Why is the fact that the Americans are helping the Russians important?

In the late author Tom Clancy’s first novel, The Hunt for Red October, the assistance rendered to the Russians by the United States is important because avoiding a nuclear war is of paramount importance.
In Clancy’s story, the Americans are put in the awkward position of having to help Captain Marko Ramius in his efforts to defect to the United States with his nuclear-powered ballistic missile submarine and having to help the Soviet Union to “find” the highly-secret submarine—in order to avoid a panicked reaction against the United States. The Hunt for Red October was written during a period of intense and extraordinarily dangerous relations between the United States and the Soviet Union. The early-1980s were characterized by a large build-up in American military capabilities after a period of degradation. Modernization of American nuclear forces was a very prominent component of that build-up. In addition, both sides were adept at carrying out sophisticated naval operations intended to both collect intelligence on the other side while enabling ship and submarine crews to prepare for an actual war—a war that thankfully never materialized. Submarines and anti-submarine warfare were among the most secretive elements of the military competition between the two superpowers, and Clancy’s novel accurately captured the technological intricacies and political machinations that played out daily in the world’s oceans and seas.
The United States, under the circumstances depicted in The Hunt for Red October, helped Captain Ramius because the Navy and intelligence community desperately wanted to know as much as possible about Soviet submarine design and construction: the Red October represented the pinnacle of Soviet engineering. In addition to the intelligence that could be gleaned from examining the submarine firsthand, enormously valuable information could be conveyed by Captain Ramius after his arrival in the United States. As Admiral Greer points out at one point regarding the Soviet naval officer, "He's about the best sub driver they have, a real charger.” Captain Ramius could provide the West with invaluable insights into the thought processes of the Soviet Navy, as well as information on Soviet naval tactics. In short, the Americans had powerful incentives to help the defecting Soviet officers aboard the Red October.
The more important question regards the decision to help the Soviet government track its missing submarine. Here is where Cold War machinations and intelligence-gathering methodologies come into play. Until Ryan can convince his superiors, all the way up to the presidency, that Captain Ramius is a genuine defector, the United States must cooperate with the Soviet Union in locating the Red October so that it can be captured or destroyed before it is able to launch an attack on the United States. It is, in short, in both sides’ interests that the submarine be located and neutralized.  As Ryan notes in his meeting with the president,

The SS-N-20 has a range of six thousand miles. That means he could have hit any target in the Northern Hemisphere from the moment he left the dock. He's had six days to do that, but he has not fired. Moreover, if he had threatened to launch his birds, he would have to consider the possibility that the Soviets would enlist our assistance to locate and sink him.

Once the American government becomes convinced, however, courtesy of Ryan’s persuasiveness, that Captain Ramius is defecting and is not a rogue officer intent on starting a war, the United States still must be seen cooperating with the Soviet Union to enable the surreptitious capture of the submarine and its commanding officers without the Soviets knowing it has succeeded. This requires a deft touch; the Navy has to be perceived as helping the Soviet Union while simultaneously conspiring against it.
American assistance to the Soviet Union is significant because it is considered mutually advantageous and because it is part of a ruse to secure possession of the submarine without the Soviets knowing it.

Single Variable Calculus, Chapter 8, 8.1, Section 8.1, Problem 8

Evaluate $\displaystyle \int x^2 \cos mx dx$ by using Integration by parts.
If we let $u = x^2$ and $dv = \cos mx dx$, then
$du = 2x dx$ and $\displaystyle v = \int \cos mx dx = \frac{1}{m} \sin mx$

So,

$
\begin{equation}
\begin{aligned}
\int x^2 \cos mx dx = uv - \int vdu &= \frac{x^2}{m} \sin (mx) - \int \left( \frac{1}{m} \sin mx \right) (2x dx)\\
\\
&= \frac{x^2}{m} \sin (mx) - \frac{2}{m} \int x \sin (mx) dx
\end{aligned}
\end{equation}
$


To evaluate $\displaystyle \int x \sin (mx) dx$, we must use integration by parts once more, so...
If we let $u_1 = x$ and $dv_1 = \sin (mx) dx$, then
$ du_1 = dx$ and $\displaystyle v_1 = \int \sin (mx) dx = \frac{1}{m} \left(-\cos (mx) \right)$

