Single Variable Calculus, Chapter 7, 7.7, Section 7.7, Problem 6

Determine the numerical value of a.) $\sin h 1$ and b.) $\sin h^{-1} 1$

a.) $\sin h 1$

Using Hyperbolic Function


$
\begin{equation}
\begin{aligned}

\sin h x =& \frac{e^x - e^{-x}}{2}
\\
\\
\sin h 1 =& \frac{e^1 - e^{-1}}{2}
\\
\\
\sin h 1 =& \frac{\displaystyle e - \frac{1}{e}}{2}
\\
\\
\sin h 1 =& \frac{e^2 - 1}{2e}
\\
\\
\sin h 1 =& 1.1752

\end{aligned}
\end{equation}
$


b.) $\sin h^{-1} 1$

Using Inverse Hyperbolic Function


$
\begin{equation}
\begin{aligned}

\sin h^{-1} x =& \ln (x + \sqrt{x^2 + 1})
\\
\\
\sin h^{-1} 1 =& \ln (1 + \sqrt{(1)^2 + 1})
\\
\\
\sin h^{-1} 1 =& \ln (1 + \sqrt{2})
\\
\\
\sin h^{-1} 1 =& 0.8814

\end{aligned}
\end{equation}
$

How does Shakespeare introduce the characters Antonio and Bassanio in the beginning of the first act of the Merchant of Venice, and in what ways does their relationship form the basis of the entire play?

Antonio and Bassanio are kinsmen and close friends, who engage in revelry with others such as Salerio, Solanio and Gratiano. In the first scene of Act I, Bassanio asks a favor of Antonio and confides in his friend.
Shakespeare's Merchant of Venice opens with Antonio talking with two friends, Salerio and Solanio. He mentions that he is overcome with a certain melancholy for which he can find no source. The other men suggest that he is anxious about his merchant ships that are at sea; furthermore, in an effort to empathize, Solanio tells Antonio that he used to worry constantly when his ships were at sea with cargo. But, Antonio declares that it is not anxiety for his ships that disturbs him. His description of his feelings are those of melancholy; it is as though Antonio has a presentiment.

I hold the world but as the world, Gratiano--A stage where every man must play a part,And mine, a sad one. (1.1.76-78)

Gratiano cautions Antonio to not cultivate melancholy in order to gain a reputation for silent wisdom:

Fish not with this melancholy baitFor this fool gudgeon, this opinion. (1.1.101-102) 

Shortly after this, Bassanio enters and jests about Gratiano, declaring that he speaks "an infinite deal of nothing" (1.1.114). Antonio asks Bassanio about his secret lady for whom Bassanio "swore a secret pilgrimage." Bassanio, then, tells Antonio that he is in debt, even as he knows that he owes Antonio money still. But, he has an idea of how he can repay his debts to Antonio, who bids him to speak, 

Then do but say to me what I should do
That in your knowledge may by me be done,
And I am pressed unto it. Therefore speak. (1.1.160-163)

Bassanio then tells his friend about the lovely and wealthy Portia. But, he has learned that she has many men who would like to marry her. So, if Antonio can provide Bassanio with some money, he can compete against these rivals. Antonio tells Bassanio that he will try to get credit and help him.
 
As it turns out, the only way that Antonio can obtain money is by going to the usurer, Shylock. He borrows three thousand ducats with the condition that if the debt is not repaid, Shylock can extract a pound of flesh from Antonio, who gambles on his ships' successful return so that he can repay the loan. 

Bassanio, who is now fortified against his rivals, goes to Belmont to try to win Portia for his wife. As Bassanio tries to decide which casket to choose so that he can marry Portia, Antonio receives word that his ships have been lost. Learning of Antonio's loss, Shylock, who hates him as a rival money-lender, is eager to collect on his loan. In the meantime, Bassanio receives word of his friend's misfortune; his marriage celebration is halted; and he hurries to help Antonio because his new wife has offered to pay Shylock much more than the original debt. Secretly, Portia and her friend Jessica disguise themselves as the doctor of law, to whom the Duke of Venice has petitioned, and his clerk.

After arriving in the Duke's court, as the doctor of law, Portia reads the contract between Antonio and Shylock and finds it binding. However, there is no provision for Shylock to spill any of Antonio's blood in taking the pound of flesh. Therefore, the payment of a pound of flesh cannot be extracted, and Shylock loses his case.
 
In the end, Bassanio's tale in Act I about his shooting the second arrow and watching it carefully in order to find the first arrow proves valid as a figurative act, both for his borrowing of money from Antonio and for the rescue of Antonio from losing a pound of flesh. 

how did the 1960's change lives

There are many ways to answer this question. Generally speaking, the 1960s are seen as an extremely turbulent decade in America where politics and social issues became more visible and more important than ever before. Part of the reason for this is that television was in more American homes than any other time previously, and as television programming grew more varied and sophisticated, television news journalism became more refined. This shift meant that people had access to more information about current events, but it also meant they shared the common experiences of watching the news on a given network.
Among the many social issues that rose to greater prominence during the 1960s were civil rights, women's rights, sexual liberation (partly spurred by the introduction of birth control pills in 1963), and the anti-war protest movement. The US became embroiled in many high-stakes political situations. The Cuban Missile Crisis was seen as a defining moment of the so-called "Cold War" and brought America to the brink of nuclear war in 1962.
The US involvement in Vietnam began to be the subject of vocal and dramatic protest by young people who thought it unfair that the government was drafting young people to fight in a war that very few Americans supported. The murder of prominent political figures like John F. Kennedy and Robert Kennedy, as well as social and spiritual leader Dr. Martin Luther King, were tragic and far-reaching events that underscored how dangerous and controversial it was to engage in efforts to make widespread change.
The culture of the protest movement spurred many artists to create literature and music inspired by the various current events. Musicians like the Beatles, Jefferson Airplane, Joni Mitchell, Bob Dylan, and Ritchie Havens wrote songs that captured the energies of the social movements taking place. The growing use of recreational drugs contributed to an atmosphere of rebellion, and the introduction of LSD was associated with the idea of consciousness expansion. Sexuality became a more prevalent topic and sex outside of marriage was seen increasingly as a norm and not a transgression. The common phrase "sex, drugs, and rock and roll" came to be associated with the 1960s, but far from being a shallow or pejorative term, it refers to enormous cultural changes occurring that would have lasting effects on countless lives for many decades.
http://www.pbs.org/opb/thesixties/timeline/timeline_text.html

The least common multiple of a number "n" and 6 is 24. What is the value for "n"?

