Why doesn't Macbeth kill Macduff? How is this significant to the play?

In Act IV, Scene 1, the apparitions warn Macbeth to be careful of Macduff, but also maintain that "none of woman born/ Shall harm Macbeth."
Macbeth is happy to hear this, but soon decides he will rid himself of Macduff anyway, just to ensure his chances of maintaining his hold on power. He soon discovers Macduff has fled England, and is miffed he didn't act sooner. To ensure no heirs follow Macduff, he orders Macduff's wife and children to be killed.

The castle of Macduff I will surprise,
Seize upon Fife, give to th' edge o' th' sword
His wife, his babes, and all unfortunate souls
That trace him in his line.

In Act V, Scene 7, Macbeth kills young Siward and proclaims he wasn't afraid to face the warrior because Siward was born of a woman. In Act V, Scene 8, Macbeth initially refuses to fight Macduff; he claims he's killed enough of Macduff's family members and wants to refrain from killing Macduff, too. Furthermore, Macbeth argues that he leads "a charmèd life, which must not yield/ To one of woman born."
 
At this point, Macbeth doesn't make any moves to kill Macduff because he thinks Macduff is no threat to him. It is only after Macduff asserts that he was "untimely ripped" from his mother's womb that Macbeth begins to take his challenger seriously.
 
So, it's significant that Macbeth doesn't initially manage to kill Macduff. Earlier in the play, Macduff escaped Macbeth's clutches when he fled England. Later, on the battlefield, Macbeth initially refuses to engage Macduff. He's only forced to fight Macduff when Macduff corners him. In the end, Macduff beheads Macbeth and proclaims Malcolm king of Scotland. In the play, Macduff fills his role as the one man who can kill Macbeth; his character fulfills the earlier prophecy that Macbeth should both beware of Macduff and be wary of the man who is not "born from a woman."

Intermediate Algebra, Chapter 1, Review Exercises, Section Review Exercises, Problem 10

Write the set $\{ x|x$ is a natural number between 3 and 9 $\}$ by listing its elements.

The natural numbers between 3 and 9 are $\{ 4,5,6,7,8 \}$.

Would you argue that the areas (i.e. Asia, the Middle East, Latin America, and Africa) can properly be called non-western? Why or why not? Would you agree or disagree with Huntington’s thesis (Huntington, S. (1993). The clash of civilizations? Foreign Affairs, 72(3), 22-49) that the differences between western and non-western cultures are an inevitable source of conflict?

Huntington defines Asia, the Middle East, Latin America, and Africa as non-western. He writes, "Civilizations are differentiated from each other by history, language, culture, tradition, and, most important, religion" (page 25). He states that economic modernization and social changes have weakened the nation-state and have made religion more important in helping people define their identities, leading to fundamentalist forms of religion. Asia, the Middle East, and Africa are partly non-western. It could be argued that they are also western in many ways, as parts of these regions also follow Christian religious traditions, which are western. In addition, their histories and languages are in part western, as the recent history of the world is interconnected, and western languages, particularly English, are now spoken almost everywhere. However, religious differences remain in these regions, particularly the Middle East. Latin America is more western in its orientation, as it has western religious traditions (mainly Catholicism), languages, and recent history.
The conflict between the western and Islamic world, as Huntington pointed out, has been going on for over 1,300 years (page 31). Increased immigration from the troubled Middle East to Europe and other areas of the west has only worsened this tension and increased forms of racism. Therefore, for the short term, it seems like the situation between civilizations will be one of conflict; however, it is difficult to predict if increased immigration to the west will eventually result in making the west more comfortable with Islam. Therefore, it is unclear whether conflict is inevitable in the long term. In addition, our relationships with China seem to be improving over time, resulting in more cooperation than conflict. 
 

Which group suffered most from European colonization of the Western hemisphere: Native Americans or enslaved Africans?

