In The Racial Contract, what is one point the author makes?

An important point Mills makes is that the "social contract" does not actually mean what it says. So, we are led to ask: "What does it really mean?"
In his book, Mills mentions traditional social contract theorists such as Hobbes, Locke, and Rousseau. He contends that the "social contract" is both political and moral in nature, where government and the governed coexist in a state of mutual consent. In other words, the governed must consent before the government gains any sort of moral or political legitimacy.
Mills argues that Locke's and Kant's versions of the social contract hold true in the "social contract" of today. Locke and Kant maintained that an intrinsic moral code is encapsulated in a political social contract. Yes, all persons are to be respected, but moral principles can be rationalized according to each person's worldview. Mills maintains that it is this "rationalization" that denies nonwhites their personhood.
On the other hand, Hobbes maintained that self-interest is our biggest motivator in life and that we act morally (and consent to the government's authority) because it benefits us.
Mills makes the point that the "social contract" is not Hobbesian but rather Lockean (or Kantian) in nature. He argues that the social contract is committed to "moral egalitarianism" or even racial equality. In fact, the intrinsic moral code in the Lockean and Kantian social contract is opposed to the "traditionalist hierarchical ideology of the old feudal order, the ideology of inherent ascribed status and natural subordination."
This idea of moral and political egalitarianism is encapsulated in the Declaration of Independence, the United States Constitution, and the Bill of Rights. However, the actual truth (according to Mills) is that our present "social contract" does not admit of the need for Lockean and Kantian notions of equality. He argues that the "social contract" is color coded and restricts the notion of equality to whites. Mills claims that today's "social contract" is Kantian or Lockean in nature, but it does not result in equality for all.

Single Variable Calculus, Chapter 4, 4.4, Section 4.4, Problem 46

Determine the horizontal asymptotes of the curve $\displaystyle y = \frac{x}{\sqrt{x^2 + 1}}$ and use them together with concavity and intervals of increase and decrease, to sketch the curve.

$\displaystyle y = \frac{x}{\sqrt{x^2 + 1}}$ has a domain $(- \infty, \infty)$, So there are no vertical asymptote

Solving for the horizontal asymptote

$\displaystyle \lim_{x \to \pm \infty} \frac{x}{\sqrt{x^2 + 1}} = \lim_{x \to \pm \infty} \frac{x}{\sqrt{x^2}} = \lim_{x \to \pm \infty} \frac{\frac{x}{x}}{\sqrt{\frac{x^2}{x^2}}} = \frac{1}{\pm \sqrt{1}} = \pm 1 $

So the horizontal asymptotes are $y = 1$ and $y = -1$

If we take the derivative of $f(x) = y$


$
\begin{equation}
\begin{aligned}

y' =& \frac{d}{dx} \left( \frac{x}{\sqrt{x^2 + 1}} \right)
\\
\\
y' =& \frac{d}{dx} \left( \frac{x}{(x^2 + 1)^{\frac{1}{2}}} \right)
\\
\\
y' =& \frac{\displaystyle (x^2 + 1)^{\frac{1}{2}} \frac{d}{dx} (x) - (x) \frac{d}{dx} (x^2 + 1)^{\frac{1}{2}} }{[(x^2 + 1)^{\frac{1}{2}}]^2}
\\
\\
y' =& \frac{\displaystyle (x^2 + 1)^{\frac{1}{2}} - \left( \frac{x}{\cancel{2}} \right) (x^2 + 1)^{\frac{-1}{2}} (\cancel{2}x) }{x^2 + 1}
\\
\\
y' =& \frac{(x^2 + 1)^{\frac{1}{2}} - (x^2) (x^2 + 1)^{\frac{-1}{2}} }{x^2 + 1}
\\
\\
y' =& \frac{\displaystyle (x^2 + 1)^{\frac{1}{2}} - \frac{x^2}{(x^2 + 1)^{\frac{1}{2}}} }{x^2 + 1}
\\
\\
y' =& \frac{x^2 + 1 - x^2}{(x^2 + 1) (x^2 + 1)^{\frac{1}{2}}}
\\
\\
y' =& \frac{1}{(x^2 + 1)^{\frac{3}{2}}}

\end{aligned}
\end{equation}
$


when $y' = 0$


$
\begin{equation}
\begin{aligned}

0 =& \frac{1}{(x^2 + 1)^{\frac{3}{2}}}

\end{aligned}
\end{equation}
$


Since $y' = 0$ does not exist, the function has no critical number

Hence, the intervals of increase or decrease are

$
\begin{array}{ccc}
\text{Interval} & y' & f \\
x < 0 & + & \text{increasing on } (- \infty, 0) \\
x > 0 & + & \text{increasing on } (0, \infty)
\end{array}
$

