Single Variable Calculus, Chapter 8, 8.2, Section 8.2, Problem 14

Determine the integral $\displaystyle \int^{\pi}_0 \sin^2 t \cos^4 t dt$


$
\begin{equation}
\begin{aligned}

\int^{\pi}_0 \sin^2 t \cos^4 t dt =& \int^{\pi}_0 \sin^2 t (\cos^2 t)^2 dt
\qquad \text{Apply half-angle formulas } \cos 2 t = 1 - 2 \sin^2 t \text{ and } \cos 2t = 2 \cos^2 t - 1
\\
\\
\int^{\pi}_0 \sin^2 t \cos^4 t dt =& \int^{\pi}_0 \left( \frac{1 - \cos 2 t}{2} \right)\left( \frac{\cos 2 t + 1}{2} \right)^2 dt
\\
\\
\int^{\pi}_0 \sin^2 t \cos^4 t dt =& \int^{\pi}_0 \left( \frac{1 - \cos 2t}{2}\right) \left( \frac{\cos^2 2 t + 2 \cos 2t + 1}{4} \right) dt
\\
\\
\int^{\pi}_0 \sin^2 t \cos^4 t dt =& \int^{\pi}_0 \left( \frac{\cos^2 2t + 2 \cos 2 t + 1 - \cos^3 2t - 2 \cos^2 2t - \cos 2 t }{8} \right) dt
\\
\
\int^{\pi}_0 \sin^2 t \cos^4 t dt =& \frac{1}{8} \int^{\pi}_0 (1 + \cos 2t - \cos^2 2t - \cos^3 2 t) dt
\\
\\
\int^{\pi}_0 \sin^2 t \cos^4 t dt =& \frac{1}{8} \int^{\pi}_0 1 dt + \frac{1}{8} \int^{\pi}_0 \cos 2t dt - \frac{1}{8} \int^{\pi}_0 \cos^2 2t dt - \frac{1}{8} \int^{\pi}_0 \cos^3 2t dt


\end{aligned}
\end{equation}
$


Let $u = 2t$, then $du = 2dt$, so $\displaystyle dt = \frac{du}{2}$. When $t = 0, u = 0$ and when $t = \pi, u = 2 \pi$. We integrate the equation term by term

@ 1st term


$
\begin{equation}
\begin{aligned}

\frac{1}{8} \int^{\pi}_0 1 dt =& \frac{1}{8} \int^{2 \pi}_0 1 \cdot \frac{du}{2}
\\
\\
\frac{1}{8} \int^{\pi}_0 1 dt =& \frac{1}{16} \int^{2 \pi}_0 1 du
\\
\\
\frac{1}{8} \int^{\pi}_0 1 dt =& \frac{1}{16} \left[ u \right]^{2 \pi}_0
\\
\\
\frac{1}{8} \int^{\pi}_0 1 dt =& \frac{1}{16} (2 \pi - 0)
\\
\\
\frac{1}{8} \int^{\pi}_0 1 dt =& \frac{2 \pi}{16}
\\
\\
\frac{1}{8} \int^{\pi}_0 1 dt =& \frac{\pi}{8}

\end{aligned}
\end{equation}
$


@ 2nd term


$
\begin{equation}
\begin{aligned}

\frac{1}{8} \int^{\pi}_0 \cos 2t dt =& \frac{1}{8} \int^{2 \pi}_0 \cos u \cdot \frac{du}{2}
\\
\\
\frac{1}{8} \int^{\pi}_0 \cos 2t dt =& \frac{1}{16} \int^{2 \pi}_0 \cos u du
\\
\\
\frac{1}{8} \int^{\pi}_0 \cos 2t dt =& \frac{1}{16} \left[ \sin u \right]^{2 \pi}_0
\\
\\
\frac{1}{8} \int^{\pi}_0 \cos 2t dt =& \frac{1}{16} (\sin 2 \pi - \sin 0)
\\
\\
\frac{1}{8} \int^{\pi}_0 \cos 2t dt =& \frac{1}{16} (0)
\\
\\
\frac{1}{8} \int^{\pi}_0 \cos 2t dt =& 0

\end{aligned}
\end{equation}
$


@ 3rd term


$
\begin{equation}
\begin{aligned}

\frac{1}{8} \int^{\pi}_0 \cos^2 2t dt =& \frac{1}{8} \int^{2 \pi}_0 \cos^2 u \cdot \frac{du}{2}
\\
\\
\frac{1}{8} \int^{\pi}_0 \cos^2 2t dt =& \frac{1}{16} \int^{2 \pi}_0 \cos^2 u du
\qquad \text{Apply half-angle formula } \cos 2 u = 2 \cos^2 u - 1
\\
\\
\frac{1}{8} \int^{\pi}_0 \cos^2 2t dt =& \frac{1}{16} \int^{2 \pi}_0 \left(\frac{\cos 2 u + 1}{2} \right) du
\\
\\
\frac{1}{8} \int^{\pi}_0 \cos^2 2t dt =& \frac{1}{32} \int^{2 \pi}_0 (\cos 2u + 1) du

\end{aligned}
\end{equation}
$


Let $v = 2u$, then $dv = 2 du$, so $\displaystyle du = \frac{dv}{2}$. When $u = 0, v = 0$ and when $u = 2 \pi, v = 4 \pi$


$
\begin{equation}
\begin{aligned}

\frac{1}{32} \int^{32}_0 (\cos 2u + 1) du =& \frac{1}{32} \int^{4 \pi}_0 (\cos v + 1) \cdot \frac{dv}{2}
\\
\\
\frac{1}{32} \int^{32}_0 (\cos 2u + 1) du =& \frac{1}{64} \int^{4 \pi}_0 (\cos v + 1) dv
\\
\\
\frac{1}{32} \int^{32}_0 (\cos 2u + 1) du =& \frac{1 }{64} \left[ \sin v + v \right]^{4 \pi}_0
\\
\\
\frac{1}{32} \int^{32}_0 (\cos 2u + 1) du =& \frac{1}{64} (\sin 4 \pi + 4 \pi - \sin 0 - 0)
\\
\\
\frac{1}{32} \int^{32}_0 (\cos 2u + 1) du =& \frac{1}{64} (0 + 4 \pi - 0 - 0)
\\
\\
\frac{1}{32} \int^{32}_0 (\cos 2u + 1) du =& \frac{4 \pi}{64}
\\
\\
\frac{1}{32} \int^{32}_0 (\cos 2u + 1) du =& \frac{\pi}{16}

\end{aligned}
\end{equation}
$


@ 4th term


$
\begin{equation}
\begin{aligned}

\frac{1}{8} \int^{\pi}_0 \cos^3 2t dt =& \frac{1}{8} \int^{2 \pi}_0 \cos^3 u \cdot \frac{du}{2}
\\
\\
\frac{1}{8} \int^{\pi}_0 \cos^3 2t dt =& \frac{1}{16} \int^{2 \pi}_0 \cos^3 u du
\\
\\
\frac{1}{8} \int^{\pi}_0 \cos^3 2t dt =& \frac{1}{16} \int^{2 \pi}_0 (\cos^2 u)(\cos u) du
\qquad \text{Apply Trigonometric Identities } \cos^2 u + \sin^2 u = 1
\\
\\
\frac{1}{8} \int^{\pi}_0 \cos^3 2t dt =& \frac{1}{16} \int^{2 \pi}_0 (1 - \sin^2 u)(\cos u) du

\end{aligned}
\end{equation}
$


Let $v = \sin u$, then $dv = \cos u du$. When $u = 0, v = 0$ and when $u = 2 \pi, v = 0$. Therefore,


$
\begin{equation}
\begin{aligned}

\frac{1}{16} \int^{2 \pi}_0 (1 - \sin^2 u)(\cos u du) =& \frac{1}{16} \int^0_0 (1 - v^2) dv
\\
\\
\frac{1}{16} \int^{2 \pi}_0 (1 - \sin^2 u)(\cos u du) =& \frac{1}{16} \left[ v - \frac{v^3}{3} \right]^0_0
\\
\\
\frac{1}{16} \int^{2 \pi}_0 (1 - \sin^2 u)(\cos u du) =& \frac{1}{16} (0)
\\
\\
\frac{1}{16} \int^{2 \pi}_0 (1 - \sin^2 u)(\cos u du) =& 0

\end{aligned}
\end{equation}
$


Combine the results of integration term by term


$
\begin{equation}
\begin{aligned}

\int^{\pi}_0 \sin^2 t \cos^4 t dt =& \frac{\pi}{8} + 0 - \frac{\pi}{16} - 0
\\
\\
\int^{\pi}_0 \sin^2 t \cos^4 t dt =& \frac{2 \pi + 0 - \pi - 0}{16}
\\
\\
\int^{\pi}_0 \sin^2 t \cos^4 t dt =& \frac{\pi}{16}
\end{aligned}
\end{equation}
$

College Algebra, Chapter 3, 3.5, Section 3.5, Problem 50

Below are the graphs of $f$ and $g$. Find a formula for the function $g$.







Based from the graph, the graph for the function $g$ can be obtained by shifting the graph of $f(x) = x^2$ two units to the right, then reflect about $x$-axis and then the result is shifted one unit upward. Thus,


$
\begin{equation}
\begin{aligned}

a(x) =& (x - 2)^2
&& \text{shifting two units to the right}
\\
\\
b(x) =& - (x - 2)^2
&& \text{reflecting about $x$-axis}
\\
\\
g(x) =& 1 - (x - 2)^2
&& \text{shift 1 unit upward}

\end{aligned}
\end{equation}
$

What type of compounds conduct heat?

