Calculus of a Single Variable, Chapter 7, 7.1, Section 7.1, Problem 70

Given ,
y = 9-|x| , y = 0
first let us find the total area of the bounded by the curves.
so we shall proceed as follows
as given ,
y = 9-|x| , y = 0
=> 9-|x|=0
=> |x| -9 =0
=> |x|=9
so x=+-9

the the area of the region is = int _-9 ^9 (9-|x| -0) dx
=>int _-9 ^0 (9+x -0) dx+int _0 ^9 (9-x -0) dx
=>[9x+x^2 /2]_-9 ^0 + [9x-x^2/2]_0 ^9
=>[0]-[-81+81/2] +[81-81/2]-[0]
=>81/2 +81/2 =81
So now we have to find the horizonal line that splits the region into two regions with area 81/2
as when the line y=b intersects the curve y=9-|x| then the area bounded is 81,so
let us solve this as follows
first we shall find the intersecting points
as ,
9-|x|=b
|x|= 9-b
x=+-(9-b)
so the area bound by these curves y=b and y=9-|x| is as follows
A= int _-(9-b) ^(9-b) (9-|x|-b)dx = 81/2
=>int _-(9-b) ^0 (9+x-b)dx +int _0 ^(9-b) (9-x-b)dx =81/2
=>[9x+x^2/2-bx]_-(9-b) ^0 +[9x-x^2/2-bx]_0 ^(9-b) = 81/2
=>[0]-[9(-(9-b))+(-(9-b))^2 /2-b(-(9-b))]+
[9((9-b))-((9-b))^2/2-b((9-b))]-[0]=81/2
=> [9((9-b))-((9-b))^2/2-b((9-b))]
-[9(-(9-b))+(-(9-b))^2/2 -b(-(9-b))]=81/2
let t= 9-b
so
=> [9(t)-(t)^2/2 -b(t)] -[9(-t)+(-t)^2/2 -b(-t)]=81/2
=>[9t-t^2/2 -bt]+[9t-t^2/2 -bt]=81/2
=>18t-t^2-2bt =81/2
but we know half the Area of the region between y=9-|x|,y=0 curves =81/2
so now ,
18t-t^2-2bt =81/2
18t-2bt-t^2 = 81/2
=>t(18-2b)-t^2=81/2
=> t^2=t(18-2b)-81/2
=>t^2 -t(18-2b)+81/2=0
this is like quadratic equation
ax^2+bx+c=0
so t = (-b+-sqrt(b^2-4ac))/2a
=(18-2b+-sqrt((-(18-2b))^2-4*(81/2)))/2
but
t=9-b
so,
9-b=(18-2b+-sqrt((-(18-2b))^2-4*(81/2)))/2
=> 18-2b=(18-2b+-sqrt((-(18-2b))^2-4*(81/2)))
=>+-sqrt((-(18-2b))^2-4*(81/2))=0
=>sqrt((-(18-2b))^2-4*(81/2))=0
=>(-(18-2b))^2-4*(81/2)=0

=>(-(18-2b))^2=4*(81/2)
=>(-(18-2b))^2=2*(81)
=>(-(18-2b))= +- sqrt(2) *9
=> -18+2b=+-9sqrt(2)
=>2b=+-9sqrt(2)+18
=>b=(18+-9sqrt(2))/2
so b= (18+-9sqrt(2))/2

I need assistance answering an Intro to Statistics question about interpreting scatterplots. I am including a pdf screenshot of the question (4.5 Weight Loss) and the scatterplot for your reference. I think I can properly describe the trend, shape, and strength of association, but the question requires that I indicate what the negative trend means. I don’t understand how to answer that part.

Statistics is one of the most challenging, and most valuable, fields of mathematics because it is a tool for directly understanding some of the most complicated problems that the world can throw at us. In this case, when the question asks you what the trend 'means,' it's not asking for a mathematical formula or rule; it's asking for you to describe a real-world relationship that appears to be represented mathematically.
The correlation in the graph is negative. That means that as one axis increases in value, the value on other axis tends to decrease. And what are the axes? Well, the x-axis represents the students' current weight, and the y-axis represents their desired weight change--positive for gain, negative for loss. 
So what we need to do is translate that relation into plain English. What does it mean that, as students initial weight increases, their desired weight change becomes more and more negative? It means something that's actually quite intuitive: that heavier people want to lose more weight.

The negative trend in this scatterplot indicates that the two variables, weight and desired weight change, are negatively correlated. That means as a student's weight goes up, his or her desired weight change goes down (meaning that it is a negative number). If you draw a line of best fit through the data (meaning a straight line that approximately goes through the center of the data), students who weight 100 pounds will want to gain about 10 pounds (positive ten), while students who weigh about 130 will not want to lose any weight (meaning a value of 0). As students' weights increase, their desired weights become negative because they want to lose weight. For example, a student who weighs 180 pounds will have a desired weight change of approximately negative 40. Therefore, as a student's weight increases, the desired weight change becomes increasingly negative, meaning that the variables are negatively correlated and there is a negative trend in the data. 

Where is the Taganac River?

The real Taganac River is a river in the Philippines. The exact location is north latitude 8 degrees, 22 minutes, and 50.3 seconds. The longitude is west 117 degrees, 14 minutes, and 40.88 seconds. This might not be the Taganac River in the story. The story doesn't give readers enough setting details for us to confirm this location, and details of the river's geographic location are not given in the story. What we do know is that the Taganac River flows fairly close to a school and has a waterfall within a mile of Morse's house. The school isn't the only building close to the river either. About halfway through the story, the narrator tells readers that the town "spent a mint" on the riverfront. The town is right up against the river, and the river now is next to a nail salon and a cafe.

y = 1/2 (xsqrt(4-x^2) + 4arcsin(x/2)) Find the derivative of the function

The derivative of y in terms of x is denoted by  d/(dx)y or y' .
 For the given problem: y =1/2[xsqrt(4-x^2)+4arcsin(x/2)] , we apply the basic derivative property:
d/(dx) c*f(x) = c d/(dx) f(x) .
Then,
d/(dx)y =d/(dx) 1/2[xsqrt(4-x^2)+4arcsin(x/2)]
y’ =1/2 d/(dx) [xsqrt(4-x^2)+4arcsin(x/2)]
 
Apply the basic differentiation property: d/(dx) (u+v) = d/(dx) (u) + d/(dx) (v)
y’ =1/2[d/(dx) (xsqrt(4-x^2))+ d/(dx) (4arcsin(x/2))]
 
For the derivative of d/(dx) (xsqrt(4-x^2)) , we apply the Product Rule: d/(dx)(u*v) = u’*v =+u*v’ .
d/(dx) (xsqrt(4-x^2))= d/(dx)(x) *sqrt(4-x^2)+ x * d/(dx) (sqrt(4-x^2))
 
Let u=x then u'= 1
    v= sqrt(4-x^2) then v' =-x/ sqrt(4-x^2)
Note: d/(dx) sqrt(4-x^2) = d/(dx)(4-x^2)^(1/2)
Applying the chain rule of derivative:
d/(dx)(4-x^2)^(1/2)= 1/2(4-x^2)^(-1/2)*(-2x)
                      =-x(4-x^2)^(-1/2)
                    =-x/(4-x^2)^(1/2)  or - –x/sqrt(4-x^2)
 Following the Product Rule, we set-up the derivative as:
d/(dx)(x) *sqrt(4-x^2)+ x * d/(dx) (sqrt(4-x^2))
= 1 * sqrt(4-x^2)+ x*(-x/sqrt(4-x^2))
= sqrt(4-x^2)-x^2/sqrt(4-x^2)
 Express as one fraction:
sqrt(4-x^2)* sqrt(4-x^2)/ sqrt(4-x^2)-x^2/sqrt(4-x^2)
=( sqrt(4-x^2))^2/ sqrt(4-x^2) –x^2/sqrt(4-x^2)
=( 4-x^2)/ sqrt(4-x^2) –x^2/sqrt(4-x^2)
=( 4-x^2-x^2)/ sqrt(4-x^2)
=( 4-2x^2)/ sqrt(4-x^2)
 
Then, d/(dx) (xsqrt(4-x^2))= ( 4-2x^2)/ sqrt(4-x^2)
 
For the derivative of d/(dx) (4arcsin(x/2)) , we apply the basic derivative property: d/(dx) c*f(x) = c d/(dx) f(x) .
d/(dx) (4arcsin(x/2))= 4 d/(dx) (arcsin(x/2))
Apply the basic derivative formula for inverse sine function: d/(dx) (arcsin(u))= (du)/sqrt(1-u^2) .
Let u =x/2 then du=1/2
4d/(dx) (4arcsin(x/2))]= 4*(1/2)/sqrt(1-(x/2)^2)
                    = 2/sqrt(1-(x^2/4))
                     =2/sqrt(1*4/4-(x^2/4))
                      = 2/sqrt((4-x^2)/4)
                     = 2/ (sqrt(4-x^2)/sqrt(4))
                   =2/ (sqrt(4-x^2)/2)
                   =2*2/sqrt(4-x^2)
                   =4/sqrt(4-x^2)
 
