Calculus of a Single Variable, Chapter 9, 9.10, Section 9.10, Problem 69

From the table of power series, we have:
(1+x)^k = 1 +kx+ (k(k-1))/2! x^2 +(k(k-1)(k-2))/3!x^3 + ...
To apply this on the given integral int_0.1^0.3 sqrt(1+x^3)dx , we let:
sqrt(1+x^3) =(1+x^3)^(1/2)
Using the aforementioned power series, we may replace the "x " with "x^3 " and "k " with "1/2 or 0.5" .
(1+x^3)^(1/2) =1 +0.5x^3+ (0.5(0.5-1))/2! (x^3)^2 + ...
=1 +0.5x^3 -0.25/2! x^6 + ...
The integral becomes:
int_0.1^0.3 sqrt(1+x^3)dx=int_0.1^0.3 [1 +0.5x^3-0.25/2! x^6 +...]dx
To determine the indefinite integral, we integrate each term using Power Rule for integration: int x^ndx =x^(n+1)/(n+1).
int_0.1^0.3 [1 +0.5x^3-0.25/2! x^6 +...]dx
=[x + 0.5 x^4/4-0.25/2! x^7/7 +...]|_0.1^0.3
=[ x +x^4/8-x^7/56]_0.1^0.3
Apply definite integral formula: F(x)|_a^b = F(b) - F(a) .
F(0.3) =0.3+0.3^4/8-0.3^7/56+ ...
=0.3 +0.0010125 -3.9054x10^(-6) + ...
F(0.1) = 0.1 +0.1^4/8-0.1^7/56+ ...
= 0.1+.0000125 - 1.7857x10^(-9) + ...
We stop at the 3rd term since we only need error less than 0.0001 .
F(0.3)- F(0.1) = [0.3 +0.0010125-3.9054x10^(-6)] -[0.1+.0000125-1.7857x10^(-9) ]
=0.2009960964
Thus, int_0.1^0.3 sqrt(1+x^3)dx ~~0.2010 .

What strong social statement is Bradbury making about the future of humans in a technological world in Fahrenheit 451? What does he feel society and its individuals must do to prevent destruction of the species? If Bradbury were to write a similar book today, what current societal dangers might he warn against that have happened in recent times?

While many science fiction writers are techno-optimists who assume the advance of technology will improve human wisdom, Bradbury is a pessimist who sees the technology of mass media and electronic entertainment as leading to stultification of the human spirit and a tyranny of the lowest common denominator of thought. He sees the very difficulty of books as their virtue and the ease of consumption of television as leading to a dumbing down of the populace.
In Fahrenheit 451, Bradbury argues that mass media becomes a narcotic (similar to Marx's notion of religion as the "opiate of the masses") and that the way it contributes to its mental pacification of the populace is by avoiding anything that might give offense to anyone and thus anything that might awake people from their mental torpor.
On these grounds, Bradbury is especially opposed to all forms of censorship. Although when he wrote this book, the dominant anti-censorship battles had to do with issues of sexuality and religion, he would have found equally abhorrent recent efforts to stymie free speech under such guises as "trigger warnings" or "hate speech". He would not feel that college students should be protected from things they found offensive. Instead, he would argue that being offended is a form of active engagement and that the job of the writer is to make readers think and react—even if some readers are offended in the process. 
He feels that to save society one must sustain the possibility of dissenting voices saying things that make us feel uncomfortable and that challenge accepted beliefs and certainties. 

Please provide a close reading of this passage from Mules and Men by Zora Hurston: The white man is always trying to know into somebody else's business. All right, I'll set something outside the door of my mind for him to play with and handle. He can read my writing but he sho' can't read my mind. I'll put this play toy in his hand, and he will seize it and go away. Then I'll say my say and sing my song. How can one of major critics, such as Dubois or Ellison, explain this controversy?

In this quote, Hurston suggests that African Americans have to always be aware of how much of themselves they reveal to the white man. The racial politics described here indicate that white people demand much of African Americans, and to maintain the peace, African Americans compromise by giving them some of that, but they still maintain their own inner lives so that they do not feel "owned" by white people.
The passage begins with the speaker saying, "The white man is always trying to know into somebody else's business." This quote establishes the idea that white people, being the race with the most power and influence at the time, expect that they can and should know everything about everyone else; the quote phrases this idea as through the white man thinks this is his right. However, the black man has a strategy to deal with this: "All right, I'll set something outside the door of my mind for him to play with and handle."
The compromise is that he will give the white man something, enough to make him think that he has some kind of inner knowledge of the black citizen. However, the black man will not allow him to truly get inside of his mind: "He can read my writing but he sho' can't read my mind." The white man can access what the speaker gives him access to, in his "writing," but there is no way for him to access the depth of the speaker's mind. The speaker continues by using some figurative language: "I'll put this play toy in his hand, and he will seize it and go away." Comparing the information given to the white man by the speaker to a "play toy" suggests that the white man's insistence on owning the black man's thoughts and inner lives is immature and that the white man is easily fooled. Once he leaves, the black man can be himself: "Then I'll say my say and sing my song." His "say" and his "song" are the forms of expression through which the black man taps into his true inner self. The white man cannot have access to this.
A useful comparison to this passage is Ellison's Invisible Man, wherein the narrator, a young African American man, joins the Brotherhood, an organization headed by powerful white men who supposedly want to help aid racial equality. The organization wants to own the narrator, giving him a new name and engineering his speaking topics and platforms. They even instigate a riot in Harlem. It turns out that the Brotherhood may not have had the narrator's best interest at heart; they only wanted to use him to further their agenda. The speaker in Hurston's story also recognizes the white man's potential to exploit African Americans.
The passage also reminds me of Langston Hughes's poem "I, Too." In the poem, the speaker describes how he is being discriminated against. He has to go "eat in the kitchen / When company comes," but while he is isolated, he is also gathering his strength. He says he will "grow strong," and that some day, the white man will "be ashamed" for not treating him like an equal human being. Hurston's speaker also distinguishes between what he can and will do in front of the white man and what he does on his own.

