How does "Ode on a Grecian Urn" by John Keats reflect the theme that art is immortal?

John Keats's "Ode on a Grecian Urn" features a speaker reflecting on the nature of art as he looks at the figures painted on an urn. Throughout the poem, the speaker observes the static figures and draws conclusions about the way in which the urn freezes the figures eternally in the states depicted on the urn.
The speaker talks directly to the urn, and he addresses it as "Sylvain historian," meaning that the urn chronicles the activities of this pastoral scene and preserves them for all time. As the speaker continues into the second stanza, he offers some examples of the figures that are frozen in time:

Fair youth, beneath the trees, thou canst not leave
Thy song, nor ever can those trees be bare;
Bold Lover, never, never canst thou kiss,
Though winning near the goal yet, do not grieve;
She cannot fade, though thou hast not thy bliss,
For ever wilt thou love, and she be fair!


The "youth" will always be listening to the silent "music" being played by "soft pipes" illustrated on the urn, and the trees will always be in flower. Though the lover will never actually get the kiss he aims for, he should "not grieve" because his beloved "cannot fade." Because the scene is frozen in time on the urn, she will always be beautiful and he will always love her. These elements are made eternal by their inscription on the urn.

The speaker continues to celebrate, though now more ecstatically, the immortality of the scene in the third stanza. He exclaims,


Ah, happy, happy boughs! that cannot shed
Your leaves, nor ever bid the Spring adieu;
And, happy melodist, unwearied,
For ever piping songs for ever new;
More happy love! more happy, happy love!
For ever warm and still to be enjoy'd,
For ever panting, and for ever young;



The trees will never lose their leaves, and the urn maintains nature and man in this eternal springtime. The musician will never get tired, and his music will always entertain. Love will always "be enjoy'd" since it can never fade or grow old. It will always be just as it is depicted here.

After more reflection on the specific details of the urn, the speaker concludes,


When old age shall this generation waste,
Thou shalt remain, in midst of other woe
Than ours, a friend to man, to whom thou say'st,
"Beauty is truth, truth beauty,—that is all
Ye know on earth, and all ye need to know."



The speaker celebrates the immortality of the urn. While the world around it will go to "waste," the urn "shalt remain." The speaker sees this as a comfort "in midst of other woe." We can always depend on the lasting nature of this beauty.

The images on the urn are frozen in time. Presumably, if the urn survives the effects of erosion and decay, those images will remain there forever. Likewise, a poem (such as "Ode on a Grecian Urn") can exist forever as long as it continues to be read. There is an immortality to poetry that visual art cannot capture because the poem can be written or passed on orally. 
Keats isn't just suggesting that art is immortal. He is philosophizing on what that immortality means and what its value is. The melodies he imagines from the pipes on the urn are "sweeter" because they are always there to be imagined. Melodies that are actually heard (in real life) might be flawed and only exist for a certain amount of time. Therefore, the unheard melodies shown on the urn are superior. On the other hand, isn't there something flawed about a melody that is not heard? The lovers are always almost kissing, but they never do. So, although the urn expresses immortal images and ideas, they do not "live" so to speak. 
In short, there are benefits and drawbacks to art's immortality. The urn shows a number of things that are immortal and frozen in time, but those images lack the lived experience that makes them come alive. The urn is immortal but "cold." Art, such as the urn or this poem, is flawless in its immortality, but also lifeless. How can something be immortal but not alive?
Keats ends with a cryptic line about beauty and truth. Given this discussion of life and art, he might be suggesting that truth and beauty manifest in life and art in different ways. In other words, the truth of art's immortality is not the same as the truth of an immortal (human) life, but there is beauty in both notions. 

2^(2x)-12*2^x+32=0 Solve the equation.

We are asked to solve 2^(2x)-12*2^x+32=0 :
Rewrite as (2^x)^2-12*2^x+32=0 and let y=2^x ; then
y^2-12y+32=0 and
(y-8)(y-4)=0 so y=8 or y=4.
y=8 ==> 2^x=8 ==> x=3
y=4 ==> 2^x=4 ==> x=2
Checking we find that these are indeed solutions.
The graph:

How significant was the Great Depression to the League of Nations failure?

