Why does Jerry say that Peter will get cancer of the mouth?

Jerry has only just sat next to Peter on a park bench before he starts talking to him without being prompted. The men are complete strangers and yet Jerry has no compunction whatsoever in addressing Peter in a somewhat brusque, presumptuous manner. When Peter prepares to light up his pipe, Jerry proceeds to give him a little lecture about the dangers of mouth cancer, which he thinks Peter will get from smoking.
Immediately, we're put on our guard by Jerry. His opening gambits in the conversation indicate what kind of person we're dealing with. Jerry is a rather peculiar individual, one of society's outcasts with a slightly weird outlook on life. A gay loner who lives in a run-down apartment, it's no wonder that Jerry appears a little strange, especially when we consider the era in which the play is set (the 1950s). In a decade of often stultifying conformity, Jerry's about as nonconformist as you can get.

What should have been the role of the United States in the world in the early twentieth century?

The answer to this question depends on what is meant by "early twentieth century." If we mean before World War One, the question would likely reference American involvement in Latin America, imperialism in the Philippines, and involvements in attempts to expand economic influence to China. Each of these, it should be fairly clear, was at odds with professions of democracy and human rights, especially the brutal suppression of a rebellion in the Philippines following the Spanish-American War. Frequent interference in the affairs of sovereign nations, it could certainly be argued, was not consistent with American ideals, nor those of any nation born in an anti-colonial struggle.
However, if "early" encompasses the entire period before the First World War, the answer becomes more complex. In the wake of this conflict, the United States attempted to isolate itself from the affairs of European nations in particular, a rejection of the internationalist values expressed in the Fourteen Points of President Woodrow Wilson. This new stance was embodied by the rejection of the Treaty of Versailles by isolationists in the Senate. They refused to ratify it on the grounds that it would have led to American membership in the League of Nations.
The fact that the United States was not involved in the League, and that the nation generally refused to openly support China, France, and Great Britain in the face of aggression by militaristic and fascist dictators, contributed to the outbreak of the Second World War. Many would argue, then, that American foreign policy in this early twentieth century crisis should have been more international in outlook.
https://history.state.gov/departmenthistory/short-history/development

Some historians might argue that the United States should have been a more progressive country in the early twentieth century, as domestically, many Presidents were moving towards a more progressive agenda. For example, Theodore Roosevelt used the Sherman Anti-Trust Act to break apart monopolies and instituted federal legislation such as the Hepburn Railroad Act and Pure Food and Drug Act to make railroad rates fairer and to regulate the quality of food and drugs, respectively.
However, internationally, Roosevelt's agenda was one of enforcing American imperialism on other countries. In 1898 (slightly before the period in the question), Roosevelt, then Assistant Secretary of the Navy, ordered Admiral Dewey to attack the Philippines if war broke out between Spain the United States (which happened after the explosion of the U.S. battleship the Maine in Havana harbor). This was the outbreak of the Spanish-American War, which led the U.S. to institute the Platt Amendment in Cuba. This legislation restricted Cuba's ability to make agreements with other countries and allowed the U.S. to intervene in Cuba's domestic affairs. Later, Teddy Roosevelt fomented a revolution in Colombia so that Panama would break away from Colombia. This revolution facilitated Roosevelt's building of the Panama Canal. His idea was to facilitate shipping from one coast of the U.S. to the other.
During the Taft administration, the U.S. pursued a policy of "dollar diplomacy" that promoted American investment abroad, particularly in Latin America and the Far East. It was not until Wilson's entry into World War I in 1917 that the U.S. supported a policy that, at least on the surface, was aimed at promoting international diplomacy and world peace. Until that time, the U.S. pursued an imperialist agenda that was aimed at aiding the American economy and facilitating American conquests abroad.
The U.S. should have taken a more progressive moral stance that provided assistance in the form of financial aid and technical knowledge to countries that were struggling to feed themselves, such as Cuba. This type of humanitarian aid would have provided greater stability than the constant military interventions conducted by the U.S. If the U.S. choose to intervene in a country such as Panama to build infrastructure that would benefit Americans, the American government could have also provided local assistance, such as setting up healthcare clinics or working to eradicate disease-carrying insects. The U.S. should have been a moral example rather than an unfettered interventionist. 

What happens at the three addresses in I Am the Messenger?

