`f(x)=e^(4x) , n=4` Find the n'th Maclaurin polynomial for the function.

Maclaurin series is a special case of Taylor series that is centered at a=0. The expansion of the function about 0 follows the formula:

`f(x)=sum_(n=0)^oo (f^n(0))/(n!) x^n`

 or

`f(x)= f(0)+(f'(0)x)/(1!)+(f^2(0))/(2!)x^2+(f^3(0))/(3!)x^3+(f^4(0))/(4!)x^4 +...`

To determine the `4th` Maclaurin polynomial from the given function `f(x)=e^(4x)` ,

we may apply derivative formula for exponential function:` d/(dx) e^u = e^u * (du)/(dx)`

Let `u =4x` then `(du)/(dx)= 4 `

Applying the values on the derivative formula for exponential function, we get:

...

Maclaurin series is a special case of Taylor series that is centered at a=0. The expansion of the function about 0 follows the formula:

`f(x)=sum_(n=0)^oo (f^n(0))/(n!) x^n`

 or

`f(x)= f(0)+(f'(0)x)/(1!)+(f^2(0))/(2!)x^2+(f^3(0))/(3!)x^3+(f^4(0))/(4!)x^4 +...`

To determine the `4th` Maclaurin polynomial from the given function `f(x)=e^(4x)` ,

we may apply derivative formula for exponential function:` d/(dx) e^u = e^u * (du)/(dx)`

Let `u =4x` then `(du)/(dx)= 4 `

Applying the values on the derivative formula for exponential function, we get:

 `d/(dx) e^(4x) = e^(4x) *4`

Applying `d/(dx) e^(4x)= 4e^(4x)`  for each `f^n(x)` , we get:

`f'(x) = d/(dx) e^(4x)`

          `=e^(4x) * 4`

         `= 4e^(4x)`

`f^2(x) = 4 *d/(dx) e^(4x)`

           `= 4*4e^(4x)`

           `=16e^(4x)`

`f^3(x) = 16*d/(dx) e^(4x)`

          `= 16*4e^(4x)`

          `=64e^(4x)`

`f^4(x) = 64*d/(dx) e^(4x)`

      `= 64*4e^(4x)`

      `=256e^(4x)`

Plug-in `x=0` , we get:

`f(0) =e^(4*0) =1`

`f'(0) =4e^(4*0)=4`

`f^2(0) =16e^(4*0)=16`

`f^3(0) =64e^(4*0)=64`

`f^4(0) =2564e^(4*0)=256`

Note: `e^(4*0)=e^0 =1` .

Plug-in the values on the formula for Maclaurin series. 

`f(x)=sum_(n=0)^4 (f^n(0))/(n!) x^n`

         `= 1+4/(1!)x+16x^2+64x^3+256/(4!)x^4`

        `=1+ 4/1x +16/(1*2)x^2 + 64/(1*2*3)x^3 +256/(1*2*3*4)x^4`

        `=1+ 4/1x +16/2x^2 + 64/6x^3 +256/24x^4`

        `= 1+4x+ 8x^2 + 32/3x^3 + 32/3x^4`

The 4th Maclaurin polynomial for the given function `f(x)= e^(4x)` will be:

`e^(4x) =1+4x+ 8x^2 + 32/3x^3 + 32/3x^4`

or `P_4(x) =1+4x+ 8x^2 + 32/3x^3 + 32/3x^4`

What were the attitudes towards witchcraft in Shakespeare's time?

Shakespeare wrote during the English Renaissance—a time in which humankind thirsted for knowledge in many areas (sciences and arts being two of the more prominent)—but it is true that the Elizabethans were a suspicious lot who often depended more on emotion than erudition. In addition, the influences of the Catholic and Anglican churches of the time encouraged people to believe in a spiritual realm, and not just one that housed peaceful, benevolent, angelic spirits. Furthermore,...