Solving for concavity and inflection points

If $\displaystyle y' = \frac{1}{(x^2 + 1)^{\frac{3}{2}}}$ or $y' = (x^2 + 1)^{\frac{-3}{2}}$, then


$
\begin{equation}
\begin{aligned}

y'' =& \frac{d}{dx} (x^2 + 1)^{\frac{-3}{2}}
\\
\\
y'' =& \frac{-3}{2} (x^2 + 1)^{\frac{-5}{2}} \frac{d}{dx} (x^2 + 1)
\\
\\
y'' =& \frac{-3}{\cancel{2}} (x^2 + 1)^{\frac{-5}{2}} (\cancel{2}x)
\\
\\
y'' =& -3x(x^2 + 1)^{\frac{-5}{2}}
\\
\\
y'' =& \frac{-3x}{(x^2 + 1)^{\frac{5}{2}}}

\end{aligned}
\end{equation}
$


when $y'' = 0$


$
\begin{equation}
\begin{aligned}

0 =& \frac{-3x}{(x^2 + 1) ^{\frac{5}{2}}}
\\
\\
0 =& -3x
\\
\\
x =& \frac{0}{-3}
\\
\\
x =& 0

\end{aligned}
\end{equation}
$


Therefore, the inflection point is

$f(0) = 0$

Thus, the concavity is..

$
\begin{array}{|c|c|c|}
\hline\\
\text{Interval} & f''(x)/y'' & \text{Concavity} \\
x < 0 & + & \text{Upward} \\
x > 0 & - & \text{Downward}\\
\hline
\end{array}
$

x=3t+5 , y=7-2t , -1

The formula of arc length of a parametric equation on the interval alt=tlt=b is:
L = int_a^b sqrt((dx/dt)^2+(dy/dt)^2) dt
The given parametric equation is:
x=3t + 5
y=7 - 2t
The derivative of x and y are:
dx/dt = 3
dy/dt = -2
So the integral needed to compute the arc length of the given parametric equation on the interval -1lt=tlt=3 is:
L = int_(-1)^3 sqrt(3^2+(-2)^2) dt
The simplified form of the integral is:
L = int_(-1)^3 sqrt13 dt
Evaluating this yields:
L = sqrt13t  |_(-1)^3
L = sqrt(13)*3 - sqrt13*(-1)
L=3sqrt13 + sqrt13
L=4sqrt13
Therefore, the arc length of the curve is 4sqrt13 units.

What are some examples of ethos, logos, and pathos in Patrick Henry's Speech to the Virginia Convention?

Patrick Henry begins the speech with pathos, saying that he regards the matter before the convention as “nothing less than a question of freedom or slavery.” The emotive language continues as he says that if he were to remain silent at such a time, he would consider himself a traitor to his country and his God. In the third paragraph, rhetorical questions add to the effect of the emotive language. These techniques of pathos continue thick and fast in the ensuing paragraphs and are to be found throughout the speech.
One of the strongest appeals to logos comes in paragraph ten. People say the colonists are too weak to take on the British. “But when shall we be stronger? Will it be the next week, or the next year?” Henry’s language becomes more ironic and closer to an emotional appeal when he asks whether the colonists will “acquire the means of effectual resistance by lying supinely on our backs,” but the initial argument is clearly a logical one. There is then an invocation of ethos when Henry claims that the colonists will not fight alone: “a just God” will take their part and supply them with allies. This comes in the midst of many more stirring appeals to pathos, enhanced by dramatic imagery.
Finally, Patrick Henry employs his own ethos in the famous final rallying cry: “as for me, give me liberty or give me death!” This would only be effective if his audience regarded Henry as a man worthy of respect.

To establish a sense of ethos, in other words, credibility, Patrick Henry writes of himself, "I have but one lamp by which my feet are guided; and that is the lamp of experience." He states that he has the necessary experience to judge the British throne because he has been watching and observing them for the last 10 years.
Henry tries to evoke a sense of pathos (or emotion) in his listeners. For example, he says, "Suffer not yourselves to be betrayed with a kiss." He tries to rouse the listeners' anger so that they feel entitled to rebel against the king. Later, he writes the following to rouse the ire of the colonists:

Our petitions have been slighted; our remonstrances have produced additional violence and insult; our supplications have been disregarded; and we have been spurned, with contempt, from the foot of the throne.

