What were challenges to each of the Four Freedoms in regards to FDR's Four Freedoms speech?

In January 1941, President Roosevelt gave a speech in which he outlined four freedoms that all people should have. These freedoms included freedom of speech and expression, freedom to worship G-d as one sees fit, freedom from want, and freedom from fear. These freedoms were being threatened by the rise of dictators to power, especially in Germany, Italy, and Japan. In essence, these freedoms refer to basic human rights that all people should have.
These freedoms or rights were disappearing in some places as World War II began. In Germany and countries controlled by Germany, people were losing their right to practice whatever religion they wanted. In Germany, Jewish people were being killed because of their religion. Freedom of speech essentially disappeared in all of these countries that were being run by dictators. People who spoke out were harshly punished. As a result, people were afraid what would happen to them if they went against the will of government in these countries. People were constantly in a state of fear. People also were living in conditions that made it hard for them to have a basic standard of living. Basic supplies and necessities were lacking in many places, especially for those people that were detained by the government.
As dictatorial governments spread throughout Europe and Asia, President Roosevelt was concerned that basic human rights were being violated.
https://www.fdrfourfreedomspark.org/fdr-the-four-freedoms/

https://www.history.com/this-day-in-history/franklin-d-roosevelt-speaks-of-four-freedoms

In "Meeting at Night" by Robert Browning, what does the lover have to cross in order to meet his lady?

The lover, perhaps a male, though it is never specified, must traverse a "grey sea" in order to arrive at the "cove" where he can dock the boat on which he has traveled. Once he makes it to land, he must cross a "mile of warm sea-scented beach" and "three fields" before he gets to the farm where his lover lives. He taps at her window, and she—presumably—lights a match with which to light a lamp or a candle so that they can see one another. The scenes depicted in the first part of the poem are rich with visual imagery, with colorful descriptions of the sea, the land, the moon, and the waves that reflect the moon's yellow light so that they appear to be fiery. The scenes depicted in the second part of the poem, however, possess many different kinds of imagery, and this makes it incredibly vibrant—perhaps echoing the feelings of the lovers when they finally unite. There are olfactory, auditory, visual, tactile, and kinesthetic images that combine to express the lovers' joy.

In the poem, the lover has to cross the ocean, a mile of "warm sea-scented" beach, and three fields in order to reach his lady's farmhouse.
The poem begins by describing the "grey sea" and how the lover travels by the light of the "yellow half-moon large and low." We get the idea that the lover is in a hurry, as he "gain(s) the cove with pushing prow." His boat presumably moves at such great speed that the waves leap like "fiery ringlets from their sleep."
At last, when he reaches the land, he finds that he only has one mile of beach and three fields to cross before the farmhouse comes into view. The rhythm of the poem, signifying the haste at which the lover travels, is reinforced by kinesthetic imagery ("A tap at the pane, the quick sharp scratch / And blue spurt of a lighted match"). The end of the poem sees the joyful reunion between the lover and his lady, with their "two hearts beating each to each!"

What does the last sentence mean

The last sentence in the story is "For the life of him he could not remember."
The word "remember" is referenced three times in the story. Old Woodifield initially has trouble remembering what he has come to tell his boss. It isn't until he gets a sip of the whisky that he remembers. The old man finally tells his boss about his daughters visiting the graves of soldiers in Belgium. Accordingly, the girls had come across the grave of the boss's son while visiting their brother's grave.
Woodifield's words remind the boss of his deep pain. The text tells us that the boss had had only one son and that this son was killed in battle during World War I. Mansfield hints that the boss and Woodifield both rely on whisky as a defense mechanism against their debilitating grief.
Even as Woodifield remembers, the boss tries to forget. So, the last line references the boss's defense against pain—forgetting or not remembering is the only way he can function in his daily life. It is the only way he can reconcile losing a well-loved child.
In the text, we learn that the boss had poured all of his energies into building up a successful business while his son was alive. His aim was to hand over the business to his son when he retired. So, working with this goal in mind lent meaning to his life. Beyond that, life has little meaning. When his son died, the boss was devastated. So, forgetting or not remembering is the only way he can cope with loss and the resultant grief.

