What political power did the Church have over people in the medieval times?

I would say that the Catholic Church had political power
over people in the Middle Ages in a few different
ways.

First of all, there were places that were directly
under the control of the Church -- places where the Pope was the official ruler of the
country.  This was in the Papal States in what is now
Italy.

Second, the Church owned a lot of land in other
parts of Europe.  Each monastery, for example, owned lots of land and the abbots of the
monasteries ran those lands.  The people on the land (if they were serfs) belonged to
the monastery.  In that way, the Church had political power because it had temporal
power over its estates.

Finally, the Church had moral
authority that could be used politically.  The Church was seen as the only way to Heaven
and so the clergy could be pretty important in making people behave in certain ways. 
They could threaten, for example, to excommunicate people who did not make the "right"
choices. 
This gave the Church more power of an indirect
sort.

What is the retrograde motion of Mars?

When observing planetary motion with respect to the stars
that are located very far away and can be considered stationary, a few unusual things
are observed. One of them is retrograde motion.

The
observance of retrograde motion is best explained by an example where you are travelling
in a car and another car is travelling beside you. If the speed of both the cars is the
same, the other car appears stationary with respect to a very distant object, if the car
is moving faster than you, you see it moving ahead. On the other hand, if you are moving
faster you would observe the car moving in the reverse
direction.

The same phenomenon occurs with the planet Mars.
When observations of the planet are taken over extended periods of time, it is observed
that the planet does not appear to move in a constant circular orbit but instead moves
backwards and creates an s-shape or a loop.

There is no
particular month of the year when the planet is seen to be in retrograde motion. The
time varies and this occurs approximately once every two
years.

The links below show the retrograde motion of the
planet in the years 2003 and 2010.

In Macbeth, why does Macbeth ask Banquo so many apparently casual questions about where he is riding?

The questions Macbeth asks Banquo in Act III, scene i appear to
be casual, as if Macbeth is only concerned that Banquo not miss that night's banquet. In fact,
they are very pointed questions. He wants to know when Banquo is leaving the castle and how far
he will be riding. This will give him an idea of when Banquo will be returning. Macbeth needs
this information so that he can pass it along to the men he has employed to attack Banquo and his
son Fleance on the road and murder them both.

After his conversation
with Banquo, Macbeth meets with the murderers. He tells them the
following:

Within this
hour at most

I will advise you where to plant
yourselves,

Acquaint you with the perfect spy o' th'
time,

The moment on 't; for 't must be done
tonight,

And something from the
palace;

This suggests that Macbeth has
a general idea as to where Banquo will be, but he wants to plan precisely where the attack should
take place to ensure success and to make sure Banquo and his son are not murdered near Macbeth's
castle. After he figures it out, he will get back to them in no more than an
hour.

Please provide a summary of Chapter 5 of To Sir with Love.

 In chapter 5, Braithwaite still
looks for a job. Here, the character of Braithwaite remains the same since all those interviews
that he has gone in this chapter and racially rejected by them because of the color of his skin.
The character also changes.

 Braithwaite keeps on searching for jobs
but he always gets the same kind of response “To black”. He then saw an advertisement of the
electrical form at Dagenham. He went to the Personnel Office and when he was called to the
manager, the manager asked him why he wanted this job. This is when Braithwaite gets irritated
because whenever he went for a job interview, he was racially rejected not by his academic skills
but his skin color. “I need the job to help me pay for little things like the food I eat, the
cloths I wear and the lodging I occupy.” But then, he was still racially rejected when the
manager politely says that he will not fit here because none of his employees are uneducated and
he also says – “They might resent the posh way you speak and….” This shows the theme of racism.
This is also the possible answer which rejects Braithwaite for every
interview.

Till here, the character of Braithwaite is in a
sad/angry/irritated mood. But then, he meets someone that picked him up from the dark to the
light. In St. James Park, he met an old man. “A big city cannot afford to have its attention
distracted from the important job of being a big city by such a tiny, unimportant item as your
happiness or mine”. He encourages Braithwaite to become a teacher in East End of London because
they desperately need teachers over there –“From all accounts it is rather a tough area and most
teachers prefer to seek jobs elsewhere”. Listening to the words of the old man who has sat with
Mr. Braithwaite for two hours talking about the situation and still not exchanged names, he tries
to apply for that job. He gave his CV to the Ministry of education. They called him for the
interview and it went well and he got a job in Greenslade School.He also says that they did not
care for his color. The East end of London, people talk openly. The area is a war effected, dirty
and messy place and not a suitable attraction for people.

