Why did Briony want to keep Robbie and Cecilia apart?

In Ian McEwan’s Atonement, Briony Tallis is a young girl whose lie prevents her older sister Cecilia and childhood friend Robbie from ever fully pursuing their burgeoning romance.
Briony is a prolific writer, even as a child, and enjoys making up stories because this gives her a sense of control and order. Like any child, Briony has a naive understanding of the world and has perhaps-false notions of the way things are. When she reads the vulgar, sexual language in Robbie’s letter to Cecilia and then walks in on their sexual encounter in the library, Briony is appalled. She views Robbie, whom she once admired, as a sexually perverted maniac who is out to hurt Cecilia. This shows that Briony doesn’t quite comprehend the adult realities of sexual attraction.
This lack of understanding motivates her to lie about Robbie raping Lola. She could have lied about this because it seemed like a logical conclusion based on her recent revelations about Robbie. In her mind, Robbie is the most likely perpetrator because of what she thinks she knows about him. On the other hand, Briony might have lied because she thought she was protecting Cecilia, even if she knew Robbie had nothing to do with Lola’s assault. This also shows that Briony sought order and control via lying, since she thought it was the best way to set things aright in her world.
While I’m not sure Briony necessarily wanted to keep Robbie and Cecilia apart, per se, she certainly had her reasons for accomplishing just that. Of course, Briony learns that what she did was wrong and misguided, but not until after the damage has been done.

How did Malcolm X's identity transition throughout his lifetime?

I will focus on two key moments in Malcolm X's life when his identity transitions—his years in prison at Norfolk Prison Colony in Massachusetts where he discovers the Nation of Islam (NOI), and his estrangement from the Nation of Islam in 1963.
Malcolm X had been involved in criminal activity since his teen years in Roxbury, Boston where he lived with his older sister, Ella. He then moved to Harlem and spent his young adulthood working as a numbers-runner (an illegitimate form of the lottery in poor black communities), a thief, and a drug dealer in Harlem. He got his nickname "Detroit Red" as a result of his upbringing in Lansing, Michigan and the natural red tint of his hair, which he straightened with a lye-based concoction called "congolene." Malcolm X, then Malcolm Little, was sent to prison for burglarizing the homes of wealthy white people. In addition to running numbers, he had become the leader of a ring of thieves which operated in Roxbury and Harlem.
Malcolm was in prison from 1946 to 1952. In 1948, he was imprisoned at Norfolk, along with his brother, Reginald, who introduced him to the NOI's philosophy. Malcolm quit smoking and gambling, refused to eat pork, and spent long hours in the prison library, memorizing the dictionary. He also read historical works, such as Will and Ariel Durant's eleven-volume work, The Story of Civilization. He also began to participate in debate classes. In his autobiography, Malcolm credited the NOI with helping him understand how American society taught him to devalue his blackness and to believe that white people were superior, despite his father's murder by white supremacists. Reflecting on his interactions with white people throughout his life, he began to espouse the NOI's belief that white people were evil. While in prison, he changed his name from "Little" to "X" to symbolize his ignorance of his true African origins.
After he left prison, he quickly became one of the NOI's most charismatic and influential spokespeople. Shortly after meeting the NOI's leader, Elijah Muhammad, Malcolm helped to organize temples in Philadelphia, New York, Boston, and even some in cities in the South. He first worked at a temple in Boston and was soon promoted to Temple No. 7 in Harlem—the largest and most prestigious in the nation, second to the NOI's headquarters in Chicago. He lectured and debated around the world and represented a contrasting vision to Dr. Martin Luther King Jr.'s messages of civil disobedience and integration, believing that black people should defend themselves against violence.
Not long after the assassination of President John F. Kennedy, Malcolm publicly commented on the tragic event as America's "chickens coming home to roost," meaning that white people's toleration of the routine violence inflicted against black people had now come to claim the life of someone they loved. The NOI reprimanded and silenced Malcolm, thereby excusing him from his duties as a proselytizer. He used the time to take a trip around the world and to go on his first pilgrimage to Mecca. While in Saudi Arabia, he has his second epiphany. He began to question the NOI's position on all whites being evil. While on his holy pilgrimage, he drank from the same cup as white men and noticed Muslim men of all colors, united in brotherhood. This realization, coupled with learning that Elijah Muhammad maintained sexual relationships with numerous young women in the NOI, some of whom bore his illegitimate children, led to Malcolm X's exit from the Nation of Islam.
When Malcolm returned to the United States in 1964, he formed the Organization for Afro-American Unity (OAAU). Its purpose was not separatism, as with the NOI, but to build self-sustaining black communities. He was willing to accept help from whites, but only after the organization became solidly sustainable.
On February 21, 1965, Malcolm X was gunned down by members of the NOI. One of the great misfortunes of his assassination is that he was unable to fulfill his vision, which became more predominant in 1965 after the radicalization of the Student Non-Violent Coordinating Committee (SNCC), by then under the leadership of Stokeley Carmichael, and the rise of the Black Panthers. Martin Luther King Jr.'s philosophies about civil rights were also becoming more radical and sought to challenge American power more directly. This was evident in his critiques of the Vietnam War, as well as his work to help low-wage workers organize. King was assassinated while trying to organize sanitation workers in Memphis, Tennessee. We can only dream of what could have been accomplished if the leaders' lives and ideas began to converge.
https://www.britannica.com/biography/Malcolm-X

How would you describe the main character in Nightjohn?

Sarny is the main character in Nightjohn. In the story, Sarny is a spunky, perceptive, and resourceful young girl.
Sarny's courage and resourcefulness is clear from the beginning of the story. She spends time listening to the conversations of Margaret and Alaine as she works in the flower beds. Margaret is Clel Waller's wife, and Alaine is Margaret's sister.
When both women express their disgust at Clel purchasing yet another slave, Sarny's ears perk up. Her unobtrusive spying nets her some interesting information. She learns that slaves are expensive and that a slave can cost over a thousand dollars.
By listening, Sarny learns about the world around her. Her determination to thrive is evident, and her efforts are soon rewarded.
The text tells us that Sarny eventually makes Nightjohn's acquaintance, and it is from him that Sarny learns how to read. For her part, Sarny understands the dangers of being literate. However, she persists, and by the end of the story, Sarny helps Nightjohn teach slaves from surrounding plantations how to read.
In all, Sarny's determination, resourcefulness, and astuteness is clear from Paulsen's text. Like Nightjohn, Sarny is determined to learn how to read and write so that slaves like her can tell the world about the cruelties they have endured.

Sarny is the protagonist in Nightjohn by Gary Paulsen. She is a twelve-year-old slave who has higher aspirations for herself. Her mother is sold, leaving her motherless. There is a misconception that she is not a bright child because her speech is impacted by a birth defect which is referred to in the book as "a stuck tongue.” Some even believe she is a witch. In reality, she is a very introspective, smart child. Nightjohn, a former slave, returns to the South, and when he gives her the opportunity, she easily learns to read and write, proving her intelligence. 
As a slave, Sarny’s literacy sets her up for problems, but she is determined, showing perseverance in the face of adversity. Despite the dangers, she helps Nightjohn as he educates other slaves because she is adamant that literacy is the key to a better life for those who were enslaved.

How might a nation try to overthrow a foreign country that reigns over it?

There have been plenty of revolutions over the course of history. Some of these occur when a country is revolting against the rule of another country that is occupying or in some way controlling it. Here a few examples of how a country in this situation might stage a revolution.
American Revolution: In this case, the American colonies were revolting against the country that founded its very existence—Britain. It involved a full-scale, long-term war that involved a third foreign power (France). This kind of overthrow is very difficult, as it requires a great deal of resources and manpower—manpower that is willing to fight and risk life and limb.
India: Britain colonized India in the 19th century (for the most part, although there had been some colonial activity before that). Eventually, in the first half of the 20th century, the Indian people began in earnest to try to throw off British rule. Gandhi led a nonviolent protest that brought about Indian independence in 1947. This is not the way revolutions usually proceed. It took an exceptional person like Gandhi to pull off a relatively bloodless revolt.
Iran: The Iranians lived under the oppressive Shah of Iran, who was supported by the United States. In this 1979 revolution, a religious group, radical Shiite Muslims, staged a revolution that included taking hostages at the U. S. Embassy. In this case, one group represented how most of the country felt, and exploited that sentiment—throwing off the power of the Shah (and America) in the process. This revolution was not nonviolent.
These are three different ways that a country might overthrow a foreign power. Doubtless there are more.  

In great detail, what are the similarities and differences between the television shows Roseanne and Will and Grace in terms of aesthetic, social implications, possible technical achievements, and relevance to its particular audience?

The similarities between the very popular television sitcoms Roseanne (which ran from 1988-1997) and Will and Grace (which ran from 1998-2006) are that they both focused on segments of the American population that are not always in the television limelight. Behind the laughter they sought to generate, the shows publicized real social issues.
Roseanne dealt with the travails of the working class and their struggles to make a living. Roseanne Conner, a working parent, has many jobs in the show, including initially working in a plastics factory and later working as a waitress and as the owner of a motorcycle repair shop. Her blue collar family and friends struggle with unplanned pregnancies, physical abuse, and illness, and the main characters are noticeably un-Hollywood (some are overweight and decidedly unglamorous). The aesthetic is Mid-American, with a plaid couch and cozy afghan center stage in the Conner family's living room. The show was one of the first that was relevant to working-class Americans and that portrayed their lives and struggles in a sympathetic way, and the show also featured openly gay characters, such as Jackie, who is a lesbian.
Will and Grace is about a Jewish interior decorator who lives with her best friend, a gay lawyer. The show was one of the first to sympathetically portray the life of a gay main character, Will, who struggles at times to tell people about his sexuality. The aesthetic of the show is quite different than that of Roseanne, as Will and Grace live in a very sophisticated, expensive New York City apartment and have glamorous clothes and lives. The show was relevant to the LBGTQ community, as it humanized the lives of gay people and showed their relationships in a sympathetic way. Will and Grace also received several technical awards for film editing, production design, and costume design. Its sets and costumes were far more lavish than those of Roseanne. 

Why does Holmes find the crime in The Hound of the Baskervilles interesting?