 
Combining the results, we get:
y' = 1/2[d/(dx) (xsqrt(4-x^2))+ d/(dx) (4arcsin(x/2))]
=1/2[( 4-2x^2)/ sqrt(4-x^2)+4/sqrt(4-x^2)]
=1/2[( 4-2x^2+4)/ sqrt(4-x^2)]
=1/2[( -2x^2+8)/ sqrt(4-x^2)]
=1/2[( 2(-x^2+4))/ sqrt(4-x^2)]
=(-x^2+4)/ sqrt(4-x^2)]
or y'=(4-x^2)/ sqrt(4-x^2)]
 
Applying Law of Exponents:   x^n/x^m= x^n-m :
y' =(4-x^2)/ sqrt(4-x^2)
=(4-x^2)^1/ (4-x^2)^(1/2)
=(4-x^2)^(1-1/2)
=(4-x^2)^(1/2)
Final answer:
y'=(4-x^2)^(1/2)
 or
y'=sqrt(4-x^2)

In O. Henry's short story "A Retrieved Reformation," why did Jimmy Valentine stop burglaries?

Jimmy Valentine decided to give up his life of crime because he falls in love with a beautiful, small-town girl named Annabel Adams. He knows such a perfect girl will not return his love if she knows how he makes his living. 
As the story opens, Jimmy is serving time in prison for one of his latest safecracking jobs. 

He had been sent to prison to stay for four years. He had been there for ten months. But he had expected to stay only three months. Jimmy Valentine had many friends outside the prison. A man with so many friends does not expect to stay in prison long.

Jimmy hardly listens to the warden when he receives the pardon the governor signed reluctantly.

“Valentine,” said the chief prison officer, “you’ll go out tomorrow morning. This is your chance. Make a man of yourself. You’re not a bad fellow at heart. Stop breaking safes open, and live a better life.”

Jimmy's friends find it increasingly difficult to help him, as he learns from one friend.
Jimmy begins to sense his career of crime isn't as easy or lucrative as he thought it would be. Ben Price, the bank detective, is always on his trail. He is building a prison record. He is getting older. He is becoming notorious as the best safecracker in the business. It looks as though it will become easier and easier for Jimmy to be sent to prison and harder and harder for him to get out. He is in danger of becoming an incorrigible recidivist and growing old behind bars.
Jimmy decides to move out of his usual area of operations in Indiana and setting up a "front" as owner and operator of a shoe business in far-away Elmore, Arkansas. That is where he falls in love.

A young lady walked across the street, passed him at the corner, and entered a door. Over the door was the sign, “The Elmore Bank.” Jimmy Valentine looked into her eyes, forgetting at once what he was. He became another man.

Annabel is an inspiration. She falls in love with Jimmy, too, and he wants to be worthy of her love. Annabel symbolizes everything that has been missing in Jimmy's criminal life. She is the catalyst for his reformation. She is meant to symbolize everything a man can obtain by living a moral life and what he is missing by committing crimes.

int (3-x) / (3x^2-2x-1) dx Use partial fractions to find the indefinite integral

int(3-x)/(3x^2-2x-1)dx
Let's use partial fraction decomposition on the integrand,
(3-x)/(3x^2-2x-1)=(3-x)/(3x^2+x-3x-1)
=(3-x)/(x(3x+1)-1(3x+1))
=(3-x)/((3x+1)(x-1))
Now form the partial fractions using the denominator,
(3-x)/((3x+1)(x-1))=A/(3x+1)+B/(x-1)
Multiply equation by the denominator (3x+1)(x-1)
=>(3-x)=A(x-1)+B(3x+1)
=>3-x=Ax-A+3Bx+B
=>3-x=(A+3B)x+(-A+B)
comparing the coefficients of the like terms,
A+3B=-1   ----------------(1)
-A+B=3      ----------------(2)
Now let's solve the above equations to get A and B,
Add the equations 1 and 2,
4B=-1+3
4B=2
B=2/4
B=1/2
Plug in the value of B in equation 1,
A+3(1/2)=-1
A+3/2=-1
A=-1-3/2
A=-5/2
Plug in the value of A and B in the partial fraction template,
=(-5/2)/(3x+1)+(1/2)/(x-1)
=-5/(2(3x+1))+1/(2(x-1))
So, int(3-x)/(3x^2-2x-1)dx=int(-5/(2(3x+1))+1/(2(x-1)))dx
Apply the sum rule,
=int-5/(2(3x+1))dx+int1/(2(x-1))dx
Take the constant out,
=-5/2int1/(3x+1)dx+1/2int1/(x-1)dx
Now let's evaluate both the above integrals separately,
int1/(3x+1)dx
Apply integral substitution:u=3x+1
=>du=3dx
=int1/u(du)/3
Take the constant out,
=1/3int1/udu
Use the common integral:int1/xdx=ln|x|
=1/3ln|u|
Substitute back u=3x+1
=1/3ln|3x+1|
Now evaluate the second integral.
int1/(x-1)dx
Apply integral substitution: u=x-1
du=1dx
=int1/udu
Use the common integral:int1/xdx=ln|x|
=ln|u|
Substitute back u=x-1
=ln|x-1|
int(3-x)/(3x^2-2x-1)dx=-5/2(1/3ln|3x+1|)+1/2ln|x-1| 
Simplify and add a constant C to the solution,
=-5/6ln|3x+1|+1/2ln|x-1|+C
 

What are some specific examples of deception in Othello that are not said by Iago?

There are certainly some instances of deception that are not perpetuated by Iago.
In Act III, Scene 4, Desdemona has lost her handkerchief and wonders aloud about its whereabouts. Emilia says she doesn't know where Desdemona's handkerchief could be. This is an act of deception on Emilia's part; she was the one who picked up Desdemona's handkerchief when Othello let it drop in Act III, Scene 3. After picking up Desdemona's handkerchief, Emilia takes it to Iago, who proceeds to plant it in Cassio's room. Iago's plan was to use Desdemona's handkerchief to frame Cassio.
Later, when Othello meets Desdemona and asks her to lend him her handkerchief, Desdemona tries to deceive him. First, she hands him a different handkerchief than the one for which he asks. When Othello notes she handed him the wrong handkerchief, Desdemona says she doesn't have his special handkerchief on her person. Othello isn't pleased to hear this, and he tells Desdemona the handkerchief is actually imbued with a special kind of magic, so it's important she never loses it.
Othello contends that the handkerchief was a gift from a witch to his mother and was the means by which his mother held the love of his father. So, he warns Desdemona to keep close tabs on the handkerchief. The implication is that, as long as she has that handkerchief, Othello will always love her. Upon hearing this, Desdemona becomes visibly agitated (she knows she misplaced the handkerchief and can't produce it), and this causes Othello to suspect she lost the handkerchief. The couple gets into a heated quarrel.
Desdemona tries to argue that the handkerchief is not lost, a deception on her part. Then, after continued demands from Othello, she tries to draw his attention away by speaking up on Cassio's behalf. This actually makes things worse for her, but poor Desdemona has no way of knowing this. Meanwhile, with each deflection by Desdemona, Othello becomes more and more convinced his wife is hiding an affair with Cassio, and he storms out in frustration. Now, Desdemona deceives Othello because she doesn't want to hurt his feelings. Ironically, her kind heart (which also leads her to speak up on behalf of Cassio) paves the path to her destruction.
Hope this helps!

College Algebra, Chapter 8, 8.4, Section 8.4, Problem 20

Find an equation for the conic whose graph is shown.


The ellipse $\displaystyle \frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1$ has center on $(h, k)$ and has vertical transverse, the length of its major axis is $2a$ while the length of its minor axis is $2b$. Based from the graph the ellipse has center on $(2, -3)$ and the length if the major axis and minor axis is $6$ and $4$ respectively. Thus, $2a = 6$ and $2b = 4$, this gives us $a = 3$ and $b = 2$. Therefore, the equation is $\displaystyle \frac{(x - 2)^2}{2^2} + \frac{(y - (-3)^2)}{3^2} = 1$

$\displaystyle \frac{(x - 2)^2}{4} + \frac{(y + 3)^2}{9} = 1$

What is the function of the description of the "Great Irish Elk" skeleton and the butter?