By killing and torturing a fly, the boss is hoping to put out of his mind the death of his son in World War I. Although it's been six years since he passed away, the boy's death still haunts his father's mind. On the particular day in which the story is set, the boss' unpleasant memories of his loss are inadvertently triggered by a visit from his friend, Old Woodifield, who makes a clumsy remark about the boy's grave. Try as he might, the boss cannot get the terrible thought of his dead son out of his mind.
That is until an unfortunate fly lands on his desk. The boss finds the torture, and eventual killing of the fly, a useful way to take his mind off from the enormous grief he still feels at the tragic loss of his only child. His sense of helplessness at the death of his son is sublimated into the power of life and death he exerts over the fly.

What were the reasons the colonists left Britain?

People left Great Britain for its American colonies for a number of reasons. Many were indentured servants, who went to places in the colonies to work for a certain period of time in return for land or other compensation awarded when they completed their terms. Some others were compelled to leave, and became indentured servants, out of debt or indigence. Others were "transported," i.e. sent to the colonies as punishment for a crime. Some came willingly, out of a desire for cheap land. A relatively small proportion came seeking religious freedom. These included, most famously, the Calvinist settlers in New England, but many Quakers also settled throughout the colonies. There were also communities of other religious dissenters, Mennonites and Moravians for example, who settled in pockets around the North American colonies. Some young men, especially those who did not receive a suitable inheritance, came to the colonies to attempt to make their fortune as land speculators, merchants, or other similar occupations. Above all, those people who had a choice came to the colonies from Great Britain because they saw economic opportunities that were simply not available at home.

Calculus: Early Transcendentals, Chapter 4, 4.9, Section 4.9, Problem 36

You need to evaluate the antiderivative of the function f'(x), such that:
int f'(x)dx = f(x) + c
int (x^2-1)/x dx = int x^2/x dx - int 1/x dx
int (x^2-1)/x dx = int x dx - int 1/x dx
int (x^2-1)/x dx = x^2/2 - ln|x| + c
Hence, f(x) = x^2/2 - ln|x| + c.
The function is indeterminate, because of the constant c, but the problem provides the information that f(1) = 1/2 and f(-1) = 0, hence, you may evaluate the constant c, such that:
f(1) = 1/2 - ln 1 + c => 1/2 = 1/2 - 0 + c => c = 0
f(-1) = 1/2 - ln 1 + c => 0 = 1/2 - 0 + c => c = 1/2
Hence, evaluating the function yields f(x) = x^2/2 - ln|x| for x >= 1 and f(x) = x^2/2 - ln|x| + 1/2 , for x <= -1.

Why was creating such a sympathetic portrait of fast-food pioneer Carl Karcher relevant to Schlosser's argument against the fast-food industry?

At the very least, it's debatable whether Schlosser's portrayal of Carl Karcher is more sympathetic than that of any of the other the fast-food entrepreneurs he describes in Fast Food Nation: The Dark Side of the All-American Meal. Most of these small-business people were very hard-working folks from modest backgrounds, whose lives, generally, seem no less sympathetic than Karcher's.
I would conjecture that the reason that Schlosser opened his book with Karcher's story is that Karcher was the most typical beneficiary of the changes that affected the fast-food industry, both in his ability to follow its trends and in his original location in the Anaheim, California area.
Schlosser's theme is the way in which the fast-food industry has completely permeated and transformed American food consumption, agriculture, and the economy in the past half-century, with much of that process engineered by corporate-government cooperation.
Returning to Carl Karcher in southern California, although he was shrewd enough to realize how the growth of the national highway system and burgeoning auto sales could make locating franchises near exits a profitable move, he was also benefiting not only from massive public investment in highway construction, but by a conspiracy of General Motors, Mack Truck, Firestone, and Standard Oil to buy up trolley lines in many U.S. cities, effectively dismantling public transportation in those areas to sell more cars, tires, and gas for the only remaining source of transportation. Nowhere was this executed with more thoroughness than California.
Karcher also benefited from the proximity of many of his franchise locations to the nascent Disneyland, located in Anaheim. From the Disney organization's innovative techniques for marketing to young, often very young, children, Karcher gained useful insights for reaching one of his industry's largest demographics.
And we may well wonder why this food can be marketed to children at all, since, as he points out, beginning in the 1970s, lobbyists have savaged fast-food safety regulations through allies in Congress.
The corporations who pay these lobbyists now control the "fast-food industrial complex," as Schlosser calls it, as well as a substantial part of American agriculture as a by-product. Through their use of flavor-enhancing chemicals, foods that look completely familiar have become completely reformulated in their incarnation as "fast-food."
In the concluding pages of his chapter on Karcher, as the former farm-boy engineers a takeover of the multi-billion-dollar corporation he's built, we see how completely he represents what the fast-food industry has become. When Schlosser asks him whether he misses the orange and lemon groves that have been replaced by fast-food places and strip malls in his beloved Anaheim, he replies, "No, I believe in progress."
https://www.thedailybeast.com/still-a-fast-food-nation-eric-schlosser-reflects-on-10-years-later