I presume that the crime in The Hound of the Baskervilles to which your question refers is the mysterious death of Sir Charles Baskerville, recounted by his friend and neighbor Dr. James Mortimer in chapters 2 and 3 of the work.
It is entirely likely that Sherlock Holmes is excited (though perhaps excitable would be a more accurate word for his reaction) about the case for two reasons. In the first place, he is well-known for being a man of science whose "methods" consist of making inferences based on empirical evidence; indeed, he is a self-described "specialist in crime" (chapter 1), and the case seems somehow to involve the supernatural via the legend of a giant, ghostly, malicious hound. It is a question that would naturally fascinate as rational a thinker as is Holmes, at least until he could prove a thoroughly rational solution.
In the second place, Holmes's fascination with the case is doubly underscored when Dr. Mortimer, whom Holmes calls "the man of science" and "a man of precise mind" (chapter 1) upon meeting him, expresses first his admiration for Holmes's being "the second highest expert in Europe" and "a practical man of affairs," appeals to Holmes for help, and intimates that his own doubts about the legend having come true cause him to "not know what to believe" (chapter 3). For a "trained man of science" (chapter 3), as Holmes calls Dr. Mortimer, to doubt his own education and rationality in the face of a frightening legend intrigues Holmes, who seems never to have been tempted to abandon his practical scientific methods and never to have met a case that could not be explained rationally. But if the doctor can be swayed, Holmes wonders, then what kind of case is it, really?
https://www.gutenberg.org/files/2852/2852-h/2852-h.htm

Why is the king relieved when he finds out that Psyche is the "Accursed"?

The King of Glome is relieved to discover that Psyche is the "Accursed," because he will not then have to die as a sacrifice to placate the Shadowbrute, the so-called god of the Mountain.
In Chapter Five, we learn that Glome has been plagued with famine, drought, and sickness. Other kings, sensing the weakness of the king of Glome, capitalize on Glome's suffering to threaten wars and rebellions. Meanwhile, the priest of Ungit tells the king that a sacrifice must be made to appease the Brute before it is too late. He maintains that the sacrifice must be "perfect" to be accepted as the Great Offering. As a warning, the priest cautions that the king must not shirk his duty, or the people of Glome will burn him alive in his palace.
By now, the king of Glome is beginning to fear that he is to be the Great Offering for the Shadowbrute. Trying to buy time, the king claims never to have heard of the Brute in his time; he tells the priest of Ungit that the Shadowbrute is a tale of his grandmother's. The priest is not to be deterred, however.
He relates that the people have already cast "holy lots" to discover the identity of the "Accursed." In the end, it has been determined that the "Accursed" resides in the king's house. Upon hearing this, the king is stricken with terror (he thinks that the priest is about to announce that he, the king, is to be the sacrifice). So, he calls on his palace guards to kill the temple guards that are gathered at the vicinity of his palace. However, his guards refuse to fight for him. The priest accuses the king of foolishness, telling him that the population is armed and ready to do violence if the king does not capitulate to their will.
Felling cornered, the king pulls out his dagger and threatens to kill the priest. However, the priest manages to repel the king by promising that he will haunt him even after his death. Defeated, the king demands to know who the "Accursed" really is. The priest eventually answers that it is the Princess Istra (or Psyche), the king's third daughter.
Upon hearing this, the king is visibly relieved that he is not to be sacrificed for the good of Glome. He then pretends to be sad that Psyche is to be the sacrifice. Meanwhile, Orual, distressed that her sister is to be the Great Offering and horrified that her father does not mean to save Psyche, tries to beg for Psyche's life. For his part, the king violently flings Orual away, as she falls at his feet in supplication. Distressed beyond endurance at the turn of events, Orual faints; she is revived in the next chapter.

In "A Rose for Emily" who/what is the antagonist?

I think an important question to ask before we discuss the antagonist is who is the protagonist? In response to your question, I would actually argue that Miss Emily Grierson is the antagonist and that the first-person plural "we" who narrates the story is the protagonist. The plot does convey portions of Miss Emily's life, but it more closely follows the thought processes and reactions of the community to Miss Emily's actions. This is one major reason why the story is not told in chronological order. If the narration followed Miss Emily's life's events, rather than the community's response to the events which it can observe, then it seems more likely that it would be written in the order in which she lived them, and it would likely have been revealed to us much sooner that she murdered Homer Barron with poison. However, because her community is the real protagonist, the narrative discusses events as and when they understand them, rather than when they actually take place.

An antagonist can refer to a person, a group of people, an institution, a situation, or any circumstance that opposes the protagonist or lead character in a literary work. In terms of this definition, the lead character, Miss Emily Grierson, is opposed by many.
The city authorities are definitely antagonists. They have constantly been harassing her about unpaid taxes. Miss Emily, however, refuses to give in to their demands and states that she has no taxes and that the authorities have to speak to Colonel Sartoris, the town mayor who originally exempted her from paying any taxes because her father had lent the town some money. The colonel has long passed away, but Miss Grierson refuses to budge. The authorities send her a tax notice every year until her death without ever getting a positive response.
The townspeople, especially Miss Emily's neighbors, are also antagonists. They perpetually gossip about her and continuously express disdain about her and the fact that the Griersons hold "themselves a little too high." The inhabitants are clearly jealous of the family and vent their resentment and ill-feeling behind closed doors. Their expressions of pity whenever she encounters some misfortune are mere platitudes and more condescending than sincere.
Some of her neighbors lodge complaints against her for the stench emanating from her house. This encourages the authorities to send out men to secretly inspect her premises and saturate them with lime. The unpleasant smell eventually disappears.
Miss Emily's two cousins can also be seen as antagonists since their sole purpose seems to be to restrict her freedom. The text also suggests that she finds their presence an unwelcome intrusion. We are informed:

... to give her a chance to get rid of the cousins. (By that time it was a cabal, and we were all Miss Emily's allies to help circumvent the cousins.)

This is patent evidence that she does not want them around and that she wants to focus on Homer Barron's interest in her. Homer's declaration that he is not the marrying kind might make him an antagonist. Miss Emily obviously likes him or even, in her own way, loves him. When she realizes that he has misled her (or perhaps that she has misled herself) and that he will not be hers, she decides to permanently claim at least his body by killing him with arsenic and sleeping next to his corpse.

Consider your relationship with a piece of technology such as a computer. How can I write an essay explaining some positive claims and normative claims about this technology?

While I cannot answer for your own personal experience with this, I can give you the difference between positive and normative claims and allow you to reach your own conclusions.  A normative claim in this case would refer to how the computer should be used.  For example, the claims "college students should use the computer primarily for research" is an example of a normative claim.  This is how the computer should be used under ideal situations--the computer as a tool for good.  
A positive claim is more fact-based--it does not necessarily have to be correct, but it has to be tested in order to be verified.  The positive claim "college students at University X spend at least half their waking hours on the computer" is a positive claim because there is no judgment attached to it.  Also, it can be proven with a survey sent out to students at the university.  
https://www.investopedia.com/ask/answers/12/difference-between-positive-normative-economics.asp

Why are the last two stanzas of "Still I Rise" different from the rest?

Most of the first seven stanzas of Maya Angelou's poem "Still I Rise" include mention of a second-person character ("you") who oppresses the narrator. The "you" the narrator addresses is the white male world that has oppressed the narrator and tried to negate her humanity. The first seven stanzas (with the exception of the third stanza) catalog some of the ways the white male world has tried to make the narrator powerless, including telling "bitter, twisted lies" about her and being offended by the narrator's pride in herself, including her "sassiness" and her "sexiness." In each of these stanzas (except the third), the narrator carries out a kind of metaphorical battle with the "you" who would like to reject and deflate her. Each of these stanzas is four lines long, and in each stanza (except the third), the second and fourth lines rhyme.
The last two stanzas are different in form and content. They are six and seven lines long, respectively, and they do not include any reference to "you." Instead, they feature ways in which the narrator will rise and include the repetition of the line "I rise." They also often begin with "I," stressing the power of the narrator to overcome all the oppression she has been subjected to. These lines are hopeful, as the narrator says she will rise "Into a daybreak that’s wondrously clear." In the last two stanzas, the narrator has broken free of the oppressive past and entered a hopeful future. 

The last two stanzas of Maya Angelou’s poem “Still I Rise” change in purpose and tone from the first seven stanzas. After the first stanza, the poem becomes interrogatory, with the narrator asking a series of provocative questions. She implores readers to consider her “haughtiness” and “sexiness,” and asks if their feelings are stirred by her ability to overcome endemic and pervasive prejudice. In the final two stanzas, she stops questioning the reader and emphatically states her case.
In the final two stanzas, she describes the inner fortitude which allows her to “rise” above the atrocities she and her ancestors endured. In addition, she describes her actions as breaking the bonds for future generations. She does not question this; she declares it to be true.

I am the dream and the hope of the slave.
I rise
I rise
I rise.

How could the ending of "Where Are You Going, Where Have You Been?" been prevented?

The ending could have been prevented by better parenting. If Connie had received more guidance and support from her parents, she may have felt less inclined to test the limits of her emerging sexuality.
The dysfunction in Connie's family is evident in the way her parents treat her. While her father largely ignores her, Connie's mother is antagonistic and critical. She constantly denigrates Connie for her looks. The impression we get of Connie's mother is that she is unhappy with her own fading looks and jealous of Connie's fresh beauty.
Connie's mother constantly compares Connie to June, her older sister. June is twenty-four and works as a secretary at Connie's high school. She has plain looks, but she is a hard worker. When she is home, June is helpful, conscientious, and dutiful. She is constantly praised by her mother and her aunts for these traits. On the other hand, Connie is persistently belittled by the same group of family members.
Because she has such an unhappy home life, Connie often sneaks across the highway to a sleazy diner to meet boys. One day, she sees a strange man watching her as she walks across the restaurant's parking lot. The man drives a gold-colored convertible jalopy. The only words he says to her are "Gonna get you, baby." These words come back to haunt her later when he turns up at her house while she is alone.
The man tells her that his name is Arnold Friend and that he knows everything about Connie and her family. He also tells her that he knows what her parents and sister are doing and how long they will be away. Arnold maintains that Connie is the only girl for him and that he means to have her. He threatens to hurt her family if she calls the police. The story ends on an ambiguous but ominous note. The implication is that Connie leaves with Arnold, but her fate is undetermined.
It is noteworthy to recognize that Joyce Carol Oates received the inspiration for this story from the case of "The Pied Piper of Tucson," a serial killer who murdered young girls and dumped their bodies in the Arizona desert in the mid 1960s. By all indications, Connie may have escaped Arnold's attention if she had never frequented the diner. From the story, we know that Connie met with boys at the diner as an act of desperate rebellion.
To recap, Connie may have been less inclined to test her limits if she had had more support and love in her life.

y = x , y = 3 , x = 0 Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line y = 4

