Notes

A binomial series is an example of infinite series. When we have a function of `f(x) = (1+x)^k` such that k is any number and convergent to `|x| lt1` , we may apply the sum of series as the value of `(1+x)^k` . This can be expressed in a formula: `(1+x)^k = sum_(n=0)^oo (k(k-1)(k-2)...(k-n+1))/(n!) x ^n`  or `(1+x)^k = 1+kx +(k(k-1))/(2!)x^2+(k(k-1)(k-2))/(2!)x^3+(k(k-1)(k-2)(k-3))/(4!)x^4+...` To evaluate the given function `f(x) = sqrt(1+x^2)` , we express it in term of fractional exponent: `f(x) =(1+x^2)^(1/2) ` or `f(x) =(1+x^2)^0.5` To apply the aforementioned formula for binomial series, we may replace "`x` " with "`x^2` " and "`k` " with "`0.5` ". We let: `(1+x^2)^0.5 = sum_(n=0)^oo (0.5(0.5-1)(0.5-2)...(0.5-n+1))/(n!) (x^2) ^n` `=sum_(n=0)^oo (0.5(0.5-1)(0.5-2)...(0.5-n+1))/(n!) x^(2n)` `=1+0.5x^(2*1) +(0.5(0.5-1))/(2!)x^(2*2)+(0.5(0.5-1)(0.5-2))/(3!)x^(2*3)+(0.5(0.5-1)(0.5-2)(0.5-3))/(4!)x^(2*4)+...` `=1+0.5x^2-0.25/(1*2)x^4+0.375/(1*2*...

At the beginning of A Separate Peace  by John Knowles, Gene returns to Devon 15 years after he graduated high school. When he visits the First Academy Building, he narrates, In through the swinging doors I reached a marble foyer, and stopped at the foot of a long white marble flight of stairs. . . The marble must be unusually hard. . . It was surprising that I had overlooked that, that crucial fact. I had more money and success and "security" than in the days when specters seemed to go up and down them with me (11-12). Not only do these hard marble stairs represent Gene's many days at school and his youth, but they are part of the reason his friend Phineas died in high school. It is in the First Academy Building that Brinker holds a mock trial to accuse Gene of breaking Finny's leg the first time. Phineas becomes angry because of the trial and storms out, slips on the marble, and breaks his leg a second time on the stairs. The fact that Gene notices how hard the marbl...

"In fair Verona, where we lay our scene," is the second line of the prologue to William Shakespeare's most famous play,  Romeo and Juliet. The prologue is intended to give the audience (or reader) a little introduction to the play they are about to see. This line is essentially telling the audience that the play is taking place in Verona, Italy. The prologue goes on to describe that the play will involve feuding families and two young lovers who will die. Sometimes, the setting of a play does not need to be said explicitly. For  Romeo and Juliet,  the setting in Italy provides a great backdrop for the main conflict of the play. There were not really noble feuds in England- where this play was first put on- to the same degree that they occurred in Italy. In order for the "fighting families" thing to really work, Shakespeare chose Italy for the setting. Presumably, Shakespeare's first audience would know a little about these feuding nobles of Italy, or at lea...

Seafloor spreading is a process that occurs at mid-ocean ridges. These ridges mark the boundaries between two plates that are diverging, or moving away from one another. As the plates move apart, the crust is stressed and weakened. This allows magma from under the surface to break up through the crust, coming out and creating new sea floor. The plates move apart, carrying the “new” sea floor with it. The idea that this happens was proposed in the 1960s. At first, seafloor spreading was not proven. However, magnetic reversals have now proven it. Magnetic reversals occur every so often. When they happen, the Earth’s magnetic field reverses its polarity. In other words, north becomes south and south becomes north. Magnetic reversal proves seafloor spreading because we can see the polarity of the Earth’s magnetic field in rocks. As magma cools, particles in it get “frozen” in the direction of the magnetic field. By looking at these particles, scientists can see the polarity of the magnetic...

Three songs that relate to Macbeth include the following: "I Shot the Sheriff" by Bob Marley. The title says it all. Although the authority figure Macbeth shot was the king, not the sheriff, the same idea holds. In Marley's song, he rationalizes his action by claiming he "did it in self-defense." Macbeth also rationalizes his action, although he does this by arguing killing the king was the quickest way to get the crown. In both cases, someone is dead and we know there will be trouble coming. "Witchy Woman" by the Eagles. This song describes Lady Macbeth, who casts a "spell" on Macbeth with her words to motivate him to kill Duncan. She also fits the line "she drove herself to madness with a silver spoon." Lady Macbeth drives herself to madness, not in pursuit of money (which is what a silver spoon signifies), but with pursuit of power.  "Don't Stop Believin'" by Journey. Macbeth, to his downfall, never stops believ...

Given` y'+ytanx=secx+cosx` when the first order linear ordinary Differentian equation has the form of `y'+p(x)y=q(x)` then the general solution is , `y(x)=((int e^(int p(x) dx) *q(x)) dx +c)/ e^(int p(x) dx)` so, `y'+ytanx=secx+cosx--------(1)` `y'+p(x)y=q(x)---------(2)` on comparing both we get, `p(x) = tanx and q(x)=secx +cosx` so on solving with the above general solution we get: y(x)=`((int e^(int p(x) dx) *q(x)) dx +c)/e^(int p(x) dx)` =`((int e^(int (tanx) dx) *(secx+cosx)) dx +c)/e^(int tanx dx)` first we shall solve `e^(int (tanx) dx)=e^(ln(secx))= secx `      so proceeding further, we get `y(x) =((int e^(int (tanx) dx) *(secx+cosx)) dx +c)/e^(int tanx dx)` =`((int secx *(secx+cosx)) dx +c)/(secx )` =`((int (sec^2 x+cosxsecx)) dx +c)/(secx )` = `((int (sec^2 x)dx +int 1 dx) +c)/secx` =` (tanx+x +c)/secx` =` sinx +(x+c)/secx` `y(x) = sinx +(x+c)/secx` to find the particular solution of the differential equation we have `y(0)=1` on substituting x=0 we get y=1 and...

