Where does Before I Fall take place?

Lauren Oliver’s novel Before I Fall details the tragic death of Samantha Kingston, who is killed in a car accident. In this first-person narrative, Samantha is forced to relive her death for seven consecutive days. For the most part, the novel takes place in Ridgeview, CT where Samantha and her friends attend Thomas Jefferson High School. Through a series of flashbacks, Samantha takes the reader to the homes of her friends and family, as well as to the accident scene where she dies.
Another perspective about the setting of this novel is that it takes place in Samantha’s memory. Through the course of the novel, Samantha experiences anger, disbelief, and denial about the details surrounding her death.
In closing, the setting of this book is the Thomas Jefferson High School, Ridgeview CT. This first-person narrative takes place in the protagonist’s memory, and the emotions she experienced in different settings forms the outline for this work.

Before I Fall is a young adult novel written by Lauren Oliver that was first published in 2010. It was later made into a film in 2017. The novel is set is a small town in Connecticut, although the film was actually shot in British Columbia. The story is set in 2008.
Sam, the young protagonist of the story, and her group of friends are from relatively prosperous families. They attend Thomas Jefferson High School. The main settings of the novel are the school, homes of various characters and their families (especially the home of Kent McFuller and Sam's home), and, crucially, the street on which the fatal car crash occurs. Although only discussed in retrospect, the summer camp which Lindsay and Juliet attended is an important part of the backstory.

What was the inciting incident in the play Proof?

Think of an inciting incident as a kind of jumping off point for a narrative's plot. If you can identify the one moment, early on, that all the subsequent action stems from, you've found the inciting incident.
In the case of the play Proof by David Auburn, arguments could be made for several different moments being the inciting incident. One might argue that the inciting incident is the death of Catherine's father, Robert, which occurred before the play begins and is revealed by Catherine's vision of him in act 1, scene 1. Certainly, his death shakes his daughters to their core, causes Catherine's sister Claire to come to town, exacerbates Catherine's doubts about her own mental stability, and leads to the discovery of Robert's brilliant mathematical proof, in that it allows those left behind to search through his things for signs of genius among the rambling.
There is also an argument to be made that the inciting incident is when Hal reveals to Catherine that he's been going through her late father's work, early in act 1, as this leads her to give him the key to Robert's desk, in which the titular proof is found.
Finally, as mentioned in the other answer, one could view the act of giving Hal the key as the inciting incident. Even though it comes a bit later in the play, towards the end of act 1, it is a clear turning point in the drama.

The "inciting incident" is the moment or occurrence that starts a problem in a work of literature or play (or in a movie). Up until that point, the audience has been introduced to the backstory, and the "inciting incident" brings up the dilemma that is going to be resolved in the rest of the work.
In the play Proof, the inciting incident takes place in Act I, Scene IV when Catherine, after kissing Hal, gives him the key to her father's desk. In the desk, Hal finds a proof, and he says, "It looks like it proves a theorem...a mathematical theorem about prime numbers, something mathematicians have been trying to prove since...since there were mathematicians, basically." Hal is not only excited about the proof because it is found after the death of Robert, Catherine's father and a famous mathematician, but also because the proof was written at a time when most people assumed Robert was far too mentally ill to be productive professionally. Just as the curtain drops to mark the end of Act I, Catherine says of the proof, "I didn't find it. I wrote it." This marks the inciting incident because there is no real proof that Catherine wrote the proof, pun intended.
Act II begins with a flashback to Robert and Catherine, and then, in Act II, Scene II, the plot continues to revolve around whether Catherine wrote the proof or not. Hal does not believe that Catherine had the capacity to write the proof, and Claire, Catherine's sister, thinks the handwriting in the proof is her father's. At the end of the play, Catherine says that even though Hal believes that the proof uses several new mathematical techniques that Robert, who was suffering from mental illness, couldn't have known, "it doesn't prove anything." Hal and Catherine begin working on the proof, collaborating, which is the greatest amount of proof Hal and the rest of the world are ever going to get that Catherine is the author of the proof. This process seems to resolve the questions that were brought up in the inciting incident to some degree. 

