Based on "The Love Suicides at Amijima" by Chikamatsu Monzaemon, provide information that can be used to write a background on the author and discuss how it influenced him to write the story.

Chikamatsu Monzaemon was a drama writer who made his name writing historically themed dramas. Rather than being performed by actors in a theatre setting, Chikamatsu's work was chanted by storytellers. In many cases, his works were also told through rudimentary puppet shows and artistic backdrops. The author was born in 1652 in what is the modern-day Fukui prefecture of Japan. Although he became a famous playwright, Chikamatsu was born into a family of samurai warriors. After his father gave up his military career, Chikamatsu's family moved to Kyoto. It is not clear how Chikamatsu came to be interested in theatre, but he began writing Kabuki plays shortly after his move to Kyoto and eventually moved to Osaka to pursue his career. There are actually two similarly titled stories by Chikamatsu, including "The Love Suicides at Sonezaki" and "Double Suicide at Amijima." "The Love Suicides at Sonezaki" marked a turning point in Chikamatsu's career. Previously, his work had focused on embellished historical tales of the nobility. "The Love Suicides at Sonezaki" and his newer works focused on everyday people, including average families, criminals and merchants. The dramatic tale of a double suicide was so popular that it inspired the author to write many others like it, effectively dividing his body of work into two distinct genres. Years later, he wrote "Double Suicide at Amijima," which focused on similar subject matter. Chikamatsu was so moved by the event that he finished his work almost immediately after he learned of it. Although the inspiration for Chikamatsu's earlier works is largely unknown, it is clear that the report of the actual double suicide "The Love Suicides at Sonezaki" was based on was responsible for sparking his interest in contemporary drama. When you consider that his early works were embellished stories of actual events, it becomes clear that Chikamatsu was more heavily influenced by the world around him than by fantasy or other works of fiction. From that point on, he continued to write about both historical and contemporary drama.
https://www.britannica.com/biography/Chikamatsu-Monzaemon

How does the play resolve the philosophical questions raised by Jocasta's judgment of the oracles?

The play resolves the philosophical questions raised by Jocasta's judgment of the oracle and of prophecy in general when all of the prophecies are shown to have come true. Laius was, in fact, killed by his child, despite his and Jocasta's best efforts to avoid the prophecy by ordering Oedipus's murder when he was a baby. Oedipus did, in fact, kill his father and marry his mother, despite his best efforts to avoid the prophecy by refusing to return to Corinth, the city he believes to be his home and the home of his parents. Teiresias also proved to be absolutely correct in his knowledge of both past and future events. Jocasta is shown to be absolutely wrong in her estimation of the value and accuracy of prophecy.

In Oedipus Rex Jocasta displays a marked skepticism towards the prophecies of the Delphic oracle. There could be more than an element of self-preservation involved here. After all, the oracle predicts that Jocasta and Laius's child would grow up to marry her and kill him. As far as Jocasta is concerned, that simply cannot happen. Her child was abandoned, exposed out in the open air; there is just no way he could have survived. It is in Jocasta's interests for the oracle's prophecies not to be true, and so it's not surprising that she should be so dismissive.
In response to Jocasta's skepticism, the Chorus makes clear just how serious a matter insolence towards the gods really is, how it brings utter ruin upon all those who display it. This feeds into a broader philosophical consideration of the relationship between gods and mortals. It's never clearly established whether Jocasta is skeptical of prophecies in general or just those that predict terrible consequences for her and her family. If it's the former, then she is indeed displaying insolence towards the gods. The resolution of this philosophical problem comes when the full, terrifying truth of the prophecy is made manifest and ruin is indeed brought crashing down upon Jocasta and her whole family.

After Soraya tells Amir about her past, she says, "I'm so lucky to have found you. You're so different from every Afghan guy I've met” (180). How do Afghan women fare in America? Are they any better off than they were in Afghanistan before the Taliban seized power? There is a noticeable absence of women in the novel. How is this significant?

