Interpret the poem "Queen-Anne's-Lace."

The key device used by the poet in this poem is personification; he describes the flower Queen Anne's Lace as if it were a lover, combining the literal (the plant is powerful, "taking / the field by force; the grass / does not rise above it") with the figurative ("wherever / his hand has lain there is / a tiny purple blemish").
We may assume that the "he" in the poem represents the sun, driving the flowers to "blossom under his touch." As the sun and the Queen Anne's Lace are characterized as lovers, so the white spread of bloom across the field, the result of their growth together, is described as "desire," spreading in the form of blossoms. Like desire, too, the spread of blossom seems to reach a climax, after which the field is "empty," the flowers scattered as the "wish to whiteness" has gone over and the flowers have receded with the end of spring.

What two jobs does Jack claim for the choirboys?

In chapter one, "The Sound of the Shell," Jack and Ralph compete to be "chief" of the boys. When Ralph gets more votes than Jack does, Ralph graciously gives command of the choir to Jack. Jack says he wants them to be the island's hunters. After that, Jack leads the hunters as they stalk and eventually kill a pig.
In chapter two, "Fire on the Mountain," Ralph tells the other boys that his father is in the navy and has told him that the Queen has maps of the world, including the island on which they're stranded. He is convinced that a ship will come near and suggests that they maintain a signal fire on top of the highest hill. Jack volunteers his choir, the hunters, to be the ones who gather wood. Later, he declares that he will divide the choir into groups that will both maintain a lookout shift and keep the signal fire burning.

In chapter 1, the boys elect Ralph as chief, and he immediately tells Jack that the choir belongs to him. The first job that Jack claims for the choir is to be the group's hunters. Throughout the novel, Jack leads his choir on numerous hunting expeditions, where they brutally kill pigs and hold feasts. Jack and his hunters become bloodthirsty and are obsessed with hunting by the end of the novel. In chapter 2, Ralph holds an assembly and attempts to create a civil, organized society. The second job that Jack claims for his choir involves maintaining the signal fire. Jack volunteers his choir to watch over and feed the signal fire on the top of the mountain. Later on, Jack relieves his hunters from their duty of maintaining the signal fire and leads them on a hunting expedition. Unfortunately, a ship passes by the island, and the hunters are not on top of the mountain to fuel the signal fire, which leads to a heated argument between Ralph and Jack over duties and priorities.

How is the parent-child relationship shown through the novel Frankenstein by Mary Shelley?

A chief way Mary Shelley shows the parent-child relationship in Frankenstein is first through Victor's loving parents, who bestow affection and every kindness on him, feeling he was given to them by God. They therefore believe that they owe him a tender upbringing. As he recalls it:

Much as they were attached to each other, they seemed to draw inexhaustible stores of affection from a very mine of love to bestow them upon me. My mother’s tender caresses and my father’s smile of benevolent pleasure while regarding me are my first recollections. I was their plaything and their idol, and something better—their child, the innocent and helpless creature bestowed on them by heaven, whom to bring up to good, and whose future lot it was in their hands to direct to happiness or misery, according as they fulfilled their duties towards me.

Victor goes on to remember an idyllic childhood provided by his caring and gentle parents:

No human being could have passed a happier childhood than myself. My parents were possessed by the very spirit of kindness and indulgence. We felt that they were not the tyrants to rule our lot according to their caprice, but the agents and creators of all the many delights which we enjoyed.

Victor himself becomes a "parent" in a very different way, playing God and bringing a creature to life through assembling and animating dead body parts. The dysfunctional relationship of Victor to his creation could not be more different than Victor's loving relationship with his own parents. Victor is repulsed by his "child" and rejects it when it comes to life:

Breathless horror and disgust filled my heart. Unable to endure the aspect of the being I had created, I rushed out of the room. . . . A mummy again endued with animation could not be so hideous as that wretch.

When Victor next encounters his creation, he is again filled with horror and wants to kill it, knowing the creature has murdered his loved ones. The creature, however, pleads with Victor to be a loving father, telling Victor that he owes him kindness, for he is the one who gave him life:

Remember that I am thy creature; I ought to be thy Adam, but I am rather the fallen angel, whom thou drivest from joy for no misdeed. Everywhere I see bliss, from which I alone am irrevocably excluded. I was benevolent and good; misery made me a fiend. Make me happy, and I shall again be virtuous.

