Why did Nathaniel Bacon believe his rebellion against the rulers of the Virginia Colony was justified?

Nathaniel Bacon's crusade was an indictment of the entire headright and indentured servitude systems.  With tobacco cultivation appearing to save the Virginia colony, wealthy planters wanted to take advantage of the strong market for tobacco.  The problem was that they did not have enough laborers for expanded production of the cash crop.  The headright system was an unfair exploitation of the working class.  Wealthy English aristocrats could pay for an indentured servant's trip to Virginia and the aristocrat would be rewarded with large tracts of land.  The indentured servant received little or nothing after the term of service was complete.  Looking for opportunities, the impoverished freemen headed to the west and the frontier where they intruded on Indian lands and were met with violence.  The governor, William Berkely needed the Indians as an ally in his lucrative fur-trading activities and was unwilling to protect the settlers from Indian attack.  Berkely believed this entire system to be unjust.  He also was angry that the freedmen were now paying taxes and not receiving adequate protection from the government they were paying taxes to.  
https://www.legendsofamerica.com/ah-nathanielbacon/

In France, how did the Declaration of the Rights of Man and the Declaration of the Rights of Woman exemplify the gender divide between those who supported revolutionary change?

The Declaration of the Rights of Man and Citizen was by no stretch of the imagination a feminist document. It was devised by men for the benefit of men. Despite its emancipatory rhetoric, the Declaration made no mention of support for women's rights. And this omission was by no means unintentional. The men of the French Revolution didn't regard women as being equal, and so didn't believe that there was any space for them in the political realm. Women were held not to be citizens, and as such were systematically excluded from public life. The role of women in the new society was to be pretty much the same as under the ancien regime.
It was in response to the blatant deficiencies of the Declaration that the radical revolutionary and feminist Olympe de Gouges drew up her own proposal, the Declaration of the Rights of Woman and of the Female Citizen. This Declaration was incredibly radical for the time, not least because de Gouges argued that women were entitled to equal civil and political rights on the basis of natural equality with men. Revolutionary opponents of de Gouges attacked her on the grounds that the whole notion of women's rights was contrary to nature. Women, they believed, were neither physically nor psychologically capable of participating in political life. De Gouges counters this assertion in the Declaration by accusing men of confusing nature with the artificial distinctions imposed by society.

In "A Jury of Her Peers," what is ironic about the method that was used to kill the victim?

With a gun available to her, it seems ironic that the oppressed and apparently unassertive Mrs. Minnie Foster Wright would choose a brutal and strenuous method like strangling to kill her husband.
Early in the narrative, a neighbor to the Wrights, Mr. Hale, tells the county attorney what happened when he came to the Wrights' house the day before Mr. Wright's death. Mr. Hale mentions that he stopped to ask Mr. Wright if he would like to "take a telephone." At the time of this story, telephones were expensive for people who were far from town because of the added costs to the company in providing the lines and other necessary equipment. So, Mr. Hale hoped that he could get other neighbors to go in with him on a party line. Further, he tells the attorney that he planned on mentioning how much the womenfolk would enjoy having a telephone, although he says he "didn't know what his wife wanted made much difference to John--"
There are other suggestions that Mrs. Wright has surrendered to a life unlike her former one. One is recounted by Mrs. Hale as she talks with Mrs. Peters, the sheriff's wife. When Mrs. Peters gathers the articles of clothing that Mrs. Wright requested, Mrs. Hale notices the worn quality of this clothing, and she tells Mrs. Peters that Mr. Wright was "close." Mrs. Hale says,

"I think maybe that's why she [Mrs. Wright] kept so much to herself. I s'pose she felt she couldn't do her part; and then, you don't enjoy things when you feel shabby. She used to wear pretty clothes and be lively....one of the town girls, singing in the choir."

The little songbird that Mrs. Wright owned may have been quite meaningful to her as it sang when, in her depressed state, she herself no longer could. Perhaps, too, the death of this songbird carries a significance known only to Mrs. Wright. So rather than use her husband's gun or another weapon, she may have repaid the death of the sweet bird in kind—a form of poetic justice. Such brutality is ironic, or unexpected, given her submissive nature and Mr. Wright's oppresive acts. Nevertheless, frightened and angered women can exhibit a strength that they normally do not possess. Minnie Wright just might have had enough strength to "wring his neck" and strangle him unexpectedly, just as he strangled her precious bird.