For a region bounded by y=x (slant line), x=0 (along the vertical axis), and y = 3 (horizontal line) revolved about the line y=4 , we may apply the Washer Method. It considers multiple disc with a hole. Basically, it can be just two disc method in which we take the difference of the volume of the bigger and smaller disc.
It follows the formula: A = pi(R_(outer))^2-(r_(i.n.n.e.r))^2]
Let R_(outer) as a function of f and r_(i.n.n.e.r) as function of g .
Then V =pi int_a^b[(f(x))^2-(g(x))^2]dx  or V =pi int_a^b[(f(y))^2-(g(y))^2]dy .
 We use a rectangular strip representation that is perpendicular to the axis of rotation as shown on the attached image. 
For the inner radius, we have:
g(x)=4-3=1
For the outer radius, we have: f(x)=4-x
The boundary values of x will be  a=0 to b=3 .
 The integral to approximate the volume of the solid is:
V=pi int_0^3 [(4-x)^2-1^2] dx
Expand using FOIL method on (4-x)^2 = (4-x)*(4-x) = 16-8x+x^2 and 1^2=1 .
The integral becomes:
V=pi int_0^3 [16-8x+x^2 -1] dx
Simplify:V=pi int_0^3 [15-8x+x^2] dx
Apply basic integration property: int (u+-v+-w) dx= int (u) dx+-int (v) dx+-int (w) dx
V=pi [ int_0^3 15 dx - int_0^3 8xd +int_0^3 x^2 dx]
Apply basic integration property: int c dx = cx and Power rule for integration: int x^n dx= x^(n+1)/(n+1).
V=pi [15x- 8*x^2/2 + x^3/3]|_0^3
V=pi [15x- 4x^2 + x^3/3]|_0^3
Apply definite integration formula: int_a^b f(x) dx = F(b)-F(a).
V=pi [15*(3)- 4(3)^2 +(3)^3/3]-pi [15(0)- 4(0)^2 + (0)^3/3]
V=pi [45- 36+9] -pi [0-0+0]
V = 18pi or 56.55 (approximated value)

sum_(n=1)^oo (-1)^(n+1)/n^2 Determine whether the series converges absolutely or conditionally, or diverges.

To determine the convergence or divergence of the series sum_(n=1)^oo (-1)^(n+1)/n^2 , we may apply Alternating Series Test.
In Alternating Series Test, the series sum (-1)^(n+1)a_n is convergent if:
1) a_n is monotone and decreasing sequence.
2) lim_(n-gtoo) a_n =0
3) a_ngt=0
For the series sum_(n=1)^oo (-1)^(n+1)/n^2 , we have:
a_n = 1/(n^2) which is a decreasing sequence.
As "n " increases, the 1/n^2 decreases.
Then, we set-up the limit as :
lim_(n-gtoo)1/n^2 = 1/oo =0
By alternating series test criteria, the seriessum_(n=1)^oo (-1)^(n+1)/n^2 converges.
The series sum_(n=1)^oo (-1)^(n+1)/n^2 has positive and negative elements. Thus, we must verify if the series converges absolutely or conditionally. Recall:
a) Absolute Convergence:  sum a_n  is absolutely convergent if sum|a_n|   is convergent.  
b) Conditional Convergence:  sum a_n  is conditionally convergent if sum|a_n|  is divergent and sum a_n  is convergent.  
We evaluate the sum |a_n| as :
sum_(n=1)^oo |(-1)^(n+1)/n^2| =sum_(n=1)^oo 1/n^2
Apply the p-series test sum_(n=1)^oo 1/n^p is convergent if pgt1 and divergent if 0ltplt=1 .
The series sum_(n=1)^oo 1/n^2 has p=2 which satisfies pgt1 . Thus, the series sum_(n=1)^oo |(-1)^(n+1)/n^2| is convergent.
 
Conclusion:
The series sum_(n=1)^oo (-1)^(n+1)/n^2 is absolutely convergent since sum |a_n| as sum_(n=1)^oo |(-1)^(n+1)/n^2| is convergent.

Calculus of a Single Variable, Chapter 5, 5.5, Section 5.5, Problem 44

f(t) = 3^(2t)/t
To take the derivative of this function, use the quotient rule (u/v)'= (v*u' - u*v')/v^2.
Applying that, f'(t) will be:
f'(t) = (t * (3^(2t))' - 3^(2t)*(t)')/t^2
f'(t) = (t*(3^(2t))' - 3^(2t) * 1)/t^2
Take note that the derivative formula of an exponential function is (a^u)' = ln(a) * a^u * u' .
So the derivative of 3^(2t) is:
f'(t) = (t*ln(3)*3^(2t) * (2t)' - 3^(2t) * 1)/t^2
f'(t)= (t*ln(3)*3^(2t) * 2 - 3^(2t) * 1)/t^2
f'(t)= (2t ln(3)*3^(2t) - 3^(2t))/t^2
f'(t) = (3^(2t)(2tln(3)-1))/t^2

Therefore, the derivative of the function is f'(t) = (3^(2t)(2tln(3)-1))/t^2 .

Single Variable Calculus, Chapter 3, 3.5, Section 3.5, Problem 26

Determine the derivative of the function $\displaystyle G(y) = \frac{(y-1)^4}{(y^2+2y)^5}$

$
\begin{equation}
\begin{aligned}
G'(y) &= \frac{(y-1)^4}{(y^2+2y)^5}\\
\\
G'(y) &= \frac{d}{dy} \left[ \frac{(y-1)^4}{(y^2+2y)^5}\right]\\
\\
G'(y) &= \frac{\left[ (y^2+2y)^5 \cdot \frac{d}{dy} (y-1)^4\right] - \left[ (y-1)^4 \cdot \frac{d}{dy} (y^2+2y)^5 \right]}{\left[ (y^2+2y)^5\right]^2}\\
\\
G'(y) &= \frac{\left[ (y^2+2y)^5 \cdot 4(y-1)^3 \frac{d}{dy} (y-1)\right] - \left[ (y-1)^4 \cdot 5(y^2+2y)^4 \frac{d}{dy} (y^2+2y) \right]}{(y^2+2y)^{10}}\\
\\
G'(y) &= \frac{\left[ (y^2+2y)^5 \cdot 4(y-1)^3 (1)\right] - \left[ (y-1)^4 \cdot 5 (y^2 +2y)^4 (2y+2)\right]}{(y^2+2y)^{10}}\\
\\
G'(y) &= \frac{\left[ 4(y^2+2y)^5(y-1)^3\right]-\left[ 5(y-1)^4 (y^2+2y)^4(2y+2)\right]}{(y^2+2y)^{10}}\\
\\
G'(y) &= \frac{(y-1)^3(y^2+2y)^4\left[ 4(y^2+2y)-5(y-1)(2y+2) \right]}{(y^2+2y)^{10}}\\
\\
G'(y) &= \frac{(y-1)^3 \left[ 4y^2 + 8y - 5(2y^2+ \cancel{2y} - \cancel{2y} - 2)\right]}{(y^2+2y)^6}\\
\\
G'(y) &= \frac{(y-1)^3(4y^2+8y-10y^2+10)}{(y^2+2y)^6}\\
\\
G'(y) &= \frac{(y-1)^3(-6y^2+8y+10)}{(y^2+2y)^6}
\end{aligned}
\end{equation}
$

Precalculus, Chapter 5, 5.3, Section 5.3, Problem 33

Find all solutions to the equation 2cos^2(x)+cos(x)-1=0 in the interval [0,2pi).
2cos^2(x)+cos(x)-1=0
(2cos(x)-1)(cos(x)+1)=0
Set each factor equal to zero and solve for the x value(s).
2cos(x)-1=0
2cos(x)=1
cos(x)=1/2
x=pi/3,x=(5pi)/3

cos(x)+1=0
cos(x)=-1
x=pi

What is the meaning of Nick's statement—"After Gatsby's death the East was haunted for me like that"?

When Nick first arrives on Long Island, he is full of hope that he can start a career and build a fulfilling life on the East Coast—having just left he Chicago area. However, by the time he has gotten caught up in breathing in the "foul dust" of Tom and Daisy's life and has suffered through Gatsby's death, he has lost his taste for the East.
Nick explains that the East (for him) has come symbolize grotesqueness, corruption, and death: it haunts his dreams. In other words, he has nightmares about his time spent there, seeing it like the distorted landscape of an El Greco painting. He knows he is not being objective—he even states that his vision is "distorted," but he can't correct this. He knows he must escape to the Midwest, which he regards as a place of purity in contrast to what he has witnessed.
Nick uses innocent, idyllic imagery to convey his perception of the Midwest:

That’s my middle west—not the wheat or the prairies or the lost Swede towns but the thrilling, returning trains of my youth and the street lamps and sleigh bells in the frosty dark and the shadows of holly wreaths thrown by lighted windows on the snow.

This perception may be idealized and inaccurate, but it reveals that the Midwest is a place of refuge for Nick, who is fleeing from the "haunting" corruptions he identifies with the East.

Following Gatsby's lonely, depressing funeral, which no one attends except Nick and Gatsby's father, Nick decides to leave the East Coast and return home to the Mid-West. Nick then mentions,

After Gatsby's death the East was haunted for me like that, distorted beyond my eyes' power of correction.

Nick's comment reveals that Gatsby's death traumatized him, and he is no longer disillusioned by the allure of the East Coast. During the summer in which Nick became acquainted with Jay Gatsby, he came to view Gatsby as a genuine friend and positive person. Despite Gatsby's many flaws, Nick felt a connection to Gatsby and could perceive infinite hope in Gatsby's character. After experiencing New York and Long Island, Nick is appalled by the immoral behavior, selfishness, and greed he's witnessed throughout the summer. In contrast, Nick views the Mid-West as a moral haven, where individuals are genuine, compassionate people who are unconcerned with maintaining appearances. Nick has no desire to return to the East Coast, and whenever he thinks about New York and Long Island, he is haunted by Gatsby's death and his experience among the upper-class citizens.

Who is the protagonist in the play Topdog/Underdog?

In Topdog/Underdog, it could easily be argued that of the two brothers, Booth and Lincoln, either is the true protagonist. However, when we consider what makes a protagonist a protagonist—progression, psychological depth, and so on—it can be more soundly argued that Lincoln is the true protagonist of the play. Lincoln is certainly the more relatable of the two, particularly since he seems to try harder to "clean-up" and make a better life for himself.
Booth (played on the stage originally by Don Cheadle and later by the MC/actor Mos Def), serves as a sort of antagonist for Lincoln. Booth challenges Lincoln literally and figuratively; he serves as a sort of foil for Lincoln, and he will eventually cause Lincoln's downfall.
Lincoln, meanwhile, struggles existentially with his own purpose. His gig as dressing up as Abraham Lincoln pays very little—far less than what the white actors make. He wants more from life while struggling with the desire to return to the streets to hustle cards. This is what makes Lincoln the true protagonist: he has a foreseeable goal and the desire—and ability—to change. He aims to find meaning and purpose in his life while avoiding the dangerous behavior that led to his friend being shot. His relatability as well as his psychological depth make him the main protagonist of the play.