For conduction of heat (or electricity), free carrier electrons are required. Generally, ionic compounds are better conductors of heat (and electricity) when in molten form or when dissolved in the water. In comparison, covalent compounds are generally bad conductors of heat (graphite is an exception). In the case of ionic compounds, in the molten state or when dissolved in water, free carriers (electrons) are available for conducting heat. In case of covalent compounds, the electrons are exchanged between the constituents, and hence, free electrons are not available for conducting heat (or electricity). Thus, the lack of free carriers causes covalent compounds to be (generally) insulators—the opposite of conductors. Ionic compounds such as sodium chloride (NaCl), on the other hand, readily dissolve in water and form ions (sodium and chloride, in this case) that can allow the solution to conduct heat (and electricity).
Metals are also good conductors of heat due to the availability of electrons that can vibrate and also move around. When one end of a metal is heated, the electrons in the hot region gain more kinetic energy and bump into the neighboring electrons, thus transferring the kinetic energy. This ensures that heat is conducted from the hot end to the colder end of the metal. Metals such as copper, silver, and aluminum are good conductors of heat.
Hope this helps.

What aspects of Things Fall Apart relate to intercultural theory, postcolonial theory, and/or theories of language contact and development?

Chinua Achebe’s 1958 novel Things Fall Apart explores the complex Igbo culture of Nigeria in the wake of European colonialism. One of Achebe’s stated purposes of the novel was to introduce the complexities and richness of African culture to Western readers who often perceive African society as primitive or backward. Things Fall Apart explores the culture clash that occurs through lack of social interaction and misunderstandings between native Africans and Western society. These misunderstandings are directly tied to the legacy of colonialism in Africa. Language similarly intersects with culture and post-colonialism because it reflects cultural points of view and is also used as a tool of control.
Achebe also wrote this novel with his Nigerian people in mind. He said he wished to “help my society regain belief in itself and put away the complexes of the years of denigration and self-abasement.” This purpose directly relates to the scars left by colonialism. The characters Reverend James Smith and the unnamed District Commissioner reveal aspects of intercultural theory and Africa’s postcolonial struggles. Smith disregards African culture and religion, rejecting the idea that Nigerians retain elements of their native heritage. The District Commissioner has a patronizing attitude toward native tribes, seeing himself as a guide to bring them into the modern era. Achebe, however, depicts African culture as a complex system of religion, government, economics, arts, and justice. He also sought to portray a realistic account of Africa’s precolonial past, an account free of the typical stereotypes and distortions found in Western depictions. This novel does not, however, idealize the Igbo’s past or present culture as perfect. For example, the Africans criticize the Christians as being foolish. Achebe believes his people also need to realign their perceptions of themselves and Westerners.
Regarding language, Achebe chose to write this novel in English in order to reach a wide audience. However, he honors the Igbo language by including Igbo words, proverbs, metaphors, speech rhythms, and cultural ideas. He incorporates these elements seamlessly into the English text to bridge the linguistic and cultural divide his narrative centers around.     

What can be inferred about the speaker's life and values from the way he imagines himself described in lines 98-108?

In Gray's famous "Elegy", the speaker is a dreamer, a man who wishes to live as the rustic souls buried in the churchyard did, "far from the madding crowd." Most of the poem is concerned with establishing that the humble, seemingly unimportant people buried here had at least as much value in life as those who achieved glory and fame in the world. Gray says that "Their lot forbade, nor circumscribed alone/Their growing virtues, but their crimes confined." This implies that a poor farmer or laborer is a greater human being, in his way, than a monarch or politician who would "wade through slaughter to a throne,/And shut the gates of mercy on mankind."
Gray's poem is a subdued expression of Enlightenment philosophy and the growing ideal of the equality of all people. Alexander Pope had put forward the same thought, though extended beyond humanity to all of creation, in his Essay on Man, in the lines "Who sees with equal eye as God of all,/A hero perish or a sparrow fall,/Atoms or systems into ruin hurled,/And now a bubble burst. and now a world." Gray, however, goes further, in celebrating the special qualities of the poor and dispossessed which place them above the fray.
Though it's dangerous to read biography into poetry, Gray's sympathy for the nameless ones buried in the churchyard was possibly due to his having seen himself as an outsider throughout his life. He probably was gay, and early in his life was devastated by the death of his friend Richard West. The "Elegy" is a masterpiece that represents a major step in the foreshadowing of the Romantic period.

What does the word "clipped" mean in this poem?

Maya Angelou's poem "Caged Bird" is full of avian metaphors and imagery. The poem itself is a metaphor for the limitations one experiences in a life of oppression. "Caged Bird" draws from Angelou's own experiences as a Black woman in the racially-segregated United States following the Civil War and even beyond the Civil Rights Movement. To this day, many Black Americans face limitations based on a systemic cycle of racial oppression which prevents class mobility. 
In talking of birds, "clipping" involves trimming a bird's wing feathers so that they cannot fly. Some bird owners or caretakers trim just one wing or enough feathers on each side, so as to render the bird unstable in flight but leaving them able to glide for a short distance. In Angelou's poem, she uses "clipped" as a metaphor for the systemic forms of oppression I have mentioned above. Being "clipped" in society on the basis of race (or other identities) prevents an individual from ever testing their capability for success. Historically, Black Americans have been denied access to schooling and certain kinds of work, and even today it is not uncommon for Black Americans to be turned down for jobs on the basis of their appearance and a failure to assimilate to a white-dominated work environment. To be "clipped," as Angelou implies, is to never be given a chance for success in life. 
https://www.poetryfoundation.org/poems/48989/caged-bird

What is the relationship between inertia and mass?

It has been said that mass is a measurement of inertia.
We can illustrate this mathematically using force, momentum, and even energy. Momentum describes how impacts and collisions will affect other objects. The equation for momentum is p=mv, where p is the momentum of an object, m is mass [a measurement of the inertia] and v is velocity. If two objects collide at equal speed, but one is twice as massive, then the two objects will move in the direction of the more massive object, at a slower speed. You can try this with marbles and a table. 
Force is defined as the rate of change of momentum, or F=d/[dt]p . This means that mass is changing how forces affect objects. It will take a greater force, for example, to move a bowling ball than a feather.
Energy is the integral of force, or E=int_()Fdx . This nest egg of formulas means that once again, mass is playing a role. Here, if two objects are moving with equal velocity, but have different masses, the one with greater mass will have greater energy. Think of how it feels to get hit by an RC car at five miles an hour versus a 200 pound linebacker at five miles an hour.
Based on all of this, it would seem that the mass of an object, in any unit, is really just a quantity of inertia.

Single Variable Calculus, Chapter 3, 3.2, Section 3.2, Problem 44

Find $f'(x)$ and $f''(x)$ on the function $f(x) = \displaystyle \frac{1}{x}$ using the definition of a derivative. Then graph $f, f'$ and $f''$ on a common screen and check to see if your answers are reasonable.

Using the definition of derivative

Using the definition of derivative


$
\begin{equation}
\begin{aligned}

\qquad f'(x) =& \lim_{h \to 0} \frac{f(x + h) = f(x)}{h}
&&
\\
\\
\qquad f'(x) =& \lim_{h \to 0} \frac{\displaystyle \frac{1}{x + h} - \frac{1}{x}}{h}
&& \text{Substitute $f(x + h)$ and $f(x)$}
\\
\\
\qquad f'(x) =& \lim_{h \to 0} \frac{x - (x + h)}{(h)(x) (x + h)}
&& \text{Get the LCD on the numerator and simplify}
\\
\\
\qquad f'(x) =& \lim_{h \to 0} \frac{\cancel{x} - \cancel{x} - h}{(h)(x)(x + h)}
&& \text{Combine like terms}
\\
\\
\qquad f'(x) =& \lim_{h \to 0} \frac{-\cancel{h}}{\cancel{(h)} (x)(x + h)}
&& \text{Cancel out like terms}
\\
\\
f'(x) =& \lim_{h \to 0} \left[ \frac{-1}{(x)(x + h)} \right] = \frac{-1}{(x)(x + 0)} = \frac{-1}{(x)(x)}
&& \text{Evaluate the limit}
\\
\\
f'(x) =& \frac{-1}{x^2}
&&

\end{aligned}
\end{equation}
$


Using the 2nd derivative of the definition


$
\begin{equation}
\begin{aligned}

\qquad f''(x) =& \lim_{h \to 0} \frac{f'(x + h) = f'(x)}{h}
&&
\\
\\
\qquad f''(x) =& \lim_{h \to 0} \frac{\displaystyle \frac{-1}{(x + h)^2} - \left( \frac{-1}{x^2} \right)}{h}
&& \text{Substitute } f'(x + h) \text{ and } f'(x)
\\
\\
\qquad f''(x) =& \lim_{h \to 0} \frac{-x^2 + (x + h)^2}{(h)(x^2)(x + h)^2}
&& \text{Get the LCD on the numerator and simplify}
\\
\\
\qquad f''(x) =& \lim_{h \to 0} \frac{-x^2 + x^2 + 2xh + h^2}{(h)(x^2)(x + h)^2}
&& \text{Expand the equation}
\\
\\
\qquad f''(x) =& \lim_{h \to 0} \frac{\cancel{-x^2} + \cancel{x^2} + 2xh + h^2}{(h)(x^2)(x + h)^2}
&& \text{Combine like terms}
\\
\\
\qquad f''(x) =& \lim_{h \to 0} \frac{2xh + h^2}{(h)(x^2)(x + h)^2}
&& \text{Factor the numerator}
\\
\\
\qquad f''(x) =& \lim_{h \to 0} \frac{\cancel{h}(2x + h)}{\cancel{(h)}(x^2)(x + h)^2}
&& \text{Cancel out like terms}
\\
\\
\qquad f''(x) =& \lim_{h \to 0} \left[ \frac{2x + h}{(x^2)(x + h)^2} \right] = \frac{2x + 0}{(x^2)(x + 0)^2} = \frac{2x}{(x^2)(x^2)}
&& \text{Evaluate the limit}
\\
\\
\qquad f''(x) =& \frac{2x}{x^4}