Shakespeare wrote during the English Renaissance—a time in which humankind thirsted for knowledge in many areas (sciences and arts being two of the more prominent)—but it is true that the Elizabethans were a suspicious lot who often depended more on emotion than erudition. In addition, the influences of the Catholic and Anglican churches of the time encouraged people to believe in a spiritual realm, and not just one that housed peaceful, benevolent, angelic spirits. Furthermore, people of Shakespeare's time considered astrology and astronomy to be the same science, so they were just as willing to accept the concept that Romeo and Juliet were "star-crossed" as they were to accept they were hormone-charged teenagers. With the combination of these elements in full force, people in Shakespeare's time were very willing to accept that there could be malevolent forces in the world and also believed that these forces could be called forth by those who wanted to utilize their dark powers. The witches in Macbeth are partial evidence of this willingness to believe in the power of witchcraft; however, Shakespeare carefully crafted the three witches so they could be viewed as "secret, black, and midnight hags" with little power of their own just as easily as they could be viewed as controlling creatures in contact with a dark, occult underworld. When it came to witchcraft during Shakespeare's time (and thus, within Shakespeare's plays), the willing suspension of disbelief was not always a necessary component. Many audience members would have considered appearances by witches and spirits to be completely plausible reasons for misfortune in the lives of real people as well as characters in a play.

5) in the story Harrison Bergeson, how does the tone of the story change once Harrison appears on the television?

The tone for much of the story before Harrison's appearance is somber, sardonic, and even fatalistic. Equality is portrayed as an established reality, the tone so insistent that we are immediately suspicious of its sanctimonious certitude.

The year was 2081, and everybody was finally equal. They weren't only equal before God and the law. They were equal every which way. Nobody was smarter than anybody else. Nobody was better looking than anybody else. Nobody was stronger or quicker than anybody else. All this equality was due to the 211th, 212th, and 213th Amendments to the Constitution, and to the unceasing vigilance of agents of the United States Handicapper General.

Yet, when Harrison appears on the screen, the tone becomes animated, anticipatory, and even defiant. The earlier passive tone becomes overshadowed, and the new tone produces a brighter, more hopeful mood.

Harrison tore the straps of his handicap harness like wet tissue paper, tore straps guaranteed to support five thousand pounds.

Harrison's scrap-iron handicaps crashed to the floor.

Harrison thrust his thumbs under the bar of the padlock that secured his head harness. The bar snapped like celery. Harrison smashed his headphones and spectacles against the wall.

He flung away his rubber-ball nose, revealed a man that would have awed Thor, the god of thunder.

The once helpless Harrison becomes a sensational figure of dissent. His rebellion is executed at a frenzied pace. He and his partner leap into the air, simultaneously abandoning the "laws of the land" and the laws of gravity and motion. The tone, although changed, invites our skeptical reaction. We are led to question how long this euphoric state can last.

The dancers leap almost thirty feet high, skimming the studio ceiling as they do so. Then, hovering in midair, both lean in for a prolonged, passionate kiss. They are brought back to earth when Diana Moon Glampers fires twice from her "double-barreled ten-gauge shotgun." The tone then reverts back to the resigned, somber, and fatalistic one we see at the beginning of the story.

What are Scout's and Jem's tones toward Calpurina when asking her about her use of English in her community?

In chapter 12 of To Kill A Mockingbird Scout and Jem accompany Calpurnia to church one Sunday. The entire congregation, like Calpurnia, is African American. Both children notice that Calpurnia speaks differently to members of her church than she does when she is at the Finch residence. Calpurnia explains that she taught her son Zeebo to read using Blackstone's Commentaries, an eighteenth century English law book:

“That’s why you don’t talk like the rest of ‘em,” said Jem.

“The rest of who?”

“Rest of the colored folks. Cal, but you talked like they did in church”

Jem's tone is one of curiosity. He picks up straight away on the fact that Calpurnia speaks differently in church than she does at home. But he still doesn't know the whole story. There is a reason why Calpurnia changes her way of speaking depending on whom she is talking to, as Scout soon finds out:

“Cal,” I asked, “why do you talk n——rtalk to the—to your folks when you know it’s not right?”

