Single Variable Calculus, Chapter 2, 2.2, Section 2.2, Problem 42

(a) Estimate the value of the $\displaystyle \lim \limits_{x \to 1} \frac{x^3 - 1}{\sqrt{x} - 1}$ by using numerical and graphical evidence.



Based from the graph, the estimated valueof the $\displaystyle \lim \limits_{x \to 1} \frac{x^3 - 1}{\sqrt{x} - 1} \approx 6$

We let the values of $x$ be..


$
\begin{equation}
\begin{aligned}

\begin{array}{|c|c|}
\hline\\
x & \lim \limits_{x \to 1} \\
\hline\\
0.9 & 5.2809 \\
0.93 & 5.4902 \\
0.94 & 5.5612 \\
0.95 & 5.6328 \\
0.99 & 5.9253 \\
0.999 & 5.9925 \\
0.9999 & 5.9993 \\
0.99999 & 5.9999\\
\hline
\end{array}

\end{aligned}
\end{equation}
$


Based on the numerical values we obtain from the table, it seems that the $\displaystyle \lim \limits_{x \to 1} \frac{x^3 - 1}{\sqrt{x} - 1} = 6$

b.) Determine how close to 1 does $x$ have to be to ensure that the function is within a distance of 0.5 of its limit.

In order to ensure that the function in part (a) is within the distance $0.5$ of its limit, the values of
$x$ should be atleast $0.06$ closer to $1$ based on the values we obtain from the table.

How did the printing press transform both the private and public lives of Europeans?

The oldest known printing press in recorded history originated in China, not Europe. However, approximately 150 years after the printing press was discovered in China, a German named Johannes Guttenberg devised a printing press in Strasbourg, France. He began working on the machine in 1440 and had it ready for commercial printing by 1450. Besides pamphlets, calendars, and other smaller projects, Gutenberg printed about 180 copies of a 1,300-page edition of the Bible. Printing spread rapidly throughout Western Europe, and it is estimated that by 1500 about 20 million books had already been printed.
Previously, every edition of a book had to be laboriously written by hand, so books were extremely expensive and very few people possessed them or could read them. Printing presses brought in an era of mass communication in which anybody who could read had access to information. Knowledge was no longer in the hands of a powerful literate elite. Instead, as books became more plentiful, many more people learned to read, and transformational ideas became available to the masses. This shook the power structures of society that then controlled Europe. For instance, it was the printing press and the mass dissemination of information that made possible the Protestant Revolution, which weakened the power of the Catholic Church.
The printing press hastened the spread of scientific discoveries through journals and books. Scientists such as Nicolaus Copernicus were able to more easily spread their then-revolutionary ideas. The printing press hastened the spread of Renaissance ideas as wealthy patrons financed the printing of classic works by Aristotle, Plato, and others.
The spread of literacy that the printing press brought about also affected people on a personal level, especially in the areas of education and freedom of thought. With the availability of books, readers were no longer dependent upon an elite group of educated teachers for knowledge. Instead, they could educate themselves. They were also not limited in their thinking by the decisions of those educated few. They were able to question long-held tenets of religious, philosophical, and scientific thought through critical reasoning.
https://www.history.com/news/printing-press-renaissance

https://www.history.com/topics/inventions/printing-press

https://www.livescience.com/2569-gutenberg-changed-world.html

Johannes Gutenberg's printing press, invented around 1440 in the midst of the Renaissance, gave people more access to written literature, including religious texts and political pamphlets.
Previously, literacy had been limited to members of the clergy, the aristocracy, and members of the merchant class. The printing press expanded the possibilities for people who had not previously had opportunities to learn to read.
The Protestant Reformation would not have happened without the printing press. In 1517, Martin Luther posted his 95 Theses on the door of the Wittenberg Castle Church. His list of grievances against the indulgences of the Catholic Church was soon published and distributed as a result of the printing press.
Gutenberg's press also allowed for the Bible to be printed and distributed for private use. Previously, worshipers were beholden to clergymen to explain to them what the Bible said and what Scripture meant. Now, Christians were able to read the Bible for themselves and interpret its meaning. This personal relationship with Scripture was a key aspect of the Protestant Reformation.
In the eighteenth-century the printing press would be used to print and distribute political pamphlets, such as Thomas Paine's Common Sense. Ideas such as Paine's would be key to the Enlightenment, which would later inspire the major Atlantic revolutions: the American Revolution, the French Revolution, and the uprising in Haiti.
Access to printed information enriched people's lives. They could enjoy literature, create a more personal relationship with God, and read political ideas. The printing press allowed people to consider what they thought about the world, thus allowing them to engage with it more constructively.

