Why did Mr. Fogg say the world had become smaller?

Phileas Fogg is at the Reform Club, discussing a recent sensational crime with a chap called Ralph. A man has stolen a considerable sum of money—£55,000, no less—from the Bank of England in London. In The Daily Telegraph, a well-known British newspaper, it's reported that the thief was a gentleman rather than a professional robber. It's this assumption that will form the basis of Inspector Fix's suspicion that Fogg himself was the gentleman thief.
During their conversation, someone remarks that the world is such a big place that the thief could be hiding pretty much anywhere. Fogg replies that the world is much smaller due to the remarkable innovations in transport that allow people to move around the globe so much more quickly than ever before. Furthermore, Fogg confidently asserts that the world has become so small, that it's actually possible to get around it in eighty days. It's only a matter of time before someone asks Fogg to put his money where his mouth is, and agree to bet that he will indeed attempt to travel round the world in eighty days.

Is the protagonist of The Spy flat or round? What purposes are served by the traits of flat or round characterization?

The answer is to this is that Isaac Bell is not developed as a round character. He is a flat character, but this is in keeping with conventions of the detective murder mystery genre, which puts the solution of the crime above other structural considerations. Having a flat character as the detective protagonist in a murder mystery serves the function of channeling focus onto the challenge of finding and understanding clues unlocking the mystery.
As a flat character, Bell is drawn as personable and sympathetic by the use of defining characteristics, but he remains at the end of the mystery story virtually as he is at the beginning: solving the mystery is his job, not an opportunity for life lessons and epiphanies (as would be the case for a round character).

"Would you mind if I waited?" Bell asked with a smile that flashed his even teeth and lighted his blue eyes.

Even though a flat character, we are given insight into Bell's moral and ethical character traits. For example, we see his ethical traits when he says that he doesn't want to "leave John Scully in a lurch" on the Frye Boys case, and we see his moral traits when he chooses to dig deeper into Dorothy Langner's request because she is "passionate" about her father's work and struck "oddly" by the handwriting in the suicide note.
In the detective murder mystery genre, a flat detective serves the purpose of structural development around solving the mystery, saving the good by-standers, saving the good reputation of the victim and bringing the perpetrator to justice. In serving the purpose of the plot in this way, it is not necessary for the detective to have personal development, to grow and change--to be a round character--while doing so.
The purpose of a round character, in contrast, is to bring the protagonist and the reader through an in-depth life experience, the result of which is change and growth ending in a deeper knowledge, appreciation and understanding of life, nature, others or self.   
https://www.britannica.com/art/flat-character

How does the protagonist behave toward the end of the story? Does her final action display hysteria? Explain.

By the end of the story, the protagonist begins to have difficulty separating herself from her visions of the woman in the wallpaper. She starts to imagine that she herself might, in fact, be the woman she has been seeing during the duration of her stay in the house. She beings to "creep" in the same fashion as the woman she's seen in the wallpaper, as well as tear down the paper that she feels has kept "her" (she is now the woman in the paper) prisoner during the story. As a final act she locks herself in her room and refuses to respond to the calls of her husband to open the door and be seen.
While it is true that the protagonist's actions fit the formal definition of hysteria - a condition characterized by "exaggerated or uncontrollable emotion or excitement" - saying that she has become "hysterical" herself is a difficult statement to make, primarily because of the purpose of the story. The author's purpose in writing the story was an indictment of the treatment of women by the patriarchal society they lived. This society blamed the "hysterical" condition of women for their illnesses, and in order to combat these illnesses they often found it necessary to seclude women in order to combat the perceived hysteria. The irony of this, something the story is attempting to point out, is the ill-treatment of women in order to cure they're "hysteria" could have actually served as the reason for the conditions to begin with. If doctors, physicians, husbands, etc. would attempt to treat the women in their lives as equals - with respect and dignity instead of cordoning them off on their own to deal with the issues they were facing, such as "hysteria" - it would lead to a better outcome for all parties involved.

Toward the end of the story, the protagonist actually comes to believe that she is the woman that she has freed from the wallpaper. She says something about not wanting to look out the windows of her room because of all the creepy creeping women in the garden outside. Then suddenly, she says,

I wonder if they all come out of that wall-paper as I did?

Earlier in the story, she works hard to free the woman she believes is trapped in the wallpaper. The narrator seems to recognize on some level that she is, herself, trapped by her "treatment," but at least she could assist this other woman to freedom. Now that she believes that she's freed this woman by ripping down all the wallpaper, she actually changes places with this woman, in her mind, perhaps because she believes this woman is now free. She even refers to someone named Jane, who we've never heard mentioned in the story before, saying to her husband,

I've got out at last . . . in spite of you and Jane! And I've pulled off most of the paper, so you can't put me back!

I believe that Jane is actually her own name, a name she no longer recognizes as hers because she believes that she is someone else.
Hysteria is not now recognized as an actual psychological condition. It was used, during this time, as a sort of catch-all term to describe any kind of emotional irregularity or mental illness (like depression or anxiety) that women had, like the postpartum depression from which the narrator probably suffers. I wouldn't describe her as hysterical, because her husband's "treatment" has actually caused her depression to develop into something much more severe: she no longer recognizes her own identity and now believes herself to be someone else.

