f(x)=e^x ,c=1 Use the definition of Taylor series to find the Taylor series, centered at c for the function.

aylor series is an example of infinite series derived from the expansion of f(x) about a single point. It is represented by infinite sum of f^n(x) centered at x=c . The general formula for Taylor series is:
f(x) = sum_(n=0)^oo (f^n(c))/(n!) (x-c)^n
or
f(x) =f(c)+f'(c)(x-c) +(f''(c))/(2!)(x-c)^2 +(f^3(c))/(3!)(x-c)^3 +(f'^4(a))/(4!)(x-c)^4 +...
To apply the definition of Taylor series for the given function f(x) = e^x , we list f^n(x) using  the derivative formula for exponential function: d/(dx) e^u = e^u * (du)/(dx).
Let u =x then (du)/(dx)= 1 .
Applying the values on the derivative formula for exponential function, we get:
d/(dx) e^x = e^x *1=e^x
Applying d/(dx) e^x= e^x   for each f^n(x) , we get:
f'(x) = d/(dx)e^x =e^x
f^2(x) =(d^2(e^x))/(dx)=e^x
f^3(x) =(d^3(e^x))/(dx)=e^x
f^4(x) =(d^4(e^x))/(dx)=e^x
All of the f^n(x) is represented by e^x .
Plug-in x=1 , we get:
f(1) =e^1 =e
f'(1) =e^1 =e
f^2(1) =e^1 =e
f^3(1) =e^1 =e
f^4(1) =e^1 =e
Note: e^1=e^1 =e .
Each f^n(x)  centered at c=1  has a value of "e ".
Plug-in the values on the formula for Maclaurin series. 
e^x= sum_(n=0)^oo (f^n(1))/(n!) (x-1)^n
     =e+e*(x-1) +e/(2!)(x-1)^2+e/(3!)(x-1)^3+ e/(4!)(x-1)^4+...
     =e+e*(x-1) +e/2(x-1)^2+e/6(x-1)^3+ e/24(x-1)^4+...
The Taylor series for the given function  f(x)=e^x centered at c=1 will be:
e^x =e+e*(x-1) +e/2(x-1)^2+e/6(x-1)^3+ e/24(x-1)^4+...
or
e^x =sum_(n=0)^oo e/(n!)(x-1)^n

Is there a message in the story the luncheon? or is it just an amusing recount

While "The Luncheon" is certainly a light and amusing story the narrator tells of a past experience, a message can also be found in the story. The narrator shares that when he was twenty years younger, he received a note from a woman regarding his book, and the rest of the story follows his response to this. Clearly flattered, the young man agrees to have lunch with the woman because she has an interest in his writing. Signs of a prideful nature become apparent in the young man. Though he does not have much money, he is unwilling to decline her request.
When the decision is made to eat at a fancy restaurant, pride again stands in the way of the young man telling the woman he simply cannot afford that particular restaurant. He comes to the conclusion that if he sacrifices drinking coffee for the next two weeks, he can "manage well enough." The narrator at first finds comfort in the woman saying, "I never eat anything for luncheon." However, she orders salmon, caviar, and champagne. The young man's worries go from trying to make ends meet to covering the bill of the luncheon.
The lesson that can be found in the humorous short story is that flattery and pride hamper the young man's ability to be truthful with his guest. Instead of explaining his financial situation, he allows her to continue to order expensive items from the menu. The narrator expresses his pleasure in seeing the woman after twenty years and realizing she has gained a substantial amount of weight.

In the novel That Was Then, This Is Now by S. E. Hinton, what did Byron find surprising about Cathy's personality?