\end{aligned}
\end{equation}
$


Graph $f, f'$ and $f''$

Calculus of a Single Variable, Chapter 8, 8.4, Section 8.4, Problem 25

Given ,
int sqrt(1-x^2)/x^4 dx
By applying Integration by parts we can solve the given integral
so,
let u= sqrt(1-x^2) ,v' = (1/x^4)
=> u' = (sqrt(1-x^2) )'
=> =d/dx (sqrt(1-x^2))
let t=1-x^2
so,
d/dx (sqrt(1-x^2))
=d/dx (sqrt(t))
= d/(dt) sqrt(t) * d/dx (t) [as d/dx f(t) = d/(dt) f(t) (dt)/dx ]
= [(1/2)t^((1/2)-1) ]*(d/dx (1-x^2))
= [(1/2)t^(-1/2)]*(-2x)
=[1/(2sqrt(1-x^2 ))]*(-2x)
=-x/sqrt(1-x^2)
so, u' = -x/sqrt(1-x^2) and as v'=(1/x^4) so
v = int 1/x^4 dx
= int x^(-4) dx
= (x^(-4+1))/(-4+1)
=(x^(-3))/(-3)
= -(1/(3x^3))

so , let us see the values altogether.
u= sqrt(1-x^2) ,u' = -x/sqrt(1-x^2) and v' = (1/x^4) ,v=-(1/(3x^3))

Now ,by applying the integration by parts int uv' is given as
int uv' = uv - int u'v
then,
int sqrt(1-x^2)/x^4 dx
= (sqrt(1-x^2)) (-(1/(3x^3))) - int (-x/sqrt(1-x^2))(-(1/(3x^3))) dx
= (sqrt(1-x^2)) (-(1/(3x^3))) -(- int (-x/sqrt(1-x^2))((1/(3x^3))) dx)
= (sqrt(1-x^2)) (-(1/(3x^3))) - int (x/sqrt(1-x^2))((1/(3x^3))) dx
= -(sqrt(1-x^2)) ((1/(3x^3))) - int (x/sqrt(1-x^2))((1/(3x^3))) dx-----(1)

Now let us solve ,
int (x/sqrt(1-x^2))((1/(3x^3))) dx
=int (1/sqrt(1-x^2))((1/(3x^2))) dx
=(1/3)int (1/sqrt(1-x^2))((1/(x^2))) dx
=(1/3)int (1/((x^2)sqrt(1-x^2))) dx
This integral can be solve by using the Trigonometric substitution(Trig substitution)

when the integrals containing sqrt(a-bx^2)then we have to take x=sqrt(a/b) sin(t)to solve the integral easily

so here , For
(1/3)int (1/((x^2)sqrt(1-x^2))) dx------(2)
x is given as
x= sqrt(1/1) sin(t) = sin(t)
as x= sin(t)
=> dx= cos(t) dt
now substituting the value of x in (2) we get
(1/3)int (1/((x^2)sqrt(1-x^2))) dx
=(1/3)int (1/(((sin(t))^2)sqrt(1-(sin(t))^2))) (cos(t) dt)
= (1/3)int (1/(((sin(t))^2)sqrt(cos(t))^2))) (cos(t) dt)
=(1/3)int (1/(((sin(t))^2)*(cos(t)))) (cos(t) dt)
=(1/3)int 1/(((sin(t))^2)) dt
=(1/3)int (csc(t))^2 dt
= (-1/3) cot(t) +c
= (-1/3) cot(arcsin(x)) +c ---(3)
[since x= sin(t) => t= arcsin(x)]

Now substituting (3) in (1) we get
(1) =>
-(sqrt(1-x^2)) ((1/(3x^3))) - int (x/sqrt(1-x^2))((1/(3x^3))) dx
=-(sqrt(1-x^2)) ((1/(3x^3))) - ((-1/3) cot(arcsin(x)) +c)
=-(sqrt(1-x^2)) ((1/(3x^3)))+(1/3) cot(arcsin(x)) +c
=(1/3) cot(arcsin(x))- (((sqrt(1-x^2))/(3x^3))) +c----(4)
cot(t) in terms of sin(t) can be given as follows
cot(t) = cos(t)/sin(t) = sqrt(1-(sin(t))^2)/sin(t)
so,
cot(arcsin(x)) = sqrt(1-(sin(arcsin(x)))^2)/sin(arcsin(x)) = sqrt(1-x^2)/x
substituting the above in (4) we get
(1/3) cot(arcsin(x))- (((sqrt(1-x^2))/(3x^3))) +c
=(1/3) (sqrt(1-x^2)/x)- (((sqrt(1-x^2))/(3x^3))) +c
=(sqrt(1-x^2)/(3x))- (((sqrt(1-x^2))/(3x^3))) +c
so,
int sqrt(1-x^2)/x^4 dx
=(sqrt(1-x^2)/(3x))- sqrt(1-x^2)/(3x^3)+c

Precalculus, Chapter 5, 5.4, Section 5.4, Problem 19

sin((13pi)/12)=sin(pi/2+pi/3+pi/4)
As we know that sin(pi/2+theta)=cos(theta)
:.sin((13pi)/12)=cos(pi/3+pi/4)
Now use the identity cos(x+y)=cos(x)cos(y)-sin(x)sin(y)
sin((13pi)/12)=cos(pi/3)cos(pi/4)-sin(pi/3)sin(pi/4)
sin((13pi)/12)=((1/2)(1/sqrt(2))-((sqrt(3)/2)(1/sqrt(2)))
sin((13pi)/12)=(1-sqrt(3))/(2sqrt(2))
sin((13pi)/12)=(sqrt(2)-sqrt(6))/4
cos((13pi)/12)=cos(pi/2+pi/3+pi/4)
We know that cos(pi/2+theta)=-sin(theta)
:.cos((13pi)/12)=-sin(pi/3+pi/4)
using identity sin(x+y)=sin(x)cos(y)+cos(x)sin(y)
cos((13pi)/12)=-(sin(pi/3)cos(pi/4)+cos(pi/3)sin(pi/4))
=-(sqrt(3)/2*1/sqrt(2)+1/2*1/sqrt(2))
=-(sqrt(3)+1)/(2sqrt(2))
=-(sqrt(6)+sqrt(2))/4
tan((13pi)/12)=sin((13pi)/12)/cos((13pi)/12)
plug in the values evaluated above,
tan((13pi)/12)=((sqrt(2)-sqrt(6))/4)/(-(sqrt(6)+sqrt(2))/4)
=(sqrt(2)-sqrt(6))/(-(sqrt(6)+sqrt(2)))
rationalizing the denominator,
=((sqrt(2)-sqrt(6))(sqrt(6)-sqrt(2)))/(-(6-2))
=(sqrt(12)-2-6+sqrt(12))/(-4)
=(2sqrt(12)-8)/(-4)
=(2*2sqrt(3)-8)/(-4)
=2-sqrt(3)

College Algebra, Chapter 4, 4.5, Section 4.5, Problem 56

Determine all the zeros of the polynomial $P(x) = x^5 + x^3 + 8x^2 + 8 $.
To find the zeros, we set $x^5 + x^3 + 8x^2 + 8 = 0$. Then,

$
\begin{equation}
\begin{aligned}
(x^5 + x^3) + (8x^2 + 8) &= 0 && \text{Group terms}\\
\\
x^3(x^2+1)+8(x^2+1) &= 0 && \text{Factor out $x^3$ and $8$}\\
\\
(x^2+1)(x^3+8) &= 0 && \text{Factor out } x^2 + 1
\end{aligned}
\end{equation}
$

If $(x^2 + 1) = 0$, then $x = \pm i$. To solve the remaining zeros of $P$, we use synthetic division to simplify $(x^3+8) = 0$. So by trial and error,
Again, by using synthetic division



Thus,

$
\begin{equation}
\begin{aligned}
P(x) &= x^5 + x^3 + 8x^2 + 8 \\
\\
&= (x^2+1)(x^3+8)\\
\\
&= (x^2+1)(x+2)(x^2-2x+4)
\end{aligned}
\end{equation}
$


Then, by using quadratic formula...

$
\begin{equation}
\begin{aligned}
x &= \frac{-(-2)\pm \sqrt{(-2)^2 - 4(1)(4)}}{2(1)}\\
\\
&= \frac{2 \pm \sqrt{-12}}{2} = 1 \pm \sqrt{3}i
\end{aligned}
\end{equation}
$

Thus, the zeros of $P$ are $-2, i, -i, 1 + \sqrt{3}i$ and $1-\sqrt{3}i$

Compare the presidencies of Roosevelt, Taft, and Wilson. What made them Progressive presidents? Identify what you believe to be the most important pieces of legislation passed during each administration. Why are these so significant? Finally, be sure to indicate what each president did to expand the meaning of freedom for Americans.