The Great Depression forced many nations to turn their attention inward in order to fix their economies.  This was especially true of the United States.  The United States never joined the League for fear of losing some of its own national sovereignty and a sense of disillusionment after the Versailles Treaty.  The worldwide depression ensured that the United States would not be a player on the world stage as it did not want to spend money on the military needed in order to ensure that nations played by the rules of the League of Nations.  The signatories to the pact, especially the victorious Allied powers of Europe, were too bloody from WWI to raise sufficient funds to equip soldiers needed to enforce the League.  Also, the Depression made more nations anxious to receive reparation funds from Germany—money Germany did not have due to the Depression and WWI.  This put a further strain on international relations and pushed Germany further into the arms of the Nazis who promised a stronger Germany and revenge on Germany's enemies.  While the League was failing before the Depression, the Depression really killed any chance it may have had to exist with or without American support.  

Precalculus, Chapter 6, 6.3, Section 6.3, Problem 34

You need to evaluate the sum of two vectors,u+v , hence you need to perform the addition of the same versors, such that:
u = <0,0> => u = 0i + 0j
v = <2,1> => v = 2i + j
u + v = <0,0> + <2,1>
u + v = <0+2,0+1> => u + v = <2,1>
Hence, evaluating the sum u + v yields u + v = <2,1>
You need to evaluate the difference of two vectors,u-v, hence you need to perform the subtraction of the same versors, such that:
u - v = <0,0> - <2,1>
u - v = <0-2,0-1> => u - v = <-2,-1>
Hence, evaluating the difference u - v yields u - v = <-2,-1> .
You need to evaluate the difference of the vectors,2u-3v , hence you need to perform first the multiplication of each vector with the indicated scalar and then you need to perform the subtraction of the same versors, such that:
2u - 3v = 2<0,0> - 3<2,1>
2u - 3v = <2*0,2*0> - <3*2,3*1>
2u - 3v = <2*0 - 3*2,2*0 - 3*1> => 2u - 3v = <-6,-3>
Hence, evaluating the difference 2u - 3v yields 2u - 3v = <-6,-3>.

In the poem "Annabel Lee," according to the narrator, why was his love taken away from him?

The narrator's lover in the famous poem by Edgar Allan Poe, "Annabel Lee," is the title character. Over the course of the poem, the narrator relates that Annabel Lee was chilled in line 14, which eventually killed her. He says, "the wind came out of a cloud by night, / chilling and killing my Annabel Lee" (25-26). As a result, her cold body was then taken away to be placed in a tomb by her family in lines 27-30.
The Merriam-Webster dictionary defines "chill" as "a cold feeling" and also as "an illness that makes you feel cold." In this context, the word "chill" is probably meant to be taken in both ways. The wind made Annabel Lee cold, which made her weak enough to catch an illness that killed her and thereby left her a corpse without the warmth of life. Alternatively, the cold wind might simply have made her so cold she died.
The narrator explicitly blames this unexpected death upon the angels in lines 11-12 and 21-23. (The word "seraph" in line 11 refers to a type of angel.) The narrator says that the angels are covetous because they are less happy in Heaven than he and Annabel Lee are together. In Christianity, angels are said to dwell in Heaven, which Merriam-Webster also defines as "a place or condition of utmost happiness." Yet, in lines 21-22, the narrator writes that they are "not half so happy in Heaven." By stating that the angels are jealous of his love with Annabel Lee, the narrator is implicitly saying that his love was greater than the greatest happiness. According to the narrator, then, Annabel Lee was killed and separated from him by angels jealous of their impossibly happy love.
https://www.merriam-webster.com/dictionary/chill