Mr. Wright was found dead in his bed with a rope around his neck. As the story unfolds, the irony of the way he died becomes clearer. Three types of irony related to the cause of death are apparent. Situational irony refers to events that are the opposite of what one might expect. Dramatic irony occurs when the reader knows something that a character doesn't know. Verbal irony refers to the use of words to mean the opposite of their literal meaning. 
First, that Mr. Wright would be strangled in his bed with a rope is ironic--unexpected--for several reasons. There was a gun in the home, and anyone who wanted to murder the man could have done so more quickly and easily with the gun. Second, that he didn't wake up while the rope was being put around his neck is unusual. Third, the method of death was like a hanging, a way to administer justice to a wrong-doer, yet the man was murdered, so the one who was administering the "justice" would be the wrong-doer--making for an ironic role reversal of the victim and perpetrator. 
Dramatic irony comes in when the women discover clues that the men--the sheriff and county attorney--don't know about. Readers learn that Minnie Wright was knotting a quilt--drawing a connection with the knot she tied around her husband's neck. The women discover the canary with a broken neck and conclude that Mr. Wright broke the innocent bird's neck, revealing that the method of Wright's execution was poetic--or ironic--justice for his brutality. The act of violence toward the bird represents the larger pattern of abuse that Wright perpetrated on his wife. The murderer killed him by "slipping a thing round his neck that choked the life out of him," but he was guilty in a different way of choking the life out of his wife over a period of twenty years. 
Finally, verbal irony surrounds the method of death. The men mock the women for debating whether Minnie was going to quilt or knot the squares of her quilt together. The innocent knot associated with quilting is ironic when juxtaposed with the deadly hangman's knot around Mr. Wright's neck. 

Beginning Algebra With Applications, Chapter 5, Review Exercises, Section Review Exercises, Problem 24

Illustrate the solution set of $3x + 2y \leq 12$

$
\begin{equation}
\begin{aligned}
3x + 2y &\leq 12
&& \text{Solve the inequality for } y \\
\\
2y &\leq -3x + 12 \\
\\
y &\leq -\frac{3}{2}x + 6
\end{aligned}
\end{equation}
$


To graph the inequality, we first find the intercepts of the line $\displaystyle y = -\frac{3}{2}x + 6$.
In this case, the $x$-intercept (set $y = 0$) is $(4,0)$

$
\begin{equation}
\begin{aligned}
0 &= -\frac{3}{2}x + 6\\
\\
\frac{3}{2}x &= 6\\
\\
x &= 6 \left( \frac{2}{3} \right)\\
\\
x &= 4
\end{aligned}
\end{equation}
$

And the $y$-intercept (set $x = 0$) is $(0,6)$

$
\begin{equation}
\begin{aligned}
y &= -\frac{1}{2} (0) + 6\\
\\
y &= 6
\end{aligned}
\end{equation}
$

So the graph is



Graph $\displaystyle y = -\frac{3}{2}x + 6 $ as a solid line. Shade
the lower half of the plane.

What are some differences and similarities between Bruno and Shmuel in The Boy in the Striped Pajamas?

One similarity that should jump out at readers is that both boys are exactly the same age. They share the same birthday, so that gives them immediate common ground. They also are both males living in essentially the same location, but that is all very surface level stuff. One similarity that probes a little deeper is that both Bruno and Shmuel are both lonely characters that are seeking friendship wherever they can find it. This actually can be used to point out some stark contrasts between the two characters. Shmuel is lonely because his family and friends are being systematically separated from each other and either worked to death or killed in one gruesome fashion or another. Bruno on the other hand is lonely and isolated due to his father's position in the Nazi hierarchy; however, Bruno's life can hardly be described as bad or even wanting. Perhaps this is why he is so quick to try and distance himself from Shmuel when they are caught together. Another striking difference exists between the ignorance level of each character. Shmuel has a decent handle on his current life situation. He's been forced to witness all kinds of atrocities, and he understands fairly well where he is. Bruno, on the other hand, is so unbelievably ignorant that he's practically an unrealistic character. He doesn't understand what is happening at the concentration camp that his own father is a major part of. This is amazing since the Hitler youth programs were designed to condition children like Bruno to adhere to Nazi beliefs and practices.

Bruno and Shmuel are both lonely and seeking friendship, and they are very excited to talk every day and build their relationship. However, Bruno understands that because of his family's position in the Nazi regime, in order to continue a friendship with a Jewish boy, he must keep their friendship a secret, thus continuing their separation by the fence of the camp. This difference—imprisonment versus freedom—is the most prominent distinguishing factor between the two boys. This is why Bruno is willing to throw Shmuel under the bus when he's caught eating chicken. Eventually, Bruno overcomes his need to preserve this difference however, and he dresses in the inmate uniform in order to help Shmuel look for his father. This choice results in Bruno losing his privileged position as a German and becoming even more like Shmuel as the two boys are led to the gas chambers together.

In many substantial ways, Bruno and Shmuel are—or rather, should be—very similar. They are both mainland Europeans from educated families; they are exactly the same age; they both have a boyish curiosity and a willingness to become friends; and they are both lonely—though for very different reasons.
The wedge driven between them by the Nazi occupation has to do with Shmuel's Jewishness. The fact of this religious/cultural distinction renders Shmuel a prisoner, and the result is that he and Bruno are placed at polar opposites on several spectra.
Shmuel is captive, while Bruno lives among his captors; they are skinny versus well-nourished; impoverished versus privileged; and dirty versus clean. According to the Nazi narrative, it is bad versus good. However, rather than reinforce such a judgement, the juxtaposition highlights one fundamental similarity, and their compatibility which results from it.
They are worlds apart, but indelibly bonded by a single shared experience: loneliness. Neither is able to access the full love of his parents. Bruno sympathizes and takes action. The boys' shared naivety is evident in their mutual decision to break Bruno into the camp.
The boys are bound together by common experience—the absence of a loving family.