The cell's endomembrane system consists of the nuclear envelope, endoplasmic reticulum, Golgi apparatus, lysosomes, vacuoles and the plasma membrane. They are continuous or can be connected via tiny vesicles that can transfer materials from one part of a cell to the next. The endoplasmic reticulum contains both smooth and rough ER. Rough ER contains bound ribosomes which are tiny factories that can manufacture proteins. The rough ER distributes transport vesicles which are secretory proteins surrounded by membranes. The Golgi apparatus are membranous sacs that can modify ER products and sort and package materials into transport vesicles. Proteins destined to be incorporated into a lysosome are formed by rough ER, travel to the Golgi apparatus to be further modified and the vesicle containing those proteins that buds off the Golgi apparatus is the newly formed lysosome.  Lysosomes are membranous sacs of hydrolytic enzymes capable of digesting large molecules.  One function of a ly...

Cicero is killed in Antony’s proscription.   Cassius seeks out Brutus to join his conspiracy against Julius Caesar because he feels that his name will lend them legitimacy.  Brutus takes the position seriously.  He takes charge right away, and when the conspirators join him at his house he makes it clear that he is in charge.  One example of this is his reaction to Cassius’s suggestion that Cicero be included.   CASSIUS But what of Cicero? shall we sound him? I think he will stand very strong with us. (Act 2, Scene 1)  There is a big discussion about whether or not Cicero should be included in the conspiracy.  After Cassius suggests him, Casca and Cinna are all for it.  Metellus Cimber says that “his silver hairs/Will purchase us a good opinion.”  They seem to think that people trust Cicero’s opinion, and with him seeming to lead the group people will be more likely to follow them.  Brutus does not like this idea.  After all, he wants to be in charge.  He doesn’t want Cicero telling hi...

The cat in "Once upon a Time" plays a significant symbolic role. First, the cat symbolizes bad fortune. Black cats are associated with bad luck, and since Gordimer also includes a "witch" in the story, the connection becomes even stronger since cats and witches are paired in folklore.  Second, the cat symbolizes the unconquerable fear the family has for most of the story. The cat is able to get into the house despite the bars and to get into the yard despite the high wall. This shows the family's fears continue to beset them despite their efforts to protect themselves. Third, the cat represents false alarm. The cat—as well as other cats in the neighborhood—routinely sets off the burglar alarms in people's homes. This suggest the fears the suburban residents have are "false alarms." The alarm they experience is not grounded in reality. In the same way that the boy who cried wolf was not believed when the wolf finally did attack the flock, so too do ...

The pervasive attitude toward smoking and smokers has definitely shifted over the last 50 years, from favorable (or at least neutral) to unfavorable. Each of the dominant three theoretical perspectives has a different way of interpreting this shift. A conflict theorist would view the change in attitudes towards smoking as a means of controlling supply, thereby keeping common goods out of the reach of the underclass. As taxes are piled on cigarettes and new legislation promotes fines for smokers, those in the underclass are forced to either quit smoking -- a solution conflict theorists would view as paternalistic -- or spend money they do not have to afford goods that are marked up at a higher percentage than other retail goods. A functionalist would view the change in attitudes towards smoking as a means of generating revenue. States, counties, cities, and townships have been able to fund projects through raising taxes on cigarettes, and levying fines on smokers. Functionalists would a...

Phillip and his family are on the island of Curacao. They are not native to the island. Phillip and his family are originally from the United States, but Phillip doesn't remember much about it. He was only seven when his family moved to the island. Phillip is now eleven, and World War II is escalating to the point where Phillip's family no longer feels safe on the island. They decide to head back to the United States. The ship that they board will take them to Miami, Florida; however, that is not their final destination. Phillip's final destination is Norfolk, Virginia. Unfortunately, the ship that he is on gets torpedoed, and Phillip is forced to survive on a small cay with Timothy.

There were a number of problems in the late 19th Century in the United States that still exist today.  There were racial tensions during this period that were very defined.  African-Americans suffered through black codes, Jim Crow Laws, and even lynchings.  There was very little economic or political opportunity for African-Americans during this time.  While African-Americans fought for civil rights for the better part of the Twentieth Century, a disconnect between the races still exists today.  Black Americans suffer from housing discrimination, unequal treatment before the law, and are behind in terms of educational and employment opportunities.   A number of other problems that existed during the Gilded and Progressive Eras are still issues today.  Urban decay, environmental protection, corporate greed and abuses, and the rights of women come to mind as obvious examples.

The main goal of Reconstruction was to reunify the North and South.  Each state in the former Confederacy had to develop a new constitution that prohibited slavery.  The federal government had somewhat ambiguous goals for extending rights to the former slaves, granting them their freedom in 1865, citizenship in 1868, and suffrage for black males over the age of 21 in 1870.  When state governments in the South tried to get in the way of the black suffrage movement, the North started military Reconstruction under Grant, and this would continue until 1877.  There was also the goal of creating stable state governments in the South that were pro-Union, and this meant granting many former Confederate leaders such as Alexander Stevens their American citizenship.  This was controversial, as many considered the first Reconstruction president, Andrew Johnson, to be soft on the ex-Confederates.  The final and most important goal of Reconstruction was that it should end--the American people did no...

Yes, Denver is a more important character in this novel than Beloved.  It is Denver whose growth from a young girl into a woman that readers follow: we see Denver struggle to develop an identity that includes her mother and her father (who she's never known), then to form a relationship with the strange Beloved (whose identity Denver figures out long before anyone else seems to), then to save her mother (and herself) when Beloved begins to consume Sethe.  Beloved is a ghost, she's the past, she's the nightmare of slavery: she is almost all these things more than she ever is a true character.  Denver, on the other hand, isn't a symbol; she's a real person in this text.  She is dynamic and complex, with a past and a future.  She is the hope for the future that there can be a life after slavery.

In Theodore Taylor's The Cay , Phillip assumes his father has ships out looking for Phillip, but what Phillip doesn't realize is that he and Timothy could be in a section of the cays surrounded by coral reefs , which make it impossible for any vessels other than fishing boats and airplanes to come near them. At first, Timothy encourages the thought that they could soon be rescued by a ship, like a schooner. He says that if they are near the cays, then they are on the ships' route to Jamaica. But, Timothy changes his tune the deeper they drift into the cays. When they reach land, though Timothy doesn't say so at first, he is fairly certain they are in Devil's Mouth , which is surrounded by coral reefs and not a place ships can reach. Timothy keeps his suspicion a secret until he knows for certain, allowing Phillip to still believe his "father had planes and ships out looking for [them]" (p. 54). After exploring the island they reach, Timothy is certain t...

