What is the meaning behind the proverb "not all that glitters is gold"?

At it‘s most basic level, this quote expresses the idea that something’s appearance is not necessarily a true representation of what it really is. In other words, things are not always what they appear to be.
A deeper analysis would show that this proverb also implies that something’s appeal (glitter) in fact may be hiding a negative or unpleasant reality (not gold after all). Gold is used here because we associate this precious metal with a high level of value. Though something else may mimic the glittering of gold, it doesn’t mean that it will hold that same value.
This can be applied to actual objects (such as fake Gucci glasses), people (someone who is handsome but then is discovered to be a liar or cruel to others), or even situations (a job promotion that seemed great but comes with far too much work).
In order to truly appreciate this proverb, one must realize that not only must they understand what phrase means, but also the lesson it is conveying. The warning “not all that glitters is gold” implies that a person much look deeper than the surface and not make assumptions based on first appearances. Critically analyze the situation or object/person and don’t make hasty decisions about them. In this way, one can appreciate both the meaning and lesson of this proverb.

The popular English expression "Not all that glitters is gold" is used to mean that sometimes things that look valuable, attractive, or promising turn out to be anything but. At an extremely literal level, one example of this can be found in iron pyrite, otherwise known as fool's gold, which is a mineral that at first glance may be mistaken for actual gold but, upon closer examination, is relatively worthless. An interpersonal example might be if you met someone who was extremely attractive and seemed superficially nice and interesting, only to find out upon spending more time with them that they were actually cruel and boring. Beauty on the outside is not the same as beauty on the inside.
To think of examples from your own life, try to remember a time when you had high expectations for something based on cursory knowledge of what it was only to dig deeper and be disappointed, or a time when you've been tricked by something's outward appearance. We've all been there!

What is Johnny's reaction when the nurse tells him that his mother is at the hospital to see him?

Readers are informed very early on in the book that Johnny's home life is not good and that his relationship with his parents is terrible. His father beats him, and his mother either ignores him completely or screams at him.

His father was always beating him up, and his mother ignored him, except when she was hacked off at something, and then you could hear her yelling at him clear down at our house.

Johnny's parents show no love for him, and that is confirmed again for readers in chapter 6. Dally, Johnny, and Ponyboy are all talking, and Dally is informing Johnny and Ponyboy of some recent news. Johnny asks if his parents have asked about him, and Dally flatly states that they have not. The news is not surprising, but Johnny is still hurt.

"My parents," Johnny repeated doggedly, "did they ask about me?"
"No," snapped Dally, "they didn't. Blast it, Johnny, what do they matter?"

Once Johnny is in the hospital, his mother does come and try to see him; however, Johnny knows that she isn't there for any kind of good reason. Johnny flatly denies his mother access to him. He tells the nurse that he does not want to see his mother. Johnny believes that she has only come to berate him for one reason or another.

"I said I don't want to see her." His voice was rising. "She's probably come to tell me about all the trouble I'm causing her and about how glad her and the old man'll be when I'm dead. Well, tell her to leave me alone. For once"—his voice broke—"for once just to leave me alone."

Johnny is so enraged and upset by the entire situation that he passes out.

Johnny has a terrible relationship with his callous mother and does not want to see her when she visits him in the hospital.
In chapter 8, Johnny Cade is lying incapacitated in his hospital bed when the nurse informs him that his mother has arrived to visit him. Johnny reacts by telling the nurse that he does not want to see her. Johnny mentions that his mother has probably come to visit him so that she can tell him how much trouble he is causing the family. Johnny then tells the nurse to tell his mother to leave him alone for once. As Pony and Two-Bit are leaving the hospital, they see Johnny's mother arguing with the nurse about her right to see her son. When she spots Pony and Two-Bit, she blames them for Johnny's injuries. Two-Bit responds by telling Johnny's mother,

No wonder he hates your guts (Hinton, 105).

Ponyboy feels sorry for Johnny and shoves Two-Bit along before he can cuss out Johnny's mother.

