What type of conflict does Brother experience as he battles with himself between love and hate for Doodle, and between longing for a normal brother versus accepting Doodle's limitations?

In “The Scarlet Ibis” by James Hurst, Brother experiences internal conflict as he struggles with his relationship with Doodle. Internal conflict occurs in the psyche of a literary character. The resolution of the conflict creates suspense and purpose in the plot of the story. In comparison, external conflict pits a character against an outside force such as a natural disaster or another character.
Brother struggles with disappointment when Doodle is unable to live up to Brother’s vision of the perfect companion. Doodle is born with developmental disabilities severe enough for the family to question whether he will live. Brother goes as far as to plan ways to cause Doodle’s death. His pride interferes when he realizes Doodle is not going to die, and he is able to respond to the family.
To resolve the internal conflict, Brother pushes Doodle past his expected physical limits by teaching him to walk, row a boat, and swim. Unfortunately, the attempt to resolve Brother’s internal conflict ultimately leads to Doodle’s demise.

I need to create a listening chart covering several elements of music for the song "Sound of Music" by Rodgers and Hammerstein. How would I do this?

A listening chart is a regular chart that has been adapted to row and column labels specifying music selection types, titles, artists, and music elements. For "The Sound of Music," your chart will reflect a single selection and a number of music elements.A chart is defined as the presentation of information displayed in boxes arranged in vertical columns and horizontal rows. You can choose between putting the selections in the vertical column, with the elements in the horizontal row, or the reverse, the elements in the vertical column and the selections in the horizontal row. With a single selection, the song will title the entire chart and you can choose between labeling the elements horizontally (across the top) or vertically (down the side). For a single selection, listing elements vertically (down the left-hand column) might prove more functional. The difference between the two methods is a visual one, with both resulting in the meeting, inside the chart boxes, of your analysis of the music elements in your music selection(s).Let's talk a little about some of the music elements that are interesting in "The Sound of Music." I'm supposing you are interested in the vocal version of "The Sound of Music," so one of the elements to analyze is lyrics and language as described by Maida Owens of the Louisiana Voices Folklife Project (Louisiana Division of the Arts, Department of Culture, Recreation & Tourism). First, lyrics tell the stories of cultures and of individuals within cultures. In "The Sound of Music," the lyrics link us with Maria's experience within her culture, helping to define her (and to set up two plot-driving internal conflicts: her love of nature's freedom and her love of song). Second, language can govern rhythm in music because languages have their own inherent rhythms in speech patterns. You may have heard it said in English poetry class that English speech fits the iambic rhythm (ba DA ba DA, con-CEDE, we COME, you GO) and that the iambic rhythm affects the meter of poetry and lyrics (prose is affected by language as cadence).A link between lyrics and dynamics can be made so that ideas expressed in language are dramatized by musical dynamics. Dynamics forcefully drive intensity and loudness in music and song. Dynamics may change abruptly or with crescendo and range from very soft, at pianissimo, to very loud, at fortissimo. Intensity shifts with expressiveness and can range anywhere from abbandono (with abandon) to grazioso (gracefully). There are a number of good examples of how dynamics dramatize the meaning in lyrics in "The Sound of Music." One of these is the link between the dolce (sweet, delicate) piano (soft) dynamic during the lyrics "sigh like a chime that flies from a church on a breeze." The link dramatizes the metaphor expressed in the lyrics by a novitiate comparing her heart's lovely sigh (sighs express beauteous feeling as often as sorrowful feeling) to a church bell chime being dispersed on a breeze. For another example, examine the contrasting dynamic tied to "laugh like a brook."Timbre and texture further support the lyrics and provide the instrumentation qualities in the music. Referring to the above example, the texture reduces from a thick (full orchestra) texture--with many instrument layers, including strings--during the introductory measures of "The Sound of Music" ("The hills are alive") to a light texture--with fewer instrument layers--during the meditative "my heart wants to..." passages. The timbre in the "my heart" passages enhances the conjunction of lyrics, dynamics and texture with a bright and brassy timber featuring trumpet, French horn and flute.Drawing on this very brief--not exhaustive--discussion, for your listening chart (supposing you choose to list music elements down the left-hand column and write your analysis across the horizontal rows), you would list in the boxes down the column Lyrics & Language, Dynamics, Timbre, Texture (in whatever order you find appropriate; some instructors specify some music elements, some specify others). You would then make your comments regarding your analysis of each (plus other elements you determine through your own listening) from left to right in the corresponding row. As an example, the listing for Timbre might say: "mellow (because of the strings) ranging to flute-y, bright, brassy."
Resources:
Louisiana's Musical Landscape Musical Elements ChartSan Mateo Community College Elements of Music
Victorian Curriculum and Assessment Authority, AU, Elements of Music Active Listening Chart
Maida Owens of the Louisiana Voices Folklife Project (Louisiana Division of the Arts, Department of Culture, Recreation & Tourism)