During the three days of the dangerous game created by General Zaroff, who is the hunter and Sanger Rainsford the prey, Rainsford learns what it is to be "a beast at bay." "You'll find this game worth playing," the general said enthusiastically." Your brain against mine. ....Your strength and stamina against mine. Outdoor chess! And the stake is not without value, eh?" At first Rainsford races in fear; then, he gets control of himself. Working under the metaphor of a chess game, Sanger Rainsford summons his expertise as a tracker as he weaves his way through the forest, covering his tracks and creating diversions meant to slow down Zaroff. When night falls, he climbs into a large joint of a thick, huge tree. Rainsford stretches out upon one of the broad limbs, trying to rest his weary body and strained mind. Reasoning that only the devil could follow his tracks, Rainsford suddenly worries that Zaroff may, indeed, be a devil. Toward dawn, Rainsford hear...

The main advice that Martin Luther King, Jr. gives to African Americans in this speech has to do with the idea of nonviolence for which King is famous.  He urges his people to make sure that they keep the moral high ground in their struggle for civil rights. King is famous for having believed in the idea of nonviolent protest.  He thought that was the only kind of action that could actually gain African Americans the rights that they wanted and deserved.  He believed that nonviolent protest would convince white people that the people who wanted to keep blacks from having equal rights were on the wrong side of the argument.  Therefore, he urged African Americans to continue to use nonviolence.  King tells African Americans that they “must not be guilty of wrongful deeds” in their attempt to gain their rights.  He said that they had to make sure that they did not let themselves give in to their anger over how whites had treated them. As he said: We must forever conduct our struggle on th...

The Chief Justice referred to in this question was John Marshall. Marshall was a Federalist, who believed in strengthening the powers of the federal government relative to the states. Jefferson, a Republican, believed in the opposite. Many of Marshall's decisions played a major role in consolidating the powers of the federal government. McCulloch v. Maryland (1819) ruled that Congress had the power, due to the Necessary and Proper Clause, to establish a national bank and that the Supremacy Clause prevented the states from taxing it. This decision was anathema to both the strict constructionist and states' rights views held by many of Jefferson's political allies (and to some extent by Jefferson himself.) In many other decisions, like Trustees of Dartmouth College v. Woodard , Martin v. Hunter's Lessee , and Cohens v. Virginia , the Marshall Court continued to rule in ways that strengthened the national government. Ironically, the most famous decision associated with th...

`(1/(3x^2-3))/(5/(x+1)-(x+4)/(x^2-3x-4))` Let's factorize the denominators of the terms in the complex fraction, `=(1/(3(x^2-1)))/(5/(x+1)-(x+4)/(x^2+x-4x-4))` `=(1/(3(x+1)(x-1)))/(5/(x+1)-(x+4)/(x(x+1)-4(x+1)))` `=(1/(3(x+1)(x-1)))/(5/(x+1)-(x+4)/((x-4)(x+1)))` LCD for all the denominators in the complex fraction is `(x+1)(x-1)(x-4)` Multiply both the numerator and denominator of the complex fraction by the LCD, `=((x+1)(x-1)(x-4)(1/(3(x+1)(x-1))))/((x+1)(x-1)(x-4)(5/(x+1)-(x+4)/((x-4)(x+1))))`   Use the distributive property in the denominator of the complex fraction, `=((x-4)/3)/(5(x-1)(x-4)-(x+4)(x-1))` `=((x-4)/3)/((x-1)(5(x-4)-(x+4)))`  `=(x-4)/(3(x-1)(5x-20-x-4))` `=(x-4)/(3(x-1)(4x-24))` `=(x-4)/(3(x-1)4(x-6))` `=(x-4)/(12(x-1)(x-6))`

The first thing the animals did was to destroy many of the things they associated with Jones and human control. They threw nose rings, castrating knives, and chains down a well, and they burned harnesses, halters, and whips into a fire. At that point they simply roamed around the farm, exercising and testing the limits of their newly-won freedom. They could not believe that the the farm was "all their own." In short, the animals began to go about the business of setting up a farm based on the principles of Animalism. The pigs wrote out the Seven Commandments on a barn wall, and they began to plan how to carry out the season's harvest. At the beginning, the animals were happy and optimistic about the future, though the reader begins to notice some evidence that their revolution has already been corrupted--the milk taken from the cows mysteriously vanished, clearly mixed in with the pigs' mash.

To evaluate the given integral problem` int e^(sqrt(2x))dx ` us u-substituion, we may let: `u = 2x` then `du = 2 dx` or `(du)/2 = dx` . Plug-in the values `u = 2x ` and `dx = (du)/2` , we get: `int e^(sqrt(2x))dx =int e^(sqrt(u))* (du)/2` Apply the basic integration property: `int c*f(x) dx = c int f(x) dx` . `int e^(sqrt(u))* (du)/2=1/2 int e^(sqrt(u)) du` Apply another set of substitution, we let: `w = sqrt(u)` Square both sides of `w =sqrt(u)`, we get: `w^2 =u` Take the derivative on each side, it becomes: `2w dw = du`  Plug-in `w =sqrt(u)` and `du = 2w dw` , we get:  `1/2 int e^(sqrt(u)) du =1/2 int e^(w) * 2w dw`                                      ` = 1/2 * 2 inte^(w) *w dw`                                      `= int e^w * w dw` . To evaluate the integral further, we apply integration by parts:`int f* g' = f*g - int g *f' Let: `f =w` then `f' = dw`        `g' = e^w dw` then `g = e^w` Applying the formula for integration by parts, we get: `int e^w * w dw = w*e^w ...

The connotative meaning of the older boys making room for Jerry is so positive because it means that they are accepting him.  Despite the fact that he is younger than they, that he is a foreigner, and that he doesn't speak their language, they allow him to hang out with them (for a while at least), and their acceptance is -- to Jerry -- the most wonderful possible thing that could have happened in that moment.  To be allowed to dive and swim and splash with these older boys that seemed "like men to Jerry," gives him a sense of belonging at the wild bay rather than the "safe beach" where his mother and all the other vacationers splash and paddle.  When the boys make room for him, it makes him feel like he is grown up, too, because they are older.  He feels more independent of his mother and more adult as a result of their acceptance.