Single Variable Calculus, Chapter 8, 8.2, Section 8.2, Problem 68

Prove that $\displaystyle \int^\pi_{-\pi} \sin (mx) \sin (nx) dx = \left\{
\begin{array}{c}
0 & \text{if} & m \neq n\\
\pi & \text{if} & m = n
\end{array}\right.
$


where $m$ and $n$ are positive integers.
If we use the sum and difference angles formula for cosine, we get


$
\begin{equation}
\begin{aligned}
\cos (mx - nx) &= \cos (mx) \cos (nx) - \sin (mx) \sin (nx) \qquad \text{and}\\
\\
\cos (mx + nx) &= \cos (mx) \cos (nx) + \sin (mx) \sin (nx)
\end{aligned}
\end{equation}
$

Then, $\cos (mx - nx) - \cos(mx+nx) = \cos (mx) \cos (nx) - \sin (mx) \sin (nx) - \cos (mx) \cos (nx) + \sin (mx) \sin (nx) = 2 \sin (mx) \sin (nx)$

Therefore,
$\displaystyle \int^\pi_{-\pi} \sin (mx)\sin(nx)dx = \int^\pi_{-\pi}\left[ \frac{\cos(mx-nx)-\cos(mx+nx)}{2} \right]dx$

if $m \neq n$

$
\begin{equation}
\begin{aligned}
\int^\pi_{-\pi} \sin (mx)\sin(nx)dx &= \frac{1}{2} \left[ \frac{\sin(mx-nx)}{(m-n)} - \frac{\sin(mx+nx)}{(m+n)} \right]^\pi_{-\pi}\\
\\
\int^\pi_{-\pi} \sin (mx)\sin(nx)dx &= \frac{1}{2} \left(\left[ \frac{\sin(m\pi - n \pi)}{(m-n)} - \frac{\sin(m\pi+n\pi)}{(m+n)} \right] - \left[ \frac{\sin(m(-\pi) - n(-\pi))}{(m-n)} - \frac{\sin(m(-\pi) + n (-\pi))}{(m+n)} \right] \right)\\
\\
&= 0
\end{aligned}
\end{equation}
$


if $m =n$,

$
\begin{equation}
\begin{aligned}
\int^\pi_{-\pi} \sin (mx) \sin(nx) dx &= \int^\pi_{-\pi} \left[ \frac{\cos (mx - mx) - \cos (mx + mx)}{2} \right] dx\\
\\
&= \frac{1}{2} \int^\pi_{-\pi} [\cos(0) - \cos(2mx)] dx\\
\\
&= \frac{1}{2} \int^\pi_{-\pi} [ 1 - \cos(mx)] dx\\
\\
&= \frac{1}{2} \left[ x - \frac{\sin(2mx)}{2m} \right]^\pi_{-\pi}\\
\\
&= \frac{1}{2} \left( \left[ \pi - \frac{\sin(2m\pi)}{m} \right] - \left[ (-\pi) - \frac{\sin(2m(-\pi))}{2m} \right] \right)\\
\\
&= \frac{1}{2} [ \pi + \pi]\\
\\
&= \frac{2\pi}{2} \\
\\
&= \pi
\end{aligned}
\end{equation}
$

What is a summary of Alfred Noyes' poem "Song of the Wooden-Legged Fiddler"?

Alfred Noyes wrote "Song of the Wooden-Legged Fiddler" in 1805. It is the tale (song) of a youngster who ran away to sea, to "fight like a sailor for country and king" (19). The poem is five stanzas, with seven lines in each. At the end of each stanza is a one-line chorus, which echoes the last line of the stanza.
In the first stanza, the singer speaks about his life as a boy. He lived in a cottage in the West, but had a restless spirit that forever called him to sea:

But I knew no peace and I took no rest
Though the roses nigh smothered my snug little nest;
          For the smell of the sea
          Was much rarer to me,
And the life of a sailor was all my joy (3-7)

The boy has a "snug" home, described as a picture-perfect cottage, surrounded by roses. However, it was not the flowers that he smelled that overwhelmed him, but the rare smell of the salty ocean water.
In the second stanza, the narrator tells of his mother's wishes for him:

My mother she wept, and she begged me to stay... (9)

He notes that she wanted to keep him close to home, tied to "her apron-string" (10), to help with the hay—we can assume with its harvest. (Hay is the crop of a season's growth of grass that is harvested before it goes to seed.) The singer relents to his mother's wishes for the cutting of the hay, but sneaks out and runs off the following spring.

