Explain the salient features of darwin's theory of natural selection

The key to Darwin's theory of natural selection is genetic variation. Genetic variation can happen for a variety of reasons, but sexual reproduction maximizes genetic diversity. As the gene pool goes through changes, different features develop on individual organisms. Those features could be a thicker fur coat, a more acute sense of smell, etc. Those are adaptations, and adaptations are key for Darwin's theory. Beneficial adaptations result in an organism being better adapted to its environment. It is more fit for its environment. Organisms that have the greatest level of fitness are then more likely to live. Darwin called this the "survival of the fittest." A fit organism that is living longer has a greater chance of reproducing. Its genes, and consequently its adaptions, are likely to be passed on to its offspring. After enough time has passed, an entire population (or even species) might have that particular adaptation. Nature has essentially naturally selected which traits get passed on by allowing the fittest organisms to survive and reproduce.
https://www.britannica.com/science/natural-selection

Single Variable Calculus, Chapter 3, 3.1, Section 3.1, Problem 36

$\displaystyle \lim_{t \to 1} \frac{t^4 + t - 2}{t - 1}$ represents the derivative of some function $f$ at some number $a$. State such an $f$ and $a$ in this case.

Using the slope equation,

$\qquad m = \lim \limits_{x \to a} \displaystyle \frac{f(x) - f(a)}{x - a}$ or $m = \lim \limits_{t \to a} \displaystyle \frac{f(t) - f(a)}{t - a} $

Referring to the formula $f(t) = t^4 + t$ and $a = 1$ by substituting value of $a$ to the function $f(x)$, the result will be the value of $f(a)$ which is $2$

Therefore,

$\qquad f(t) = t^4 + t$ and $a = 1$

Single Variable Calculus, Chapter 4, 4.7, Section 4.7, Problem 38

Find the height and radius of the cup that will use the smallest amount of paper of a cone shaped paper drinking cup that is to be made to hold 27cm$^3$ of water.
Recall that the volume of the cone is $\displaystyle V = \frac{1}{3} \pi r^3 h = 27$, so...
$\displaystyle h = \frac{81}{\pi r^2}$

And, the surface area of the cone is equal to $s = \pi r \ell$



By Pythagorean Theorem,
$\ell = \sqrt{r^2 + h^2}$
So, $ S = \pi r \left(\sqrt{r^2 + h^2}\right)$

Substituting the value of $h$, we get...
$\displaystyle S = \pi r \sqrt{r^2 + \left( \frac{81}{\pi r^2} \right)^2}$

We can rewrite $S$ as...

$
\begin{equation}
\begin{aligned}
S &= \sqrt{\pi^2 r^2 \left( r^2 + \left( \frac{81}{\pi r^2} \right)^2 \right)}\\
\\
S &= \sqrt{\pi^2 + r^4 + \frac{81^2}{r^2}}
\end{aligned}
\end{equation}
$


Taking the derivative of $S$ with respect to $r$, we get...

$
\begin{equation}
\begin{aligned}
S' &= \frac{4\pi^2 r^3 + 81^2 \left( \frac{-2r}{r^4} \right)}{2 \sqrt{\pi^2 r^4 + \frac{81^2}{r^2}}}\\
\\
S' &= \frac{4\pi^2 r^3 + 81^2 \left( \frac{-2}{8r^3} \right)}{2 \sqrt{\pi^2 r^4 + \frac{81^2}{r^2}}}
\end{aligned}
\end{equation}
$


when $S' =0$

$
\begin{equation}
\begin{aligned}
0 &= 4\pi^2 r^3 + 81^2 \left( \frac{-2}{r^3} \right)\\
\\
\frac{13122}{r^3} &= 4 \pi^2 + r^3\\
\\
r &= \sqrt[6]{\frac{6561}{2\pi^2}} = 2.6319\text{cm}

\end{aligned}
\end{equation}
$


So when $r = 2.6319$cm, then
$\displaystyle h = \frac{81}{\pi r^2} = \frac{81}{\pi(2.6319)^2} = 3.7221$cm

Therefore, the radius and height of the cup that will use the smallest amount of paper will be $r = 2.6319$cm and $ h = 3.7221$cm respectively.

What language devices does Shakespeare use in act 3, scene 4, lines 17 to 53, to create dramatic effect in Hamlet?

In Hamlet 3.4.17–53, Hamlet confronts his mother, Gertrude, for betraying his father and marrying Claudius, while she is alone in her room. In the middle of this confrontation, Hamlet hears a noise and stabs someone who is moving behind the curtain. He assumes that the person who is spying on their conversation is Claudius, but he soon discovers that he has accidentally stabbed Polonius instead. Tension is high throughout this scene.
As Hamlet begins to confront his mother, he tells her, "you shall not budge. / You go not till I set you up a glass / Where you may see the inmost part of you" (3.4.23–25). Hamlet uses a metaphor here to convey that he intends to show his mother the nature of her true character.The passage is also rife with alliteration to draw attention to important details. In line 20, when Hamlet stabs Polonius, Hamlet cries, "Dead for a ducat, dead!" (3.4.29), and in lines 36 and 37, both Hamlet and Gertrude repeat the line "kill a king" (3.4.36–37).Lines 34 and 35 also are written as a heroic couplet. Hamlet remarks "A bloody deed—almost as bad, good mother, / As kill a king and marry with his brother" (3.4.34–35). This change to a rhyming couplet here when before the text was unrhymed emphasizes these lines and Hamlet's meaning.The last few lines of this passage are also full of metaphor, simile, and personification as Hamlet throws accusations at his mother.