In Chapter 3, Bryon takes Cathy to a school dance, where he runs into his ex-girlfriend, Angela Shepard. Angela approaches Bryon and he dismisses her by asking if she came with Ponyboy Curtis. Angela responds to Bryon's comment by cursing at him while she walks away. When Angela leaves, Cathy asks Bryon about Angela, and he tells Cathy that Angela was someone he used to date. Cathy then makes a sarcastic comment about Angela's "nice language" and refers to her as a "real lady." Bryon is shocked at Cathy's response because for the first time he realizes that Cathy wasn't jealous of Angela at all. Bryon mentions that Cathy was the first girl he's ever dated who wasn't scared that some other girl would show her up. Initially, Bryon doesn't know what to make of it but then decides that Cathy has a lot more sense than most girls.

Assume that it is 5 years in the future. The Armstrong suits are settled. 1. What major business law issues do you believe Lance Armstrong faced? 2. How do you believe he was able to resolve them (or not)?

Armstrong's fraud against USPS might be seen as having caused USPS to violate major US marketing and advertising laws prohibiting deception, illegal activity in marketing and untested claims of health benefits.
Imagining that USPS loses its suit against Armstrong because Armstrong himself was not a signatory of the marketing/sponsorship agreement USPS had with Tailwind Sports, then Armstrong might be rightly sued for causing USPS to violate US marketing and advertising laws requiring truthful claims that are not deceptive: Armstrong, associated with the USPS brand, deceived USPS and the public by hiding sports doping. 
US marketing and advertising laws prohibit doing anything illegal in the course of marketing. It might be said that the fact that Armstrong broke doping laws (according to the Department of Justice) while acting as part of USPS brand advertising--which they associated with "Lance Armstrong: American hero"--provides significant grounds for action against Armstrong since he admittedly violated US drug and doping laws.
Armstrong may also be connected to violation of US marketing and advertising laws regulating advertised health benefits since association with Armstrong--cancer survivor and founder of Livestrong--may imply health benefits even without overtly claiming health benefits.
https://www.cyclingnews.com/news/armstrong-accused-of-withholding-data-in-false-claims-act-lawsuit/

https://www.usatoday.com/story/sports/cycling/2016/03/07/lance-armstrong-false-claims-act-lawsuit-cooper-ruling/81439814/

Five years in the future, Lance Armstrong would face some major business laws issue one being civil fraud lawsuit. Lance Armstrong is a seven times cyclist winner who confessed in Oprah winfrey show that he was doping. This would be taken against him by the United States Post office and Floyd Landis, his former team mate.   In a whistle blower case,  Landis would present the doping allegations to court   where Lance Armstrong would be required to pay the USA government one hundred million dollars  as  he was given by the USPS  approximately thirty two million dollar  in a series of payments from the years 1998 to 2004.  The government  is entitled to the payment as under  the “reverse” false claims law, In this case, it can argue that  it was right for the team to refund the money as  it there was breach of  sponsorship contract due to the  use of  banned drugs and blood transfusions which helped  in enhancing the performance of the athlete. Under the false claims act in the principle of restitution and contract law, the owner of the cyclist was obligated by the law to repay the funds which are link to any breach of the sponsorship where the damages could sum to one hundred million dollars.
In his defense, Armstrong can argue that the lawsuit which is filed by the government is worth nothing as the USPS was not affected in any way by the doping allegations but rather they benefited a lot in terms of sponsorship when compared to what it paid the cycling team.  The benefits include but not limited to brand exposure and positive publicist from the USPS gear that was worn by Armstrong as a cancer survivor who worked hard against all odds to win the Tour de France." In addition to that, Armstrong can argue that the court should dismiss the case as the contracts for the sponsorship were between the USPS and Tailwind. Nevertheless, if found guilty, he could lose all his fortunes and livelihood in addition to a bad and damaged reputation.

What is a summary of the poem "The Haystack in the Floods"?

