College Algebra, Chapter 1, 1.5, Section 1.5, Problem 46

Find all real solutions of the equation $\displaystyle \sqrt{x} - 3 \sqrt[4]{x} - 4 = 0$


$
\begin{equation}
\begin{aligned}

\sqrt{x} - 3 \sqrt[4]{x} - 4 =& 0
&& \text{Given}
\\
\\
(\sqrt[4]{x})^2 - 3 \sqrt[4]{x - 4} =& 0
&& \text{Let } w = \sqrt[4]{x}
\\
\\
(w - 4)(w + 1)=& 0
&& \text{Factor}
\\
\\
w - 4 =& 0 \text{ and } w + 1 = 0
&& \text{Zero Product Property}
\\
\\
w =& 4 \text{ and } w = -1
&& \text{Solve for } w
\\
\\
\sqrt[4]{x} =& 4 \text{ and } \sqrt[4]{x} = -1
&& \text{Substitute } w = \sqrt[4]{x}
\\
\\
(\sqrt[4]{x})^4 =& (4)^4 \text{ and } (\sqrt[4]{x})^4 = (-1)^4
&& \text{Raise both sides by } 4
\\
\\
x =& 256 \text{ and } x = 1
&& \text{Solve for } x
\\
\\
x =& 256
&& \text{The only solution for the equation } \sqrt{x} - 3 \sqrt[4]{x} - 4 = 0

\end{aligned}
\end{equation}
$

How did Andrew Jackson attempt to argue that "Indian Removal" was in the Native Americans' best interest, and how might opponents of this policy respond to this?

Andrew Jackson believed that the Indians should not have the same rights as United States citizens and observed that States had already established jurisdiction over lands that previously belonged to the Indians. According to him, Indians were non-citizen subjects with rights that were subject to overriding interests of the settlers and the State governments. Thus, he expected them to yield to these interests lest they suffer the consequences of resisting.
Jackson supported removal because the continued settlement of the Indians within State borders was bound to create conflicts, as the Indians had their own governments, which could not coexist with the State governments. The situation would threaten security and stability of entire regions, and it was also unconstitutional because the Indians would be establishing a foreign and independent government within the States’ jurisdiction.
In order to avoid the impending conflicts and to supposedly guarantee Indian peace, happiness, prosperity, Jackson supported the removal of Indians from jurisdictions considered to be in the United States.
Those against removal argued against Jackson’s assertions that the Indians were an impediment to civilization or a threat to security. They stated that the Indians had established a country that they lived in for hundreds of years and it was morally wrong to displace them. They argued for integration of the different communities because displacement was unsustainable and eventually the different groups would have to coexist.
https://guides.loc.gov/indian-removal-act

https://scholarworks.umb.edu/cgi/viewcontent.cgi?referer=&httpsredir=1&article=1045&context=masters_theses

Single Variable Calculus, Chapter 3, 3.6, Section 3.6, Problem 2

a.) Determine $y'$ by Implicit Differentiation.
b.) Find the equation explicitly for $y$ and differentiate to get $y'$ in terms of $x$.
c.) Check that your solutions to part (a) and (b) are consistent by substituting the expression for $y$ into your solution for part (a).

a.) Given: $4x^2+9y^2=36$
$\displaystyle \frac{d}{dx} (4x^2) + \frac{d}{dx} (9y^2) = \frac{d}{dx} ( 36 )$
$\displaystyle 4 \frac{d}{dx} (x^2) + 9 \frac{d}{dx}(y^2) = \frac{d}{dx}(36)$
$\displaystyle (4)(2x) + (9) (2y) \left( \frac{dy}{dx}\right) = 0 $

$
\begin{equation}
\begin{aligned}
8 x + (18y) \frac{dy}{dx} & = 0 \\
\\
(18 y) \frac{dy}{dx} & = -8x\\
\\
\frac{\cancel{18y} \frac{dy}{dx}}{\cancel{18y}} & = \frac{-8x}{18y}\\
\\
\frac{dy}{dx} &= \frac{-8x}{18y}\\
\\
\frac{dy}{dx} &= \frac{-4x}{9y} \qquad \text{ or } \qquad y' = \frac{-4x}{9y} && \text{(Equation 1)}
\end{aligned}
\end{equation}
$


