Notes

We are asked to graph the function `y=-6/x+4 ` : Note that this is a transformation of the graph of y=1/x with a translation of 4 units up, a reflection over the horizontal axis, and a vertical dilation of factor 6. There is a vertical asymptote at x=0, and a horizontal asymptote at y=4. The x-intercept is at 3/2. The graph:

In Act III, Scene 1, both Mercutio and Tybalt are killed in a street fight. Mercutio is stabbed by Tybalt while Romeo is attempting to break up the fight. Once Mercutio dies, Tybalt, who had run away, inexplicably returns to the scene where he is promptly challenged and killed by the grieving Romeo. When the Prince arrives, he listens to Benvolio, who had been a witness to the violence, give an account of the fight. Benvolio explains truthfully, although Lady Capulet says he is lying. Her version, however, is actually full of lies, claiming that the Montagues had ganged up on Tybalt. She demands the death penalty for Romeo. The Prince listens and his judgement is that Romeo should be banished from Verona. This verdict is unacceptable to Lady Capulet because earlier in the play the Prince had decreed that the penalty for fighting in the streets would be death, with no exceptions. The Prince's decision is fair for three reasons. First, Tybalt instigated the fight by challenging Romeo...

At first glance, it might appear as if the characters in  The Outsiders  by S. E. Hinton and  Romeo and Juliet  by William Shakespeare do not have much in common; however, when one isolates the core of the conflicts, it becomes clear that the relationships in both stories are rooted in rivalry and secrecy.  In both  The Outsiders  and  Romeo and Juliet , an overarching theme is rivalry. While   socioeconomic status separates the greasers from the Socs in The Outsiders , last names separates the Montagues from the Capulets, who appear to be relatively equal in terms of socioeconomic status. Due to these rivalries, there are instances of deadly interactions in both stories. In  The Outsiders , Johnny kills Bob (whether his actions were justified or not is debatable). In  Romeo and Juliet , Romeo kills Tybalt and Paris, and Tybalt kills Mercutio. These deaths have profound effects on the main characters. In Ponyboy's case, the Johnny's death and other events in the novel inspire h...

At the beginning of Chapter 3, Nick observes that each Friday, a fruiterer from New York sends five crates of lemons and oranges to Gatsby's house. Each Monday, after a weekend of parties, "these same oranges and lemons left his back door in a pyramid of pulp-less halves" (43). Gatsby owns a device by which the juice of 200 lemons can be removed in half an hour. The guests are similar to the fruit because, after a weekend of carousing, they also leave Gatsby's house in a state of waste and destruction. Just like the fancy food that arrives at Gatsby's house, the guests are just for show. Gatsby doesn't care much which guests or what food arrives at his house. He only cares that they present a facade of having a good time and meeting the expectations of the upper crust, particularly Daisy. 

Magma is another name for molten rocks. Due to the heat of Earth's interior, rocks melt and rise to the surface. If there is an opening in the Earth's crust, the molten rocks flow over the surface and are known as lava. This is observed during volcanic eruptions. The melting of rocks is a function of temperature. The higher the temperature, the more rock melts and the less viscous the magma is. It should be noted that there are a variety of rocks in the Earth's mantle and they melt at different temperatures. If the temperature is lower, only some rocks (generally the ones with more silicon content) will melt. Thus, at lower temperatures, we will observe magma with high silicon content. Temperature also affects the final magma composition and structure. For example, at lower temperatures, the magma crystallizes faster.  Hope this helps. 

Baldwin's "Sonny's Blues" takes place in the pre-civil rights era in Harlem, which was and still is a largely African-American part of New York City. The title of the story is a play on words, since "blue" is a synonym for depression and the blues is a form of music based upon people's problems, a form generally credited as African-American in origin.  The narrator tells the story of his younger brother, Sonny, who is a gifted pianist, but who has struggled with drug addiction and has been incarcerated as a result. We learn of the boys' early lives and the tragic legacy of the boys' father, who watched his guitar-playing brother get run down and killed by a group of drunken white men. This is a story of despair and hope that has a great deal to teach us about darkness and pain, about the failure and endurance of fraternal love, about the curse of drug addiction, and about the gift of musical genius. The boys' early lives are lived in poverty,...