The Great Irish Elk and butter description serves to illustrate the archaeological value of the bog in terms of Irish identity, culture, and history.
In the poem, the reference to the Great Irish Elk also highlights Heaney's own childhood memory of an elk fossil found in a bog near his hometown. To Heaney, the archaeological find is evidence that the bog is a repository of memories gleaned from centuries of Irish culture and history. The Great Irish Elk is at present time extinct; however, skeletons and fossils of these elks have been found in numerous locations around the world. The elk skeleton serves to underline the vastness and depth of Irish history; centuries upon centuries of Irish history lie in the bogs of Heaney's poem.
Additionally, the preserved butter in the poem ("sunk under/ More than a hundred years/...recovered salty and white") references the social practice of preserving or storing food in peat bogs for extended periods of time. It is remarkable that the hundred-year old butter is so well preserved that it emerges "white" from the bog.
So, Heaney uses the two items to describe how bogs are able to preserve generations of rich Irish history within the depths of their "kind, black butter."

Consider Fitzgerald's use of symbolism. Choose one symbol, and analyze it in detail.

One significant symbol in The Great Gatsby is the billboard, featuring the eyes of Doctor T.J. Eckleburg. The billboard is first mentioned in chapter 2, seemingly just an addition to the in-depth description of the gloomy and repressive area of the valley of ashes. Fitzgerald writes, “The eyes of Doctor T.J. Eckleburg are blue and gigantic…they look out of no face, but, instead, from a pair of enormous yellow spectacles, which pass over a non-existent nose” (Fitzgerald 23). At this point, the reader is aware of the strangeness of the valley of ashes, and it seems that these gigantic eyes merely add to the imagery of the lost and forgotten place located between the West Egg and New York.
However, as the novel continues, the image begins to evolve into an important symbol of an omniscient, powerful force, such as God, watching and judging the sinful actions of these characters.
The second mention of the billboard occurs when Tom and Nick pick up Myrtle from Wilson’s garage. Tom, looking around the room with distaste, exchanges a frown with Doctor Eckleburg (Fitzgerald 26). This moment reveals the billboard as a character with the ability to interact with others and evaluate situations. Later, Nick comments that Doctor Eckleburg was keeping a vigil over the valley of ashes (Fitzgerald 124). “Vigil” is a word heavy with religious connotations and is indicative of the watchfulness kept by Doctor Eckleburg over the valley of ashes.
By the end of the novel, the full revelation of the significance of this symbol is shown. After Myrtle’s death, Wilson points to the billboard and declares, “God sees everything” (Fitzgerald 160).
Throughout the novel, these characters are dishonest, adulterous, judgmental, and, by the end, murderous people. Although some characters, Tom and Daisy specifically, emerge from the events of the novel without serious punishment, the reader can trust that God, through the eyes of Doctor T.J. Eckleburg, has witnessed everything and that they will receive proper punishment in time.

One of the most recognizable, prominent symbols throughout the novel is the green light situated at the end of Daisy's dock, which is across from Gatsby's home. Jay Gatsby ritually stands at the edge of his lawn and gazes at the green light across the water. At the end of chapter 1, Nick Carraway witnesses Gatsby standing with his arms outstretched staring in the direction of the green light coming from Daisy's dock. The green light symbolizes Gatsby's hopes and dreams. Gatsby's goal is to win Daisy's affection and eventually live a fulfilling life with her, which is what motivates him to become wealthy and successful. His obsession with staring at the green light represents his hope in a future with Daisy. In chapter 5, Gatsby reintroduces himself to Daisy and shows her around his mansion, which impresses her. When Jay Gatsby believes he has attained his goal of winning Daisy's affection, Nick says,

Compared to the great distance that had separated him from Daisy it had seemed very near to her, almost touching her. It had seemed as close as a star to the moon. Now it was again a green light on a dock. His count of enchanted objects had diminished by one (Fitzgerald, 56).

In addition to symbolizing Gatsby's hope in a future with Daisy, the green light also represents everything that Gatsby craves, particularly money. The light connotes money, and the color green also holds symbolic significance throughout the novel. At the end of the novel, Nick reminisces about Gatsby staring at the green light dreaming of one day attaining his goals. Unfortunately, Gatsby was too naive to realize that his goals were unattainable. Nick ends the novel by saying,

Gatsby believed in the green light, the orgastic future that year by year recedes before us. It eluded us then, but that’s no matter—to-morrow we will run faster, stretch out our arms farther. . . . And one fine morning— (Fitzgerald, 87).

Why is war the way to settle a dispute?

Though war is almost universally frowned upon by most civilized nations, this doesn't change the fact that many of these nations have been—or are currently involved in—some sort of violent conflict.
There are many who oppose war without reservation; however, others feel that war is a means to an end when negotiations and other peace-keeping efforts fail. Most free countries look at war as a means of preserving their own freedoms and ways of life. Once a country feels that its rights are infringed upon, the country becomes more likely to engage in a violent conflict to protect its interests.
Some have argued that war ends up preserving more lives than it costs. It has been argued that war allows for the separation of societies and people who otherwise would not get along. Instead of forcing these people to live together, war allows for like-minded people to live in separate societies. 
Others argue that war allows for nations to better control their own financial interests. This ultimately leads to more prosperity and supposedly more happiness in the countries who are victorious at keeping outside influences out of their countries.
https://fee.org/articles/why-war/

What was Encolpius' purpose in Satyricon by Petronius?

Satyricon, written by Gaius Petronius of Nero's court and narrated by Encolpius, is a fragment. The section featuring Encolpius' adventures is fragmented, with no ending: "the exit of the two principal characters is not fixed at the time our fragments come to an end" (W. C. Firebaugh, Translator). Consequently, there is some difficulty in attributing purpose to Encolpius' narrative and actions.If it is true, however, that Petronius' story had a deep and lasting influence on literature, as Firebaugh posits ("[Satyricon's] powerful influence upon the literature of the world"), then we ought to be able to borrow a literary model and a Shakespearean technique to overlay what we do have in order to discover a purpose attributable to Encolpius.The literary model is that of laying out in the early pages of a work the key elements that will take shape through the course of the narrative. The Shakespearean technique is that of putting wisdom into the lines of seemingly insignificant characters and speeches: Shakespearean Clowns and Fools often carry the lines, in seemingly insignificant speeches, that establish key elements of plot, characterization and theme; King Lear's Fool comes readily to mind.By using this approach—by borrowing the model and technique as an overlay—we can say that Agamemnon's recitation of verse in the fifth paragraph (called Chapter the Fifth)—being a foundational speech early in the fragment and being seemingly insignificant verse—holds the clue to both the direction of the plot development and to Encolpius' purpose.We find in Agamemnon's recitation that he warns young men against "riff-raff," "evil companions" in dining and drinking, "poetry" and "sirens." We go on to find through Encolpius' narration that these are precisely the things that Encolpius, Ascyltos and Giton pursue, engage in and run from in their adventures. We also find in Encolpius' opening speech that he decries false rhetoric of "empty discord" and admonishes a "dignified ... a chaste, style" that "rises supreme by its own natural purity" (Chapter the Second), which he associates with the learning and wisdom of Sophocles, Euripides, Plato and Demosthenes.When we put these two together—Agamemnon warning against the things that deter a young man from acquiring clear thinking and Encolpius admonishing "supreme" rhetoric—we can deduce that a reasonable purpose attributable to Encolpius would be that of illustrating the need for pursuit of wisdom, such as Agamemnon admonishes, through the pursuit of clear thinking expressed in "supreme" rhetoric, such as Encolpius admonishes.
If this suggested purpose proves plausible, then we might expect the story would have ended, being a satirical comedy, with Encolpius, Ascyltos and Giton returning to where they began, sadder but wiser young men, having indulged in all Agamemnon warned against and now newly devoted to the right course, that which was followed by Socrates, Demosthenes and Cicero.

Agamemnon: ...later, [after] the loreOf Socrates' school he has mastered, the reins let him fling,And brandish the weapons that mighty Demosthenes bore.Then, steeped in the culture and music of Greece, let his tasteBe ripened and mellowed by all the great writers of Rome.At first, let him haunt not the courts; let his pages be gracedBy ringing and rhythmic effusions composed in his home [not to the Court] [...]In eloquent words such as undaunted Cicero chose.Come! Gird up thy soul! Inspiration will then force a ventAnd rush in a flood from [the] heart... (Chapter the Fifth)

What is the theme of "Men of England"?