How are Frida Kahlo and Georgia O'Keefe alike?

Modern painters Frida Kahlo and, Georgia O'Keeffe are among the top 20th century artists. And they were similar in many ways.
Kahlo and O’Keefe’s work were both intensely personal. Similarly, they created art that expressed intimacy and was heavily inspired by their own perspective of femininity.
While Kahlo did mostly small work, using surrealist techniques for her self-portraits, O’Keefe used large canvasses to paint flowers that hinted at female sex organs.
Kahlo and O’Keefe did, in fact, know each other.
A letter written from Kahlo to O’Keefe in 1933 stated,
I thought of you a lot and never forget your wonderful hands and the color of your eyes. I will see you soon. I am sure that in New York I will be much happier. If you still in the hospital when I come back I will bring you flowers, but it is so difficult to find the ones I would like for you. I would be so happy if you could write me even two words. I like you very much Georgia ... (Kahlo, 1933)
Both were married to well-known artists and both endured complex, and at times troubling, marriages. Many would argue that both Kahlo and O'Keefe are more widely known and appreciated than their husbands now.
Kahlo and O’Keefe both share iconic status today, thanks to their work that changed the way people talk about Modernist art.

While the work of Georgia O'Keeffe (1887-1986) differs quite a bit in style from that of Frida Kahlo (1907-1954,) the two artists have more in common than meets the eye. Both women were married to famous artists – O'Keeffe to Alfred Stieglitz and Kahlo to Diego Rivera – and both often found themselves living in their husbands’ shadows. O'Keeffe is considered a semi-abstractionist and Kahlo a semi-surrealist, both with an air of sensuality and – one might say – daring feminism for the time in which they lived, a time when women had to fight to be taken seriously in the art world. Perhaps it was this fight that lead them both to create such personal and non-conformist work.
O’Keeffe and Kahlo were more than just contemporaries; they actually knew one another, although the extent and nature of their closeness is left largely to conjecture. There is, however, a documented letter from Kahlo to O’Keefe, penned in 1933, offering comfort in the wake of a nervous breakdown. O’Keeffe had failed to successfully complete a mural commissioned by Radio City Music Hall and – in the aftermath of her breakdown – had relocated to Bermuda for recovery. She didn’t paint for a year after the incident. In her letter, Kahlo wrote:
“I would like to tell you every thing that happened to me since the last time we saw each other, but most of them are sad and you mustn't know sad things now. After all I shouldn't complain because I have been happy in many ways though. Diego is good to me, and you can't imagine how happy he has been working on the frescoes here. I have been painting a little too and that helped. I thought of you a lot and never forget your wonderful hands and the color of your eyes.”

Frida Kahlo (1907-54) and Georgia O'Keeffe (1887-1986) were both Modern painters of the early to mid-twentieth century.
Both artists are linked to powerful male artists: Frida Kahlo was married, twice, to Mexican muralist Diego Rivera, and O'Keeffe was married to Alfred Stieglitz, American photographer and gallerist. Rivera's success and access enabled Kahlo to meet other influential artists such as Andre Breton, who championed her work, and Stieglitz gave O'Keeffe her first show, in Manhattan.
Neither woman had children, and each was quite independent of her successful and powerful husband. Each developed a distinctive style that moved away from traditional forms and subjects. Kahlo was drawn to Mexican folk art, surrealism, and self-portraiture. O'Keeffe leaned toward abstraction and often expressed it through extreme close-ups of flowers as well as interpretive landscapes.

Why must Jonas have courage and bravery to be a Receiver?