Both Bruno and Shmuel are young boys living in Auschwitz, and they become close friends after having numerous conversations between the large fence that separates them from each other. Both boys were forced to move from their homes and do not like Auschwitz. Bruno and Shmuel are also both curious boys, who are depicted as relatively friendly individuals. They are both portrayed as selfless individuals, and they make sacrifices to help each other. Both boys share the same birthday and enjoy talking to each other through the fence. Despite their drastically different situations, Bruno and Shmuel become close friends and tragically die together in a gas chamber.
Despite their similar personalities, Bruno is the son of a Nazi commandant, while Shmuel is a Jewish prisoner. Bruno lives a luxurious life outside of the concentration camp while Shmuel has to endure the dangerous, horrific environment inside the death camp. Bruno is also portrayed as a more naive character who does not fully understand the nature of the concentration camp, while Shmuel is much more aware of his surroundings. Bruno also makes several blasé remarks, while Shmuel is a more humble, quiet boy.

In the book The Boy in the Striped Pyjamas, the two protagonists, Bruno and Shmuel, have a number of similarities and differences. They are age-mates, as they were both born on April 15th, 1934. They are also both forced to move from their homes, which they had been fond of: Bruno, from Berlin to Auschwitz due to his father’s role with the Nazis, and Shmuel from his home in Poland to the concentration camp in Auschwitz as a Jewish prisoner. Additionally, both boys were educated: Bruno had a private tutor named Herr Liszt and Shmuel was taught by his mother who was a teacher by profession.
In terms of differences, the first obvious one is the fact that Bruno was born in Germany and was therefore German while Shmuel was born in Poland to his Jewish parents. Whereas Bruno lived in opulence, in a house with food, resources, and servants, Shmuel languished in impoverished conditions at the concentration camp. Finally, Bruno was ignorant of the activities that took place in the concentration camp and only wondered why and how the emaciated, pajama-clad people ended up there. Shmuel, on the other hand, was fully aware of what was happening and experienced the deplorable conditions firsthand.

Single Variable Calculus, Chapter 7, 7.7, Section 7.7, Problem 2

Determine the numerical value of a.) $\tan h 0$ and b.) $\tan h 1$

a.) $\tan h 0$

Using Hyperbolic Function


$
\begin{equation}
\begin{aligned}

\tan h x =& \frac{\sin h x}{\cos h x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}
\\
\\
\tan h 0 =& \frac{e^0 - e^{-0}}{e^0 + e^{-0}}
\\
\\
\tan h 0 =& \frac{\displaystyle e^0 - \frac{1}{e^0}}{\displaystyle e^0 + \frac{1}{e^0}}
\\
\\
\tan h 0 =& \frac{1 - 1}{1 + 1}
\\
\\
\tan h 0 =& \frac{0}{2}
\\
\\
\tan h 0 =& 0

\end{aligned}
\end{equation}
$


b.) $\tan h 1$

Using Hyperbolic Function


$
\begin{equation}
\begin{aligned}

\tan h x =& \frac{\sin h x}{\cos h x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}
\\
\\
\tan 1 =& \frac{e^1 - e^{-1}}{e^1 + e^{-1}}
\\
\\
\tan 1 =& \frac{\displaystyle e - \frac{1}{e}}{\displaystyle e + \frac{1}{e}}
\\
\\
\tan 1 =& \frac{\displaystyle \frac{e^2 - 1}{\cancel{e}}}{\displaystyle \frac{e^2 + 1}{\cancel{e}}}
\\
\\
\tan 1 =& \frac{e^2 - 1}{e^2 + 1}
\\
\\
\tan 1 =& 0.7616

\end{aligned}
\end{equation}
$

Beginning Algebra With Applications, Chapter 3, 3.2, Section 3.2, Problem 196

An adult and a child are on a seesaw that is 14 ft long. The adult weighs 175 lb and the child weighs 70 lb. How many feet from the child must the fulcrum be places so that the see saw balances?

Using the Lever System Equation

$F_1 x = F_2 (d-x)$

Solving for $x$,


$
\begin{equation}
\begin{aligned}

F_1 x =& F_2 (d-x)
&& \text{Apply Distributive Property}
\\
\\
F_1 x + F_2 x =& F_2 d
&& \text{Add } F_2 x
\\
\\
x(F_1 + F_2) =& F_2 d
&& \text{Factor out } x
\\
\\
x =& \frac{F_2 d}{F_1 + F_2}
&& \text{Divide by } F_1 + F_2
\\
\\
x =& \frac{175(14)}{70+175}
&& \text{Substitute } F_1 = 70, F_2 = 175 \text{ and } d = 14
\\
\\
x =& \frac{2450}{245}
&& \text{Simplify}
\\
\\
x =& 10 \text{ ft}
&&

\end{aligned}
\end{equation}
$


The child must be 10 ft from the fulcrum so that the seesaw balances.