Since the protagonist is the central character in a literary work, Booth fits this role because he becomes more developed as a character than the others. Unlike some protagonists, however, Booth is no hero; instead, he is morally weak, and even possesses evil traits.
For one thing, Booth manufactures his own antagonist in Lincoln since he often perceives his brother as an adversary when he is not. Certainly, he blames his dissatisfaction with his life on his brother--not unlike his namesake and that man's attitude about Abraham Lincoln. In one scene, for instance, as the brothers are having supper, Booth recalls the day his mother packed her things and abandoned them. He irrationally places blame upon his brother for his sense of abandonment since he was alone after she left. (He was truant that day whereas Lincoln had gone on to school.) Further, when Booth wants to become a dealer of three-card monte, he blames his brother for not helping him learn how to "hustle," despite knowing that Lincoln lost all his money this way, as well as having witnessed an acquaintance murdered over this game. Yet, when Lincoln asks Booth to help him practice his death scene in a more dramatic manner after he is "shot" by those who come to the arcade, Booth refuses.
Clearly, it is Booth's moral weakness--not unlike that of classic protagonists--which causes his downfall. He continues to blame others for his lack of success in life and does not try to obtain honest work. Instead, he tries to manipulate others. In another scene in which Link refuses to teach him how to hustle cards, Booth tells his brother,

Here I am trying to earn a living and you standing in my way. YOU STANDING IN MY WAY, LINK.

So much does Booth project his own failures onto his brother that he eventually shoots him in a tragic and ironic imitation of the historical scene of John Wilkes Booth and Abraham Lincoln.
https://literarydevices.net/protagonist/

Explain how the attitudes toward Great Britain and France shaped American politics in the late eighteenth century.

Attitudes toward Great Britain and France served to polarize the already bitter and fractious world of American politics in the late eighteenth century. Democratic-Republicans such as Thomas Jefferson were generally sympathetic to France, seeing it as the home of republican liberty.
The French had wholeheartedly supported the Americans in their revolution against the British, leading to the development of close ties between the two countries. Even when the French Revolution took a violent turn, Jefferson and his fellow Democratic-Republicans were reluctant to give up their emotional and intellectual attachment to Republican France.
Jefferson's opponents in the Federalist Party were more sympathetic toward the British. They believed that the British political system, for all its faults, provided a fair measure of stability as well as protection for private property. This led to their being accused by Democratic-Republicans of supporting a restoration of the monarchy.
There was no truth to such accusations, but the Federalists were unashamed elitists with a profound, abiding distrust of democracy. This made it all too easy for their opponents to label them as crypto-monarchists, and their leader, President Adams, as wanting to make himself king.
In due course, these substantial differences between the two parties would define the contours of American politics for generations to come. In the ensuing decades, it would be the intellectual heirs of Republican France, the Democratic-Republicans, and after them, the Democrats, who would dominate the national political scene.

American policies toward Great Britain and France were impacted by differing beliefs toward these countries and by the actions of these countries. Under the leadership of President Washington, the United States remained neutral in foreign affairs. Even though Great Britain and France were fighting against each other, President Washington knew the United States was in no position to make enemies by taking sides in a conflict that the United States couldn’t afford to fight.
Toward the end of Washington’s presidency, two political parties were forming in the country. The Federalist Party believed the United States should be friendly with Great Britain and support them while the Democratic-Republican Party felt the United States should work closely with France and support them. President Adams continued the American policy of neutrality, even though it cost him politically. When France refused to meet with American negotiators regarding the seizing of ships and the impressment of sailors and instead demanded a bribe and a loan in order to meet, many Americans felt the United States should go to war against France. However, President Adams felt going to war was not in the best interests of the United States, and he worked out a negotiated settlement with France.
Both political parties were upset with the actions of France and Great Britain. The United States continuously demanded that American ships not be seized and that the impressment of American sailors come to an end. This was an ongoing concern the United States had with both countries as the 1800s approached.

By the late eighteenth century, Britain and France were fighting a war that would soon engulf Europe and ultimately the world.  The United States, following the presidency of George Washington, was beginning to form political parties.  These two parties were the Democratic-Republicans and the Federalists.  
The Democratic-Republicans, under the leadership of Thomas Jefferson, favored France in the war between Britain and France.  They saw the French Revolution as an extension of the American Revolution and a demonstration of the principle that everyone desires freedom.  Even as the French Revolution grew bloody and anti-clerical, many in the party still had warm feelings toward France.  They also wanted to give France aid in the war, as France was the colonists' primary ally during the American Revolution.  
On the other hand, the Federalists favored Britain in their war with France.  The Federalists saw chaos in the Reign of Terror and thought that all of the killings would be the undoing of civilization.  They were also repulsed by the confiscation of church property in France and the killings of the clergy, even though many in the Federalist party were Protestant.  They saw in Britain a source of political stability and also the United States' main trading partner.  John Adams, president during this time period, even signed off on the Alien and Sedition Acts, which were an attempt to keep anti-British radicals out of the country and curtail anti-government sentiment.  
Both parties did not approve of the ship seizures by Britain and France after United States sailors tried to run the blockade and trade in Europe.  John Adams waged the "Quasi-War" with the French navy during this time period—it would continue until Napoleon came to power and agreed to stop seizing American shipping.  Britain would continue to do so and also pressed American sailors into its own merchant marine.  This would ultimately lead to the war of 1812.

College Algebra, Exercise P, Exercise P.1, Section Exercise P.1, Problem 22

Suppose at a certain car rental agency a compact car rents for \$30 a day and 10 cents a mile.
a.) How much does it cost to rent a car for 3 days if the car is driven 280 miles?
Since the rental charge for a car per day is \$30 and 10 cents per mile. The total cost would be (suppose that $T$ is the total cost).

$
\begin{equation}
\begin{aligned}
T &= \$ 30 (3) + 10 \text{cents} (280 \text{mi})\\
\\
T &= \$ 30(3) + \$ \frac{10}{100} (280 \text{mi})\\
\\
T &= 90 + 0.1(280)\\
\\
T &= 90 + 28\\
\\
T &= \$ 118 && \text{Cost to rent a car for 3 days and 250 miles}
\end{aligned}
\end{equation}
$


b.) Find a formula that models the cost $C$ of renting this car for $n$ days if it is driven $m$ miles.
$C = \$ 30(n) + 10 \text{cents} (m)$ model

c.) If the cost for a 3-day rental was \$ 140, how many miles was the car driven?
From the model in part(b), we solve $C = \$ 30(n) + 10 \text{cents}(m)$ for $m$.

$
\begin{equation}
\begin{aligned}
C - \$ 30(n) &= \$ 30(n) + 10 \text{cents} (m) - \$ 30(n) && \text{Subtract both sides by \$30 }(n)\\
\\
\frac{C- \$30 (n)}{10\text{cents}} &= \frac{\cancel{10\text{cents}}(m)}{\cancel{10\text{cents}}} && \text{Divide both sides by 10cents}\\
\\
m &= \frac{C - \$30 (n)}{10 \text{cents}} \text{ model}\\
\\
m &= \frac{\$ 140 - \$ 30(3)}{10 \text{cents}} && \text{Substitute } C = \$ 140 \text{ and } n = 3\\
\\
m &= \frac{\$ 140 - \$ 30(3)}{\$ 0.1} && \text{Recall that } \$ 1 = 100 \text{cents}\\
\\
m &= \frac{140-90}{0.1} && \text{Simplify}\\
\\
m &= 500 \text{miles}
\end{aligned}
\end{equation}
$

How does the War Department feel about Geronimo’s story being told?

The War Department is surprisingly relaxed about the publication of Geronimo's story. It openly acknowledges that his manuscript is "an interesting autobiography of a notable Indian...;" an important work that tells the story from the side of the Native-Americans. However, the department does object to a number of individual passages in the manuscript. The first one concerns an attack by U.S. soldiers upon an Indian tent at Apache Pass, yet Geronimo's account is confirmed by contemporary news reports.
The War Department also objects to Geronimo's criticism of General Crook, but the publishers of the book maintain that it's simply a matter of private opinion and doesn't materially concern the history of the Apaches. Geronimo criticizes another officer General Miles, accusing him of bad faith. The publishers recognize that General Miles concluded a treaty with the Apaches, one whose terms were subsequently violated when Geronimo and other Apache warriors were actively prevented from returning to their tribal lands. However, they conclude that the unfair treatment meted out to the Apaches was the responsibility of the government and not of General Miles personally.

3^(x+4)=6^(2x-5) Solve the equation.

3^(x+4) = 6^(2x-5)
To solve, take the natural logarithm of both sides.
ln (3^(x+4)) = ln (6^(2x-5))
To simplify each side, apply the logarithm rule ln (a^m) =m*ln(a) .
(x+4)ln(3) = (2x-5) ln (6)
xln(3)+4ln(3) = 2xln(6) - 5ln(6)
Then, bring together the terms with x on one side of the equation. Also, bring together the terms without x on the other side of the equation.
xln(3) - 2xln(6) = -4ln(3) -5ln(6)
At the left side, factor out the GCF.
x(ln(3) - 2ln(6)) =-4ln(3) -5ln(6)
And, isolate the x.
x = (-4ln(3) - 5ln(6))/(ln(3)-2ln(6))
x~~5.374
Therefore, the solution is x~~5.374 .

Beginning Algebra With Applications, Chapter 5, 5.2, Section 5.2, Problem 118

Graph $3x-4y = 4$ using a graphing device. Verify that the graph has the correct $x-$ and $y$-intercepts.







$x$-intercept:


$
\begin{equation}
\begin{aligned}

3x-4y =& 4
&& \text{Given equation}
\\
3x-4(0) =& 4
&& \text{To find the $x$-intercept, let } y = 0
\\
3x =& 4
&& \text{Divide by } 3
\\
x =& \frac{4}{3}
&&

\end{aligned}
\end{equation}
$


The $x$-intercept is $\displaystyle \left( \frac{4}{3},0 \right)$

$y$-intercept:


$
\begin{equation}
\begin{aligned}

3x-4y =& 4
&& \text{Given equation}
\\
3(0)-4y =& 4
&& \text{To find the $y$-intercept, let } x=0
\\
-4y =& 4
&& \text{Divide by } -4
\\
y =& -1
&&

\end{aligned}
\end{equation}
$



The $y$-intercept is $(0,-1)$

What are two examples of deus ex machina in The Man who Came to Dinner by George S. Kaufman and Moss Hart?