There are many themes in this novel, including:  innocence, friendship, and the Holocaust.  The main techniques are dialogue and characterization.  Innocence is revealed through Bruno's interaction with his own family as well as his interaction with Shmuel.  Bruno is constantly mispronouncing names (such as "the Fury" for "the Führer" and "Out-With" for "Auschwitz").  Bruno also does not understand why his family had to move from Berlin to Poland or why his friend Shmuel is on the other side of the fence and, as a result, is not being taken care of.  The theme of friendship is definitely apparent through the interactions between Bruno and Shmuel.  The two are friends even though Bruno is German (and the son of a Nazi commandant) and Shmuel is a Polish Jew living in a concentration camp.  In fact, Bruno continually calls Shmuel his friend, brings him food, talks with him, and eventually risks his life when he enters the camp to help him find h...

There are at least two main disadvantages of judge-made law relative to statute law.  One disadvantage has to do with the legitimacy of such law while the other has more to do with pragmatic issues. In pragmatic terms, law made by judges will not be able to touch on as many issues at as many times as statute law can.  Judges can only make law when cases come before them.  They cannot go out and change laws simply because they want to.  In addition, they cannot rewrite laws based on whatever they feel is best at a given time.  Instead, they can only rule on specific issues raised in their cases.  For example, let us look at “Obamacare.”  Judges could strike down certain portions of the law if they wanted, but they only if plaintiffs brought cases that touch on those portions of the law.  In addition, judges would not be able to replace the law with something they liked better.  They would only be able to invalidate the law as it currently stands. In terms of legitimacy, the disadvantage...

Binomial series is an example of an infinite series. When it is convergent at `|x|lt1` , we may follow the sum of the binomial series as `(1+x)^k` where `k` is any number. The formula will be: `(1+x)^k = sum_(n=0)^oo (k(k-1)(k-2) ...(k-n+1))/(n!) x^n ` or `(1+x)^k = 1 + kx + (k(k-1))/(2!) x^2 + (k(k-1)(k-2))/(3!)x^3 +(k(k-1)(k-2)(k-3))/(4!)x^4+...` To evaluate the given function `f(x) = 1/sqrt(4+x^2)` , we may apply `4+x^2=4(1+x^2/4)` . The function becomes: `f(x) =1/sqrt(4(1+x^2/4))` `f(x) =1/(2sqrt(1+x^2/4))` Apply radical property: `sqrt(x) = x^(1/2)` . The function becomes: `f(x) =1/(2(1+x^2/4)^(1/2))` Apply Law of Exponents: `1/x^n = x^(-n)` to rewrite  the function as: `f(x) =1/2(1+x^2/4)^(-1/2)` or `f(x)=1/2(1+x^2/4)^(-0.5)` Apply the aforementioned formula on `(1+x^2/4)^(-0.5)` by letting: `x=x^2/4 ` and `k =-0.5.` `(1+x^2/4)^(-0.5) = sum_(n=0)^oo (-0.5(-0.5-1)(-0.5-2) ...(-0.5-n+1))/(n!) (x^2/4)^n` `= sum_(n=0)^oo (-0.5(-1.5)(-2.5) ...(-0.5-n+1))/(n!) x^(2n)/4^n` `= 1 + (-0.5)...

`sum_(n=0)^oo(1/2^n-1/3^n)` `=sum_(n=0)^oo1/2^n-sum_(n=0)^oo1/3^n` `=(1/2^0+1/2^1+1/2^2+.......+1/2^oo)-(1/3^0+1/3^1+1/3^2+......+1/3^oo)` `=(1+1/2+1/2^2+.......1/2^oo)-(1+1/3+1/3^2+........+1/3^oo)` Now both of the above are geometric series,having first term 1 and common ratios `1/2,1/3` respectively, Geometric series converges to the sum: `sum_(n=0)^ooar^n=a/(1-r),0<|r|<1` Using the above, `=(1/(1-1/2))-(1/(1-1/3))` `=(1/(1/2))-(1/(2/3))` `=2-3/2` `=1/2`

In W.E.B. DuBois's seminal work,  The Souls of Black Folk , the problem of the twentieth-century is, indeed, that of the "color-line." The phrase "the color-line" first appeared in an article called "The Color Line" written by Frederick Douglass for the North American Review in 1881. It became better-known when DuBois used the phrase to expand its relevance for the issues confronting black Americans in the twentieth-century. The color-line is a direct reference to the segregation that existed in the United States -- both the legal,  de jure , segregation in the South, and the  de facto , or "in effect," racism in the North and the West. Segregation made it so that blacks and whites lived in such disparate worlds that their lives generally did not look at all the same. Though black people were very often aware the rights and privileges that were kept from them, particularly since working-class blacks were very often employed in white homes as ...

The relationship between the pressure and volume of a gas is  inverse . This means that when the pressure of a gas increases, the volume of the gas decreases. The reverse is also true - when the volume of a gas increases, the pressure of the gas decreases.  The relationship between the pressure and volume of a gas is described by Boyle's Law :    `~P_1V_1 ` = `~P_2V_2` ` ` `~P_1` = initial pressure `~V_1` = initial volume `~P_2` = final pressure `~V_2` = final volume Example : A 2.0 L sample of gas has a pressure of 3.0 atm. What will the volume of the gas be if the pressure is increased to 5.0 atm? (Assume that the temperature and number of moles of the gas remains constant.)    `~P_1` = 3.0 atm    `~V_1` = 2.0 L    `~P_2` = 5.0 atm    `~V_2` = ? L When these values are plugged into Boyle's Law we get the following result:    (3.0 atm)(2.0 L) = (5.0 atm)(? L)        = 1.2 L Therefore, when the pressure of the gas was increased from 3.0 atm to 5.0 atm, the volume of the gas ...