Let's consider the prime factorization of both given numbers, 6 and 24.
It is clear that 6 = 2^1 * 3^1  and  24 = 2^3 * 3^1.
Hence the number n must contain 2 exactly in degree 3 in its prime factorization. If it would have 2 in greater degree, the LCM of 6 and n would have 2 in that greater degree, and if in less, then in less.
Also n may contain 3 in degree not greater than 1. It may contain 3 in degrees 0 or 1, because 6 already have 3^1 and 24 also.
And it cannot have any other prime factors.
This gives us two options for n:  2^3 * 3^0 = 8 and 2^3 * 3^1 = 24.

What is the first spirit called?

The first spirit is the Ghost of Christmas Past. He takes Scrooge back to some scenes from his earlier years. This part of the book provides us with a useful bit of exposition concerning Scrooge's story, helping us to get a better understanding of him and how he became such an unpleasant old miser.
We learn that Scrooge was desperately unhappy as a boy and was sent away to a boarding school he thoroughly detested. We also learn that Scrooge was very close to Fan, his kind, loving sister who pleaded with her father to allow young Ebenezer to return home and escape his miserable schooldays.
The Ghost of Christmas Past shows us another side of Scrooge, one more human and vulnerable. He even used to like Christmas in the past years, enjoying the seasonal hospitality of his kindly employer, Mr. Fezziwig. However, Scrooge became greedy and ambitious, putting money ahead of all his relationships. He was engaged to be married to Belle, but she broke off the engagement when she realized what kind of a man her fiancé was becoming. The Ghost of Christmas Past presents Scrooge with a scene of Belle, now happily married, enjoying Christmas Eve with her family on the night that Jacob Marley passed away.

"My child is yet a stranger in the world, she hath not seen the change of fourteen years" What language technique did Shakespeare use here?

Lord Capulet is speaking to Paris in reply to the young lord's request to marry Juliet in Act 1, scene 2. Shakespeare employs a metaphor, which is a figure of speech used to apply to a thing regarded as symbolic or representative of something else.
In this instance, Capulet is comparing Juliet's naivety, immaturity, inexperience, and youth to the lack of knowledge one has as a stranger to something. Juliet, he suggests, is much too young to consider marriage. She still has to learn about the world and gain the necessary knowledge to understand what it means to be wed. Furthermore, one can infer that Lord Capulet might be afraid that she, because of her youth, might embarrass the family and bring them into disrepute if she is married too young.  
Lord Capulet suggests that Paris should give her another two years and win her affections in the interim before he extends his suit again. He declares his love and affection for his daughter and states that "she is the hopeful lady of [his] earth," which means that his hopes are grounded in her to make him proud. He trusts that Juliet will not disappoint him. 
It is, therefore, ironic that Lord Capulet soon changes his mind and acquiesces to Paris' request in Act three, scene four.

Sir Paris, I will make a desperate tenderOf my child's love: I think she will be ruledIn all respects by me; nay, more, I doubt it not.

Furthermore, he is immensely upset when Juliet later rejects his instruction to get married. He rants and raves and calls her ungrateful. He also threatens to disown her and throw her into the street if she refuses.  
More irony lies in the fact that Juliet has already married Romeo at this juncture. In spite of her father's fear about her immaturity, she has taken it upon herself to get married. Her actions, therefore, shatter all hope that her father has had. She and her love's impulsive decisions create greater complications and eventually result in their tragic and untimely demise.  

In these lines, Capulet tells Paris that as Juliet is only 13 years old ("she has not yet seen the change of 14 years" means 'she is not yet 14 years old'), she is too young and naive to be married. 
The language technique used here is a metaphor, as Capulet calls Juliet "a stranger" to the world. In this way, he compares Juliet's naivety and lack of knowledge about the world to the lack of knowledge we have about a stranger. Over time, a stranger becomes familiar to us, just as over time, one becomes familiar with the world, and naivety is replaced by experience and knowledge. 
Capulet's words are also rather visual and give us a real sense of the passage of time. Juliet has not "seen the change" of many years, and these words give us the sense of passing seasons, or the passage of time. This imagery is reinforced in the following line, where Capulet asks Paris to allow Juliet "two more summers" before she is married. 

In The Good Thief by Hannah Tinti, what happened to Ren's Mother?

The protagonist of Hannah Tinti's The Good Thief is Ren, an orphan with a missing left hand who lives in St. Anthony's orphanage until being adopted by Benjamin Nab. Ren grows up with no knowledge of his family and with the only evidence of his past being his name: the letters R-E-N had been sewn into his baby clothing when he was abandoned at the gates of the orphanage.
Ren eventually learns the true story of his family from his mother's brother, Uncle McGinty. Ren's grandfather sexually abused his daughter, Margaret (Ren's mother), which led the woman to fall into a deep depression and attempt to commit suicide several times. Later on, Margaret became pregnant with Ren and died shortly after giving birth to him.