In The Signalman by Charles Dickens, what scares the narrator when he looks down the mouth of the train tunnel?

For the first time, the narrator sees the sight that the signal-man had described to him earlier:

Close at the mouth of the tunnel, I saw the appearance of a man, with his left sleeve across his eyes, passionately waving his right arm.

This seizes the narrator with a "nameless horror" because he believes he is seeing the ghost whose appearances have tormented the signal-man. This quickly passes when he realizes he is seeing one of a group of men investigating an accident scene: The signal-man has been killed by a train.
Dickens has laid the groundwork for this terror by his eerie, sympathetic, and lonely portrayal of the signal-man. Reviewing this background helps us understand why the narrator felt fear.
The signal-man works in a bleak and desolate place, and believes he has been visited by a ghostly apparition on repeated occasions. The signal-man believes the ghost is attempting to announce impending disaster or death. Twice, the ghost's appearance was followed by death on the railway.
The narrator is moved by the signal-man's devotion to duty, sincerity, and feelings of helplessness to prevent future deaths. He is also affected by the creepiness of the signal-man's stories. He repeatedly shakes off his own fear and attempts to find rational explanations for the signal-man's experiences:

Resisting the slow touch of a frozen finger tracing out my spine, I showed him how that this figure must be a deception of his sense of sight. . .
A disagreeable shudder crept over me, but I did my best against it. It was not to be denied, I rejoined, that this was a remarkable coincidence, calculated deeply to impress his mind. But it was unquestionable that remarkable coincidences did continually occur.

After the two men peer into the tunnel together and see nothing, the narrator seems to shake off his doubts entirely. His concern is to weigh public safety against his loyalty to the signal-man: It seems the signal-man is suffering from psychological problems that could lead to a mistake on the job, which could put train passengers at risk. He doesn't want to report the signal-man to his superiors and get him into trouble, so he resolves to help him meet with a doctor.
This, then, is the context in which the narrator finally sees the apparition, or at least momentarily believes he is seeing the apparition. Suddenly he sees the very thing he had convinced himself doesn't exist.
As it turns out, the man he sees isn't the ghost, but the circumstances lead the narrator (and reader) to believe that eerie, supernatural forces have indeed been at work.

What kind of a reception did Scrooge get in Fred's home?

When the reformed Scrooge arrives at Fred's house the afternoon of Christmas Day, it is after he had rudely turned down his nephew's invitation the day before to spend Christmas with him. He therefore startles the family when he shows up and asks if he can be let in to the party. However, his niece is glad to see him, as is Fred. Fred shakes his hand very heartily and makes his uncle feel welcome in no time. As Dickens puts it:

It is a mercy he [Fred] didn’t shake his arm off. He was at home in five minutes. Nothing could be heartier.

Everyone there greets Scrooge happily. In his newly joyful mood, Scrooge can't help but find everyone and everything at his nephew's home wonderful.
Dickens shows that Scrooge didn't have to be home all alone on Christmas Day, for he has loving and forgiving relatives ready to respond to his overtures and make room for him in their lives.

At the start of A Christmas Carol, Fred, Scrooge's nephew, graciously invites his uncle to Christmas lunch with him and his family. At this stage, however, Scrooge absolutely loathes Christmas; he thinks it's all just a load of humbug. And besides, Fred's too poor to be merry. So old Ebenezer rudely rejects Fred's kind invitation.
After his remarkable transformation, Scrooge has a massive change of heart and decides to pay a visit to Fred and his family. Not surprisingly, they're absolutely astonished when he turns up. They're completely unaware that Scrooge is a different man from the ornery old skinflint they've come to know. Yet it soon becomes obvious to everyone that Scrooge has indeed changed, and that he genuinely wants to celebrate Christmas. Scrooge receives a warm welcome and is made to feel right at home. The air is filled with joy and merriment, and Scrooge throws himself enthusiastically into the celebrations.