Can you explain how/why this is an external force conflict in "The Lottery" by Shirley Jackson? Old Man Warner brags about how many lotteries he has survived and scolds those who think it is time to rethink this. He obviously believes the lottery serves a common good, only because he has survived them all. Perhaps his tune would change if his moment came.

What you have placed in quotation marks is not from "The Lottery" itself. I am going to assume this is your writing prompt, a brief summary of what might be an external conflict, and I've revised it somewhat. Since any conflict that is not an internal conflict must be some sort of external conflict, I would have to say that there would be a conflict between Old Man Warner and the town if he were to be the one chosen to die, and there would be a conflict between him and tradition in that event, too.
The sole protesting voice is Tessie Hutchinson. She first protests when it is her family group chosen, then protests when her daughter is not considered part of the family group, thus increasing her own odds of being chosen, and then protests again when she is selected for stoning. It is fair to say that there is a conflict between her and the town at this point, and the same conflict would be between the person chosen and the town, no matter who is selected. If that were Old Man Warner, he would probably be complaining loudly. One might be able to lead a willing scapegoat out into the desert, but once one is sacrificing a human, a conflict between the sacrifice and those doing the sacrificing is going to be inevitable.
Another way to view this as an external conflict might be to view it as a conflict between the person chosen and tradition, with similar reasoning. Everyone, including Old Man Warner, is completely on board with this tradition of stoning someone to death once annually. But as we can see from Tessie Hutchinson, this respect can dissipate very quickly when the tradition is about to kill a person. Old Man Warner is fully in favor of this tradition because it has not cost him a thing. If he were the winner of the lottery, he would quite likely find himself in conflict with tradition.

What new weapons and tactics emerged from World War II?

Necessity, it is said, is the mother of invention, and the developments of warfare that took place during World War Two bear this out. While the twentieth century witnessed a boom in advances in military technology, by the end of World War Two, the war in the air, in particular, had been entirely transformed from the early days of World War One, when soldiers shot at each other using unanchored guns from propeller-driven biplanes.
The invention of the jet engine, certainly, was one of the advances that changed the course of the war and flying in general. The jet engine was actually invented and patented in 1930 by Frank Whittle, but it was the Second World War which spurred Whittle into action; a flight test was performed in 1941. This technology was shared with the Americans by the Royal Air Force, meaning that the two nations were able to independently develop faster and more powerful airplanes. At the same time, German engineers were independently developing similar advances in jet engine technology. The result was that airplanes were able to cover longer distances, fly faster, and stay airborne for longer. Massive battles in the air, such as the Battle of Britain, and long flights, such as the US expedition to Hiroshima, would not have been possible without the growth of aeronautical technology.
This leads us on to the second key advancement in military technology resulting from World War Two: the atom bomb. Again, Allied scientists were not the only ones who were making advances in this arena—indeed, scientists were propelled by fears that German scientists had already developed a nuclear weapon. Albert Einstein, a refugee in the United States, convinced President Roosevelt to embark on research in this area, and by 1945, a weapon was ready to be tested. The first bomb was tested in New Mexico, and the dropping of the atom bomb on Hiroshima, Japan, is generally thought to have hastened the end of the war. This massive shift in weapons technology led the world into a new age of fear and stalemate, which would become known as the Cold War: weapons and scientific developments were pitted against each other, but strikes were more generally feared rather than enacted.

Why and how did Europeans openly challenge the Catholic Church, church doctrine, and the bible?

By the time of the Reformation, European society was experiencing greater social mobility. The rising middle class was challenging the established social order. Humanists stressed the central importance of individual talents and performance regardless of noble descent. Printing made books widely available and facilitated access to the Bible for literate lay people. People were increasingly looking for a personalized approach to religion to accommodate diverse spiritual needs and guide individuals in their quest for salvation.
Meanwhile the Catholic Church as an institution remained deeply committed to medieval ritualism. To satisfy its financial needs it relied on payment for various religious ceremonies as well as on the sale of indulgences, which promised their buyers forgiveness of sins if they repented sincerely in addition to making their purchase.
Martin Luther, a pious German Augustinian monk and a professor of theology who was of modest middle class origin, resented the indulgences, which he found incompatible with a personal connection to God. He posted his objections. Because he was a forceful and effective writer in both German and Latin, he soon became very popular with the German reading public. When the Papacy denounced him as a dangerous heretic, he responded by denying the notion of clergy as a separate estate and the Church’s claims to be the sole mediator between people and God. He stressed the importance of the individual believer's conscience and faith. He also translated the Bible into German and argued that Scripture as the Word of God was much more reliable than Church traditions. His numerous followers formed their own religious communities, creating Protestantism as an independent religion.

f(x) = arctan(e^x) Find the derivative of the function

The given function: f(x) =arctan(e^x)  is in a form of inverse trigonometric function.
It can be evaluated using the derivative formula for inverse of tangent function:
d/(dx)arctan(u) = ((du)/(dx))/(1+x^2) .
 We  let u = e^x then  (du)/(dx)= d/(dx) (e^x)= e^x .
Applying the the formula, we get:
f'(x)= d/(dx) arctan(e^x)
          =e^x/(1 +(e^x)^2)
Using the law of exponent: (x^n)^m=x^(n*m) , we may simplify the part:
(e^x)^2 = e^((x*2)) = e^(2x)
The derivative of the function f(x) = arctan(e^x) becomes:
f'(x)= e^x/(1 +e^(2x)) as the Final Answer.