Would the Biblical allusions in "Ransom of Red Chief" be ironic or just further describe the ironic situation that a child is terrorizing his kidnappers?

The biblical allusions that O. Henry employs in "The Ransom of Red Chief" do reinforce the overall irony of the story's plot—a child getting the better of his kidnappers—but also feature delightful irony in their own right.
Comparing Red Chief's flinging the stone at Bill to the story of David and Goliath is ironic because by this time in the story, readers likely already feel sorry for Bill and dislike Red Chief. It's the most natural thing in the world for people to take the side of the underdog (the David against the Goliath), yet readers know Red Chief is no hero. He has already pitched rocks at a kitten, after all. As a result, we're tempted to take Goliath's side.
When Bill asks Sam if he knows who his favorite Bible character is, it's ironic on a couple of levels. First, we don't expect an outlaw to have a favorite Bible character. Even Sam thinks the question is a result of Bill's being temporarily knocked senseless. Second, though, Bill's choice of a favorite is one of the worst villains in the Bible—King Herod. Herod tried to kill the Christ Child. Once readers get the joke, though, they can understand Bill's choice. If King Herod had been around at the right time in Summit's history, Bill's tormenter would never have been able to launch the rock that could have killed Bill. 
The third allusion isn't strictly biblical, but it is religious. Bill compares himself to "martyrs in old times. . . that suffered death rather than give up the particular graft they enjoyed." This passage is ironic because it presents martyrs as if they were petty criminals, like Bill and Sam. Bill correlates the tortures he has endured at the hands of Red Chief with the persecution stalwarts of the faith endured for their commitment to God. Bill is no saint, but Red Chief does seem to possess "supernatural" ability to wear down his enemy.
The clever, multi-layered irony of the biblical allusions adds to the humor of "The Ransom of Red Chief."

How is The Great Gatsby a commentary on the decay of social and moral values that came about in the 1920s?

The novel parallels the context of society during that time, which creates a backdrop of realism that serves to emphasize the social and moral decay of the 1920s. During the time in which the novel was written in, the 18th Amendment was passed and it outlawed the "manufacture, sale, or transportation of intoxicating liquors" on a national level. Prohibition, however, had little effect on the hedonism of the liquor-loving public, and speakeasies, a type of illegal bar, cropped up everywhere.

In the Great Gatsby, many, such as the protagonist Gatsby, took advantage of the 18th amendment and opened up a lot of “side-street drug-stores”, that “sold grain alcohol over the counter”, confirming Tom’s suspicion in chapter 6 that “A lot of these newly rich people are just big bootleggers”.

Additionally, Gatsby also has other stores in other states such as “Chicago” and “Philadelphia” that were “calling him on the wire”.

There is also the use of synecdoche, “Chicago” and “Philadelphia” to represent Gatsby’s illegal activities in other states also serves to highlight the immoral activities that have spread throughout America like a plague. Ultimately, highlighting the decay of social and moral values that came about in the 1920s as people used whatever means they could to attain wealth.

Fitzgerald portrays the decay of social and moral values during the 1920s by illustrating the opulence, greed, deceit, and reckless behavior of the wealthy citizens living in New York City and Long Island. Fitzgerald is highly critical of American citizens' unrestrained desire for status, wealth, and pleasure during the 1920s, which is most significantly depicted during Gatsby's wild parties. Fitzgerald's poignant commentary of the reckless social atmosphere of the 1920s also depicts the corruption and vain pursuit of the American Dream. Essentially, Jay Gatsby reaches the apex of American society through illegal means only to feel empty and lonely. The same can be said for Daisy Buchanan, who marries Tom simply because he is wealthy but suffers in her terrible marriage. Fitzgerald also portrays the hypocrisy of society by illustrating how the most wealthy, physically attractive people are morally debased. Despite Daisy's outward beauty, she is extremely superficial, careless, and shallow. Overall, Fitzgerald illustrates the dark side of the Roaring Twenties by depicting the vain pursuit of the American Dream, emphasizing the corrupting effects of materialism, and portraying the consequences attached to leading a life full of debauchery and pleasure.