The theme of "Men of England" is that the majority of the workforce in England is being exploited. The poem is a call to action to the ordinary worker to stop allowing himself to be exploited and to work for his own profit instead of benefitting someone else. The poem focuses on the conflict between the upper class and the working class. It recognizes the difficulty and backlash that revolution creates, but it warns the working class that if they do not throw off this class oppression, they are working to create their own graves.
The first five stanzas of the poem point out all the inequities that exist in the British social system. The first five stanzas have a series of questions and statements like "Wherefore weave with toil and care / The rich robes your tyrants wear?" (lines 3–4). Each instance in the first five stanzas points out that the working class performs the labor, but the rich profit from it.
The sixth stanza tells the men what to do and gives them a call to action. It says,

Sow seed—but let no tyrant reap:
Find wealth—let no imposter heap:
Weave robes—let not the idle wear:
Forge arms—in your defence to bear (lines 21–24).


The seventh stanza recognizes the difficulty of revolution, in the lines "Why shake the chains ye wrought? Ye see / The steel ye tempered glance on ye" (lines 27–28). The narrator recognizes here that if you shake chains that you yourself made, they might hurt you in the backlash. However, the poem concludes that if the working class does not throw off its oppression, "fair / England be your Sepulchre" (lines 31–32).
https://www.poetryfoundation.org/poems/52304/a-song-men-of-england

y'-y = 16 Solve the first-order differential equation

y' - y = 16
To solve, rewrite the derivative as dy/dx .
(dy)/dx - y = 16
Then, express the equation in the form N(y)dy = M(x) dx .
(dy)/dx = y+16
(dy)/(y+16) = dx
Take the integral of both sides.
int (dy)/(y+16) = int dx
For the left side of the equation, apply the formula int (du)/u = ln|u|+C .
And for the right side, apply the formula int adx =ax + C.
ln |y+ 16| + C_1 = x + C_2
Then, isolate the y. To do so, move the C1 to the right side.
ln|y+16| = x + C_2-C_1
Since C1 and C2 represent any number, express it as a single constant C.
ln|y+16| = x + C
Then, convert this to exponential equation.
y+16=e^(x+C)
And, move the 16 to the right side.
y = e^(x+C) - 16
 
Therefore, the general solution is y = e^(x+C)-16 .

Intermediate Algebra, Chapter 2, 2.7 summary exercises, Section 2.7, Problem 10

Evaluate the equation $|3x - 7| - 4 = 0$. Then give the solution in set notation.

$
\begin{equation}
\begin{aligned}
|3x - 7| - 4 &= 0\\
\\
|3x - 7| &= 4 && \text{Add 4 on each side}
\end{aligned}
\end{equation}
$


By using the property of Absolute value, we have

$
\begin{equation}
\begin{aligned}
3x - 7 &= 4 && \text{and} & 3x - 7 &= -4\\
\\
3x &= 11 && \text{and} & 3x & = 3 && \text{Add 7 on each side}\\
\\
x &= \frac{11}{3} && \text{and} & x &= 1 && \text{Divide each side by $3$ and solve for $x$.}
\end{aligned}
\end{equation}
$



Thus, the solution set is $\displaystyle \left\{ \frac{11}{3}, 1 \right\}$

In John Keats's ode "To Autumn," who are the "bosom-friends"? Why are they "conspiring"?

Keats's ode "To Autumn" is an apostrophe, which means it directly addresses someone or something absent or merely rhetorical. In this case, as the title of the poem indicates, the speaker addresses the season of autumn:

Season of mists and mellow fruitfulness,Close bosom-friend of the maturing sun (ll.1-2)

The image describes the close friendship of autumn and the sun that has helped the crops to grow; together, they have "conspired" to produce the abundant harvest detailed in the rest of the first stanza.
In establishing a personal relationship between inanimate entities, Keats introduces the personification of autumn that will run throughout the poem. Indeed, his use of apostrophe already endows the season with a degree of humanity, but it is enhanced in later stanzas as we see him sitting on the floor amidst the grain, napping in a meadow, and engaging in various harvest-time tasks. These actions depict autumn as a productive figure who can nevertheless stop to appreciate the beauty around him in the "winnowing wind" (l.15) or the "fume of poppies" (l.17).
This picture of a vibrant individual seems meant to contradict the hint of anxiety the speaker feels at the waning of the year that autumn brings, and the coming of winter, the metaphorical season of death. This hint is already present in the "maturing" sun of the second line - autumn and his best friend are not exactly young anymore. Yet the speaker urges autumn, "Think not of [the songs of spring], thou hast thy music too" (l.24). Although autumn is not the time of new birth symbolized by spring, the poem insists on the value of its different beauty. This assertion is perhaps especially poignant considering that this was one of the last poems Keats wrote before his early death of tuberculosis. So near the end of his life, the poem articulates the beauty and richness still to be found in maturity and endings.

What is the significance of Wounded knee? Where is it?

Wounded Knee refers to a location in the state of South Dakota as well as a massacre which occurred there in the year of 1890. There is a body of water called Wounded Knee Creek in what is now known as the Lakota Pine Ridge Indian Reservation, but prior to the establishment of Reservations in the United States, this was territory of the Oglala Nation. 
In 1889, the Pine Ridge Reservation was established out of territory previously part of the Great Sioux Reservation. There was much conflict surrounding the founding of reservations, and many First Nations (American Indian) people were slaughtered or uprooted from their homes. Lands were seized by the United States government and the American Bison (a very important resource for the First Nations) were killed in such large numbers by American settlers that they almost went extinct. Reservations were intended to be spaces where First Nations people could maintain their traditional lifeways independent from the United States government. Unfortunately, the government was the entity appointing the location of the reservations and forcing First Nations people to move onto them. 
A religious and political movement was founded in this time by the Paiute Nation (in the region of present-day Nevada) prophet Wovoka. Wovoka had a vision that the Christian messiah Jesus Christ would return to Earth as First Nations man, and believed that if every First Nations person were to practice the Ghost Dance, all of the white settlers would leave First Nations land and the reincarnated Christ would restore these territories to splendor. Wovoka's vision included the promise that traditional resources like the Bison would be restored and the spirits of dead ancestors would be risen from the grave to teach the living.
The Ghost Dance movement gained a following, and the last Ghost Dances were held on the Pine Ridge Reservation in 1890. The U.S. government wanted to suppress the practice of Ghost Dancing and tried to make negotiations with First Nations chiefs to prevent people from practicing the dance. When this failed to play out as the government had hoped, they resorted to coercion by violence. The government intended to surround disarm the Lakota people on the reservation so as to prevent any further uprising. On the 29th of December, during rounds to seize weapons from the Lakota camp, a deaf man named Black Coyote refused to give up his rifle. In the struggle to take Black Coyote's rifle from him, the gun went off and fighting broke out. 
After the initial, accidental gunshot, both United States troopers and Lakota men opened fire. Though women and children tried to take shelter in camp or flee the gunfire, the Lakota camp was surrounded and the United States soldiers fired directly into the camp. In the span of less than an hour, hundreds of First Nations people were killed, with estimates ranging between 150 and 300. Survivors were forced to move to the reservation. On the side of the United States, 25 men were killed and several more were injured during the fight. 
It is important to put these numbers into perspective by understanding the percentage of people who were killed or injured during the Massacre. Of the perhaps 350 Lakota who were camped at Wounded Knee, even the most conservative estimate of 150 deaths indicates that nearly half the group was killed. In contrast, of the 500 soldiers and troopers sent by the U.S. government, only 25 (or 5%) died. The Lakota people were outnumbered, surrounded, and indiscriminately killed. 
The Wounded Knee Massacre is historically significant not only for the intensity of violence which occurred there, but also as an archetypal event for the long legacy of the colonization of the Americas and the eradication of First Nations people and culture. The Wounded Knee Massacre was initially called the Wounded Knee Battle, but this was a gross misnomer. A battle implies two (or more) groups of an equal mind to engage in combat-- the Massacre at Wounded Knee was an attack committed by the United States government against the Lakota People. 
The conflict did not end in 1890. In 1973, around 200 members of the American Indian Movement staged a protest at Wounded Knee. The activists were there to protest the corrupt tribal president Richard Wilson and the United States government's failure to negotiate and uphold treaties with the First Nations. Again, the Lakota people were surrounded by U.S. law enforcement and though there was only a total of two deaths, the occupation lasted 71 days. During this time, electricity, water, and food were cut off from Wounded Knee in an attempt to starve out the protesters. The U.S. law enforcement officials repeatedly opened fire on the Lakota people and both of the two people who died during the Incident were First Nations protesters.
I encourage you to read up on the current Dakota Access Pipeline protest to learn more about the continuing legacy of the oppression of First Nations people in the United States.
http://plainshumanities.unl.edu/encyclopedia/doc/egp.war.056

https://woundedkneemuseum.org/

https://www.britannica.com/place/Wounded-Knee

In the poem "On Children" by Khalil Gibran, the narrator states that children's souls live "in the house of tomorrow, For which you [parents] cannot visit even in your dreams." What is an interpretation of these words? Give a specific explanation.

