Who are the heroes of Shakespeare's historical plays and why?

The hero in Shakespeare's historical plays can be any
character that Shakespeare choses to lift up in his writing. It might be a king, or a
king's son or a citizen. (While you refer specifically to the historical plays, please
note that Shakespeare's tragedies also include "the history plays with tragic designs,
such as Richard II.")

Many of
Shakespeare's heroes were tragic heroes. These are great ("accomplished") man, who
ultimately die, and their deaths are their own fault caused by their tragic flaw. In
Hamlet, for instance, Hamlet is the son of the
murdered king. He is the tragic hero of the story, who dies (with everyone else); his
tragic flaw is indecision.

In one of Shakespeare's
historical plays, Brutus truly loves Julius Caesar, in the play by the same name.
However, he becomes our tragic hero. It is not Mark Antony who
seems to love Caesar as well, but uses Caesar's death to promote
his own future as one of the Triumvirate (leaders) of the Roman Empire. Brutus is a
Roman nobleman. Some say that his flaw is his "philosophical commitment to principle,"
while others argue that it could be "self-deception" in making himself believe that what
he is doing (his part in Caesar's assassination) is noble and
acceptable.

On the other hand, not all of Shakespeare's
heroes die. In Henry V, Henry is one of Shakespeare's heroes who
not only lives, but is triumphant at the play's end in defeating the French in a
decisive battle during The Hundred Years War. Henry is wise, and a valiant king and
warrior. He encourages his men to fight when the odds seem impossible in the favor the
the French. Henry's men respect him and respond to his battle cry; meanwhile, Henry
enters the battlefield as his men do—they fight side-by-side, while other kings would
have been waiting, somewhere safe, until the fighting was over. Henry V was a true hero,
refusing to ask of his men what he could not give himself: his life, if
necessary.

There are different kinds of heroes in the
historical plays. Sometimes they are tragic heroes who give their lives for a greater
cause, or lose their lives because of a tragic flaw, while in other of Shakespeare's
historical plays, the hero is still a great man, but he overcomes obstacles and comes
out triumphant at the play's
end.

Additional
Source:

http://extension.berkeley.edu/oxford/seminars/heroes_long.html

How does the mood of Romeo and Juliet change during act 3?

The mood of the play in Acts 1 and 2 very greatly from Act
3, at which point the plot is shocked back into the world of the feud. In Act 1, Romeo
is heartbroken at the thougt of not having his Rosaline, while Juliet tells her mother
that marriage is "an honor" she has not thought of (Act 1).  This full distance between
their lives serves to enhance the passionate exchanges they hold in the final scene of
Act I into the balcony scene in Act 2. . In Act 2, we see the two arranging their
wedding with the help of Friar Laurence and the Nurse. If nothing else, the mood can
best be described as hopeful here, as stated by Friar Laurence when he states that he
hopes the union will serve to end the feud.

Act 3 presents
the dramatic twist of events. The harsh reality of the feud is brought back to the
forefront with the deaths of Mercutio and Tybalt, and the tragedy is further enhanced by
Romeo's banishment by the Prince for his role in the deaths. The hopeful mood has been
washed away and replaced by the violent and desperate mood of Act 3's
events.

What are three falling actions in The Devil's Arithmetic?

Let us remember that the falling action comes straight
after the climax of a work of literature, or the point of greatest emotional intensity,
and leads towards the resolution. In this great holocaust novel, the climax is
definitely when Hannah and her friends go into Lilith's Cave to face their death like so
many other Jews during the Holocaust. The falling action, therefore, occurs straight
after this event in Chapter 19. Three events of note would be Hannah's return to the
present and her sense of loss at not having said goodbye to Shifre and Esther. Then,
when Hannah sits next to her Aunt Eva, she is able to decode the tatoo on her arm and
the meaning of the letters and numbers, understanding finally the identity of her aunt
and Grandpa Will. Hannah, thanks to her experiences in the past, is now able to
understand her relatives, their horrific experiences, and the importance of such
ceremonies as Passover that commemorate both the suffering of the Jewish people and
their deliverance.

