Calculus: Early Transcendentals, Chapter 7, 7.2, Section 7.2, Problem 47

int(1-tan^2(x))/(sec^2(x))dx
Rewrite the integrand as,
int(1-tan^2(x))/(sec^2(x))dx=int(1-(sin^2(x))/(cos^2(x)))/(1/(cos^2(x)))dx
=int(cos^2(x)-sin^2(x))dx
Now use the following identities:
cos^2(x)=(1+cos(2x))/2
sin^2(x)=(1-cos(2x))/2
=int((1+cos(2x))/2-(1-cos(2x))/2)dx
=int(1+cos(2x)-1+cos(2x))/2dx
=intcos(2x)dx
=sin(2x)/2
add a constant C to the solution,
=sin(2x)/2+C

After witnessing the narrator's emotional collapse through the story's end, do you conclude that she displays herself as legitimately psychotic, or do you think the “treatment” by her physician-husband has made her so?

Charlotte Perkins Gilman has included many aspects in her own life, mirroring the narrator; namely, PPD. At the time when "The Yellow Wallpaper" was written, 1892, women were still seen as delicate creatures. Perkins Gilman herself was given a "rest cure" by Dr. S. Weir Mitchell. The narrator's treatment prescribed by her husband parallels Perkins Gilman's "cure." In fact, Dr. Mitchell is even mentioned in the text itself.
While the narrator does suffer from mental illness, it is only made worse by her husband's idea of a cure. From the start of the short story, the narrator is locked away in a room with a bolted down bed, "rings and things" on the walls, and a barred window. John treats his wife as a childish mental patient, as he calls her juvenile pet names, rather than a spouse. Indeed, John's sister, Jennie, even watches over the narrator, encouraging her not to do anything too strenuous, including writing.
The narrator is isolated, alone with her own thoughts and the "horrid" yellow wallpaper. As one does when one is bored, she traces the patterns in the wallpaper. These patterns soon take form, a hallucination brought on by her illness. The madness in the wallpaper is a reflection of her own spiral into insanity. The woman who shakes the wallpaper is the projection of narrator herself, trapped, alone, and striving to escape. Clearly at this point of the story, the rest cure John prescribed is anything but effective.
As the narrator continues to succumb into her madness, we see less and less of John, or even Jennie; it's simply the narrator and the wallpaper. The woman in the wallpaper needs freedom, just as the narrator needs freedom from her illness. The climax of the story where we see the complete breakdown of the narrator as she "frees" the woman in the wallpaper by tearing it down, serves as the peak state of her psychosis. She finds a rope and throws away the key to her room. John comes and faints at the sight of her. The narrator questions why he should faint, giving the assumption that the narrator's appearance reflects her inner turmoil or even that she committed suicide and tells the end of her story from a voyeuristic perspective.
So while the narrator does suffer from a psychotic breakdown, her lack of proper treatment ultimately led to her assumed suicide. One cannot completely blame her husband; however, if we take into consideration Perkins Gilman's own life, perhaps her implied intention was to showcase the misunderstanding of her own husband and doctors at the time. Perkins Gilman "escaped" from her own husband and baby to travel west. She got her freedom just as the narrator did.

The protagonist in Charlotte Perkins Gilman's "The Yellow Wallpaper" does experience an emotional collapse at the end of the short story. That said, to determine if her breakdown is the result of psychosis or treatment by her husband, the reader must look at how her breakdown "speaks" to him or her individually. This type of answer is subjective.
Literary critics have speculated that the protagonist suffers from postpartum depression. Her husband, a doctor, has "prescribed" the treatment plan he believes to be the best for her: isolation and time to relax. Unfortunately, many people do not do well in isolation. Given that people are social creatures, humans need social interaction. Isolation and lack of social interaction may cause one to lose one's grasp on reality. Due to the protagonist's isolation, one could infer that she went insane (when compounded with her PPD).
Therefore, one could argue that her undiagnosed PPD could have led to her breakdown. One could also argue that her husband's treatment, or lack of treatment, led to her emotional collapse. In the end, a solid argument can be made for either point.

How is Cassius able to sway characters away from their moral code?