Chapter three is pivotal chapter in the book. This is the chapter where Lockwood tries to spend the night in the crazy old fashioned bed that had been Catherine’s, sees her carvings in the window sill, reads from her journal, and has the horrible dream about the sermon and Catherine at the window trying to get in. Even though Lockwood has only dreamed about Catherine’s ghost “walking the earth these twenty years,” in practical terms Catherine’s spirit has haunted him – her spirit clearly dominates the room. This is no doubt the reason, as Zillah says when she leads Lockwood to the room at the start of the chapter, that Heathcliff had “an odd notion” about the room. But then, she says, there had been “so many queer goings on” she couldn’t begin to explain. Zillah’s resignation suggests that she is an unwilling spectator to Heathcliff’s psychological battle with Catherine, in the same way that Lockwood becomes an unwilling (and unwitting) participant. Heathcliff’s reaction to Lockwood’s ...

By the end of Act 1, Scene 5, Macbeth and his wife have met and figured out that they are both on board with a plan to eliminate Duncan.  However, Lady Macbeth is the one giving all the instruction to her husband.  She tells him, "Look like th' innocent / flower, / But be the serpent under 't" (1.5.76-78).  In other words, she tells him to appear loyal and kind as he has always done, but prepare his dark and disloyal thoughts beneath the friendly facade.  Further, when he says that he wants to speak more with her about their plan, she takes it all into her own hands, saying, "Leave all the rest to me" (1.5.86).  He hardly gets a word in edgewise because she is so determined and controlling, just as she'd planned to be before he came home when she said that she wanted to "pour [her] spirits into [his] ear" and make him feel as ruthless as she does. After Duncan's arrival, however, Macbeth has changed his mind and says, "We will proceed ...

In Chapter Four of The Lightning Thief , Percy faces the challenge presented by the fact that he saw the three Fates in the chapter prior.  Both his mother and best friend, Grover, are terrified for him, and his mother drives frantically in an attempt to get Percy safely to Camp Half-Blood.  The fact that Percy saw the three Fates, Grover tells him, means that he is about to die: a challenge indeed.  As the group nears the camp's boundary, Percy realizes that they are being pursued by the Minotaur: a bullish beast that is frightening and terribly strong.  Percy's mother tries to hurry Percy toward the camp boundary, and the monster snatches her, and she "dissolved before [Percy's] eyes."  He gets angry now, and he takes on the Minotaur himself, eventually stabbing the monster with its own horn, defeating him.  Therefore, Percy's challenges in this chapter are largely physical, though he does have to begin to contend with his grief over his perceived loss of hi...

Portia in The Merchant of Venice and Portia in Julius Caesar are both admirable female characters who display many of the same traits of courage, intelligence, loyalty, and ingenuity. They are some of the more attractive female characters in Shakespeare and seem intended as positive role models for women. Both are members of the upper classes. There are, however, several differences between them: The women lived in different eras.  The Merchant of Venice is set in Renaissance Venice; Julius Caesar , in ancient Rome. Thus, although both plays were set in what would eventually become Italy, a region Shakespeare never visited, The Merchant of Venice is set in Shakespeare's own period (around 1600 CE) and Julius Caesar over 1,500 years before he was born. Julius Caesar 's Portia is the wife of Brutus and mother of his children, while The Merchant of Venice 's Portia is initially an unmarried young woman being courted by various suitors. Thus,  The Merchant of Venice 's...

Survival of the fittest is part of Charles Darwin's mechanism for evolutionary changes.  It's part of a larger concept called "natural selection."  It works like this.  Random changes in the genetic code lead to individuals in a species having various adaptations.  A particular adaptation might be good for a particular time and environment, or it might be bad.  Individuals with an adaptation that is not beneficial are more likely to die or be killed.  Individuals with a beneficial adaptation are more likely to survive.  The ability of an organism to survive and pass on their genes is called "fitness."  In other words, organisms with beneficial adaptations are the fittest organisms within a given population; therefore, the fittest tend to survive.  Survival of the fittest.   Because the fittest are surviving, they are passing on their genetic traits.  Nature is essentially selecting which traits are getting passed down from generation to generation and which ...

Given to solve, `lim_(x->0) (sqrt(25-x^2)-5)/x` upon Rationalizing numerator we get =` lim_(x->0) ((sqrt(25-x^2)-5)/x) ((sqrt(25-x^2)+5)/(sqrt(25-x^2)+5))` =`lim_(x->0) (((sqrt(25-x^2)^2-5^2)/(x(sqrt(25-x^2)+5)))` =`lim_(x->0) ((((25-x^2)-25)/(x(sqrt(25-x^2)+5)))` =`lim_(x->0) ((-x^2)/(x(sqrt(25-x^2)+5)))` =`lim_(x->0) ((-x)/((sqrt(25-x^2)+5)))` Now plugging the value of `x =0 ` we get `((-x)/((sqrt(25-x^2)+5)))` = `((-0)/((sqrt(25-0^2)+5)))`  =`0`       