 Till this
point, Braithwaite’s character starts to change. His character is now in a relaxed and in a happy
mood. So in this chapter, the character of Braithwaite is
developed.

The plot if this chapter is when he meets the old man in
the park feeding the ducks. The plot is developed when they both have a two hour talk. The old
man is the main person of the plot and he is the one who changes Braithwaite and helped him
continue to search for a job and finally, Braithwaite got the job in Greenslade
School.

The theme that is introduced in this chapter is racism;
mainly racism. Every single time when Braithwaite goes for an interview, the theme of racism is
introduced. They racially reject him because of the color of his skin. Also, the theme of
friendship is introduced. It is introduces when Braithwaite has the talk between the old man in
the park. Usually people don’t get closer to Negros in London just like it happened in the first
chapter when the lady refused to sit with Braithwaite in the bus. But this old man was freely
talking to Braithwaite and through their talk; we can find that they were
friendly.     

What is the inverse of the function f(x) = (e^3x)*14 - 21

We'll suggest another method of getting the inverse
function.

We know that the product of derivative and it's inverse is
1.

f'(x)*[f^-1(x)]' =1

We'll divide
both sides by f'(x):

[f^-1(x)]' =
1/f'(x)

We'll calculate the integral of functions both
sides:

Int [f^-1(x)]'dx = Int
dx/f'(x)

We'll differentiate the function
f(x):

f'(x) = [(e^3x)*14 - 21]'

f'(x) =
42*e^3x

f^-1(x) = Int dx/42*e^3x

Int
dx/42*e^3x = (1/42)*Int e^-3xdx

Int dx/42*e^3x =
-e^-3x/3*42

Int dx/42*e^3x =
-1/126*e^3x

Therefore, the inverse function is:
f^-1(x) = -1/126*e^3x.

Give 2 examples of cruelty his brother did to Doodle.

A first way that Doodle's brother is cruel is when he
shows Doodle his coffin from when Doodle was an infant. Not only does the sight of a
coffin scare a young child, but to consider that it was determined for Doodle himself
made it that much more vulgar. He admits the cruelty in these
lines:

One
time I showed him his casket, telling him how we all believed he would die. When I made
him touch the casket, he screamed. And even when we were outside in the bright sunshine
he clung to me, crying, "Don't leave me, Brother! Don't leave
me!"

A second way he
demonstrates this cruelty happens in the very end. When Doodle's brother leaves him
after it had started to rain, we see abandonment have dire, deadly
consequences:

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The rain came, roaring through the pines. And
then, like a bursting Roman candle, a gum tree ahead of us was shattered by a bolt of
lightning. When the deafening thunder had died, I heard Doodle cry out, "Brother,
Brother, don't leave me! Don't leave me!"

The knowledge
that our plans had come to nothing was bitter, and that streak of cruelty within me
awakened. I ran as fast as I could, leaving him far behind with a wall of rain dividing
us. Soon I could hear his voice no
more.

This section shows that
Doodle's brother understood his cruelty. It also demonstrates that each time Doodle's
brother responded this way, Doodle interpreted the act as
abandonment.

For a character in Doodle's condition, no
greater affliction could be done to him by another than that, to be completely
abandoned.

How is Montag a hero in the book Fahrenheit 451?

Montag is heroic in that he reacts against a controlling society
that prohibits individuality and imagination. He finds such mindless conformity damaging to the
spirit.

Montag rebels against his dehumanizing society in the hope
of becoming a real person, one who truly thinks and acts without drugs. Having witnessed the
deterioration of his wife Mildred, who has become no more than the visions on the parlor screens,
a mere automaton who laughs on cue, and who listens to the buds in her ears, never hearing her
heart or that of her husband, Montag makes supreme efforts to re-engage her in life, but she
insists upon dwelling in the artificial world of her society. In fact, she is so insistent about
choosing this world that she reports her husband to the authorities. But, he defies these powers
of society, setting fire to Beatty and fleeing to Faber, who connects him to the people who dwell
in another place where they preserve the texts of famous works, the recordings of the thoughts of
philosophers, scientists, theologians, satirists, and historians of the past.

While Montag hides from the authorities, he talks with Granger, who
tells him about his grandfather, a sculptor, who created things. "He shaped the world. He
did things to the world," Granger
reflects.

"Everyone
must leave something behind when he dies, my grandfather said....Something your hand touches some
way so your soul has somewhere to go when you die....It doesn't matter what you do ...so long as
you change something from the way it was before you touched
it."

Montag changes his society by
rebelling against it, by burning what is evil rather than what is good, and he will leave
something behind as he has memorized the Book of Ecclesiastes.