How did the Cold War and economic opportunities during it increase conformity and consensus in the 1950s?

The Cold War against the Soviet Union gave Americans in the 1950s a common cause around which to rally. It also contributed to the economic boom, as the U.S.entered what some have called a "permanent war economy" meant to ensure the U.S. was prepared to go to "hot" war at any moment—and prepared to step in to prevent the spread of communism into other regions.
The Cold War led to McCarthyism. Although McCarthy's hunt to weed out communists and communist sympathizers ended in 1954, the effects of this campaign lingered through the 1950s. Many leftwing intellectuals and artists had seen their careers destroyed and many were not able to bounce back. People who had been threatened with McCarthy's purges but escaped tended to take home the following lesson: conformity was a good way to survive.
The growing economic prosperity fostered conformity. Radicalism had been encouraged in the 1930s by the economic crisis of the Depression. Many people were desperate because of lack of work and felt a revolution was the only way to improve the country. In the 1940s, a war had captured people's attention. Now, with the economy working at full throttle, people had jobs and could purchase consumer goods again. Society seemed to be delivering on its promises. Life was good. Going to work was rewarded with raises and a rising standard of living. People had no reason not to conform and were more interested in success than challenging norms. 
Prosperity also meant more people could buy televisions. Television sitcoms and dramas of the 1950s fostered conformity. By watching this almost exclusively white, middle-class world, people across the country learned what was expected behavior. Social norms were taught by televisions show  such as Ozzie and Harriet, Leave it to Beaver, and Perry Mason. Television advertising also enforced conformity. People were exposed to continuing messages about what foods to eat, what cars to buy, and what standards of bodily and household cleanliness were acceptable.
Between fears of communism uniting people, suspected communists being silenced, a growing, prospering consumer society satisfying people's material desires after a long period of deprivation, and television enforcing norms, it is not surprising that the 1950s was characterized by a high degree of conformity.

Calculus of a Single Variable, Chapter 10, 10.3, Section 10.3, Problem 46

The formula of arc length of a parametric equation on the interval alt=tlt=b is:
L = int_a^b sqrt((dx/dt)^2+(dy/dt)^2) dt
The given parametric equation is:
x=6t^2
y=2t^3
The derivative of x and y are:
dx/dt= 12t
dy/dt = 6t^2
So the integral needed to compute the arc length of the given parametric equation on the interval 1lt=tlt=4 is:
L= int_1^4 sqrt ((12t)^2+(6t^2)^2) dt
The simplified form of the integral is:
L= int_1^4 sqrt (144t^2+36t^4)dt
L=int_1^4 sqrt (36t^2(4+t^2))dt
L= int_1^4 6tsqrt(4+t^2)dt
To take the integral, apply u-substitution method.

u= 4+t^2
du=2t dt
1/2du=tdt
t=1 , u =4+1^2=5
t=4 , u = 4+4^2=20

Expressing the integral in terms of u, it becomes:
L=6int_1^4 sqrt(4+t^2)* tdt
L=6 int _5^20 sqrtu *1/2du
L=3int_5^20 sqrtu du
L=3int_5^20 u^(1/2)du
L=3*u^(3/2)/(3/2) |_5^20
L=2u^(3/2) |_5^20
L = 2usqrtu |_5^20
L=2(20)sqrt20 - 2(5)sqrt5
L=40sqrt20-10sqrt5
L=40*2sqrt5 - 10sqrt5
L=80sqrt5-10sqrt5
L=70sqrt5
Therefore, the arc length of the curve is 70sqrt5 units.