Afghan women are treated with sexism in America. For example, Amir says that while fathers and sons can speak openly about women, no Afghan daughter can ever openly discuss a man with her father. In addition, a father would not bring up a man with his daughter unless the man were an approved suitor whose father has asked the girl's parents' permission for marriage (147). While boys run around and largely do as they wish, girls are expected to remain virtuous and prepare for their weddings. Soraya tells Amir that she was living with a man, and her father threatened to shoot the man and himself if Soraya did not return home. One might argue that, in the book, women are not much better off in America than they were in Afghanistan before the Taliban.
The absence of women in the novel shows that women are cloistered away. Afghan boys spend most of their time with other boys and men. The role of women in the context of this society has been reduced to the point at which they are barely noticeable. Instead, men are figures who take part in society, while women are behind closed doors.

what are some examples of research questions about mathematics in elementary grades?

Here are some areas of interest and questions that arise in mathematics teaching in the elementary grades:
Curriculum -- What content should be taught? When should it be taught? What are the objectives to teaching a specific idea or group of ideas? How should the content be arranged within a particular year and across the elementary grades? When should algebraic ideas be introduced?
Learning -- How important is discovery learning? How can discovery learning best be utilized in the classroom? What is the most effective use of manipulatives? Which fosters better retention - teaching for understanding or practical applications?
General -- How effective is homework? How much time should be devoted to practicing skills? How should classes be organized? How can we implement differentiated instruction especially as related to gifted or remedial students? Should we insist that elementary teachers be math specialists? Should we integrate computer assisted instruction, and if so what are the long term ramifications and limitations? How should gender inequality be addressed at the elementary level?
https://eric.ed.gov/?id=ED052033

I have to write an essay using Aristotle's definition of a tragic hero to compare the tragic stories of Othello and Oedipus, and then after that, I have to state which is a better tragic hero than the other. How do I start off this essay, and how do I compare them?

A good place to start such an essay might begin with a definition of the term, “tragic hero,” as defined by Aristotle in Poetics, and how that particular character elicits an emotional connection in and builds a relationship with his audience. While doing this, one might also touch upon the reasons why audiences are drawn to tragedies and what they gain from indulging in an instance of fictional schadenfreude.
With the definition of tragic hero firmly in place, one might next begin an examination of the characters, Othello and Oedipus, noting how each embodies (or perhaps does not embody) the Aristotelian ideal: how do they elicit respect, fear, and pity in their audiences? What fatal flaws lead to their respective downfalls, what do they learn as a result, and lastly, how does each accept or reject their fate? A thorough examination of the character arc--the journeys that Othello and Oedipus make--as well as a discussion of these, might also prove very useful when arguing the last section of your question, that of tragic degree.
Although some scholars and critics consider the character of Oedipus the penultimate tragic hero, your experience might differ. It might be helpful to think about whom you respected or related to more at the start of each play, whose journey engaged you more, and ultimately, which ending you found more satisfying.  
 
 
http://www.classics.upenn.edu/myth/php/tragedy/index.php?page=oedhero

Single Variable Calculus, Chapter 3, 3.1, Section 3.1, Problem 10

a.) Determine the slope of the tangent to the curve $\displaystyle y = \frac{1}{\sqrt{x}}$ at the point where $x = a$

Using the equation

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

Let $\displaystyle f(x) = \frac{1}{\sqrt{x}}$. So the slope of the tangent to the curve at the point where $x = a$ is