This play by Shakespeare not only describes the conflicts between love and hate but also shows the grave consequence such hatred can have (to help you see the whole story, you can check out this link , which provides a brief summary). Love: Romeo and Juliet love each other despite the intense feud between their families. They marry, even though it is forbidden. In the end, they portray that they would rather die than live without each other, even going so far as to kill themselves to avoid the reality of life alone. Hate: The Montagues and Capulet families hate each other, dueling and killing one another. Neither side will repent or make amends. Conflict: Romeo and Juliet, each from one of the opposing families, quickly meet, fall in love, and are married. Love springs up in the midst of heavy hatred all around. Although they love each other, neither gives thought to his/her family. Romeo does not think about how his actions (which cause him to be banished) will affect his relatives...

According to the shepherd, if his love agrees, they will enjoy all the pleasures of nature, including valleys, woods, hills, fields, and mountains. While they are enjoying nature, they will sit on rocks and watch shepherds tending to their sheep, and they will sit by rivers where birds will serenade them. The shepherd will make his beloved a bed of roses, a hat of flowers, and a gown (or "kirtle") sewn with myrtle leaves. The shepherd will also make his lover a gown made of the best wool from his sheep, and he will make his beloved slippers and a gold belt buckle. He will give his lover a belt of straw with decorations made of amber and coral, and they will be serenaded by swains, or young people in the country. The delights that Marlowe describes are pastoral and represent the idyllic aspect of nature. 

Mercutio was never supposed to become involved in the altercation that ended his life.  Tybalt came for Romeo, and Romeo alone.  However, when Romeo refused to fight Tybalt, Mercutio felt that his behavior brought him dishonor.  Rather than permit this, Mercutio fought Tybalt himself, and Tybalt slays him when Romeo comes between them.  When Mercutio curses the houses of Capulet and Montague with plague, it recalls the line from the Prologue where the Chorus calls the love between Romeo and Juliet's love "death-marked."  Further, the "fearful passage" that their love takes spells destruction for so many more than just the two of them.  Their love is, to be sure, death-marked; consider everyone that dies during the "fearful passage" of their relationship: Mercutio, Tybalt, County Paris, their two selves.  When Mercutio sends "a plague" on the lovers' families, he helps to fulfill the Prologue's promises.

Hello! 1 . When constructing the Koch snowflake, we start with an equilateral triangle (stage `0`). On each subsequent stage the figure remains a closed polygon, all segments of this polygon have the same length. Denote the perimeter at the `n`-th stage as `P_n,` it is given that `P_0 = 3.` After `n`-th stage we take each segment of the polygon and broke it onto three equal sub-segments. Two sub-segments at the ends remain at their places, while the middle sub-segment is replaced by two segments of the same length looking outwards of the center. Thus, each segment of length `x` is replaced with `4` segments of the length `x/3` each, the new length becomes `4/3 x.` The same ratio applies to the perimeters because they are the sums of the lengths. This way we see that  `P_(n+1) = 4/3 P_n.` A sequence whose next term is fixed times more than the previous is called geometric progression, and its `n`-th term is `P_n = P_0 * (4/3)^n = 3* (4/3)^n.` For first `n`'s the perimeters are `P_1 ...

The conflict that I would like to discuss here is actually the "supposed" conflict, or one that would be expected of a Nazi commandant's son and a Jewish boy:  Bruno vs. Shmuel.  This conflict can be seen in this quote that I will discuss later in the paragraph: We're not supposed to be friends, you and me. We're meant to be enemies. Did you know that?  It is ironic to label it as a conflict at all in that the two are not really against each other, except in the idealism of Nazi Germany.  Still, it is interesting to see how other characters would react to that friendship.  Bruno's father, the Nazi commandant, (as well as Lieutenant Kotler) would be in favor of the conflict and forbid the friendship completely.  I believe Bruno's mother as well as characters such as Pavel and Maria would look upon Bruno with sad eyes whether the two boys were in conflict or friends.  All three characters would discourage any communication between the boys and feel sad for t...

When preparing a report or project on the threat of a terrorist attack utilizing biological weapons, there is a wealth of material in the public domain that any student can access.  Because the use of germs to destroy or demoralize an enemy has been around since at least the 14th century, there is a long history of efforts by myriad governments and nongovernmental organizations to develop biological weaponry and, conversely, to protect against the use by adversaries of such weapons. From the moment a dead animal or human corpse was deliberately dropped into the water supply of an enemy village or town during medieval times for the purpose of contaminating those water supplies, biological weapons and the threat of their use has been a part of civil defense planning. In the United States, concerns about the use of biological and chemical weapons by the Soviet Union during the Cold War -- and the Soviet Union was known to maintain a massive program designed to develop biological weapons a...