Look at Napoleon's actions--he was absent at the Battle of the Cowshed, he had Snowball exiled, he had naysayers killed after mock trials, and he never did any work.  While no animals but Benjamin could read the Commandments of Animalism, they seemed always to change in favor of the pigs.  The pigs lived like humans at the end of the story, and the name of the farm reverted back to what it was before the revolution.  There are many reasons not to like Napoleon.  Your job here is to give exact reasons on why Napoleon is worse than Jones.  Remember, these animals cannot read, so when you compose your speech, try to focus on the conditions on the Farm.  The animals may be illiterate, but they can remember if they were better off than they were at the time of the rebellion.  You could even tell the story of Boxer the horse and how his death is a lesson in what loyalty to Animal Farm causes.  There are many directions for this speech.  I would pay close attention to the length requireme...

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 1/(x^2+5)^(3/2)dx` , it resembles one of the formula from integration table.  We may apply the integral formula for rational function with roots as: `int 1/(u^2+a^2)^(3/2)du= u/(a^2sqrt(u^2+a^2))+C` By comparing "`u^2+a^2` " with "`x^2+5` " , we determine the corresponding values as: `u^2=x^2` then `u = x` and `du = dx` `a^2 =5` then `a = sqrt(5)` . Plug-in the corresponding values on the aforementioned integral formula for rational function with roots, we get:  `int 1/(x^2+5)^(3/2)dx =x/(5sqrt(x^2+5))+C`

First, the hunter calls Phoenix "Granny." She is very old and this might have been a term of endearment. But given the hunter's overall treatment of Phoenix, the term "Granny" is probably intended with condescension. He does help her up, but he also belittles her. He treats her like a child:  I know you old colored people! Wouldn't miss going to town to see Santa Claus!  In his ignorance, he assumes she is simple-minded. It would never occur to him that she is a clever, determined individual. Phoenix manages to steal a nickel that falls out of his pocket. When he points the gun in her face, she is stoic and shows her courage. He then lies and says he'd give her some money if he had any. Everything about the hunter's behavior with Phoenix shows his pathetic attempts to assert his superiority. He helps her up but treats her like a child. He pointlessly aims his gun at her to test his ability to intimidate. His rudeness is based upon racism and ignoranc...

I would agree with this statement, to the extent that the film industry frequently looks toward literature for new material and timeless tales. Because the film industry is the primary field of entertainment which uses material from literature, I will focus my response on literature adapted for film. Shakespearean plays, such as Hamlet, Macbeth, and Othello, have been produced and reproduced over the twentieth century. Laurence Olivier directed and starred in the most critically acclaimed version of Hamlet in the 1940s. The play was re-adapted by Mel Gibson in 1990, who also directed and starred. The latter was not as highly regarded by critics, but was a bigger hit with audiences. Ethan Hawke revamped the play in 2000, transforming the brooding prince into a young filmmaker from New York City, struggling against a production company called Denmark Corp. in the wake of his father's death.  There have been as many, if not more, productions of Macbeth produced in both the United King...

The book contains 29 chapters, originally marked by Roman numerals. Here is a summary of the main points of each one. I – The three men -- narrator J., George, and Harris -- discuss their health issues. As a remedy for being overworked, they decide to go on a boat trip along the River Thames. II – The friends begin to make plans. They decide to camp out in good weather and to use hotels and inns and pubs in wet weather. J. tells us more about Montmorency, the fox terrier. III – J. relates the story of his Uncle Podger. The men make a list of supplies for the trip. They decide to use a boat with a cover as a tent. They decide about clothing. IV – They decide about food to take, and they choose to take a “methylated spirit stove” and whiskey. J. relates a lengthy story about cheese. J. packs the equipment and clothes, and George and Harris packs the food hamper. V – The first day of the trip. They get up late. J. talks about weather and barometers. He and Harris take the train and get in...

Everyone in the world faces scarcity.  There are no exceptions.  Even Bill Gates cannot escape scarcity.  Scarcity is endemic to the human condition. Scarcity exists because people have limited resources but, at the same time, have unlimited desires.  We people tend to want many things, but we do not have unlimited resources with which to fulfill those wants.  This is true even of Bill Gates.  Let us imagine that Bill Gates wants to buy a certain island.  He might not be able to because the island is not for sale.  The owner of the island may love the island and may not want to sell it for any price.  In addition, even Bill Gates could not buy absolutely everything he might want to buy.  For example, Bill Gates could not come anywhere near to buying up all the shares of Apple or Google.  Those companies are worth much more than Gates’ net worth.  When Gates invests in the stock market, he cannot buy everything because even he does not have enough money to do so. Now imagine that Gates ...

Given parametric equations are: `x=3t^2,y=t^3-t` We need to find the second derivative, to determine the concavity of the curve. `dy/dx=(dy/dt)/(dx/dt)` Let's take the derivative of x and y with respect to t, `dx/dt=3*2t=6t` `dy/dt=3t^2-1` `dy/dx=(3t^2-1)/(6t)` `dy/dx=(3t^2)/(6t)-1/(6t)` `dy/dx=t/2-1/(6t)` `(d^2y)/dx^2=d/dx[dy/dx]` `=(d/dt[dy/dx])/(dx/dt)` `=(d/dt(t/2-1/(6t)))/(6t)` `=(1/2-1/6(-1)t^(-2))/(6t)` `=(1/2+1/(6t^2))/(6t)` `=((3t^2+1)/(6t^2))/(6t)` `=(3t^2+1)/(6t^2(6t))` `=(3t^2+1)/(36t^3)` Curve is concave upwards if second derivative is positive and concave downwards if it is negative, So, the curve is concave upward for `t>0` Curve is concave downward for `t<0`

The first quotation makes use of a classical literary allusion, and makes us think of the dawn is more than a natural phenomenon: It is a goddess who helps ensure the knight's victory. The second makes use of a simile that evokes several meanings, including the idea that a youthful appearance is fleeting. I explain and develop these points below. 1. Classical allusion: Spenser's words allude to Aurora, the goddess of the dawn. This quote is taken from Book I, Canto XI. For background, consider what has been recounted just before these lines. The knight has fallen for a second time, but luckily he has landed in a balm that has flowed from sacred tree. The dragon "durst not approach" because the life-giving force of the balm is anathema to it. So the fight comes to a temporary standstill, and night falls.   The balm is healing the knight's wounds as he sleeps. But Una is still understandably frightened for him. The night is intrinsically worrisome, "noyous,...