The Latin phrase deus ex machina was originally used to describe a specific plot device used in Roman and Greek theater. Many Greek and Roman tragedy writers used this device to resolve complicated plots, which could not be resolved otherwise. Today, the same phrase is used to describe a fictional situation where we are presented with something completely unexpected but also implausible, which suddenly helps resolve the plot. The resolution usually comes in the form of a new event or character. 
The first example of a deus ex machina is the arrival of Banjo. Sheridan Whiteside, up to this point, has been an insufferable tyrant to everyone around him, including his assistant Maggie. He has, in fact, broken up Maggie's relationship with Jefferson by bringing Lorraine Sheldon into the picture, a vain and superficial (albeit beautiful) actress. There is no apparent way to solve the situation, and, in any case, Whiteside appears not to care about anyone but himself, so there really would be no reason for him to suddenly become a decent human being. However, Banjo, his friend who is also a famous Hollywood comedian, pushes Whiteside to be a better person. Together, they decide to do something to eliminate Lorraine, as she now represents a big problem for Maggie's relationship.
At this point we encounter the second deus ex machina. Banjo and Whiteside decide that the best way to get rid of Lorraine is to trap her in an ancient mummy case, which conveniently has been delivered to Whiteside earlier in the story (although not for that purpose). Even more conveniently, Banjo happens to have a plane, so the next step is to put the mummy case containing Lorraine on his plane and then fly her out to Nova Scotia. As is plain to see, this is a totally implausible situation; however, it allows the plot to detangle and find its resolution.

Deus ex machina refers to a plot device where a complex problem or tricky situation is rapidly resolved via the swift involvement of some unanticipated turn of events, or via the appearance of a character or item. This plot device is called deus ex machina because the turn of events is so sudden and unexpected, it’s as if the hand of God is intervening in the narrative. The play The Man Who Came to Dinner, by George S. Kaufman and Moss Hart contains several moments of deus ex machina. One example would be the sudden unexpected arrival of the film comedian Banjo, who comes to the immediate aid of Sherry, acting as an important force in resolving the third act. Another incident of this device is when Sherry suddenly notices the picture of Harriet Stanley in her youth and suddenly realizes she is an infamous killer and thus blackmails her successfully, allowing Sherry and Banjo to move the Egyptian mummy case onto Banjo’s plane. The final and most memorable incident of this device is when all the most pressing elements of the plot are resolved and Sherry slips on a patch of ice again, mimicking the inciting incident of the first act.

A deus ex machina is an unexpected event in the final act of a play or movie that could be considered contrived and a way to wrap up the plot. One example in the third act of The Man Who Came to Dinner is the use of the Egyptian mummy case sent to Sherry from the Khedive of Egypt. Sherry, who is repentant about having allowed Lorraine to try to seduce Bert, gets Banjo to help him lock Lorraine in the mummy case. The mummy case is a deus ex machina that the playwright uses to get rid of Lorraine so that Maggie and Bert can get married. The second example of a deus ex machina is the photograph of Harriet Stanley that Sherry catches sight of. He then suddenly realizes that she was a murderess and uses this information to convince Mr. Stanley to help him get the mummy case, which contains Lorraine, onto a plane. A final example of a deus ex machina is when Sherry slips on the stairs on the way out and threatens the Stanley family with another lawsuit unless they let him stay. 

College Algebra, Chapter 8, Review Exercises, Section Review Exercises, Problem 18

Determine the center, vertices, foci and asymptotres of the hyperbola $\displaystyle \frac{x^2}{49} - \frac{y^2}{32} = 1$. Then, sketch its graph

The hyperbola has the form $\displaystyle \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ with center at origin and horizontal transverse axis since the denominator
of $x^2$ is positive. This gives $a^2 = 49$ and $b^2 = 32$, so $a = 7, b = 4\sqrt{2}$ and $c = \sqrt{a^2 + b^2} = \sqrt{49+32} = 9$
Then, the following are determined as

$
\begin{equation}
\begin{aligned}
\text{center } (h,k) && \rightarrow && (0,0)\\
\\
\text{vertices } (\pm a,0)&& \rightarrow && (\pm 7,0)\\
\\
\text{foci } (\pm c, 0) && \rightarrow && (\pm9, 0)\\
\\
\text{asymptote } y = \pm \frac{b}{a}x && \rightarrow && y = \pm \frac{4\sqrt{2}}{7}x
\end{aligned}
\end{equation}
$

Therefore, the graph is

Why did Buck follow Thornton around everywhere?

Buck follows Thornton around for the simple reason that he's become deeply attached to him. Although Buck is beckoned to the forest by the call of the wild, he always returns to Thornton's fireside whenever he gets the opportunity. He doesn't have to do this; he does it because he wants to. This demonstrates that his bond with Thornton is based upon a deep, abiding affection rather than fear, as if often the case with relationships between man and dog.
At the same time, however, this relationship, though built upon great love and affection, isn't really good for Buck in the long-term. However kind Thornton may be, he's still in a position of dominance over Buck. This stands in contrast with Buck's natural domain in the forest, where he can truly be his own master, and not beholden to anyone.
Though a genuinely good man on the whole, Thornton's only human. And we see him abuse Buck's love and loyalty when he makes a wager that his dog can pull a thousand-pound sled, break it out of its runners, and "walk off." In effect, Thornton is exploiting Buck; risking his health and welfare for the sake of a bet. But Buck goes along with the stunt, not because he's forced to, but because he wants to out of loyalty to his master. Even so, Buck's true nature is being suppressed by his relationship to Thornton, and he needs to hark the call of the wild before it's too late, before he loses his identity completely.

Calculus: Early Transcendentals, Chapter 5, 5.2, Section 5.2, Problem 11

You need to use the midpoint rule to approximate the interval. First, you need to find Delta x , such that:
Delta x = (b-a)/n
The problem provides b=2, a=0 and n = 5, such that:
Delta x = (2-0)/5 =2/5
Hence, the following 5 intervals of length 2/5 are: [0,2/5], [2/5,4/5], [4/5,6/5], [6/5,8/5],[8/5,2].
Now, you may evaluate the integral such that:
int_0^2 x/(x+1) dx = Delta x(f((0+2/5)/2) + f((2/5+4/5)/2) + f((4/5+6/5)/2) + f((6/5+8/5)/2) + f((8/5+2)/2) )
int_0^2 x/(x+1) dx = 2/5(f(2/10) + f(6/10) + f(1) + f(14/10) + f(18/10))
int_0^2 x/(x+1) dx = 2/5(2/(10(2/10+1)) + 6/(10(6/10+1)) + 1/2 + 14/(10(14/10+1) + 18/(10(18/10+1) )
int_0^2 x/(x+1) dx = 2/5(2/12 + 6/16 + 1/2 + 14/24 + 18/28)
int_0^2 x/(x+1) dx = 2/5(1/6 + 3/8 + 1/2 + 7/12 + 9/14)
int_0^2 x/(x+1) dx = 2/5(1/6 + 3/8 + 1/2 + 7/12 + 9/14)
int_0^2 x/(x+1) dx = 2/5(0.1666 + 0.3750 + 0.5 + 0.5833 + 0.6428)
int_0^2 x/(x+1) dx = 0.9070
Hence, approximating the definite integral, using the midpoint rule, yields int_0^2 x/(x+1) dx = 0.9070.

Calculus of a Single Variable, Chapter 9, 9.3, Section 9.3, Problem 22

sum_(n=1)^oon/(n^4+2n^2+1)
The integral test is applicable if f is positive, continuous and decreasing function on the infinite interval [k,oo) where k>=1 and a_n=f(x) . Then the series converges or diverges if and only if the improper integral int_k^oof(x)dx converges or diverges.
For the given series a_n=n/(n^4+2n^2+1)
Consider f(x)=x/(x^4+2x^2+1)
f(x)=x/(x^2+1)^2
From the attached graph of the function, we can see that the function is continuous, positive and decreasing on the interval [1,oo)
We can also determine whether f(x) is decreasing by finding the derivative f'(x) such that f'(x)<0 for x>=1 .
Apply the quotient rule to find the derivative,
f'(x)=((x^2+1)^2d/dx(x)-xd/dx(x^2+1)^2)/(x^2+1)^4
f'(x)=((x^2+1)^2-x(2(x^2+1)2x))/(x^2+1)^4
f'(x)=((x^2+1)(x^2+1-4x^2))/(x^2+1)^4
f'(x)=(-3x^2+1)/(x^2+1)^3
f'(x)=-(3x^2-1)/(x^2+1)^3<0
Since the function satisfies the conditions for the integral test, we can apply the integral test.
Now let's determine the convergence or divergence of the improper integral as follows:
int_1^oox/(x^2+1)^2dx=lim_(b->oo)int_1^bx/(x^2+1)^2dx
Let's first evaluate the indefinite integral intx/(x^2+1)^2dx
Apply integral substitution:u=x^2+1
=>du=2xdx
intx/(x^2+1)^2dx=int1/(u^2)(du)/2
=1/2int1/u^2du
Apply the power rule,
=1/2(u^(-2+1)/(-2+1))
=-1/(2u)
Substitute back u=(x^2+1)
=-1/(2(x^2+1))+C where C is a constant
Now int_1^oox/(x^2+1)^2dx=lim_(b->oo)[-1/(2(x^2+1))]_1^b
=lim_(b->oo)-1/2[1/(b^2+1)-1/(1^2+1)]
=-1/2[0-1/2]
=1/4
Since the integral int_1^oox/(x^4+2x^2+1)dx converges, we conclude from the integral test that the series sum_(n=1)^oon/(n^4+2n^2+1) converges.

John Proctor admits he does not go to church regularly. What reason does he give?

In act 1, Rebecca suggests that Parris send Reverend Hale back to Beverly in order to avoid conflict among the citizens, and she warns that there is "prodigious danger in the seeking of loose spirits." Thomas Putnam disagrees with Rebecca Nurse and demands that Reverend Parris send for Reverend Hale. John Proctor challenges Thomas Putnam, who brings up the fact that John Proctor has not attended a Sabbath meeting since the winter. Proctor responds by saying that he has trouble traveling five miles into town to listen to Reverend Parris only preach about hellfire and damnation. He also tells Reverend Parris that there are many other citizens who refuse to attend Sunday services because of his harsh, depressing sermons.
In act 2, Reverend Hale visits Proctor's home in order to search for evidence of witchcraft and to follow up on recent accusations. Hale also brings up the fact that John has only been to church twenty-six times in seventeen months and asks Proctor to explain his absences. Proctor responds by mentioning that his wife has been sick all winter and brings up the fact that Parris only preaches about golden candlesticks and hellfire. Overall, John Proctor admits that he does not go to church regularly because it is inconvenient for him to travel five miles to hear Reverend Parris preach about miscellaneous objects and hellfire.

Who is the "blonde assassin"?