Atticus agrees with Scout that they will still read together.  Atticus was in the habit of reading to Scout, following the words with his finger.  Because of this, and because of Calpurnia's giving Scout sentences to copy when she wanted to keep her busy, Scout is already able to read fluently when she starts school.   The young, inexperienced teacher is horrified that Scout has been taught to read at home.  (She assumes that Scout's father used the wrong methods to teach her.)  She tells Scout not to read outside of school anymore, and she, the teacher, will try to "undo the damage." Scout does not want to give up one of her favorite pastimes, so she mounts a campaign to get Atticus to let her stop attending school.  When Atticus finds out that one of Scout's main concerns is that she won't be allowed to read outside of school, he tells Scout that he'll still let her read at home as long as she agrees to go to school. 

 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(...

One of the subjects and themes of The Phantom of the Opera that is still relevant today is the choice that people must make between professional success and personal happiness. Christine agrees to be beholden to Erik, the phantom, so that she can achieve success in the opera, but Erik forces her to be  a virtual prisoner to him and to give up her true love, Raoul. The parallel to today's society is that many people sacrifice personal happiness, such as time spent doing what they love or time spent with their families and loved ones, to achieve success in the professional world. Another theme is the meaning of true love. At first, Erik feels that he wants to entrap Christine and kill off her lover, Raoul, but he comes to realize that true love means letting Christine be free to marry as she wishes and allowing her to live the life she wants. 

In "The Demon Lover," Mrs. Drover has returned to her London home to collect some belongings for herself and her family. (They fled this house when the war began and moved out to the country where the German bombs could not reach them.) Mrs. Drover has clearly not visited this house for some time: it is shuttered, the air is stale, and the remaining possessions are covered in a film of dust. While collecting the items she requires, Mrs. Drover notices that somebody has left a letter for her on the hall table. This letter appears to come from her former fiancé, a soldier who was reported missing during the war, and who has returned to keep an appointment that they made twenty-five years ago.  Mrs. Drover flees the house with the items she has come to collect, without realizing that she has walked straight into the arms of her demon lover.  

In Chapter 2, Ponyboy tells the story about when Johnny was severely beaten by a group of Socs to Cherry while they are waiting in line at the concession stand. He says that four months ago, while he and several Greasers were walking back from the DX station, Steve noticed Johnny's jean jacket lying in the middle of the street. When they looked at the collar of the jacket, they saw rust colored stains. They then noticed that there was blood all over the ground leading to an open field where they could hear moans. The boys ran across the field and found Johnny lying face down. Soda turned Johnny over, and there were huge bruises, cuts, and gashes on his face. Ponyboy says the he felt sick after seeing Johnny's face. Johnny told the boys that four Socs jumped him, and one of them was wearing a fist full of rings, which is why his face is so torn up. Ponyboy then tells Cherry that Johnny hasn't walked alone ever since and carries a six-inch switchblade. Pony says, " He...

On a street in downtown Dublin during the Irish Civil War a Republican sniper is posted on a rooftop in Liam O'Flaherty's short story "The Sniper." Opposite him on the other side of the street is a Free State sniper. The street which separates them is a symbol of the division between the Irish who wanted complete freedom from the British, Republicans, and those that were willing to accept being part of the British Commonwealth, Free Staters. More than that it is symbolic of the disagreement which brewed between people who had once been allied. After all, the Free Staters were once part of the Irish Republican Army and the division in the army often involved a clash of those from the same family. It is noteworthy that the enemy armored car stops on "the opposite side of the street" from the Republican sniper. The lines between the two foes is symbolically drawn, and when an old woman appears to tell the armored car commander the position of the Republican, bo...

