Who is Dame Clarice?

Dame Clarice is the innkeeper's wife at the Burford Bridge Inn on Westhumble Lane. She is described as a diminutive, energetic woman. Dame Clarice has an extroverted and commanding personality. She frequently barks orders at her husband and the employees of the inn. Adam describes her as a busy worker: she is either cooking, giving orders, or nudging people out of her way. Dame Clarice seems to supply all the energy in the inn.
Adam bristles when Dame Clarice calls him a "little fellow." He worries that she will strip him of his muddy clothes and scrub him down in the kitchen. However, Dame Clarice doesn't do this; she merely hands him a towel and shows him a shed in the stableyard, where he and Roger (his father) can wash in privacy.
Adam doesn't visit the inn again until four months later. However, this time Roger is not with him. Dame Clarice tells Adam that Roger has been hunting for him all over the south of England. The innkeeper eventually discloses that Roger is in London.

Single Variable Calculus, Chapter 7, 7.7, Section 7.7, Problem 12

Show that $\cos h(x + y) = \cos h x \cos h y + \sin h x \sin h y$

Solving for the left-hand side of the equation

Using Hyperbolic Function


$
\begin{equation}
\begin{aligned}

\cos h x =& \frac{e^x + e^{-x}}{2}
\\
\\
\cos h (x + y) =& \frac{e^{(x + y)} + e^{- (x + y)} }{2}
\\
\\
\cos h (x + y) =& \frac{e^{(x + y)} + e^{(-x-y)} }{2}
\\
\\
\cos h (x + y) =& \frac{e^x e^y + e^{-x} e^{-y}}{2}

\end{aligned}
\end{equation}
$


Using the Identities


$
\begin{equation}
\begin{aligned}

\cos h x + \sin h x =& e^x \text{ and } \cos h x - \sin h x = e^{-x}
\\
\\
\cos h (x + y) =& \frac{(\cos h x + \sin h x)(\cos hy + \sin h y) + (\cos hx - \sin hx)(\cos hy - \sin hy) }{2}
\\
\\
\cos h (x + y) =& \frac{\cos h x \cos hy + \cancel{\cos h x \sin h y} + \cancel{\cos hy \sin hx} + \sin hx \sin hy + \cos hx \cos hy - \cancel{\cos h x \sin h y} -\cancel{\cos hy \sin hx} + \sin hx \sin hy }{2}
\\
\\
\cos h (x + y) =& \frac{2 \cos hx \cos hy + 2 \sin hx \sin hy}{2}
\\
\\
\cos h (x + y) =& \frac{\cancel{2} (\cos hx \cos hy + \sin hx \sin hy)}{\cancel{2}}
\\
\\
\cos h(x + y) =& \cos hx \cos hy + \sin hx \sin hy


\end{aligned}
\end{equation}
$

f(x) = xe^x , n=4 Find the n'th Maclaurin polynomial for the function.

Maclaurin series is a special case of Taylor series that is centered at c=0 . The expansion of the function about 0 follows the formula:
f(x)=sum_(n=0)^oo (f^n(0))/(n!) x^n
 or
f(x)= f(0)+(f'(0))/(1!)x+(f^2(0))/(2!)x^2+(f^3(0))/(3!)x^3+(f^4(0))/(4!)x^4 +...
To determine the Maclaurin polynomial of degree n=4 for the given function f(x)=xe^x , we may apply the formula for Maclaurin series.
To list f^n(x) up to n=4 , we may apply the Product rule for differentiation: d/(dx) (u*v) = u' *v +u*v' and derivative property: d/(dx) (f+g) = d/(dx) f +d/(dx) g .
f(x)=xe^x
Let: u =x then u' = 1
      v = e^x  then v' = e^x
d/(dx) (xe^x) =(1*e^x) + (x*e^x)
                  =e^x +xe^x
f'(x)=d/(dx) (xe^x)
            = e^x +xe^x
f^2(x) = d/(dx) (e^x +xe^x)
           =d/(dx) e^x + d/(dx) xe^x
           = e^x + (e^x+xe^x)
           = 2e^x+xe^x
f^3(x) = d/(dx) ( 2e^x +xe^x)
            =d/(dx) 2e^x + d/(dx) xe^x
            = 2e^x + (e^x+xe^x)
            = 3e^x+xe^x
f^4(x) = d/(dx) ( 3e^x +xe^x)
            =d/(dx) 3e^x + d/(dx) xe^x
            = 3e^x + (e^x+xe^x)
            = 4e^x+xe^x
Plug-in x=0 for each f^n(x) , we get:
f(0)=0*e^0
         =0*1
          =0
f'(0)=e^0+0*e^0
            =1 +0*1
           =1
f^2(0)=2e^0+0*e^0
            =2*1 +0*1
            =2
f^3(0)=3e^0+0*e^0
            =3*1 +0*1
            =3
f^4(0)=4e^0+0*e^0
            =4*1 +0*1
            =4
Plug-in the values on the formula for Maclaurin series, we get:
sum_(n=0)^4 (f^n(0))/(n!) x^n
       = f(0)+(f'(0))/(1!)x+(f^2(0))/(2!)x^2+(f^3(0))/(3!)x^3+(f^4(0))/(4!)x^4
       = 0+1/(1!)x+2/(2!)x^2+3/(3!)x^3+4/(4!)x^4
       = 0+1/1x+2/2x^2+3/6x^3+4/24x^4
       = 0+x+x^2+1/2x^3+1/6x^4
       = x+x^2+1/2x^3+1/6x^4
The Maclaurin polynomial of degree n=4 for the given function f(x)=xe^x will be:
P(x)=x+x^2+1/2x^3+1/6x^4 

Can there be any other title for the Story of My Life by Helen Keller?