Thus,

$
\begin{equation}
\begin{aligned}
\int x \sin (mx) dx &= u_1 v_1 - \int v_1 du_1 = \frac{-x}{m} \cos (mx) - \int \frac{-\cos (mx) dx}{m}\\
\\
&= \frac{-x \cos (mx)}{m} + \frac{\sin(mx)}{m^2} + c
\end{aligned}
\end{equation}
$


Therefore,



$
\begin{equation}
\begin{aligned}
\int x^2 \cos mx dx &= \frac{x^2}{m} \sin (mx) - \frac{2}{m} \left[ \frac{-x \cos (mx)}{m} + \frac{\sin (mx)}{m^2} + c\right]\\
\\
&= \frac{x^2 \sin (mx)}{m} + \frac{2x \cos (mx)}{m^2} - \frac{2 \sin (mx)}{m^3} + c
\end{aligned}
\end{equation}
$

Single Variable Calculus, Chapter 7, 7.8, Section 7.8, Problem 92

Prove that $\displaystyle f''(x) = \lim_{h \to 0 } \frac{f(x+1)-2f(x) + f(x-h)}{h^2}$ suppose that $f''(x)$ is continuous.

By applying L'Hospital's Rule
$\displaystyle f''(x) = \lim_{h \to 0 } \frac{f(x+1)-2f(x) + f(x-h)}{h^2} = \lim_{h \to 0} \frac{f'(x+h)(1)+f'(x-h)(-1)}{2h}$

Again, we must apply L'Hospital's Rule since the limit is an indeterminate form

$
\begin{equation}
\begin{aligned}
\lim_{h \to 0} \frac{f'(x+h)(1)+f'(x-h)(-1)}{2h} &= \lim_{h \to 0} \frac{f''(x+h) - f''(x-h)(-1)}{2}\\
\\
&= \lim_{h \to 0} \frac{2f''(x+h)}{2}\\
\\
&= \lim_{h \to 0} f''(x+h)\\
\\
&= f''(x+0)\\
\\
&= f''(x)
\end{aligned}
\end{equation}
$

What is a theme of Touching Spirit Bear and textual evidence that explains the theme?

Touching Spirit Bear by Ben Mikaelsen is a coming-of-age story about a young man named Cole who undergoes a healing treatment on a deserted island as punishment for violently attacking another boy. Cole is taken under the wings of two men: 
Garvey—Cole’s parole officer who got him into the rehabilitation program called Circle Justice.
Edwin—He is one of the Native Americans in charge of Circle Justice.
It’s not surprise that isolation would become a central theme to the book since Cole is put on an island alone to work out his problems. The ideal goal for Cole is to reflect on his actions and reenter society as a changed person. Isolation is explored in several ways—mostly through Cole’s experiences.
 First, Cole feels social isolation from the community. For example, Cole believes that people in his community try to help him be a better person only because they want to get rid of him. Mikaelsen writes, “He hated their fake concern. They didn't really care what happened to him. They were gutless—he could see it in their eyes. They were afraid, glad to be rid of him. They pretended to help only because they didn't know what else to do.” Do you see how Cole is isolated from society? He does it to himself, but his community also pushes him away too.
Second, Cole experiences physical isolation. He is physically placed on an island alone. The separation from others really gets to him. Mikaelsen says, “Some nights he cried himself to sleep from loneliness. He couldn't help it. The silence became overpowering, and he longed to hear another human voice.” It’s not that Cole even wants to interact with others; he just wants to hear them to know that he isn’t alone. 
Isolation persists throughout the book until the end when Cole returns to society. From there, he must use his new skills to reenter his community.

While the "circle" theme (or, "karma") is a great and prominent theme in Mikaelsen's Touching Spirit Bear (as outlined above), there are also a myriad of other themes Mikaelsen weaves through his narrative that are worthy of exploration.
Healing is a major theme of the novel. The urge and necessity to "heal" spans across dimensions, both physical and spiritual. Some textual examples are:

Cole's initial assault on Peter. This attack is a necessary step towards healing, and is a physical and spiritual endurance on both characters thereafter.

When the Spirit Bear attacks Cole. This instance explores Cole's pain and path to healing that parallels his initial assault on Peter.

Edwin and Gavey's atonement for their past sins. This highlights a prominent struggle towards peace that promotes healing.