Given to solve, `lim_(x->oo) x^3/(e^(x/2))` as `x->oo` then the `x^3/(e^(x/2))=oo/oo` form so upon applying the L 'Hopital rule we get the solution as follows, as for the general equation it is as follows `lim_(x->a) f(x)/g(x) is = 0/0` or `(+-oo)/(+-oo)` then by using the L'Hopital Rule we get  the solution with the  below form. `lim_(x->a) (f'(x))/(g'(x))` so , now evaluating `lim_(x->oo) (x^3)/(e^(x/2))` =`lim_(x->oo) ((x^3)')/((e^(x/2))')` = `lim_(x->oo) (3x^2)/((e^(x/2))(1/2))` again `(3x^2)/((e^(x/2))(1/2))` is of the form `oo/oo` so , applying the L'Hopital Rule we get = `lim_(x->oo) (3x^2)/((e^(x/2))(1/2))` = `lim_(x->oo) ((3x^2)')/(((e^(x/2))(1/2))')` =`lim_(x->oo) ((6x))/(((e^(x/2))(1/2)(1/2)))` =`lim_(x->oo) ((6x))/(((e^(x/2))(1/4)))` again `((6x))/(((e^(x/2))(1/4))) ` is of the form `oo/oo` so , applying the L'Hopital Rule we get =`lim_(x->oo) ((6x))/(((e^(x/2))(1/4)))` =`lim_(x->oo) ((6x)')/...

These particular lines are from a Robert Louis Stevenson poem entitled “The Vagabond” which is included in a collection of poetry called Songs of Travel and Other Verses . A vagabond is one who travels, one who wanders, without the benefit of a stable home. Stevenson, in this poem, asks for the life of a vagabond or wanderer. He is asking for only “the heaven above and the road below me.” His desire is not to have a stable home, but to have the sky above him and road beneath him on which to walk. The first lines of the poem say, “Give to me the life I love,” the life of a wanderer, and “Give the jolly heaven above/And the byway nigh me.” This is the same sentiment as asking for the sky or “heaven” above and the “road” or “byway” beneath. His home, his place, is to wander, with sky above and road below, on which he continues his wandering. It replaces the home he does not have. If you examine the rest of the poem, you will see many other references to the life of a wanderer. Robert Loui...

Othello's tragic downfall is in his gullibility and lack of good judgement. While he is easily swayed in his opinions about others, his opinions are seemingly fixed unless an outside voice leads him to question his own beliefs. In this, Othello never questions the "goodness" of Iago--nobody calls Iago's character into question until the end of the play. Therefore, Othello never sees "cause" to question Iago's character. Iago's entire plot hinges on the knowledge that Othello is gullible. We see this in Act 1, scene 3 when, plotting to slander Cassio and Desdemona, Iago says of Othello, "The Moor is of a free and open nature, That thinks men honest that but seem to be so, And will as tenderly be led by the nose As asses are." Without this character trait, Iago's plot would never have worked on Othello. And while the relationship between Desdemona and Othello seems solid and loving at the beginning of the play, it is very easy for Iago to...

When Peter describes himself as just "Peter Pan," Wendy is surprised at his lack of names outside of a given name and surname. It is possible she also thinks that "Pan" is quite short in syllables as compared to her own surname-- "Darling." Wendy, herself, has four names in all--a given (or first) name, two middle names, and one surname. For the time period and assumed social status of the Darling family, it would not have been uncommon for Wendy to know people with very long names. The book  Peter Pan was first published in 1904, so we can assume the setting to be around around this time, possibly earlier. Based on the fact that Mr. and Mrs. Darling are dressing nicely for an evening social event and can afford to employ household staff, they are likely of the upper or upper-middle class.  During both the Victorian and Edwardian periods, people of the upper classes of society often had very long names as a means of carrying on their heritage. There could...

The basic principle underlying current rules of evidence is that of probable cause ; the police must have a preponderance of the evidence that this suspected person is guilty of a crime in order to search them or seize their property, and with rare exceptions must acquire a warrant on this basis. If evidence is not obtained according to this rule, the exclusionary rule applies, disallowing the use of that evidence in court. In theory, this is not the only way one could penalize illegal searches, but it is the one we generally use. The exclusionary rule was first applied in the case Boyd v. United States (1886), and actually applied not to a search per se but the requirement to furnish business documents. It was originally argued that this constituted a Fifth Amendment violation, requiring self-incriminating statements. Adams v. New York (1904) actually specifically rejected the claim of a general exclusionary rule applied to all illegally-seized evidence, but this precedent did not las...

The first issue here is definitional. All types of communication are media. One can imagine, for example, reading a play, seeing a live performance of a play, watching a film of a play in a movie theater, playing a DVD, or live streaming a play to a computer or tablet. All of these are media.  Each type of medium has its own advantages. Print, for example, allows for quiet, reflexive concentration, deeply engaging with the content of a work. Live performance can be a deeply communal experience, with the interaction of performer and audience creating emotional depth. Large screens and surround sound can enhance the visual and aural aesthetics of certain films. Live streaming can be a cheap and convenient way of relaxing, although it does not allow for the sort of concentration necessary for appreciating complex works. 