On a night of delight in the following spring,
                With a pair of stout shoon
                And a seafaring tune
     And a bundle and stick in the light of the moon,
                Down the long road
                To Portsmouth I strode... (13-18)

He takes a pair of strong shoes ("shoon"), and he whistles a sea chantey, having packed a few belongings in a piece of cloth (like a kerchief), carried on the end of a stick—walking to Portsmouth to follow his dream. We can infer this is some distance if his home does not afford him the smell of the sea.
In the third stanza, the sailor is returning home, though the sea still calls him away. The memories of sailing the Spanish Main—chasing French and Spanish frigates—are still fresh in his mind. Perhaps the most poignant line is found in this section of the song:

For at heart an old sailor is always a boy... (25)

This is the line that reminds the listener, even until the end of the song, that the ocean's call never loses its power as the boy grows into a man, and eventually an aged sailor who can no longer make his living on the sea.
The listener (or reader) can believe that a sailor's heart never ages and the draw of the profession never fades: even after all of his experiences, the sea still calls to him like a mythical siren. The smells of gunpowder (for the canon) and pitch (tar to waterproof the ship) are still in his nose. His heart will belong to the sea (he notes) until he cannot tell the difference between either of them—and even when (as he looks out over the ship's side) he cannot distinguish the "grin o' the guns from a glint o' the sea" (27). At this point he will lack the mental capacity to tell the difference between the legendary Nelson and some other jack-tar like himself.
The fourth stanza lets the listener know that the sailor is no longer young:

Ay! Now that I'm old I'm as bold as the best... (32)

By now he has lost his leg—a wooden peg has taken its place; and he is as bald as an egg. Even in light of these drastic changes, his love of the sea has not diminished:

The smell of the sea
              Is like victuals to me... (37-38)

In other words, living on the ocean is as necessary to him as food ("victuals"). Even in his grave, he is certain he will still be calling "Ahoy!" When his body is "ready to rest" (die), he echoes the sentiment he shared earlier:

At heart an old sailor is always a boy. (41)

Interestingly, while he recalls whistling a sea song when he left home, his story has become a sea chantey, based upon his life rather than another sailor's, as he has lived the life he was called to so many years before. It would also seem that his choice was true to his heart and he has no regrets—other than, perhaps, having to leave his life on the sea because he can no longer sail. For the boy's heart in him is still passionately in love with the sea.

Intermediate Algebra, Chapter 2, 2.1, Section 2.1, Problem 22

Solve the equation $5(x + 3) + 4x - 5 = 4 - 2x$, and check your solution. If applicable, tell whether the equation is an identity or contradiction.


$
\begin{equation}
\begin{aligned}

5(x + 3) + 4x - 5 =& 4 - 2x
&& \text{Given equation}
\\
5x + 15 + 4x - 5 =& 4 - 2x
&& \text{Distributive property}
\\
9x + 10 =& 4 - 2x
&& \text{Combine like terms}
\\
9x + 2x =& 4 - 10
&& \text{Add $(2x-10)$ from each side}
\\
11x =& -6
&& \text{Combine like terms}
\\
\frac{11x}{11} =& \frac{-6}{11}
&& \text{Divide both sides by $11$}
\\
x =& \frac{-6}{11}
&&

\end{aligned}
\end{equation}
$


Checking:


$
\begin{equation}
\begin{aligned}

5 \left( \frac{-6}{11} + 3 \right) + 4 \left( \frac{-6}{11} \right) - 5 =& 4 - 2 \left( \frac{-6}{11} \right)
&& \text{Substitute } x = \frac{-6}{11}
\\
\\
5 \left( \frac{27}{11} \right) + 4 \left( \frac{-6}{11} \right) - 5 =& 4 - 2 \left( \frac{-6}{11} \right)
&& \text{Add inside the parentheses}
\\
\\
\frac{135}{11} - \frac{24}{11} - 5 =& 4 + \frac{12}{11}
&& \text{Multiply}
\\
\\
\frac{56}{11} =& \frac{56}{11}
&& \text{True}

\end{aligned}
\end{equation}
$

How to prove Abigail is immoral and unchristian? What will be the quotation to support it (quotation must be related with the relationship between sexual behavior and reputation)

Abigail Williams is by definition an immoral individual throughout the play. She not only engages in an affair with John Proctor, but continues to tempt him while they are alone, threatens to kill the other girls if they do not follow her lead, and falsely accuses innocent citizens of witchcraft. When Reverend Parris and the adults leave Betty's room in Act One, Mary Warren begs Abigail to tell the truth and Betty mentions that Abigail drank blood to put a curse on Elizabeth Proctor. Abigail responds by saying,

"Now look you. All of you. We danced. And Tituba conjured Ruth Putnam’s dead sisters. And that is all. And mark this. Let either of you breathe a word, or the edge of a word, about the other things, and I will come to you in the black of some terrible night and I will bring a pointy reckoning that will shudder you. And you know I can do it; I saw Indians smash my dear parents’ heads on the pillow next to mine, and I have seen some reddish work done at night, and I can make you wish you had never seen the sun go down!" (Miller, 30).

