Calculus: Early Transcendentals, Chapter 6, 6.2, Section 6.2, Problem 13

y=1+sec(x),y=3
Refer the image. From the graph, the curves intersects at x=-pi/3 and x=pi/3.
Using washer method,
A cross section is a washer of cross sectional area A(x) with,
Inner radius=(1+sec(x))-1=sec(x)
Outer radius=3-1=2
A(x)=pi(2^2-(sec(x))^2)
A(x)=pi(4-sec^2(x))
Volume of the solid obtained by rotating the region bounded by the given curves about y=1 (V) is,
V=int_(-pi/3)^(pi/3)A(x)dx
V=int_(-pi/3)^(pi/3)pi(4-sec^2(x))dx
V=2piint_0^(pi/3)(4-sec^2(x))dx
V=2pi[4x-tan(x)]_0^(pi/3)
V=2pi((4*pi/3-tan(pi/3))-(4*0-tan(0)))
V=2pi(4*pi/3-sqrt(3)-0)
V=2pi((4pi)/3-sqrt(3))

Single Variable Calculus, Chapter 4, 4.5, Section 4.5, Problem 36

Use the guidelines of curve sketching to sketch the curve. $\displaystyle y = \sec x + \tan x \quad ,0 < x < \frac{\pi}{2}$

The guidelines of Curve Sketching
A. Domain.
The function has domain $\displaystyle \left( 0, \frac{\pi}{2} \right)$

B. Intercepts.
Solving for $y$-intercept, when $x=0$
$y = \sec 0 + \tan 0 = 1 + 0 = 1$

C. Symmetry.
By using symmetry test, we can say that the function is not symmetric to either $y$-axis or origin.

D. Asymptotes.
We can rewrite $\displaystyle y = \sec x + \tan x \text{ as } y = \frac{1}{\cos x} + \frac{\sin x}{\cos x} \quad y = \frac{1 + \sin x}{\cos x}$
when $\cos x = 0$,
$\displaystyle x = \frac{\pi}{2} + 2\pi n$, where $n$ is any integet
For interval $\displaystyle 0 < x < \frac{\pi }{2}$, the vertical asymptote is none.

E. Intervals of Increase or Decrease.

$
\begin{equation}
\begin{aligned}
\text{if } f(x) &= \sec x + \tan x, \text{ then}\\
\\
f'(x) &= \sec x \tan x + \sec^2 x\\
\\
f'(x) &= \frac{\sin x}{\cos ^2 x} + \frac{1}{\cos^2 x} = \frac{\sin x + 1}{\cos^2 x}\\
\\
\\
\text{when } f'(x) &= 0 \\
\\
0 &= \sin x + 1\\
\\
\sin x &= -1\\
\\
x &= \frac{-\pi}{2} + 2 \pi n \qquad \text{where } n \text{ is any integer}
\end{aligned}
\end{equation}
$

For interval $\displaystyle 0 < x < \frac{\pi }{2}$,
The function has no critical numbers.
Thus the interval of increase or decrease is...

$
\begin{array}{|c|c|c|}
\hline\\
\text{Interval} & f'(x) & f\\
\hline\\
0 < x < \frac{\pi}{2} & + & \text{increasing on } (0,\frac{\pi}{2})\\
\hline
\end{array}
$


F. Local Maximum and Minimum Values.
Since $f'(x)$ is always increasing on interval $\displaystyle 0 < x < \frac{\pi }{2}$. It means that we have no local maxima and minima at this interval

G. Concavity and Points of Inflection.
Since $f(x)$ is always increasing, we can say that the function has no inflection points. Therefore, the function has no upward concavity at $\displaystyle \left( 0, \frac{\pi}{2} \right)$

H. Sketch the Graph.

What is a good thesis statement for A Christmas Carol?