Scout's tone, like Jem's, is curious, but she is also being presumptuous in assuming that there is a "right" way of talking. Scout thinks that if you know how to talk in the "proper" way, then there is no excuse for speaking in the kind of dialect used by African Americans.

Scout's tone might also be said to be a little arrogant as she automatically assumes that Calpurnia does not have a life or identity of her own outside the Finch household.

Calpurnia explains that people at church would think she was putting on airs and graces if she talked like she did at home. Yes, she knows better, but sometimes it's best not to let on, otherwise people might think you're trying to make yourself feel superior to them.

This brief episode in the story provides us with another insight into the various hierarchies that exist in Maycomb—different races, different classes, even different ways of speaking that keep people apart.

What is the purpose of the dream at the end of the story Battle Royal?

It seems clear that the grandfather's deathbed warning combined with the narrator's grueling battle experience has unsettled the narrator; his dream contains information about life's beginnings and endings, and the knowledge he now carries about his life as a black man. 

The grandfather's refusal to laugh at the dream circus clowns seems to symbolize the seriousness of the narrator's situation. Graduation from high school is a milestone, usually a time for optimism; for the narrator,...

It seems clear that the grandfather's deathbed warning combined with the narrator's grueling battle experience has unsettled the narrator; his dream contains information about life's beginnings and endings, and the knowledge he now carries about his life as a black man. 

The grandfather's refusal to laugh at the dream circus clowns seems to symbolize the seriousness of the narrator's situation. Graduation from high school is a milestone, usually a time for optimism; for the narrator, instead, it is a mirthless time. Graduating from childhood to adulthood means accepting the full weight of society's treatment of black men.

The envelopes the narrator dreams himself opening are symbolic of the endless endurance tests he has been put through, and will be put through as he ages. Like the "battle royal" of the previous evening, the narrator must perform a number of arbitrary tasks for permission to do what he wants to do, to achieve anything greater for himself. (This unending rabbit hole of envelope-opening seems to reference Booker T. Washington's "politics of respectability" argument.)

We can only guess Ellison's purpose in using a dream sequence to comment on these themes. Perhaps he means to highlight that a black man's agency can be taken away from him even in the subconscious, so punitive is society's judgment. 

What are the ways that ecological theory supports culturally sensitive work with Hispanic children and families?

On the most basic level, ecological theory concerns how individuals interact or adapt to their environment.

Thus, the ecological approach to social work among Hispanic families centers on acculturation strategies. Acculturation (as related to Hispanic families) refers to how immigrants assimilate or adapt to a culturally foreign environment. Ecological theory supports culturally sensitive work with Hispanic children and families by focusing on solutions within an ecosystem. In ecological or ecosystem theory, the ecosystem is a multi-connected network or matrix that consists of individuals, families, groups, and communities.

Bronfenbrenner (father of the ecological systems theory) hypothesized that the ecological or ecosystems framework allows social workers to study how interrelated systems within an ecosystem contribute to social dysfunction. Thus, analyzing the psycho-social factors that affect the individual allows social workers to find effective solutions to acculturation issues. The ecological/ecosystem approach allows social workers to focus on five main ecosystems that may affect any one immigrant: the microsystem, mesosystem, exosystem, macrosystem, and chronosystem.

The microsystem consists of the immigrant's closest relationships (parents, friends, teachers, caregivers, etc). The esosystem comprises the interaction between the different microsystems in an immigrant's life. For example, a Hispanic child may have supportive parents who attend PTA conferences. Here, the child's microsystems (parents and teachers) engage to provide the support he/she needs to thrive in a foreign environment. 

Meanwhile, the exosystem is not directly connected to the immigrant. It may consist of the child immigrant's parents' employers or members of the larger community. The macrosystem consists of the social, religious, and political systems that influenced the immigrant prior to his/her arrival on foreign soil. Last, but not least, the chronosystem involves changes in the immigrant's life, such as divorces, job transitions, or other types of social upheaval. 