Cassius is able to convince Brutus that his close friend, Julius Caesar, is a threat to the Roman Republic and persuades Brutus to join the conspirators. Cassius is aware of Brutus's honorable nature and forms a brilliant argument that depicts Caesar as an enemy of Rome. Cassius tells Brutus that Caesar already believes that he is more exalted, accomplished, and better than any citizen or senator and plans to rule Rome as a monarch. Cassius also cites Caesar's ambitious nature to sway Brutus into joining the conspirators. In addition to presenting a moving argument that Caesar will destroy the Roman Republic for personal gain, Cassius also has false letters written by "concerned" citizens strategically placed where Brutus will find them. Cassius then appeals to Brutus's sense of duty by mentioning his revered ancestors, who defeated Rome's last emperor to establish the Republic. In summary, Cassius appeals to Brutus's honorable nature, gives numerous examples of Caesar's arrogance, plants false letters in his home, and argues that Caesar is an enemy of the Roman populace to convince Brutus to join the conspirators and assassinate Julius Caesar.

College Algebra, Chapter 5, Review Exercise, Section Review Exercise, Problem 48

Expand the Logarithmic Expression $\displaystyle \log \left( \frac{4x^3}{y^2 (x - 1)^5} \right)$

$
\begin{equation}
\begin{aligned}
\log \left( \frac{4x^3}{y^2 (x - 1)^5} \right) &= \log 4x^3 - \log y^2 (x - 1)^5 && \text{Laws of Logarithm } \log_a \frac{A}{B} = \log_a A - \log_a B\\
\\
\log \left( \frac{4x^3}{y^2 (x - 1)^5} \right) &= \log 4 + \log x^3 - \left( \log y^2 + \log (x - 1)^5 \right) && \text{Laws of Logarithm } \log_a AB = \log_a A + \log_a B\\
\\
\log \left( \frac{4x^3}{y^2 (x - 1)^5} \right) &= \log 4 + 3 \log x - ( 2 \log y + 5 \log (x - 1)) && \text{Laws of Logarithm } \log_a A^c = C\log_a A\\
\\
\log \left( \frac{4x^3}{y^2 (x - 1)^5} \right) &= \log 4 + 3 \log x - 2 \log y - 5 \log (x - 1) && \text{Distributive Property}
\end{aligned}
\end{equation}
$

What role does setting play in William Faulkner's “A Rose for Emily” and Ernest Hemingway's “Hills Like White Elephants"? How does it affect the plot and characters? What quotes support this?

In William Faulkner's "A Rose for Emily," the setting/environment of the South plays an intricate cultural and existential role in the life of the main character, Miss Emily Grierson. But in Ernest Hemingway's "Hills Like White Elephants," the setting of the story takes on a symbolic role for the main characters.
"A Rose for Emily"
The renowned British writer W. Somerset Maugham wrote, 

It is very difficult to know people. For men and women are not only themselves, they are also the region in which they are born, [where] they learned to walk, the games they played as children, the old wives' tales they overheard...the schools they attended...the poets they read, and the God they believed in. 

This insight of Maugham's is certainly applicable to the character of Miss Emily Grierson of "A Rose for Emily." It is difficult for some of the townspeople to understand Miss Emily. Time has passed her by; she is a relic of the region and culture of the Old South, a culture controlled by patriarchy and Jim Crow. Using an example of this perception of Emily Grierson as a product of her culture, the narrator reflects that after she dies, she is "a fallen monument."

...only Miss Emily's house was left, lifting its stubborn and coquettish decay.... Miss Emily had gone to join the representatives of those august names where they lay in in the cedar-bemused cemetery among the ranked and anonymous graves of Union and Confederate soldiers who fell at the battle of Jefferson. 

That Emily is so much a "monument" of the post–Civil War culture is further demonstrated by her attitude toward the aldermen as they approach her for payment of her taxes. Still dwelling in the Old South in her mind, she tells them, "I have no taxes," because her father had made an arrangement with Colonel Sartoris, who was then the mayor.That she lived under the patriarchy of her father is further demonstrated by Miss Emily's not having married; her stern patriarch "vanquished" any suitors because he felt they were not suitable. Indeed, the portrait of Emily and her father when she was a girl reveals their relationship: she is in the background and her father a looming figure in the foreground, "his back to her and clutching a horse whip."
When Emily's father dies, the ladies of town come to her house to offer condolences. However, Emily meets them at the door and tells them that her father is not dead. After three days of ministers and doctors visiting her, Miss Emily finally breaks down, and her father, under whose shadow she lived, is quickly buried.
With the death of her father, Emily begins to be seen with a Northerner, a Mr. Homer Barron. As she continues to see Mr. Barron, the townspeople gossip. Finally, the Baptist minister "calls upon" Emily because some of the citizens find Emily's behavior disturbing. Even relatives from Alabama are sent for in order to dissuade Miss Emily from being seen with the likes of Barron--an action "unbecoming" to a Southern lady. Further, after "the old Negro servant" admits Homer at the back entrance one evening, Barron is seen no more and Miss Emily becomes very reclusive. Then, a strange odor emanates from the house. However, when the neighbors complain, a younger alderman suggests that word be sent to Miss Emily to get rid of the smell. One old Southerner is angered at this affront to Miss Emily Grierson:

"Dammit, sir," Judge Stevens said, "will you accuse a lady to her face of smelling bad?"