There are a number of reasons for why characters lie in this play, but the simplest way to answer the question is to say that the lies serve their desires and protect their interests. Every character has a different set of desires and interests. One very prominent liar is Abigail Williams. Abigail lies about her activities in the woods because she doesn't want her uncle to think she is disobedient, particularly because she was recently fired from her job as a servant for the Proctors. Abigail lies about seeing various townswomen "with the devil" because she knows that showing she is caught up in the hysteria will help destabilize the town. Abigail believes she will get her way if the status quo is breached. Abigail wants John Proctor to leave his wife for her, and thinks this can happen if she lies about Elizabeth Proctor and impugns her good name. Abigail also lies in the courtroom, pretending to see things that support her narrative of bewitchment. All the lies Abigail ...

`sum_(n=1)^oo(1/n^2-1/n^3)` Apply the series sum/difference rule: `=sum_(n=1)^oo1/n^2-sum_(n=1)^oo1/n^3` Observe that both the series are p-series of the form`sum_(n=1)^oo1/n^p` Recall that the p-series test is applicable for the series of the form `sum_(n=1)^oo1/n^p` ,where `p>0`   If `p>1` , then the p-series converges If `0<p<=1` , then the p-series diverges Now, both the series have `p>1` As per the p-series test , both the series converge and so their sum/difference will also converge. Hence the series `sum_(n=1)^oo(1/n^2-1/n^3)` converges.

An integral in which one of the limits of integration is infinity is an improper integral.  Because we cannot find the definite integral using infinity (since it is not an actual value), we will need to rewrite the improper integral as a limit, shown below: `int_1^oo 6/(x^4)dx=lim_(n->oo)int_1^n 6/(x^4)dx` We can now take the integral first and the limit second.  We can find the integral using the power rule: `lim_(n->oo)int_1^n 6/(x^4)dx=lim_(n->oo)int_1^n 6x^-4dx=lim_(n->oo)[6*x^-3/-3]_1^n=lim_(n->oo)[-2/x^3]_1^n` Now we simply need to evaluate the limit. As n approaches infinity, the fraction will approach 0: `lim_(n->oo)-2/n^3-(-2)/1^3=lim_(n->oo)-2/n^3+2=0+2=2` Thus the integral converges to a value of 2.

Okonkwo is both representative of the values of his community and a rebel against them at different times in the novel. The obvious point in the novel when Okonkwo's role and position in his society change is when he is exiled by the tribe.  Before his exile, Okonkwo is a leader and a well-respected man of the tribe. He is renowned for his hard work, as noted in his success in growing yams; his physical strength, beginning with his famous wrestling victory over Amalinze the Cat; and his respect and reverence of most of the beliefs and traditions of the tribe. Okonkwo does set himself apart a couple of times, though. For one, he beats his wife during the Week of Peace. The narrator notes that the beating was a result of Okonkwo's "justifiable anger," but the tribe's tradition does not allow for violence of any kind during the Week of Peace. Even more importantly, Okonkwo's actions are called into question when he kills Ikemefuna with his machete. Ikemefuna was ...

The two men, who are shooting a film ‘‘about food stamps’’ for the county move away after Granny comes outside, but they continue filming anyway as they have no respect for her wish to be left alone. As the grandchildren play outside, they notice the filmmakers, who have been moving around and then enter the property of the Cains. After Granny comes outside one of the men asks her if she minds if they "shoot a bit around here?" "I do indeed," said Granny with no smile. Smilin man was smiling up a storm....But he didn't seem to have another word to say, so he and the camera man backed on out the yard, but you could hear the camera buzzin still. "Suppose you just shut that machine off," said Granny, real low through her teeth, and took a step down the porch.... Then, the man with the camera points the camera directly at her and says, "Now, Aunty...." This use of the term "Aunty" is meant to be a respectful address to an older African-...

The tricuspid and bicuspid valves of the heart are also known as atrioventricular valves because they are located between the atria and the venticles of the heart.  They prevent blood from flowing back into the atria when the ventricles contract to move the blood either into the pulmonary trunk or the aorta to send the blood through the pulmonary or systemic circulation, respectively.  You might also see that the bicuspid valve is called the mitral valve. Deoxygenated blood returning from the tissues of the body (systemic circulation) returns to the right atrium of the heart via the superior and inferior vena cava.  On the left side of the heart, the blood is returning to the atrium from the pulmonary circulation where the blood is reoxygenated at the lungs.  The atria (on the right and left sides) then contract and move the blood from atria through the tricuspid (right) and bicuspid (left) valves and into the ventricles.  The blood from the right atrium moves into the right ventricle ...