Single Variable Calculus, Chapter 8, 8.3, Section 8.3, Problem 36

Evaluate $\displaystyle \int \frac{dx}{x^4 \sqrt{x^2-2 }}$, check your answer if its reasonable by grouping the integrand and its indefinite integral on the same screen.
By trigonometric substitution,



$\displaystyle\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\sqrt{2}}{x}$ and $\displaystyle \tan \theta = \frac{\sqrt{x^2 - 2}}{\sqrt{2}}$
$\displaystyle x = \frac{\sqrt{2}}{\cos \theta} = \sqrt{2} \sec \theta$

So,

$
\begin{equation}
\begin{aligned}
\int \frac{dx}{x^4 \left( \sqrt{x^2 - 2} \right) } &= \int \frac{\sqrt{2}\sec \theta \tan \theta d \theta}{ \left( \sqrt{2} \sec \theta \right)^4 ( \sqrt{2} \tan \theta )}\\
\\
&= \int \frac{d \theta}{4 \sec^3 \theta}\\
\\
&= \frac{1}{4} \int \cos^3 \theta d \theta
\end{aligned}
\end{equation}
$

By using integration by parts,
If we let $u = \cos^2 \theta$ and $dv = \cos \theta d \theta$, then
$du = 2 \cos \theta ( - \sin \theta d \theta)$ and $\displaystyle v = \int \cos \theta d\theta = \sin \theta$

So,
$\displaystyle \int \cos^3 \theta d \theta = uv - \int v du = \sin \theta \cos^2 \theta + 2 \int \sin^2 \theta \cos \theta d \theta$


$
\begin{equation}
\begin{aligned}
\text{by substituting, if we let } z &= \sin \theta \text{ , then } dz = \cos \theta d \theta\\
\\
&= \sin \theta \cos^2 \theta + 2 \int z^2 dz\\
\\
&= \sin \theta \cos^2 \theta + 2 \left[ \frac{z^3}{3} \right]\\
\\
&= \sin \theta \cos^2 \theta + 2 \frac{(\sin \theta)^3}{3}
\end{aligned}
\end{equation}
$


So,
$\displaystyle \frac{1}{4} \int \cos^3 \theta d \theta = \frac{1}{4} \left[ \sin \theta \cos^2 \theta + \frac{2(\sin \theta)^3}{3}\right] + c$

From the triangle,

$
\begin{equation}
\begin{aligned}
&= \frac{1}{4} \left[ \left( \frac{\sqrt{x^2-2}}{x} \right) \left( \frac{\sqrt{2}}{2} \right)^2 + \frac{2}{3} \left( \frac{\sqrt{x^2-2}}{x} \right)^2 \right]\\
\\
&= \frac{1}{4} \left[ \frac{2\sqrt{x^2-2}}{x^3} + \frac{2}{3} \left( \frac{\sqrt{x^2 - 2}}{x^3} \right)^3 \right]\\
\\
&= \left[ \frac{\sqrt{x^2 - 2}}{2x^3} + \frac{\left( \sqrt{x^2 - 2 } \right)^3}{6x^3} \right]\\
\\
\int \frac{dx}{x^4 \sqrt{x^2-2}} &= \frac{3\sqrt{x^2 - 2}+\left( \sqrt{x^2 - 2} \right)^3}{6x^3} + c\\
\end{aligned}
\end{equation}
$




Based from the graph, we can say that our answer is reasonable because $f$ is increasing whenever $f$ is positive. On the other hand, $f$ is decreasing whenever $f'$ is negative.

Single Variable Calculus, Chapter 7, 7.3-2, Section 7.3-2, Problem 78

Find the integral $\displaystyle \int \frac{(1 + e^x)^2}{e^x} dx$


$
\begin{equation}
\begin{aligned}

\int \frac{(1 + e^x)^2}{e^x} dx =& \int \left( \frac{1 + 2e^x + e^{2x}}{e^x} \right) dx
\\
\\
\int \frac{(1 + e^x)^2}{e^x} dx =& \int \left( \frac{1}{e^x} + \frac{2e^x}{e^x} + \frac{e^{2x}}{e^x} \right) dx
\\
\\
\int \frac{(1 + e^x)^2}{e^x} dx =& \int (e^{-x} + 2 + e^x) dx
\\
\\
\int \frac{(1 + e^x)^2}{e^x} dx =& e^{-x} (-1) + 2 \left( \frac{x^{0 + 1}}{0 + 1} \right) + e^x + C
\\
\\
\int \frac{(1 + e^x)^2}{e^x} dx =& -e^{-x} + 2x + e^x + C

\end{aligned}
\end{equation}
$

In the poem "The Day Zimmer Lost Religion" what is the theme and idea?