After threatening the girls, John Proctor arrives and Abigail reveals her emotions for him. She proceeds to tempt John Proctor by saying,

"I have a sense for heat, John, and yours has drawn me to my window, and I have seen you looking up, burning in your loneliness. Do you tell me you’ve never looked up at my window?" (Miller, 32).

In Act Two, Mary Warren says that she saved Elizabeth's life in court after Abigail falsely accused her of witchcraft. Elizabeth, who is an innocent Christian, understands Abigail's motives to remove her in order to get to John. Elizabeth tells John,

"She wants me dead. I knew all week it would come to this!" (Miller, 56).

Abigail's unrestrained sexual appetite, her violent temper, and her manipulative personality illustrate that she is an immoral individual.

How does Mr. Darcy fall in love with Elizabeth? What aspects of her character does he fall for, and where is this shown in the novel?

Mr. Darcy fell in love with Elizabeth gradually but intensely. The first time he is shown to be attracted to her is in chapter 6, where he notices the "beautiful expression of her dark eyes." At this point, he has been eavesdropping on her conversations, which are having an impact on him. Though Darcy tries to appear indifferent to Elizabeth, his eavesdropping betrays his desire to get to know her better. In spite of his pride and "critical eye" he must admit to himself that she has a "light and pleasing" figure and manners that show an "easy playfulness." Even when she appears at Netherfield having walked through muddy puddles to reach Jane in chapter 8, he notices that her eyes were "brightened by the exercise." By chapter 11, he is thoroughly impressed by Elizabeth's personality, conversation, wit, and liveliness—so much so that "he began to feel the danger of paying Elizabeth too much attention." In chapter 34, he finally declares his ardent love for her.

Even Darcy himself struggles to answer this question. In chapter 60, Elizabeth asks Darcy to explain how, when, or why he fell in love with her:

she wanted Mr. Darcy to account for his having ever fallen in love with her. “How could you begin? . . . what could set you off in the first place?”

At this point in the novel, Elizabeth has already agreed to marry Mr. Darcy. Neither Elizabeth nor Mr. Darcy greatly enjoyed each other's company early in the book. Mr. Darcy, however, steadily grows in admiration of Elizabeth. He proposes to her, for the first time, midway through the novel. She refuses this first proposal, shocked that he would ever consider such a match, but she learns to appreciate and love Mr. Darcy as they continue to interact in the second half of the book. He responds to Elizabeth's question:

I cannot fix on the hour, or the spot, or the look, or the words, which laid the foundation. It is too long ago. I was in the middle before I knew that I had begun.

She then recounts their initial observations of one another and early feelings toward each other. At first, he had refused an opportunity to dance with her with his infamous line,

She is tolerable, but not handsome enough to tempt me (chapter 3).

Clearly, Darcy does not fall in love with Elizabeth simply for her appearance. She also doubts that he could have fallen in love with her early manners and actions:

my behaviour to you was at least always bordering on the uncivil, and I never spoke to you without rather wishing to give you pain than not. Now be sincere; did you admire me for my impertinence? (chapter 60)

Impertinence refers to disrespect or discourtesy. In our modern language, we might call impertinence an extreme form of sassiness. Darcy takes this negative word, impertinence, and rephrases it in a positive manner:

For the liveliness of your mind, I did (chapter 60).

He tells her that he appreciated her clever words (or, perhaps, her well-planned sass). Additionally, he mentions the "affectionate behavior" that Elizabeth showed Jane when she got ill and was forced to stay at the Bingley's house until she got better. Mr. Darcy learned to look past mere physical appearance. Yes, it does seem that Mr. Darcy grew to think that Elizabeth is beautiful as time progressed. In a conversation with Miss Bingley he announces,

I have been meditating on the very great pleasure which a pair of fine eyes in the face of a pretty woman can bestow.

When she asks him whose eyes captured his attention, he responds:

Miss Elizabeth Bennet (chapter 6).

However, it isn't Elizabeth's physical beauty alone that draws his love and affection. Darcy admits to appreciating Elizabeth's active mind and her passionate concern for her friends and family. Her love and concern for others is seen not only when Elizabeth cares for her sister Jane when she got ill, but is also seen when Elizabeth goes to visit her friend Charlotte Lucas after Charlotte marries an irksome husband for money and comfort rather than love.