b.) Solving for $y$

$
\begin{equation}
\begin{aligned}
9y^2 &= 36-4x^2\\
\\
\frac{\cancel{9}y^2}{\cancel{9}} &= \frac{36-4x^2}{9}\\
\\
y^2 &= \frac{36-4x^2}{9}\\
\\
y &= \pm \sqrt{\frac{36-4x^2}{9}}\\
\\
y &= \pm \frac{\sqrt{4(9-x^2)}}{\sqrt{9}}\\
\\
y &= \pm \frac{2 \sqrt{9-x^2}}{3} && \text{(Equation 2)}\\
\\
\frac{d}{dx} (y) & = \pm \frac{d}{dx} \left( \frac{2\sqrt{9-x^2}}{3} \right)\\
\\
\frac{dy}{dx} & = \pm \left( \frac{2}{3} \right) \frac{d}{dx} (9-x^2)^{\frac{1}{2}}\\
\\
\frac{dy}{dx} & = \pm \left( \frac{\cancel{2}}{3} \right) \left( \frac{1}{\cancel{2}}\right) (9-x^2)^{\frac{-1}{2}} \cdot \frac{d}{dx} (9-x^2)\\
\\
\frac{dy}{dx} & = \pm \frac{1}{3}(9-x^2)^{\frac{-1}{2}} (0-2x)\\
\\
\frac{dy}{dx} & = \pm \frac{1}{3} (9-x^2)^{\frac{-1}{2}} (-2x)\\
\\
\frac{dy}{dx} & = \pm \frac{2x}{3\sqrt{9-x^2}} \qquad \text{ or } \qquad y' = \pm \frac{2x}{3\sqrt{9-x^2}}
\end{aligned}
\end{equation}
$


c.) Substituting Equation 2 in Equation 1

$
\begin{equation}
\begin{aligned}
y' &= \frac{-4x}{9y} && \text{(Equation 1)}\\
\\
y' &= \pm \frac{2}{3} \sqrt{9-x^2} && \text{(Equation 2)}\\
\\
y' &= \frac{-4x}{ 9 \left( \pm \frac{2}{3} \sqrt{9-x^2}\right)}\\
\\
y' &= \frac{4x}{\pm \frac{18}{3} \sqrt{9-x^2}}\\
\\
y' &= \frac{4x}{\pm 6 \sqrt{9-x^2}}\\
\\
y' &= \frac{2x}{3 \sqrt{9-x^2}}
\end{aligned}
\end{equation}
$


Results from part (a) and part (b) are equivalent

In The Pioneers by James Fenimore Cooper, identify Natty's values and what can be said about the paradox of Natty's life given that he is one of the first pioneers in this valley.

Natty represents an interesting paradox in the novel. While he implicitly values all life, he is also an avid hunter. In the novel, Natty is the face of imperialism as well as the Other. In terms of racial heritage, Natty is white. Yet, culturally, he relates more to the Other than to his own kind. Natty is one of the first pioneers in the valley; yet, he favors the preservation of the virgin land rather than the progress civilization promises.
As we enjoy Natty's story (for The Pioneer really is his story), we are led to ponder the position of someone who is neither fully entrenched in the expectations of his kind nor fully accepting of every aspect of the culture he has adopted. Natty is certainly full of contradictions. He prefers to live a simple life in an untouched wilderness but can see no virtue in intermarriage between whites and Native Americans, a practice he considers miscegenation.
The novel is set in Templeton, a town that represents a new world civilization dependent on homesteading and modern agriculture. This new way of life is seen as an invasive venture by Natty. Unaccustomed to the laws of the wilderness, white settlers often kill game indiscriminately, with no thought of waste. In chapter 22, the townsfolk of Templeton fire a multitude of volleys into a flock of pigeons. Natty Bumppo stands apart from the crowd in disapproval. Meanwhile, Billy Kirby chides Natty for his uncompromising stance, maintaining that if the latter had to sow his wheat "twice and three times," he would be less sympathetic toward the birds.
Natty understands Kirby's position but argues that it is wasteful to "kill twenty and eat one." Natty acknowledges that the people must hunt to survive but still maintains that life in every form should be cherished. To prove his point, Natty kills a pigeon and stuns his fellow pioneers with the accuracy of his aim. He selectively picks out his game (whether fish, deer, or bird) and kills only what is necessary for survival. Among his fellow Americans, Natty is a paradox; he is white but rejects the ideological basis for western agricultural and civilizational expansion. At the same time, Natty is fiercely independent, unwilling to entertain romantic entanglements with either a white or native woman.
Natty's paradoxical nature precipitates a kind of postcolonial anxiety among his people. The challenge of how one classifies a man like Natty becomes a matter of obsessive concern in Templeton. Things come to a head when Natty is accused of hunting a buck out of season. Men are sent to search his hut, but the fierce nonconformist resists them. Gleefully, Mr. Lippet (the lawyer) announces that Natty must stand trial because he assaulted the magistrate and the constable who presented him with a search warrant.
Meanwhile, the sheriff gathers a posse of men to arrest Bumppo for these alleged crimes. In the end, Natty is imprisoned and fined. However, in a last act of defiance, he burns his hut so that no one can infringe upon his private space (without his express permission). Because Natty does not conform to the expectations of the occupying force (from which he derives his racial heritage), he is persecuted and marginalized. For more, please refer to the links below.
https://jfcoopersociety.org/

