How did Winnie feel about her home? How did her mother and grandmother feel about her home?

Additionally, Winnie's mother and grandmother view their home as a place of safety and well-being. When Winnie is playing outside in chapter 3, talking to a nearby toad, her mother calls out to her, "Come in now, Winnie. Right away. You'll get heat stroke out there on a day like this" (14).
Her mother and grandmother are not only worried that Winnie might get dirty playing outside, but they are also worried that she might get ill from the heat.
Soon after, in chapter 4, Winnie talks to a stranger described as the man in the yellow suit. Winnie is pleasant while conversing with this man and answering his questions. After, her grandmother overhears her talking to him, a stranger, and she comes outside to see what is happening. She shows signs of being worried about this man's presence. The book describes how she "squinted suspiciously" at him (19). Overall, she seems concerned for the safety of her granddaughter as she questions him: "We haven't met, that I can recall. Who are you? Who are you looking for?" (20). He, in turn, asks the grandmother questions about how long they had lived in their home and who the family knows in the neighborhood. She responds, with distrust, "I don't know everyone . . . And I don't stand outside in the dark discussing such a thing with strangers."
Later, after hearing some peculiar music coming from the woods and suggesting that the sound came from elves, the grandmother leads Winnie back inside the house, which she views as a place of safety:

"She shook the gate latch under his nose, to make sure it was locked, and then, taking Winnie by the hand once more, she marched up the path into the cottage, shutting the door firmly behind her" (21).

While Winnie feels trapped and restricted by the walls of her house and the fence around her yard, her mother and grandmother view their home as a place of safety and well-being.
The man in the yellow suit ultimately uses the family's concern for Winnie's safety to gain ownership of the Foster's woods. He tells Winnie's family, "Why, the little girl and I, we're friends already. It would be a great relief to see her safely home again, wouldn't it? . . . Dreadful thing, kidnapping" (74). He knows that they want Winnie returned to her safe home and that they would be willing to sacrifice almost anything to get her back with them. The Foster family views the home as a place of safety and the woods and places beyond the home as places of unknown dangers.

At the beginning of Tuck Everlasting, Winnie feels as if her home is a cage. Her mother and her grandmother don't let her go outside of her fenced yard, and they constantly watch her and give her reminders and warnings. Because she wants more freedom and independence, she decides to run away. When the Tucks kidnap her and take her to their "homely little house" in the woods, she is unprepared for how untidy it is. She is used to the "pitiless" order of her home, so she is amazed "that people could live in such disarray." Later, when she returns home, she seems to take comfort in her own little rocking chair in her room. Even though she's what we would call grounded, as punishment for helping Mae escape from jail, she seems more satisfied with her home since she's had an experience of her own.
The passage in the novel that gives the most insight into how Winnie's mother and grandmother feel about their home is the first paragraph of chapter 10. Here we learn that the two women kept their cottage spotless: They "mopped and swept and scoured [it] into limp submission." They never procrastinate in their duties of keeping the home clean. Indeed, they "had made a fortress out of duty." We also know from chapter 1 that the home is a "touch-me-not" cottage that was forbidding to visitors. And in chapter 25, we learn that the whole family, including the women, were proud. From these descriptions, we can infer that Winnie's mother and grandmother take great pride in keeping their home pristine, but they don't open their home to strangers willingly. They seem to care more about order and rules than making their home a place that communicates love and emotional warmth.
The home is not presented as a welcoming place from the perspective of any of these three female characters. Readers feel little remorse when they read in the Epilogue that the house is no longer there when the Tucks finally return decades later.

There is one image in "Rip Van Winkle" that is unchanged. What is it? And what is the meaning of the image?