One interpretation is that the "blond assassin" in the poem refers to the frost mentioned two lines previously. The frost is personified as beheading "any happy Flower," and this personification is continued in the line, "The blond assassin passes on." The frost is described as blond perhaps to compound the impression that it is cold, not just literally but emotionally. It "beheads" the flower with no hesitation or compassion. The word assassin also emphasizes this idea that the frost is a cold, unemotional killer.
A second interpretation is that the "blond assassin" refers to the sun. The word "blond" would be a more fitting description of the yellow color of the sun. The sun also might be an assassin in the sense that it looks on seemingly with indifference ("The Sun proceeds unmoved") rather than prevent the frost from beheading the flower. It thus might be said to contribute to the death of the flower, indirectly. The sun also presumably kills the frost, in accordance with the natural cycle of seasons that Dickinson alludes to in the poem, and so might be considered an "assassin" in this sense too.

In this very short poem, the "blond assassin" is winter or, at least, the frosty cold that comes ahead of the actual season. In the first stanza, the narrator says that the "frost beheads" the happy flower with "accidental power" while the unsuspecting flower plays (as in summer or early fall); in other words, the frost doesn't mean to kill the flower, but it cannot help it because this is simply what frost, what the cold, does. As a killer, the narrator refers to the frost as an "assassin" in line 5, and the adjective "blond" seems to refer to the frost's coloring. It does not refer to the frost's blond hair, but, rather, the frost is very pale and fair, like blond hair is. This paleness or fairness is what is referred to by the word "blond."

One of the reasons The Tempest is considered a comedy is because Prospero forgives the men who have betrayed him. When do you think Prospero makes the decision to forgive them? Are there any characters that deserve an apology from Prospero?

Throughout the play, Prospero has been dictatorial and controlling. But in the final act he becomes forgiving. The play offers three reasons for this change.
First, Ariel motivates Prospero to forgive. Ariel reports to Prospero that his enemies are sorrowful, pointing out particularly that Gonzalo is crying so hard that tears run down his beard. Ariel says that if Prospero were to see them as Ariel has, Prospero would forgive them. Ariel gives the following description of them to his master:

The King,His brother and yours, abide all three distracted,And the remainder mourning over them, Brimful of sorrow and dismay; but chieflyHim that you termed, sir, the good old Lord Gonzalo.His tears run down his beard like winter's dropsFrom eaves of reeds. Your charm so strongly works 'emThat if you now beheld them, your affections Would become tender.

Prospero asks Ariel if he really thinks his heart would soften if he saw his enemies. Ariel doesn't say what he thinks Prospero would feel but instead states:

Mine would, sir, were I human.

By saying this, Ariel communicates that forgiveness is a human quality. This encourages Prospero to say his own heart will become forgiving. How can he, he says, not forgive his fellow human beings when a spirit, which is "but air," feels compassion towards them? Prospero thus declares that he cannot be less kind and forgiving than a nonhuman. Ariel, by modeling merciful feelings, shames Prospero into forgiveness. Prospero states:

Hast thou, which art but air, a touch, a feeling Of their afflictions, and shall not myself,One of their kind, that relish all as sharplyPassion as they, be kindlier moved than thou art?

A second reason Prospero offers for forgiveness is that it is the rarer, and hence nobler, path. We see in Prospero's forgiveness a man who wants to be thought well of by others. He will take the high road, and in so doing, set himself apart from most people:

The rarer action isIn virtue than in vengeance

Finally, Prospero forgives his brother Antonio for stealing his kingdom, because he knows Antonio must return it to him. Prospero says to Antonio:

For you, most wicked sir, whom to call brotherWould even infect my mouth, I do forgive Thy rankest fault, all of them, and requireMy dukedom of thee, which perforce I knowThou must restore.

This isn't the fullest form of forgiveness, as we note that Prospero still feels it would "infect" his mouth to call Antonio his brother. Obviously, he still has harsh feelings towards his sibling. It could also be called a transactional kind of forgiveness to forgive because you are going to get something in return. Nevertheless, Propero is making an attempt to do the right thing. He may still be angry at his brother, but he will no longer seek revenge against him. This is a step in the right direction.
However, Prospero never forgives Caliban. Caliban does deserve an apology. He helped Prospero when Prospero first shipwrecked on the island, teaching him to survive. Once Prospero learned this, rather than respond gratefully, he enslaved Caliban. Many critics have read this as colonialism in microcosm: whites coming to a foreign place, relying on the native dwellers to survive, and then betraying, harming, and enslaving them—and justifying doing so by labeling them subhuman. Caliban does strike back against Prospero, but one could argue that Caliban is more sinned against than sinning.

How can a make a thesis statement out of these four points? My first language and how I learned it New languages learned when and how I use them Why I want to learn English, and how other languages help me to learn new ones. What would be a good topic sentence for each paragraph? I am not so good at this, and feel I need to understand how you think around the making of both thesis and topic sentences. Please use my selection of points as an example.

Your thesis statement provides the reader of your essay with an overall statement about what you will be proving. It could be something along the lines of, "I learned Spanish as my first language by growing up in Colombia, and when I learned English to attend high school and college in the United States, my knowledge of Spanish grammar, the differences between spoken and written language in Spanish, and my ability to write in Spanish all helped me learn English." The first paragraph in your essay would include more background about your first language and when and why you started to learn English.
Each later paragraph in your essay would examine in further depth one of these points. For example, in your first body paragraph (which comes after the introduction), the topic sentence might be something along the lines of, "My knowledge of Spanish grammar helped me learn English." The topic sentence contains the main idea that you will discuss in that paragraph--in this case the reason why learning your first language (or languages) helped you learn other languages. The paragraph would flesh out this idea and include more details about how learning grammar in your first language helped you learn English grammar.
The next body paragraph might begin with the topic sentence "Learning the differences between spoken and written language in Spanish when I was younger helped me understand these differences in English." The rest of the paragraph would include details about this point. The third body paragraph might start with the topic sentence, "My knowledge of how to write in Spanish helped me learn to write in English," and then the rest of the paragraph would expand on this idea and provide details to support it. Finally, you would conclude your essay with a concluding paragraph that restates your main idea.

Single Variable Calculus, Chapter 8, 8.2, Section 8.2, Problem 20

Determine the integral $\displaystyle \int \cos^2 x \sin 2x dx$


$
\begin{equation}
\begin{aligned}

\int \cos^2 x \sin 2x dx =& \int \cos^2 x (2 \sin x \cos x) dx \qquad \text{Apply Trigonometric Identity } \sin 2x = 2 \sin x \cos x
\\
\\
\int \cos^2 x \sin 2x dx =& 2 \int \cos^3 x \sin x dx

\end{aligned}
\end{equation}
$


Let $u = \cos x$, then $du = - \sin x dx$, so $\sin x dx = -du$. Thus,


$
\begin{equation}
\begin{aligned}

2 \int \cos^3 x \sin x dx =& 2 \int u^3 \cdot -du
\\
\\
2 \int \cos^3 x \sin x dx =& -2 \int u^3 du
\\
\\
2 \int \cos^3 x \sin x dx =& -2 \left( \frac{u^{3 + 1}}{3 + 1} \right) + c
\\
\\
2 \int \cos^3 x \sin x dx =& \frac{-2u^4}{4} + c
\\
\\
2 \int \cos^3 x \sin x dx =& \frac{-u^4}{2} + c
\\
\\
2 \int \cos^3 x \sin x dx =& \frac{-(\cos x)^4}{2} + c
\\
\\
2 \int \cos^3 x \sin x dx =& \frac{-\cos^4 x}{2} + c


\end{aligned}
\end{equation}
$

Where did the earliest forms of life come from?

The Descent of Man by Charles Darwin one a landmark work in what became the science of evolutionary biology. In this work, Darwin noticed the correspondences among various species. He also was aware of how selective breeding worked. Farmers had for millennia been breeding animals for desired traits to develop horses that ran faster or could haul heavier weights, dogs that were efficient hunting partners, or food animals that were hardy, placid, and able to efficiently convert fodder to meat. By combining these types of knowledge with his encounters with primitive peoples, he argued that humans, and in fact, all animals, were the product of evolution.
The key mechanisms he used to explain evolution were random mutation and natural selection. Random mutation is something we can all observe when we see, for example, polydactyl cats or other such minor variations in species. Natural selection ensures that if a mutation helps the species survive, it will propagate through the species, eventually creating a distinct subspecies or even a new species. Thus Darwin argues that humans evolved from non-human primates through precisely this mechanism of random mutation and natural selection.
The very earliest forms of life, of course, were not human but were unicellular organisms that formed nearly four billion years ago, probably from chemical reactions.

What food did Brian find to eat?

In the book Hatchet, the thirteen-year-old protagonist, Brian Robeson, is in some serious trouble when the pilot of the small plane he is on has a heart attack and dies. Brian somehow manages to land the plane into a lake and survive with only some scratches and bruises. But now what? How will this boy from New York City survive alone in the Canadian wilderness? Luckily he has a hatchet that his mother gave him, and this tool will be key to his survival.
Brian recognizes that the first thing he needs to survive is food and shelter. The first food he locates consists of some bright red berries, which he eats despite their bitter taste. Later in the book, a raspberry patch helps alleviate some hunger but puts him face to face with a bear. The next nourishment he finds is turtle eggs, which he eats raw. Once Brian learns how to build a tool to catch fish, he has a feast. His final success is catching a foolbird, which he kills with a spear. Brian also finds some freeze-dried food in the plane before he is rescued. Food is not that plentiful for Brian, but using his wits, patience, and hatchet, he learns how to survive long enough to stay alive.

How old is a rock if it contains a radioactive element with a half life of 100 million years?

There isn't enough information in the question in order to provide a definitive answer. We would need to know how much of the rock contains the radioactive element and/or how much of the stable "daughter atoms" are present from the decaying radioactive "parent atoms." Knowing the half-life gives you the rate of decay, but without knowing how much radioactive decay has already happened, determining the age of the rock isn't realistically possible. A half-life is an interesting thing, because no matter how much decay has occurred, there will always be some radioactivity present. This is because the half-life determines how long it will take for half of the radioactivity to decay, and it's possible to continue dividing by half forever, because you just use smaller numbers.
http://www.geologyin.com/2015/02/radiometric-dating.html

I need your help to answer the questions below from the book Sam Patch, the Famous Jumper by Paul E. Johnson. How did Sam Patch achieve fame? Please include location(s) in your answer. What distinguished his first act of fame from his second? What type of job did he have and where did he work? How does the first question reflect the community in which he worked? What job opportunities were there for uneducated people? What overarching issues does his life reflect? What was the political context of the time and location in which Sam Patch lived?