Cole's Mother, and her broken heart. In "letting go" of her troubled marriage, Cole's Mother embarks on a very tenuous path to healing.
Along with healing, comes forgiveness. This is another theme Mikaelsen explores, and almost works hand-in-hand with "healing". Each of the examples above insight pain, healing, and forgiveness as the ultimate goal. A vital asset in healing is forgiveness, and each character is on their own path towards enlightenment.
In choosing a theme above (or creating your own) I'd advise you to consider which theme speaks to you, and investigate your own reasoning into why the author uses this theme. This will help you create a strong thesis statement and argument.

There are several wonderful themes in Ben Mikaelsen's novel. Touching Spirit Bear. 
One dominate symbol and theme in the novel is the circle and traveling in circles. In literature, circles can mean wholeness or healing.  Cole Matthew's life moves in a full circle as he is sent to the Alaskan island as a 15-year old juvenile delinquent who assaulted a boy for telling on him and later returning to the island and living with Peter Driscal, the boy he assaulted.
Some additional textual evidence of circles as a theme are:
Cole keeps getting himself in trouble and his parents and their high-priced lawyers usually rescue him.
Cole burns down the hut and later rebuilds it.
Cole continuously tries to swim off the island, running from his problems, only to return to the island to complete his healing.
Through flashbacks, Mikaelsen shows events that led Cole to being on the island and how his parents' actions contributed.  Later during physical therapy he reconciles with his mother.
Cole blames everyone for his anger and later learns to forgive.
Peter beats up on Cole in anger and later heals
Cole and Peter carve a circle symbol as the last symbol on the totem pole symbolizing their healing.

Edna Pontellier could not have told why, wishing to go to the beach with Robert, she should in the first place have declined, and in the second place have followed in obedience to one of the two contradictory impulses which impelled her. A certain light was beginning to dawn dimly within her—the light which, showing the way, forbids it. At that early period it served but to bewilder her. It moved her to dreams, to thoughtfulness, to the shadowy anguish which had overcome her the midnight when she had abandoned herself to tears. In short, Mrs. Pontellier was beginning to realize her position in the universe as a human being, and to recognize her relations as an individual to the world within and about her.This may seem like a ponderous weight of wisdom to descend upon the soul of a young woman of twenty-eight-- perhaps more wisdom than the Holy Ghost is usually pleased to vouchsafe to any woman. But the beginning of things, of a world especially, is necessarily vague, tangled, chaotic, and exceedingly disturbing. How few of us ever emerge from such beginning!How many souls perish in its tumult! The voice of the sea is seductive; never ceasing, whispering, clamoring, murmuring, inviting the soul to wander for a spell in abysses of solitude; to lose itself in mazes of inward contemplation. The voice of the sea speaks to the soul. The touch of the sea is sensuous, enfolding the body in its soft, close embrace. 1. The “it” in line 9 refers to: a) “ the two contradictory impluses” b)“a certain light...beginning to dawn” c) “the way” d) ”the shadowy anguish” 2. In line 23,”vouchsafe” can be best replaced by: a) bestow b) inform c) refuse d) deign 3. The final paragraph of the passage(lines 34-36) foreshadow a) Edna’s affair with Arobin b) Robert’s shipwreck on his way to Mexico c) the sucide at the end of the novel d) doctor Mandalet’s disapperance 4. Which of the following lines best describes Edna’s journey from this point to the end of the novel? a) “the voice of the sea speaks to the soul.” b)”this may seem like a ponderous weight of wisdom” c)”it moved her to dreams, to thoughtfullness, to shadowy anguish...” d)”of a world especially,is necessarily vague, tangled, chaotic and exceedingly disturbing.” 5. Line 30 contains an example of which literacy device? a) polysyndeton b)asyndeton c)anthropomorphism D) metonymy 6.the final two paragraphs depict a sea that is: a) anthropomorphised b) didactic c) personified d) dynamic 7.which of the following best describes edna’s characterization in the passage: 1. She is firm in her resolve to do as she pleases. 2. She is confused and lost regarding who she is and what she wants. 3. She is young and a bit naive. a) 2 only b) 1 and 3 c) 2 and 3 d) 1,2, and 3 8. The shift in perspective in the sentence beginning on line 26 and ending on line 27 does all of the following except a) give the reader a direct connection to the emotions Edna is feeling. b) imply that the situation is not uncommon c) give the narrator the opportunity to insert her commentary and experiences d) describe a situation for Edna to be a part of something greater than she. 9. The ”anguish in line 12,most likely refers to a) Edna’s distress over the creole culture and her lack of understanding with it. b)the heartbreak Edna felt when she listened to Madame Ratignolle reveal the intimate details of her delivery, knowing she would never be a “mother-women” c) the torment of knowing that is on the beach with Robert and the moment,though beautiful to her, would have repercussions. d) the sadness Edna felt when she realized, after being scolded by Leonce and checking on Raoul, that she is trapped. 10. The tone of the passage the best be described as a) sardonic b)reverent c) didactic d)solemn What is a literacy foil? Which characters serve as foils in The Awakening and how does their relationship illuminate the text? Protagonist Edna Pontellier is said to possess “that outward existence which conforms, the inward life that questions.” Identify a character who outwardly conforms while questioning inwardly. Analyze how this tension between outward conformity and inward questioning contributes to the meaning of the work. Avoid mere plot summary.