I can think of a few reasons why Jonas needs to be brave in order to be the next Receiver. The first reason that Jonas is brave is that he is chosen to do a job that he knows nothing about. Before being given a final job, each child spends time volunteering in various positions. The child has a chance to learn about and participate in various working fields. Jonas has a fairly good idea about what many jobs in his community involve; however, he doesn't know anything about being the Receiver. The unknown can be scary, and Jonas needs to be brave to step into that role.  
Jonas also needs to be brave because he's alone. He's the only child chosen to be the next Receiver; this is the nature of the position, but it's still scary to take on an unknown task by yourself. To compound the issue, the Giver's rules forbid Jonas from discussing what he learns with anybody. Jonas is literally taking on the emotional load of an entire community by himself, and he can't even talk about it with a friend. He needs to be brave.  
Finally, Jonas needs to continue to be brave because he learns what happens to Rosemary. She was the previous Receiver, and she couldn't cope with all of the memories and feelings that she was being given. Even the Giver couldn't give her enough happy memories to undo the emotional toll that the pain and suffering memories had put on her. 

The Giver continues, "I backed off, gave her more little delights. But, everything changed, once she knew about pain. I could see it in her eyes."

The emotional weight eventually became too great for Rosemary, and she asked to be "released" from the community. Her death affected the entire community because her knowledge was released to the community as well. The Giver tells Jonas that "It was chaos." 

"They really suffered for a while. Finally it subsided as the memories were assimilated. But it certainly made them aware of how they need a Receiver to contain all that pain. And knowledge." 

Jonas needs to be brave because he knows that the previous person to hold his position failed and asked to be killed, and he knows that his death would cause problems for an entire community. Jonas needs to be brave because he has a huge responsibility in the community. 

How did Uncle Tom's Cabin by Harriet Beecher Stowe influence American literature?

Uncle Tom's Cabin had a very profound effect on American history, as it convinced many moderate Americans that slavery was wrong by personalizing slavery and showing its effects on people. While some abolitionists thought the book was too lenient, defenders of slavery thought Stowe, a northerner, was off base in her assessment and portrayal of slavery. As the legend goes, even Abraham Lincoln thought Stowe's book had had a major effect on the nation and said to her in 1862, "So you're the little woman who wrote the book that started this great war." 
Stowe's work was in the tradition of the slave narrative. Freed slaves such as Frederick Douglass wrote famous autobiographies about their experiences in slavery, but Stowe, a white woman, wrote a work that outsold all the previous slave narratives. She changed literature by taking a genre that had traditionally been written by African-Americans as autobiography and turning into a novelistic form. While Stowe intended the character of Uncle Tom to be a virtuous, Christian man, modern critics have charged that her characterization is racist in many ways. For example, they have criticized Tom for being too submissive. At the time, however, it was novel for a white author to depict a slave as the hero, while the white plantation owner, Simon Legree, was the villain. 

Beginning Algebra With Applications, Chapter 8, 8.2, Section 8.2, Problem 72

Factor $2a^3 + 6a^2 + 4a$


$
\begin{equation}
\begin{aligned}

2a^3 + 6a^2 + 4a =& 2a(a^2 + 3a + 2)
&& \text{Factor out } 2a
\\
=& 2a(a+1)(a+2)
&& \text{Factor } a^2 + 3a + 2

\end{aligned}
\end{equation}
$

Calculus of a Single Variable, Chapter 3, 3.4, Section 3.4, Problem 36

Given: f(x)=-x^4+4x^3+8x^2
Find the critical x values by setting the first derivative equal to zero and solving for the x value(s).
f'(x)=-4x^3+12x^2+16x=0
-4x(x^2-3x-4)=0
-4x(x-4)(x+1)=0
x=0, x=4,x=-1
If f'(x)>0 the function is increasing over an interval.
If f'(x)>0 the function is decreasing over an interval.
Choose a value for x that is less than -1.
f'(-2)=48 Since f'(2)>0 the function increases in the interval (-oo,-1).
Choose a value for x that is between -1 and 0.
f'(-.5)=-4.5 Since f'(-.5)<0 the function decreases in the interval (-1, 0).
Choose a value for x that is between 0 and 4.
f'(1)=24 Since f'(1)>0 the function increases in the interval (0, 4).
Choose a value for x that is greater than 4.
f'(5)=-120 Since f'(5)<0 the function decreases in the interval (4, oo).
Because the function changes direction from increasing to decreasing a relative maximum will exist at x=-1 and at x=4. The relative maximum points are
(-1, 3) and (4, 128).
Because the function changes direction from decreasing to increasing a relative minimum value will exist at x=0. The relative minimum point is (0, 0).