So, ecological theory supports culturally sensitive work with Hispanic children and families by focusing on solutions that incorporate the interrelation between these five systems. Let's take the example of Hispanic elders who must rely on US-oriented caregivers. Many Hispanic elders experience difficulty in reconciling the virtues and values of their native culture with that of the foreign culture they encounter. To be efficient caregivers, nursing and medical staff must communicate an appreciation for the Hispanic patient's culture and values as well as acknowledge the validity of their perspective. Thus, attentive care at the microsystem level can lessen the development of mood disorders such as depression or anxiety among Hispanic elders.

Attention to stress-contributing factors at the exosystem or chronosystem levels is also important. Social workers who use the ecological approach understand the need to address the psycho-social factors that contribute to high poverty levels among Hispanic immigrants. Even though Hispanic immigrants often harbor strong work ethics, the lack of education prevents many from realizing the American Dream. An ecological approach would include on-the-job training, English proficiency classes, and GED classes. It may also include cultural awareness classes, where Hispanic children and families learn to appreciate and understand American culture.

At the exosystem level, many states require those on welfare to fulfill specific work requirements. Because Hispanic families often have low levels of English proficiency, they continue to live subsistence lives. An ecological approach would factor in these language challenges in formulating effective solutions to poverty issues. For example, state agencies may hire and assign Spanish-speaking social workers to work with Hispanic families. Ideally, these workers will have a knowledge of and an appreciation for Hispanic culture. Such workers can facilitate culturally sensitive discussions about the need for English-language proficiency classes, on-the-job training, high school diplomas, college educations, etc. 

For more, please refer to the links below. 

What is some sensory language in The Most Dangerous Game?

"The Most Dangerous Game" is a short story written by Richard Connell, in which a man named Sanger Rainsford finds himself trapped on an island, being hunted by a fierce predator—another man. General Zaroff, who felt he was not challenged enough with hunting animals, has been bringing people to his island to hunt for sport.

The sensory language in the story adds to the sense of suspense; the narrative wouldn't be nearly as effective...

"The Most Dangerous Game" is a short story written by Richard Connell, in which a man named Sanger Rainsford finds himself trapped on an island, being hunted by a fierce predator—another man. General Zaroff, who felt he was not challenged enough with hunting animals, has been bringing people to his island to hunt for sport.

The sensory language in the story adds to the sense of suspense; the narrative wouldn't be nearly as effective if readers weren't able to take in Rainsford's surroundings.

Some good examples of sensory language in the story are: 

1) "It's like moist black velvet."

Rainsford uses this phrase to describe the "moonless Caribbean night" that he and Whitney are sailing through near the beginning of the story. This appeals to both the sense of sight and touch—we understand that what Rainsford is seeing is pitch black, like a rich black velvet, and that the night air feels humid and moist.

2) "The cry was pinched off short as the blood-warm waters of the Caribbean Sea closed over his head." 

This sentence gives the reader a great sense of the feeling of the sea against Rainsford's skin—it's as warm as blood, like falling into a hot bath.

3) "The lights from the windows sent a flickering illumination that made grotesque patterns on the courtyard below, and Rainsford could see moving about there a dozen or so huge black shapes; as they turned toward him, their eyes glittered greenly."

This quote is all about sense of sight—it describes Rainsford's first impression of seeing Zaroff's fearsome hunting dogs. We not only know how they look to Rainsford but also understand the intimidating aura they give off.

4) "He knew his pursuer was coming; he heard the padding sound of feet on the soft earth, and the night breeze brought him the perfume of the general's cigarette."

This quote appeals to the sense of hearing and sense of smell, which are wrapped up together in one moment of tension as Rainsford waits for the General to fall into his trap.

There are plenty more examples in the story, but I think those are a few good ones to get you started!