Explain the concepts of utility and its importance in economics.

Utility is in one sense extremely simple, and in another tremendously complex. The basic idea of some things being better than other things is something we intuit, probably from birth, wired into our brains by evolution: we like eating sugar, we don't like getting burned. But actually formalizing utility into something that can be applied in economic theory to understand human behavior is fiendishly complicated.In general what we do is abstract away from all this, and simply take it as given that we can assign some number to each thing (each good we might buy, or each event that might happen to us), where higher numbers are better than lower numbers, and call that the utility. Then, we assume that people seek to maximize that utility, choosing things that will lead them to higher numbers.Economists generally distinguish between two concepts of utility, a "weak" or ordinal utility which only says whether A is better than B, but cannot say how much, and a "strong" or cardinal utility that actually says that A is some number X units of utility better than B. Cardinal utility is much easier to work with, especially when dealing with matters of risk; but it is also much harder to measure in the real world. We can relatively straightforwardly determine whether people like A versus B by giving them the choice and seeing which one they go for; but in order to determine how much more, we need to devise far more complicated and clever experiments. Worse, it's actually quite common for people to behave in experiments in such a way that they can't possibly be acting by a coherent utility function, but instead must be using some sort of heuristics to make their judgments; the field of behavioral and cognitive economics arose from such findings.The units of measure for cardinal utility are also quite strange, and there is no standard unit for utility currently used by economists. Hypothetically we may speak of "utiles" or "utilons", but in reality there is no way of actually measuring such a thing. Most real work in welfare economics tries to assign utility as dollar values of "willingness-to-pay", which is simplest but carries many pitfalls; to avoid these some economists have borrowed welfare measures from public health and use measures such as Disability Adjusted Life Years or Quality Adjusted Life Years---but these too are not without their flaws. For these reasons, sometimes ordinal utility is more useful, and we can simply try to determine what people prefer, without worrying about the magnitudes of the differences.Still, the essential concept that people try to maximize their own welfare and seek what they like over what they do not like is a very fundamental and useful one. Humans do not behave randomly, nor do they generally seek out things that hurt them without some commensurately large benefit. If we could actually precisely measure utility, it would provide us with very detailed information about how people make decisions.
http://economicsconcepts.com/cardinal_utility_analysis.htm

http://staffwww.fullcoll.edu/fchan/Micro/3utility_maximization_model.htm

What are the basic principles of Confucian philosophy? How did the formulation of the principles constitute a response to China’s disarray at the time? Why do you think Confucianism, rather than Legalism or Daoism (Taoism), eventually became the philosophical basis of the Chinese empire?

China was in a state of what Wright refers to as "chaos and moral decline" during the Eastern Zhou (Wright 20). Confucius's two main principles of li, or correct ritual, and ren, or "humanness," brought order to the chaos by stressing the importance of carrying out certain actions but also carrying them out with the proper degree of humanity. Without this sense of ren, Confucius thought the practice of rituals would be hollow. By following correct actions with the correct approach, order could be brought to a disorderly society. 
Taoism, which allowed for greater freedom than Confucianism, stressed following the tao, loosely translated as "the way." However, this philosophy was not as powerful as Confucianism as a philosophical basis for China because leaders often lost their sense of the tao if their empires became too advanced. Taoism works best in a simple, peaceful society, and is not always amenable to the demands of a quickly evolving civilization, which China's was at this point. Confucianism and Taoism were often practiced simultaneously, as people used Confucian ideas to guide their public lives and Taoist ideas to guide their private lives.
Legalism, which instituted a system of rewards and punishments, did not necessarily create moral leaders. Only Confucianism, with its emphasis on li and ren, was able to create the type of moral leadership that could create an orderly society. That is why Confucianism became the philosophical basis for Chinese life.

In the film Primal Fear, why is the defense attorney and the psychiatrist convinced that the defendant is insane? How does the lawyer prove that he is legally insane?