Again, Emily retreats into her Old South environment as she gives China painting lessons. But when these girls who have been her students grow up, no one else comes to the old Grierson home. Not long afterwards, Emily dies. In attendance at her funeral are very old men--

...some in their brushed Confederate uniforms...talking of Miss Emily as if she had been a contemporary of theirs, believing that they had danced with her and courted her, perhaps....

Miss Emily, an anachronism, the narrator notes, has passed from generation to generation in a setting that has been impervious to time.
"Hills Like White Elephants"
Symbolism is intrinsic to the narrative of Hemingway's short story. Certainly the dialogue is minimal and the characters limited. In fact, one of the characters is not even given a name.
The narrative revolves around a conversation between lovers, the American and a girl named Jig, as they sit at a table outside the railroad station building. The train tracks that go between Barcelona and Madrid, Spain, create a separation of the terrain. On one side "there was no shade and no trees"; on the other "were fields of grain and trees along the bank of the Ebro." The contrast between barren land and that which is fertile on the two sides of the tracks are symbolic of the outcome of the decision the girl faces. She and the man discuss her having an abortion.
Because the young man does not want to be a father, he tells his lover, "We can have the whole world." However, she disagrees, "No, we can't." She contends that the world is not theirs any more because "once they take it away, you never get it back." She, of course, contemplates having a baby or not having a baby as well as the selfishness of her lover, while the man thinks only of a procedure. Truly, they are divided, just as the land around them is divided.
The division between the fertile land and the barren white hills symbolizes the dichotomy of death and life that the girl's choice involves. Interestingly, while the man remains at the table on the side with the hills that are like white elephants, a term used at times for objects no one wants, the girl named Jig walks around so that she can view the scenery on the other side where there are fields of grain and trees. Significantly, "a shadow of a cloud moved across the field of grain..." as she looks out.
This shadow signifies to the girl that whether or not she has the "operation," everything will not be fine, although the man says it will. She "looks across at the hills on the dry side of the valley..." knowing that the relationship is over. Thus the setting has taken on a symbolic role in this narrative.

Compare the productive and allocative efficiency of monopolistic and perfect competition.

A monopolistic market has less productive and allocative efficiency than a perfectly competitive market.
In terms of productive efficiency, a monopolistic firm operates at a price that is higher than its minimum average cost. The firm can achieve minimum average cost by increasing its output, but it will not do so because it achieves maximum profit when its output is less than the point where it achieves minimum average cost. In addition, a monopolistic firm wastes resources that could have been used for production on other things like influencing government policies in order to maintain its monopolistic market status.
In terms of allocative efficiency, a monopolistic firm charges a price that is higher than its marginal costs at its optimum output (Marginal Cost = Marginal Revenue). The firm can achieve maximum social benefit by increasing its output, but it will not do so because it achieves maximum profit when its output is less than the point where it achieves maximum social benefit (Marginal Cost = Demand). This results in net loss of consumer surplus and producer surplus, compared with a perfectly competitive firm at its optimal output.

What is an example of a linguistic or structural technique used in Act III, Scene 3 of Macbeth?

Though Act III, Scene 3 of Shakespeare's Macbeth is short, it has some remarkable structural and linguistic devices at play that are worth examining. The component I want to focus on is the regular rhyming pattern used within the dialogue, as it is both a linguistic and structural device that lends considerable rhythm to the text. As an example, check out the first six lines of the scene:

FIRST MURDERER: 
But who did bid thee join with us?
THIRD MURDERER: 
Macbeth.
SECOND MURDERER: 
He needs not our mistrust, since he deliversOur offices and what we have to doTo the direction just.
FIRST MURDERER: 
Then stand with us.

As you can see, Shakespeare utilizes a repeated rhyme within the murderers' dialogue to create a poetic rhythm and pattern. This decision is both linguistic and structural because it ties the language of the scene more closely together, making the violent event to come (Banquo's murder) more compact and ruthless. Indeed, the brutal events of the scene are hauntingly juxtaposed by the murderers' rhythmic, poetic language. 
Shakespeare uses a similar linguistic and structural device when he depicts the Weird Sisters. Just as the murderers' dialogue is glued together by regular rhymes, so too is the witches' conversation ordered by a poetic, rhyming pattern. Throughout the play, Shakespeare used tightly crafted poetry to hint at (and juxtapose) the dark deeds and the villainous sides of human nature.