In The Freedom Writers Diary, how was the importance of self-worth shown through the thoughts, words, and actions of the students in Ms. Gruwell's class?

The importance of self-worth to Ms. Gruwell's students can be seen in how they use her content to formulate emotionally strong positions about the world and their place in it.
Ms. Gruwell understands her students' profound deficit of self-worth.  She recognizes that years of being overlooked and maligned both educationally and socially have prevented them from recognizing meaning in themselves and their world.  This is clear when Erin notes how "it doesn't take a rocket scientist to figure out that if you tell kids they're stupid--directly or indirectly--sooner or later they start to believe it.”  Ms. Gruwell recognizes that connecting content to the experiences of her students can increase their self-worth, evident in her students' thoughts, words, and actions.
Tommy Johnson's diary entry displays self-worth.  His opening sentence insists that he is better than the violence dominating his world. When Tommy writes, "They say America is the 'Land of the Free and Home of the Brave,' but what's so free about a land where people get killed," he shows an acquisition of self-worth. His opening question reflects how he believes that the world in which he lives is wrong and must be questioned.  The purpose of his diary entry is to make a connection with Zlata Filipovic, whose diary he has read as a part of Ms. Gruwell's class.  When he asks Zlata how to handle a situation where violence is so prevalent, his self-worth compels him to turn to someone else who can help. Writing a diary entry to Zlata is an action that shows Tommy's self-worth.  When he closes his diary entry with "Your Friend," self-worth is evident in his affirmation of friendship to share and learn from someone in a similar situation.
In Diary 36, we see another example of self-worth's importance to one of Ms. Gruwell's students.  This particular student did not expect to be emotionally impacted while reading Anne Frank's diary.  However, the book "came alive" for the student and a connection reflecting self-worth was made: "I cried when she cried, and just like her I wanted to know why the Germans were killing her people.  Just like her, I knew the feeling of discrimination and to be looked down upon based on the way you look.  Just like her 'I sometimes feel like a bird in a cage and just want to fly away."  Diary 36 shows Ms. Gruwell's content imparting self-worth in the thoughts, actions, and words of her student.  There is an empathy between the student and Anne Frank.   Like Tommy Jefferson, the student affirms self-worth in the insistence that what is happening in the world is wrong and must be repudiated. 
The words and thoughts of Diary 37 are also rooted in self-worth.  This particular entry connects the student's own experience with Zlata's and Anne's.  The diary entry is focused on an abusive and emotionally absent "sperm donor" of a father.  Self-worth is powerfully evident in the diary entry's conclusion of how "I won't die or get taken advantage of.  I'm going to be strong."  The student has appropriated the examples of Anne Frank and Zlata Filipovic as behavior models.  The diary entry reflects self-worth in the awareness that they deserve better and merit more than what is around them.  
The affirmation of strength and self-worth in the three diary entries reminds us of the power that education can have on a child. Content such as Ms. Gruwell's can transform their perception of the world and their place in it.  It enhances self-worth because students recognize that their experiences are mirrored in the realities of other people.  They realize they are not alone.  Ms. Gruwell's students realize this in their actions of composing diary entries.  They also display self-worth in words that communicate powerful thoughts of strength and refusal to accept injustice in their lives and the world around them.

What causes his second fire to go out ?

In Jack London’s “To Build a Fire,” the man builds three fires. The second fire goes out because the man makes a mistake: he builds the fire under a pine tree. Although this makes it easier for him to collect sticks to feed the flames, it ultimately proves fatal. Snow has been collecting on the top of the tree’s branches, and every time the man has pulled a stick, the snow has slowly worked its way down the tree, falling from one branch to the next:

Each time he pulled a stick he shook the tree slightly. There had been just enough movement to cause the awful thing to happen. High up in the tree one branch dropped its load of snow. This fell on the branches beneath.

Eventually, this snow falls onto the fire itself, extinguishing it and leaving in its place “a pile of fresh snow.”
This is how the second fire goes out. He does manage to a light a third fire, which is also referred to as “the second time.” This provides a sense of repetition in the man’s struggle against the harsh environment. At this point, however, the man is hardly in control of his own body, and he ends up badly burning himself. He is unable to start a fire with the small pieces of burning grass. Both of these failures depict a man that does not truly understand nature. The man is aware of facts, but he lacks the form of knowledge that the dog has—an innate knowledge of how to survive. Because of this, he mistakenly builds the second fire under the tree, and he pays the ultimate cost.

x=4cos^2theta , y=2sintheta Find all points (if any) of horizontal and vertical tangency to the curve.