You ask what a literary foil is, which characters serve as foils in the text, and what their relationships with Edna might illuminate.  To begin, a foil is a character who possesses traits that contrast the traits of another character, often the protagonist, in order to illuminate the qualities possessed by that other character.  We might consider both Adele Ratignolle as well as Mademoiselle Reisz foils to Edna, each accomplishing something different. 
The contrast with Adele shows us how short Edna falls when it comes to doing her duty to her husband, Leonce, and behaving in accordance with her Creole community's mores.  Adele is quite feminine, a "sensuous Madonna," and a "mother-woman," who spends her time ministering to her children and performing her wifely duties as though they were a divine calling.  She keeps up her music to beautify her home and to bring joy to her husband.  She would do nothing that might embarrass him or subject him to ridicule by his peers.  
Mademoiselle Reisz, on the other hand, shows us how very far short Edna falls when it comes to becoming an artist and giving up the need to be accepted by society.  Reisz is invited to parties in order to be the entertainment, not because she is accepted as an equal.  However, Edna is unwilling to give up ties to society completely, and she still craves acceptance and praise, especially from Adele.  Reisz says Edna must have strong wings to soar above the plain of tradition, but Edna is not strong enough to do so alone.  
Both of these characters illuminate Edna's inability to choose a role.  Initially, she chooses to be a Creole wife and mother, but she finds that unfulfilling.  Then, she wishes to be an artist. She dabbles without any real, serious talent, but she finds that unfulfilling as well.  She must be willing to give something up in order to have each one, but neither option is, in the end, sufficient to satisfy her.  
To address your next open-ended question, the tension between Edna's outward conformity and inward questioning illuminates the theme that the individual is almost always at odds with society in some way.  Further, the tension conveys the idea that the price of total freedom from society's rules is alienation from that society.  Edna cannot break the rules of society and still be accepted by that society.  No person can.

Is the Commander sterile in The Handmaid's Tale?

The short answer is yes, probably, the Commander is sterile. When Offred goes for her routine appointment at the gynecologist, he says as much to her and offers to get her pregnant himself. She is horrified by the doctor's suggestion and is sure that he could be an Eye (a spy for the government) or that he might report her as disloyal and disobedient if she declines his offer—but the fact is that Offred has had a child before, and so we know it is most likely possible for her to become pregnant again. However, the Commander has never fathered a child, and, because of his age, it is extremely likely that he cannot do so. The doctor implies that this is the case himself.

The reader is never explicitly told whether or not the Commander is sterile, but it can be inferred that he is. The Commander is an older officer of the “Sons of Jacob,” so because he is in a respectable position in this dystopian society, it is forbidden to even insinuate that he is sterile. Actually, it is forbidden to insinuate that any man in power is sterile, even though it is a fact that most of the civilization has become sterile from toxicity in the environment. Sterility is publicly blamed only on the women because of the patriarchal nature of Gilead.
The handmaids, then, are a group of women that are forced to reproduce by these men in power because they have been proven to have healthy reproductive systems, which is difficult to find. Childbirth has become so rare that any single healthy child is revered and celebrated while the handmaids continue to live a life of reproductive slavery. In this society, however, even being in this position is a “blessing” because there are many other worse positions to be in, such as cleaning up toxic waste until eventual death.
Offred, having not produced a child for the Commander yet, is in grave danger of being sent to the toxic wastelands because the handmaids have a “deadline” that they must produce a child within. The reader can assume that the Commander’s wife, Serena Joy, is aware that her husband is most likely sterile. It is difficult for her to watch the monthly “ritual” between the Commander and Offred (essentially a ritual of rape), so she arranges for Offred to sleep with her husband’s chauffeur, Nick, in secret. This is a risky move that could get all three of them in trouble, but both Nick and Offred are afraid of what Serena Joy may do should they not follow her instructions so they oblige.

