Summarize the major research findings of "Toward an experimental ecology of human development."

Based on findings of prior research, the author, Bronfenbrenner proposes that methods for natural observation research have been applied in a manner that fails to consider the changing environments that affect human behavior. Observation in one natural setting is not enough to fully understand human behavior. The examples the author offers relate to observation studies of child behavior and development. Bronfenbrenner suggests that researchers cannot fully understand child development within the context of a single setting, with a single set of adults.
The child may behave one way at home with both parents but display different behavior and react differently to each of his/her parents in a setting outside the home, such as at school, at a medical appointment, or in another setting outside the home. The author suggests that age, socioeconomic status, and other demographics may influence how the child responds to the father and to the mother. Bronfenbrenner gives an example of a child from a middle-class family, whose interactions favor the mother in a laboratory setting and the father in a home setting. At the same time, the child from a lower economic class tends to favor the father in the laboratory setting and the mother in the home setting. Therefore, unless the researcher is seeking to answer a single, specific question related to child development and the specific environment, a richer, more robust methodology is needed to make general observations related to child development and parent interactions across all settings. 
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.458.7039&rep=rep1&type=pdf

Single Variable Calculus, Chapter 3, 3.5, Section 3.5, Problem 79

Show that $\displaystyle a(t) = v(t) \frac{dV}{ds}$ of a particle that moves along a straight line with displacement $s(t)$, velocity $v(t)$ and the acceleration $a(t)$. Explain the difference between the meanings of the derivatives $\displaystyle \frac{dv}{dt}$ and $\displaystyle \frac{dv}{ds}$

Recall that $a(t) \displaystyle = \frac{dV}{dt}$ and $v(t) = \displaystyle \frac{ds}{dt}$


$
\begin{equation}
\begin{aligned}

a(t) =& \frac{dV}{dt} = \frac{dV}{ds} \left( \frac{ds}{dt} \right) = \frac{dV}{ds} v
\\
\\
& \text{So,}
\\
\\
a(t) =& v \frac{dV}{ds}
\\
\\


\end{aligned}
\end{equation}
$


$\displaystyle \frac{dV}{dt}$ represents how the velocity changes with respect to time. On the other hand, $\displaystyle \frac{dV}{ds}$ represents how the velocity changes with respect to displacement of a certain particle.

Does the quote "a plague on both your houses" have any significance in the play of Romeo and Juliet?

Mercutio utters this line -- "A plague o' both your houses!" -- after he has been killed by Tybalt.  Tybalt came looking for Romeo in order to challenge him to a fight because he was offended by Romeo's presence at the Capulets' big party the night before.  When Tybalt eventually finds Romeo, Romeo has just come from marrying Juliet -- Tybalt's cousin -- and so Romeo refuses to fight this man who he is now related to, by marriage.  Mercutio, not knowing this, interprets Romeo's refusal to fight as a "dishonorable, vile submission" (3.1.72), and he fights Tybalt on Romeo's behalf.  Romeo comes between them in a well-meaning but ill-advised attempt to stop the fight and promote peace, and Tybalt is able to stab Mercutio under Romeo's arm.  For several reasons, then, Mercutio feels that his death is actually the fault of both Romeo and Tybalt: as he lays dying, he curses both "houses," or families, for their roles in his death.  He curses each family to have some tragedy, and, obviously, both families do experience tragedy when their children -- Romeo and Juliet -- kill themselves rather than live without one another.

int_0^(1/2) arctan(x^2) dx Use a power series to approximate the value of the integral with an error of less than 0.0001.