Parametric curve (x(t),y(t)) has a horizontal tangent if its slope dy/dx is zero i.e when dy/dt=0 and dx/dt!=0
It has a vertical tangent if its slope approaches infinity i.e it is undefined, which implies that dx/dt=0 and dy/dt!=0  
Given parametric equations are:
x=4cos^2(theta) ,y=2sin(theta)
Here the parameter is theta
Let's take the derivative of x and y with respect to theta
dx/(d theta)=4(2cos(theta)d/(d theta)cos(theta))
dx/(d theta)=4(2cos(theta)(-sin(theta)))
dx/(d theta)=-4(2sin(theta)cos(theta))
Use trigonometric identity: sin(2theta)=2sin(theta)cos(theta)
dx/(d theta)=-4sin(2theta)
dy/(d theta)=2cos(theta)
For Horizontal tangents, set the derivative of y equal to zero
dy/(d theta)=2cos(theta)=0
=>cos(theta)=0
=>theta=pi/2,(3pi)/2
Let's check dx/(d theta) for the above angles,
For theta=pi/2
dx/(d theta)=-4sin(2*pi/2)=-4sin(pi)=0
For theta=(3pi)/2
dx/(d theta)=-4sin(2*(3pi)/2)=-4sin(3pi)=0
So, there are no horizontal tangents.
Now for vertical tangents, set the derivative of x equal to zero,
dx/(d theta)=-4sin(2theta)=0
=>sin(2theta)=0
=>2theta=0,pi,2pi,3pi
=>theta=0,pi/2,pi,(3pi)/2
Let's check for the above angles,
For theta=0
dy/(d theta)=2cos(0)=2 
For theta=pi/2
dy/(d theta)=2cos(pi/2)=0
For theta=pi
dy/(d theta)=2cos(pi)=-2
For theta=(3pi)/2
dy/(d theta)=2cos((3pi)/2)=0
So, the curve has vertical tangents at theta=0,pi
Now let's find the corresponding x and y coordinates by plugging theta in the parametric equation,
For theta=0
x=4cos^2(0)=4
y=2sin(0)=0
For theta=pi
x=4cos^2(pi)=4
y=2sin(pi)=0
So, the given parametric curve has vertical tangent at (4,0).

Compare and contrast the presidential reaction to the Great Depression of Herbert Hoover and Franklin D. Roosevelt.

While many tend to fault Herbert Hoover for doing nothing to address the Great Depression, he and Franklin Delano Roosevelt actually had similar approaches to the Depression. Hoover was president when the stock market crashed in 1929. Unfortunately, Hoover, like most economists at the time, thought that the crash was just another recession that would pass with time. Recessions, or economic "panics," were considered natural parts of the business cycle that would, for the most part, be fixed with time as the economy readjusted. However, by 1931, it was becoming increasingly clear that the economy was not rebounding and that the years of unregulated speculation and credit spending in the 1920s was taking its toll. Contrary to popular belief, Hoover did not simply sit back and do nothing to fix the economy. In fact, he founded government agencies like the Reconstruction Finance Corporation to loan money to banks, railroads, and other large companies to help them avoid bankruptcy. Hoover also signed the Emergency Relief and Construction Act to provide public works projects and jobs, like the building of the Hoover Dam. However, Hoover also believed in what he called "rugged individualism," the ability of individuals to persevere, help themselves, and help others during difficult times. While he made efforts to provide relief and jobs, Hoover mostly relied on local and state agencies, along with charities, to solve poverty and unemployment. Hoover, like many of his contemporaries, feared that federal programs that were too expansive would cause people to become overly dependent on the government, which would undermine their "rugged individualism" and work ethic. State and local governments were soon overwhelmed by the number of homeless and unemployed, finding themselves without the finances and resources needed to address the Depression. By 1932, "Hoovervilles" were springing up—camps of makeshift houses and shacks filled with the homeless.
Franklin Delano Roosevelt easily won the election of 1932, especially after Hoover dispatched the National Guard to disperse the Bonus Army, a group of veterans who marched on Washington to demand their bonus pay from World War I. Two veterans were killed after fighting broke out.
Franklin Delano Roosevelt preferred a direct and government-involved approach to the Depression. Like Hoover, he did not want people to become dependent on the government. His New Deal was essentially an expansion of some of Hoover's programs, and it focused on providing jobs through public works programs. Roosevelt differed from Hoover in his willingness to use direct financial aid and in his propensity for more government control and oversight of industries and programs. Roosevelt believed in Keynesian economics, a new theory at the time, which advocated deficit spending. The key idea behind this theory was that spending money on government programs such as the Civilian Conservation Corp and Works Progress Administration would stimulate the economy. Roosevelt also attempted to reform the economy by creating agencies like the FDIC, which insured bank deposits and provided financial security to millions of Americans through Social Security.