What causes Giovanni to sigh in "Rappaccini's Daughter"?

Giovanni sighs in the opening of the story from a mixture of emotions. He is “a young man for the first time out of his sphere,” so he has never been away from home before, and he is homesick. He knows the history, or “the poem,” of his country, and he knows something of the history of the family whose ancient mansion he has taken lodgings in, and he has been reminiscing about what he knows. He remembers that one of the ancestors of the family whose crest is above the door was shown as being in one of the circles of Hell in Dante’s Inferno. He also notes that the family is long extinct. All of this melancholy nostalgia coupled with missing his home leads to his heavy sigh.

Giovanni Guasconti has moved to northern Italy in order to attend classes at the University of Padua, though he is originally from the southern part of the country. It sounds as though he finds the north somewhat dreary in comparison to his home. First, the climate is very different: the sun does not seem to shine as brightly in his eyes. Second, Giovanni clearly misses his home, and this is the first time he has ever really been away. Third, he recollects that a member of the family to whom the armorial crest above the door to his building once belonged was mentioned in Dante's Inferno, adding to the dismal mood of this place. Finally, the narrator describes Giovanni's apartment as "desolate and ill-furnished," unlike his home. Therefore, he has a number of reasons for which to sigh.

In what ways did the United States expand its territory and industries during the 1800s?

The United States enjoyed a period of rapid growth during the nineteenth century. In terms of territory, the United States more than doubled in size with the Louisiana Purchase. The young nation then gained the Southwest and California in a controversial yet successful war against Mexico. The United States then bought a strip of land south of the Gila River in Arizona less than ten years after the end of the Mexican War. The United States settled land disputes with Britain in order to gain clear title to the Oregon Territory and then bought Alaska from the Russians after the Civil War. American sugar and pineapple growers overthrew the rightful queen of Hawaii in order to annex the land for the United States and to have their sugar and pineapple imported duty-free. In another controversial yet successful war, the United States gained Puerto Rico, Guam, and the Philippines from Spain.
Most of these land acquisitions were done under the guise of Manifest Destiny, which was the belief that God chose the American people to rule the North American continent, but many of the land acquisitions also had economic underpinnings, such as the need to acquire mineral wealth and cheaper farmland for a growing population.
Industrially, American factories performed mostly well during this period, with the exception of a few major recessions that were mainly caused by factory overproduction. The cotton gin made the United States a key player in the textile industry, and Southern cotton powered the economies of the Northeastern United States as well as Britain. Interchangeable parts, also pioneered by Eli Whitney, helped to make American goods cheaper for people all over the world.
The Civil War demonstrated the power of the telegraph and the railroad in supplying soldiers—after the war, railroads become more common and people of the United States started to depend on store-bought goods more than what was traditionally made in the home. The boom in the railroad market as well as the need to create bigger factories and places to house workers made possible a boom in the steel industry—by the turn of the century, U.S. Steel was a household name all over the world. The need for workers to work longer hours led to the rise of the kerosene lamp, later to be replaced by electricity.
All of these things in industry were powered by American innovators such as Carnegie, Rockefeller, and Edison. While most Americans did not get rich from the industrialization of the nation, most Americans did at least receive some of the benefit of the new consumer goods produced by the United States.

Single Variable Calculus, Chapter 7, 7.2-1, Section 7.2-1, Problem 16

Find the domain of

a.) $f(x) = \sin (e^{-x})$

b.) $g(x) = \sqrt{1 - 2^x}$

a.) Recall that $f(x)$ is a trigonometric function that is defined for all values of $x$. Thus, the domain is $(- \infty, \infty)$.

b.) Since $g(x)$ contains square root,


$
\begin{equation}
\begin{aligned}

& 1 - 2^x \geq 0
\\
\\
& 1 \geq 2^x
\end{aligned}
\end{equation}
$


If we take the natural logarithm of both sides..

$\ln 1 \geq \ln 2^x$

By the property of logarithm


$
\begin{equation}
\begin{aligned}

& \ln 1 \geq x (\ln 2)
\\
\\
& x \leq \frac{\ln 1}{\ln 2}

\end{aligned}
\end{equation}
$


Thus, the domain of $g(x)$ is

$\displaystyle \left(- \infty, \frac{\ln 1}{\ln 2}\right]$ or $\displaystyle \left( -\infty, 0 \right]$

How does Romeo deal with his banishment in act 3?

The always emotional Romeo is hiding out in Friar Laurence's cell when he finds out he has been banished from Verona for killing Tybalt. Rather than experiencing gratitude that he has been spared the death penalty, he reacts very badly, feeling his life is over because he will be separated from his beloved Juliet. All the woes of the world seem to have to have fallen on his head. The friar scolds him and admonishes him to get a grip.
Juliet's nurse shows up with tokens of esteem from Juliet, but Romeo—perhaps not surprisingly, given he has just killed her cousin—fears she is angry at him. He threatens to stab himself and end it all. The friar tells him if he does, then his love for Juliet is a lie—"perjury"—for his death would kill Juliet, the beloved he had "vowed to cherish." Romeo's "wit," or intelligence, would be warped ("misshapen") if he acted so rashly:

Thy dear love sworn but hollow perjury,
Killing that love which thou hast vowed to cherish;
Thy wit, that ornament to shape and love,
Misshapen in the conduct of them both
Romeo sees reason and listens to the friar's counsel, but his impulsive and emotional behavior foreshadows how he will react when he thinks Juliet is dead near the end of the play.