, directrix, and axis of symmetry of the parabola.

y^2=28x
Take note that one of the vertex form of parabola is
(y - k)^2 = 4p(x-h)
where
(h,k) is the vertex and,
p is the distance between vertex and focus and also the same distance between the vertex and directrix.
So to graph it, first find the vertex.
Rewriting the given equation in exact form as above, it becomes:
(y-0)^2 =28(x-0)
So the vertex of the parabola is (0,0).
Then, determine the points of the parabola. To do so, isolate the y.
y^2=28x
y=+-sqrt(28x)
y=+-2sqrt(7x)
Then, assign a value to x. And solve for the y values.
x=7,y=+-2sqrt(7*7) =+-2*7=+-14
Plot these three points (0,0), (7,-14) and (7,14). And, connect them.
Therefore, the graph of the given equation is:

To determine the focus and directrix, consider the coefficient of the unsquared portion of the given equation and set it equal to 4p.
4p=28
And solve for p.
p=28/4
p=7
So both the focus and the directrix are 7 units from the vertex.
Since the parabola opens to the right, the coordinates of the focus is:
(h + p, k) = (0+7, 0) = (7,0)
And the equation of directrix is:
x=h-k
x=0-7
x=-7
Therefore, the focus is (7,0) and the directrix is x=-7 .
Take note that the axis of symmetry of a parabola is a line that passes the vertex and the focus. And it is perpendicular to the directrix. (See attachment.)
Therefore, the equation of its axis of symmetry is y=0 .

Describe the differences and similarities between the societies and economies of the southern, middle, and New England colonies.

By 1750, there were about one million colonists living in the Thirteen Colonies, and the structure of the society and economy was not the same throughout.
For instance, the southern colonies had large numbers of slaves. By the middle of the eighteenth-century, there were as many as a quarter million enslaved Africans in these colonies. Previously, these colonies had relied on indentured servitude for labor. However, as a result of improving conditions in England, the number of indentured servants rapidly diminished by mid-century. As a result, more slaves were forcibly brought to the southern colonies. This created a dual society with white plantation owners enjoying the benefits of the labor of a subjugated slave class.
The middle colonies had some slaves, but not nearly in the numbers of the colonies to the south. Many of these colonies were dominated by descendants of Dutch settlers who were granted land by the English crown. They controlled large estates and dominated local governments. These colonies were often marked by practicing considerable religious tolerance compared to the other colonies.
New England society was characterized by strict religious devotion. Puritanism was commonly practiced in these colonies, and other sects were not tolerated. The line between church and state was very thin in this theocratic society.
The economy of the South was based mostly on agriculture. The fertile soil meant that much could be grown there. Tobacco, corn, and wheat were produced in abundance. Much was used to feed the other colonies, as well as for export to England and the West Indies. The middle colonies had a mix of agriculture and commercial centers. This diversified economy brought a lot of wealth to the region, especially around the Chesapeake Bay. New England's poor soil and short growing season meant the region could not support much more than subsistence agriculture. Instead, the economy of New England focused on trade, whaling, fishing, and manufacturing. Indeed, the many natural harbors and mill creeks in the region were ideal for this type of economy.
https://www.benfranklinsworld.com/category/colonial-america/

There were differences and similarities between the southern, middle and New England colonies. One difference was economic. The New England colonies had lots of industries. Because of the rocky soil and the cooler climate, many people worked in the shipping, lumber, fishing, and manufacturing industries. The southern colonies were mainly agricultural. With fertile soil and a mild climate, a lot of farming was done. Crops such as rice, tobacco, and indigo were grown. The middle colonies had both manufacturing and farming. The colonies closer to New England had more industry, while those closer to the South had more farming.
The societies were also different. Southern colonies depended on slaves to help them with the farming. The slaves had no rights and were not free to do what they wanted to do. The white plantation owner dominated southern society. The middle colonies had some slaves. Religious freedom was more common in the middle colonies. In New England, there were few slaves. People tended to live in towns, and the town meeting was a common part of the government structure.
There were similarities between the regions. Each region was fairly self-sufficient economically. Trade was an important activity of each area. Each region depended on trade with Europe, especially with Great Britain. Religion also played a role to some degree in the lives of the people in each region. Men, unlike women, were more likely to get a formal education. Each region eventually had some form of government.
There were differences and similarities between the southern, middle, and New England colonies.
https://blablawriting.com/similarities-and-differences-of-the-new-englandmiddle-and-southern-colonies-essay

https://thomashagen.wordpress.com/history-papers/major-differences-between-the-colonies/