What literary devices are there in "The Ones Who Walk Away from Omelas"?

Polysyndeton refers to using several conjunctions in a row to achieve a dramatic effect. That can be seen in this sentence about the child:

The child used to scream for help at night, and cry a good deal, but now it only makes a kind of whining, "eh-haa, eh-haa," and it speaks less and less often.

Comparing this long string of clauses to the ones above it show a much longer sentence structure. It is at this point that the child is dehumanized, moving from actual speech in the sentences prior to losing the ability to communicate due to lack of human stimulation. Therefore, polysyndeton is used to mimic this loss of semantic control as language devolves in structure as well.
Imagery is used to show the contrasting lives between the citizens of Omelas and the child held captive. Consider the following imagery used to describe the processions:

A marvelous smell of cooking goes forth from the red and blue tents of the provisioners. The faces of small children are amiably sticky; in the benign grey beard of a man a couple of crumbs of rich pastry are entangled. The youths and girls have mounted their horses and are beginning to group around the starting line of the course. An old woman, small, fat, and laughing, is passing out flowers from a basket, and tall young men wear her flowers in their shining hair.

The imagery here connotes a lightness, a carefree existence of friendly faces and hair adorned with flowers. Of course, this carefree lifestyle is only possible because the townspeople are willing to sacrifice one child. Consider the contrasting imagery provided to show the life of this child:

It is feeble-minded. Perhaps it was born defective, or perhaps it has become imbecile through fear, malnutrition, and neglect. It picks its nose and ... sits hunched in the corner farthest from the bucket and the two mops. It is afraid of the mops. It finds them horrible. It shuts its eyes, but it knows the mops are still standing there; and the door is locked; and nobody will come. The door is always locked; and nobody ever comes, except that sometimes—the child has no understanding of time or interval—sometimes the door rattles terribly and opens, and a person, or several people, are there. One of them may come in and kick the child to make it stand up.

These two scenes are juxtaposed with only authorial intrusion (when the narrator steps away from the story and speaks directly to the reader) between them to bring into sharp contrast the startling ways the townspeople are able to live at the expense of the great suffering of one child. The close proximity of these very differing images show the reader both all the people have gained and at what expense they have achieved it.

Le Guin writes in a poetic style in this short story. In the opening paragraph, as the narrator describes the beauties of Omelas, she uses alliteration to create a sense of rhythm with repeated "p" sounds in the following:

past great parks and public buildings, processions moved.

Throughout the story, she uses similes (comparisons using the words "like" or "as"). For example, she likens the voices of children at the festival in the opening scene to:

high calls rising like the swallows' crossing flights, over the music and the singing

Later she compares nudes to soufflés:

Surely the beautiful nudes can just wander about, offering themselves like divine souffles

The horses are personified when it is said that they:

boasted to one another

Le Guin uses an exclamation as she writes of the people of Omelas as "not wretched. O miracle!" Additionally, the word "wretched" is an example of an archaic term in the context of 1970s American English.
The narrator also steps into the text to directly address the audience, pointing to the fictiveness of the story. The direct address below also uses repetition, a literary device that adds emphasis:

If so, please add an orgy. If an orgy would help, don't hesitate

The poetic diction of the story's description of happiness has a tongue-in-cheek or ironic quality, foreshadowing that the fact that all of this happiness is founded on a horrific crime.