In order to get a better understanding of those particular lines, I recommend looking at the previous few lines that lead into them.  

You may give them your love but not your thoughts, For they have their own thoughts.You may house their bodies but not their souls,For their souls dwell in the house of tomorrow, which you cannot visit, not even in your dreams.

The poem's narrator is speaking to parents about their children, and the narrator bluntly tells parents that they are allowed to give children their love but not their thoughts. This means that parents should not force their kids to have the same opinions, likes, dislikes, and so on as the parents do. Children are people, too. They have their own thoughts, wishes, desires, and opinions. Those five lines of the poem really stress the idea that each child is a unique individual, and parents shouldn't try to force a mold on them.  
If a reader considers the five lines that are listed above, the first line tells parents what they cannot do, and the second line is the reason why. Line three again tells parents what they cannot do, and lines four and five contain the reason. Parents cannot contain the soul or spirit of their children, because that part of the child is the part that dreams of his or her own future. That future is one that does not involve the parents. It is a future that the parents cannot even fathom, and that is why parents cannot visit it in their dreams.  

int_0^ln5 e^x/(1+e^(2x)) dx Evaluate the definite integral

For the given integral problem: int_0^(ln(5))e^x/(1+e^(2x))dx , it resembles the basic integration formula for inverse tangent:
int_a^b (du)/(u^2+c^2) = (1/c)arctan(u/c) |_a^b
where we let:
u^2 =e^(2x) or (e^x)^2   then u= e^x
c^2 =1 or 1^2 then c=1
For the derivative of u =e^(x) , we apply the derivative of exponential function:
du =e^x dx .
 
Applying u-substitution: u = e^x and du = e^x dx , we get:
int e^x/(1+e^(2x))dx =int (e^xdx)/(1+(e^x)^2)
                              =int (du)/(1+(u)^2)
Applying the basic integral formula of inverse tangent, we get:
int (du)/(1+(u)^2) =(1/1)arctan(u/1)
                           = arctan(u)
Express it in terms of x by plug-in u=e^x :
arctan(u) =arctan(e^x)
 
Evaluate with the given boundary limit:
arctan(e^x)|_0^(ln(5)) =arctan(e^(ln(5)))-arctan(e^0)
                          =arctan(5)-arctan(1)
                                =arctan(5) -pi/4

In "Once Upon a Time," how does the cat symbolize and support the theme?

The cat in "Once Upon a Time" symbolically supports three themes of the story. 
First, the cat represents the couple's unquenchable fear. For most of the story, the couple keeps making "improvements" to their home that they think will make them more secure so they won't have to be afraid of break-ins and burglaries from the "people of another color." The cat is still able to get through the barred windows and over the extended wall. When the cat manages to get through each new security upgrade, it shows that the couple's fears cannot be assuaged by creating physical barriers. 
Next, the cat symbolizes that the couple's fears are baseless. The cats in the neighborhood continually set off the burglar alarms. This represents the fact that the couple's fears are "false alarms." If anything, they are worried about the wrong thing. Instead of worrying about protecting their possessions and status, they should be worried about bringing justice and stability to their society by addressing the underlying causes of the social unrest that is wracking their community. This supports the theme that fear of "the other" is baseless; once people get to know each other, they can live in mutual respect in a win-win relationship.
At the end of the story, the cat represents wisdom and foresight. The husband assures his wife that the cat will not attempt to get over the wall with the Dragon's Teeth in place because "cats always look before they leap." This supports the theme that people must evaluate their actions in terms of those actions' future repercussions. Being short-sighted and caring only for one's own tribe at the expense of the larger society will produce tragedy in the end, as the couple finds out when their son dies by becoming enmeshed in the trap his parents created.
The cat symbolizes fear, false alarm, and foresight, reinforcing several themes in the story.

In “Tickets, Please!” by D.H. Lawrence, why are the travelers reluctant to dismount from their coach?

In the story, the travelers are often reluctant to dismount from their coach because it is usually very cold outside: to have no protection against the elements is an extremely uncomfortable experience.
The passengers would rather stay in the safe haven of their coach than be exposed to howling winds in the dead of night. Also, waiting out in the cold for another coach is often a treacherous exercise in patience. Often, other coaches which pass by are full of passengers and can admit no more travelers; to add insult to injury, the passing travelers often howl in "derision" at those stranded passengers who dare to brave the elements. Additionally, a long wait may yield nothing more than another coach that is unfit to take passengers, a "Depot Only" tram.
To reiterate his point, the narrator points out that it is quite common for passengers to stay in their coach until the last minute, even in the event of a fire. Essentially, the passengers won't disembark until their lives are truly in danger ("till flames actually appear").

What are some values from Holes by Louis Sachar?

Justice, of course, is a major theme of the book, but Sachar's treatment of this theme calls into question the validity of criminal justice institutions—the camp—and suggests that true justice is the product of personal morality. This can be seen by comparing the treatment of the boys at the hands of the corrupt "professionals" who run the camp with the boys' treatment of one another. More specifically, the boys' ability to treat each other with genuine kindness can be seen in the bond that develops between Stanley and Zero after Stanley begins to teach Zero to read.
Truth, or personal integrity, is another major theme in the book. While Stanley's decision to steal the water truck is, in one sense, a crime, in a larger sense the risk he takes in doing so is an expression of his own moral code: he feels an obligation to search for Zero. On another level, the "truth" about the camp and why the boys are digging the holes is another example of how appearances can be deceiving. The camp is not really concerned about juvenile corrections; the warden is not really concerned about the inmates; the boys are not criminals but, in actuality, victims.
A final theme would be fate. The story suggests that what happens to Stanley and Zero has been preordained somehow. Sam's lynching is the primary act of injustice that sets the plot in motion: it not only causes Kate to turn into a bandit and bury her loot, it actually creates the landscape of the camp (after the lynching, the lake mysteriously dries up). It seems only fitting that both Stanley and the warden had ancestors who searched for the treasure, that it is Kate's hundred-year-old preserves that keep Stanley alive in the desert, and that Stanley ultimately is the one to prevail over the warden.

One value reflected in the book Holes is the importance of friendship. In the book, there is a tremendous value placed on friendship. Stanley actually becomes quite happy with his situation (despite it still being dangerous) once his friendship with Zero is solidified.  
Another value present in the book is justice. That might seem odd at first because the book starts with Stanley being punished for a crime that he didn't commit. There doesn't seem to be any working legal justice. As the story continues, though, readers see a strong sense of moral justice among many of the boys at Camp Green Lake. They know they are being unfairly treated, and there is a sense of moral outrage at the severity of the punishments based on the minor crimes they committed. They know justice is not being properly served.  
Perhaps I can take the justice value further. By the end of the book, everything is set right. Good people are rewarded and evil people are punished. It's as if the universe itself willed justice to be done. Call it fate or cosmic justice, but it still happened.

Othello describes himself as "one who loved not wisely, but too well." How well do you understand Othello's behavior at the end of the play? Is he still a hero in your eyes? Why or why not?

At the end of the play, Othello smothers Desdemona because he thinks she is unfaithful. Iago has successfully manipulated him into believing that Desdemona and Cassio were having an affair. When Othello realizes Desdemona was innocent, he commits suicide.
While Othello is no longer a hero in my eyes after he kills his wife, I do have sympathy for him. He lived in a society run by an honor code, in which it was expected a man would avenge perceived wrongdoing (such as adultery) with violence; on top of that, Othello was insecure from the start about his ability truly to attract the love of Desdemona. He was a middle-aged black soldier and had to convince others he had won Desdemona's heart honestly, not through magic. He trusted an evil man, Iago, who manipulated him by playing on his weak spot. He lived in a patriarchal culture in which men often relied upon other men more than women, and Iago constantly suggested to him that all women are unfaithful. 
All of the above, however, are explanations and not excuses. Othello acted too rashly; he acted out of his own insecurity and wounded heart, thinking of his own pain more than his wife. Even if it had turned out that Desdemona had been unfaithful, killing her would have been excessive. It is not heroic to murder a defenseless, weaker woman.
Some critics, such as Rene Girard, contend that Shakespeare critiques the honor (or revenge) code throughout his plays, and I agree. 

What character traits best describe Steve Harmon in Monster by Walter Dean Myers?

I'll start by saying that Steve Harmon is not a bad person.  He writes exactly that in his own journal too.  

"I know that in my heart I am not a bad person." 