Sam Patch was a daredevil/stuntman who grew up in extreme poverty, working as a child in a textile mill alongside other family members and friends. Many textile factories in the early 1800s used water falls to provide the power necessary to run the machinery. The factory where Sam Patch worked as a boy was located near Pawtucket Falls in Rhode Island. The boys dared each other to jump into the falls from the top of the mill. It was a pass time for bored child laborers, but they perfected the jumping craft. 
In his 20s, Sam Patch moved to Paterson, New Jersey, taking a job in a textile mill there. He began jumping into the Passaic Falls, where Paul E. Johnson speculates his first two jumps were of a socio/political nature, i.e., class consciousness. His first jump was purportedly to protest the opening of a private park for the upper middle class and elite to enjoy without having the working class in their midst. Sam's jump drew attention from the aristocrats opening the bridge to the park as people watched him instead of paying attention to the elitist opening ceremony. His second jump was also a supposed political statement in support of factory workers protesting a change in their lunch hour. 
Whatever the reasons for Sam Patch’s jumping, he became famous as a hero of the working man after his first two jumps into the Passaic Falls. He used that fame to make his stunts into a commercial enterprise.  However, his career as a stuntman ended after a few short years when he jumped to his death at Genesee Falls in New York in November, 1829. He turns up as a legendary hero for the working class in children’s books and literature as well as poetry. Most notably, he was a subject in the works of Ralph Waldo Emerson, Herman Melville, and Nathaniel Hawthorne.
Some may say the overarching theme of Sam Patch’s life was the struggle of the working class versus the middle class and elites. As a poor, uneducated millworker, Patch likely could never have been anything but a poor working class American without his stuntman feats. There was little hope for upward mobility in the small, impoverished factory towns of early 19th century America.
At the time, however, factions in the Democrat-Republican party were calling for more democracy for the common man. Opposition to elitist politics was championed by Andrew Jackson who promoted equal rights for white men in America. Jacksonian Democracy is pitted against the monied interests of corporations, commercial banks, and private interest. In the Northeast, where Sam Patch lived and worked, the yeoman farmer and artisan economy was being squashed by cash-crop farming and capitalist factories. Sam’s father was supposedly an artisan shoemaker who was forced out of business and into the factories by capitalists. 
The story of Sam Patch delves into American socio-economic culture and history so that American history becomes a personal story, not just the story of a nation.
http://www.historyinreview.org/sam_patch.html

https://www.history.com/topics/19th-century/jacksonian-democracy

What causes Scout to think the world is coming to an end? What does Atticus mean when he says, “Looks like all of Maycomb was out tonight, in one way or another”? Who or what is the “Absolute Morphodite”?

In Chapter 8, it begins to snow. Scout has never seen snow before, so she thinks something horrible is happening. According to Eula May, it hasn't snowed since 1885, so Scout would never have had the opportunity to encounter it. Jem and Atticus know enough to know what snow is, but they don't have experience with it either. Atticus says, "Looks like all of Maycomb was out tonight, in one way or another," because he realizes that it was Boo Radley who put the blanket on Scout without her noticing. Everyone else in town was helping to tame the spreading fire at Miss Maudie's, including Nathan Radley. Jem and Scout were standing in front of the Radley property, which means they were in prime position for Boo to see how cold they were through his window. The "Absolute Morphodite" is how Miss Maudie refers to the snowman in the Finch's yard. Morphodite is another word for hermaphrodite, meaning something that has traits of both sexes. The snowman was originally built to look like Mr. Avery and then disguised with Miss Maudie's hat and shears, making it both a man and a woman. When Scout hears Miss Maudie call the snowman a morphodite, she only hears it as an insult, and later uses the word against Jem in Chapter 14.

“You damn morphodite, I’ll kill you!”

Scout doesn't understand the true meaning of the word, and instead of asking, she absorbs it, incorrectly, into her vocabulary.

Single Variable Calculus, Chapter 2, 2.3, Section 2.3, Problem 16

Determine the $\displaystyle \lim \limits_{x \to -1} \frac{x^2 - 4x}{x^2 - 3x - 4}$, if it exists.


$
\begin{equation}
\begin{aligned}
& \lim \limits_{x \to -1} \frac{x^2 - 4}{x^2 - 3x - 4} = \lim \limits_{x \to -1} \frac{x \cancel{(x - 4)}}{\cancel{(x - 4)}(x +1)}
&& \text{ Get the factor and cancel out like terms. }\\
\\
& \lim \limits_{x \to -1} \frac{x}{x + 1} = \frac{-1}{-1 + 1} = \frac{-1}{0}
&& \text{ Result will be undefine }\\
\\
& \fbox{$\lim \limits_{x \to -1} \displaystyle \frac{x^2 - 4}{x^2 - 3x -4} \text{ Limit does not exist }$ }

\end{aligned}
\end{equation}
$

What is the plot of Burn Marks by Sara Paretsky? Who are the main characters?

Burn Marks is Paretsky's sixth novel with the main character V.I. (Victoria Iphigenia, or Vic) Warshawski, a Chicago-born private investigator. 
The characters include V.I., her on-again, off-again boyfriend, cop Michael Furey, her alcoholic Aunt Elena, and an old friend, Rosalyn Fuentes, who is running for a Cook County office.
Aunt Elena comes to Vic looking for a place to stay after arson has destroyed the downscale, single-room occupancy (SRO) hotel where she has been living.  Vic finds a room for her and her companion, Cerise.  Not long after Vic settles them in a new room, Cerise is found dead elsewhere, and Elena disappears.  Vic is hired by an insurance company to find the arsonist before it makes payment to the building's owner.
In the course of her investigation Vic is discouraged and threatened by most officials she encounters, including the head of the arson division, a building developer, and the owner of the burned SRO. She isn't sure who presents the biggest threat: someone in the police department, Cook County politicians, or construction developers. Her relationship with Rosalyn (Roz) Fuentes is strained because of issues of trust.
Ultimately, Vic finds her aunt and rescues her from another arson, but not until she has survived various pursuits and attempts on her life. She uncovers corruption in which politicians and developers collude.
Burn Marks hits many social criticism notes: paternalism in policing and politics, the greed of real estate developers, and anger at the living conditions of the poor in urban America. 

Does Neoclassicism still influence modern architecture?

Neoclassicism does still influence architecture and will continue to influence the design and understanding of buildings still to come. Neoclassicism harkens back to the Classical periods of Greek and Rome, and buildings of this style feature characteristics like columns and geometric forms. This style of architecture grew out of a reaction or distaste for the excess of Rococo design, which was rich in detail. In turn, many architectural styles have developed as interpretations of or reactions to Neoclassical style. 
As an example, let's consider a building which might be considered a reaction or rejection of Neoclassical style— Fallingwater. Neoclassical architecture is very upright, with a sturdy base, strong columns, and typically a triangular roof or the appearance thereof. This mimics the iconic temples of Greece, and government buildings all over the West bear this style. Homes built in the Neoclassical style also bear a very upright appearance, with all elements of the home contained in a neat, four-sided space. In contrast, Frank Lloyd Wright's Fallingwater has elements which seemingly jut out at odd angles. What's more, Fallingwater was built to interact and co-exist with the surrounding environment, rather than dominate it as a Neoclassical structure might. I cannot say with any certainty that Frank Lloyd Wright had this rejection clear in his mind when he designed the structure, but his knowledge of architecture is built upon the thousands of years of design which preceded him.
In a more explicit sense, many buildings today (especially governmental) are built in the Neoclassical style in order to emulate Classical societies and their values. 
https://www.britannica.com/art/Neoclassicism

https://www.metmuseum.org/toah/hd/neoc_1/hd_neoc_1.htm

Calculus of a Single Variable, Chapter 6, 6.1, Section 6.1, Problem 52

The given problem: (dy)/(dx) = 5e^(-x/2) is in form of a first order ordinary differential equation. To evaluate this, we may follow the variable separable differential equation: N(y) dy= M(x)dx
Cross-multiply dx to the other side, we get:
dy= 5e^(-x/2)dx
In this form, we may now proceed to direct integration on both sides:
int dy= int 5e^(-x/2)dx
For the left side, we apply basic integration property: int (dy)=y .
For the right side, we may apply u-substitution by letting: u = -x/2 then du =-1/2 dx or -2du= dx .
Plug-in the values: -x/2=u and dx=-2du , we get:
int 5e^(-x/2)dx=int 5e^(u)* (-2 du)
=int -10e^(u)du
Apply the basic integration property: int c*f(x)dx= c int f(x) dx .
int -10e^(u) du=(-10) int e^(u) du
Apply basic integration formula for exponential function:
(-10)int e^(u) du= -10e^(u)+C
Plug-in u=-x/2 on -10e^(u)+C , we get:
int 5e^(-x/2) dx=-10e^(-x/2)+C
Combining the results from both sides, we get the general solution of differential equation as:
y=-10e^(-x/2)+C

What did Karl Marx mean when he called religion "the opiate of the people"? How does his view of religion fit into his thinking about modern society?

By writing that religion was "the opiate of the people," Marx meant that religion was a tool that the ruling class, or bourgeoisie, used to keep the proletariat, or working class, suppressed. Marx referred to the way that the poor turned to religion for comfort rather than starting a revolution against the upper classes, who kept them down. Like an opiate, religion offered the poor a kind of immediate comfort or release from reality, but it did not offer lasting comfort or solutions to their pain. Marx believed that only a full-scale revolution by the working class, or proletariat, could better their position.
Marx believed that modern times, including the growth of capitalism (which he wrote about in the mid-1800s), had resulted in a society in which the bourgeoisie exploited the proletariat. In capitalism, the proletariat became a commodity to be used by the bourgeoisie. The bourgeoisie had an interest in preventing the proletariat from carrying out a revolution, as this would disturb the privileged position of the upper class. Therefore, the upper class had an interest in maintaining the religion of the working class, which promised them better conditions in the afterlife, so that they wouldn't revolt in this life.

In the short story "Thank You, Ma'm," why did Mrs. Jones have Roger wash his face?