Both Native Americans and Africans suffered at the hands of European colonizers. When we consider the question of who suffered more, it's important to understand perspectives.  In my answer, I'll look at a variety of perspectives and compare the impact of European colonization on both groups.
Social
The most immediate answer as to which group suffered more would be... well, both.  But if we look at the Caste System (la Sociedad de las Castas), I would argue that enslaved Africans were most negatively impacted by European colonies in the New World.  When colonies were created, Europeans brought with them a social system that organized the inhabitants (both free and enslaved) based on the percentage of European blood in them, or "whiteness."  At the top of the caste system were the Europeans, both Spanish/Portuguese born as well as Europeans born in the New World.  Underneath those Europeans were the mixed races, with preference to mixes that had higher percentages of "whiteness."  Underneath the mixes were the Native Americans and freed blacks, and at the bottom were African slaves.  As far as opportunity is concerned, African slaves definitely suffered the most.  While Native Americans were also denied opportunities, the higher presence of African slaves on the Eastern coast as well as in Caribbean islands would suggest that African slaves were the lowest (and therefore suffered most) socially.
Political
Politically, I would argue that Native Americans suffered the most from European colonization of the New World.  Prior to European arrival, strong empires like the Aztecs in Mexico and the Inca in the Andes Mountains of South America ran efficient, organized empires.  When Europeans made contact, better weapons and smallpox transmission (due to Native Americans' lack of immunity) wiped out large percentages of the populations, allowing Europeans to settle and ultimately colonize.  While it's true that both Native Americans AND Africans did not have a say in colonial government (captaincies for Portuguese territories, viceroyalties for Spanish territories, and colonial governments and charters in the British colonies), I would argue that because the political foundation set by Native Americans was destroyed and replaced with European institutions, and that Native Americans were forced to suffer labor and discrimination from the new political leaders, that the Native Americans suffered more than the enslaved Africans.
Economic
For similar reasons as my political analysis, I would argue that Native Americans suffered more--because their governments and empires were replaced, and the new European colonies profited off the natural resources of Native American land, the Native Americans suffered more.
Labor
Without a doubt, I would argue that enslaved Africans suffered more at the hands of European colonial governments.  While it is true that Native Americans were also forced to work (in the Andes Mountains, the old Inca mit'a system of lottery-style labor drafts was used against the indigenous populations for silver mining), a diminished population of Native Americans due to diseases like smallpox meant that a new labor force had to be brought in to the plantations of the Americas.  This is where we see the intensification of slave trading from Africa to the New World.  A majority of the slaves that were shipped over the Atlantic went to either Brazil or the Caribbean to work on sugarcane plantations, where conditions were notoriously brutal and life expectancy was low.  As a result, more slaves were shipped in to replace the loss of labor when a slave died.  This callous exchange of bodies leads me to believe that the African slaves suffered more at the hands of European colonizers, even though Native American labor was also exploited.  
Conclusion
While both Native Americans and Africans suffered at the hands of Europeans, based on sheer brutality and physical suffering, I would argue that enslaved Africans had the harder time of living in New World colonies.  While Native Americans were also forced to give up their land, lose possession of resources, and were exploited, the sheer brutality of slave trade as well as the poor treatment of African slaves puts them higher up in my analysis. To a significant extent, though, comparison is a futile exercise: the important thing is to recognize what both groups endured and carry this awareness into the future, where we hope to prevent atrocities like these.

Explain how the Era of Good Feelings emerged and what factors led to its demise.

The "Era of Good Feelings" emerged in 1815 as a result of the disintegration of the Federalist party after the Hartford Convention and a lull in partisan politics between the Democratic-Republics and Federalists.
This era was named as such because it represented a time in which the sense of national purpose and desire for unity in America was at a high as a result of the War of 1812 and Napoleonic Wars. The country had begun to envision the federal government as having a permanent and significant role in the well-being of the nation, and, following suit, Madison authorized a protective tariff on manufactures and measures for a national bank. Optimism seemed to be overshadowing previous political conflicts.  
Despite the title, factions of the Monroe administration and the Republican Party were still experiencing the weight of divisive politics. Monroe attempted to include top prospective presidential cabinets and leaders in his own cabinet to manage factional disputes; however, amalgamation resulted in his failure to appeal to Republican solidarity, resulting in a loss of party discipline.
The Panic of 1819 and McCullock v. Maryland ultimately resurrected the political debates about state sovereignty, federal power, and loose versus strict interpretation of the Constitution. The Missouri Crisis of 1820 also contributed to this conflict, and the country saw a sudden decline in the mutual agreement that had marked the era. Thus, the modern Democratic Party was born as an answer to this division, and the Era of Good Feelings came to an end.

Calculus: Early Transcendentals, Chapter 1, 1.1, Section 1.1, Problem 26

The amount of air required to inflate a spherical balloon from radius r to radius (r+1) is required:
The difference in the volumes is:
V_2-V_1=4/3 pi (r+1)^3-4/3 pi r^3
=4/3 pi[r^3+3r^2+3r+1-r^3]
=4/3 pi[3r^2+3r+1]
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Given an initial radius r, the amount of air A needed to increase the radius of a spherical balloon is:
A=4/3 pi(3r^2+3r+1)
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Intermediate Algebra, Chapter 1, 1.2, Section 1.2, Problem 28

Find the difference $\displaystyle \frac{3}{14} - \left( \frac{-3}{4} \right)$.


$
\begin{equation}
\begin{aligned}

\frac{3}{14} - \left( \frac{-3}{4} \right) =& \frac{3}{14} + \frac{3}{4}
&& \text{To subtract, add the additive inverse(opposite)}
\\
\\
=& \frac{6}{28} + \frac{21}{28}
&& \text{Get the LCD}
\\
\\
=& \frac{27}{28}
&&

\end{aligned}
\end{equation}
$