Yes, there is some foreshadowing in  Lyddie  by Katherine Patterson. One early instance of foreshadowing occurs while Lyddie is working at Cutler's Tavern. Lyddie engages in a very brief conversation with the woman in the pink, silk dress. The woman tells Lyddie about the potential earnings for factory girls. This foreshadows Lyddie's eventual move to the city to become a factory girl. Another example can be found in chapter six. Lyddie meets Ezekial during this chapter, and he tells Lyddie that "a little reading is an exceedingly dangerous thing." This foreshadows Lyddie's desire to increase her reading level and overall education. In a way, Ezekial's comment foreshadows Lyddie's choice to go to college instead of return home. Lyddie's conversation with Ezekial also brings up a bit of thematic foreshadowing. Ezekial is a runaway slave, and he subtly hints that Lyddie is a slave too. Lyddie fiercely defends the notion that she is not  a slave; however,...

Maclaurin series is a special case of Taylor series that is centered at a=0 . The expansion of the function about 0 follows the formula: `f(x)=sum_(n=0)^oo (f^n(0))/(n!) x^n`  or `f(x)= f(0)+(f'(0)x)/(1!)+(f^2(0))/(2!)x^2+(f^3(0))/(3!)x^3+(f^4(0))/(4!)x^4 +...` To determine the Maclaurin polynomial of degree `n=4` for the given function `f(x)=e^(-x/2)` , we may apply the formula for Maclaurin series.. To list `f^n(x)` , we may apply derivative formula for exponential function: `d/(dx) e^u = e^u * (du)/(dx)` . Let `u =-x/2` then `(du)/(dx)= -1/2` Applying the values on the derivative formula for exponential function, we get: `d/(dx) e^(-x/2) = e^(-x/2) *(-1/2)`             ` = -e^(-x/2)/2 or -1/2e^(-x/2)` Applying `d/(dx) e^(-x/2)= -e^(-x/2)/2`   for each `f^n(x)` , we get: `f'(x) = d/(dx) e^(-x/2)`           `=-1/2e^(-x/2)` `f^2(x) = d/(dx) (-1/2e^(-x/2))`           `=-1/2 *d/(dx) e^(-x/2)`            `=-1/2 *(-1/2e^(-x/2))`            `=1/4e^(-x/2)`  `f^3(x) = d/(dx) (1/4e^(-...

DNA (deoxyribonucleic acid) holds all of our genetic information. Our genes are what allows our bodies to function and makes us who we are. DNA is found inside the nucleus of all of our cells. Eventually, all cells die and so they need to replicate themselves through the process of mitosis and meiosis. In this process DNA is replicated. DNA will first make a copy of itself in a process called replication with the aid of proteins. As a cell is dividing the chromosomes (tightly wound DNA) move to the center of the cell and then are pulled apart. So, half of a chromosome will be in each cell. Once the cell is separated the DNA is able to replicate itself so that a whole chromosome will be formed again. This process of cells dividing is called mitosis, which leaves you with two identical cells. Sex cells undergo meiosis, which is similar to mitosis except it undergoes division twice. The result is four cells that are not identical. Copied DNA is transmitted from one cell to the next but fr...

In Toni Cade Bambara's "Raymond's Run," Squeaky is largely responsible for taking care of her older brother Raymond. Although Raymond is older and bigger than Squeaky, his mental development does not match his age and physical size. Squeaky never specifies what makes Raymond developmentally behind many of his peers; instead, she merely says he has a "big head" that people sometimes comment on or ask him about. Squeaky takes running very seriously and competes in a lot of races. To prepare for a race, Squeaky practices her breathing exercises and tries to strengthen her knees while going around her neighborhood. Raymond accompanies Squeaky as she practices her running, which means he must also run with her. Squeaky says Raymond has good motivation to keep up with her: "if he hangs back someone’s liable to walk up to him and get smart, or take his allowance from him, or ask him where he got that great big pumpkin head."

The Old Man and the Sea is set in Cuba in the middle of the twentieth century. The setting is a small village. There are several reasons why setting is important. First, the narrative depends on the old man living in extreme poverty without a social safety net. If the story were set in a village in the Canadian maritime provinces in the present, the old man would have access to first rate free medical care, disability insurance, and welfare payments so that he would not be in danger of starving when injured or unlucky. Next, the issue of luck and superstitions about it would not be of such importance in a more developed part of the world with a more educated populace. Also, in a more developed country, the boy would be in school and have many job options; apprenticeship would be less crucial. The sort of ethos of machismo is also a part of Cuban culture that might not be found to as great a degree in Scandanavia or other countries where Hemingway could have set the story. The small vi...