https://www.merriam-webster.com/dictionary/heaven

https://www.owleyes.org/text/annabel-lee/read/annabel-lee

This narrative poem by Poe expresses the fictitious narrator's grief over the loss of his bride. (Whether the poem is autobiographical is uncertain.) He seems in his mind to have decided that "the winged seraphs of heaven" killed Annabel Lee because they were jealous of the love between himself and her.
Throughout the poem, the narrator mentions how great the love was between them. According to the narrator, "she lived with no other thought" than to love and be loved by him. They "loved with a love that was more than love." It was "stronger by far than the love" of those older and wiser than they.
At the end of the second stanza, the narrator states that the seraphs (a biblical type of angel) coveted his bride and him. In the third stanza, he goes on to explain that this is the reason that "a wind blew out of the cloud," chilling Annabel Lee.
He repeats this in the fourth stanza, stating that the angels were "not half so happy in heaven" and were envious of the couple. He states that "all men know" that this is the reason his Annabel Lee died.

Why do you think it's important to learn about the physical features of other countries?

Understanding geography helps understand other peoples, cultures, and nations. How important are knowing about deserts to an understanding of the Middle East? How important are the Andes in the history of Colombia, Ecuador, Peru, Bolivia, and Chile? How does knowing about the vast regions of cold tundra help us understand Russia and Russian people?
A knowledge of the physical features of a nation could have changed the course of history in many ways, helping prevent any number of peoples or leaders from making disastrous mistakes. What if Napoleon or Hitler had not been defeated in large part by Russian winters? What if Americans knew better just how mountainous Afghanistan is, or how dense the jungles of Vietnam are? Personally, I think Americans might view American Indian origins differently if they only understood just how narrow the Bering Strait is and how easily Native-made boats might have crossed it for thousands of years.

Within the context of the social sciences, the physical features of a country matter a great deal more than most people realize.  Physical features are a country's geography and climate, and these have a bearing on the history, culture, and collective behavior of a people. Even the mythology of a people is influenced by its geography, as, for example, the ocean features prominently in Japanese myths. How people get their food, how and what they build, how they worship, and how they raise their children are inevitably influenced by the terrain and the weather.  Malcolm Gladwell offers an excellent example of this in The Outliers, in which he discusses how growing rice in China has produced generations of hard-working, problem-solving, math-savvy students.  This culture is to some degree a consequence of the physical terrain in many parts of China.  Sociologists take note of how different cultures are in warmer climates as opposed to colder climates and how people who live at the shore are different in many ways from those who live inland. In my own city, which comprises mostly hills and valleys, neighborhoods preserve their separate identities and cultures more than in other cities because of the physical features.  Geography and climate are powerful influences on people, and the social sciences need to take this into account.

What is the connection between pederasty and education?

Pederasty describes a relationship between an older and younger man, typically of a consensual, sexual nature. In Ancient Greece, the practice of pederasty was quite common and heavily tied to education and training. An older, well-established man would form a romantic relationship with a boy who was somewhere between the ages of twelve and eighteen. This relationship was more a dynamic of master and apprentice than a romantic couple, and the older man was at least partly responsible for the education, socialization, and skill training of his younger partner. In some parts of Greece, the older man would also provide gifts to his younger partner to help him establish a life. One such tradition involved presenting the young man with military clothing as a gift to begin his military training.
Though such a relationship strikes discomfort in the heart of many a modern person, it is important to note that these relationships were fundamentally different from pedophilic behavior. The pederastic relationship involved much more than sexual intimacy, and though there was an age-based power dynamic involved, the younger partner is understood to have consented to the relationship. Some historians debate the degree of consent or affection involved in pederasty, but we can glean knowledge from the term itself. The root word erastes translates as "love" or "lover" and was used in terms describing relationships of both hetero- and homosexual natures. In contrast to the nurturing and romantic connotations of pederasty, pedophilia is a pathological, consumptive, and abusive sexual impulse which has no regard for whether a younger partner consents.
In short, the pederastic relationships of Ancient Greece were not only romantic or sexual, but also involved the tutelage and training of a younger man by someone who was older and more experienced.
http://www.inquiriesjournal.com/articles/175/examining-greek-pederastic-relationships