Of course perhaps that is not solid evidence because most people would probably believe that they are not a bad person; however, other people in the book that truly know Steve's personality attest to the fact that Steve is a good kid.  Take Mr. Sawicki for example.  He is Steve's favorite teacher and he says that Steve is "talented, bright, and compassionate."  Those character traits are absolutely true about Steve, and they are shown concretely in the way that he loves and treats his brother, Jerry.  
Readers are provided with further evidence of just how tenderhearted Steve is when we see him jail.  Steve is terrified of jail, and he's terrified of all of the violence that he sees around him in the jail.  He never considers using violence against others.  
I would definitely call Steve a good kid; however, that doesn't mean he is problem-free.  A contributing factor for his legal predicament is the fact that he is insecure.  Steve lives in a rough neighborhood.  There are gangs, theft, and violence.  Although deep down, Steve isn't like those guys, he still feels the need to posture and emulate those guys.  He wants to be cool and tough like the people that he sees around him.  That's because he's insecure in who he is to begin with.  

"I had looked at him [James King] and wanted to be tough like him."

Throughout the novel Monster, Steve Harmon is portrayed as a shy, introspective individual who is extremely self-conscious. He gets involved with a group of thugs who are planning a robbery because he wants to be viewed as cool and tough throughout his community. Myers does not specifically state whether Steve participates in the crime, but Steve ends up being accused of aiding James King and Richard "Bobo" Evans in robbing a local drugstore. During his time in jail, Steve contemplates and questions his own morals. He also struggles with his identity after the prosecuting attorney calls him a monster. Steve comes across as an innocent individual who made the terrible mistake of associating himself with criminals. He expresses his fear throughout the novel and regrets his past decisions. Steve is also a loving brother and son. He enjoys his family and shares a close relationship with his parents and brother. Overall, Steve is an intelligent, sensitive individual who struggles with his personal identity after being on trial for murder. 

Beginning Algebra With Applications, Chapter 3, 3.1, Section 3.1, Problem 80

Solve the equation $-0.813 + x = -1.096$ and check
if your answer is correct.

$
\begin{equation}
\begin{aligned}
-0.813 + x + 0.813 &= -1.096 + 0.813 && \text{Add $0.813$ from each side} \\
\\
x &= -0.283
\end{aligned}
\end{equation}
$

By checking,

$
\begin{equation}
\begin{aligned}
-0.813 + ( - 0.283 ) &= -1.096 && \text{Replace the variable by the given number, } -0.283\\
\\
-0.813 - 0.283 &= -1.096 && \text{Use the assigned variable to write the variable expression}\\
\\
-1.096 &= -1.096 && \text{Compare the results}
\end{aligned}
\end{equation}
$


The results are same; Therefore, $-0.283$ is a solution of the equation $-0.813 + x = -1.096$

In "An Occurrence at Owl Creek Bridge," how does the interaction between Farquhar, his wife, and the soldier contribute to the plot of the story?

The part of the story that the question is asking about is part two of the story.  I've always liked how "An Occurrence at Owl Creek Bridge" doesn't tell the story is a linear, chronological manner.  Part two is a flashback.  Part one explains that there is some guy about to be hanged from a bridge.  Part two gives the backstory to that guy on the bridge.  
His name is Peyton Farquhar.  He's married, and he is a plantation owner.  Readers are told that he longs to partake in the war in some manner.  

. . . he chafed under the inglorious restraint, longing for the release of his energies, the larger life of the soldier, the opportunity for distinction.

The "gray-clad soldier" gives Farquhar just such an opportunity.  Farquhar and his wife are sitting outside of their house when the soldier rides up.  Mrs. Farquhar quickly goes to get the solider a drink.  While she is inside, the solider tells Farquhar that Union troops are at the Owl Creek bridge.  The solider also tells Farquhar that the bridge is lightly defended and susceptible to sabotage. 

"Only a picket post half a mile out, on the railroad, and a single sentinel at this end of the bridge. . . I was there a month ago," he replied. "I observed that the flood of last winter had lodged a great quantity of driftwood against the wooden pier at this end of the bridge. It is now dry and would burn like tinder."

Mrs. Farquhar returns, the soldier drinks, he thanks them, and he rides off.  I don't really think Mrs. Farquhar has much to do with propelling the plot forward, but the soldier definitely does.  The final sentence of section two tells readers that the southern soldier was actually a Federal scout in disguise.  This means that the man being hanged on the bridge in section one is actually Peyton Farquhar.  Additionally, Farquhar was deviously lured to the bridge.  The soldier played on Farquhar's desire to be a part of the war, and Farquhar took the bait.  Unfortunately, it cost Farquhar his life. 

Calculus of a Single Variable, Chapter 9, 9.6, Section 9.6, Problem 17

To determine the convergence or divergence of a series sum a_n using Root test, we evaluate a 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.
We may apply Root test on the given series sum_(n=1)^oo n(6/5)^n when we let: a_n=n(6/5)^n .
Then, set-up the limit as:
lim_(n-gtoo) |n(6/5)^n|^(1/n) =lim_(n-gtoo) (n(6/5)^n)^(1/n)
Apply Law of Exponents: (x*y)^n = x^n*y^n and (x^n)^m = x^(n*m).
lim_(n-gtoo) (n(6/5)^n)^(1/n)=lim_(n-gtoo) n^(1/n) ((6/5)^n)^(1/n)
=lim_(n-gtoo) n^(1/n) (6/5)^(n*1/n)
=lim_(n-gtoo) n^(1/n) (6/5)^(n/n)
=lim_(n-gtoo) n^(1/n) (6/5)^1
=lim_(n-gtoo) 6/5n^(1/n)
Evaluate the limit.
lim_(n-gtoo) 6/5n^(1/n) =6/5lim_(n-gtoo) n^(1/n)
=6/5 *1
=6/5 or 1.2
The limit value L =6/5 or 1.2 satisfies the condition: Lgt1 since 6/5gt1 or 1.2gt1 .
Therefore, the series sum_(n=1)^oo n(6/5)^n is divergent.

How do you start a cover letter?

There are several things to consider when writing a cover letter. I will assume this is a cover letter for a job. At the top of the page, in the center, you should list your name, address, city, state, and zip code, and some contact information. The next step is to write out the date.
Then you begin the body of the cover letter. Be sure to start with a greeting. You should then indicate that you are applying for a specific job. Be very clear about which job you are applying for. You should indicate what materials are being sent or where those materials can be accessed. It might be helpful to indicate the name of a person who may have steered you to the job. Make sure you show you are passionate and excited in the opening.
Here is a sample cover letter. It could look like this:
                                           John Smith
                                           1111 Willow Way
                                           My City, CA 91609
                                           555-555-5555
                                           Email address
                                                                    October 24, 2016
 
Dear Ms. Smith,
     I am applying for the seventh and eighth grade Social Studies position at Happy Hollow School. I have been teaching Social Studies for the past five years, and I have always been hoping to teach at your school. I am very passionate about this subject. I have developed many lesson plans that have been published. I also am active in state and national Social Studies organizations. Jim Dooling, one of your Science teachers, recommended that I apply for this job. I look forward to discussing this position with you in person. I have enclosed my resume and two sample lessons that I have published. I am very excited about this opportunity!
                                                                    Sincerely,
                                                                    Your Name
https://www.thebalancecareers.com/cover-letter-opening-sentences-examples-2061030

Calculus of a Single Variable, Chapter 5, 5.5, Section 5.5, Problem 49

Derivative of a function h with respect to t is denoted as h'(t).
The given function: h(t) = log_5(4-t)^2 is in a form of a logarithmic function.
From the derivative for logarithmic functions, we follow:

d/(dx)log_a(u) =((du)/(dx))/(u*ln(a))

By comparison: log_5(4-t)^2 vs.log_a(u) we should let:
a=5 and u = (4-t)^2
For the derivative of u, recall the Chain Rule formula:
d/(dx)(f(g(x)))= f'(g(x))*g'(x)
Using u=(4-t)^2 , we let:
f(t) = t^2
g(t) = 4-t as the inner function
f'(t)= 2t
f'(g(t))= 2*(4-t)
g'(t)= (-1)
Following the Chain Rule formula, we get:
d/(dx) (4-t)^2= 2 *(4-t)*(-1)
d/(dx) (4-t)^2= -2*(4-t)
or
(du)/(dx)=-2*(4-t)
Plug-in the values:
u =(4-t)^2 , a=5 , and (du)/(dx)=-2*(4-t)
in the d/(dx)log_a(u) =((du)/(dx))/(u*ln(a)) , we get:
d/(dx) (log_5(4-t)^2) = ((-2)*(4-t))/((4-t)^2ln(5))
Cancel out common factor (4-t):
d/(dx) (log_5(4-t)^2) = -2/((4-t)ln(5))
or h'(t)= -2/((4-t)ln(5))

How to write a good theme

A good theme should be written in the form of a complete sentence.  It should also be a statement for which you can provide textual evidence in the form of quotations from the book.  Further, a theme should be universal in nature, not written in such a way that it is application only to one particular book or character but is, rather, something that applies to people or life more generally.  
For example, you might say, "The individual is always at odds with society," rather than, "This particular character is at odds with his or her society."
You might say, "Nature does not care about human suffering," rather than, "Nature does not care about a particular character's suffering."
Finally, you might say, "It takes a crisis of confidence in order for individuals to grow," rather than, "It takes a crisis of confidence in order for a particular character to grow."