First, I will mention what happens at the three addresses and then provide a little more detail on each.
The first address is 45 Edgar Street. Ed has to go to this address at midnight. There, he sees a man yell at his wife and then rape her. Their daughter, helpless to intervene, cries on the porch.
The second address is 13 Harrison Avenue. Ed has to go to this address at 6 p.m. There, he meets Milla Johnson. She is a harmless old lady, but it is quite obvious that she is senile. Milla keeps referring to Ed as "Jimmy."
The third address is 6 Macedoni Street at 5:30 am. There, he meets a fifteen-year-old girl named Sophie. Sophie is a runner, and that is her passion in life. Ed isn't sure what to do with her except to watch her run.
Now, I'll provide a little more detail about the addresses.
First, Ed is a nineteen-year-old under-aged cab driver. He has three main friends. They are Marv, Ritchie, and Audrey. Ed has a crush on Audrey, but she doesn't seem to be remotely interested in him.
One day, he comes home from work to find an unfamiliar-looking piece of mail waiting for him. Inside is an Ace of Diamonds with three addresses and corresponding times. Ed is unsure about how he should proceed. He gets cold feet and tries to put off his assignments. Eventually, he does make his way to the first address on the list: 45 Edgar Street. He gets there and waits for midnight. When the time comes, a huge, mean-looking man enters the house at 45 Edgar Street. He is obviously drunk.
The man yells for his wife, who soon makes her appearance in her nightgown. Immediately, the man wrenches the nightgown off her and proceeds to rape her. Her screams ring out into the night, but none of the neighbors hear. Finally, Ed sees a little girl cry on the porch, as her mother is being raped. Later, her own mother comes out and weeps on the same porch. Ed suddenly realizes that he's been sent to save the mother and her daughter. However, he is afraid: the man is heavily-built and intimidating. Ed decides to leave it alone for now.
Next, he makes his way to 13 Harrison Avenue. He's quite relieved when he gets there because nothing eventful happens. An old lady lives at the address. Ed sits in his cab and watches her make dinner. She eats alone; Ed thinks that she looks like a gentle and kind old lady. He is comforted just by watching her. Ed returns the next day at about the same time: 6 pm. This time, he knocks on the door. The old lady answers and addresses him as "Jimmy." Ed decides to get into the act and have dinner with her. Before he leaves, Ed comes to realize that he has been sent to comfort a lonely old lady. Eventually, Ed discovers that the old lady is Milla Johnson, whose husband (Jimmy) died serving his country in World War Two.
The third (and last) address Ed visits is 6 Macedoni Street. He gets there at 5:30 am one day. Macedoni Street is quite hilly, and a two-story house sits at the top of the hill. At about 5:30 am, a young girl leaves the house. She is dressed in red, sporty shorts and a hooded sweatshirt. Ed notices that she has no shoes on. She is quite obviously a runner. As he watches her, he realizes that the girl is beautiful. She is about fifteen and is five-foot-nine. Ed feels ashamed as he experiences feelings of lust for the girl.
At this point, he isn't sure what he should do for the girl. After all, she doesn't seem to need him. He finally decides that her problem has something to do with running.

Intermediate Algebra, Chapter 3, 3.4, Section 3.4, Problem 20

Illustrate the linear inequality $x - 5y \leq 0 $ in two variables.
To graph $x - 5y \leq 0$ we must graph the boundary line $x - 5y = 0$ first. To do this, we need to find the
intercepts of the line

$x$-intercept (set $y = 0$):

$
\begin{equation}
\begin{aligned}
x -5(0) &= 0\\
\\
x &= 0
\end{aligned}
\end{equation}
$


$y$-intercept (set $x = 0$):

$
\begin{equation}
\begin{aligned}
(0) - 5y &= 0 \\
\\
-5y &= 0 \\
\\
y &= 0
\end{aligned}
\end{equation}
$


The $x$ and $y$-intercepts are the origin or at point $(0,0)$

Now, by using test point, let's say point $(-6,1)$ for the left of the boundary line,

$
\begin{equation}
\begin{aligned}
x - 5y &\leq 0 \\
\\
-6 - 5 (1) &\leq 0 \\
\\
-6 - 5 &\leq 0 \\
\\
-11 &\leq0
\end{aligned}
\end{equation}
$


Since the inequality symbol is $\leq$, then the boundary line must be solid.
Moreover, since the test point satisfy the inequality, then we must shade the left
portion of the boundary line. So the graph is,

What will happen if New York stays loyal to England in the story Chains?

New York City was a vital, military location during the revolutionary war.  It is a defensible location, it has ample space to house troops, it has land access, and it has a harbor for water access.  In addition to being a strategic location, it would be demoralizing to the Patriots if New York stayed in British control.  It's pure conjecture, but it's possible that the Patriots might never have won the war if New York had remained under British control.  
Perhaps the question is asking about what specifically might happen to characters within the novel Chains if New York remains under Loyalist control.  If New York stays under British control, then the Loyalists can safely remain in the city.  That means the Locktons can continue to live in the city and covertly work to undermine the Patriot army.  Isabel does manage to escape from the Locktons, but as long as the Locktons are able to stay in New York, she knows that her chances of freedom are very low.  Isabel can continue to spy on the Loyalists, but her risks go up the longer that she does that.  