 From a table of power series, recall that we have:
arctan(x) = sum_(n=0)^oo (-1)^n x^(2n+1)/(2n+1)
To apply this on the given problem, we replace the "x " with "x^2 ".
We get:
arctan(x^2) =sum_(n=0)^oo (-1)^n (x^2)^(2n+1)/(2n+1)
                        =sum_(n=0)^oo (-1)^n x^(2*(2n+1))/(2n+1)
                        =sum_(n=0)^oo (-1)^n x^(4n+2)/(2n+1)
                        = x^2 -x^6/3+x^10/5-x^14/7 +...
The integral becomes:
int_0^(1/2) arctan(x^2)dx = int_0^(1/2) [x^2 -x^6/3+x^10/5- ...]
To determine the indefinite integral, we integrate each term using Power Rule for integration: int x^n dx = x^(n+1)/(n+1) .     
int_0^(1/2) [x^2 -x^6/3+x^10/5-...]
=[x^3/3 -x^7/21+x^11/55-...]_0^(1/2)  
Applying definite integral: F(x)|_a^b = F(b)-F(a) .
F(1/2) or F(0.5)=0.5^3/3 -0.5^7/21+0.5^11/55- ...
                            = 0.0416667 - 0.0003720+0.0000089-...
F(0)=0^3/3 -0^7/21+0^11/55-...
          =0 -0+0 -...     All terms go to zero.
 We stop at the 3rd term since we only need error less than 0.0001 .
int_0^(1/2) arctan(x^2)dx =0.0416667 - 0.0003720+0.0000089
                                = 0.0413036
Thus, int_0^(1/2) arctan(x^2)dx~~0.0413 .

How is the neighborhood described?

The narrator of "Sonny's Blues" describes the neighborhood as "filled with a hidden menace which was its very breath of life." Though the place where he lives is a relatively new housing project, it has become, in his words, "already rundown." The playground is not populated in the daytime with children playing the innocent games of childhood; it is populated at night. Though Baldwin's narrator does not say what goes on there, readers understand that it is far from wholesome.
The narrator observes that the neighborhood where he lives as an adult is not much different from where he and Sonny grew up; it is, he says, as if "I was simply bringing him back into the danger he had almost died trying to escape." The streets of Harlem he describes are filled with subway stations, bars with jukeboxes spilling music into the streets, and neighborhood people gathering outside a barbecue joint to hold an impromptu street revival. The narrator remembers his and Sonny's mother telling their father they should try "to move to a neighborhood that might be safer for the children." He also remembers times when their apartment was filled with friends and neighbors on Sunday afternoons after big dinners when he felt safe and loved, but also aware of the dangers outside that lay waiting.

which process changes sedimentary rock into igneous rock

In the rock cycle, the thing that determines the type of a rock is the way the rock was formed. Igneous rocks are formed by the cooling and solidifying of molten rock (magma or lava), sedimentary rock is formed by deposition of sediments which are either compacted or cemented together, and metamorphic rock is formed when other rocks are subjected to extreme pressure and/or heat that is not enough to actually melt the rock. The heat and/or pressure can deform the rock and also cause different minerals to form, or else change the shape and distribution of existing minerals.
In the determination of whether a rock’s type is igneous, sedimentary, or metamorphic, it doesn’t matter what knd of rock it came from. It only matters how it was made. The rock in the question is igneous rock, so it had to solidify from molten rock. To become molten, the original sedimentary rock had to melt. Once molten, of course, the structure of the previously existing rock would disappear. The process that would transform a sedimentary rock, or any other rock, into an igneous rock, then, is melting or, to be more specific, melting followed by cooling and solidifying.

How is the following quote ironic, and what theme does it show? "We've got to have rules and obey them. After all, we're not savages."

In chapter 2, the boys fail to make a sustainable signal fire, and Ralph mentions that they need to put people in charge of maintaining the fire at all times. Ralph also says that they should have more rules and Jack agrees by saying,

"We've got to have rules and obey them. After all, we're not savages. We're English, and the English are best at everything." (Golding, 33)

Jack's comment is ironic because as the novel progresses, Jack and his band of hunters reveal their savage, primitive nature. Jack and his hunters paint their faces and become bloodthirsty barbarians as they hunt pigs, murder Simon and Piggy, and even attempt to kill Ralph.
After landing on the island, they rapidly descend into savagery and live like uncivilized beings. Jack's comment also emphasizes the theme of civility vs. savagery. As English schoolboys, Jack and the others have been taught that they are inherently civil and superior to other races, ethnicities, and cultures. However, Golding portrays all humans as inherently evil, wicked individuals without the restraints and influence of society to prevent them from acting like savages and indulging in their true desires.