What significant outcomes did the War of 1812 have on the United States?

Though no territory changed hands during the War of 1812, the war was quite significant for the United States.  The most important development from the war was a new wave of nationalism in the United States.  The United States took on the largest empire in the world and survived without any territory being lost and with its trade networks intact.  The Federalist Party ceased to exist after its members tried to take the Northeast out of the Union during the Hartford Convention.  When it became public that the Federalist Party tried to secede, they were considered disloyal Americans, and this led to the rise of the Democratic-Republican Party.  While the nation would always have political debates, they became less contentious and the period after the war was known as "The Era of Good Feelings."
In terms of diplomacy, the war brought about American superiority in the region around the Great Lakes.  Britain abandoned its forts there, and they stopped arming the Native Americans there.  Britain also agreed to stop impressment, which was the practice of taking American sailors for its own merchant marine.  The war also brought a new generation of heroes who would go on to achieve political fame, such as Andrew Jackson, William Henry Harrison, and Sam Houston.

Single Variable Calculus, Chapter 7, 7.3-2, Section 7.3-2, Problem 48

Differentiate $\displaystyle f(t) = \sin^2 (e^{\sin^2 t})$


$
\begin{equation}
\begin{aligned}

f'(t) =& \frac{d}{dt} [\sin ^2 (e^{\sin ^2 t})]
\\
\\
f'(t) =& \frac{d}{dt} [\sin (e^{\sin ^2 t})]^2
\\
\\
f'(t) =& 2 \sin (e^{\sin ^2 t}) \frac{d}{dt} [\sin (e^{\sin ^2 t})]
\\
\\
f'(t) =& 2 \sin (e^{\sin ^2 t}) \cos (e^{\sin ^2 t}) \frac{d}{dt} (e^{\sin ^2 t})
\\
\\
f'(t) =& 2 \sin (e^{\sin ^2 t}) \cos (e^{\sin ^2 t}) e^{\sin ^2 t} \frac{d}{dt} (\sin ^2 t)
\\
\\
f'(t) =& 2 e^{\sin ^2 t} \sin (e^{\sin ^2 t}) \cos (e^{\sin ^2 t}) (2 \sin t) \frac{d}{dt} (\sin t)
\\
\\
f'(t) =& 4 e^{\sin ^2 t} \sin t \sin (e^{\sin ^2 t}) \cos (e^{\sin ^2 t}) \cos t





\end{aligned}
\end{equation}
$

Which of the two brothers in The Scarlet Ibis do you understand and feel sympathy for?

“The Scarlet Ibis” presents two brothers, the narrator and his baby brother Doodle. Doodle is born when the narrator is six years old. He is born with a deformity of sorts, as he “seemed all head, with a small shriveled body” at birth. In fact, it is thought that he would not live long and a “mahogany coffin” is made for him in advance. His parents call him William Armstrong, a name that the narrator thinks as most appropriate “on a tombstone”.
The reader cannot help but sympathize with Doodle. Even his mother is moved to tears when she explains to the narrator Doodle’s situation—the baby boy is an invalid who would never be able to play with his older brother as other normal children do.
Therefore, when Doodle learns to crawl, the whole family is awestruck! The reader feels happy that all is not lost. There is hope that something may eventually come of the boy’s difficult life, and that he would not, after all, spend all his years lying in a bed, unable to move.
Since Doodle cannot walk, his father makes him a go-cart. The narrator finally gets the playmate he has always yearned for, because he is now able to take his brother with him outdoors, to his favorite place, Old Woman Swamp. When the narrator explains their trips to Old Woman Swamp, and how they’d “collect flowers of all kinds to weave into necklaces and crowns”, the reader is happy that the brothers are content in each other’s company. The reader is happy that Doodle is able to enjoy the beautiful things that life has to offer, in spite of his difficult health problems. We put ourselves in Doodle’s shoes and imagine what life would be like if we were always sick and had to be careful with every single thing that we did. Even when the narrator is mean to Doodle, as all siblings are sometimes to each other, the reader is still able to see the underlying love between the two.
Doodle’s brother determines to teach him how to walk at five years of age. This is because he is embarrassed by having a brother who cannot walk. After a lot of practice, Doodle does learn how to walk against all odds. The reader is drawn to Doodle’s determination at self-improvement. However, his brother does not stop at that. His pride makes him want to teach Doodle much more, perhaps more than Doodle is really able to achieve. He unintentionally works Doodle to his death. The reader weeps for the dead boy.

What is the theme of the "The School Story"?

Themes of The School Story include independence and creativity.
Natalie is a very creative girl who writes her own book called The Cheater. Her best friend, Zoe, is also very creative. She comes up with the idea to be Natalie's agent so that she can get her book published without Natalie's mother, who works in the publishing industry, knowing. By having faith in themselves and meeting obstacles with creative solutions, the girls are able to achieve their goals.
Independence is another theme in the book. It's important to Natalie that her mother doesn't know she's gotten a book contract for The Cheater. She wants her mother to approach it as if anyone wrote it. She still wants her mother's editorial skills; she just wants to go through the process like any other author would. Only when Natalie and Zoe run into problems that can't be tackled without the help of an adult do they get help from their teacher. The two girls are very independent throughout the story. Zoe even devises a campaign to get the book more attention to help ensure that it sells a lot of copies.