In the film Primal Fear (1996), an altar boy by the name of Aaron Stampler is charged with the first-degree murder of a senior Catholic clergyman, Archbishop Rushman. Martin Vail, a hotshot defense attorney, agrees to represents Aaron for nothing. Martin likes the limelight and knows that the upcoming murder case will be a media sensation. At the same time, he has a genuine belief that every defendant deserves the best possible legal representation. As lawyers go, Martin believes he is just about the best there is.
However, even for an excellent trial lawyer like Martin, the Stampler case is a real challenge. Stampler is captured by police after a televised chase on foot through the streets of Chicago. When he is finally caught, he is covered in the murdered Archbishop's blood. The media quickly dubs the young man "The Butcher Boy of St. Mike's." It would seem that Martin Vail has a real struggle ahead of him.
In a case like this, when the odds are stacked so heavily against you, sometimes the only line of defense is insanity. Initially, however, Vail has no reasonable grounds for using this defense. Aaron cannot remember a thing about the night of the murder, but he does not display any outward signs that might convince anyone that he was insane at the time he committed the killing.
Vail begins to snoop around and discovers some information that he hopes will strengthen a very weak case. He comes into possession of a pornographic video which shows Aaron being sexually abused by Rushman. After confronting Aaron with the evidence, Vail brings in a psychologist, Dr. Arrington, to examine his client. During the examination, a violent, sociopathic alter ego called Roy appears to emerge. "Roy" tells the psychologist that he killed the Archbishop in order to protect Aaron.
Dr. Arrington, for one, is convinced that Aaron is suffering from multiple personality disorder. (This is also known as dissociative identity disorder.) All the elements seem to be there, including repeated black-outs and sexual abuse. Dr. Arrington describes Aaron's case as "textbook multiple personality disorder" and concludes he is insane. Vail, however, is initially unconvinced. However, he is thinking like a lawyer; he knows that he cannot introduce an insanity plea midway through a trial, even if he wanted to.
It should be remembered here that multiple personality disorder was a highly contentious diagnosis at the time and a subject of great controversy. Many trained professionals dismissed multiple personality disorder as lacking in scientific rigor. Dr. Arrington, however, appears to be one of the true believers. She has clearly conducted a lot of research into the issue and feels confident enough to make a diagnosis, staking her professional reputation as she gives testimony in court on behalf of the defense.
One should also mention that Dr. Arrington is a neuropsychologist rather than a forensic psychiatrist. In simple terms, this means that her field of expertise relates to brain injuries and their subsequent effects on a patient's cognitive abilities. Yet, there is no suggestion that Aaron has suffered any kind of physical trauma to the brain. At the very least, we may, along with the Assistant District Attorney, conclude that Dr. Arrington is not qualified to make such a diagnosis, especially given its contentiousness at that particular time.
As for Vail, he comes to accept Dr. Arrington's diagnosis when Aaron suddenly takes on the persona of "Roy" in open court and lunges violently at the Assistant DA, threatening to break her neck. Vail was skirting around the issue of insanity because he knew he could not introduce it as a defense in the middle of the trial. Instead, he relied on trying to establish the medical existence of amnesia in Aaron's mind. However, when Aaron, in the guise of Roy, attacks Assistant DA Venable, Vail seems as certain as he can be that Aaron really is suffering from multiple personality disorder. Seeing is believing. Fortunately for him, the judge agrees and finds Aaron not guilty by reason of insanity, remanding him to a secure psychiatric facility.
https://trauma.blog.yorku.ca/2013/02/dissociative-identity-disorder-drawing-the-line-between-fact-and-fiction/

How does money influence notions of gentility in Great Expectations by Charles Dickens?

This really has to do with the issues of class structure in Britain. In earlier centuries, there was presumed to be a fixed social hierarchy with the monarch at the summit, followed by the nobility, then the gentry, and finally commoners. Wealth was expected in most cases to follow this social hierarchy, with those on top being wealthier than those on bottom. Rich merchants or craftspeople could gradually become assimilated into the gentry by means of marriage. An alternative but quite rare way mode of moving up in class for the exceptionally talented was through education, followed by a well-regarded career (usually in the Church). As these modes of class mobility were gradual and relatively limited before the Industrial Revolution, families which rose in class had time to assimilate themselves to the manners and beliefs of upper-class culture.
With increased opportunities for great wealth to be obtained in the colonies and in manufacturing in the late eighteenth and nineteenth centuries came a decoupling of the relationships among birth, manners, and money. Magwitch is an example of one of the new rich; Mrs. Pocket and Miss Havisham, of an older social tradition of gentility. Pip is caught between these worlds and value systems. 

How does the setting of the book affect Sam Gribley's decisions?