In Fahrenheit 451, part 2, what's one quote that shows Faber is brave, and what page number is it on?

Faber proclaims himself a coward. However, there are a number of quotations in part 2 of Fahrenheit 451 which suggest that Faber is, in some respects, brave. I have given two examples below. The page references refer to the 2018 Simon and Schuster Paperback edition of the book.

Then the old man grew even more courageous and said something else and that was a poem, too.

This quotation is on page 71, which is about two pages in to part 2. It suggests that Faber is brave because he seems to have a book in his possession or at least has access to books. He risks having his house burned to the ground if he is found to have books or access to books. It is, therefore, brave to even take this risk.

Then his eyes touched on the book under Montag's arm and he did not look so old any more and not quite as fragile. Slowly his fear went.

This second quotation can be found on page 77, which is approximately five pages in to part 2. The quotation suggests that Faber can be brave when he recognizes a comrade, or someone he can trust. The book under Montag's arm is a sign that Montag can be trusted.

Faber demonstrates courage when Montag comes to visit him in the novel's section called "The Sieve and the Sand." Page numbers will vary according to what addition you're reading.
Faber admits to Montag that he had been an English professor, and when they meet in a park, he recites to Montag a "rhymeless poem." Faber gives Montag his address, and when they later meet there, Faber is emboldened when he notices that Montag is carrying a book. He explains to Montag his theories about why books became "hated and feared." Faber tells Montag what is important is "quality of information," "leisure to digest it," and "the right to carry out actions based on what we learn from the interaction of the first two," but objects when Montag tells him that he "can get books." Faber tells Montag that printing books would be too dangerous, but he does suggest a plan to burn the firemen's houses, and then insists that he is joking.
Faber not only entertains subversive ideas, he gives Montag a listening device he has built. These are the brave acts of an independent thinker, but Faber still considers himself a coward.

In Part Two, Montag visits Faber's home in hopes of gaining insight into how to comprehend the texts that he is reading. Throughout their conversation, Faber explains to Montag how and why society has rejected the literary arts. Faber clearly values literature and even suggests how Montag could possibly topple the firemen structure from within. Montag is intrigued after he hears Faber's plan, but Faber insists that it would be certain suicide to challenge the government. Before leaving, Montag begins to rip pages out of the Bible that he brought. Faber responds by urging him to stop immediately before finally agreeing to help Montag. Faber then opens his bedroom door and leads Montag into a small chamber, where the green bullet is sitting on a table. Faber then says,

"I've waited, trembling, half a lifetime for someone to speak to me. I dared speak to no one. That day in the park when we sat together, I knew that some day you might drop by, with fire or friendship, it was hard to guess" (Bradbury, 43).

Faber's comment proves that he is a courageous man. Giving Montag his address in the park that fateful day was an extremely dangerous thing to do. Fortunately, in Montag, Faber found an ally he could trust.

Single Variable Calculus, Chapter 5, 5.3, Section 5.3, Problem 36

Find the integral $\displaystyle \int^2_{-2} f(x) dx$ where $\displaystyle f(x) = \left \{ \begin{array}{c}
2 & \text{if } & -2 \leq x \leq 0 \\
4 - x^2 & \text{if } & 0 < x \leq 2
\end{array} \right. $

Using Properties of Integral


$
\begin{equation}
\begin{aligned}

& \int^c_a f(x) dx + \int^b_c f(x) dx = \int^b_a f(x) dx
\\
\\
& \int^2_{-2} f(x) dx = \int^0_{-2} f(x) dx + \int^2_0 f(x) dx
\\
\\
& \int^2_{-2} f(x) dx = \int^0_{-2} 2 dx + \int^2_0 (4 - x^2) dx

\end{aligned}
\end{equation}
$


Using 2nd Fundamental Theorem of Calculus

$\displaystyle \int^b_a f(x) dx = F(b) - F(a)$, where $F$ is any anti-derivative of $f$.

