What are the men about to do on page 31 of A Gathering of Old Men, and what role does this action have in the larger context outside of Marshall Plantation?

Two old men named Mat and Chimley are out fishing when they are told about Mathu and what Candy wants them to do. After they consider the situation, they decide to help.
Mat and Chimley, two men in their seventies, are fishing on a favorite bank when a young boy rushes up to them and tells them that Clatoo says that Miss Merle has told him to tell the two older men that Miss Candy Marshall wants them "on the place" right away. She wants them to bring twelve-gauge shotguns and to have some empty number five shells and get to the quarter right away. When the two old men ask what all this is about, the boy replies that it has something to do with Mathu, as well as something to do with Beau Boutan, who is dead in Mathu's yard. 
After the boy departs, the two old friends sit where they are, and they ponder the situation, wondering what would happen to them if they were in the position of Mathu. The two men do not look at each other, for they are deep in their own thoughts of what happens to a black man who has a fight with a white man, a man he kills.
"I'm seventy-one," Chimley finally says to his friend. Then, he recalls that Mathu is the only one of them who has ever stood up to white men. In response, Mat tells his friend, "I don't have the strength to climb under the bed," meaning that he no longer wants to run and hide from the conflicts they have always had in life. Mat ponders things a while and responds,

"I have to go. This could be my last chance. Now when we are old men, we have to be brave."

The two old men have decided to gather together with others and protect Mathu by pretending that they have all shot Beau, a Cajun farmer, a white man. These old black men hope to transform themselves with this powerful act of group assertion. This powerful act of theirs is one of self-affirmation, restoring in them their manly pride. In addition, this act by the old men may encourage others outside the old plantation to find strength in unification with others of their ilk.

What three events cause Miss Stoner to fear for her life and prompt her urgent visit to Holmes?

The most important event was the death of Helen Stoner's twin sister two years earlier. It is still vividly imprinted on Helen's mind, especially because the mystery of Julia's death was never solved. When Sherlock Holmes enters the case in order to protect Helen, he inevitably solves the two-year-old "locked-room murder mystery" as well as saving Helen from the same fate as her sister's. 
Another event that causes Helen Stoner to fear for her life and to go to Sherlock Holmes for help is her being only recently moved into the room that was occupied by Julia at the time of her agonizing death. Helen doesn't like being there. She can't sleep well in the strange room which keeps reminding her of the night her sister died in her arms. It is because Helen suffers from insomnia in this room, and possibly because she has intuitive suspicions of her stepfather in the room next-door, that she is awake at around three o'clock in the morning when she hears the low whistle.
It is this third event, the whistle, that causes her greatest fear and prompts her to leave for London as soon as it is daylight in order to bring her problem to the great detective. As she tells him during their interview:

"Imagine, then, my thrill of terror when last night, as I lay awake, thinking over her terrible fate, I suddenly heard in the silence of the night the low whistle which had been the herald of her own death." 

Helen doesn't know it, but the "speckled band," the deadly snake, must have been in bed with her at the time she heard the whistle. It would have been slithering silently up the dummy bell-rope while she remained in bed wide awake and only waiting for daylight before she could leave for London.

Calculus of a Single Variable, Chapter 10, 10.3, Section 10.3, Problem 26

The parametric equations are:
x=t^3-6t ------------------(1)
y=t^2 -----------------(2)
From equation 2,
t=+-sqrt(y)
Substitute t=sqrt(y) in equation (1),
x=(sqrt(y))^3-6sqrt(y)
=>x=ysqrt(y)-6sqrt(y) ----------------(3)
Now substitute t=-sqrt(y) in equation (1),
x=-ysqrt(y)+6sqrt(y) ----------------(4)
The curve will cross itself at the point, where x and y values are same for different values of t.
So setting the equations 3 and 4 equal will give the point,
ysqrt(y)-6sqrt(y)=-ysqrt(y)+6sqrt(y)
=>ysqrt(y)+ysqrt(y)=6sqrt(y)+6sqrt(y)
=>2ysqrt(y)=12sqrt(y)
=>2y=12
=>y=6
Plug in the value of y in equation 4,
x=-6sqrt(6)+6sqrt(6)
=>x=0
So the curve crosses itself at the point (0,6). Note that,we can find this point by plotting the graph also.
Now let's find t for this point,
t=+-sqrt(y)=+-sqrt(6)
The derivative dy/dx is the slope of the line tangent to the parametric graph (x(t),y(t))
dy/dx=(dy/dt)/(dx/dt)
y=t^2
=>dy/dt=2t
x=t^3-6t
=>dx/dt=3t^2-6
dy/dx=(2t)/(3t^2-6)
For t=sqrt(6) , dy/dx=(2sqrt(6))/(3(sqrt(6))^2-6)=(2sqrt(6))/(18-6)=sqrt(6)/6
Equation of the tangent line can be found by using point slope form of the line,
y-6=sqrt(6)/6(x-0)
=>y=sqrt(6)/6x+6
For t=-sqrt(6) , dy/dx=(2(-sqrt(6)))/(3(-sqrt(6))^2-6)=(-2sqrt(6))/(18-6)=(-sqrt(6))/6
Equation of the tangent line will be:
y-6=(-sqrt(6))/6(x-0)
=>y=(-sqrt(6))/6x+6
Equations of the tangent line where the curve crosses itself are:
y=sqrt(6)/6x+6 and y=-sqrt(6)/6x+6

How do Benedick and Beatrice insult each other, mainly in act 1, scene 1, but throughout the play as well?

