College Algebra, Chapter 1, 1.5, Section 1.5, Problem 50

Find all real solutions of the equation $\displaystyle x^{\frac{1}{2}} + 3x^{\frac{-1}{2}} = 10x^{\frac{-3}{2}}$


$
\begin{equation}
\begin{aligned}

x^{\frac{1}{2}} + 3x^{\frac{-1}{2}} =& 10x^{\frac{-3}{2}}
&& \text{Given}
\\
\\
x^{\frac{1}{2}} + 3x^{\frac{-1}{2}} - 10x^{\frac{-3}{2}} =& 0
&& \text{Subtract } 10x^{\frac{-3}{2}}
\\
\\
x^{\frac{1}{2}} + \frac{3}{x^{\frac{1}{2}}} - \frac{10}{x^{\frac{3}{2}}} =& 0
&& \text{Multiply both sides by } x^{\frac{3}{2}}
\\
\\
x^{\frac{1}{2}} \cdot x^{\frac{3}{2}} + \frac{3x^{\frac{3}{2}}}{x^{\frac{1}{2}}} - 10 =& 0
&& \text{Use the Properties of Exponents}
\\
\\
x^2 + 3x - 10 =& 0
&& \text{Factor}
\\
\\
(x + 5)(x - 2) =& 0
&& \text{Zero Product Property}
\\
\\
x + 5 =& 0 \text{ and } x - 2 = 0
&& \text{Solve for } x
\\
\\
x =& -5 \text{ and } x = 2
&&
\\
\\
x =& 2
&& \text{The only solution that satisfy the equation}

\end{aligned}
\end{equation}
$

Single Variable Calculus, Chapter 7, 7.3-1, Section 7.3-1, Problem 46

Determine the limit $\displaystyle \lim_{x \to 2^-} \log_3 \left( 8x - x^4 \right)$
By using table of values as $x$ approaches 2 from the left.

$
\begin{array}{|c|c|}
\hline\\
x & y \\
\hline\\
1.99 & -0.8929\\
1.999 & -2.3180\\
1.9999 & -3.7481\\
1.99999 & -5.1788\\
\hline
\end{array}
$

We set that the as $x$ approaches 2 from the left, $y$ becomes a negatively large value, and by substituting 2 in the equation. We get...
$\displaystyle \lim_{x \to 2^-} \log_5 \left[ 8(2) - (2)^4 \right] = \lim_{x \to 2^-} \log_5 0$
And by the property of log
$\displaystyle \lim_{x \to 0^+} \log_a x = - \infty$
Therefore, $\displaystyle \lim_{x \to 2^-} \log_5 \left( 8x - x^4 \right) = -\infty$

How would you describe and analyze the Prince of Arragon in The Merchant of Venice?

The first reference to the prince is in Act I, Scene 2, when Portia and Nerissa, her lady in waiting, are discussing Portia's suitors. Each one is individually mentioned by Nerissa, with Portia providing insight into her sentiments about each man. Nerissa refers to the Prince of Arragon as the 'Neapolitan prince.' It is clear Portia does not like him much, as we can read in her comment:

Ay, that's a colt indeed, for he doth nothing buttalk of his horse; and he makes it a greatappropriation to his own good parts, that he canshoe him himself. I am much afeard my lady hismother played false with a smith.

Portia seems to believe the prince is obsessed with his horse. He talks about it incessantly and is very proud he can shoe it himself. Portia mocks the prince and says she believes his peculiar behavior could only be because he was most likely the result of an affair between his mother and a blacksmith.
In Act II, Scene 9, we learn more about the Prince of Arragon when he chooses a casket. His name seems to suit his personality and one can easily assume 'Arragon' is a deliberate play on 'arrogant,' since the prince displays a conceited opinion of himself as can be inferred from his response to the inscription on the lead casket:

'Who chooseth me must give and hazard all he hath.'You shall look fairer, ere I give or hazard.

The Prince of Arragon's remark is quite offensive, since he states that Portia should either be more beautiful, lighter in complexion, or more generous before the prince will be moved to give and risk everything he has. When he looks at the inscription on the gold casket, his response is supercilious:

'Who chooseth me shall gain what many men desire.'What many men desire! that 'many' may be meantBy the fool multitude, that choose by show...
...I will not choose what many men desire,Because I will not jump with common spiritsAnd rank me with the barbarous multitudes.