Evaluating the integral $\displaystyle \int^0_{-2} 2dx,$

Let $\displaystyle f(x) = 2$, then


$
\begin{equation}
\begin{aligned}

& F(x) = 2x + C
\\
\\
& \int^0_{-2} 2dx = F(0) - F(-2)
\\
\\
& \int^0_{-2} 2dx = 2(0) + C - [2(-2) + C]
\\
\\
& \int^0_{-2} 2dx = 0 + C + 4 - C
\\
\\
& \int^0_{-2} 2dx = 4

\end{aligned}
\end{equation}
$


Evaluating the integral $\displaystyle \int^2_0 (4 - x^2) dx$,

Let $f(x) = 4 - x^2$, then


$
\begin{equation}
\begin{aligned}

F(x) =& 4x - \left( \frac{x^{2 + 1}}{2 + 1} \right) + C
\\
\\
F(x) =& 4x - \frac{x^3}{3} + C

\end{aligned}
\end{equation}
$




$
\begin{equation}
\begin{aligned}

\int^2_0 (4 - x^2) dx =& F(2) - F(0)
\\
\\
\int^2_0 (4 - x^2) dx =& 4(2) - \frac{(2)^3}{3} + C - \left[ 4(0) - \frac{0)^3}{3} + C \right]
\\
\\
\int^2_0 (4 - x^2) dx =& 8 - \frac{8}{3} + C - 0 + 0 - C
\\
\\
\int^2_0 (4 - x^2) dx =& \frac{24 - 8}{3}
\\
\\
\int^2_0 (4 - x^2) dx =& \frac{16}{3}


\end{aligned}
\end{equation}
$


Solving for $\displaystyle \int^2_{-2} f(x) dx$


$
\begin{equation}
\begin{aligned}

\int^2_{-2} f(x) dx =& 4 + \frac{16}{3}
\\
\\
\int^2_{-2} f(x) dx =& \frac{12 + 16}{3}
\\
\\
\int^2_{-2} f(x) dx =& \frac{28}{3}

\end{aligned}
\end{equation}
$

Single Variable Calculus, Chapter 4, 4.7, Section 4.7, Problem 32

Suppose that a poster with 1-inch margins as the bottom and sides and a 2-inch margin at the top have an area of 180in$^2$. What dimensions will give the largest printed area?







$ A = xy = 180;\\
\displaystyle y = \frac{180}{x}$


$A_1 = (x-2)(y-3)$

Substituting the value of $y$, we have

$
\begin{equation}
\begin{aligned}
A_1 &= (x - 2) \left( \frac{180}{x} - 3\right)\\
\\
A_1 &= 180 - 3x - \frac{360}{x} + 6
\end{aligned}
\end{equation}
$


Taking the derivative,

$
\begin{equation}
\begin{aligned}
A_1' &= -3 + \frac{360}{x^2}\\
\end{aligned}
\end{equation}
$


when $A_1' = 0,$

$
\begin{equation}
\begin{aligned}
0 & = - 3 + \frac{360}{x^2}\\
\\
\frac{360}{x^2} &= 3\\
\\
x^2 &= \frac{360}{3}\\
\\
x &= 2\sqrt{30}\text{ inches}
\end{aligned}
\end{equation}
$

when $x = 2\sqrt{30}\text{ inches,}$

$
\begin{equation}
\begin{aligned}
y &= \frac{180}{x} = \frac{180}{2\sqrt{30}} = 3 \sqrt{30}\text{ inches}
\end{aligned}
\end{equation}
$


Therefore, the dimensions that will give the largest printed area is $2 \sqrt{30}$in by $3\sqrt{30}$in