It might be interesting to argue the idea that Scrooge wasn't always such an unpleasant and greedy person, that the conditions of his childhood paved the way for him to become avaricious later in life. There's a good deal of evidence to suggest that a difficult childhood, where Scrooge was abandoned at a miserable boarding school and alienated by his family, made him prioritize money over everything else. When the Ghost of Christmas Past takes Scrooge back to his past, the old man "wept to see his poor forgotten self as he had used to be." His younger self has only a "feeble fire" and is all alone in this school, where there is "too much getting up by candle-light, and not too much to eat." An empty belly and small fire could certainly contribute to Scrooge's acquisitiveness; he certainly would not want to return to this position later in life. Further, when his sister comes to collect him, she tells him, "Father is so much kinder than he used to be." This is a rather troubling line: was Scrooge sent away by an unkind, even abusive, father? How bad was his very early childhood? How did he come to be totally abandoned in this sad place? We cannot know, but the conditions in which he lived at the boarding school certainly seem to betray either a lack of interest or neglect on his father's part. In thinking of this, we might grow more understanding of Scrooge's desire to acquire money—since money wouldn't abandon him and would, instead, provide him with the security he lacked as a child—especially after Belle breaks it off with him as well.

When writing a thesis statement it's always important to bear in mind that you're putting forward an argument. You need to have something you want to say about the story, then be prepared to argue your case, backing it up with evidence from the text. There are a number of potential arguments you could use in relation to A Christmas Carol. For example, you could argue that Scrooge has only really changed his ways for selfish reasons. In that sense, he remains as selfish at the end of the story as he did at the beginning, albeit with a radically different effect upon other people. The Ghost of Christmas Future gives Scrooge a frightening glimpse of what lies in store if he doesn't mend his ways. He's absolutely terrified at the prospect, genuinely concerned at the fate of his soul. So perhaps it could be argued that Scrooge's dramatic conversion is motivated, not by a genuine desire to improve the lot of his fellow man, but by fear and self-interest. For Scrooge, it's all about doing whatever he can to save himself.

Why does Shylock nurse a grudge against Antonio in The Merchant of Venice?

Shakespeare “The Merchant of Venice” profoundly assembles on the platform upon laws and rules that are usually manipulated for brutal or gratuitous purposes. This ‘ancient grudge’ corrupts the transmission of bitterness and hatred between Christianity and Judaism pouring the aura of hypocrisy everywhere. We find Antonio, a Venetian merchant, habitual of criticizing all Jews for the inflated rates of interest on credit. Shylock, who nurses a long-standing grudge against Antonio, draws him into surprise by favoring for a loan. Shylock behavioral sense of gifting him three thousand ducats without interest tends to predict his single-minded pursuit towards brutality- by sanctioning a ‘will’ to a ‘pound of Antonio’s own flesh’. Despite Bassanio’s warnings, Antonio agrees to take the risks.
Shylock dictatorship of hideous expansion is worn out up by lifting up to multiple interpretations of ‘the pound of flesh’. The fact about their friendship is so noteworthy that Bassanio’s debt was to be rewarded with Antonio’s flesh. The blood of Antonio symbolizes Shylock’s own flesh and blood, as a repay for the loss of his daughter, Jessica. It shows a steady cue of the strictness in Shylock’s world, the numerical mind of Shylock demands revenge in exchange for his three thousand ducats. The other characters measured it in emotions with long metaphors and words, whereas Shylock resembles it more of numerical quantities, to evaluate revenge rather than forgiveness.

Shylock is a Jewish usurer who makes profits on his loans, so he resents the merchant Antonio, who lends money interest-free to people, which undercuts Shylock.

How like a fawning publican he looks!I hate him for he is a Christian,But more for that in low simplicityHe lends out money gratis and brings downThe rate of usance here with us in Venice.If I can catch him once upon the hip,I will feed fat the ancient grudge I bear him (Act I, Scene 3, lines 41-47).

As a Jew in Venice, Shylock is restricted to a certain area of the city called the ghetto, a crowded neighborhood where all Jews are required to live. During the Renaissance, Jews were perceived as a threat to Christians, so they were isolated in this city. There was also resentment toward their skills in medicine and banking, so they were prevented from entering such fields. As a result, Shylock is resentful because he feels he could rise much higher in Venetian society if he were not prohibited from certain opportunities and places, avenues open to someone like Antonio.