"A School Story" is a horror story written by English author M.R. James near the turn of the twentieth century. It seems unlikely that your question is about a story that is not well-known in America and in a fairly obscure literary genre, but the theme of this ghost story is concerned with the reach of history (in the form of a Byzantine coin) and the intervention of unwelcome supernatural elements into the lives of two men and their teacher.
What seems more likely is that you're asking about the theme of The School Story, which is actually a 2001 young adult novel by Andrew Clements with illustrations by Brian Selznick. Themes in this novel explore the difficulties in getting published, as twelve-year-old Natalie discovers; it is also about finding one's own voice as an author and self-advocate as Natalie does not want help from her mother, an editor at a publishing house. The novel also touches on the value of friendship, seen between Natalie and Zoe, and how caring, invested mentors, such as Ms. Clayton, can help young people realize their potential.

What's borsch? What symbolically significant about this being Zaroff's meal when we first meet him?

Borscht (spelled 'borsch' in the story) is a sour soup made from beetroots. It's an ethnic cuisine common to many Eastern European countries. 
Borscht is introduced in the story in the following passage, in which Rainsford is speaking with General Zaroff and Ivan:

They were eating borsch, the rich, red soup with whipped cream so dear to Russian palates. Half apologetically General Zaroff said, "We do our best to preserve the amenities of civilization here. Please forgive any lapses. We are well off the beaten track, you know. Do you think the champagne has suffered from its long ocean trip?"

Zaroff's borscht serves two purposes in this role. The first is easy to spot, since Zaroff explains it himself. It's a sign of civilization--an ironic sign, since Zaroff is a killer who hunts and murders humans. 
The second is subtler in the reading, but it's unmistakable if you look at borscht for yourself. Dyed crimson by the beetroots that give it its characteristic sour flavor, a bowl of borscht looks much like a bowl of blood. Zaroff dining on thick red liquid symbolizes his bloodthirsty nature as a hunter of men.

Why was witchcraft a crime in colonial Massachusetts?

In order to understand why witchcraft was seen as a crime, it is necessary to consider the reasons upon which the Massachusetts Bay Colony was founded. In the early 1600s, a religious minority group hoped to "purify" the Church of England of any beliefs that were not based on Scripture. These Puritans left England with the hopes of forming their own community that would be built upon these strong religious beliefs. In 1630, they left England and founded the Massachusetts Bay Colony.
Many factors led to cries of witchcraft within the colony. In 1692, a group of young girls was said to be "afflicted." They were described as having fits or seizures. Unable to find anything physically or visibly wrong with the girls, a doctor in the village labeled the girls as "bewitched." As a result, three women were accused of witchcraft. It was because of these three women's "magic" that the three young girls were ill. This series of events led to a sort of chain reaction and mass hysteria. Around 200 were accused and twenty executed for being found guilty of witchcraft.
Scripture was the law in the colony. The daily lives of the colonists consisted of prayer and other ways in which they could fulfill God's plan. Anything thought to be the work of the devil, including witchcraft, would have been considered a crime.
https://www.smithsonianmag.com/history/a-brief-history-of-the-salem-witch-trials-175162489/

Why are Jody's dad and Billy Buck going to Salinas in Steinbeck's "The Gift"?

Billy Buck and Jody's father leave early in the morning with older dairy cows that they are taking to the butcher. As they depart on horseback, Jody sees them

...drive six old milk cows out of the corral and start over the hill toward Salinas. They were going to sell the old cows to the butcher.

The two men turn the trip into more than a delivery because when they return home in the darkness of evening, Jody knows that they enjoyed some leisure activities before departing. At supper his father tells Jody that he better get to sleep because he is going to need him in the morning. Of course, Jody wonders what is going to occur the next day. As he lies in bed, trying to hear his parents as they talk downstairs, the boy cannot detect many words said beneath him. However, he hears his father tell his mother, “But, Ruth, I didn’t give much for him.” So, Jody knows that some animal has been purchased.
In the morning Jody dresses hurriedly, eats quickly, and eagerly anticipates what he will see as he follows respectfully behind the two men who head to the barn. Because it is darker in the barn than outside, Jody's eyes must adjust to the dim light. Soon, however, he can make out a red colt in the stall. "Jody’s throat collapsed in on itself and cut his breath short" because of his excitement. His father tells Jody to take good care of him or he will sell the colt. Every day Jody devotes time to his prized gift.
Tragically, though, Jody's little horse dies after becoming too cold in a winter rain and then running out one night when he is not yet well. This experience teaches Jody about death, and he passes from innocence to experience.