As Leonato in the first scene of this play, in reference to Beatrice's facetious queries as to how many men Benedick has killed and eaten, "there is a kind of merry war betwixt Signior Benedick" and his niece. This scene serves to give the audience an insight into the nature of the relationship between the two, and its structure allows us to see that Beatrice "insults" Benedick habitually when he is not present, as at the very beginning of the scene, and Benedick also speaks critically of Beatrice to his friends, as at the end of the scene. It is not only when they are together that the "skirmish of wit" takes place: both Benedick and Beatrice make sure to continue their war even when they are not together.
Of course, their preoccupation with thinking up witty criticisms of each other leads those around them to question what this interaction really implies, and whether the insults are truly meant as such. Benedick remarks on Beatrice's "fury" to Claudio, but he also says that she "exceeds" Hero "as much in beauty as the first of May doth the last of December." In this scene, Shakespeare is setting up his hero and heroine as two people who very much enjoy pretending to dislike each other, but the foreshadowing at the end of the scene from Don Pedro—"I shall see thee, ere I die, look pale with love"—is an early suggestion to the audience that, before the play is over, the two will be reconciled.

The insults the pair direct to each other, importantly, are intended to amuse, not only each other but those around them, as Leonato has evidently recognized. Beatrice makes a comment that would not be out of place in a romantic comedy to this day when she says, "I wonder that you will still be talking, Signior Benedick: nobody marks you." Benedick responds by calling her "my dear Lady Disdain," and Beatrice's reciprocal quip is that disdain cannot possibly die while it can feed on such fit matter as Benedick provides. The interchange is like a parry, or a dance: Beatrice offers an outright insult—"scratching could not make it worse, an 'twere such a face as yours were"—and Benedick calls her "a rare parrot teacher." The audience is supposed to interpret that this is an argument that has been much played out between the two of them, and it certainly is playing. Note how Beatrice says that Benedick "always end[s] with a jade's trick," alluding to a poorly bred horse (jade) which cannot finish a race. Between Benedick and Beatrice, their witty interchanges are a competition, and Beatrice resents being denied the full extent of them.

We know that the interchanges between these two are born out of a long association. We see this also in act 2, scene 1, in which Beatrice is completely unfooled by Benedick's disguise and happily insults Benedick to his face, but pretending she does not know it is him, calling him "a very dull fool" in order to see his reaction. In seeing through Benedick so quickly, however, what Beatrice is really telling the audience is that the two of them are, in their own way, close; they know each other and they know what to say to rouse each other's humor and anger. Through the wordplay of their insults, Beatrice and Benedick enact their own kind of courtship.

Glencoe Algebra 2, Chapter 2, 2.4, Section 2.4, Problem 36

Given an equation of a line L1 is y = 1/4 x + 7
y = 1/4 x + 7

so the slope of the line L1 be m_1 is = 1/4
as we know that the product of the slopes of the two perpendicular lines is equal to -1
let the slope of the required line is m_2
so ,
(m_1)(m_2) = -1
=> m_2 = -4
As,the slope-intercept form of the required line is
y= (m_2)x+b
from the above we know m_2 = -4 , so the line equation is
y= (-4)x+b --------------(1)
we need to find the value of b , as the line passes through the point
(x,y)= (2, -5 ) , then on substituting we get
-5 =(-4)*(2)+b
=> b = -5 +8 = 3
so the equation of the line is
y= (-4)x+ 3

Single Variable Calculus, Chapter 6, 6.4, Section 6.4, Problem 6

Use the Midpoint Rule to estimate the work done by the force $f(x)$ (in newtons) in moving an object $x$ meters from $x = 4$ to $x = 20$.

$\begin{array}{|c|c|c|c|c|c|c|c|c|c|}
\hline\\
x & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 & 20 \\
\hline\\
f(x) & 5 & 5.8 & 7.0 & 8.8 & 9.6 & 8.2 & 6.7 & 5.2 & 4.1\\
\hline
\end{array} $

Since the range is separated into 4 segments in Midpoint, we have..


$
\begin{equation}
\begin{aligned}

W = \int^{20}_4 f(x) dx \approx & \Delta x [f(6) + f(10) + f(14) + f(18)]
\\
\\
\approx & \frac{20 - 4}{4} [5.8 + 8.8 + 8.2 + 5.2]
\\
\\
\approx & 112 \text{ Joules}

\end{aligned}
\end{equation}
$

How did the Hatter try to repair his watch?

In the "Mad Tea-Party" chapter of Carroll's Alice's Adventures in Wonderland, the Hatter, the March Hare, and the Dormouse are all crowded at one part of the table when Alice approaches. When the Hatter's watch is first mentioned, he shakes it and seems puzzled that it does not work:

he had taken his watch out of his pocket, and was looking at it uneasily, shaking it every now and then, and holding it to his ear (68).

The Hatter obviously does not know what is wrong with his watch and also does not know what to do to fix it. When the March Hare takes the watch, he has a silly potential solution:

The March Hare took the watch and looked at it gloomily: then he dipped it into his cup of tea, and looked at it again (68).

Alice is astounded to find that the watch tells the day of the month but not the hour. The Hatter claims to know time instinctively. The Hatter tells Alice that it's always six o'clock; therefore, it is always tea time. The members of the tea party simply move to the next place settings when they are finished with each cup of tea and plate of bread and butter. The entire scene is nonsensical, as is much of what happens in Wonderland, so we should not expect the Hatter or the March Hare to have a good solution to the broken watch.