Roosevelt, Taft, and Wilson shared a belief that the federal government, and the president more specifically, should play a role in improving the lives of the American people. They were especially concerned with curbing the power of monopolies, which they saw as potentially dangerous. Roosevelt earned a reputation as a "trustbuster" by using the powers conferred on the federal government by the Sherman Antitrust Act to break up the Northern Securities company, a railroad titan. He also signed the Hepburn Act, which boosted the power of the Interstate Commerce Commission to regulate the railroads. He signed the Pure Food and Drug Act as part of a campaign to regulate the mass production of foods, another move against powerful businesses that controlled the packing and canning industries. William Howard Taft continued to break up trusts, empowering the ICC to battle the railroad companies in particular. Though he took on powerful trusts, including the Standard Oil Trust, he was less ideologically committed to restraining the monopolies than Roosevelt and, unlike his predecessor, was sensitive to criticism from business leaders. Still, his record on this crucial Progressive reform arguably surpasses that of Roosevelt. Wilson also took direct aim at big business, attacking the trusts, and, unlike Taft, signed a significantly lower tariff, long a cornerstone of Progressive policy. Wilson pushed for the Federal Reserve, the centralized organization that would eventually control the nation's monetary policy, and promoted and signed federal law to establish shorter hours and better conditions for workers.
https://millercenter.org/president/roosevelt/domestic-affairs

https://millercenter.org/president/taft/domestic-affairs

https://millercenter.org/president/wilson/domestic-affairs

The presidencies of Roosevelt, Taft, and Wilson are termed progressive because of the focus and emphasis they had on empowering the American masses. These presidents were against trusts that took undue advantage of the country’s economy and adversely influenced the nation’s politics to fuel business interests. They were also pro-conservation and supported stronger environmental conservation policies. Additionally, they led governments that were focused on protecting the rights of the consumers and workers from the exploitative, giant corporations.
The Elkins Act and the Hepburn Act were important pieces of legislation passed during the Roosevelt administration. The laws were an onslaught on railroad trusts and companies engaged in unfair practices. The Elkins Act granted authority to the Interstate Commerce Commission (ICC) to punish railroad companies that offered rebates and businesses that accepted these rebates. The Hepburn Act also strengthened the ICC by granting it the authority to scrutinize financial statements of railway companies and setting maximum railroad rates. The laws ensured that large corporations did not enjoy undue advantage and made it possible for other small businesses to engage and compete in trade.
Taft signed the first piece of legislation that required employers to cover their employees engaged in interstate trade. The laws were aimed at protecting the workers and ensuring they were compensated when they suffered injuries in the line of duty. The legislation affirmed citizens’ freedom and rights to work in different states under the protection of their employers and the government.
The Clayton Antitrust laws were passed during the Wilson administration and sought to strengthen the Sherman Antitrust laws by making additional clarifications aimed at regulating big business. These laws dealt with exploitative pricing, monopolistic mergers, and exclusive business engagements. These practices were detrimental to the existence of smaller businesses and exposed the consumer to exploitation by large corporations. The new laws ensured that the people enjoyed products and services that were fairly priced.
https://millercenter.org/president/william-taft/key-events

https://millercenter.org/president/wilson/domestic-affairs

https://www.ushistory.org/us/43b.asp

Presidents Roosevelt, Taft, and Wilson are considered progressive presidents. Each president did several things to further the cause of the Progressives.
President Roosevelt took several actions and had several laws passed while he was in office. He wasn’t afraid to go after businesses that were acting unfairly or that were harming the public’s interests. He went after the Northern Securities Company when its actions in a stock battle nearly caused a financial panic. He also threatened to intervene in the coal strike of 1902 when the owners of the coal mines refused to negotiate. There was concern that there wouldn’t be enough coal to heat homes in the winter. Several laws were passed to protect consumers while Theodore Roosevelt was president. The Meat Inspection Act required federal inspection of the meat industry. Cleanliness standards were established to help try to ensure that the meat was being prepared in clean conditions. The Pure Food and Drug Act made it illegal to falsely label food and medicines. These actions and laws helped Americans know that their interests were being protected, which can be interpreted as a form of freedom.
President Taft also did several things that are considered progressive. President Taft broke up more trusts than President Roosevelt did. He also established the Children’s Bureau to investigate problems with child labor. The Mann-Elkins Act gave the federal government more power to regulate businesses. The Bureau of Mines was created to monitor the actions of the mining companies. These actions also helped people believe the federal government was on their side and would help them if abusive conditions developed in the workplace, especially regarding children.
For President Wilson, there were some progressive changes, as well. The Underwood Tariff reduced tariffs. The Federal Reserve Act brought stability to the banking industry. The Federal Reserve Board would set banking policy. The decisions of the Federal Reserve Board would also impact the economy since the board could change interest rates. The Federal Trade Commission could require businesses to stop unfair business practices. The Clayton Antitrust Act allowed unions to exist and prevented businesses from charging different prices to different people. The Adamson Act created an eight-hour day for railroad workers. The Keating-Owen Child Labor Act made it illegal to hire kids under the age of fourteen in certain industries. These laws also reassured Americans that the government was working to protect their interests.
These actions by these presidents helped free Americans from the fear that they were powerless and that the government would not work to protect their interests.
https://millercenter.org/president/roosevelt/impact-and-legacy

https://millercenter.org/president/wilson/impact-and-legacy

https://www.whitehouse.gov/about-the-white-house/presidents/william-howard-taft/

In the poem "The Demon Lover," the poet clearly shows how a woman's desire for wealth ultimately ends in misery. Explain.

In the poem, the woman initially declines her former lover's invitation to fulfill a previous marriage vow. She tells him that she is now married and has two children; therefore, she can't go with him.
The woman's former lover is devastated and maintains that he only returned for the hope of claiming her. Upon hearing this, the woman poses a hypothetical question: if she decides to leave her husband and children, what will her former lover give her? Can he give her anything more valuable than her husband and children?
Her former lover surprises her by saying that he currently has seven ships on the sea; additionally, he was brought to land on the eighth. Upon hearing this, the woman immediately says goodbye to her two children.
Her decision made, the woman steps foot on her former lover's ship and delights in the fact that the sails are made of taffeta and the masts, of beaten gold. Before long, however, her former lover's facial expression changes ("dismal grew his countenance"). Soon, the woman spies her lover's "cloven foot." Here, the cloven foot is a reference to the Devil; the implication is that the woman has been lured to her damnation by him.
The devilish identity of the former lover is confirmed when he tells the woman:

"Oh, yon are the hills of heaven," he said, "Where you will never win."
Oh, yon is the mountain of hell," he cried, "Where you and I will go."

The last stanza reiterates the theme that a woman's consuming desire for wealth will ultimately end in misery.

He struck the top-mast with his hand,/The fore-mast with his knee;/And he broke that gallant ship in twain,/And sank her in the sea.

 
 

In the Friar's opening soliloquy in Romeo and Juliet, what does he say about the flower that he inspects?

In Romeo and Juliet, when Friar Laurence is introduced, he is holding a basket in his hand (into which he places his medicinal herbs), and this action establishes a core aspect to his characterization and his role in the within the play. Indeed, the dual nature by which these plants an herbs can simultaneously serve as curative and poison is a key theme across the soliloquy, along with the dualistic nature of life and death, all of which are significant themes within the larger arc of Romeo and Juliet.
While inspecting the flower, he states:

"Within the infant rind of this small flower / Poison hath residence, and medicine power: / For this, being smelt, with that part cheers each part; / Being tasted, slays all senses with the heart. / Two such opposed kings encamp them still / In man as well as herbs—grace and rude will/ And where the worser is predominant, / Full soon the canker death eats up that plant."

However, these lines intertwine with larger themes within the play and within the soliloquy itself. This example illustrates a far larger pattern. Earlier, as part of that same soliloquy, he says:

"I must up-fill this osier cage of ours / With baleful weeds and precious-juiced flowers. / The earth, that's nature's mother, is her tomb; / What is her burying gave, that is her womb."

There is illustrated here a kind of dualism about life and death, poison and curative, one which is present at the heart of medicine and at the heart of nature itself. Laurence himself is well aware of this tension, and his example of the plant serves to illustrate that tension, through which the beneficial and harmful properties exist side-by-side. In so doing, he is also foreshadowing later events in the story, and the role he himself and his medicinal knowledge will play within the tragedy of Romeo and Juliet.

When we first encounter Friar Lawrence in Act II Scene iii, he's gathering herbs and flowers into a basket, with which he intends to make medicine. He makes no distinction between beautiful flowers and weeds; they're both equally effective in terms of their medicinal properties. Shakespeare is using a metaphor here; the variety of flowers picked by Friar Lawrence represents both the good and bad elements that coexist in human nature. Just after Romeo enters, Friar Lawrence inspects another flower, one that illustrates the point:

Within the infant rind of this small flower Poison hath residence and medicine power. For this, being smelt, with that part cheers each part; Being tasted, stays all senses with the heart. Two such opposèd kings encamp them still, In man as well as herbs—grace and rude will.

