What figurative language is in "Mom had grown up in the desert. She loved the dry, crackling heat" from page 21 of The Glass Castle? How do you know? What does this language tell you about the desert and Jeannette's mother?

The words "crackling heat" are an example of onomatopoeia. "Crackling" refers to the short, sharp sounds a fire makes when it's burning. Onomatopoeia is a word that describes or mimics the sound something makes. As a literary device, onomatopoeia is an effective means of making an author's prose more vivid.
In the book, the author uses onomatopoeia to highlight Jeannette's mother's love for the desert. "Crackling" describes the scorching desert heat perfectly. "Crackling" is also an emblem of Jeannette's mother's resourcefulness, tenacity, and fierce independence. From the text, we learn Jeannette's mother has the skills necessary to survive in the desert and thrive in her surroundings. Jeannette proclaims that her mother knows which plants are toxic and which are good for eating. Her mother also knows where to find water and how to make do with what she finds.
Jeannette maintains that her mother is courageous enough to drink unpurified water, as "long as animals were drinking from it." To make toothpaste, Jeannette's mother teaches her children to mix hydrogen peroxide with a little baking soda. In short, Jeannette's mother welcomes the "emptiness and severity" of the open desert. Her love of the "crackling heat" testifies to her courage, independence, and resourcefulness.

What is the impact when there is a lack of research on a diagnosis when clinicians are trying to diagnose a patient?

When there is a lack of research on a specific disorder, patients suffer, as they are not able to receive the most tailored and up-to-date treatment. As the article from the National Institute of Mental Health in the link below states, diagnoses in the DSM (Diagnostic and Statistical Manual of Mental Disorders), which clinicians use to diagnose mental disorders, are based on clusters of symptoms. Disorders are not based on objective measures from the laboratory. In other words, this is like diagnosing medical conditions based on symptoms such as fever rather than on laboratory measurements. 
Refining the research about diagnoses will ensure validity--that is, making sure that the diagnosis is a condition, not just a set of symptoms, and that it has measurable differences from other conditions. In addition, additional research about diagnoses will help clinicians make more accurate diagnoses and ensure that people who should be diagnosed are diagnosed. If patients receive correct diagnoses, they can then access proper treatment. For example, the diagnosis of Autism Spectrum Disorder (ASD) is currently made based on a set of symptoms. However, if research showed ways to diagnose autism based on objective measurements (such as brain scans, for example), clinicians could be sure that people who were diagnosed definitely had that disorder and could receive effective treatments. 
https://www.nimh.nih.gov/about/directors/thomas-insel/blog/2013/transforming-diagnosis.shtml

Calculus: Early Transcendentals, Chapter 4, 4.9, Section 4.9, Problem 15

The most general antiderivative G(t) of the function g(t) can be found using the following relation:
int g(t)dt = G(t) + c
int (1 + t + t^2)/(sqrt t)dt = int (1)/(sqrt t)dt + int (t)/(sqrt t)dt + int t^2/(sqrt t) dt
You need to use the following formula:
int t^(-n) dt = (t^(-n+1))/(-n+1)
int (1)/(sqrt t)dt= int t^(-1/2) dt = (t^(-1/2+1))/(-1/2+1) + c = 2t^(1/2) + c= 2sqrt t + c
int (t)/(sqrt t)dt= int (sqrt t*sqrt t)/(sqrt t)dt = int sqrt t dt = (t^(1/2+1))/(1/2+1) + c = (2/3)*t^(3/2) + c = (2/3)*tsqrt t + c
int t^2/(sqrt t) dt = int t^(2-1/2)dt = int t^(3/2)dt = (t^(3/2+1))/(3/2+1) + c
int t^2/(sqrt t) dt = (2/5)*t^(5/2) + c = (2/5)*t^2*sqrt t + c
Gathering all the results yields:
int (1 + t + t^2)/(sqrt t)dt = 2t^(1/2) + (2/3)*tsqrt t + (2/5)*t^2*sqrt t + c
Hence, evaluating the most general antiderivative of the function yields G(t) = 2t^(1/2) + (2/3)*tsqrt t + (2/5)*t^2*sqrt t + c.

What are the differences between Phoneme, Phone and Allophone?

Speaking in phonetic terms, a phone is quite simply a speech sound. This can be absolutely any sound or gesture, and it is the broadest of these terms since its meaning does not hinge on whether or not it changes the descriptive meaning of the language in which it is used. A phoneme, on the other hand, is a phonetic sound that would change the meaning of the word that it is used in if it were swapped with another. For example, pear and bear have two completely different meanings, though they are only separated by the phonemes /p/ and /b/. Allophones are different spoken sounds for the same phoneme, and typically do not have bearing on the meaning of the word in language. In fact, speakers of native language and laymen in the field of phonetics typically are not aware of different allophones within a phoneme.

All of these words and their corresponding definitions have to do with the study of phonetics and linguistics. Both the study of phonetics, which focuses on the details of human speech, and the study of linguistics, which focuses on the structural elements of language, require an understanding of these terms.
A phone is any sound made during speech in any language at all, but it does not necessarily mean anything specific. In contrast, a phoneme is specific to a language, and different phonemes in English and other languages do actually mean different things. Allophones are different versions of specific phonemes, or spoken sounds in speech. They do not necessarily change the meaning of the phoneme, and in fact, sometimes people use different allophones in their own speech without even realizing it.

These concepts in linguistics are often confused, but they are quite distinct.