I believe that the question is asking if it is possible that there could be an alternative title to Helen Keller's autobiography The Story of My Life.
Yes, there could be alternative titles.  
I'm sure Helen Keller thought of multiple titles before choosing the final title.  
Helen Keller did not have an easy life because of her multiple disabilities, but she overcame a great deal.  A possible title could have been My Struggle. Perhaps then Hitler would have been forced to rethink his autobiography's title.  
Since Helen Keller used "my" in the title, it appears that she wanted the title to focus on herself.  Perhaps How I See the World would have worked for a title.  Or How I See Things would work, but it is very similar to the previous title.  
The alternative title could remove the "I," "me," or "my" from the title and go for something mysterious and catchy.  I had a student suggest Through Darkness and Silence as a possible title.  I like that title a lot, but it does make me think of submarine warfare.  The story of Louis Zamperini, written by Laura Hillenbrand, went with a one word title -- Unbroken.  The title captures the unbreakable spirit of the man.  An alternative title to Helen Keller's book could do the same one word idea.  I like Perseverance as a possible title.   

In "The Canterville Ghost" by Oscar Wilde, where had the ghost taken the bottle of oil?

In Chapter Two of "The Canterville Ghost," Mr Otis is disturbed by the sound of the ghost rattling his chains in the early hours of the morning. In response, Mr Otis offers the ghost some oil called Tammany Rising Sun Lubricator. He leaves it by "the bedroom candles" and offers to bring the ghost more oil, should he require it. When Mr Otis returns to his room, the ghost picks up the oil but then throws it "violently" to the floor and flees the corridor.
By Chapter Four, however, the ghost has a fresh bottle of oil. According to the narrator, the ghost retrieved this new bottle of oil from Mr Otis's bedroom while the family is downstairs having dinner. This time, he takes Mr Otis's advice and the ghost oils his chains so that he can walk around the house without being detected.

Discuss the importance of the Benjy section in The Sound and the Fury.

Benjy's section in The Sound and the Fury is one of the most famous chapters in all of 20th-century American literature, and there is really nothing else like it. It is a work of "high modernism," a phrase often used to describe the dense and complicated novels of the early 20th century which were concerned with mimetically expressing human consciousness.
Benjy, the youngest of the Compson children, has a severe mental disability and, as such, he is unable to talk. As one would expect, this severely limits his ability to communicate effectively and often his only outward expressions are fits of crying.
However, even though Benjy cannot talk, Faulkner wants to express what it is like to be Benjy, and therefore he narrates the chapter using free-indirect discourse to give us Benjy's point of view. In doing so, Faulkner quickly shows us that Benjy is more complicated than he initially seems. Even though he cannot communicate verbally, Benjy has a rich internal life that, while confusing, allows the reader to access and understand the world of the novel in a unique way.
There are many notable qualities of Benjy's narrative, but what readers often find most disorienting are the constant temporal shifts. Often, Benjy will have an experience in the present which reminds him of something that happened in the past, and then suddenly the reader is sucked into the past with Benjy even though the novel continues to be narrated in the present tense. Usually, Faulkner warns us when such transitions are coming by utilizing italics, such as in the following passage:

"Wait a minute." Luster said. "You snagged on that nail again. Cant you never crawl through here without snagging on that nail."
Caddy uncaught me and we crawled through. Uncle Maury said to not let anybody see us, so we better stoop over, Caddy said (Benjy's section, page 4 in Vintage International edition).

These transitions are frequent and it takes awhile to piece together the various temporalities through which Benjy's mind drifts. Such confusion can make the chapter frustrating, but it is ultimately rewarding once you are able to fit the puzzle pieces together.

Benjy's section sets the tone for the rest of the book. As with most modernist works of literature, The Sound and the Fury, a highly complex, experimental novel, is concerned with the impossibility of ideal communication. That explains why Faulkner chooses to begin the novel with Benjy, a seriously disabled invalid without the ability to articulate his experiences in speech. The difficulty of this first section is exacerbated by the fact that Benjy has no concept of time; all the events he recalls take place on one day: Easter Saturday, 1928.
Inevitably, Benjy's narrative is confusing and disjointed. But this serves a very useful purpose in relation to the book's overall structure. The puzzling prose that Faulkner employs creates a sense of mystery and suspense, foreshadowing the events to come, making us want to find out what's really going on. Trying to figure out the reasons for the Compson family's decline will take some doing, and Benjy's section, the first in The Sound and the Fury, gives us fair warning.

What are Hobbes’s views on the divine right of the monarchy?

Hobbes did not believe in monarchy by divine right for one simple reason: he was an atheist. He could not come right out and admit this, however, because it would have caused him serious trouble with the authorities. However, there is little doubt that Hobbes was, at best, an agnostic. In any case, Hobbes' political philosophy has no need to speculate about the existence of God. His entire system is completely secular and naturalistic; it has no need for divine intervention whatsoever.
The omnipotent monarch of Hobbes's philosophy does not occupy the throne because God somehow supernaturally intervened to put him there. He is there because men in a state of nature got together, gave up their natural rights, and invested them in a sovereign with absolute power to maintain order and stability.
The sovereign does have an important religious role to play, however. He gets to decide the outward signs of religious observance. He cannot control men's minds, but he can control which doctrines are taught. He can also actively censor and suppress those he deems injurious to public safety. This is an important foundation of the royal sovereignty. Religious differences were a major source of the social and political disorder that plagued England in Hobbes's lifetime and were a source he particularly abhorred. Putting the all-powerful sovereign at the head of a national church is, for Hobbes, the only way to heal these dangerous divisions.