The passage of the Prohibition of Mixed Marriages Act in 1949 banned marriages between white and non-white South Africans and nullified existing mixed race marriages. It was amended in 1968 to invalidate South African citizens' interracial marriages even if the marriage had been officiated in another country. The South African public largely agreed with anti-miscegenation laws, and existing Apartheid legislation already ensured that racial groups were largely segregated and rarely able to interact as equals. Prior to the passage of the law, only 0.23% of all marriages in the country were mixed race as a result of this segregation. Most resistance to the law came from churches, who argued against state intervention in the institution of marriage. However, the law was unchanged until 1985, when the government passed the Immorality and Prohibition of Mixed Marriages Amendment Act to repeal the ban.

Faber is a retired English professor who serves as Montag's mentor and friend. Faber encourages Montag to engage in intellectual pursuits and assists Montag in fleeing the dystopian society. Whenever Montag cannot understand the meaning of various texts that he is reading, he consults Faber for answers. Faber teaches Montag the significance of literature and has several discussions regarding the ills of their dystopian society. Faber aids Montag by giving him a two-way communication device that Montag puts in his ear. Faber is able to listen to Montag's conversations and speaks to Montag via the earpiece. After Montag kills Beatty and flees to Faber's house, Faber gives Montag directions to locate and join the traveling group of hobo intellectuals. Montag follows Faber's directions and escapes the Hound. Montag's life permanently changes for the better after he meets Granger and joins the traveling intellectuals. Faber's comradery, wisdom, and assistance saves M...

The majority of Edwards's sermon details the depth of God's disgust with sinners. The eternal repercussions of God's wrath are made exquisitely clear:  "'Tis everlasting Wrath. It would be dreadful to suffer this Fierceness and Wrath of Almighty God one Moment; but you must suffer it to all Eternity: there will be no End to this exquisite horrible Misery..." Moreover, Edwards uses compelling imagery to describe the tortuous nature of eternal damnation: "Damnation don’t slumber, the Pit is prepared, the Fire is made ready, the Furnace is now hot, ready to receive them, the Flames do now rage and glow" Edwards disabuses the congregation's notion that they will somehow sidestep damnation through the strengths that they assume they have: "Natural Men’s Prudence and Care to preserve their own Lives, or the Care of others to preserve them, don’t secure ‘em a Moment." But because the purpose of the sermon was to reinvigorate religious commitme...

In Judith Ortiz Cofer's "American History," the protagonist, Elena, faces a moral dilemma on November 22, 1963. This is the day President John F. Kennedy is shot and when the country mourns the shocking loss of a beloved man. Elena and the other students from school are sent home out of respect for the country's loss. Elena notices on her way home that her apartment building is completely silent. This silence is unusual because there are usually different types of music blasting from the windows and people socializing out front. On this day in history, however, Elena's building is silent to show respect for a time of mourning. The moral dilemma occurs when Elena decides to go to Eugene's house for their study date rather than to church with her mother. Elena's mother attempts to change Elena's mind by saying the following: "Hija, the president has been killed. We must show respect. He was a great man. Come to church with me tonight."  Elena...

The second part of the statement is best supported by using the banker.  When the story begins, he is an obscenely wealthy individual.  The money that is on the line when making the bet with the lawyer is trivial to him.  He makes the bet of two million in the same casual manner that I would make a bet using two dollars.   But at the end of the story, the banker has lost almost all of his net worth.  His wealth was temporary and fleeting.   As for the first part of the statement, the best support is found through the lawyer's character.  By the end of his fifteen years of confinement, he is incredibly well-educated.  He's read up on just about every possible topic, and he has taught himself multiple languages; however, he's completely jaded with mankind by the time that he leaves his cell.   "To prove to you in action how I despise all that you live by, I renounce the two millions of which I once dreamed as of paradise and which now I despise." Despite his vast kn...

Cultural identity, and the related value system, of any group is based on fundamental principles, which guide worldviews. Worldviews, or ways of interpreting the surrounding world, play a huge role in communication. Fundamental assumptions drive interpersonal interactions, so when these assumptions are different, communication can go haywire. For example, consider the differences between individualistic cultures (of western countries like the US) and collectivistic cultures (of eastern countries like Japan). In individualistic cultures, people make decisions based on their own self worth and personal goals. The idea in cultures that are self-focused is that, while society is important, it is more important to succeed as an individual. On the other hand, collectivistic cultures tend to value the success of a group over individual success. What's important to understand is that the responses are very different in their basic assumptions. While a person from an individualistic culture...

Waves can be characterized by highs and lows. The highest point in a wave is known as the crest, while the lowest point is known as the trough. There are few properties of a wave that are used to characterize it. These parameters include wavelength, frequency and amplitude.  The distance between any two successive crests is known as the wavelength. It is also equal to the distance between any two troughs. In fact, wavelength is equal to the distance between any two corresponding points (such as crests or troughs, etc.) on successive waves. The frequency is the number of completed cycles (crest to crest is one cycle, for example) in a given time period. The amplitude of a wave is the distance between the midline and the crest or trough.  Hope this helps. 

To solve for the height of the desk, apply the formula that relates power to work which is: `P = W/ (Delta t)` where  P is power, W is the work done, and `Delta t` is the time interval. Plugging in the given values P=3.5W and `Delta t ` = 2.8s, the formula becomes: `3.5 = W/2.8` Isolating the W, it yields: `3.5*2.8=W/2.8*2.8` `9.8=W` So, the work done in lifting the object from the floor to the desk is 9.8J. Take note that work is calculated by multiplying the force  by the amount of movement. `W = F * Delta x` where F is the force and `Delta x` is the displacement. In this problem, the only force present in lifting the object is its weight. `W = mg * Delta x` Plugging in the values W=9.8J, m=1.2kg and g=9.81 m/s^2, the formula becomes: `9.8=1.2*9.81*Delta x` Isolating Delta x results to: `9.8=11.772*Delta x` `9.8/11.772=(11.772*Delta x)/11.772` `0.8324 = Delta x` Therefore, the desk is 0.8324 meters high. To solve for the time it took to lift the new object that has a mass of 2kg, the...