Although Langston Hughes does not explicitly tell us why Mrs. Jones makes Roger wash his face in “Thank you, Ma’m,” we can certainly infer the reason from the story.  We can infer that Mrs. Jones makes Roger wash his face in order to help instill in him a sense of pride and self-worth.
In this story, Mrs. Jones is clearly trying to reform Roger by making him respect himself.  After she initially kicks and shakes him, she stops trying to punish him much and instead tries to rehabilitate him.  When she takes him home, she repeatedly leaves him alone in such a way that he could both take her money and escape.  She is clearly trying to make him feel better about himself so that he will stop trying to commit crimes.
Once we understand this, we can see why Mrs. Jones has Roger wash his face.  His dirty face is a symbol of his life in general.  It shows that he does not respect himself enough to keep clean and it shows that his upbringing has been poor because he has no one at home to teach him to keep clean.  Mrs. Jones wants to reverse this situation.  She wants him to clean his face (and comb his hair) so that he will be presentable and look like someone who can be respected.  If he does this, perhaps he can learn to respect himself as well.
In this story, Roger’s face symbolizes the state of his life and his self-respect.  Mrs. Jones wants him to wash it as a first step to regaining his sense of self-worth so that he can stay out of trouble.

f(x) = ln(x - 3) Use the derivative to determine whether the function is strictly monotonic on its entire domain and therefore has an inverse function.

f(x)=ln(x-3)
Take note that a function is strictly monotonic if it is increasing on its entire domain or decreasing on its entire domain.
For our function,
f(x)=ln(x-3)
to determine if it is strictly monotonic, let's first figure out its domain.
Take note that in logarithm, its argument should be above zero. So to get its domain, set its argument x-3 greater than zero.
x-3gt0
xgt3
So the domain of the given function is (3, oo) .
Then, let's apply the derivative. It will be strictly monotonic if there is no sign change in the value of f'(x).
The derivative of the function
f(x) = ln(x-3)
is
f'(x) = 1/(x-3)
Notice that the derivative of the function can never be zero. Because of that, the function f(x) has no critical numbers. This means that there will be no sign change in the value of f'(x). To verify, assign values to x falls within the domain of the function and plug-in them to f'(x).
x=4
f'(x) = 1/(4-3)=1
x=10
f'(x) = 1/(10-3)=1/7
x=23
f'(x) =1/(23-3)=1/20
x=42
f'(x)=1/(42-3)=1/39
Notice that on the interval (3,oo) , the value of f'(x) is always positive. There is no sign change in the value of f'(x). So the function is entirely increasing on its domain.
Therefore, the function f(x)=ln(x-3) is strictly monotonic on its entire domain.

In American Sniper, describe Christopher Kyle's job, family, friends, hobbies, religion, and spirituality through an ecological perspective.

An ecological perspective involves how individuals adapt and react to their social and cultural environments. Several constructs explain how the individual interacts successfully (or unsuccessfully) with their environments:
1) Adaptation: This explains an individual's capacity to adapt to changes such as occupational stresses, relationship difficulties, and other types of personal tragedies.
In the book, Chris discusses his time working for David Landrum at his Hood County ranch; Chris worked after-school shifts and summers for Landrum. Here, Chris experienced firsthand the challenges inherent in the ranching business; by all indications, Chris's later ability to adapt to difficult surroundings as a SEAL was honed from his early training as a ranch hand. At the ranch, he roomed in a six-by-twelve bunkhouse, and he braved the cold of winter with only a gas stove and electric heater. At the same time, he battled raccoons and armadillos in his quarters. Chris learned patience (a great prerequisite for succeeding as a SEAL) while training horses at Landrum's ranch.
Later, Chris jumped at the opportunity when the Navy offered him a chance to try out for the SEALS. From BUD/S training through Hell Week, Chris learned to do whatever it took to succeed. It was this ability to adapt and to overcome (in the face of challenges) that propelled Chris to a successful career as a sniper. This ability also proved useful when, later in his career, he decided not to reenlist in order to devote more time to his duties as a father and husband.
2) Goodness-of-Fit: This explains the degree a person's temperament is suited to his cultural and social environments.
From his earliest youth, Chris loved to fight on behalf of the underdog. He was fiercely protective of his younger brother and never lost an opportunity to defend those who were weaker and smaller than him. Chris also enjoyed a good fight, and his aggressive temperament largely explained his ease in combat environments during his tours of duty. According to Taya, Chris's sense of duty and caring also made him a phenomenal SEAL. When the stakes were high, Chris always upheld the trust of soldiers who depended upon him.
3) Niche: This explains how an individual's social position determines success. If the individual enjoys equal access to economic and educational advantages, his environment is a good niche (or place) for him.
In the book, we learn that Chris attended Tarleton State University after he graduated from high school. Tarleton State became part of the Texas A&M University system in 1917. By all indications, Chris had equal access to the educational and economic opportunities his peers enjoyed. When the Navy announced that it would accept him for BUD/S (Basic Underwater Demolition/ SEAL training), Chris was further able to access the necessary training that would aid his success as a future SEAL.
4) Habitat: An individual is said to live in an empowering habitat if the sociopolitical, economic, and cultural forces that surround him are largely supportive of his mental and physical health.
In the book, Chris Kyle tells us that he was raised in the Christian faith in Odessa, a small town in north-central Texas. There, he was taught the elements of patriotism, independence, and loyalty. God, country, and family were the three most important things to him; he also loved hunting and enjoyed helping his family raise cattle during his childhood years.
Chris had supportive parents and had an equally close relationship with his younger brother. Chris credits his parents for instilling in him a good work ethic and his father specifically for bequeathing him his sense of "justice and fair play." Chris's father was also largely instrumental in encouraging him to pursue his ambitions; both parents convinced Chris to finish college before serving in the military.
As a SEAL, Chris enjoyed the support of good neighbors and friends. Often, after returning from a mission, Chris's supportive network would allow him time to decompress before they threw him barbecue parties. During his absence, neighbors, family members, and close friends helped Taya and the children to survive and thrive. It can be said that Chris lived in an empowering habitat, where the sociopolitical, economic, and cultural forces were largely supportive of his mental and physical health.

What is the theme of "To Autumn"?

Ostensibly, Keats's "To Autumn" is a paean of praise to this most inspirational of seasons. But, as is always the case with Keats, there is considerably more to this than meets the eye—a richer, more complex vision lurking beneath the opulent pleasures of nature, bursting to shine forth.
A recurring theme in Keats's odes is the fragility and transience of the natural world. And we encounter it once again here. Keats delights in providing us with lush descriptions of this "season of mists and mellow fruitfulness," while at the same time recognizing that the season, like each one of us, must one day pass, no matter how beautiful it is. But this shouldn't cause worry; new life will emerge from the old in a never-ending cycle of death and rebirth:

And full-grown lambs loud bleat from hilly bourn; 
   Hedge-crickets sing; and now with treble soft 
   The red-breast whistles from a garden-croft
      And gathering swallows twitter in the skies.

The season is drawing to a close, but nature is blossoming into full maturity as it points toward the onset of winter. The lambs are now "full grown," and the swallows are starting to gather in the skies.
 
Nature is so remarkably fruitful in all its variety. At times, it threatens to overwhelm us with the sheer scale of its fecundity. Man is the junior partner here; in his relationship to nature it is the world of the animals, the clouds, and the sweet, luscious fruit that dominates. In the midst of this endless cycle of seasonal change, there is nothing we can do but stand and admire. We must simply sit back and, in our reverie, enjoy the joyous bounties of nature, our sadness at their passing tinged with a realization that they will one day return.

Single Variable Calculus, Chapter 2, 2.5, Section 2.5, Problem 10

Show that the function $f(x) = x^2 + \sqrt{7-x}$ is continuous at the given number $a = 4$ using the definition of continuity and the properties of limits.

By using the properties of limit,


$
\begin{equation}
\begin{aligned}
& \lim \limits_{x \to 4} (x^2 + \sqrt{7 -x}) && = \lim \limits_{x \to 4} x^2 + \sqrt{\lim \limits_{x \to 4} 7 - \lim \limits_{x \to 4} x}
&& \text{ Apply sum, difference}\\
& \phantom{x} && = (4)^2 + \sqrt{7-4} && \text{ Substitute the given value}\\
& \phantom{x} && = 16 + \sqrt{3}
\end{aligned}
\end{equation}
$


By using the definition of continuity,
$\lim \limits_{x \to a} f(x) = f(a)$


$
\begin{equation}
\begin{aligned}
& \lim \limits_{x \to 4} (x^2+ \sqrt{7-x})&& = f(4) = (4)^2 + \sqrt{7-4}\\
& \phantom{x} && = 16 + \sqrt{3}
\end{aligned}
\end{equation}
$




Therefore, by applying either of the two, we have shown that the function is continuous at 4 and is equal to $16 + \sqrt{3}$

Write a note on Katherine Mansfield as a short story writer.

Katherine Mansfield, like her friend Virginia Woolf, was trying in the years during and after World War I to write in a new way, a way that would use language to reveal truths the Edwardians usually kept hidden. She used simple, childlike language and usually, but not always, a stream-of-consciousness style that showed the impressions flowing through a character's mind. She captured the thoughts (the interiority) of her characters—often shown later to be in error—as they were happening. This is typical of the experimental, modernist prose of her era.
Katherine Mansfield's long short story "The Prelude" was the first publication of the Hogarth Press, Virginia and Leonard Woolf's publishing company. They were looking for new, edgy, state-of-the-art writing that would challenge old norms. They published this groundbreaking and influential story in 1917 as a stand-alone volume.
It is told in a stream-of-consciousness way, largely through the eyes of female children and women, though there is a section showing the family patriarch's point-of-view as he comes home from work. It challenges Victorian portraits of the idyllic, loving family by showing that the two younger daughters are treated indifferently and seemingly are unloved. It breaks taboo ground in having the wife openly think about how much she dislikes sex with her husband. As with her other stories, Mansfield uses simple prose and striking images to convey her ideas.
Other famous Mansfield stories are "The Garden Party," an exploration of the British class divide; "Miss Brill," a study of an elderly and marginalized single woman; and "The Fly," a cry of rage at the senseless slaughter of World War I, which Mansfield likened to slowly torturing a fly to death.

Katherine Mansfield, born Kathleen Mansfield Beauchamp, left her native New Zealand for London when she was nineteen and lived in Europe for most of her life until she died in 1923, at 34, from tuberculosis.
Her early exposure to the Maori led her to write sympathetically of them in her later years. Stories like "Prelude" and "How Pearl Button was Kidnapped" observe the repressive nature of colonialism.
Mansfield lived as a bohemian and was influenced by artistic movements like Fauvism, with its emphasis on individual expression, and the literary movement of Modernism, which rejected traditional structural aspects of stories like expositions and conclusions and sought to capture the disillusionment of life after WWI.
One of her most famous stories, "The Garden Party," revisits and refines themes found in her earlier works, especially the divisions of social class along both racial and economic lines. Despite a comfortable upbringing, Mansfield was sensitive to people whose circumstances were reduced through no fault of their own.
http://www.katherinemansfieldsociety.org/short-stories-by-katherine-mansfield/

Each corner of a right angled triangle is occupied by identical point charges "A", "B", and "C" respectively. Draw a sketch of this arrangement. "A" exerts force F on "B". An equal force F is exerted by "C" on "B" (/_ ABC= 90 degrees). Determine an expression for the net force on "B".

Hello!
Because the charges are identical, they repel each other. Therefore the direction of the force that A exerts on B is directed from A to B (outside the triangle ABC ). The direction of the force that C exerts on B is from C to B (also outside the triangle ABC ).
The net force is the vector sum of these two forces. Both forces have equal magnitude F and the angle between them is 90 degrees, and the magnitude of the net force is the length of the hypotenuse of a right triangle with the legs of length F. It is F*sqrt(2) (this is the answer for "an expression").
The direction of the net force is 45 degrees to both forces outside of the triangle ABC, because the parallelogram of forces is a square in this case. Please look at the picture attached.