The use of words such as "slighted," "insult," "spurned," and "contempt" are designed to make the colonists angry toward the British and color their emotional response to recent events so that they are motivated to start a revolution.
Lastly, Henry uses logos (logic). For example, he asks of the British forces gathering in the New World:

Can gentlemen assign any other possible motive for it? Has Great Britain any enemy, in this quarter of the world, to call for all this accumulation of navies and armies?

In other words, he states that it is only logical to conclude that the British are about to launch a war against the colonies and that the colonies have no reason to remain loyal to Britain. 

Ethos is argument from the character of the speaker. It can be of two types, intrinsic, meaning created within the context of the speech itself, and extrinsic, or existing independent of or prior to the speech. For extrinsic ethos, Henry was a well educated lawyer who had served in House of Burgesses and had a solid track record of pushing back against British rule. His intrinsic ethos appears at the start of the speech, where he affirms his patriotism and argues that his actions are motivated precisely by loyalty rather than disloyalty.
Argument from logos implies reference to logic rather than simple citation of facts. It is a method of emphasizing the abstract principles of reasoning such as the rules of inference. A typical example of this is:

... in proportion to the magnitude of the subject ought to be the freedom of the debate ...

This is an argument from analogy, following what in ancient rhetoric was termed the topos of greater and lesser, which is considered by Aristotle to be among the common topics of both forensic and deliberative oratory. 
The peroration, with its use of highly charged emotional language and vivid imagery, is an example of pathetic argument which attempts to sway the emotions of its hearers. 

Appeals to ethos help a speaker or writer establish credibility.  Since Patrick Henry had been elected to the Virginia House of Burgesses, he had some built-in credibility.  He adds to that by addressing the House formally and demonstrating his respect for the men who were advocating further negotiation with Britain--the opposite of what Henry believed was the right course of action.  He builds ethos when he asserts, "different men often see the same subject in different lights."
Appeals to logos signal to your audience that reason and rationality are present in your thinking.  Henry points out that the colonies have tried negotiating with Britain for a decade and that it would be illogical to expect a different outcome if nothing different is attempted.  Henry's logic is made clear when he declares "I know of no way of judging of the future but by the past."  Britain has set a precedent by rebuffing the colonies' petitions and protests.
There are many appeals to emotion, or pathos, in Henry's speech.  When Henry alludes to skirmishes between patriots and the British in Boston, he observes: "Our brethren are already in the field! Why stand we here idle?" It seems he might be trying to make the men of the Virginia colony feel guilty for not taking up arms to support the northern colonies and doing nothing substantive to repel the aggression of the British.
 

For each of the primary sources provided explain why reform would NOT result in equality. https://www.marxists.org/archive/lenin/works/1901/witbd/i.htm https://www.marxists.org/archive/kollonta/1909/social-basis.htm

In What Is To Be Done? Lenin vigorously attacks those within the socialist movement who strive for what he calls "freedom of criticism" of Marxism's revolutionary aims. Critics such as Eduard Bernstein wanted to modify and soften Marx's theories. They argued that socialist reforms could come without violent revolution. They wanted to pursue the gradual reform of capitalism through peaceful changes to the existing system.
To Lenin, this reformism would not lead to the equality the working classes aimed for under socialism. In fact, it would play into the hands of the enemy bourgeoisie by radically watering down socialism. It would turn socialism from the total overthrow of a corrupt capitalist system into a weak trade union movement that would become merely an "appendage" of bourgeois liberalism. Reformism would fatally compromise socialism because of its denial of a fundamental Marxist principle: that working class interests are the complete opposite of bourgeois interests. To advocate working with the enemy would demoralize and undermine the whole movement. As Lenin put it,

But an essential condition for such an alliance must be the full opportunity for the socialists to reveal to the working class that its interests are diametrically opposed to the interests of the bourgeoisie. However, the Bernsteinian and “critical” trend, to which the majority of the legal Marxists turned, deprived the socialists of this opportunity and demoralised the socialist consciousness by vulgarising Marxism, by advocating the theory of the blunting of social contradictions, by declaring the idea of the social revolution and of the dictatorship of the proletariat to be absurd, by reducing the working-class movement and the class struggle to narrow trade-unionism and to a “realistic” struggle for petty, gradual reforms. This was synonymous with bourgeois democracy’s denial of socialism’s right to independence and, consequently, of its right to existence; in practice it meant a striving to convert the nascent working-class movement into an appendage of the liberals.

Lenin very strongly asserted that workers would get nowhere—would not achieve the equality of a classless society—unless they rejected compromise and the old, worn-out idea that they could work with the very group out to destroy them. The only way to win a real victory over a "monster" (the bourgeoisie) was by holding firm to fundamental Marxist principles. Lenin wrote the following:

If you must unite, Marx wrote to the party leaders, then enter into agreements to satisfy the practical aims of the movement, but do not allow any bargaining over principles, do not make theoretical “concessions”. This was Marx’s idea, and yet there are people among us who seek-in his name to belittle the significance of theory!
Without revolutionary theory there can be no revolutionary movement.