"The Haystack in the Floods" is an account of the death of an English knight, Robert, in a roadside ambush during the Hundred Years' War, seen from the perspective of his French mistress, Jehane.
The action takes place after the Battle of Poitiers. This battle was a decisive victory for the English; however, after the battle, Robert and his small retinue of soldiers are left behind enemy lines. He, his men, and Jehane are fleeing French-held territory for the English province of Gascony. They have been riding for several days through a rain-drenched late-autumn landscape, and everyone is cold, exhausted, and soaked through, tense with the expectation of a French attack.
The reader sees all this from Jehane's point of view: the "dripping leafless woods," the "mud [which] splash'd wretchedly," the anxious "murmuring from [Robert's] men." She is numb with cold and fatigue, and

often for pure doubt and dreadShe [sobs], made giddy in the headBy the swift riding.

Robert tries to rally his small company with promises that they will soon reach safety, but their hopes are dashed and their fears are realized when they are ambushed by the Frenchman Godmar, who has thirty soldiers to assist him. Robert's company is outnumbered two to one, and Jehane "[sees] at once the wretched end" that awaits them all. Robert tries to encourage her, reminding her of the English victory at Poitiers to make the ambush seem less threatening and promising again that safety is near:

Nay, love, 'tis scarcely two to one,At Poictiers where we made them runSo fast—why, sweet my love, good cheer,The Gascon frontier is so near.Naught after this.

Jehane is not at all convinced by this. She knows that if Robert dies, she, a Frenchwoman, will be taken to Paris to be executed as a traitor. "Would God that this next hour were past!" she prays. Robert has no answer for her fears but shouts a battle cry for his men to charge Godmar's soldiers. His men do not respond; instead, one of them ties Robert up and takes him to Godmar, perhaps hoping for safe passage if he turns over their leader.
Godmar tells Jehane that she must have sex with him or he'll kill Robert:

Now, Jehane,Your lover's life is on the waneSo fast, that, if this very hourYou yield not as my paramour,He will not see the rain leave off.

Jehane is horrified and immediately refuses. Godmar is furious; he demands to know why she has refused when she knows very well that he could murder Robert this instant and rape her anyway:

What hinders me from taking you,And doing that I list to doTo your fair wilful body, whileYour knight lies dead?

Jehane smiles grimly and replies that if he does this, she will kill him in his sleep; she will rip out his throat with her teeth, so help her God. Godmar retorts that if she does not submit to rape, he will "tell/All that [he] know[s]" to the French and Jehane will be executed as a traitor—either drowned or burnt at the stake:

Do you know, Jehane, [the French] cry for you:"Jehane the brown! Jehane the brown!Give us Jehane to burn or drown!"

He tells Jehane she has one hour to decide whether she will have sex with him or not. The price of refusing is Robert's death and her own. Jehane, overwhelmed, staggers over to "the haystack in the floods" and lies down, but she is so exhausted and numbed with horror she cannot consider her decision; she simply passes out and sleeps the full hour.
When she awakens, she goes to Godmar and again refuses to be raped. Godmar is stunned:

[She] strangely childlike came, and said:"I will not." Straightway Godmar's head,As though it hung on strong wires, turn'dMost sharply round, and his face burn'd.

Robert says nothing but looks deeply into Jehane's eyes, and they reach out to each other one last time. Godmar, furious, pulls them apart and hacks Robert's head off on the spot. He tells Jehane that her "way/Lies backwards towards the Chatelet," back into French territory, where she will meet a horrible death. Jehane is silent:

She shook her head and gazed awhileAt her cold hands with a rueful smile,As though this thing had made her mad.

She no longer cares what happens to her now that Robert is dead.

"The Haystack in the Floods," Morris’s stark, 160-line poem, which takes place against the backdrop of the Hundred Years’ War in mid-14th century France, is a fictional account of Sir Robert de Marny, an English knight, and his mistress, Jehane.
While en-route to Gascony, an English safe-hold in France, the two are intercepted by enemy troops. The sense of foreboding, established in the first five lines of the poem, indicates that their fate at the hands of enemy leader Godmar will not be a happy one. The descriptions in further passages, vividly detailing the dreariness of both weather/landscape and its effects on Jehane, further underscore the pitiless, bleak mood of Morris’s work: the skeletal woods, the rain and tears that stain Jehane’s face, her heavy hair, and the mud that splatters as much as it impedes forward progress, for instance. The lovers seemed doomed from the outset.
Nor does much improve for the couple after Godmar’s arrival. Like two haystacks in the rain, the traitors stand, naked and exposed. After a brief but pointed exchange, Godmar decapitates Robert and takes Jehane prisoner. Her fate awaits in Paris, where she will either burn at the stake or suffer a trial by water.

