How did the seven years of rain affect the children from "All Summer in a Day"?

The children have been somewhat traumatized by the constant rain.  They desperately want the sun to come out.  You could say this made them somewhat bitter and mean, based on the way they treat Margot.

They were all nine years old, and if there had been a day, seven years ago, when the sun came out for an hour and showed its face to the stunned world, they could not recall. 

The children desperately want the sun to come out.  They are jealous of Margot because she has come more recently from Earth than they have, and she says she remembers it. For this they relentlessly bully her.
Margot does not fit in.  In addition to being an outsider and more of an Earthling, she does not relate to the other children.  She doesn't try to be one of them.

They edged away from her, they would not look at her. She felt them go away. And this was because she would play no games with them in the echoing tunnels of the underground city.

The sun coming out and the stopping and then restarting of the rain seems to do something to the children.  They lock Margot in the closet so she misses the sun's brief appearance, a particularly cruel trick that seems to make even the children realize it's terrible once they have done it.
Rain can be depressing.  There is even a name for the feeling people get when the weather makes them depressed.  It is called Seasonal Affective Disorder.  Imagine how bad this disorder would get if the seasons were seven years long.  Even when the sun comes out, it is very brief.  People need sunshine.
 
https://www.mayoclinic.org/diseases-conditions/seasonal-affective-disorder/symptoms-causes/syc-20364651

Calculus and Its Applications, Chapter 1, 1.6, Section 1.6, Problem 24

Differentiate $\displaystyle y = \frac{3x^4 + 2x}{x^3 - 1}$

By applying Long Division first before differentiating, we get



$\displaystyle y = \frac{3x^4 + 2x}{x^3 -1} = 3x + \frac{5x}{x^3 - 1}$


Then,

$
\begin{equation}
\begin{aligned}
y' &= \frac{d}{dx} \left[ 3x + \frac{5x}{x^3 - 1} \right] = \frac{d}{dx} (3x) + \frac{d}{dx} \left( \frac{5x}{x^3 - 1} \right)\\
\\
&= 3 + \left[ \frac{(x^3 - 1) \cdot \frac{d}{dx} (5x) - (5x) \cdot \frac{d}{dx}(x^3 - 1) }{(x^3 - 1)^2} \right]\\
\\
&= 3 + \left[ \frac{(x^3 - 1)(5) - (5x) (3x^2) }{(x^3 - 1)^2} \right]\\
\\
&= 3 + \left[ \frac{5x^3 - 5 - 15x^3}{(x^3 - 1)^2} \right]\\
\\
&= 3 + \left[ \frac{-10x^3 - 5}{(x^3 - 1)^2} \right]\\
\\
&= \frac{3(x^3 - 1)^2 - 10x^3 - 5}{(x^3 - 1)^2}\\
\\
&= \frac{3\left[(x^3)^2 - 2(x^3)(1) + (-1)^2 \right] -10x^3 -5 }{(x^3 - 1)^2}\\
\\
&= \frac{3 \left[ x^6 - 2x^3 + 1 \right] - 10x^3 - 5}{(x^3 - 1)^2}\\
\\
&= \frac{3x^6 - 6x^3 + 3 - 10x^3 - 5}{(x^3 - 1)^2}\\
\\
&= \frac{3x^6 - 16x^3 - 2}{(x^3 - 1)^2}
\end{aligned}
\end{equation}
$

What historical context can help me understand the line "So have I also, often in wretchedness" from "The Wanderer"?

You are right to seek out the historical context for this ancient, Anglo Saxon poem in order to understand the meaning of that line. First, let us put the line in context and then talk about its historical significance. The following is from the Old English translation from my own textbook:

So I,
often wretched and sorrowful,
bereft of my homeland,
far from noble kinsmen,
have had to bind in fetters my inmost thoughts

We learn from the first stanza that the speaker feels lonely, "wretched and sorrowful" because he seems destined to wander the open ocean on a sea voyage. During Anglo-Saxon times, sea voyages were common, long, and treacherous. There were no modern conveniences. Sails and stars were the useful implements. Sailors were at the mercy of the weather. Obviously, from the lines in the quote above, the wanderer is without his "noble kinsmen"—his friends and family. The wait to see family and friends again will most likely be "for a long time," as the speaker tells us in the poem.  
Another important part of the historical context (although not contained in the quote you mention) is the Anglo-Saxon transition from paganism to Christianity. Note that even in your quotation, old paganism does not provide enough hope for the wanderer. The only thing that does is hinted at in the first and the last lines of the poem. The first lines mention "the mercy of the Lord" while the last lines reveal the following:

It is better for the one that seeks mercy, consolation from the father in the heavens where, for us, all permanence rests.