In the second paragraph, the narrator describes the way we tend to think of happiness as "something rather stupid." We think that pain and evil are more intellectual and more interesting. The narrator says, "This is the treason of the artist: a refusal to admit the banality of evil and the terrible boredom of pain." Thus, the narrator uses a metaphor to compare the tendency of artists and writers to depict pain and suffering rather than happiness as a kind of treason against their fellow human beings.
The narrator employs another metaphor when they say that the citizens of Omelas are "not naive and happy children." The narrator compares these "mature, intelligent, passionate adults" to children—whose happiness is often thought of as the result of their innocence and naivety—to show that the citizens' happiness is more legitimate somehow, that it isn't a default state that results from their ignorance (because they are not, in fact, ignorant).
The narrator uses a simile when they describe the "beautiful nudes"—people that can wander around, offering themselves "like divine souffles to the hunger of the needy." The narrator compares these people to an elegant and often beautiful type of food.
The narrator also uses metonymy when they say that "A boundless and generous contentment . . . is what swells the hearts of the people of Omelas." Now, the citizens' hearts do not actually swell up and get bigger, but rather, the citizens feel emotionally filled up, satisfied, and happy. As hearts are often connected to emotion, they stand in here, figuratively, for those feelings.

The opening paragraph of this short story has a few good examples of various literary devices. When the story begins, readers are introduced to a large celebration that is happening. It is called the "Festival of Summer." People are gathered and having a great time, and music is playing along with their procession.
Readers are told a great metaphor here. We are told that the "procession was a dance." The very next sentence has a simile that further develops the happy gathering of people. The children's calls rose "like the swallows' crossing flights." A little later in the same paragraph, the author provides readers with some personification of the horses that are in the procession. We are told that they wear minimal gear and have manes braided with streamers. They look amazing, and the horses know it. They flare their nostrils and "boasted to one another." Horses are perfectly capable of flaring their nostrils, however, boasting and bragging is a human trait, so this personification provides a great, concrete image to readers of how the horses are behaving.

Comment on the end

The Second Mrs. Tanqueray ends with the death of Paula Tanqueray, second wife to Aubrey Tanqueray. What's interesting about this ending is that it has brought up various interpretations regarding the author's intent. Was it meant to be the natural progression of events? Or was the author not willing to delve into the psychological implications of the story events that would lead to a reasonable conclusion?
I believe that while this ending felt a little contrived, it was clear that Paula became bored of her marriage to Aubrey and the new life that he has provided for her. While Aubrey was committed to loving Paula despite social classes, Paula still felt like an outsider and the circumstances surrounding her transition only heightened that sense of perpetual loneliness. I would have picked a less drastic way of concluding her character arc, but I'm not entirely surprised by the outcome.

Calculus of a Single Variable, Chapter 5, 5.7, Section 5.7, Problem 36

We have to evaluate the integral: \int \frac{2x-5}{x^2+2x+2}dx
We can write the integral as:
\int \frac{2x-5}{x^2+2x+2}dx=\int\frac{2x-5}{(x+1)^2+1}dx
Let x+1=t
So, dx=dt
Now we can write the integral as:
\int \frac{2x-5}{(x+1)^2+1}dx=\int \frac{2(t-1)-5}{t^2+1}dt
=\int \frac{2t-7}{t^2+1}dt
=\int \frac{2t}{t^2+1}dt-\int\frac{7}{t^2+1}dt --------------->(1)

Now we will first evaluate the integral \int \frac{2t}{t^2+1}dt

Let t^2+1=u
So, 2tdt=du
Hence we can write,
\int \frac{2tdt}{t^2+1}=\int \frac{du}{u}
=ln(u)
=ln(t^2+1)

Now we will evaluate the second integral : \int \frac{7}{t^2+1}dt
\int \frac{7}{t^2+1}dt=7\int \frac{1}{t^2+1}dt
=7tan^{-1}(t)

Substituting both these integral results in (1) we get,
\int \frac{2x-5}{x^2+2x+2}dx=ln(t^2+1)-7tan^{-1}(t)+C where C is a constant
=ln((x+1)^2+1)-7tan^{-1}(x+1)+C
=ln(x^2+2x+2)-7tan^{-1}(x+1)+C