Single Variable Calculus, Chapter 2, 2.3, Section 2.3, Problem 8

Determine the $\lim\limits_{u \rightarrow -2} \quad \sqrt{u^4+3u+6}$ and justify each step by indicating the appropriate limit law(s).


$
\begin{equation}
\begin{aligned}
\lim\limits_{u \rightarrow -2} \quad \sqrt{u^4+3u+6} & = \sqrt{\lim\limits_{u \rightarrow -2} (u^4+3u+6)} && \text{(Root Law)}\\
\lim\limits_{u \rightarrow -2} \quad \sqrt{u^4+3u+6} & = \sqrt{\lim\limits_{u \rightarrow -2} u^4 + \lim\limits_{u \rightarrow -2}3u + \lim\limits_{u \rightarrow -2} 6} && \text{(Sum Law)}\\
\lim\limits_{u \rightarrow -2} \quad \sqrt{u^4+3u+6} & = \sqrt{\lim\limits_{u \rightarrow -2}u^4 + 3 \lim\limits_{u \rightarrow -2} u + 6 } && \text{(Constant Multiple and Constant Law)}\\
\lim\limits_{u \rightarrow -2} \quad \sqrt{u^4+3u+6} & = \sqrt{(-2)^4+3(-2)+6} && \text{(Power Special Limit Law)}
\end{aligned}
\end{equation}\\
\boxed{ \lim\limits_{u \rightarrow -2} \quad \sqrt{u^4+3u+6} = 4}
$

y = arcsec(4x) , (sqrt(2)/4, pi/4) Find an equation of the tangent line to the graph of the function at the given point

Equation of a tangent line to the graph of function f at point (x_0,y_0) is given by y=y_0+f'(x_0)(x-x_0).
The first step to finding equation of tangent line is to calculate the derivative of the given function. To calculate this derivative we will have to use the chain rule  (u(v))'=u'(v)cdot v'.
y'=1/(|4x|sqrt((4x)^2-1))cdot4=1/(|x|sqrt(16x^2-1))  
Now we calculate the value of the derivative at the given point.
y'(sqrt2/4)=1/(|sqrt2/4|sqrt(16(sqrt2/4)^2-1))=1/(sqrt2/4sqrt(16cdot1/8-1))=1/(sqrt2/4)=4/sqrt2=2sqrt2  
We now have everything needed to write the equation of the tangent line.
y=pi/4+2sqrt2(x-sqrt2/4)
y=2sqrt2x+(pi-4)/4
Graph of the function (red) along with the tangent line (blue) can be seen in the image below.                                                                
https://en.wikipedia.org/wiki/Chain_rule

https://en.wikipedia.org/wiki/Tangent

College Algebra, Chapter 2, 2.3, Section 2.3, Problem 56

Find all real solutions of the equation $x^4 - 8x^2 + 2 = 0$

$
\begin{equation}
\begin{aligned}
x^4 - 8x^2 + 2 &= 0\\
\\
w^2 - 8w + 2 &= 0 && \text{Let } w = x^2\\
\\
w^2 - 8w &= -2 && \text{Subtract } 2\\
\\
w^2 - 8w + 16 &= -2 + 16 && \text{Complete the square: Add } \left( -\frac{8}{2} \right)^2 = 16\\
\\
(w - 4)^2 &= 14 && \text{Perfect Square}\\
\\
(w - 4) &= \pm \sqrt{14} && \text{Take the square root}\\
\\
w &= 4 \pm \sqrt{14} && \text{Add }4\\
\\
w = 4 + \sqrt{14} \text{ and } w &= 4 - \sqrt{14} && \text{Solve for } w\\
\\
x^2 = 4 + \sqrt{14} \text{ and } x^2 &= 4 - \sqrt{14} && \text{Substitute } w = x^2\\
\\
x = \pm \sqrt{4+\sqrt{14}} \text{ and } x &= \pm \sqrt{4 - \sqrt{4}} && \text{Take the square root}
\end{aligned}
\end{equation}
$


Thus, the solutions are
$x = \sqrt{4+\sqrt{14}},\quad x = - \sqrt{4+\sqrt{14}},\quad x = \sqrt{4-\sqrt{14}},$ and $ x = -\sqrt{4-\sqrt{14}}$

Is IT defeated in the end of the book?

IT is defeated in the sense that the Murrys are freed from the hold IT had over their minds and/or bodies. Further, at the end of the book, the Murrys are all returned to earth. At the same time, we have no assurance that IT's power over the planet Camazotz has been lessened or broken.
This could be understood as a flaw in the novel. The Murrys have saved each other and gone through, along with Calvin, a personal family drama that has ended happily for them all, but as far as we know, the people of Camazotz are still under the tyranny of IT. They are not allowed to think or make decisions for themselves. We have to imagine that the planet is still shrouded in darkness and that the people are still subjected to extreme conformity and punished severely for noncompliance. A more satisfying ending might have led to the Murrys freeing the people of the planet as they freed themselves.