In this quote from Chapter 17 of Charlotte Brontë's Jane Eyre, Jane describes Miss Blanche Ingram. Jane essentially lists Miss Ingram's good qualities, for Jane is slightly intimidated by her, but also knows that Miss Ingram is self-aware and proud of her own accomplishments, which is unappealing to Jane. Jane describes Miss Ingram as "self-conscious," and says that she plays on the ignorance of the other people in the room by being clever and promoting her talents. Jane also describes Miss Ingram as "majestic," and "the very type of majesty." It is Jane's opinion of Miss Ingram and her tone while describing her that influence the structure of the quote. The alternating use of colons and semi-colons make the quote read like a list. The quote therefore suggests through the use of repetition that everything about Miss Ingram is equally fine. For if her playing, her voice, and her French are "brilliant," "fine," and "well,...

The touch-me-not cottage is the house that the Foster family lives in.  It is called the touch-me-not cottage because everything appears so incredibly neat and ordered that to touch anything might mess up the perfection.  That's the main reason why people actively avoid the Foster cottage.  The house is uninviting because of its perfection.   On the left stood the first house, a square and solid cottage with a touch-me-not appearance, surrounded by grass cut painfully to the quick and enclosed by a capable iron fence some four feet high which clearly said, "Move on—we don't want you here." Winnie herself feels stifled in her own home.  It's why she is contemplating running away.  Later, when she finally arrives at the Tuck household, she is amazed as the disarray of everything.  She is not disgusted or appalled at the Tuck household though.  On the contrary, she is amazed at how welcoming it feels.  It feels like a family lives there, loves living there, and loves...

I think that songs by The Beatles could be used in a Romeo and Juliet  soundtrack.  For example, "When I'm Sixty-Four" is about a couple wondering if they will be around for one another when they get older. Such a song would be ideal for Romeo and Juliet, two young people who are wondering if they will last to see the joys of old age or even tomorrow. Another Beatles song, "Yesterday," addresses the feelings of how things look in retrospect. When Romeo is about to take the apothecary's potion and before Juliet takes her own life, both of them could use the song to think about how "yesterday/ love was such an easy game to play/ now I need a place to hide away." "She's Leaving Home" is a song that could convey how Juliet's parents might feel upon realizing the folly of their actions.  This song is about a mother and father lamenting their daughter having left home to pursue her own life. Juliet would fit such a description and the s...

Hello! Yes we can:) All atoms of each type have their specific masses. They are known, particularly in grams per mole. By the definition, one mole of any set of particles contains fixed number of particles, called Avogadro number. It is denoted `N_A` and is equal to about `6 * 10^(23).` The mass of one mole of hydrogen `H` is about `1 (g)/(mol),` the mass of one mole of sulfur `S` is about `32 (g)/(mol)` and the mass of one mole of oxygen `O` is about `16 (g)/(mol).` Therefore the mass of one mole of sulfuric acid `H_2SO_4` is about `2*1 + 32 + 4*16 = 98 (g)/(mol).` Finally, one molecule of sulfuric acid has the mass about `98/N_A approx 98/(6*10^(23)) approx 16.3*10^(-23)=1.63*10^(-22) (g).` So your book says the truth.

The poem describes many different ways of looking at rain and includes perspectives of people in different situations and occupations: a sick man in his room, schoolboys on the street, a farmer in his fields, and finally the Poet (Longfellow likely means himself, and it is noteworthy that "Poet" is capitalized). The descriptions of the rain can be read metaphorically as the ways in which people see the power, beauty, and benefits of the natural world. Just prior to mentioning the farmer, Longfellow describes the patient oxen in the fields and their reaction to the rain, with a suggestion of the rain emanating from the divine: "their large and lustrous eyes seem to thank the Lord, more than man's spoken word." The section about the Poet implies that he can see meanings in the rain that may not be observed by others, such as the farmer. By the poem's end, we understand the rain and the water it delivers to all of the waterways serves as a metaphor for the univ...

Your question is asking for a bit of speculation on the reader’s part. If the internal conflict is removed from Jack Finney’s short story “Contents of the Dead Man’s Pocket” it becomes a man versus elements story. Finney’s purpose with his original story is to demonstrate how Tom’s need for recognition and wealth interfere with his understanding of the importance of his relationship with his wife, and an appreciation for what they already have. If Tom were not dealing with his inner conflict, he may not have gone out on the ledge to retrieve the piece of paper containing his notes for a work project. The project would not hold the same importance for him, so losing it out the window would not be quite as significant. If Tom did go out on the ledge after the paper, the portions of the story that emphasized Tom’s inner feelings would be eliminated, while his physical tribulations would have been highlighted to a greater extent.  The author would have a different focus or theme for the st...

`sum_(n=2)^oo n/(ln (n))` To determine if the series is convergent or divergent, apply the nth-Term Test for Divergence. It states that if the limit of `a_n` is not zero, or does not exist, then the sum diverges. `lim_(n->oo) a_n!=0`      or      `lim_(n->oo) a_n =DNE` `:.` `sum` `a_n`   diverges Applying this, the limit of the term of the series as n approaches infinity is: `lim_(n->oo) a_n` `=lim_(n->oo) n/ln(n)` To take the limit of this, use L’Hospital’s Rule. `=lim_(n->oo) (1)/(1/n)` `=lim_(n->oo) n` `=oo` Therefore, by the nth-Term Test for Divergence, the series diverges.

"The Signal-Man" takes place in a signal box somewhere in England. From the narrator's observations, the reader learns that the signal box is accessed via a "rough zigzag" path which passes through a cutting (an open passage for a railway). This cutting is very damp and dark and rarely receives any sunlight, as the narrator comments:  So little sunlight ever found its way to this spot, that it had an earthy, deadly smell. Beyond the cutting, the box itself is an office which contains the basic items needed to carry out this occupation, including a desk and a fire. The box is noted for being small and dark.  By describing the setting in this manner, Dickens creates a gloomy and oppressive atmosphere. Moreover, by employing certain words and phrases, like "I had left the natural world," Dickens suggests that the signal box is its own world, distinct from the world above in which people, like the narrator, exist. By creating this other world, Dickens adds...