In My Side of the Mountain, Sam Gribley has run away from his home in New York City in order to live off the land in the Catskill mountains. This setting is key. It could be considered another major character in the book. Everything Sam does and learns about is based on his close attention and reaction to the forest and all of the plants and animals that live in it. When he needs a place to live, he hollows out a large hemlock tree. When he needs a door for it, he finds a dead deer that a hunter has abandoned. Then he teaches himself how to process the hide to make a deerskin covering for his doorway. His diet is whatever he can find on the mountain, sometimes through trial and error. He trains a falcon he names Frightful in order to help him hunt rabbits and small game. He takes some of the mussels that raccoon Jessie Coon James digs out of the river, before she can eat all of them herself. When winter comes, Sam learns how to keep warm from a fire properly ventilated in his tree-home. He discovers how much work it is to break out of a thick ice coating. The theme here should not be seen as Man versus Nature; but rather, Man connecting, cooperating, and surviving in and with Nature. Imagine how much you would learn and know about your own natural habitat if you had to live with its resources alone, and not with anything you could buy at a store. This is how important the mountain environment is to Sam. It has become his life and his livelihood.

What is a pontoon bridge? Why are so many people crossing the bridge?

Quite simply, a pontoon bridge is a bridge that floats on water. It utilizes small barges or boats to stay afloat. The design of the pontoon bridge is reflective of its temporary nature. Pontoon bridges are used predominantly during wartime. The bridges allow one group of people to cross a body of water without necessarily allowing an opposing group to do the same thing, as the bridge is typically destroyed or collapsed after satisfying its intended use. The people crossing the bridge are refugees that are fleeing from their small village which has become part of a war zone. They fear losing their lives to stray gun or artillery fire. This was one of the unfortunate realities of the Spanish Civil War and, indeed, every war in history.

"Old Man at the Bridge" by Ernest Hemingway is a story set during the Spanish Civil War, a conflict occupying the years from 1936 to 1939 in which the conservative and protofascist Nationalists were fighting against the liberal Republicans. It was an extremely violent conflict with atrocities perpetuated on both sides--but especially by the Nationalists. Hemingway worked as a reporter for the North American Newspaper Alliance during the war.
A pontoon bridge is a temporary structure using air-filled floats or small boats to support a path across a river. Because these structures are cheap and quick to erect but flimsy and unable to bear large weights, they are normally used only in emergencies or military operations.
The people crossing the bridge are fleeing from a war zone. They know that if they are trapped when the fighting starts, they may well end up as "collateral damage." They are residents of a small village and fear for their lives.

Calculus of a Single Variable, Chapter 9, 9.10, Section 9.10, Problem 66

From the table of power series, we have:
cos(x) = sum_(n=0)^oo (-1)^nx^(2n)/((2n))!
= 1-x^2/(2!)+x^4/(4!)-x^6/(6!)+ ...
To apply this on the given integral int_0^1 cos(x^2) dx , we may replace the "x " with "x^2 ".
cos(x^2) = sum_(n=0)^oo(-1)^n (x^2)^(2n)/((2n)!)
=sum_(n=0)^oo (-1)^n x^(4n)/((2n)!)
= 1-x^4/2+x^8/24-x^(12)/720+ x^16/40320 -...
The integral becomes:
int_0^1 cos(x^2) dx =int_0^1 [ 1-x^4/2+x^8/24-x^(12)/720+ x^16/40320-...]dx
To determine the indefinite integral, we integrate each term using Power Rule for integration: int x^ndx =x^(n+1)/(n+1) .
int_0^1 [ 1-x^4/2+x^8/24-x^(12)/720+ x^16/40320-...]dx
=x-x^5/(2*5)+x^9/(24*9)-x^(13)/(720*13)+ x^17/(40320*17)- ...
=x-x^5/10+x^9/216-x^(13)/9360+ x^17/685440- ...
Apply definite integral formula: F(x)|_a^b = F(b) - F(a) .
F(1)=1-1^5/10+1^9/216-1^(13)/9360+ 1^17/685440- ...
=1-1/10+1/216-1/9360+ 1/685440- ...
F(0) =0-0^5/10+0^9/216-0^(13)/9360+ 0^17/685440- ...
= 0-0+0-0+0- ...
All the terms are 0 then F(0)= 0 .
We can stop at 5th term (1/685440~~ 0.0000014589) since we only need an error less than 0.0001 .
Then,
F(1)-F(0)= [1-1/10+1/216-1/9360+ 1/685440-]-[0]
= 0.9045242509
Thus, the approximated integral value is:
int_0^1 cos(x^2) dx ~~0.9045