College Algebra, Chapter 9, 9.2, Section 9.2, Problem 48

Determine the partial sum $s_u$ of the arithmetic sequence that satisfies $a_2 = 8, a_5 = 9.5$ and $n = 15$.

First, we need to find the common difference of the sequence to find the first term. So,
For $a_2$,

$
\begin{equation}
\begin{aligned}
a_n &= a + (n - 1) d\\
\\
a_2 &= a + (2 - 1) d\\
\\
8 &= a + d && \text{Equation 1}
\end{aligned}
\end{equation}
$

For $a_5$,

$
\begin{equation}
\begin{aligned}
a_5 &= a + (5 -1 ) d\\
\\
9.5 &= a + 4d && \text{Equation 2}
\end{aligned}
\end{equation}
$

Using Equations 1 and 2, we get

$
\begin{equation}
\begin{aligned}
8 -d &= 9.5 - 4d \\
\\
3d &= 1.5\\
\\
d &= 0.5
\end{aligned}
\end{equation}
$

Thus, the first term is

$
\begin{equation}
\begin{aligned}
8 &= a + 0.5\\
\\
a &= 7.5
\end{aligned}
\end{equation}
$

Therefore, the sum is

$
\begin{equation}
\begin{aligned}
s_n &= \frac{n}{2}[2a + (n - 1) d]\\
\\
s_{15} &= \frac{15}{2} [ 2 (7.5) + (15 -1) (0.5)]\\
\\
&= 165
\end{aligned}
\end{equation}
$

Why is Atticus so insistent that Jem killed Bob Ewell?

In chapter 30, Atticus and Sheriff Tate are discussing Bob Ewell's death, and Atticus mentions that he believes Jem was responsible for murdering Bob in self-defense. However, Sheriff Tate disagrees with Atticus and insists that Bob Ewell fell on his own knife and died. Atticus initially believes that Sheriff Tate is attempting to cover up Jem's role in Bob Ewell's death and refuses to allow Tate to protect Jem. As a morally upright man, Atticus has tried his best to raise his children with integrity and does not want to let them down as a parent. Atticus explains his reasoning for not allowing Sheriff Tate to cover up Jem's involvement by saying,

"Before Jem looks at anyone else he looks at me, and I’ve tried to live so I can look squarely back at him . . . if I connived at something like this, frankly I couldn’t meet his eye, and the day I can’t do that I’ll know I’ve lost him. I don’t want to lose him and Scout, because they’re all I’ve got" (Lee, 277).

Essentially, Atticus does not want his children to discover that he attempted to manipulate the law in any way to protect Jem, because that would make him a hypocrite like the other Maycomb citizens. Eventually, Sheriff Tate indirectly informs Atticus that Boo Radley was the person who stabbed and killed Bob Ewell during the struggle.

What is Takaki's thesis about American history in chapter 1?

In chapter 1, "A Different Mirror: The Making of Multicultural America," Ronald Takaki's thesis about American history is that it's always been told wrongly as a story about how European settlers and their descendants are the true Americans.
Let me restate that thesis in a more detailed way, borrowing some of Takaki's wording: Embedded deep within American history is a "Master Narrative" that is "powerful and popular but inaccurate," acting as a "filter" that still affects our perceptions of each other as either American or "Other," and that false narrative is this: ". . . our country was settled by European immigrants, and Americans are white."
In other words, Takaki's thesis is that American history has always been told as a story with the wrong structure, the wrong basic belief. "Not all of us came originally from Europe!" he points out, citing how a third of us have no European ancestry and even how major cities—in addition to the state of California—are predominated by people with other ancestries.
In chapter 1, Takaki goes on to develop this thesis by spotlighting the historians whose work created the false narrative and allowed it to propagate. Takaki challenges the false narrative, pointing out how it's crumbling today in the face of reality. (The reality is that "our expanding racial diversity is challenging the Master Narrative.")
He strikes an optimistic tone when he traces how multiculturalism is now being acknowledged and embraced in universities across the nation, but he asserts that we must "study race and ethnicity" in an inclusive manner, rather than missing the point by studying one race at a time in isolation.
Next, he relates the history of the various ethnic groups who constitute America, explaining how and when they arrived en masse—and, in the case of Native Americans, how they were here all along. Takaki ties these groups together, conceptually, by arguing that they all shared "experiences and dreams" as well as a love of America's principle of "equality for everyone, regardless of race or religion."
He summarizes the history of all these multicultural Americans, tracing their participation in wars and major cultural upheavals such as the Civil Rights Movement.
Takaki concludes by reiterating his thesis: that the false Master Narrative must be replaced by the true narrative that America has been settled by "the people of all nations."

This question is referring to historian Ronald Takaki's 1993 book A Different Mirror: A History of Multicultural America. The title is significant, because it basically outlines Takaki's main argument. In the first chapter, Takaki takes aim at what he calls the "Master Narrative of American History." This, he argues, is a sort of assumption that pervaded the writing and teaching of American history for more than a century, described by Takaki as follows:

. . . [O]ur country was settled by European immigrants, and Americans are white. "Race," observed Toni Morrison, has functioned as a "metaphor" necessary to the "construction of Americanness": in the creation of our national identity, "American" has been defined as "white." Not to be "white" is to be designated as the "other"—different, inferior, and unassimilable.

In A Different Mirror Takaki offers a different narrative, one based on the conclusion of decades of historical scholarship that the United States is "a nation peopled by the world, and we are all Americans." He attempts to go beyond scholarship that has focused on one minority group in order to emphasize the multicultural nature of the United States.
Throughout the first chapter, he outlines the narrative, which includes, by way of analysis and comparison, the experiences of Americans of African, Asian, Irish, Jewish, Mexican, Muslim, and Native ancestry or cultural identification. United States history looks very different when seen from these multiple perspectives. It is Takaki's argument in chapter 1 that this understanding is more accurate, relevant, and explanatory than the so-called "Master Narrative."