Such an act That blurs the grace and blush of modesty, Calls virtue hypocrite, takes off the rose From the fair forehead of an innocent love And sets a blister there, makes marriage vows As false as dicers’ oaths (3.4.49–54)

Hamlet tells his mother that she makes modesty "blush" and that she "takes off the rose" from the image of innocent love and replaces it with a "blister." He essentially tells her that she has made the name of innocent love shamed and dirtied and that she treats marriage promises like gamblers' bets.

All of these language devices create and build dramatic effect throughout the scene.
https://www.folgerdigitaltexts.org/html/Ham.html

Which narrative point of view uses a narrator outside of the story who knows the thoughts, feelings, futures, and pasts of every character?

As indicated in the other answer, this narrative point of view is called third person omniscient. It is only fully possible in fiction writing, for in writing nonfiction, even what is called narrative nonfiction, the writer cannot possibly know every detail of the thoughts and lives of either historical or living people.
Other points of view include first person, which involves a character telling a story from his or her own point of view, using "I" and "me" in description of the action. In this form of narration, the reader can only see, hear, sense, or understand what the first person narrator experiences. Third person limited narration is similar, although it is told from the point of view not of the narrator but of one particular character, a person who the narrator refers to by his or her name and as "he" or "she." In this case, we are allowed into the mind of that character and can only gauge what others are thinking through that single character's reactions and perceptions.
Second person narration, which uses "you" and "your" to tell a story, is much more unusual but can be used effectively to create a sense of distance between a narrator and the events he is describing.

The method of storytelling in which a narrator is privy to the thoughts and feelings of characters and can move freely through time to reveal information is known as a third person omniscient point of view. This is different from a third person limited point of view, in which knowledge is filtered through the perspective of a character, thus limiting the scope of information. In a work of literature narrated through a third person omniscient viewpoint, the author has a great amount of freedom with which to construct a story, and readers can easily be given relevant background information regarding any aspect of the story. This point of view is particularly useful in developing characters, since an omniscient, all-knowing narrator can provide intimate insight into a character's history and feelings. Omniscient narration is often more objective than first person or third person limited narration, since information is revealed directly by the author as opposed to a character involved in the story.

Calculus of a Single Variable, Chapter 5, 5.2, Section 5.2, Problem 29

To apply u-substitution , we let u = sqrt(x)-3 .
Then du = 1/(2sqrt(x) dx .
Rearrange du = 1/(2sqrt(x)) dx into dx =2sqrt(x) du
Substituting dx=2sqrt(x) du and u =sqrt(x)-3 :
int sqrt(x)/(sqrt(x)-3)dx = int sqrt(x)/u*2sqrt(x) dx
Simplify: sqrt(x)*sqrt(x) = x
int sqrt(x)/u *2sqrt(x) du = int (2x)/u du
Rearrange u=sqrt(x)-3 into sqrt(x)=u+3
Squaring both sides ofsqrt(x)=u+3 then
x=u^2+6u+9
int (2x)/u du = 2 int (u^2+6u+9)/u du
= 2 int (u^2/u + 6u/u + 9/u) du
= 2 int (u + 6 + 9/u) du
=2 *(u^2/2+6u+9lnabs|u|) +C
Substitute u =sqrt(x)-3:
2 *(u^2/2+6u+9ln|u|)+C =2 *((sqrt(x)-3)^2/2+6(sqrt(x)-3)+9ln|(sqrt(x)-3)|)+C
=(sqrt(x)-3)^2+12(sqrt(x)-3)+18ln|(sqrt(x)-3)| +C
= x-6sqrt(x)+9+12sqrt(x)-36 +18ln|sqrt(x)-3|+C
= x + 6sqrt(x)-27 +18ln|sqrt(x)-3|+C

Where do the boys go to in chapter 6 for the first time

Chapter 6 begins in a very familiar way. Two people are talking about Ender, how he's doing, and their expectations of the young child. One specific thing that is pointed out is that Ender's launch group isn't bonding together very well.

"I don't care. His fault or not, he's poisoning that group. They're supposed to bond, and right where he stands there's a chasm a mile wide."

After this particular conversation, the chapter begins its narration about Ender. We find out in the first line that Ender's launch group is going into the "battleroom" for the very first time. The group's experience doesn't start well. The null gravity is scary and confusing to the boys, and the suits make movement difficult.

Worse, the suits were confining. It was harder to make precise movements, since the suits bent just a bit slower, resisted a bit more than any clothing they had ever worn before.

Ender, doing what Ender does best, starts experimenting with the suit and moving throughout the battleroom. As Ender experiments, he watches the other boys very closely. He analyzes everything and begins thinking tactically almost right from the start. Alai is also doing the same kind of thinking, and he and Ender start working together. This is important because Alai and Ender begin a friendship at this point, and Ender can't claim too many of those. The entire group begins figuring the battleroom out and discovers that the guns are designed to lock up the suit of the person that was "shot." The entire groups enjoys their time in the battleroom, and they begin bonding through this fun process of trial and error and making fart jokes.

"You win."
"I want to see your fart collection," Alai said.
"I stored it in your locker. Didn't you notice?"
"I thought it was my socks."