$
\begin{equation}
\begin{aligned}

\displaystyle m =& \lim \limits_{h \to 0} \frac{f(a + h) - f(a)}{h}
&& \\
\\
\displaystyle m =& \lim \limits_{h \to 0} \frac{\frac{1}{\sqrt{a + h}} - \frac{1}{\sqrt{a}}}{h}
&& \text{Substitute value of $a$}\\
\\
\displaystyle m =& \lim \limits_{h \to 0} \frac{\sqrt{a} - \sqrt{a + h}}{(h)(\sqrt{a}) (\sqrt{a + h})}
&& \text{Get the LCD on the numerator and simplify}\\
\\
\displaystyle m =& \lim \limits_{h \to 0} \frac{\sqrt{a} - \sqrt{a + h}}{(h)(\sqrt{a}) (\sqrt{a + h})} \cdot
\frac{\sqrt{a} + \sqrt{a + h}}{\sqrt{a} + \sqrt{a + h}}
&& \text{Multiply both numerator and denominator by $(\sqrt{a} + \sqrt{a + h})$}\\
\\
\displaystyle m =& \frac{a - (a+ h)}{(h)(\sqrt{a})(\sqrt{a + h})(\sqrt{a} + \sqrt{a + h})}
&& \text{Combine like terms}\\
\\
\displaystyle m =& \frac{-\cancel{h}}{\cancel{(h)}(\sqrt{a})(\sqrt{a + h})(\sqrt{a} + \sqrt{a + h})}
&& \text{Cancel out like terms}\\
\\
\displaystyle m =& \frac{-1}{(\sqrt{a}) (\sqrt{a + h})(\sqrt{a} + \sqrt{a + h})} = \frac{-1}{(\sqrt{a}) (\sqrt{a + 0}) (\sqrt{a} + \sqrt{a + 0})}
&& \text{Evaluate the limit}\\
\\
\displaystyle m =& \frac{-1}{2a \sqrt{a}}
&& \text{Slope of the tangent}

\end{aligned}
\end{equation}
$


b.) Determine the equations of the tangent lines at the points $(1, 1)$ and $\displaystyle\left(4, \frac{1}{2}\right)$

Solving for the equation of the tangent line at $(1, 1)$

Using the equation of slope of the tangent in part (a), we have


$
\begin{equation}
\begin{aligned}

a =& 1
&& \text{So the slope is }\\
\\
m =& \frac{-1}{2a \sqrt{a}}
&& \\
\\
m =& \frac{-1}{2(1) \sqrt{1}}
&& \text{Substitute value of $a$}\\
\\
m =& \frac{-1}{2}
&& \text{Slope of the tangent line at $(1, 1)$}\\
\end{aligned}
\end{equation}
$

Using point slope form

$
\begin{equation}
\begin{aligned}

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

Therefore,
The equation of the tangent line at $(1,1)$ is $ y = \displaystyle \frac{-x + 3}{2}$

Solving for the equation of the tangent line at $\displaystyle \left(4, \frac{1}{2}\right)$

Using the equation of slope of the tangent in part (a), we have $a = 4$. So the slope is


$
\begin{equation}
\begin{aligned}

m =& \frac{-1}{2a \sqrt{a}}
&& \\
\\
m =& \frac{-1}{2(4)\sqrt{4}}
&& \text{Substitute the value of $x, y$ and $m$}\\
\\
m =& \frac{-1}{16}
&& \text{Slope of the tangent line at $\left(4, \frac{1}{2}\right)$}

\end{aligned}
\end{equation}
$


Using point slope form



$
\begin{equation}
\begin{aligned}

y - y_1 =& m ( x - x_1)
&& \\
\\
y - \frac{1}{2} =& \frac{-1}{16} (x - 4 )
&& \text{Substitute value of $x, y$ and $m$}\\
\\
y =& \frac{-x + 4}{16} + \frac{1}{2}
&& \text{Get the LCD}\\
\\
y =& \frac{-x + 4 + 8}{16}
&& \text{Combine like terms}
\\
y =& \displaystyle \frac{-x + 12}{16}
\end{aligned}
\end{equation}
$


Therefore,
The equation of the tangent line at $\displaystyle\left(4, \frac{1}{2}\right)$ is $ y = \displaystyle \frac{-x + 12}{16}$


c.) Graph the curve and both tangent lines on a common screen.

Compare and contrast the subject matter of the Iliad and Dante's Inferno.