What impact did Christopher Columbus have upon the New World?

Columbus had a profound impact on the New World.  Even though he thought that he had found Asia, Columbus's voyage still brought Spain a great deal of wealth.  Thousands of young Spaniards dreamed of going to America and striking it rich either in the spice trade or in gold.  With the exception of sugar production, the spice trade never really materialized in America the way the Spaniards had hoped, but thousands of Spaniards grew rich on the gold trade.  Of course, this was often done by subjugating the indigenous people who lived in the New World.  The arrival of Columbus marked the end of many native cultures in the New World.  European diseases such as smallpox, typhoid, and malaria wiped out over ninety percent of the indigenous people who lived in the New World in the two hundred years after Columbus's voyages.  Also, European livestock, such as hogs, escaped their owners and quickly became feral.  These hogs ate many crops Native Americans needed for survival and are still a nuisance in the United States today.  The horses that escaped from their pens were domesticated by the Plains Indians; the horse was soon used as a type of wealth.  It could be used on the battlefield as well as the buffalo hunt.  Columbus's arrival in the New World brought mainly bad things for the natives who already lived there, but the horse was the one positive thing brought from Europe to the New World.  

Why does Mercutio curse both the Capulets and Montagues when he dies?

Mercutio curses both the Capulets and Montagues as he dies in Act 3, Scene 1. His curse "a plague o' both your houses!" shows that he blames the two families for killing him.
Mercutio believes that he had to fight Tybalt to protect the Montague reputation. By extension, Mercutio also blames Romeo for his fatal sword wound. In Mercutio's mind, Romeo's intervention caused him to sustain the bloody injury.
Additionally, Mercutio is angry that Tybalt managed to get the best of him. It's obvious from Mercutio's comments (to Romeo) that he doesn't think very much of Tybalt's sword-fighting skills. In fact, Mercutio thinks that Tybalt fights like he learned all his moves from a manual of sorts. Yet, to Mercutio's humiliation, Tybalt managed to get away without so much as an injury.
So, the curse serves a few purposes. First, it is one way for Mercutio to voice his distress; after all, he is dying. Second, he wants Romeo to know that he blames the feuding families for his fate. Third, Mercutio wants Romeo to know that he includes him in the insult; after all, it was Romeo's intervention that caused him to sustain his fatal wound.

What are examples of stereotypes in The Elegance of the Hedgehog?

There are certainly some clear stereotypes in the novel. You might be interested in the ones below.

And since it has been written somewhere that concierges are old, ugly, and sour, so has it been branded in fiery letters on the pediment of that same imbecilic firmament that the aforementioned concierges have rather large dithering cats who sleep all day on cushions covered with crocheted cases. 

In Chapter 2, Renee candidly discusses stereotypes. The above quote highlights the stereotypical description of a concierge. With sardonic humor, Renee spends the rest of the chapter reiterating why she fits the stereotype perfectly. She tells us that she is "short, ugly, and plump" and that she has bunions on her feet and terrible breath.
Because of her detached attitude, she has few friends. She admits that she is merely "polite" but "rarely friendly." Here's another example of a stereotype Renee discusses in this chapter:

Similarly, it has been decreed that concierges watch television interminably while their rather large cats doze, and that the entrance to the building must smell of pot-au-feu, cabbage soup, or a country-style cassoulet.

She tells us that she quite happily embodies all the stereotypical traits of the concierge:

I then obligingly flaunt these pauper's victuals . . . because I am a pauper in a house full of rich people and this display nourishes both the consensual cliche and my cat, Leo, who has become rather large by virtue of these meals that should have been mine . . . while I am free—without any olfactory disturbances and without anyone suspecting a thing—to indulge my own culinary proclivities.

It appears that Renee goes to great trouble to create a stereotypical persona that her rich tenants can feel comfortable with. However, there is a reason for her strange behavior. Putting up a facade allows her to indulge her intellectual pursuits and decadent taste for elegant cuisine without recriminations from the wealthy residents in her building. Thus, the stereotypical persona Renee flaunts is actually her attempt at preserving her individuality and personal agency in a class-conscious society.