College Algebra, Chapter 7, Review Exercises, Section Review Exercises, Problem 44

Determine the determinant of the matrix $\displaystyle A = \left[ \begin{array}{cc}
2 & 2 \\
1 & -3
\end{array} \right]$ and if possible, the inverse of the matrix.

Using the formula

$\displaystyle |D| = \left[ \begin{array}{cc}
2 & 2 \\
1 & -3
\end{array} \right] = 2 \cdot (-3) - 2 \cdot 1 = -8$


$
\begin{equation}
\begin{aligned}

A^{-1} =& \frac{1}{ad - bc} \left[ \begin{array}{cc}
d & -b \\
-c & a
\end{array} \right]
\\
\\
A^{-1} =& \left[ \begin{array}{cc}
2 & 2 \\
1 & -3
\end{array} \right]^{-1} = \frac{1}{(2)(-3)-(2)(1)} \left[ \begin{array}{cc}
-3 & -2 \\
-1 & 2
\end{array} \right] = \frac{-1}{8} \left[ \begin{array}{cc}
-3 & -2 \\
-1 & 2
\end{array} \right]
= \left[ \begin{array}{cc}
\displaystyle \frac{3}{8} & \displaystyle \frac{1}{4} \\
\displaystyle \frac{1}{8} & \displaystyle \frac{-1}{4}
\end{array} \right]

\end{aligned}
\end{equation}
$

Is sugar a mixture?

I believe that this question is asking whether or not table sugar is a mixture.  
Table sugar, or sucrose, is not a mixture.  It is a compound.  
Let me start with a pure substance and work up from there.  Elements are pure substances because they cannot be broken down any more.  An element is made up of only a single kind of atom.  The element gold is made up of gold atoms.  Carbon is made up of carbon atoms.  
A compound is made of two or more elements that have been chemically bonded/joined to each other.  Compounds cannot be separated through physical means. Water is a good example of a compound.  It is made of hydrogen and oxygen. A compound is what table sugar is.  Sucrose is made of three elements: carbon, hydrogen, and oxygen.  However, this information is true of a lot of sugars, so I will clarify.  Sucrose is identified as C12H22O11.    
A mixture consists of two or more elements that are not chemically bonded to each other.  It could be separated by physical means. "Lucky Charms" cereal is a good example.  It is a mixture composed of the cereal part and the awesome marshmallow part. Each "element" can be identified and separated from the other through physical means.  Sugar can be mixed into all kinds of things to create a mixture or a solution, but the sugar itself is a compound.  

What feeling of the persona is likened to an apple tree in the first stanza?

The apple is specifically mentioned in stanza 3; however, stanza 2 is the stanza about the actual tree that this question is referring to. I would like to make mention of the fact that stanza 2 does not specifically name an "apple" or a "tree." Stanza 1 tells readers that the narrator's wrath "did grow." The narrator then waters his wrath and bathes it in sunlight as well. Stanza 3 then tells us that "it" grew until it bore an apple. "It" is two things at the same time: it is the narrator's wrath as well as the apple tree that bore the poisoned apple. The persona of the apple tree is wrath, anger, bitterness, hatred, etc. The narrator lets his anger grow and grow until it is all consuming, and it eventually causes the death of his friend. This poem is about the danger of holding on to anger and nurturing it with continued feelings of fear and deceit, because the anger won't stay static. It will grow like any plant, and the fruit that it bears will be just as poisoned as the relationship that initiated the angry feelings.

It is not until stanza three of William Blake's poem "The Poison Tree" that the comparison to an apple is made. The direct quote is,

"And it grew both day and night.
Till it bore an apple bright."

This is part of an extended metaphor carried throughout the poem likening the speaker's "wrath" to a plant that the speaker cultivates by keeping it a secret and allowing it to grow deeper and darker over time. The apple in stanza three represents the fruits of this anger and in stanza four it is revealed to be a poison apple: when the speaker's enemy attempts to steal and eat the apple, it kills him.
So the apple is an instrument of revenge and can be said to represent the speaker's (or persona's, to use the terms of your question) anger at his foe.