Recall that indefinite integral follows the formula: `int f(x) dx = F(x) +C`  where:` f(x)` as the integrand            `F(x)` as the anti-derivative function             `C`  as the arbitrary constant known as constant of integration For the given problem` int sqrt(5x^2-1) dx` , it resembles one of the formula from integration table.  We may apply the integral formula for function with roots as: `int sqrt(u^2+-a^2)du=1/2usqrt(u^2+-a^2)+-1/2a^2ln|u+sqrt(u^2+-a^2)| +C` . Take note the sign inside the root is "`(-)` " then we follow the formula as: `int sqrt(u^2-a^2)du=1/2usqrt(u^2-a^2)-1/2a^2ln|u+sqrt(u^2-a^2)| +C` By comparing "`u^2-a^2` " with "`5x^2-1` " , we determine the corresponding values as: `u^2=5x^2` or `(sqrt(5)x)^2` then `u =sqrt(5)x` `a^2 =1` or `1^2` then `a=1` For the derivative of `u` , we get `du = sqrt(5) dx` or `(du)/sqrt(5) =dx` . Plug-in on the values `u^2=5x^2` and `(du)/sqrt(5) =dx` on the integral problem, we get:  `int sqrt(5x^2-1)...

The main reason why some American colonists turned to smuggling (more common) and piracy (less common) was economic.  Simply put, it was a good way to make money.  In addition, many people had so little respect for the laws they were breaking that they did not feel that it was morally wrong to do so. In many (perhaps all) times and places, people have been motivated by the desire to get rich.  When people find a way that they can make money, they will want to pursue it.  Smuggling and piracy were good ways to make money.  Goods that were legally imported into the colonies tended to be expensive.  American goods sold through legal channels tended to get a lower price than smugglers could get if they sold directly to foreign buyers.  In other words, people could make a lot more money by smuggling or engaging in piracy than they could by obeying the law.  This is always a strong incentive to break the law. It is also easier for most people to break the law if they do not have much respect...

In Fences, social problems along with a lack of emotional connection all represent obstacles that prevent Troy from having what he wants. Troy believes he is socially marginalized.  Racial prejudice and institutional bias are factors that prevent him from achieving what he wants.  Troy feels that there is a challenge in being a man of color in America.  For example, he feels that racial integration of baseball helped to ruin his chances of pursuing his dream as a major league ballplayer. Additionally, he feels that racial bias prevents him from advancing in his job with the sanitation department. For example, Troy confronts his boss as to why whites drive the trucks but "the colored" are lifting the trash.  Troy feels that racial discrimination will never allow him to get a "paper job," a profession that affords him more money and greater social respect.  He believes that racial prejudice impedes him from the wealth and social status that could allow him to be happy...

The letters stand for the Latin phrase “curriculum vitae,” which translates generally into “the progression of one’s career over the course of a lifetime.” This term is used most often in academic and business circles to refer to someone’s professional résumé. This document includes at least the following components, as they relate to the individual: (a) academic degrees earned, listed from the highest ranked to the lowest (Ph.D., M.Ed., BA, for example), with the name of the school and years for each one; (b) academic or official positions held, listed chronologically with the newest at the top, and years and locations for each one; (c) any publications written by the person, including full bibliographic citations, from newest to oldest (and this section could be broken into categories of “articles” and “books,” etc., if applicable); (d) any professional talks or presentations given at conferences or workshops, from newest to oldest; (e) honors and awards earned, from newest to oldest...

The Three Weird Sisters may be considered culpable in Duncan’s death only by giving Macbeth the prophecy. However, they did not specify how Macbeth would be king. There are many possible paths for the fulfillment of this prophecy. It was Macbeth’s hubris to assume that he had to be active in his accession to the throne. In this way, the major guilt lies with Macbeth, not the witches. They simply spoke the truth as Fate would bring it about. Initially, Macbeth, while he considers the possibility of murder, is not sure if he should do anything. It is his wife, Lady Macbeth, who pushes him over the edge. She accuses him of being weak in not taking a darker path to achieve his destiny. Left alone, he may have waited to see how Fate played out the prophecy. Overall, it is Macbeth who is culpable of the death of Duncan and Banquo, as well as the others. Lady Macbeth is also culpable of “aiding and abetting” these murders. She herself killed the guards that Macbeth was too frightened to kill....