Explain why the properties of a pure substance do not vary from sample to sample.

When you take a sample of something—a dirt sample from an archaeological site, a sap sample from a tree, an ice sample from the Arctic—you are taking a statistical sample of the substance's compositional atoms in the process. So if a certain region of Arctic ice is 95% water and 5% air bubbles on average, then on average your sample will match that, but there's no guarantee that you won't hit a patch that has a particularly thick or thin concentration of air. You must take multiple samples to determine the average across the entire region.
In contrast, if a pure substance is truly pure—100% unadulterated concentration of whatever you're sampling—then there is no need to worry about the properties changing. Unless you taint the substance in the sampling process, its properties cannot change, because every sample will be equally as pure as the source from which you are sampling.

I need help to sketch the vector field F(x,y) = 2xi - yj by choosing four relevant vectors in each quadrant. Then I need to find and sketch the equation of the field line that passes through the point (x,y) = (2,2). It is possible to choose vectors where x = 0 and y = 0. Could you please help?

For each x,y the point (x, y) is the point where a vector starts and F(x, y) is a pair of its components. For x=y=0 we have F(x,y)=vec0 and we cannot draw it with an arrow because it has no direction.
For the given point (2,2) its field vector is F(2,2) = (4, -2), it is the direction vector of the corresponding line. This means the slope of this line is y/x = -2/4 = -1/2 and the equation is  y-2=-1/2 (x-2), or y=-1/2 x+3.
To draw relevant vectors, choose some integer arguments of F(x,y). It is simple to start a vector F(x,y) at the point (x,y): we just need add (x,y) and (2x, -y) and obtain (3x, 0). This way all vectors point to the x-axis.
The attached picture uses the starting points (1,1), (1,2), (1,3), (1,4) for the first quadrant, (-1,1), (-2,2), (-3,3), (-4,4) for the second, (-1,-1), (-2,-1), (-1,-2), (-2,-2) for the third and (1,-1), (2,-1), (3,-1), (4,-1) for the fourth. Actually, the field is symmetric across x and y axes.
http://tutorial.math.lamar.edu/Classes/CalcIII/VectorFields.aspx

Why isn't Salerio present in the Act III, Scene iii of The Merchant of Venice by Shakespeare? Where is he?

In Act 3, Scene 3, Shylock stands in his doorway alongside Antonio, Solanio, and a Jailer. Antonio tries his best to reason with Shylock, but Shylock refuses to listen to him. Shylock insists that he will have his bond, and Solanio comments that he is an obstinate dog. Antonio then mentions that he only wishes that Bassanio will come see him pay his debt before leaving with the Jailer. The reason Salerio is not present in the Act 3, Scene 3 is because he traveled to Belmont to give Bassanio a letter from Antonio. In Act 3, Scene 2, Salerio arrives at Belmont with the terrible news that Antonio has lost all of his merchant ships at sea and Shylock plans to collect Antonio's flesh. Bassanio becomes extremely upset and explains the situation to Portia before leaving for Venice. During Act 3, Scene 3, Salerio is traveling with Bassanio to return to Venice before Antonio is forced to pay his bond.

What is the characterization of Tessie Hutchinson?

Tessie Hutchinson is an ordinary housewife who accepts life as it comes to her without questioning its values until it is too late.
She arrives late at the annual lottery and chats easily with her neighbors, saying she had almost forgotten it. This suggests it is not an event she wants to think about. Nevertheless, it is a tradition in her village, and she apparently goes along with it every year without raising any problems. She doesn't seem to feel the least bit of fear that she might be the chosen sacrifice.
Tessie Hutchinson is an "everyman," a conventional person willing to put up with evil or barbarism and not rock the boat until the evil touches her. Then, it is another story. When her family's name and then her name is drawn in the lottery, suddenly the procedure, and her fate, are "unfair." She apparently has lacked the empathy to understand how others might feel as the victims of an arbitrary stoning. By the time it is her turn, she has participated too long to expect any mercy.
Jackson's story suggests that we speak up against injustice rather than assume, like Tessie, that it will never happen to us.

In Shirley Jackson's "The Lottery," Tessie Hutchinson is first characterized as a harried housewife who has simply forgotten that it is lottery day until she notices her husband and children are gone. It is possible that her forgetfulness is either intentional or a subconscious effort to avoid the lottery. She hurries to the gathering, greets her neighbors, and makes a lighthearted joke about her tardiness. She urges her husband to quickly take his slip of paper, cracking another joke that amuses those standing near. Her levity quickly comes to an end, however, once she sees that the lottery will claim one of her family members. She adopts a protesting, defensive tone. Getting no support from her husband or any of the crowd, she becomes quieter, yet insistent that the proceedings aren't being conducted fairly. Her defiance intensifies, and her husband has to pry her paper from her hand. Tessie Hutchinson does not accept her fate uncomplainingly, which makes her death by stoning all the more unsettling at the story's conclusion.

Calculus of a Single Variable, Chapter 9, 9.6, Section 9.6, Problem 20

To determine the convergence or divergence of a series sum a_n using Root test, we evaluate a 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.
We may apply Root test on the given series sum_(n=1)^oo 5^n/n^4 when we let: a_n =5^n/n^4 .
Applying the Root test, we set-up the limit as:
lim_(n-gtoo) |5^n/n^4|^(1/n) =lim_(n-gtoo) (5^n/n^4)^(1/n)
Apply Law of Exponent: (x/y)^n = x^n/y^n and (x^n)^m= x^(n*m) .
lim_(n-gtoo) (5^n/n^4)^(1/n) =lim_(n-gtoo) (5^n)^(1/n)/(n^4)^(1/n)
=lim_(n-gtoo)5^(n*1/n)/n^(4*1/n)
=lim_(n-gtoo)5^(n/n)/n^(4/n)
=lim_(n-gtoo)5^1/n^(4/n)
=lim_(n-gtoo)5/n^(4/n)
Evaluate the limit.
lim_(n-gtoo) 5/n^(4/n)=5 lim_(n-gtoo) 1/n^(4/n)
=5 *1/oo^(4/oo)
=5 *1/oo^(0)
=5 *1/1
= 5*1
=5
The limit value L =5 satisfies the condition: Lgt1 since 5gt1 .
Conclusion: The series sum_(n=1)^oo 5^n/n^4 is divergent.

Why did Western nations use appeasement with the Axis powers?

One reason the Allied powers pursued a policy of appeasement is that they were weary of war after the bloodletting of the First World War and were reluctant to pursue any policy that seemed likely to bring about another conflict. Having lost vast swaths of their male populations, they were understandably fearful about entering into any new war.
Another reason was that Hitler, while repugnant to many Westerners, was seen by many as the lesser of two evils when compared to Joseph Stalin in the Soviet Union. Communists advocated global revolution in a way that Hitler did not, and many saw Hitler as a valuable if dangerous bulwark against the Soviets. This is why they were so horrified at the announcement of the Nazi-Soviet non-aggression pact in 1939. In fact, many British, American, and Frenchmen shared Hitler's views on race and anti-Semitism.
In 1938, with the crisis over the Sudetenland breaking out, the British army was not seen as capable of dealing with the threat posed by the Nazis. The Sudetenland was not worth going to war over, but without it Czechoslovakia was seen as impossible to defend. So essentially, the Munich Conference, where the much-reviled Neville Chamberlain claimed to have achieved "peace in our time," was held over a territory seen as a lost cause by many strategists.
Another reason that Europeans pursued appeasement is that they underestimated Hitler's capacity for boldness and dishonesty. When he promised no further aggressions at Munich, some believed him, if only because, as mentioned above, they had little choice.
Finally, the United States, which, as a massive industrial and financial power, was essential to any war against the Germans, adopted an isolationist stance in the 1930s. "America First" advocates, many of whom were genuinely sympathetic to the Nazis, were vocal and prominent in the United States. Others in Congress attempted to restrain President Franklin Roosevelt, whose instincts were always toward involvement in global affairs. Without the backing of the United States, the European powers played with a weaker hand in dealing with Hitler's aggression.
https://www.iwm.org.uk/history/how-britain-hoped-to-avoid-war-with-germany-in-the-1930s

https://www.spiegel.de/international/europe/the-road-to-world-war-ii-how-appeasement-failed-to-stop-hitler-a-646481.html

To what extent is Christopher's condition responsible for these three types of conflict that arise in the book? Identify and analyze the following: character versus character, character versus self, character versus nature.