In the first chapter of his book Fast Food Nation, Eric Schlosser details the lives and careers of the fast food industry's founding fathers. Carl Karcher, unlike the McDonald brothers, Karcher was able to take his business from a small upstart all the way to a multi-million dollar corporation with locations nationwide.
Karcher's story is quintessentially American. He and his wife started their restaurant business with a single hot dog stand. Karcher quickly realized that he could take advantage of the booming car culture and cater to the increasing flexibility and transience of American drivers. Simply put, Americans had fallen in love with their cars. The automobile had, for many Americans, become their second space—a place away from home where they could relax, explore new ideas and places, and try new experiences.
Schlosser uses Karcher as an example of the American dream, told by the dreamer himself (Karcher). Schlosser knows that the thesis of Fast Food Nation is loaded. Many Americans cannot imagine life without fast food restaurants. The chains provide quick, low-cost food and also employ millions of Americans. By using Karcher's story as a jumping-off point, Schlosser paints a picture of American success. He then moves on in chapter two to illuminate the less-than-honest practices of many fast food chains. This comparison creates a juxtaposition that is raised repeatedly throughout the book. Schlosser knows that the history of fast food in America isn't a simply a tale of evil corporations taking advantage of a captive audience. Rather, the story is much more nuanced. By relying on facts and interjections of personal narratives, Schlosser allows the reader to come to their own conclusions about the nature, intent, and future of fast food in America.

While Schlosser includes the story of rags-to-riches fast food entrepreneur Carl Karcher, the founder of Carl's Drive-in Barbeque and later Carl's Jr. restaurants, he makes it clear that Karcher's successful beginning did not last. Karcher's story is in many ways symbolic of the hopeful rise and inevitable fall of the fast food industry that Schlosser writes about in his book.
Karcher was a hard-working, poor entrepreneur who capitalized on the meteoric growth of southern California and its car culture to build a fast food empire at the end of World War II. In the 1980s, however, Karcher and other members of his family were accused of insider trading, and he came to an agreement with the SEC (Securities and Exchange Commission) in which he paid more than half a million dollars. He was eventually also pushed out as head of the company, though he was later able to return to lead it. While he recovered his position and fortune, his company had become a huge business that was missing a great deal of the personal touch and hopeful dreams of his early enterprise. Therefore, Schlosser's inclusion of Karcher's story does not negate his argument that the fast food industry has declined over the years. 

What fear caused Penelope to hold back her belief it was Odysseus for so long?

Penelope must contend with her fear that she is being duped by someone who is not actually her husband. After all, Odysseus has been gone for twenty years, and he would likely look very different after such a length of time; he would have gone from being a young man to a middle-aged one. When he arrives at their home, claiming to be himself, he is able to string his bow and shoot an arrow through a row of axe handles, but still Penelope doubts his identity. She issues an additional test; she tells a servant to remove the bed she shared with Odysseus and place it in another room so that he can sleep in it. The real Odysseus would know that this is impossible because he had carved their bed into the trunk of a tree that grew up through their house. When Odysseus becomes angry, insisting that such a thing would not be possible, she knows that it is, indeed, him, because no one else knows the details concerning the bed.