Inside the little flower there is both good and bad, medicine and poison. It has a lovely smell, but if you try to eat it, you'll die. There are two opposing elements in everything: human beings, as well as flowers.

int root(3)(x)/(root(3)(x) - 1) dx Find the indefinite integral by u substitution. (let u be the denominator of the integral)

Solving indefinite integral by u-substitution, we follow:
int f(g(x))*g'(x) = int f(u) *du where we let u = g(x) .
By following the instruction to let "u" be the denominator  of the integral,
 it means we let: u = root(3)(x) -1
Find the derivative of "u" which is du = 1/(3x^(2/3))dx
 Then du =1/(3x^(2/3))dx can be rearrange into 3x^(2/3)du =dx .
Applying u-substitution using u =root(3)(x)-1 and 3x^(2/3)du =dx .
int root(3)(x)/(root(3)(x)-1) dx = int root(3)(x)/u*3x^(2/3)du
                    = int (x^(1/3)*3x^(2/3))/udu
                    =int (3x^(1/3+2/3))/udu
                  =3 int x/udu
Note: x^(1/3+2/3) = x^(3/3)
                           =x^1 or x
Algebraic techniques:
From u = root(3)(x)-1 , we can rearrange it into root(3)(x)=u+1 .
Raising both sides by a power 3:
(root(3)(x))^3 =(u+1)^3
x = (u+1)*(u+1)*(u+1)
By FOIL: (u+1)*(u+1) = u*u +u*1+1*u+1*1
                                = u^2+u+u+1
                                 = u^2+2u+1
Then let (u+1)(u+1) = u^2 +2u +1 in (u+1)(u+1)(u+1) :
(u+1)(u+1)(u+1) = (u+1)*(u^2+2u+1)
Applying distributive property:
(u+1)(u^2+2u+1) = u *(u^2+2u+1) + 1*(u^2+2u+1)
                           = u^3 +2u^2+u +u^2+2u+1
                           =u^3+3u^2+3u+1
 then x = (u+1)*(u+1)*(u+1) is the same as 
x =u^3+3u^2+3u+1
 Substitute x=u^3+3u^2+3u+1 in 3 int x/udu :
3 int x/udu = 3 int (u^3+3u^2+3u+1 )/u du
                  = 3int (u^3/u+(3u^2)/u+(3u)/u+1/u) du
                 =3int (u^2+3u+3+1/u) du
Evaluating each term in separate integral:
3 * [ int u^2 *du+ int 3u*du+int 3*du+ int 1/u du]
where: 
int u^2 *du = u^3/3
int 3u*du =(3u^2)/2
int 3*du = 3u
int 1/u du= ln|u|
3 * [ int u^2 *du+ int 3u*du+int 3*du+ int 1/u du] becomes:
3*[u^3/3 +(3u^2)/2 +3u+ln|u|] +C= 3u^3/3 +(9u^2)/2 +9u+3ln|u|+C
Substitute u = root(3)(x)-1:
3u^3/3 +(9u^2)/2 +9u+3ln|u| +C = (root(3)(x)-1)^3 +(9(root(3)(x)-1)^2)/2 +9(root(3)(x)-1)+3ln|(root(3)(x)-1)| +C
 
 
 
 
 
 

Describe and analyze Mao’s philosophy of revolution. Was it applicable to China? Would it have been applicable to Britain or the United States?

In the 1920s, the Soviets provided support to organize urban workers in China’s wealthier coastal cities, but these efforts were violently crushed. Mao Zedong, by contrast, wanted to unite the country’s poverty-stricken peasant population and began to enact his plan in the 1930s. His view of revolution differed from what had occurred in Russia’s communist revolution, where peasants were viewed as secondary in importance to workers in industry.
Mao’s decision was critical to the ultimate success of the Chinese Communist Party. It’s crucial to understand that China is an extremely large country, and in the early twentieth century, it had been divided into spheres of influence by Western powers and Japan. Any effort to start a communist revolution would have to unify hundreds of millions of people. By focusing on the rural population, he was concentrating his efforts on the most people. In 1930, China's population was roughly 500 million, and it was overwhelmingly rural. By working to improve peasants' quality of life, Mao won them to the communist cause and ultimately prevailed in establishing a unified communist country in 1949.
It's doubtful that such a strategy would have been successful in the US or Britain. When Mao began his revolution in the 1930s, both the US and Britain were already industrialized countries that had higher urban populations than rural. Additionally, both the US and Britain had well-established centralized governments that likely would have been more effective at resisting a revolution. Mao had the luxury of operating in a fragmented country with no centralized government.
http://afe.easia.columbia.edu/special/china_1900_farmers.htm

What are the similarities between Harris and J.'s Uncle Podger?

Like J's Uncle Podger, Harris is a brash man who has an over-inflated sense of his own importance. He tends to assume a stance of pompous confidence when faced with challenges. Both Uncle Podger and Harris presume that their wisdom, analytical abilities, and powers of observation are equal to the difficulties before them. However, they are far less talented than they imagine themselves to be. Both men invariably end up needing the assistance of others.
In the story, Uncle Podger tries to hang up a picture frame for Aunt Podger. He assures her that she won't have to worry: after all, he's going to do it. Before long, however, Uncle Podger wrecks chaos on the whole household due to his short temper and even shorter attention span. He misplaces his hammer and his coat and makes the whole household responsible for locating them. His ineptness is further illustrated when he smashes his thumb with the hammer while driving a nail into the wall. Meanwhile, the women of the household complain about his poor attitude.
For his part, Uncle Podger steadfastly ignores their complaints. He pats himself on the back for a job well done and neglects to mention that he had assistance in completing the task. Harris behaves similarly in the Hampton Court maze fiasco. He portrays himself as the capable savior of those who are lost. Yet, he fails to lead the group out of the maze. Over and over again, Harris leads everyone in a circle. No one is able to get out until one of the old keepers comes to the group's assistance.
Later in the story, Harris tries to fix scrambled eggs for his friends. He proclaims his culinary prowess and assures everyone that he is a master at making the dish.

People who had once tasted his scrambled eggs, so we gathered from his conversation, never cared for any other food afterwards, but pined away and died when they could not get them.

However, Harris proves so inept at cooking up scrambled eggs that the party has to go without them for their breakfast. Due to his clumsiness and poor cooking skills, Harris is only able to produce a teaspoonful of very burnt eggs (he had originally cracked six eggs into the frying pan). So, we can see the similarities between Harris and Uncle Podger. Both are brash and prone to over-inflate their abilities. Neither is willing to admit that their individual rhetoric often fails to match reality.

When the narrator J. tells us the story of his Uncle Podger in Chapter III, he says that his friend Harris always reminds him of his relative, because he is “so ready to take the burden of everything himself, and put it on the backs of other people.” But we hardly know Harris at this point in the book. So we must have faith that J. can see a similarity between the two men. Later, we have at least two chances to make the comparison ourselves. Go to Chapter VI and read Harris’s own account of how he and his country cousin got lost in the Hampton Court maze. And go to Chapter XI and read about Harris’s attempt to cook scrambled eggs for his friends, on the second day out. In both instances, Harris thinks he knows what he’s doing. He thinks he’s got the job under control. But he has to bring in and involve other people in his process. And in the end, he fails at the task.

How is genetic information stored in the base sequences in DNA?

DNA has four nitrogen bases.  They are abbreviated A, T, C, and G.  A and T can pair up, and C and G pair up.  This is why they are referred to as base pairs.  Human DNA has approximately 3 billion base pairs.  Regardless of the species, though, it is the order of the base pairs that matters.  A strand of DNA contains the stored information that determines how to make proteins.  A sequence of DNA will get "read" and "transcribed" into a strand of RNA.  RNA does not have the "T" nitrogen base.  Instead it uses uracil, which is abbreviated "U."  The RNA strand will then get translated at a ribosome.  Think of the ribosome like a protein factory.  The RNA tells the ribosome which piece goes where and when it goes there.  The "pieces" are amino acids, and a string of amino acids is a protein.  Those proteins eventually determine an organism's phenotype as determined by thousands of traits. 
When the RNA strand makes it to the ribosome, the ribosome will read the base pairs in groups of three.  This is called a codon.  Let us use a random sequence of 9 bases.  AUCGGCAGU.  That strand of 9 bases contains 3 codons.  AUC-GGC-AGU.  Each codon corresponds to a specific amino acid.  AUC = isoleucine.  GGC = glycine.  AGU = serine.  In reality, the amino acid chain will be hundreds of pieces long.  Once the chain is complete, the protein is finished.  If the order of the bases was different, the corresponding codons would be different.  That would then change which amino acids get placed into the chain, producing an entirely different protein.  
https://learn.genetics.utah.edu/content/basics/transcribe/

Intermediate Algebra, Chapter 2, 2.7, Section 2.7, Problem 48

Solve the inequality $|-2x - 4| < 5$, and graph the solution set.

The expression $-2x - 4$ must represent a number that is less than or equal $5$ units from on either side of the number line. That is, $-2x - 4$ between $-5$ and $5$ (inclusive). So we have


$
\begin{equation}
\begin{aligned}

-5 < & -2x - 4 < 5
&&
\\
-1 < & -2x < 9
&& \text{Add each side by } 4
\\
\frac{1}{2} > & x > - \frac{9}{2}
&& \text{Divide each side by $-2$. Change signs $$}

\end{aligned}
\end{equation}
$



The solution set is $\displaystyle \left( - \frac{9}{2}, \frac{1}{2} \right)$.

Explain how a bill passed in 1854 could be credited with sending the nation into civil war seven years later.