“Rip Van Winkle” is the story of a henpecked husband who mysteriously falls asleep for twenty years. When he awakes, everything in the village is seemingly different. The only things that remain the same are the Catskill Mountains (called the “Kaatskill Mountains” in the text), revealing the significance of the mystical setting and indicating that the mountains have a profound influence over the events of the plot.
The story opens with an explanation of the setting. The opening line states, “Whoever has made a voyage up the Hudson must remember the Kaatskill mountains” (Irving). This is followed by an in-depth description of the “magical hues and shapes of these mountains” (Irving). The beginning of the story emphasizes the setting, calling it memorable and magical and effectively foreshadowing the strange events to come.
Later, after another disagreement with his wife, Rip escapes to the mountains. On his journey, he has “unconsciously scrambled to one of the highest parts of the Kaatskill mountains” (Irving). It is at this place that Rip meets the strange group of men and drinks from their flagon. He then falls asleep for twenty years. It is the enchanting backdrop of the Catskill Mountains that helps the reader suspend disbelief and instead see the possibility of the supernatural properties found in the mountains.
When Rip awakes, the village is completely altered, and he is very confused. Soon, however, he looks up, and “there stood the Kaatskill mountains—there ran the silver Hudson at a distance—there was every hill and dale precisely as it had always been” (Irving). It is only because of the Catskill Mountains that Rip can unquestionably recognize his home village and return to his life there.
Throughout the story, the Catskill Mountains are an important element, greatly affecting characterization and plot development. It is through the magical mountain range that Rip escapes from his miserable life, skips twenty years, and, in the end, regains a happy existence near the enduring Catskill Mountains.

What is a quote that shows why the monster put the locket in Justine's pocket?

The creature describes the way he murdered William and how he found the valuable portrait of the beautiful woman, William's mother, in the boy's possession.  He says that he knew such a woman would never look upon him and smile that way, and this thought makes him bitterly angry.  The creature soon finds Justine, asleep in a barn nearby; he recognizes her beauty, as well, and that she is "'one of those who joy-imparting smiles are bestowed on all but [him].'"  He realizes that if he wakes her, she would scream in terror and curse him.  The creature thinks,

not I, but she, shall suffer; the murder I have committed because I am forever robbed of all that she could give me, she shall atone.  The crime had its source in her; be hers the punishment!  Thanks to the lessons of Felix and the sanguinary laws of man, I had learned now to work mischief.  I bent over her and placed the portrait securely in one of the folds of her dress.  (Chapter 16)

Thus, he decides that Justine is, indirectly, the source of all his misery (as she becomes a sort of representative of all humanity, who has or will reject him), and his misery compelled him to murder, and so the murder of William can be traced back to her.  Therefore, he thinks, she should bear the punishment, not he.  He believes human laws to be bloodthirsty and cruel, and he has likewise learned some cruelty from hearing the histories taught by Felix, and so he knows how to wreak havoc.  He purposely frames Justine for William's murder by placing the valuable bauble in her dress.

Calculus of a Single Variable, Chapter 9, 9.6, Section 9.6, Problem 44

To apply the Root test on a series sum a_n , we determine the limit as:
lim_(n-gtoo) root(n)(|a_n|)= L
or
lim_(n-gtoo) |a_n|^(1/n)= L
Then, we follow the conditions:
a) Llt1 then the series is absolutely convergent.
b) Lgt1 then the series is divergent.
c) L=1 or does not exist then the test is inconclusive. The series may be divergent, conditionally convergent, or absolutely convergent.
To apply the Root Test to determine the convergence or divergence of the series sum_(n=0)^oo e^(-3n) , we let a_n = e^(-3n) .
Apply Law of Exponent: x^(-n) = 1/x^n .
a_ n = 1/e^(3n).
Applying the Root test, we set-up the limit as:
lim_(n-gtoo) |1/e^(3n)|^(1/n) =lim_(n-gtoo) (1/e^(3n))^(1/n)
Apply the Law of Exponents: (x/y)^n = x^n/y^n and (x^n)^m= x^(n*m) .
lim_(n-gtoo) (1/e^(3n))^(1/n) =lim_(n-gtoo) 1^(1/n)/(e^(3n))^(1/n)
=lim_(n-gtoo) 1^(1/n)/e^(3n*1/n)
=lim_(n-gtoo) 1^(1/n)/e^((3n)/n)
=lim_(n-gtoo) 1^(1/n)/e^3
Apply the limit property: lim_(x-gta)[(f(x))/(g(x))] =(lim_(x-gta) f(x))/(lim_(x-gta) g(x)) .
lim_(n-gtoo) 1^(1/n)/e^3 =(lim_(n-gtoo) 1^(1/n))/(lim_(n-gtoo)e^3 )
= 1^(1/oo) /e^3
=1^0/e^3
=1/e^3 or 0.0498 (approximated value)
The limit value L = 1/e^3 or 0.0498 satisfies the condition: Llt1 since 0.0498lt1.
Thus, the series sum_(n=0)^oo e^(-3n) is absolutely convergent.