"The Wasteland" takes place in a South African town at night. More specifically, the setting is a lot with old cars, wire, and iron. The story takes place during apartheid, the system of legal racial segregation that was in place in South Africa from 1948 to 1994. The protagonist, an old man pursued by robbers, must escape to this lot because the only other place in sight is a convent, which has been barred to the public. In this lot, the old man at first runs into wire and cars, and the lot is so dark that he cannot see his attackers. In this darkness, he hits one of the men pursuing him with a stick. He then has the insight to hide beneath a truck, where he realizes that the man he has killed is his son, who urged the other men to attack his own father.

`int_1^2 (4y^2-7y-12)/(y(y+2)(y-3))dy` To solve this, apply partial fraction decomposition.  When converting the integrand to sum of proper rational expressions, set the equation as follows: `(4y^2-7y-12)/(y(y+2)(y-3))= A/y + B/(y+2)+C/(y-3)` Multiply both sides by the LCD. `4y^2-7y-12=A(y+2)(y-3)+By(y-3)+Cy(y+2)` `4y^2-7y-12=Ay^2-Ay-6A+By^2-3By+Cy^2+2Cy` `4y^2-7y-12=(A+B+C)y^2+(-A-3B+2C)Y-6A` For the two sides to be equal, the two polynomials should be the same. So set the coefficients of the polynomials equal to each other. y^2: `4=A+B+C`     (Let this be EQ1.) y: `-7=-A-3B+2C`     (Let this be EQ2.) Constant: `-12=-6A`     (Let this be EQ3.) To solve for the value of A, consider EQ3. `-12=-6A` `2=A` Plug-in this value of A to EQ1. `4=A+B+C` `4=2+B+C` `2=B+C` Then, isolate the C. `2-B=C` Plug-in this expression and the value of A to EQ2. `-7=-A-3B+2C` `-7=-2-3B+2(2-B)` `-7=-2-3B+4-2B` `-7=2-5B` `-9=-5B` `9/5=B` And plug-in the value of A and B to EQ1. `4=A+B+C` `4=2+9/5+C` `4=19/5+C`...

The protagonist of The Ponder Heart is Edna Earle, who has to manage the family's hotel and prevent her eccentric Uncle Daniel from giving away what remains of the family's fortune. As she says, "It's always taken a lot out of me, being smart" (10). While Edna Earle loves many aspects of her uncle, Daniel, and calls him the "sweetest, most unspoiled thing in the world" (11), Daniel is the antagonist of the novella. Working in opposition to Edna Earle, he gives away the family's possessions without restraint, including trips to Memphis, a string of hams, a pick-up truck, and even a heifer. Eventually, Daniel's largesse becomes so unrestrained that Edna Earle and Daniel's father, Sam, try to have Daniel put in an institution. They later coerce Daniel into marrying a wealthy widow, but Daniel foils Edna Earle's plans at every step.

You can find the names of the two feuding families very early in the short story “The Interlopers.”  The two families are the von Gradwitz family and the Znaeym family. These two families have been feuding with one another for a few generations now.  The two protagonists in the story are the current heads of the families.  They are continuing a feud that began in the days of their grandfathers.  In those days, their grandfathers engaged in a lawsuit with one another regarding the control of a certain parcel of land.  Both men claimed that it was theirs, but the court awarded the land to the von Gradwitz family. Ever since, the two families have been feuding, though the feud has “come to a head” in recent times. The answer, then, is that the two families are the von Gradwitz family and the Znaeym family.

To apply the  Root test  on a series ` sum a_n` , we determine the limit as: `lim_(n-gtoo) root(n)(|a_n|)= L` or `lim_(n-gtoo) |a_n|^(1/n)= L` Then, we follow the conditions: a) `Llt1` then the series is  absolutely convergent . b) `Lgt1` then the series is  divergent . c) `L=1` or  does not exist   then the  test is inconclusive . The series may be divergent, conditionally convergent, or absolutely convergent. To apply the  Root Test  to determine the convergence or divergence of the  series  `sum_(n=0)^oo e^(-3n)` , we  let `a_n = e^(-3n)` . Apply Law of Exponent: `x^(-n) = 1/x^n` .  `a_ n = 1/e^(3n).` Applying the Root test, we set-up the limit as: `lim_(n-gtoo) |1/e^(3n)|^(1/n) =lim_(n-gtoo) (1/e^(3n))^(1/n)`  Apply the Law of Exponents: `(x/y)^n = x^n/y^n` and `(x^n)^m= x^(n*m)` . `lim_(n-gtoo) (1/e^(3n))^(1/n) =lim_(n-gtoo) 1^(1/n)/(e^(3n))^(1/n)`                            ` =lim_(n-gtoo) 1^(1/n)/e^(3n*1/n)`                           `=lim_(n-gtoo) 1^(1/n)/e^((3n)/n) `          ...

As Juliet awaits Romeo, she muses on love and lovers, wishing for night to fall so that Romeo can safely ascend to her bedroom and they can spend the night together. As Juliet eagerly anticipates Romeo's arrival, she considers what it will be like to lose her virginity to her husband, thinking of how handsome he is.  She says, in part, "Oh, I have bought the mansion of a love, / But not possessed it, and though I am sold, / Not yet enjoyed" (3.2.28-30). What she means is that she begun the process of marriage, but she has not completed it. Though she belongs to Romeo now, he hasn't made her his yet. In other words, Juliet is married, but she and Romeo have not consummated their marriage yet. Via metaphor, she compares love to a mansion, a luxurious and beautiful home, but she has not moved into it yet.  Although Juliet no longer belongs to her father, having been "sold" to her husband, the metaphorical transaction is not complete until Romeo has "enjoy...

Some historians might argue that the United States should have been a more progressive country in the early twentieth century, as domestically, many Presidents were moving towards a more progressive agenda. For example, Theodore Roosevelt used the Sherman Anti-Trust Act to break apart monopolies and instituted federal legislation such as the Hepburn Railroad Act and Pure Food and Drug Act to make railroad rates fairer and to regulate the quality of food and drugs, respectively. However, internationally, Roosevelt's agenda was one of enforcing American imperialism on other countries. In 1898 (slightly before the period in the question), Roosevelt, then Assistant Secretary of the Navy, ordered Admiral Dewey to attack the Philippines if war broke out between Spain the United States (which happened after the explosion of the U.S. battleship the Maine in Havana harbor). This was the outbreak of the Spanish-American War, which led the U.S. to institute the Platt Amendment in Cuba. This l...