While we never know for sure, the text strongly suggests that the Commander is sterile. The reason handmaids exist at all is that pollution has rendered much of the population sterile. The Commander is an older man who has never fathered a child. Given the high levels of infertility in the culture, the Commander's age, and his lack of other children, the odds are very high that he can't procreate.
However, in this dystopia, a fundamentalist Christian world, a man can never be considered sterile. The myth that protects the men is that it is always the woman's fault if a pregnancy doesn't occur. If a handmaid doesn't eventually get pregnant, she is blamed, even if every likelihood is that the problem is with the man.
Serena and Offred both know this. Serena, a pragmatist, understands that her husband is most likely sterile. (Her thoughts on the subject help convince us that he is sterile.) Therefore, she arranges for Offred to have sex with the younger, very possibly fertile Nick. Although this is a risky move, Offred agrees to it, knowing that if she doesn't get pregnant, she will be sent to die cleaning up toxic waste.

In what ways are the current problems facing the United States similar to and/or different from the problems identified during the founding ?

The founders of the American constitution were challenged with an immensely bold undertaking: to address the problems faced by the colonies but do so in such a way that their solutions would pave the path for "a more perfect union" for centuries to come. Therefore they had to build a rigid but flexible structure, able to handle both the crises of the day and those of the future. In many ways they created a framework that has done just that—in other ways, at best, there were obstacles no one could have foreseen.
The largest of the discrepancies have stemmed from three main things: technology, political norms, and geography. Technology may be the largest unforeseeable factor: today our communication, governance, social glue, journalism, news reporting, and, therefore, democracy are all fundamentally changed by technology. New questions about the second amendment and gun rights are raised by entirely new forms of weapons in civilian hands. The internet has brought us self-publishing, which has shaken trust in established central news outlets, and instantaneous transmission, whereas communication in late-18th century America was done via horseback or stagecoach. Drones, missiles, and AI are changing our warfare and national security. The existence of smart phones and big data affect our privacy and central intelligence. When drafting both the protections of individuals under the Bill of Rights and the powers of the three branches of government, we have seen how simple wording in the Constitution can have profound consequences. Take, by example, how the wording "right to assemble" can be interpreted in an online context or how "a well regulated Militia" can be understood in the context of assault weapons, bombs, and gases designed to kill crowds. Granted, the Bill of Rights followed the founding of America in 1791, but the examples still stand as disparities in problems faced across time.
The second difference, political norms, is one that has changed in part because of the US constitution. The founders were responding to the central power of a king, whereas today we have presidents, prime ministers, and congresses whose powers are more distributed than monarchs of old. The difference, more broadly, is due to a general shift in regional powers from colonialism to tighter nation states and the constraining realities of nuclear power, economic warfare, cyber warfare, and global alliances such as the UN, EU, and NATO that shape global hegemonies.
The third major difference is a simple matter of geography. Some argue it was easier for Americans to fight for independence as colonial entities in America because they were separated from their oppressors by an ocean and had at their disposal a wealth of resources and land (after they had committed terrible genocide against the many indigenous populations existing before them). Any separatists and secessionists of today's America will find it much harder, as the South discovered in the mid-19th century.
I chose to end with the similarities between the problems the founders faced and those that exist now because I find them far more compelling. Of the similarities of great significance are the troubles imposed by a tyranny of the majority, the danger of factions to democracy, the power of money to corrupt, the threats misinformation and alienation of news sources bring to the people's right to free press and sharing of information, the damage to national security caused by degradation of trust in intelligence agencies, and the conflicts of interest held by a governing entity having lasting consequences. Many of the beliefs held by the founding fathers were to thwart these various threats against democracy and to create institutions that could withstand them. By drafting checks and balances between the executive, judicial, and legislative branches of government and by preserving freedom of speech and the press as a fourth check on the government, the founders hoped that the nation would continue to build on its good assets while fortifying against the corruption that has so often toppled former regimes. They also hoped to eventually purge the country's inherited habits of oppression, as was the case with institutionalized slavery (a slow process). The current dilemmas we face today that we share with the colonies are more important than ever.