Calculus of a Single Variable, Chapter 7, 7.4, Section 7.4, Problem 14

The arc length of a function is the length of the described curve within some interval on the x-axis (and a corresponding interval on the y-axis). If we were to lay a piece of string over the line of the graph in this window, the graph going from corner to diagonally opposite corner in a curve, the arc length is the length of the piece of string used.
In this case the function to find the arc length of is
y = ln ((e^x+1)/(e^x - 1))
where ln(x) is the natural logarithm of x (the inverse function to e^(x) , recall).
The window over which to find the length is the rectangle given by ln 2 <= x <= ln3 and ln2 <= y <= ln3.
The function for the arc length of a generic function y = f(x) is
s = int_a^b sqrt(1+((dy)/(dx))^2) \quad dx
This integrates along the line of the graph, adding infinitesimal sections together where each very small section added is a straight line diagonal over the small window x_0 <=x <=x_0 + dx , y_0 <=y <= y_0 + dy . Though y = f(x) may be a curve, the point is that these windows over which the length is integrated (added up) are so small that the graph is a straight line within them making it a simple thing to add the small (straight line) sections together. This concept is the basis of calculus.
In this example, since y = ln ((e^x+1)/(e^x-1)) = ln(e^x + 1) - ln(e^x-1) then its derivative (dy)/(dx) is given by
(dy)/(dx) = (1/(e^x+1))e^x - (1/(e^x-1))e^x = e^x(1/(e^x+1) - 1/(e^x-1)) = e^x((e^x-1-e^x-1)/((e^x+1)(e^x-1))) = -(2e^x)/((e^x+1)(e^x-1))
Now working in steps to find the integrand in the formula for the arc length s , we have that
1+ ((dy)/(dx))^2 = 1 + (4e^(2x))/((e^x+1)^2(e^x-1)^2) = ((e^x+1)^2(e^x-1)^2 + 4e^(2x))/((e^x+1)^2(e^x-1)^2)
and that
sqrt(1+((dy)/(dx))^2) = (sqrt((e^x+1)^2(e^x-1)^2 + 4e^(2x)))/((e^x+1)(e^x-1))
= sqrt((e^(2x)+2e^x +1)(e^(2x)-2e^(x)+1)+4e^(2x))/((e^x+1)(e^x-1))
= sqrt((e^(4x)-2e^(3x)+e^(2x)+2e^(3x) - 4e^(2x) + 2e^x + e^(2x)-2e^x + 1)+4e^(2x))/((e^x+1)(e^x-1))
Adding up that very long set of exponential terms, finding that most cancel, we finally have that
sqrt(1+((dy)/(dx))^2) = sqrt((e^(4x)-2e^(2x)+1)+4e^(2x))/((e^x+1)(e^x-1)) = sqrt(e^(4x)+2e^(2x)+1)/((e^x+1)(e^x-1))
= sqrt((e^(2x)+1)^2)/((e^x+1)(e^x-1)) = (e^(2x)+1)/((e^x+1)(e^x-1))
= (e^(2x)+1)/(e^(2x)-1)
Integrating the integrand over the required interval, we have that
s = int_(ln2)^(ln3) ((e^(2x)+1)/(e^(2x)-1)) \quad dx
= int_(ln2)^(ln3) (2e^(2x)- e^(2x) + 1)/(e^(2x)-1) \quad dx = int_(ln2)^(ln3) (2e^(2x))/(e^(2x)-1) -1 \quad dx
= ln(e^(2x)-1)|_ln2^(ln3) - x |_(ln2)^(ln3)
= ln (e^(2ln3)-1) - ln(e^(2ln2)-1) - ln3 + ln2
= ln(9-1) - ln(4-1) - ln3 + ln2 = ln((8/3)(2/3))
= ln(16/9) is the final answer

Why does Hurston devote this chapter to Janie’s jealousy of Nunkie?