Calculus of a Single Variable, Chapter 2, 2.2, Section 2.2, Problem 60

So I will start by rewriting this equation in the form:
f(x)=x^2+9
We know that if we have f'(x) then we can plug in any value for x, and the result will be what the slope of the function is at that given point.
So if we find the derivative of this original function, we should be left with the following:
f'(x)=2x
If the tangent line to this graph is a horizontal line, then the slope should be 0/x or just 0.
So we can set what we calculated for f'(x) equal to zero and we can solve for our x-value:
2x=0
x=0
So, this tells us that if our "x" value is zero, then the slope should be horizontal. Now that we know the x-value of the coordinate, lets plug this into the original equation to find the y-coordinate!:
y=0^2+9=9
Therefore, our coordinates for the point where the tangent line to the graph is horizontal would be (0,9)

Consider the power of a lie. Oedipus grew up thinking he was someone else's son and the truth unfurled destiny. Othello was happy in his marriage until he was plagued by the lies and schemes of Iago. How much of a role did fate play in the fault of men? Argue whether in those plays it is men's will and their words which will shape the tragic hero's destiny, the hand of fate, or both. Take into account Aristotle's definition of what makes a tragic hero.

Aristotle defined the tragic hero as one whose error in judgment leads to his downfall. While fate plays the major role in Oedipus's downfall, both Oedipus and Othello make mistakes due to a tragic flaw in themselves that leads to errors in judgment.
Oedipus is destined to murder his mother and marry his mother. That is his inescapable fate, and why initially, he is meant to be killed as an infant. He doesn't know this fate, because people have lied to him about his real identity. On some fundamental level, the lies don't matter--his fate is going to be the same regardless of what people say to him. And the lies are well meant, part of a hope he can defy his destiny.  
Although his fate is preordained, his own flaw, his hubris or pride, makes the situation worse. He simply can't imagine that he might be the cause of the plague in Thebes. Had he had less pride, the truth might have been easier for him to bear; the consequences less dire. He might, for example, have been able to truly see with his inner eyes without having had to blind himself.
In Othello, there is no sense that Othello's fate is preordained. No gods or oracles have  decreed he will murder his wife. In this case,  therefore, Iago's lies and Othello's own tragic flaws loom larger in the tragedy that unfolds.
Othello's flaw is his insecurity. For all his success as a military leader, he can't truly believe that Desdemona could love him. His error in judgment is in trusting Iago. Iago's lies are not well-meant. Iago wants to destroy him. If Othello had had more confidence in his ability to be loved by a woman and more ability to discern Iago's falseness, this tragedy could have been averted. 
 
 

What are some things that have changed when the group of men came back from the safari? (A sound of Thunder)

When the safari participants return, many things have changed in the year 2055, due to an act of carelessness on Eckels's part.
In Ray Bradbury's short story "A Sound of Thunder," Eckels, the main character, pays for a time travel safari to the Cretaceous period to hunt a tyrannosaurus rex. The safari guide, Travis, warns the participants not to leave the metal, anti-gravity path that has been laid by Time Safari, Inc., to ensure that nothing is touched in the past. When Eckels asks why Travis explains the chain reaction of events in great detail: 

"Eventually, it all boils down to this: Fifty nine million years later, a cave man, one of a dozen in the entire world, goes hunting wild boar or saber toothed tiger for food. But you, friend, have stepped on all the saber toothed tigers in that region. By stepping on one single mouse. So the cave man starves. And the cave man, please note, is not just any expendable man, no! He is an entire future nation. From his loins would have sprung ten sons. From their loins one hundred sons. And thus onward toward civilization." 

Despite the detailed warnings, Eckels becomes so awestruck and afraid when he sees the tyrannosaurus rex, that he steps off the path and into the jungle moss. In doing this, he killed a butterfly, which changed the course of human history through a chain reaction of events. 
Some things that changed as a result of Eckels killing the butterfly were: the chemical taint to the air when he returned, the man sitting at the desk was "not quite the same," the spelling on the sign at Time Safari, Inc., had changed, and of course, the biggest change was that Deutscher, the anti-everything man, had been elected president instead of Keith. 

In "Everyday Use" by Alice Walker, how is Mama's parenting effective?

In this story, Mama's parenting is effective because the right daughter ends up finally receiving the recognition she deserves.  It seems as though Mama has spent a lot of time getting Dee the things she wants and dreaming of being enough to satisfy (and not embarrass) her worldly daughter. Humble Maggie has been left somewhat in the shadows.  Toward the end of the story, when Dee insists that she get to take the family quilts—quilts she actually rejected when she went off to school—Mama has a sort of epiphany. She recalls,

When I looked at her like that something hit me in the top of my head and ran down to the soles of my feet. Just like when I'm in church and the spirit of God touches me and I get happy and shout. I did something I never done before: hugged Maggie to me, then dragged her on into the room, snatched the quilts out of Miss Wangero's hands and dumped them into Maggie's lap.

Mama finally seems to understand that Maggie has never felt special or deserving in the way her sister seems to have always felt. She recognizes that Maggie truly deserves the quilts because she's the one who truly appreciates the quilts, uses them, and knows her heritage; Dee only seems to want the quilts in order to show them off. She doesn't know the stories the way Maggie does. Finally, Mama rewards Maggie by giving her all the quilts and denying them to the less-appreciative and somewhat exploitative Dee.