The creature, feared and hated by everyone, recognizes that loving parenting will bring out his benevolent side and make him good. On the inside, Frankenstein's creation desires to be a caring member of society. It is simply his outward shell that is hideous. He only becomes a murderer because everyone has rejected him: the love he tried to show was met with fear, hatred, and disgust.
Nevertheless, Victor is unable to get past the creature's outer appearance and, now, the havoc the creature has brought to Victor's family through acting out his rage and grief in murder. Victor is never able to transcend his repulsion toward his "child." He is never able, ironically, to replicate the loving and caring behavior he most appreciated from his own parents.
Shelley is critiquing playing God, but she is also suggesting that having played God, Victor is responsible for the outcome of his actions and should have treated his creation as a decent parent would.

What is the theme of the play The Admirable Crichton?

The major theme in the play is social class and how it's determined in and out of English society.
Social class and the various ways it's maintained, as well as its effects on people, is an important factor in the story. When a group of English aristocrats and their servants are marooned on an island, none of the pampered people know how to care for themselves. Though they try to conform to English standards at first, the only person who is able to thrive in the environment is the butler Crichton. He quickly becomes the leader of the group and this is affirmed several times despite power struggles from Lord Loam and the other aristocrats. When they're rescued and return to traditional society, however, everyone reassumes their normal place. Only Lady Mary says that perhaps England itself is wrong for the way it puts people in social classes; Crichton is a product of English society, though, and disagrees with her. He says, "My lady, not even from you can I listen to a word against England."

One of the most abiding themes of the play is the fluidity of social class, and especially how easily it can change under certain environmental conditions. Crichton, a humble butler, becomes the dominant character on the desert island, due to his intelligence and resourcefulness. In the so-called civilized world, Crichton would never get the opportunity to show his true character. But here on a desert island he flourishes among the hapless aristocrats who are supposedly his social superiors.
Social class is presented in the play as being highly artificial, a product of an equally artificial society. It says something about the rigidities of class structure in Edwardian England that someone from a relatively humble background can only really achieve a position of authority on a remote desert island, far away from his native shores. There is a natural aristocracy in this world, it would seem, and Crichton is a part of it, but only—appropriately enough—in the natural world.

College Algebra, Chapter 9, 9.1, Section 9.1, Problem 70

Define the sequence
$\displaystyle G_n = \frac{1}{\sqrt{5}} \left( \frac{(1+\sqrt{5})^n - (1-\sqrt{5})^n}{2^n} \right)$
Find the first 10 terms of this sequence using a calculator. Compare to the Fibonacci Sequence $F_n$

$
\begin{equation}
\begin{aligned}
G_ 1 = \frac{1}{\sqrt{5}} \left( \frac{(1 + \sqrt{5})^1 - (1 - \sqrt{5})^1}{2^1} \right) = 1\\
\\
G_ 2 = \frac{1}{\sqrt{5}} \left( \frac{(1 + \sqrt{5})^2 - (1 - \sqrt{5})^2}{2^2} \right) = 1\\
\\
G_ 3 = \frac{1}{\sqrt{5}} \left( \frac{(1 + \sqrt{5})^3 - (1 - \sqrt{5})^3}{2^3} \right) = 2\\
\\
G_ 4 = \frac{1}{\sqrt{5}} \left( \frac{(1 + \sqrt{5})^4 - (1 - \sqrt{5})^4}{2^4} \right) = 3\\
\\
G_ 5 = \frac{1}{\sqrt{5}} \left( \frac{(1 + \sqrt{5})^5 - (1 - \sqrt{5})^5}{2^5} \right) = 5\\
\\
G_ 6 = \frac{1}{\sqrt{5}} \left( \frac{(1 + \sqrt{5})^6 - (1 - \sqrt{5})^6}{2^6} \right) = 8\\
\\
G_ 7 = \frac{1}{\sqrt{5}} \left( \frac{(1 + \sqrt{5})^7 - (1 - \sqrt{5})^7}{2^7} \right) = 13\\
\\
G_ 8 = \frac{1}{\sqrt{5}} \left( \frac{(1 + \sqrt{5})^8 - (1 - \sqrt{5})^8}{2^8} \right) = 21\\
\\
G_ 9 = \frac{1}{\sqrt{5}} \left( \frac{(1 + \sqrt{5})^9 - (1 - \sqrt{5})^9}{2^9} \right) = 34\\
\\
G_{10} = \frac{1}{\sqrt{5}} \left( \frac{(1 + \sqrt{5})^{10} - (1 - \sqrt{5})^{10}}{2^{10}} \right) = 55\\
\end{aligned}
\end{equation}
$