Chapter 15 of Their Eyes Were Watching God features the story of Janie's jealousy of Nunkie. At this point in Hurston's novel, Janie and Tea Cake are working in the Everglades (the muck), and Nunkie is also working there. Janie thinks Tea Cake and Nunkie are flirting with each other and sees them in some kind of physical act that she imagines is sexual or about to become so. Janie and Tea Cake fight about his relationship with Nunkie (or, as Tea Cake argues, the lack of a relationship), but then their anger turns into passion. Tea Cake reassures Janie that he loves her and insists that Nunkie means nothing to him; Nunkie could never measure up to Janie.
Hurston may choose to incorporate the chapter about Nunkie for a couple of reasons. For one, it shows the passion Janie and Tea Cake feel for one another. Their relationship is unlike Janie's other two marriages. When she was married to Logan, it was simply for financial security, and there was no love or desire on her part. Her marriage to Joe Starks began as a playful flirtation but became a one-sided power play when Joe became mayor of Eatonville; he began treating Janie as a possession and separated her from the townspeople, breeding resentment in Janie. With Tea Cake, Janie feels that she can be carefree and playful again. She feels that he treats her as an equal (financially, she is actually his superior, due to Joe's inheritance); for example, he plays checkers with her. They go to the muck together and work in the Everglades. Janie's reaction to Nunkie indicates her feelings for Tea Cake in the sense that her jealousy means she is afraid to lose him. This was not the case with her other husbands.
A second possible reason for the Nunkie chapter is that it foreshadows some darker moments in Janie and Tea Cake's relationship. Although the marriage is much better than her previous two, it is not perfect. A couple of chapters after this, Tea Cake beats Janie to show his dominance over her. Later, he contracts rabies from a dog during the hurricane (he was trying to save Janie), and he goes insane. He tries to kill Janie, and she feels she must kill him. Their relationship ends up being violent and tragic, and the Nunkie episode is one of the early signs of the potential volatility in their marriage. Once she is cleared of responsibility in Tea Cake's death, Janie moves back to Eatonville and lives her life for herself.

What is the atmosphere in "After Twenty Years?"

In the short story "After Twenty Years," the author O. Henry creates an atmosphere of mystery and uncertainty to set up the surprise ending. He does this by describing the weather, the time, and the lack of pedestrians.


The time was barely 10 o'clock at night, but chilly gusts of wind with a taste of rain in them had well nigh depeopled the streets.



The vicinity was one that kept early hours. Now and then you might see the lights of a cigar store or of an all-night lunch counter; but the majority of the doors belonged to business places that had long since been closed.


From these descriptions, the reader is immersed in a setting of a street that is dark, windy, and almost deserted. This creates the atmosphere of mystery that the author is trying to convey. When the policeman sees the lone man standing in the doorway of a shop, the atmosphere of mystery is heightened, for the reader becomes curious about who this solitary person in such a bleak setting could be. To add to the atmosphere, the description of the stranger in the doorway is framed by the lighting of a match, so that the policeman catches only a glimpse of the man's face. The author adds further to the mystery by explaining that the man has a slight scar on his face and a scarfpin and watch that suggest that he is very rich. It's impressive that with a few well-placed words and phrases, O. Henry is able to create such a mysterious and atmospheric setting for this fascinating and surprising story.

The atmosphere of "After Twenty Years" establishes the mood of mystery and suspense that sets up the story's surprise ending. It is night as the story opens. The cop is on patrol and the weather is pretty unpleasant. It's cold, the wind is blowing, and the gales are flecked with a little rain. The part of the city where the cop is walking is mainly shrouded in darkness. Most of the properties in the area belong to businesses that have long since closed for the day. As such, there aren't too many people around, except for the occasional patron of a small restaurant.
All of the various ambient elements provide the perfect setting for the story. The darkness and relative isolation of the location allow us to concentrate more easily on the two characters and their fateful meeting. And by setting the story at night and in inclement weather, Henry is cleverly hinting that the meeting won't have a very happy ending for at least one of the two men.

int sec^2(3x) dx Find the indefinite integral

Indefinite integral are written in the form of int f(x) dx = F(x) +C
 where: f(x) as the integrand
           F(x) as the anti-derivative function 
           C  as the arbitrary constant known as constant of integration
For the given problem: int sec^2(3x)dx , the integrand function: f(x)=sec^2(3x) is in a form of trigonometric function.
To solve for the indefinite integral, we may apply the basic integration formula for secant function:
int sec^2(u)du=tan(u)+C
We may apply u-substitution when by letting:
u= 3x then du =3 dx or (du)/3 = dx .
Plug-in the values of u =3x and dx= (du)/3 , we get:
int sec^2(3x)dx =int sec^2(u)*(du)/3
                          =int (sec^2(u))/3du
Apply basic integration property: int c*f(x)dx= c int f(x)dx .
int (sec^2(u))/3du =(1/3)int sec^2(u)du
Then following the integral formula for secant, we get:
(1/3)int sec^2(u)du= 1/3tan(u)+C
Plug-in u =3x to solve for the indefinite integral F(x):
int sec^2(3x)dx=1/3tan(3x)+C