Theme is defined as the universal, underlying message an author tries to communicate about people, society, nature, or another broad category.
In Paul Zimmer’s “The Day Zimmer Lost Religion,” the speaker and the author are conflated because of the title. If one examines the title, it is clear that one should focus on the thematic idea of religion when writing a thematic statement.
In each of the three stanzas, Zimmer seems to define his understanding of religion. In the first, Zimmer worries about abandoning religion, because he fears God’s retribution. In the second stanza, Zimmer describes rebelling against the religious traditions he used to hold dear. By the third stanza, Zimmer experiences an epiphany that religion and a spiritual connection with God are not the same thing. Instead of punishing Zimmer for rejecting religious traditions, Jesus knows that Zimmer “is grown up and ready. . . now.” This means that Zimmer, no longer attached to the hollow ceremonies of organized religion, is finally free to pursue faith.
Therefore, one could argue that the theme of this poem is that the institution of religion is an unnecessary tradition based on fear, that creates barriers for those seeking spiritual fulfillment.

What are three similarities that the characters Willy Loman from Death of a Salesman and Hamlet of Shakespeare share? Please elaborate with examples.

Though it would seem like a 63-year-old American traveling salesman in the middle of the twentieth century and a young Danish prince from the late middle ages would not have much in common, they are similar in a few ways.
First of all, both men entertain thoughts of suicide. The reasons for Hamlet's existential despair are not fully articulated, but readers can understand his unhappiness at his father's murder, his mother's swift remarriage (to his uncle, the murderer, no less), losing out on the throne that should be his, and his perception that his girlfriend has become a spy. Hamlet's contemplation of suicide is discussed in his famous "to be or not to be" soliloquy in act 3, scene 1. Willy Loman's reasons for committing suicide are not fully articulated either, but readers understand that he is deeply discouraged over his lack of success. Additionally, he is at an age when most people are thinking of retiring. He still faces debt, and he is let go at his job because he has outlived his usefulness, a major blow to his psyche. He is not a role model to his sons, he has not amassed enough wealth to have a comfortable retirement, and he has betrayed his wife. His shame is overwhelming. He surrenders his dignity to Howard by begging to be kept on at his job by saying, "You can’t eat the orange and throw the peel away—a man is not a piece of fruit!"
Secondly, both Hamlet and Willy have "visitations" from people who have passed on. In Hamlet's case it is his father's ghost who reveals to him the circumstances of his murder and urges him to seek vengeance with the words "revenge his foul and most unnatural murder." The ghost appears in the play and recites lines, but it is only Hamlet who hears them, opening up the possibility that Hamlet only imagines it. Likewise, Willy Loman engages in conversation with his late brother Ben, a man who had been adventurous and had great success in life, two qualities that Willy lacks. In the end, Willy understands Ben to have given him the go-ahead for his suicide.
Lastly, both men have fractious relationships with others. Hamlet is angry with his mother because of her hasty marriage to Claudius; he castigates her in act 3, scene 4, ultimately asking her "O shame, where is thy blush?" Hamlet also rejects Ophelia for her disloyalty; when she seeks to return some of his love tokens in act 3, scene 1, he denies ever giving them to her, saying "I never gave you aught." Willy's relationship with his son Biff is troubled. Biff is a ne'er-do-well that makes Willy feel like a failure as a parent. Biff and Willy's difficult relationship is further strained because Biff discovers that Willy has been having an affair. Willy also has trouble getting along with his much younger boss, Howard, whose callousness in the way he fires Willy is especially humiliating.

Precalculus, Chapter 6, 6.5, Section 6.5, Problem 9

You need to find the absolute value of the complex number, using the formula |z| = sqrt(a^2+b^2) , hence, you need to determine a and b.
a = 4, b = -6.
Replacing 4 for a and -6 for b in formula of absolute value, yields:
|z| = sqrt(4^2+(-6)^2)
|z| = sqrt(16+36)
|z| = sqrt52 => |z| = 2 sqrt 13
Hence, the distance of the complex number z = 4 - 6i from the origin is given by the aboslute value |z| = 2 sqrt 13 .
The complex number z = 4 - 6i is displayed as the point (4,-6) in a coordinate plane, or as a vector from the origin to the point (4,-6).