What was the task that Zero and Stanley had to help each other with?

Zero may not be over-endowed in the brains department, but this boy sure knows how to dig a hole. The cruel Mr. Pendanski says that's all he'll ever be good for, and though that's an unfair assessment, Zero is inarguably a skilled digger. Stanley's none too thrilled about this soul-destroying toil, so it's perfectly understandable that he should get Zero to dig his holes for him.
On its face, it sounds quite exploitative that Stanley's taking advantage of Zero. But the boys's burgeoning friendship isn't just a one-way street; in return for Zero's digging, Stanley agrees to help Zero with his reading. This reciprocity will form the basis of their relationship from here on, and in due course, the boys will show incredible bravery and selflessness in helping each other, and together they will break free from the camp's cruel system of control.

In the book Holes, by Louis Sachar, the main character, Stanley Yelnats, is sent to a juvenile camp, Camp Green Lake, where boys are forced to dig holes in order to build character. While he is there, Stanley makes an unlikely friend in Zero, the least popular boy at the camp. Zero is seen as odd because he rarely speaks. Stanley and Zero become unlikely friends when Zero helps Stanley by digging his day's hole for him. In chapter 22, their friendship further develops when they realize they can help each other get by at Camp Green Lake. Stanley knows Zero is illiterate: he cannot read. He proposes a deal in which Stanley will teach Zero to read, and in return, Zero will help him dig holes. Both boys benefit from sharing the work of these tasks. Later in the novel, it is revealed that there is one final task the boys must complete together: in order to break a curse on the Yelnats family, Stanley must carry Zero, who is really a descendent of Madame Zeroni, up a mountain.

How did Native Americans resist or accommodate European colonialism in North America? How did Native Americans' resistance or accommodation shape the development of the European colonies?