Lenin reiterated a common refrain of his time: communists were fearful from past experience that the upper classes would offer compromise as a way to trap and destroy the socialist movement
In The Social Basis of the Woman Question, Alexandra Kollontai critiques the idea that working-class women can achieve equality by allying themselves with feminists. She defines the feminists as bourgeois women whose class interests are diametrically opposed to those of working-class women. She argues that once bourgeois women achieve the equality they desire, they will unite with bourgeois men to throw working-class women under the bus. As Kollontai puts it,

The feminists seek equality in the framework of the existing class society, in no way do they attack the basis of this society. They fight for prerogatives for themselves, without challenging the existing prerogatives and privileges.

She also says the following:

Once the barrier is down and the bourgeois women have received access to political activity, the recent defenders of the “rights of all women” become enthusiastic defenders of the privileges of their class, content to leave the younger sisters with no rights at all.

Further, "reforms" of the current capitalist system to give women more rights cannot work, for the system itself is corrupt. For example, giving women more sexual freedom will only mean that men will more fully exploit them sexually. The only way for women to truly gain equality is through a complete upheaval and transformation of the system into socialism. As long as a class system exists, women cannot truly be united, for class will divide them. The divide is not between men and women but between the classes:

A woman can possess equal rights and be truly free only in a world of socialised labour, of harmony and justice. . . . But each new concession won by the bourgeois woman would give her yet another weapon for the exploitation of her younger sister and would go on increasing the division between the women of the two opposite social camps.
A colossal upheaval of the entire social and economic structure was required before women could begin to retrieve the significance and independence they had lost.

Lenin and Kollantai are united in the view that no real equality can be won for the working classes through alliance or compromise with the existing bourgeois system. To do so would prove disastrous. It plays into enemy hands and is a betrayal of socialism. "Reform" will not work. Only revolutionary change will bring about equality.

Calculus: Early Transcendentals, Chapter 7, 7.3, Section 7.3, Problem 9

intdx/sqrt(x^2+16)
Let x=4tantheta for -pi/2dx/[d(theta)]=4sec^2theta
dx=4sec^2thetad(theta)

sqrt(x^2+16)=
sqrt((4tantheta)^2+16)=
sqrt(16tan^2theta+16)=
sqrt[16(tan^2theta+1)]=
sqrt(16sec^2theta)=
4|sec(theta)|


intdx/[4sec(theta)]=
int(4sec^2(theta)d(theta))/(4sec(theta))=
intsec(theta)d(theta)=
ln|sec(theta)+tan(theta)|+C_1=
ln|sqrt(x^2+16)/4+x/4|+C_1=
ln|(sqrt(x^2+16)+x)/4|+C_1=
ln|(sqrt(x^2+16)+x)|-ln4+C_1=
ln|sqrt(x^2+16)+x|+C
where C is the constant C_1-ln4 .

The final answer is

ln|(sqrt(x^2+16)+x)+C
where C is the constant C_1-ln4.

What happens to Gloria because she laughs so hard at the story of Otis's arrest?

Opal learns about Otis’s arrest from Otis himself, on her first day of work at Gertrude’s Pets. On this day, Opal and her dog Winn-Dixie chance on Otis playing his guitar to the pets. Otis explains to Opal that he sometimes likes to make music to the caged animals because he understands what it feels like to be locked up all of the time; he once was a jailbird.
Many days after this incident, Opal decides to tell her friend, Gloria Dump, about what Otis told her, that he had once been in jail. Gloria advises Opal to learn to “not always judge others by the things that they have done, rather by their present deeds”.
One day, Otis divulges to Opal the real reason why he was jailed. He says that he had been playing music on the streets when he was accosted by a group of policemen. The policemen wanted him to stop playing his guitar along the streets. However, he refused to comply and when they tried to handcuff him, he became violent and hit at one of the policemen. For this violent act, Otis was arrested.
In Chapter 20 of the story, Opal tells Gloria the reason why Otis went to prison. When Gloria hears the story, “she laughs so hard that she has to grab hold of her false teeth so that they do not fall out of her mouth”. Gloria laughs because the story given to her by Opal does not present Otis as a tough, dangerous criminal who belongs behind bars. In fact, it is hilarious that Otis really was jailed just for his passion for music. Clearly, Otis had not wanted to hurt any of the policemen. He had just wanted to play music for other people’s enjoyment. Gloria laughs at the irony of it all.