Calculus: Early Transcendentals, Chapter 7, 7.3, Section 7.3, Problem 18

intdx/[(ax)^2-b^2]^(3/2)
Let's use the integral substitution,
Let u=ax
du=adx
=>dx=(du)/a
=int(du)/(a(u^2-b^2)^(3/2))
=1/aint(du)/(u^2-b^2)^(3/2)
Now let's use the trigonometric substitution,
Let u=bsec(theta)
so du=bsec(theta)tan(theta)d theta
Plug these in the integrand,
=1/aint(bsec(theta)tan(theta))/(b^2sec^2(theta)-b^2)^(3/2)d theta
=1/aint(bsec(theta)tan(theta))/(b^2(sec^2(theta)-1))^(3/2)d theta
=1/aint(bsec(theta)tan(theta))/((b^2)^(3/2)(sec^2(theta)-1)^(3/2))d theta
Now use the identity:tan^2(theta)=sec^2(theta)-1
=1/aint(bsec(theta)tan(theta))/(b^3(tan^2(theta))^(3/2))d theta
=1/aint(sec(theta)tan(theta))/(b^2tan^3(theta))d theta
=1/(ab^2)intsec(theta)/(tan^2(theta))d theta
=1/(ab^2)int(1/cos(theta))/((sin^2(theta))/(cos^2(theta)))d theta
=1/(ab^2)int(1/cos(theta))*(cos^2(theta))/(sin^2(theta))d theta
=1/(ab^2)intcos(theta)/(sin^2(theta))d theta
Now let v=sin(theta)
=>dv=cos(theta)d theta
=1/(ab^2)int1/v^2dv
=1/(ab^2)(v^(-2+1)/(-2+1))
=1/(ab^2)(-1/v)
substitute back v=sin(theta)
=-1/(ab^2sin(theta))
We have used the substitution u=bsec(theta)
So,cos(theta)=b/u
using pythagorean identity,
sin^2(theta)+cos^2(theta)=1
sin^2(theta)+(b/u)^2=1
sin^2(theta)=1-b^2/u^2
sin^2(theta)=(u^2-b^2)/u^2
sin(theta)=sqrt(u^2-b^2)/u
Also recall we have used u=ax,
:.sin(theta)=sqrt((ax)^2-b^2)/(ax)
=-1/(ab^2sqrt((ax)^2-b^2)/(ax))
=(-1/(b^2))(x/sqrt((ax)^2-b^2))
Add a constant C to the solution,
=(-1/b^2)(x/sqrt((ax)^2-b^2))+C

What are the poetic and literary devices used in the poem "Crazy Kate" by William Cowper?

In "Crazy Kate," Cowper uses personification to express the vicissitudes of fate to which Kate has been subjected. For example, in the beginning of the poem, he writes of a person who "better days / saw better clad, in cloak of satin trimmed / With lace." He makes "better days" the observer of Kate (and he also repeats the word "better" to emphasize Kate's decline). Later, he writes, "Her fancy followed him through foaming waves / To distant shores." In these lines, another example of personification, Cowper makes Kate's "fancy," or imagination, the subject of the action in which she is following her lost love, a sailor, to his destination. Cowper's use of personification stresses that Kate herself is hapless and can't follow her lost love, though she wants to. Instead, only her imagination can follow him. "Fancy" also "would oft anticipate his glad return," meaning that her imagination also believes falsely that he will come back. The poet's use of personification expresses the idea that her emotions and delusions are coming to take over her mind.
Later in the poem, Kate must spend her days alongside the shore and, "unless when Charity forbids, / The livelong night." In another use of personification, only "Charity" (meaning people's good will) can prevent Kate from being homeless day and night. The poet's use of personification to express charity also expresses that Kate is subject to the vicissitudes of fate and the whims of her fellow man to live. 
Cowper uses symbols, such as the "cloak of satin trimmed / With lace, and hat with splendid ribbon bound," to stand for Kate's sanity and better luck in earlier days. After her love is lost, her attire is described: "A tattered apron hides, / Worn as a cloak, and hardly hides a gown / More tattered still." Her clothes symbolize her decline, her inner pain, and her mental disintegration. As her clothes decline, so does she, mentally. Her clothes are the outward manifestation of her inner state. The "idle pin" that Kate begs from others is also a symbol of her mental decline, as she hordes the pins as if they were gold, but they serve no purpose. 
At the end of the poem, Cowper uses anaphora, or repetition of words at the beginning of clauses, in the lines: "Though pressed with hunger oft, or comelier clothes, / Though pinched with cold, asks never. Kate is crazed." The poet's repetition of the word "though" expresses that Kate is really oppressed by cold and hunger, but she never asks for food. Her extreme want, compared with what she asks for, shows her mental distress.