To a great extent, Christopher's condition is responsible for three main conflicts in the book: character versus character, character versus self, and character versus nature.
In the story, 15-year-old Christopher is mildly autistic, and his symptoms are typical of someone with this condition. The first indication of Christopher's autism is shown in his preoccupation with numbers, specifically prime numbers. In fact, the chapters of his book are ordered accordingly. For example, we begin with Chapter 2, progress to Chapter 3, and proceed accordingly. In chapter 19, Christopher explains why he likes prime numbers (they are logical) and how he calculates subsequent numbers (beginning from 2).
Christopher's conflicts with his father, Mrs. Shears, and, later, Mr. Shears and his mother (in London) are often precipitated by his condition. A large part of Christopher's challenges derive from his inability to correctly interpret nonverbal forms of communication such as facial expressions, gestures, and body language. His difficulties are briefly alluded to in chapter 3. Here, Christopher explains that he can tell when a person is sad or happy. However, he runs into problems deciphering other types of facial expressions.
In chapter 11, Christopher is approached by the police. During their interaction, he becomes confused by an officer's rapid questions. At this point, it is clear that Christopher is overwhelmed, and he protects himself in the only way he knows how: he begins groaning heavily. Alarmed by Christopher's behavior, the officer takes hold of the teenager in an attempt to bring him to his feet. Because many autistic individuals are hypersensitive to touch (especially during a stressful situation), Christopher hits back at the officer. Additionally, Christopher's inability to correctly interpret the officer's body language also contributes to his outburst.
In chapter 17, we learn that Christopher has been arrested for assaulting an officer. However, he shows little indication of being distressed by his arrest. In fact, Christopher's mind continues to focus on what he likes best: numbers. He happily notes that his jail cell is a perfect cube and contains exactly 8 cubic meters of air. Like many autistic individuals, Christopher harbors intense obsessions; in Christopher's case, prime numbers and scientific facts give him a feeling of control and normalcy.
It can be seen that Christopher's struggles with himself and others largely stem from his condition. In chapter 67, Christopher admits that he does not like strangers, as he has a difficult time understanding them. We learn more about Christopher's condition in chapter 73, where he describes some of his "behavioral problems." In chapter 97, Christopher learns from Mrs. Alexander (a neighbor) the real reason for his father's explosive dislike of Mr. Shear.
Despite Mrs. Alexander's careful and empathetic manner, Christopher is distrustful of her. Again, his difficulties arise from his condition. Throughout the novel, Christopher has difficulty relating to others, and he struggles to process his experiences. When overwhelmed, he falls back to reciting facts and statistics.
Later, in the book, we learn that Christopher also has unconventional feelings about nature. In chapter 103, he describes his impression of clouds. He tells us that sometimes clouds are boring: they are grey or blue and have little personality. At other times, a big, grey cloud may be a rain cloud, and it may take the shape of an alien spaceship "hundreds of kilometers long." In the same chapter, we learn that Christopher smells "nothing" when he sniffs the air in the garden. Again, Christopher's conflict with nature stems from his condition; many autistic individuals have difficulty processing sensory information, such as sights, sounds, and smells.
In all, Christopher's condition is directly responsible for the three types of conflict in the book.
https://www.autismspeaks.org/what-are-symptoms-autism

https://www.autismspectrum.org.au/about-autism/what-is-autism

Is there a quote to show how the littlun with the birthmark dies? Thanks.

In chapter 2, one of the littluns with a mulberry-colored birthmark on his face tells the group of boys during an assembly that he saw a snakelike "beastie" the previous night, and the older boys assure the littluns that he was simply experiencing a nightmare. Later on, Ralph proposes that the boys climb to the top of the mountain and create a signal fire so that passing ships will stop to rescue them. The boys get excited and gather a large bundle of dead wood for the signal fire. They then use a lens from Piggy's glasses to magnify the sun and start the fire. Suddenly the fire bursts into flames, and the wind carries the flames down the side of the mountain, causing a small forest fire. Piggy immediately begins criticizing the older boys for their lack of responsibility and asks if anyone thought to take count of how many littluns are in the group. Piggy then mentions that some of the littluns were playing in the location of the forest fire by saying,

and them little 'uns was wandering about down there where the fire is. How d'you know they aren't still there? (Golding, 36).

One of the boys then says, "Him that talked about the snakes. He was down there," and Golding writes,

A tree exploded in the fire like a bomb. Tall swathes of creepers rose for a moment into view, agonized, and went down again. The little boys screamed at them (36).

When the boys continue to ask if they saw the littlun with the mulberry-colored birthmark, Golding writes,

Ralph muttered the reply as if in shame. "Perhaps he went back to the, the—" Beneath them, on the unfriendly side of the mountain, the drum-roll continued (36).

While Golding does not explicitly describe what happens to the littlun with the mulberry-colored birthmark, he implies that the littlun dies in the forest fire, which is created when the boys initially attempt to create a signal fire.

Single Variable Calculus, Chapter 3, 3.1, Section 3.1, Problem 3

a.) Determine the slope of the tangent line to the parabola $y = 4x - x^2$ at the point $(1, 3)$

$(i)\text{ Using the definition (Slope of the tangent line)}$

$\displaystyle \lim \limits_{x \to a} \frac{f(x) - f(a)} {x - a}$

Here, we have $a = 1$ and $f(x) = 4x - x^2$, so the slope is


$
\begin{equation}
\begin{aligned}

\displaystyle m =& \lim \limits_{x \to 1} \frac{f(x) - f(a)}{x - 1}\\
\\
\displaystyle m =& \lim \limits_{x \to 1} \frac{4x - x^2 - [4(1) - (1)^2]}{x - 1}
&& \text{ Substituting value of $a$ and $x$}\\
\\
\displaystyle m =& \lim \limits_{x \to 1} \frac{4x - x^2 - 3}{x - 1}
&& \text{ Factor the numerator}\\
\\
\displaystyle m =& \lim \limits_{x \to 1} \frac{-1(x - 3) \cancel{(x - 1)}}{\cancel{x - 1}}
&& \text{ Cancel out like terms and simplify}\\
\\
\displaystyle m =& \lim \limits_{x \to 1} (-x+3) = -1+3= 2
&& \text{ Evaluate the limit }\\

\end{aligned}
\end{equation}
$

Therefore,
The slope of the tangent line is $m=2$
$(ii)$ Using the equation

$\displaystyle m = \lim \limits_{h \to 0} \frac{f (a + h) - f(a)}{h}$

Let $f(x) = 4x - x^2$. So the slope of the tangent line at $(1, 3)$ is


$
\begin{equation}
\begin{aligned}

\displaystyle m =& \lim \limits_{h \to 0} \frac{f(1 + h) - f(1)}{h}\\
\\
\displaystyle m =& \lim \limits_{h \to 0} \frac{4(1 + h) - ( 1 + h)^2 - [4(1) - (1)^2]}{h}
&& \text{ Subsitute value of $a$} \\
\\
\displaystyle m =& \lim \limits_{h \to 0} \frac{4 + 4h - ( 1 + 2h + h^2) - 3}{h}
&& \text{ Expand and simplify }\\
\\
\displaystyle m =& \lim \limits_{h \to 0} \frac{2h - h^2}{h}
&& \text{ Factor the numerator}\\
\\
\displaystyle m =& \lim \limits_{h \to 0} \frac{\cancel{h} ( 2 - h)}{\cancel{h}}
&& \text{ Cancel out like terms }\\
\\
\displaystyle m =& \lim \limits_{h \to 0} (2 - h) = 2 - 0 = 2
&& \text{ Evaluate the limit}\\
\end{aligned}
\end{equation}
$

Therefore,
The slope of the tangent line is $m=2$
b.) Write an expression of the tangent line in part (a).

Using the point slope form


$
\begin{equation}
\begin{aligned}

y - y_1 =& m ( x - x_1) && \\
\\
y - 3 =& 2 ( x - 1)
&& \text{ Substitute value of $x, y$ and $m$}\\
\\
y =& 2x - 2 + 3
&& \text{ Combine like terms }\\
y =& 2x + 1
\end{aligned}
\end{equation}
$

Therefore,
The equation of the tangent line at $(1,3)$ is $y = 2x + 1$
c.) Illustrate the graph of the parabola and the tangent line. As a check on your work, zoom in toward the point $(1,3)$ until
the parabola and the tangent line are indistinguishable.