Homer's Iliad and Dante's Inferno are both epic poems that examine the negative consequences of unbridled human emotion—of giving in to unethical personal desires that harm others.
A big difference between the two works is the role of the gods and of destiny. A background story to The Iliad explains the role of the gods in setting up the conditions that lead to the Trojan War, the subject of the poem. Why does Paris of Troy abduct Helen, Queen of Sparta and wife of Menaleus, setting into motion the Trojan War? The goddesses Eris and Aphrodite had already set him up to carry out this brazen and selfish act. It was his destiny.
Sometime before the Trojan War begins, Zeus holds a banquet for the wedding of the goddess Thetis and King Peleus (parents of Achilles) and does not invite Eris, goddess of discord. She shows up anyway, determined to make trouble in revenge for the snub. She brings a golden apple for "the fairest." Zeus gives the touchy assignment of judging who is the fairest to Paris, a Spartan prince. When he chooses Aphrodite over Athena and Hera, the goddess of love rewards him with the most beautiful woman in the world: Helen, Queen of Sparta (later known as Helen of Troy). So Helen, though already married, and Paris become entwined by destiny as well as by unethical desires, thereby bringing about the Trojan War and all the loss, death, and brutality that ensues. Helen herself is a child born (hatched) as a result of one of Zeus's many indiscretions—the seduction of the mortal Leda while in the form of a swan.
In the ancient Iliad and its background stories, we see deities who can be both wanton and selfish, and we see flawed humans who, like the deities they worship, sometimes make choices based on unbridled desire, without thought given to the greater implications of their choices and the harm that could be caused to others.
Dante, on the other hand, writes from a worldview that is highly informed by the precepts of Christianity and the concept of free will. The Christian God and Christ are nothing like the deities of Homer's age. Christians must uphold a code of behavior that is ever-mindful of controlling passions that can harm others. In fact, Helen, Paris, and Achilles appear in the first of the nine circles of Hell in The Inferno (where each circle is increasingly more punitive) for the sins of lust and infidelity, while Odysseus is placed in the eighth circle for his far more serious sins of premeditated deception.
One might explore the question of why Dante did not place the gods and goddesses of the Greek pantheon in the circles of Hell as well.

Homer's Iliad is the story of part of the Trojan War concerned with the "wrath of Achilles." At the beginning of the epic, Achilles has a disagreement with Agamemnon concerning a war prize and withdraws from the fighting. Being the greatest of the Greek warriors, Achilles's absence limits the abilities of the Greeks to succeed on the battlefield. At the end of the epic, he returns to the fighting and kills Hector, the greatest of the Trojan heroes. It is a pagan poem, grounded in polytheism, and a story of a war, describing in detail many battles, discussions of strategy, and issues of honor in battle.
Dante's Inferno is a Christian poem detailing an imaginary journey through the circles of Hell as the narrator is escorted by the Latin poet Virgil. We encounter many famous historical people on the journey within the context of Christian religion. The narrator at the end of the poem emerges from his journey on Easter day.
While both poems are important works that became both standard reading and symbols of cultural greatness, they are very different, with one focusing on war and the other on religion and one belonging to oral tradition and the other to literary practice.
The major similarity between them is that ancient Greece and Renaissance Italy were not actually nations but rather collections of city states and territories united by cultural and linguistic ties. Thus poetry and other cultural performances gave them a sense of unity that did not exist in the political realm.

Homer's other great epic, the Odyssey, probably has more in common with Dante's Inferno than the Iliad. Indeed, the Odyssey, like the Inferno, involves the hero descending into the underworld and returning after encountering its inhabitants. But both the Iliad and The Divine Comedy in general are epic poems, emphasizing the complex relationship between humanity and the divine. Perhaps the most obvious connection between the two poems is not really thematic, though. Dante populates hell with many of the figures from Homer's poems, including the blind poet himself. (Remember that the Roman epic poet Virgil is actually Dante's guide in Inferno). In Canto IV we discover Homer, along with other poets, including Horace, Ovid, and Lucan—all poets who would have been familiar to Dante's readers—and Hector, one of the main characters in the Iliad. These men are in the First Circle of Hell, reserved for those who were not evil people, but had been born before Christ and were therefore not able to be saved. One circle deeper into Hell, Dante encounters Achilles and Paris, two major figures in the Iliad, who are there for the sin of lust. Helen of Troy is there, too, for her betrayal of her husband, Menelaus. Their actions essentially set the events portrayed in the Iliad in motion. Odysseus, one of the most important characters in the Iliad, is in the Eighth Circle of Hell for his trickery. So there are many connections between Dante's Inferno and the Iliad, and a good essay might take this a step further by considering the reasons that Dante chose to portray these characters in Hell.