Calculus of a Single Variable, Chapter 8, 8.3, Section 8.3, Problem 24

inttan^3(pix/2)sec^2(pix/2)dx
apply integral substitution: u=(pix)/2
=>du=(pi/2)dx
=>dx=(2/pi)du
inttan^3(pix/2)sec^2(pix/2)dx=inttan^3(u)sec^2(u)(2/pi)du
Take the constant out,
=(2/pi)inttan^3(u)sec^2(u)du
Again apply integral substitution: v=tan(u)
=>dv=sec^2(u)du
=2/piintv^3du
Apply the power rule,
=2/pi(v^(3+1)/(3+1))
=2/pi((v^4)/4)
Substitute back v=tan(u) and u=(pix)/2
=1/(2pi)tan^4((pix)/2)
Add a constant C to the solution,
=1/(2pi)tan^4((pix)/2)+C

Were African-Americans free during Reconstruction?

The answer to this question depends on how one defines the word "freedom." African-Americans in the South were free in the sense that they were no longer enslaved. Slavery was formally abolished by the Thirteenth Amendment at the end of the Civil War. One might also argue that African-American men, at least, enjoyed a measure of political freedom during part of Reconstruction as they gained the right to vote in most states. In many areas, African-American sheriffs, legislators, and other political officials were put in office. But the end of Reconstruction brought an end to large-scale political participation, as Southern states enacted poll taxes, literacy tests, and other measures intended to disfranchise African-Americans. They also experienced the rise of Jim Crow laws that rigidly segregated Southern society. But throughout Reconstruction, most African-American people never experienced freedom in the sense that many of them understood it. For many black men in particular, freedom meant land ownership and the ability to provide for one's own family. Reconstruction, through its failure (or unwillingness) to enact land reforms, never made this possible, and the majority of African-American families lived meager existences as sharecroppers. So "freedom" was a relative term.
https://www.ushistory.org/us/35.asp

Why do you think Crane indented stanzas 2 and 4?

Based on the "what do you think" wording of the question, you can feel free to take a personal stance on why Crane organized this poem in this manner and defend it. I think a good place to start is by pointing out that the indentations of stanzas 2 and 4 really call them to attention. Essentially, Crane is forcing readers to notice that stanzas 2 and 4 are different than stanzas 1, 3, and 5. They are indented, and their length is longer than the other stanzas as well. Savvy poetry readers should then look deeper into those stanzas. They aren't simply structurally different. They focus on a different topic, audience, and location. Stanzas 1, 3, and 5 are being spoken to specific people. The people are loved ones of soldiers who have died in battle. The indented stanzas then move readers to a wider angle. We have been transported to a battlefield filled with booming noises and thousands of soldiers. The indented stanzas are much wider in scope than the other more intimate stanzas.

The indented second and fourth stanzas of "War is Kind" serve as a kind of chorus. They provide a generalized account of the horrors of war and the terrible effect they have on ordinary soldiers. Crane subtly cross-cuts these anti-war stanzas with the ironic first, third, and fifth stanzas, in which the narrator tries to offer comfort to the bereaved women by describing the heroism of their menfolk as they died in battle.
The indented stanzas are used by Crane to provide a withering commentary on the commonly accepted notion of war as glorious and heroic. The eye-catching paraphernalia of battle—the fluttering flag of the regiment and the impressive eagle insignia—cannot hide the cold reality of war, with all the horrifying bloodshed it entails:

Swift blazing flag of the regiment,Eagle with crest of red and gold, These men were born to drill and die. Point for them the virtue of slaughter, Make plain to them the excellence of killing And a field where a thousand corpses lie.
http://www.theotherpages.org/poems/crane01.html