The bill that was passed in 1854, and which helped cause the Civil War to begin, is called the Kansas-Nebraska Act.  It was proposed by Senator Stephen A. Douglas (who opposed Abraham Lincoln in the Lincoln-Douglas Debates).  Its main immediate effect was to overturn the Missouri Compromise and allow people in the territories of Kansas and Nebraska to vote for themselves as to whether their territories should allow slavery.  This increased the amount of tension and hatred between the North and South and helped bring about the Civil War.
Before the Kansas-Nebraska Act, the Missouri Compromise had settled the issue of what parts of the Louisiana Purchase would allow slavery and which would not.  The Missouri Compromise had drawn a line across the Purchase.  Anything south of that line would have slaves and anything north of the line would be free.  The area that now makes up both Kansas and Nebraska was north of this line and was therefore going to be free territory.  This law was passed in 1820 and it helped keep the peace between North and South because it prevented any further conflicts over which areas would be slave and which would be free.
In 1854, Senator Douglas proposed a bill that would throw out the Missouri Compromise (follow the link below to read about why he wanted to do this).  Under the Missouri Compromise, both Kansas and Nebraska were free areas.  Under Douglas’s bill, this was called into question.  Instead of being free areas, these two territories would be up for grabs.  People in the territories would get to vote on the issue of slavery.
This helped lead to the Civil War in two main ways.  First, it helped to convince the North that the South had too much power.  Northerners feared the political power of the South and the Kansas-Nebraska Act did more to convince them that their fears were justified.  The South, they felt, had been able to overturn the Missouri Compromise, take land that had been free, and make it possible for slavery to be introduced there.  The North wondered what else the South might do with this power and it resented the idea that it had to fight again for land that had already been set aside as free soil.  By making the North resent and fear the South more than it had, the Kansas-Nebraska Act helped bring on the Civil War.
Secondly, the act brought about a period of violence in Kansas that we now call “Bleeding Kansas.”  Both North and South wanted to win when the people of Kansas voted on whether to allow slavery.  Therefore, both sides sent settlers into the area.  In addition, people within Kansas started to fight one another, hoping to drive the opposition out of the territory before the vote.  Terrible atrocities were committed by both sides.  This was the first really serious violence over the issue of slavery.  Of course, having people killing each other over this issue made both the North and the South hate each other more.
In these ways, the Kansas-Nebraska Act helped bring on the Civil War.  It did not cause it to happen immediately, but it increased the amount of hatred and distrust between the two regions of the country.  This is how a bill passed in 1854 could help lead to the start of the war seven years later.
https://www.history.com/topics/19th-century/kansas-nebraska-act

How many African Americans became United States senators and representatives by 1900?

After Congress and the federal government effectively lost interest in Reconstruction the Southern states enacted what were known as the Jim Crow laws. This was legislation that systematically excluded African Americans from the political process, both as voters and as elected representatives. Inevitably, the Jim Crow laws had a disastrous effect upon the number of African Americans in the US Congress. In December 1887, for the first time in decades, Congress convened without a single African American representative. Although three black men served in the 51st Congress from 1889–91, the long-term political representation of African Americans substantially declined as the national focus on civil rights issues faded.
In the decade 1891–1901 only five African Americans were elected to the House of Representatives: Henry Cheatham; George White; Thomas Miller; George Murray; and John M. Langston.

Single Variable Calculus, Chapter 3, 3.4, Section 3.4, Problem 50

Find $\displaystyle \lim\limits_{\theta \to 0^+} \frac{A(\theta)}{B(\theta)}$ for a semicircle with diameter $PQ$ that sits on a isosceles triangle $PQR$ to form a region shaped like a two dimensional ice cream cone, as shown in the figure. Where $A(\theta)$ is there area of the semicircle and $B(\theta)$ is the area of the triangle.




For the area of semicircle $A(\theta)$. We let $r$ be the radius forming this triangle





$
\begin{equation}
\begin{aligned}
\sin \frac{\theta}{2} &= \frac{r}{10}\\
\\
r &= 10 \sin \frac{\theta}{2}
\end{aligned}
\end{equation}
$

Solving for $A(\theta)$

$
\begin{equation}
\begin{aligned}
A(\theta) &= \frac{\pi r^2}{2} &&; \text{where } r = 10 \sin \frac{\theta}{2}\\
\\
A(\theta) &= \frac{\pi(10 \sin \frac{\theta}{2})^2}{2}\\
\\
A(\theta) &= 50 \pi \left( \sin^2 \left( \frac{\theta}{2}\right)\right)

\end{aligned}
\end{equation}
$


For the area of triangle $B(\theta)$ given two sides and an included angle we have,

$
\begin{equation}
\begin{aligned}
B(\theta) &= \frac{1}{2} ab \sin \theta\\
\\
B(\theta) &= \frac{1}{2} (10)(10) \sin \theta\\
\\
B(\theta) &= 50 \sin \theta
\end{aligned}
\end{equation}
$


Thus,

$
\begin{equation}
\begin{aligned}
\lim\limits_{\theta \to 0^+} \frac{A(\theta)}{B(\theta)} &= \lim\limits_{\theta \to 0^+} \frac{\cancel{50} \left( \sin \frac{\theta}{2}\right)^2}{\cancel{50}\sin \theta} && \text{We can introduce a factor } \frac{\theta}{\theta} \text{ and } \frac{\frac{\theta}{2}}{\frac{\theta}{2}} \text{ to use the property of limit.}\\
\\
\lim\limits_{\theta \to 0^+} \frac{A(\theta)}{B(\theta)} &= \lim\limits_{\theta \to 0^+} \frac{\pi \left( \sin \frac{\pi}{2}\right)^2}{\sin \theta} \left( \frac{\theta}{\cancel{\theta}} \right) \left( \frac{\frac{\cancel{\theta}}{2}}{\frac{\theta}{2}}\right)\\
\\
\lim\limits_{\theta \to 0^+} \frac{A(\theta)}{B(\theta)} &= \lim\limits_{\theta \to 0^+} \frac{\pi}{2} \left[ \left( \frac{\theta}{\sin \theta} \right) \left( \frac{\sin \frac{\theta}{2}}{\frac{\theta}{2}}\right) \left( \sin \frac{\theta}{2}\right)\right] && \text{recall that } \lim\limits_{\theta \to 0 } \frac{\sin \theta}{\theta} =1\\
\\
\lim\limits_{\theta \to 0^+} \frac{A(\theta)}{B(\theta)} &= \lim\limits_{\theta \to 0^+} \frac{\pi}{2} (1) (1) \left( \sin \frac{\theta}{2}\right)\\
\\
\lim\limits_{\theta \to 0^+} \frac{A(\theta)}{B(\theta)} &= \frac{\pi}{2} \left( \sin \frac{\theta}{2}\right)\\
\\
\lim\limits_{\theta \to 0^+} \frac{A(\theta)}{B(\theta)} &= \frac{\pi}{2} \sin \frac{0}{2}\\
\\
\lim\limits_{\theta \to 0^+} \frac{A(\theta)}{B(\theta)} &= 0

\end{aligned}
\end{equation}
$

I need help with a paper about "Two Kinds" that focuses on June as a character.

Amy Tan’s novel “The Joy Luck Club” has four sections, with the first being “Two Kinds.” The whole story is a touching tale about a mother and daughter trying to survive together despite significant ideologies. Writing a paper about June, the daughter, will require an informative thesis and several body paragraphs that analyze her character. Your thesis is important because it tells your readers what your essay is about and helps you keep your essay from veering off-topic.
Perhaps the best way to write a comprehensive essay about June is to create an outline that details the important aspects of June’s character. If I were writing the essay, I’d include paragraphs about:
June’s character traits, both internal and external
June’s relationship with her mother
June’s change in character over the span of the book.
The major conflicts in the book that impact June.
 To fill out these talking points, you will need to have concrete evidence from the book to match up with your own statements. For example, when writing a paragraph about the tension between June and her mother, you should include a quote about how June fights to keep her identity even though her mother is pushing her to become someone else. You could use the following quote:
 “But underneath the scarf I still knew who I was. I made a promise to myself: I would always remember my parents’ wishes, but I would never forget myself.”
 Remember, that each paragraph needs to be a mixture of your ideas and quotes from the book that support those ideas. When you are finished writing your character essay, add a witty title to it and you will be good to go.

It sounds like you want your paper to be a character analysis of Jing-Mei (June).  The first thing to do is to come up with a thesis statement.  You want to do this because it will guide your writing.  Without the thesis statement, your paper ends up reading like a string of loosely connected paragraphs about June.  I would pick a thesis that allows you to discuss multiple aspects of her character.  Something like the following might be good.  "Although June appears to be a rebellious young girl, she does deeply desire to work hard and please her mother."   
This thesis allows you to discuss why June is so eager in the beginning of the story to try to be the prodigy that her mother thinks that she can be.  It also alerts your reader to the fact that you will be discussing parts of the story where June completely rebels against the pressure that her mother is putting on her.  
One thing to definitely include in this kind of character analysis paper is supporting quotes from the text.  A quote that shows June's desire to please her mother and work hard toward becoming a prodigy is the following quote. 

In fact, in the beginning I was just as excited as my mother, maybe even more so. I pictured this prodigy part of me as many different images, and I tried each one on for size....  In all of my imaginings I was filled with a sense that I would soon become perfect: My mother and father would adore me. I would be beyond reproach.

It's here at this early point in the story that readers see June's excitement at being loved and adored by her parents.  She wants to work hard for it.  Perhaps you could claim that there is some selfish motivation here to be famous, but the quote shows June's emphasis on her parents being the people that adore her.  June is a good kid, she loves her parents, and she wants to make them happy.  I would even claim that June is a hardworking young girl as well.  That's why she is so willing to try so many different versions of being a child prodigy. 

I was a dainty ballerina girl standing by the curtain, waiting to hear the music that would send me floating on my tiptoes. I was like the Christ child lifted out of the straw manger, crying with holy indignity. I was Cinderella stepping from her pumpkin carriage with sparkly cartoon music filling the air

Of course, that all completely changes as June realizes that being a prodigy is just as difficult as fully satisfying her mother's hopes. 

And after seeing, once again, my mother's disappointed face, something inside me began to die.