As a last mention of historical context, we should note that the Anglo-Saxon life—especially an Anglo-Saxon sailor's—is one of uncertainty. Disease, weather, famine, and death are all realities for the Anglo Saxon. Mercy from God is all Anglo-Saxon wanderers can hope for.

In Lee's To Kill a Mockingbird, how does Scout help Boo Radley?

In the novel, Boo Radley saves Scout and Jem from Mr. Ewell, and he earns the gratitude of both Atticus and the children.
When Sheriff Tate and Atticus discuss the aftermath of Mr. Ewell's attack and Jem's possible culpability for Mr. Ewell's death, Boo Radley is present. However, because of his shy ways, Boo doesn't join in the conversation. Scout instinctively knows the reason for his reticence, and she's respectful of Boo's quiet ways. With a compassion and sensitivity beyond her years, Scout quietly leads Boo to the farthest corner of the porch, where he will feel more comfortable in the shadows. Scout doesn't judge Boo; she helps him by extending him the gentlest of courtesies, but she doesn't make any effort to change who he is or to cause him discomfort.
After Sheriff Tate and Atticus decide on the best course of action regarding Mr. Ewell's death, Boo gets ready to leave. Before he leaves, however, he nods towards the front door. Instinctively, Scout pipes up, "You’d like to say good night to Jem, wouldn’t you, Mr. Arthur? Come right in." Scout readily anticipates Boo's desires; quietly, she takes Boo by the hand to Jem's bed.
For her part, Scout is able to see Boo for who he is: though a socially-awkward individual, Boo is a man of courage. He put his life on the line to save Jem and Scout from certain death. It was Boo who stabbed Mr. Ewell, an action at once incongruent with his outward demeanor as well as characteristic of his deeper personality. Recall Miss Stephanie's anecdote about Boo, who, when he was thirty-three years old, drove a pair of scissors into his father's leg and then comfortably resumed his scrapbook activities after retrieving the scissors.
Scout realizes that Boo is more complex than most of the adults make him out to be. She definitely understands his tendency to withdraw into himself and makes allowances for it. Yet, she also understands his ability to react in unexpected ways. Her receptivity to Boo can be seen at the moment Boo stands over Jem's bed. Almost imperceptibly, Boo lifts his hand, only to drop it again. Scout instinctively knows what he needs and wants to do, and she gently gives Boo permission to touch Jem.
After touching Jem lightly on the hair, Boo quietly asks Scout to take him home. With great kindness and tact, Scout asks Boo to bend his arm a little so that she can put her hand into the crook of his arm. Scout does this because she wants anyone who's watching to think that Boo is the one escorting her, "as any gentleman would do." Scout essentially protects Boo's dignity and image in the eyes of the community.
So, throughout their interactions, Scout helps Boo by treating him with sensitivity, kindness, and compassion. In Scout's eyes, this is the only way she can repay a man who's done so much for her and Jem.

What advantages did Britain have that helped it be the first to industrialize?

Since the Norman invasion in 1066, or perhaps the Viking attack on Lindisfarne in 793, the British peoples have relied on world trade via the oceans. William the Conqueror and his Norman nobles led expeditions deep into the Mediterranean. English shipwrights were historical leaders in innovation and development. The Magna Carta (1215) endowed English citizens with more rights and freedoms than the citizens of virtually any other country until the American and French revolutions nearly four hundred years later.
Coupling the English spirit of adventure with an emerging market economy and a governmental system that encouraged entrepreneurship, the British developed a world empire—truly, one which the “sun never set upon.” The military adventures of the British in the New World, Europe, Africa, and India made Britain a superpower by 1815. The exchange of goods, ideas, commerce, and access to appropriate raw materials—most significantly coal and iron—allowed the British to absorb, adapt, improve, and capitalize on (sell) their products in a global market.
The English and Scottish invented the incandescent light (1802), the steam locomotive railway (1804), the electromagnet (1825), a pedal bicycle (1839), and the first steam-powered, propeller-driven passenger liner (1843). By harnessing the power of steam, created in boilers forged from English iron and burning English coal, entrepreneurs soon recognized that powered machinery could be placed in specialized facilities and production could vastly exceed the scale of the historical “cottage industry.”