Using Fibonacci Sequence $F_n = F_{n-1} + F_{n -2}$
Since $F_1 = 1$ and $F_2 = 2$, then

$
\begin{equation}
\begin{aligned}
F_3 &= F_2 + F_1 = 1 + 1 = 2 &&& F_7 &= F_6 + F_5 = 8 + 5 = 13\\
\\
F_4 &= F_3 + F_2 = 2 + 1 = 3 &&& F_8 &= F_7 + F_6 = 13 + 8 = 21\\
\\
F_5 &= F_4 + F_3 = 3 + 2 = 5 &&& F_9 &= F_8 + F_7 = 21 + 13 = 34\\
\\
F_6 &= F_5 + F_4 = 5 + 3 = 8 &&& F_{10} &= F_9 + F_8 = 34 + 21 = 55
\end{aligned}
\end{equation}
$

Calculus of a Single Variable, Chapter 7, 7.4, Section 7.4, Problem 7

Arc length (L) of the function y=f(x) on the interval [a,b] is given by the formula,
L=int_a^bsqrt(1+(dy/dx)^2) dx, if y=f(x) and a <= x <= b,
Now let's differentiate the function,
y=3/2x^(2/3)
dy/dx=3/2(2/3)x^(2/3-1)
dy/dx=1/x^(1/3)
Now let's plug the derivative in the arc length formula,
L=int_1^8sqrt(1+(1/x^(1/3))^2)dx
L=int_1^8sqrt(1+1/x^(2/3))dx
L=int_1^8sqrt((x^(2/3)+1)/x^(2/3))dx
L=int_1^8(1/x^(1/3))sqrt(x^(2/3)+1)dx
Now let's evaluate first the indefinite integral by using integral substitution,
Let t=x^(2/3)+1
dt=2/3x^(2/3-1)dx
dt/dx=2/(3x^(1/3))
dx/x^(1/3)=3/2dt
intsqrt(x^(2/3)+1)(1/x^(1/3))dx=int3/2sqrt(t)dt
=3/2(t^(1/2+1)/(1/2+1))
=3/2(t^(3/2)/(3/2))
=t^(3/2)
=(x^(2/3)+1)^(3/2)
L=[(x^(2/3)+1)^(3/2)]_1^8
L=[(8^(2/3)+1)^(3/2)]-[(1^(2/3)+1)^(3/2)]
L=[5^(3/2)]-[2^(3/2)]
L=11.18033989-2.828427125
L=8.351912763
Arc length (L) of the function over the given interval is ~~8.352

Precalculus, Chapter 9, 9.4, Section 9.4, Problem 72

You need to remember what a quadratic model is, such that:
a_n = f(n) = a*n^2 + b*n + c
The problem provides the following information, such that:
a_0 = 3 => f(0) = a*0^2 + b*0 + c => c = 3
a_2 = 0 => f(2) = a*2^2 + b*2 + c => 4a + 2b + c = 0
a_6 = 36 => f(6) = a*6^2 + b*6 + c => 36a + 6b + c = 36
You need to replace 3 for c in the next two equations, such that:
4a + 2b + 3 = 0 => 4a + 2b = -3
36a + 6b + 3 = 36 => 36a + 6b = 33 => 18a + 2b = 11
Subtract the equation 4a + 2b = -3 from the equation 18a + 2b = 11 :
18a + 2b - 4a - 2b = 11 + 3
14a = 14 => a = 1
Replace 1 for a in equation 4a + 2b = -3 , such that:
4 + 2b = -3 => 2b = -7 => b = -7/2
Hence, the quadratic model for the given sequence is a_n = n^2 - (7/2)*n + 3.