June quickly moves from being a willing and adaptable young girl to a stubborn and willful daughter.  She purposefully tries to wreck her mom's latest ideas and intentions for her.  It's not that June doesn't want to be brilliant and amazing.  It's just that she wants to be brilliant and amazing for being who she wants to be.  She wants to be her own independent and strong person.  She doesn't want to be who her mom wants her to be.  
As for a supporting quote that shows June as rebellious and willful, I would use this next quote.  

I won't let her change me, I promised myself. I won't be what I'm not. So now when my mother presented her tests, I performed listlessly, my head propped on one arm. I pretended to be bored.

Why should early childhood educators know about Erik Erikson's psychosocial theory?

In addition to understanding Erikson’s theories of development for their intrinsic value, it is also important to understand them because they are a foundation upon which further research and theory has been built.
Early Childhood Educators will, for the most part, be working with children who are in the first three stages of Erikson’s psychosocial development framework. These correspond to Freud’s oral, anal, and genital psychosexual stages of development. Erikson’s stages can also be viewed alongside Piaget’s theories of cognitive development which are sensorimotor and preoperational, with some children around seven years old moving into the concrete stage at Piaget’s “age of reason”.
By understanding Erikson’s theories of development, we are better able to critically analyze and interpret other developmental frameworks. It is not that early childhood educators should understand Erikson’s framework in isolation and apply it rigidly; it is important to understand it as it relates to other theories and frameworks. A good example of this is when we look at Noam Chomsky’s ideas about language acquisition. While it is possible to study and understand Chomsky in isolation, we get a broader understanding of his ideas about “biological pre-programming” when we consider other nativist theories of development.
As well as correlative theories, if we understand Erikson’s stages we can also understand the criticisms that have been made. One critique of Erikson’s theory is that it is based on the development of male children in Western Europe and North America. This criticism does not mean we should not learn about Erikson’s theory at all but instead that we should have a good understanding of it so we can see the ways in which it might be modified to be useful—or so that we can know which parts of the theory can help us in our work. It would not be possible to fully appreciate and address these criticisms without having a firm grasp of what it is they are responding to.
Finally, it is important for early childhood educators to understand the professionalization of early childhood education and the history of the occupation. It is important to maintain professional development throughout your career, and having a solid framework of understanding is the best way to begin this endeavor.

That is like asking why English majors should study Shakespeare or physicists should study Einstein: very simply, Erik Erickson has been an extremely influential figure in the field of child development. He is especially important to early child educators because he moved the conversation about the psychological development of children away from the home and psycho-sexual development. Instead, he put the emphasis on the wider social factors that influence the healthy psychological growth of children and adults.
As early childhood educators will be working with children anywhere between two- to six-years-old, many stages of Erickson's stages of development come in to play. Children in that age range are still negotiating building trust through developing positive relationships with caregivers, gaining independence and autonomy, developing a sense of purpose, and learning to have pride and a sense that they can achieve. A good educator is doing more than imparting information: he must use consistency, praise, and encouragement to help a child develop. He must also avoid blaming, shaming, or making a child feel inferior. Erickson's theories provide a clear roadmap for guiding a child safely through early childhood so that he or she can become an intact and productive adult.

Erikson's psychosocial theory is one of the few developmental theories that spans an entire lifetime. Erikson developed his theory in relation to his training in psychoanalysis, and his first stages are mirrors to Freudian stages. 
The key to Erikson's theory is that it combines an individual's psychological characteristics with that person's interaction with other people. It also takes different developmental periods of life and identifies them with essential questions or crises. The earliest stage, for example, is Trust vs. Mistrust, which infants experience. If they have attentive parents, they learn to trust people. If they are neglected or abused, they learn not to trust. 
It is important for teachers of young children to be aware of these stages in part because how different children resolve these crises can have a lot to do with their behavior in a classroom and also because the teacher becomes a significant person in a child's life and can influence the resolution of developmental crises. 
During pre-school, children will address the issue of being able to take initiative or feeling guilty about taking initiative. Teachers need to help children take initiative and make choices themselves. Pre-school teachers need to support the development of independence in their students. 
In the early grades, children are learning how to work effectively (Industry vs. Inferiority). If children do not learn how to do their schoolwork, then they will have a hard time buckling down to it later in life. For example, sometimes gifted students will be able to handle elementary school-type work without any studying, but then will have a hard time knowing how to learn when they enter higher grades. 
Erikson's theory is one of several theories that early childhood educators need to know so they can appropriately support their students' growth and development.

What does Judy mean when she tells Jess and Leslie that she is stuck?

Judy means that she has currently run out of ideas on how to proceed in one of her novels.
In the story, Judy is Leslie's mother; she is a novelist, writing under the pseudonym of "Judith Hancock." When Judy is writing, she often seems distracted. In chapter 9, the rain threatens to spoil Leslie and Jess's week-long vacation. The two children spend a short time moping on the porch of Leslie's home. Eventually, Leslie admits that she wants to visit Terabithia, even though it is raining.
Jess concurs, and Leslie begins to collect her boots and raincoat. She suggests that they go to Jess's house to collect his as well. However, Jess says that he does not have boots or a raincoat that fit him at home. He insists that he will go to Terabithia just as he is. Leslie then suggests that she will fetch one of her father's old raincoats for Jess.
Just as Leslie is making her way upstairs, Judy appears in the hallway. She seems preoccupied, and Leslie immediately apologizes for disturbing her mother. Judy replies that she is "stuck" anyway and may as well stop for a break. This means that Judy is out of ideas on how to proceed on one of her stories. Some people will say that she is suffering from a mental block.
After a brief, distracted conversation with the children, Judy eventually returns to typing furiously on her typewriter. This prompts Leslie to comment that her mother has come "unstuck."

In The Outsiders, what are the rules of the rumble? (pages 140–142)

The rules of the rumble are simple: no weapons are to be used (just fists), and the first to run loses. This is not how the Socs normally fight, so the rules of the rumble are an important concession to the Greasers. To be sure, the Greasers do indeed carry weapons such as knives, but they're only to be used for self-defense in the event of an attack. An organized rumble obviously wouldn't fall into that category.
The very idea of having a gang fight without weapons is highly unusual; to have a gang fight on the basis of rules is positively bizarre, but that's just the way the Greasers are. The rules of the rumble are an expression of their self-image as a class apart. They may not be as socially prominent as the Socs, but they do regard themselves as more honorable and more noble than their hated rivals.

The only official rule of the rumble is that no one is allowed to use weapons during the fight. This rule is brought up several times in the novel. Cherry Valance initially informs the Greasers that the Socs are willing to abide by the rules and will not bring weapons to the rumble. Ponyboy mentions that other gangs, notably Shepard's gang, were known to use "bicycle chains, blades, pop bottles, pieces of pipe, pool sticks," and occasionally guns in their fights—but the greasers "never went in for weapons." The other unwritten rule of the rumble is that the number of gang members on both sides needs to be roughly the same. Since the Greasers are short-handed, they enlist help from the Brumly boys and Tim Shepard's gang. Despite the rule forbidding the use of weapons during the rumble, the leader of the Brumly boys uses a pipe to beat up a Soc member towards the end of the brawl.

What were the major events in the life of Frederick Douglass that formed his identity?

The first major life event that formed Frederick Douglass's identity was his witnessing his Aunt Hester being brutally whipped. This event occurred when Douglass was very young. At the time he did not understand why his aunt was being beaten. However, the event frightened him and taught him that the world was fundamentally unsafe for him and those he loved.
A second event in early childhood that shaped Douglass's identity was his being sold away at the age of seven from his plantation of birth. Douglass was taken from his friends and family and placed in an unfamiliar and decidedly more brutal environment. Douglass would later note that this was when he began to understand what slavery was. He began to understand how the system of American slavery worked to dehumanize both enslaved people as well as those who held them captive.
A final defining moment in Douglass's early childhood was his learning to read. Upon learning to read, Douglass began to realize that the oppression of black people is a matter of social hierarchy and racial brutality. The oppression of black people was not due to innate inferiority (as he had previously believed). Learning to read marked a turning point when Douglass began to believe that he could eventually achieve emancipation and help abolish the system of slavery itself.

Why did Brian's mother give him a hatchet?

In chapter 1, Brian is flying on a small bush plane to stay with his father, who is a mechanical engineer working in the oil fields of Canada. Brian's parents recently divorced, and his father has custody of him throughout the summer. When Brian opens his satchel on the plane, he pulls out a hatchet with a rubber handgrip. Brian then remembers when he received the gift from his mother before he left to stay with his father in Canada. She had given Brian the hatchet as a gift so that he could use it while out in the woods with his father. Initially, Brian felt like the gift was useless, but he eventually learns that it will be essential to his survival following the plane crash. Fortunately, Brian survives the crash and learns to use his hatchet for a myriad of purposes. Throughout the novel, Brian becomes an expert woodsman and survives alone in the expansive, challenging Canadian wilderness.

Brian's mother gives him a hatchet because she thinks he might be able to use it while out in the woods with his father.  
When the story begins, Brian is on his way to Canada, where his father now lives. His mother and father recently went through a divorce, and the courts set up a system that has Brian living with his mom during the school year and with his dad during the summer. Dad now lives in a remote, wooded area. Brian's mom buys Brian the hatchet as a going-away present. She tells Brian he might be able to use it while out with his dad in the woods.

Brian took the sack and opened the top. Inside there was a hatchet, the kind with a steel handle and a rubber handgrip. The head was in a stout leather case that had a brass-riveted belt loop.
"It goes on your belt." His mother spoke now without looking at him. There were some farm trucks on the road now and she had to weave through them and watch traffic. "The man at the store said you could use it. You know. In the woods with your father."

Unfortunately for Brian, the plane crashes, and he never makes it to his father's place. Fortunately for Brian, he has the hatchet. 