It is obvious the prince deems himself superior to other people, whom he refers to as 'barbarous multitudes.' As a result, the prince rejects the gold casket, claiming he is not a commoner who is easily led by what he sees on the outside and ignores the true value of what is inside.
When the prince reads the inscription on the silver casket, he reveals a pompous attitude, for he is quite condescending and patronizing. He boorishly sermonizes about the value of being rewarded with what one deserves and not obtaining merit by corrupt means. Apparently, the prince forgets he was born into wealth and privilege without having to work for those advantages. His remarks, therefore, come across as quite hypocritical.
When he decides on the silver casket, the prince clearly indicates his superiority complex, as he assumes he is entitled to Portia's hand. He says, "I will assume desert," which means he thinks he deserves to be the one who gets Portia. In the end, though, the prince is honest enough to admit having been foolish. He states he has become twice as foolish as he was when he arrived.

With one fool's head I came to woo,But I go away with two. 

Finite Mathematics, Chapter 1, 1.1, Section 1.1, Problem 44

Graph the equation: $y = x - 1$
To graph the line, we must first determine the intercepts of the line.
$x$-intercept (set $y = 0$):

$
\begin{equation}
\begin{aligned}
0 &= x - 1\\
\\
x &= 1
\end{aligned}
\end{equation}
$


$y$-intercept (set $x = 0$):

$
\begin{equation}
\begin{aligned}
y &= 0 - 1\\
\\
y &= -1
\end{aligned}
\end{equation}
$

The line pass through $(1,0)$ and $(0,-1)$. So the graph is

What do Julia, Winston, and O'Brien discuss?

In Part 2, Chapter 8, Winston and Julia go to see O'Brien at his luxury apartment. O'Brien is a member of the Inner Party and so enjoys special privileges unavailable to those outside this charmed circle. When they finally meet Winston confesses to O'Brien that he and Julia are both enemies of the Party. Winston's trust in Julia has given him the confidence to trust O'Brien as well. All three drink wine and make a toast to the past, demonstrating their hatred for the Party and its rule.
O'Brien reveals the existence of a secret Brotherhood, an underground insurrectionary group dedicated to the overthrow of the Party. The Brotherhood is inspired by the ideas of Emmanuel Goldstein, a mysterious oppositional figure who represents the antithesis of Big Brother. Yet, no one knows for sure whether he actually exists. But O'Brien assures Winston that Goldstein is indeed a real person. He promises to send Winston and Julia a book written by Emmanuel Goldstein which will spell out in precise detail how the Party can be destroyed. Then, they will become members of the Brotherhood.
As Winston leaves he asks O'Brien if they will meet again "in the place where there is no darkness," a suitably ambiguous expression which foreshadows much of what is about to happen. O'Brien also knows an old nursery rhyme which indicates to Winston that he's genuinely opposed to the Party, as such historical knowledge is strictly forbidden in the totalitarian state of Oceania.

What is the difference between prokaryotes and eukaryotes?