According to the book Weapons of Mass Destruction and Terrorism by James Forest and Russell Howard, the second edition, describe the implications of a lack of wide standardization of testing methods and facilities in the CBRNE community (pg. 809-820).

According to the National Science and Technology Council Committee on Homeland and National Security (NSTC), the lack of standardization of testing methods leads to grave consequences. In fact, such a lack is dangerous.
First, let us define some terms. CBRNE is an acronym for chemical, biological, radiological, nuclear and high-yield explosives. Terrorists use these weapons to cause mass casualties during attacks. For their part, CBRNE teams are experts who use self-contained breathing apparatuses and respiratory protective devices (RPDs) to protect themselves while analyzing scenes potentially linked to terrorist attacks. By extension, uniform test procedures ensure that products perform according to expectations during such missions.
Without wide standardization of testing methods and facilities, the implications are dire:
1) Lack of testing method standardization leads to costly redundancy: When government agencies and contractors don't have uniform standards for tests, testing facilities, infrastructure, and methodologies, manufacturers won't have a clear idea of expectations regarding their products. In turn, government agencies must repeat expensive testing procedures and replace previously purchased items that fail to meet current requirements.
2) Wide disparity in standards leads to dangerous inefficiencies. For example, government agencies currently use differing terminology for test methods, performance standards, and accreditation. For example, each government agency has differing missions. The NSTC maintains that a soldier and a first responder have differing missions. So, both will differ in how they rely on gear such as self-contained breathing apparatuses. By extension, performance requirements and testing procedures must differ for that same gear. Even the testing infrastructure required to conduct the testing of the gear will differ.
Essentially, the gear used is similar but the process to certify the accuracy and precision of the equipment will differ based on the mission of its users. A lack of uniform standards leads to the danger of gear performing at sub-par levels during catastrophes.
Without standardization of testing and facility infrastructure, it will essentially be difficult to produce equipment that withstands the requirements of each unique mission.
3) Lack of standardization means that local emergency responders and State and local purchasing agents cannot readily analyze the capabilities and limitations of commercial CBRNE systems.
Source: National Science and Technology Council Committee on Homeland and National Security Subcommittee on Standards, May 2011.

What was Miss Trunchbull's behavior like towards small kids in school?

Miss. Trunchbull treats children in a very poor manner. She asserts her dominance in anyway possible and does not like to see joy or creativity through learning. She would prefer that children not exist and that they grow up quickly. One of her favorite ways to assert her power over the children in her school is the threat of the chokey. It is a small closet turned into a torture chamber.
One student described is as such. ''It is a very tall, but very narrow cupboard. The floor is only ten inches square so you can't sit down or squat in it. You have to stand.''
Miss. Trunchbull does not like children at all and is the opposite of what a good principal should be. She is cruel and unkind to anyone that she meets.

In a word: appalling. Miss Trunchbull positively hates children. In fact, she hates them so much that she even denies she used to be one. She's a harsh, psychotic tyrant who strikes mortal fear into the hearts of children and adults alike. But her innate viciousness and cruelty come out most strongly in relation to her pupils, whom she terrorizes mercilessly, imposing extreme discipline for the most minor of infractions. Boys are forbidden to have long hair; girls are forbidden to wear pigtails. One unfortunate girl who did so, Amanda Thripp, was picked up by her pigtails and thrown over a fence by Miss Trunchbull. Indeed, Miss Trunchbull is so psychotically violent she once nearly killed a boy by throwing him out of a fifth-story window for eating Liquorice allsorts in class.
But Miss Trunchbull hates small children most of all. She's such a big bully that she really loathes anyone smaller than her. She just can't understand why small children don't just grow up more quickly. They stay small on purpose, she reckons:

I have never been able to understand why small children are so disgusting. They are the bane of my life. They are like insects. They should be got rid of as early as possible.