Intermediate Algebra, Chapter 2, Cumulative Exercises, Section Cumulative Exercises, Problem 2

Suppose that $\displaystyle A = \left\{ -8, -\frac{2}{3}, -\sqrt{6}, 0, \frac{4}{5}, 9, \sqrt{36} \right\}$. Simplify the elements of $A$ as necessary and then list the elements that belongs to Whole numbers.

The whole numbers from the list are $\{ 0,9,\sqrt{36} \text{ or } 6\}$

Discuss the significance of the title of "Man and Superman."

In Shaw's play it is the character of Ann Whitefield who encourages and cajoles John Tanner—the would-be superman of the title—to become "that which he is," to paraphrase the subtitle of one of Nietzsche's books. She is the animating spirit of the action, the life-force which incites Tanner to develop into something more substantial than the unthinking anarchist, the impetuous, hot-headed would-be revolutionary.
In doing so, Ann takes on the role of a female Don Juan, cleverly seducing Tanner into an arrangement with which he feels rather uncomfortable, largely on account of his disdain for the niceties of bourgeois social convention. Ann's vigorous assertion of her individuality presents a model for Tanner to emulate. In Nietzschean terms this isn't so much a creative evolution of personality as a rediscovery of what's truly inside. The greatest spirits of every age have superman within them, both men and women.

The title "Man and Superman," which is a play by British dramatist George Bernard Shaw, is a reference not to the comic book superhero but to German philosopher Friedrich Nietzsche's concept of the superman, or "Ubermensch." Nietzsche conceived of the "Ubermensch" as the ultimate human being, an ideal to which each individual person and society as a whole should aspire. In working towards this goal, Nietzsche posited that each generation of humans would get closer and closer to "perfection."
Shaw references Nietzsche in the title of his play to place his work in conversation with philosophers of the past. The play itself, especially the third act, centers on philosophical debate. Shaw's stance on Nietzsche's argument (that each generation of humans became closer and closer to "superman" through a process similar to Darwin's concept of natural selection) was that this process is driven through attraction to one another's "life force," which would drive the next generation further along this path of evolution.

Single Variable Calculus, Chapter 4, 4.3, Section 4.3, Problem 42

Sketch the curve of $y = x ^3 - 3a^2x+2a^3$; where $a$ is a positive constant.
If $y = x^3 - 3a^2 x + 2a^3$, then

$
\begin{equation}
\begin{aligned}
y' &= 3x^2 - 3a^2\\
y'' &= 6x
\end{aligned}
\end{equation}
$


Solving for critical numbers, when $y'=0$

$
\begin{equation}
\begin{aligned}
0 & = 3x^2 - 3a^2\\
3x^2 &= 3a^2\\
x &= \sqrt{a^2}\\
x &= \pm a
\end{aligned}
\end{equation}
$

Hence, we can divide the interval by;

$
\begin{array}{|c|c|c|}
\hline\\
\text{Interval} & f'(x) & f\\
\hline\\
x < -a & + & \text{increasing on } (-\infty,-a)\\
\hline\\
-a < x < a & - & \text{decreasing on } (-a,a)\\
\hline\\
x > a & + & \text{increasing on } (a,\infty)\\
\hline
\end{array}
$


Solving for the inflection point, when $y'' = 0$

$
\begin{equation}
\begin{aligned}
y'' = 0 &= 6x\\
x &= 0&
\end{aligned}
\end{equation}
$


$
\begin{array}{|c|c|c|}
\hline\\
\text{Interval} & f'(x) & f\\
\hline\\
x < 0 & + & \text{Upward } \\
\hline\\
x > 0 & - & \text{Downward }\\
\hline
\end{array}
$


For the coordinates of the graph,

$
\begin{equation}
\begin{aligned}
\text{when } x &= a, &&& \text{when } x &= -a,\\
\\
f(a) &= a^3 - 3a^2(a) + 2a^3 &&& f(-a) &= (-a)^3 - 3a^2 (-a) + 2a^3\\
\\
f(a) &= 0 \text{ (Local Minimum)} &&& f(-a) &= 4a^3 \text{ (Local Maximum)}\\
\\
\text{when } x &= 0,\\
\\
f(0) &= (0)^3 - 3a^2 (0) + 2a^3\\
\\
f(0) &= 2a^3
\end{aligned}
\end{equation}
$

Therefore, we can illustrate the function as