A phoneme is a unit of sound that differentiates one word from another. For example, the phoneme “t“ in “Tim” differentiates it from “him”. Likewise, “h” is the distinguishing phoneme in “him.”
A phone is any unit of sound in English. It need not change the meaning of a word if replaced.
An allophone is a variety of a phoneme, pronounced slightly differently to other varieties but having the same outcome and representing the same thing. For example, the letter “p” in “push” is pronounced with aspiration, more strongly than the “p” in “spit,” but these remain two versions of the same phoneme. Exchanging one for the other will not actually alter the meaning of the word.

List two important groups of organisms that appeared during each of the three most recent geologic eras.

The three most recent geologic era include Cenozoic (also known as the age of mammals), Mesozoic (also called the age of reptiles), and Paleozoic, with Cenozoic being the current era. All these eras are a part of the Phanerozoic eon and each is further divided into periods, for example, the Cenozoic era is divided into the Tertiary and Quarternary periods.
The Paleozoic era lasted from 251-542 million years ago, while the Mesozoic era lasted from 65-251 million years ago. The Cenozoic era started around 65 million years ago and is currently underway.
The most important groups of organisms that appeared during each of these eras are:
1) Cenozoic: Human beings, rodents
2) Mesozoic: Mammals, birds, flowering plants, primates
3) Paleozoic: Fish, reptiles, amphibians
The Cenozoic era is most famous for the diversity of the mammals and the evolution of human beings. The Mesozoic era is known for the evolution and extinction of the dinosaurs, and evolution of birds, mammals, primates and flowering plants. The Paleozoic era is most famous for the evolution of the fish, chordates (animals with a backbone), vascular plants, amphibians, etc.
Hope this helps.
https://www.aoml.noaa.gov/outreach/floridaseagrant/pdf_files/TropicalConnections_GeologicalTimeWithMajorEvolutionaryEventsInFossilRecord_KruczynskiFletcher.pdf

Hi, I was wondering how Brutus in the Tragedy of Julius Caesar contributed to the downfall of the conspirators. I know one is that he let Antony live and speak at Caesar's funeral, but are there any other mistakes he made? Thank you!

Brutus undoubtedly made a huge mistake in allowing Cassius to talk him into the plot of assassinating Caesar. Cassius is motivated purely by thoughts of personal gain. Brutus, however, has a genuine commitment to the cause of republican liberty and a genuine belief that Caesar's political ambitions represent a threat to that cherished ideal.
However, Brutus is too trusting and too ready to take other people at their word. This tragic flaw is what enables the scheming, devious Cassius to lure him into a plot to murder a man who is supposed to be a close friend. Cassius subtly manipulates Brutus, playing on his sense of honor and nobility to convince him that some of the finest people in Rome are clamoring for him to take over. Brutus is so fixated on what he believes to be his duty that he does not realize that someone else is using him for their own ends.
Brutus's naivety and sense of duty cause him to make another mistake. He blithely assumes that there is no need for the conspirators to pledge an oath of loyalty to each other. This blinds him to the obvious fact that the conspirators have different motives for their participation in the assassination plot, some of which are far from disinterested or noble. Brutus seems to think that showing commitment to the plot is somehow a sufficient sign of loyalty. In this, he is tragically mistaken. Some of the conspirators talk openly about what they are up to; this is how Artemidorus finds out about the plot. He tries to warn Caesar as he is about to the enter the Senate chamber on the Ides of March.
On the field of battle, Brutus makes a huge tactical blunder in marching his troops to Philippi. Cassius is a lot more shrewd in this situation, recognizing that it is much better to allow the enemy to come to them, tiring themselves out in the process:

Tis better that the enemy seek us.
So shall he waste his means, weary his soldiers,
Doing himself offense, whilst we, lying still,
Are full of rest, defense, and nimbleness (act 4, scene 3).

Brutus is so blinded by what he sees as his historical destiny that he is not prepared to listen to reason. Ironically, this is one occasion on which he should have listened to Cassius:


The enemy increaseth every day.
We, at the height, are ready to decline.
There is a tide in the affairs of men,
Which, taken at the flood, leads on to fortune;
Omitted, all the voyage of their life
Is bound in shallows and in miseries.
On such a full sea are we now afloat,
And we must take the current when it serves
Or lose our ventures (act 4, scene 3).



Brutus fervently believes that the tide of history is turning in the conspirators' favor; there must be no retreat and no backing down; now is the right time to strike. He is so intoxicated by this grand vision, however, that he neglects basic military tactics, leading to defeat and, ultimately, his own death.

College Algebra, Chapter 8, 8.3, Section 8.3, Problem 42

Find an equation for the hyperbola with foci $(0,\pm 1)$ and length of the transverse axis of 1.
The hyperbola $\displaystyle \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ has length of transverse axis of $2a$ and foci $(0,\pm c)$ where $c^2 = a^2 + b^2$.
Thus, gives us $2a = 1$ and $c = 1$. So, $\displaystyle a=\frac{1}{2}$. Then, by substituting the values, we obtain

$
\begin{equation}
\begin{aligned}
1^2 &= \left( \frac{1}{2} \right)^2 + b^2\\
\\
1 &= \frac{1}{4} + b^2 \\
\\
b^2 &= \frac{3}{4}\\
\\
b &= \frac{\sqrt{3}}{2}
\end{aligned}
\end{equation}
$


Therefore, the equation is

$
\begin{equation}
\begin{aligned}
\frac{y^2}{\left( \frac{1}{2} \right)^2} - \frac{x^2}{\left( \frac{\sqrt{3}}{2} \right)^2} &= 1\\
\\
\frac{y^2}{\frac{1}{4}} - \frac{x^2}{\frac{3}{4}} &= 1\\
\\
4y^2 - \frac{4x^2}{3} &= 1
\end{aligned}
\end{equation}
$