Single Variable Calculus, Chapter 8, 8.1, Section 8.1, Problem 36

Evaluate $\displaystyle \int^\pi_0 e^{\cos t} \sin 2t dt$ by making a substitution first, then by using Integration by parts.
Recall that $\sin 2t = 2 \sin t \cos t$ so,
$\displaystyle \int^\pi_0 e^{\cos t} \sin 2 t dt = \int^\pi_0 e^{\cos t} (2 \sin t \cos t) dt$

if we let $z = \cos t$, then $dz = - \sin t dt$
Make sure that the upper and lower limits are also in terms of $z$, so...
$\displaystyle \int^\pi_0 e^{\cos t} (2 \sin t \cos t) dt = -2 \int^{\cos \pi}_{\cos 0} z e^zdz = -2\int^{-1}_1 ze^z dz$

By using integration by parts,
If we let $u = z$ and $dv = e^z dz$. Then,
$du = dz$ and $\displaystyle v = \int e^z dz = e^z$

So,

$
\begin{equation}
\begin{aligned}
-2 \int^{-1}_1 ze^z dz = uv - \int v du &= -2 \left[ ze^z - \int e^z dz \right]\\
\\
&= -2 \left[ ze^z - e^z\right]\\
\\
&= -2e^z [z-1]
\end{aligned}
\end{equation}
$


Evaluating from 1 to -1,
$\displaystyle = \frac{-4}{e}$

What limits the narrator's perceptions in the Harry Potter series?

The Harry Potter series is written in the third-person limited point of view rather than the third-person omniscient. While Harry Potter is the main character in the series, he does not tell the story himself, and J. K. Rowling mostly limits her narration to what Harry would know and experience in the moment. Neither the character nor the narrator is all-knowing in this style, so they cannot perceive the thoughts and feelings of others. They can only perceive the actions of other characters or events. For example, Hermione might be perceived by Harry to be thinking, but we don’t know what she is thinking about unless she speaks. We can’t get inside her head, because Harry can’t. The notable exception to this is with the series’ main villain, Lord Voldemort. He and Harry share a connection that allows them to enter each other’s minds, although Voldemort uses this far less frequently than Harry does. There are very few scenes in the series where either Harry or Voldemort (with Harry mentally present in some capacity) is not present. Harry is present in nearly every scene, or at least in the general vicinity of the action, and every character is filtered through the prism of his perception of them. This limits the knowledge of the narrator to that of an eleven- to seventeen-year-old boy wizard.

J.K. Rowling's Harry Potter series is written from the third person limited perspective. This means that, though the narrative is written in the third person, the perspective is limited to one specific character's experiences. Thus, Rowling's narrator's perception is limited in that the narrative is pretty much only told from Harry's perspective (there are a few exceptions to this rule in the series, but they are rare). As heroic as he is, Harry can't possibly be everywhere at once, so he is unable to know everything that's going on in the world of witchcraft and wizardry. Moreover, he can't know other characters' exact thoughts. As a result, the narrator's perception is limited in the Harry Potter series in that the narrative is essentially confined to Harry's personal experiences. 

How does Shakespeare assure his friend that his beauty will ever remain undying?

Shakespeare asserts that his friend will be immortalized through Shakespeare's verse. The words in his poems will insure that his friend's beauty will remain undying. As Shakespeare writes in Sonnet 55:

So till the judgment that your self arise, You live in this [the poem], and dwell in lovers' eyes.

Shakespeare also notes in the above quote that the friend will live in the memories [eyes] of his lovers.
Shakespeare repeats the idea of immortality through poetry in Sonnet 81:

Your monument shall be my gentle verse, Which eyes not yet created shall o'er-read.

As indicated in the first quote, this is not a pagan concept, as it might be in Beowulf, where the hero's only immortality comes through verse or art. Shakespeare is careful to note that the verses he writes simply act as a placeholder to keep the friend's memory alive until the Second Coming raises him again in bodily form, a Christian concept.
The concept of art as immortal in contrast to the decay and death of the human body is a common one in poetry. Often, however, as with the Roman poet Horace, writers thought of the lines they wrote as immortalizing themselves. Shakespeare, however, puts the emphasis on his poems keeping alive his friends and lovers. We may nevertheless suspect that he was also thinking of gaining his own immortality through his writing.

I assume you are referring to Shakespeare's "Fair Youth" sonnets (1–126) with this question. These sonnets were written to an unidentified young man beloved of the poet, and many of them focus upon the themes of youth, reputation, death, and beauty. The poet does assure his beloved on several occasions that, although "summer's lease hath all too short a date" (Sonnet 18), his beauty will not disappear with his youth. In Sonnet 18, the poet states, "thy eternal summer shall not fade," for the simple reason that "so long as men can breathe, or eyes can see, / So long lives this [poem], and this gives life to thee." That is, the poet is sure that his verses will live on forever, and therefore the youth and beauty of the young man are immortalized in them.
This theme is continued in Sonnet 54, for example, where the poet compares the beauty of his beloved to that of roses, which, in dying, simply produce beautiful odours to replace their beautiful flowers. Here, Shakespeare assures his "beauteous and lovely youth" that "when that shall fade, my verse distills your truth."
Shakespeare's sonnets repeatedly revisit this theme of verse as an epitaph. The poet is concerned that his works should allow the continuance of his own reputation, as well as of the beauty and reputation of his beloved.