All life on earth is divided into three domains - bacteria, archaea, and eukarya. The domains archaea and bacteria make up the prokaryotes, single-celled organisms that lack a nucleus. In contrast, the eukaryotes (eukarya) are single or multi-celled organisms, the cells of which contain a nucleus and membrane-bound organelles. Eukaryotes include everything from the unicellular algae and yeasts to the multicellular plants and animals. We’ll discuss more of the defining features of prokaryotes and eukaryotes below.
Prokaryotic cells are typically smaller than eukaryotic cells, with a diameter generally between one to 10 micrometers (though there are some exceptions!). These cells have a relatively simple structure, consisting of a membrane around the cytoplasm, ribosomes (molecules that help make proteins) freely floating in the cytoplasm, and a nucleoid, an area within the cytoplasm that contains the cell’s DNA. This DNA is typically arranged as a single, circular chromosome. Most bacteria also have a cell wall made out of peptidoglycan (a protein/carbohydrate polymer), which maintains the cell’s shapes and protects it from dehydration. Some prokaryotic cells also have a flagellum (a whip-like extension used for movement); if present, the flagellum is made of the protein flagellin. The surface of prokaryotic cells can also contain a hair-like extension called a pilus, which allows cells to transfer DNA directly with one another.
In contrast, eukaryotic cells are larger (generally, between 10 and 100 micrometers) and more complex than prokaryotic cells. One defining feature of eukaryotic cells is the presence of a membrane-bound nucleus which houses the cell’s genetic material. Within the nucleus, the DNA of eukaryotes is typically arranged in many linear chromosomes (in contrast to the single circular chromosome of prokaryotes) - a eukaryotic cell can have up to 1260 chromosomes, though this number varies widely among species. Unlike prokaryotic cells, eukaryotic cells have membrane-bound organelles, which are specialized compartments within the cytoplasm that perform specific functions (for example, the mitochondria of eukaryotic cells are involved in energy production). Like prokaryotes, eukaryotes cells also have ribosomes for protein production, but, in addition to being freely floating in the cytoplasm, the ribosomes are also found attached to the endoplasmic reticulum, an organelle involved in protein and lipid modification and transport. Some eukaryotic cells also have flagella for movement, but these flagella are made out of microtubules, not flagellin. If a cell wall is present (for example, as it is in fungi and plant cells), it’s made out of chitin (a type of polysaccharide found in fungi) or cellulose (in plants). Eukaryotic cells do not have pili, and cannot directly exchange DNA the same way prokaryotes can.
While the the presence of membrane-bound organelles, organization of the genetic material, and presence of a membrane-bound nucleus are the major features distinguishing prokaryotes and eukaryotes, these groups also differ in the size of their ribosomes (70S in prokaryotes and 80S in eukaryotes), how they reproduce (by binary fission in prokaryotes, and by mitosis and meiosis/sexual reproduction in eukaryotes), and some aspects of how their DNA is translated into proteins. Comparing features like these between prokaryotes and eukaryotes is one way scientists try to understand evolution of life on earth: the general view (called the endosymbiotic theory) is that eukaryotes were created when one prokaryote engulfed another (in fact, the mitochondria of eukaryotic cells is thought to have descended from the “eaten” prokaryote), but it’s important to note that the exact relationship between the two is an area of ongoing research.
https://evolution.berkeley.edu/evolibrary/article/_0/endosymbiosis_03

https://www2.estrellamountain.edu/faculty/farabee/biobk/BioBookmito.html

Prokaryotic cells do not have a nucleus. They do have genetic material, but it is not as organized. In the theory of evolution, prokaryotic cells are considered the beginning of life. Prokaryotic cells have their genetic material inside of a nucleoid. Bacteria are examples of prokaryotic cells. Transcription, the process by which new DNA is formed, takes place inside the cytoplasm. This DNA is circular. Prokaryotic cell walls, if applicable, contain peptidoglycan.
Eukaryotic cells have a nucleus which contains their genetic material. This allows for greater complexity, as these cells also have different organelles within them, such as mitochondria and ribosomes. Transcription in eukaryotic cells takes place inside of the nucleus. DNA in eukaryotic cells is linear. Eukaryotic cell walls are composed of cellulose. This is applicable for plant cells only.
https://www.majordifferences.com/2013/08/difference-between-prokaryotic-cell-and_27.html

Was the collapse of the USSR in the late 1980’s-1991 highly probable? Why or why not? Explain carefully.

To some students and analysts of the Soviet Union, its collapse was probable and, to a large extent, inevitable. To others, it was highly improbable. The difference between the two schools of thought centers on the debates during the 1970s and 1980s as to whether the actual strength of the Soviet Union's economy, the burden on that economy of exorbitant spending on its armed forces, and the ruthlessness of the empire's internal security apparatus were all being accurately gauged. The Soviet Union was viewed by many as a powerful monolith whose military power and hold on its people, as well as on the nations it occupied in Eastern and Central Europe, was so strong as to preclude the center's collapse. However, while it fielded enormous military forces of considerable power, the Soviet Union's economy was far more precarious than many American analysts estimated. Additionally, the increasing ability of citizens of Soviet-occupied and satellite nations to compare their economic situations and cultures with those of the far more prosperous West, especially the stark contrast between East and West Germany, reflected very poorly on the Soviet Union.
Certainly, in retrospect, the collapse of the Soviet Union was probable; the fragility of its economy, however, was insufficiently appreciated by many in the West, so the number of analysts and others who predicted the collapse was far fewer than should have been the case.