What does the last quote "Nothing like that could happen again. Not in this day and age" represent?

I think that the last quote is where Boyne's voice emerges
throughout a narrative where Bruno's voice was heard.  At the end, when Bruno dies, the
eyes and perception that led us throughout the narrative is also gone.  It is up to
Boyne to be able to provide some level of both closure and testament to Bruno in the
process.  The last quote is able to do this and also strike the reader on two levels. 
The first is that there has to be some reckoning between Boyne's statement, which
presents consciousness as it should be, and our own reality, which sadly does fall short
of this standard.  The conditions in which we live are ones where there are still
examples of genocide happening.  There are situations where children are being murdered,
no different than Bruno and Shmuel, and we, as the reader, must account for it in
reflecting on the last quote.  The second level in which the quote impacts the reader is
that while it is Boyne's voice speaking, he has managed to continue the child's faith in
the world and the child's viewpoint that has constructed the novel. For Bruno, his
transcendent views of the world are able to rise above the horrible contingency of
Nazism in the closing words of the novel.  In suggesting that something like the
Holocaust or what happened to Bruno and Shmuel should never happen again, we, as the
reader, are left to see the world, our world, as Bruno would.  While we have to reckon
with the differences between Bruno's vision and how our world is, Bruno gives us a
standard to which we are compelled to aspire.

What is y if 8^x = 2 and 3^(x+y) = 81?

You need to use the property of logarithms such that:
`log_a (b^c) = c*log_a b` , hence you should take logarithms to base 3 both sides
of `3^(x+y) = 81`  such that:

`log_3(3^(x+y)) = log_3 81
=gt (x+y)*log_3 3 = log_3 (3^4)`

Substituting 1 for `log_3
3`  yields:

`(x+y)*1 = 4*1 =gt x+y =
4`

Notice that you should find x using the exponential `8^x
= 2`  such that:

`8 = 2^3 =gt 8^x = (2^3)^x =gt 8^x =
2^(3x)`

Hence, you need to substitute `2^(3x)`  for `8^x `
in equation `8^x = 2`  such that:

`2^(3x) =` 
`2` 

Notice that bases of exponentials both sides are
alike, hence you should equate exponents such that:

`3x = 1
=gt x = 1/3`

You need to substitute `1/3`  for x in `x+y =
4`  such that:

`1/3 + y = 4 =gt y = 4 -
1/3`

Bringing the terms to a common denominator
yields:

`y = (12-1)/3 =gt y =
11/3`

Hence, evaluating y under
given conditions yields `y=11/3` .

What kind of lure or bait did Santiago use in The Old Man and the Sea?

When Santiago baited the collection of poles that included
the pole on which he caught the "big fish", he used a combination of baits. The main
part of the hook was covered by a tuna, which had been wrapped around the shank of the
hook and sewed in place in preparation for the fishing trip. Sardines were hooked
through the eye sockets to cover and disguise the pointed ends of the double hook. The
lure of the fresh fish enticingly coming through the water at one hundred fathoms depth
was what attracted the marlin to take the bait.

What is the differential of (tanx) . Can you prove it please?

We can solve the differential of tan x using the first
principle, but an easier way to do it is to use the quotient rule as we know that tan x
= sin x / cos x.

[tan x]' = [sin x / cos
x]'

=> [(sin x)'*cos x - (sin x)*(cos x)']/(cos
x)^2

the derivative of sin x = cos x and that of cos x is -
sin x

=> [cos x* cos x  + sin x * sin x]/(cos
x)^2

=> [(cos x)^2 + (sin x)^2]/(cos
x)^2

=> 1 / (cos
x)^2

=> (sec
x)^2

The derivative of tan x is (sec
x)^2