The idea of land ownership was not understood by Native Americans - they were a migratory people that followed the herds and fair weather. They were accustomed to lands that were open to everyone and established certain areas where they set up temporary camps for the summer or winter. Very rarely did they set up a permanent building. So when the first European colonies established themselves, their ways were an alien concept. There was no need to own land or animals or suffer in one place for the winter when there was plenty available for all.
During the early years of Virginia, colonists were encouraged to mark their territory with certain markings that came to be known as "tomahawk improvements". This required the settler to make certain improvements and place stakes to mark one-thousand-acre plots of land. As simple as that, ownership was established. The Native Americans didn't understand this concept of ownership.
When Simon Kenton was establishing territory in the Ohio River area, he attempted to (illegally) purchase land from the Shawnee chief, Tecumseh. Tecumseh sold him the land, laughing to himself that selling land was much the same as selling the moon. No one could own the land! Simon Kenton made a lot of tomahawk improvements in this area and came to be considered a wealthy man.
Later, more and more colonists arrived and claimed these same marked lands for themselves, and due to the lack of proper surveying and written records, Simon Kenton lost much of his established wealth during that time. When he met with Tecumseh again, the chief reminded him, "You see? You cannot own the land." Simon Kenton was one of the few colonists who actually understood what it was like to have his land taken out from under him.
At first, the Native Americans agreed to move to the west side of the Ohio River and let the European colonists settle to the east of it. This was a tenuous truce, as new colonists often ignored the treaties and would use the Ohio River to travel to lands to the south and west of the river, land the Native Americans had not agreed to give over to the colonies. Native Americans had many different nations, or tribes, and communication between the tribes only occurred sporadically, when someone traveled or there was an annual gathering. Some tribes didn't get along or communicate at all. So when the Colonists met with Tribal Leaders and a Treaty was established, the European Colonists believed that these treaties applied to all Native American Nations. The Native Americans understood that it only applied to the tribes whose chiefs had attended the treaty signing.
Also during these meetings, European leaders attempted to establish dominance over the tribes. They called King George III of England "the great white father across the sea" and referred to him in a way that made it sound like he was the great, benevolent father to the Native Americans as well. Some tribes were charmed by the tools and implements the Europeans brought with them and were very happy to adjust their ways to suit those of the Europeans, who also taught them about their religion. The Cherokee were one of the first tribes that the colonists deemed to be "civilized", learning European religion and farming methods.
Rules were easily broken simply because of ignorance that any even existed. Colonists set out to explore the country, never realizing that treaties had been set, and outbreaks of violence broke out, instigated by both sides. Forts were built by the colonists on lands across the river, and Native Americans came to realize that the treaties they signed held no real authority. The Europeans revised and rewrote the treaties, again and again, inching further and further to the West and South. During negotiations, the Europeans often made displays of military might. Although they offered gifts as payment for the lands, it was made very clear, using intimidation, that the lands were going to be taken. Many Chiefs signed the treaties simply to prevent bloodshed.
There was an actual turning point in the relationship between the settlers and the Native Americans. King George III of England, "the great white father across the sea" actually did attempt to establish serious boundaries for the Native Americans and tried to abide by the treaties, but policing the effectiveness of this is difficult when you are trying to rule from a far away country. He established British Indian Agents to negotiate terms between the Natives and the Europeans, and these agents also attempted to keep the boundaries in place.
Some Natives actually braved the oceans and traveled to meet with the King in person.
It is possible that things would have been very different for the Native Americans if there hadn't been an American Revolution. When the "thirteen fires" broke away from the great white father across the sea, all of the established treaties made with the King of England became null and void in the eyes of the newly established United States.
England tried to retain a foothold, using the British Indian Agents to negotiate with the Native Americans as allies, but it was at this point that the colonists broke negotiations with the Natives and began forcefully removing them from their lands.
Native Americans may also have had more success if they had united as one nation; one chief, Tecumseh, tried very hard to accomplish this, but was unsuccessful. In the end, England failed them as an ally and retreated to Canada.

Native Americans resisted European colonial expansion in many different ways. They sometimes boycotted white settlers by refusing to trade with them or to provide them with relevant information, and they sometimes denounced the Europeans as intruders and criticized their way of life and noxious influence on Native Americans. The many instances of such resistance included the temporary boycott of the Jamestown settlers by the Powhatan federation and the critique of white influence by religious Native American leaders immediately preceding the Pontiac rebellion in 1763.
The ultimate form of Native American resistance was, of course, armed conflict, which sometimes took the form of large scale war or sporadic attacks on enemy settlements. The colonists engaged in brutal revenge attacks, often on innocent populations, as they regarded all Native Americans as guilty for any attack by any Native American. This brutal warfare played an important role in shaping the political positions of the colonists. For example, the Delawares and other Native American peoples attacked white settlements in the Ohio valley during the 1760s and 1770s. They believed that the whites there intended to take away their land, as an earlier group of European settlers had in the seventeenth century. When the British colonial authorities proved unable to protect those white settlers who had defied the official prohibition on white settlement to the west of the Appalachian Mountains by settling in the Ohio Valley, the whites there felt betrayed by their own colonial government. These events contributed to the settlers’ support of the American Revolution.
The ongoing low-grade warfare between the whites and the Native Americans was an integral part of the western frontier experience throughout most of the nineteenth century. It shaped American social and cultural attitudes and contributed to the suspicion and rejection of the “other” on the one hand and to the romanticizing of arms, armed struggle, and the self-reliance of pioneers on the other hand. The ongoing American love of guns may derive from this romanticization; this is a persistent feature of US culture throughout history, despite routine shootings of innocent people, suicides, and gun-related accidents.