Describe at least a dozen potential moral, ethical, clinical, and legal issues in the movie Good Will Hunting? What legal standard or ethics code were broken in this movie as well? Also what were the consequences of his decision/behaviors on the therapy?

Here are some issues raised by the film Good Will Hunting:
1) Child abuse: Will is a survivor of child abuse, and the resulting trauma this has caused in his life is where much of the conflict of the film centers.
2) Lying (Will is a liar): Issues of trust and truth come up repeatedly in the film. Will constantly misleads people about his past, although the reasons for his lying vary. Because of his background as an abused chilld, Will is reluctant to trust anyone.
3) Foster care: Will is an orphan. His childhood, spent in a series of abusive foster homes, suggests that the state bears some responsibility for his abuse.
4) The judicial system: Similarly, the judicial system is suspect in the film, since the law (as it applies to Will) seems incapable of recognizing his potential. It takes the intervention of Lambeau to get Will out of a jail term that would undoubted prove disastrous to him emotionally like his earlier court-ordered foster care.
5) Marginalization of women: The film is about men caring for other men. The one female character, Will’s girlfriend Skylar, is little more than a reward for his emotional healing.
6) Transgressive therapy techniques: Sean utilizes a number of non-standard or problematic techniques in his sessions with Will. For one thing, Sean assaults Will during their first session; for another, Sean gives regular reports to Lambeau about his confidential sessions with Will.
7) Fighting: Physical conflict is a recurring theme in the film. The story begins with Will’s participation in a bar brawl, and the sense in much of the film is that violence is lurking just under the surface of Will’s personality.
8) Use of the abuse photos: Sean’s use of Will’s sealed records in the climactic scene is potentially problematic legally and ethically.
9) Elitism: Will’s journey from laborer to intellectual is shepherded along by elite males: first, Lambeau, who gets him into therapy, and then Sean, the therapist of last resort. The idea that Will is marked for greatness is underlined by his friend Chuckie, who tells Will that he would be wasting his potential by remaining a laborer.
10) Therapy: The film has a complex relationship to therapy and therapists. Will goes through five therapists before finding Sean; he is able to “game” the system by reading up on the therapists and saying things he thinks they want to hear. Sean, on the other hand, is able to reach Will because of his willingness to break the “rules” of therapy and reveal his own vulnerability to his patient.
11) Freedom: The movie suggests that Will’s freedom is tied to his emotional self-awareness and his ability to connect to others (Sean, in particular). It is only after he is able to confront the trauma of his past that he is able to move towards personal fulfillment, symbolized by his drive to California.
12) Isolation: Will is emotionally isolated. Because of the shame and anger he feels about his childhood abuse, he is unable to be emotionally open with anyone. Although Will sees this as an expression of strength and control, the movie works to show that these feelings are harmful and limiting.
13) Genius: Will is a genius. His intellect is a kind of superpower that he must learn to harness by developing his emotional intelligence. In this sense, the film supports the idea of the “genius” as an exceptional person whose talents cannot be wasted.
There are so many issues the film raises.
While everyone talks about the cathartic final “not your fault” sequence between Will and Sean, the film’s ideology (in particular, its uncritical embrace of patriarchy) can be very problematic.
Hope this helps!
 
 

Asked to give a lecture on the "topic" of women and fiction to students at a women’s college, Virginia Woolf begins not with a complicated analysis of the "problem" but rather with a bit of practical advice: she advises them to drink wine and have a room of their own. Why?

Woolf is making the argument that women are not producing literature at the same rate as men—especially good literature—because they lack economic resources. In the 1920s, when she was writing her essay, it was often argued that women didn't achieve as much as men because of natural inferiority and lack of intelligence. Woolf emphatically wants to show this is untrue. She does this, in part, by comparing, step by step, a well-endowed men's college, that can afford good food, good wine, and comfortable quarters, to a much more austere women's college, stretching its resources to cover the basics.
Woolf's argument is that creating and producing good literature takes uninterrupted time, which includes privacy. Until women get rooms of their own in which to work and some money (symbolized by the wine the men drink at their college), they can't fully develop their talents. Woolf wants women to come out of the idealistic clouds and to face up to and battle their collective poverty in comparison to men so that they have the resources to become the people they can be. Though she controls it, Woolf is very angry at the way men have appropriated most of society's resources and fail to share them equitably with the women who helped make gathering the resources possible.

College Algebra, Chapter 3, 3.4, Section 3.4, Problem 18

A function $\displaystyle g(x) = \frac{2}{x + 1}$. Determine the average rate of change of the function between $x = 0$ and $x = h$.


$
\begin{equation}
\begin{aligned}

\text{average rate of change } =& \frac{g(b) - g(a)}{b - a}
&& \text{Model}
\\
\\
\text{average rate of change } =& \frac{g(h) - g(0)}{h - 0}
&& \text{Substitute } a = 0 \text{ and } b = h
\\
\\
\text{average rate of change } =& \frac{\displaystyle \frac{2}{h + 1} - \frac{2}{0 + 1} }{h}
&& \text{Simplify}
\\
\\
\text{average rate of change } =& \frac{\displaystyle \frac{2}{h + 1} - 2}{h}
&& \text{Get the LCD}
\\
\\
\text{average rate of change } =& \frac{2 - 2 (h + 1)}{h(h + 1)}
&& \text{Apply Distributive Property}
\\
\\
\text{average rate of change } =& \frac{2 - 2h - 2}{h (h + 1)}
&& \text{Combine like terms}
\\
\\
\text{average rate of change } =& \frac{-2 \cancel{h}}{\cancel{h} (h + 1)}
&& \text{Cancel out like terms}
\\
\\
\text{average rate of change } =& \frac{2}{(h + 1)}
&& \text{Answer}

\end{aligned}
\end{equation}
$

Single Variable Calculus, Chapter 5, 5.4, Section 5.4, Problem 32

Find the integrals $\displaystyle \int^9_1 \frac{3x-2}{\sqrt{x}} dx$

$
\begin{equation}
\begin{aligned}
\int\frac{3x-2}{\sqrt{x}} dx &= \int \left( \frac{3x}{\sqrt{x}} - \frac{2}{\sqrt{x}} \right) dx \\
\\
\int\frac{3x-2}{\sqrt{x}} dx &= \int \left( 3x^{\frac{1}{2}} - 2x^{\frac{-1}{2}} \right) dx \\
\\
\int\frac{3x-2}{\sqrt{x}} dx &= 3 \int x^{\frac{1}{2}} dx - 2 \int x^{\frac{-1}{2}} dx\\
\\
\int\frac{3x-2}{\sqrt{x}} dx &= 3 \left( \frac{x^{\frac{1}{2} +1 }}{\frac{1}{2} +1 } \right) - 2 \left( \frac{x^{\frac{-1}{2}+1}}{\frac{-1}{2}+1 } \right) + C\\
\\
\int\frac{3x-2}{\sqrt{x}} dx &= 2 x^{\frac{3}{2}} - 2 \left( 2x^{\frac{1}{2}} \right) + C\\
\\
\int\frac{3x-2}{\sqrt{x}} dx &= 2x^{\frac{3}{2}} - 4x^{\frac{1}{2}} + C\\
\\
\int^9_1 \frac{3x-2}{\sqrt{x}} dx &= 2(9)^{\frac{3}{2}} - 4 (9)^{\frac{1}{2}} + C- \left[ 2(1)^{\frac{3}{2}} - 4(1)^{\frac{1}{2}} + C \right]\\
\\
\int^9_1 \frac{3x-2}{\sqrt{x}} dx &= 2\left[(9)^{\frac{1}{2}} \right]^3 - 4 (3) +C - 2 + 4 -C \\
\\
\int^9_1 \frac{3x-2}{\sqrt{x}} dx &= 2(3)^3 - 12 + 2\\
\\
\int^9_1 \frac{3x-2}{\sqrt{x}} dx &= 54 - 12 + 2\\
\\
\int^9_1 \frac{3x-2}{\sqrt{x}} dx &= 44

\end{aligned}
\end{equation}
$

Why was the purchase of Alaska so controversial?

As vice president serving under then-President Abraham Lincoln, Andrew Johnson was already a controversial figure, the lone Southern senator to choose to remain in Congress rather than resign when the state he represented, Tennessee, seceded from the Union. Ascending to the presidency following Lincoln’s assassination, Johnson remained a controversial figure for his support of racist policies in the American South, policies codified in what became known as “Jim Crow” laws, and for his opposition to the Fourteenth Amendment to the Constitution, which assured citizenship to those born or naturalized in the United States. Johnson would later be impeached as a result of his protracted confrontations with the Republican-led Congress over Executive authorities, especially regarding the handling of Cabinet-level personnel decisions—an issue that led to his impeachment by the House of Representatives (followed by his acquittal by the Senate by one-vote margin).
So, what does any of this have to do with the purchase of Alaska? Plenty. President Johnson’s secretary of State, William Seward, successfully negotiated the purchase of a vast tract of land from the Russian government. That tract of land, Alaska, was viewed by many Americans as a worthless and inhospitable region that, while certainly increasing the aggregate size of the United States, contributed nothing to the country’s overall development. The combination of Johnson’s unpopularity among many in Washington, D.C. and across the North and questionable judgments on the part of many of those same people regarding Alaska’s potential led to wide-spread criticism of Seward and, by extension, Johnson for the region’s purchase.
https://www.archives.gov/publications/prologue/1994/winter/alaska-check

https://www.history.com/news/why-the-purchase-of-alaska-was-far-from-folly

https://www.whitehouse.gov/about-the-white-house/presidents/andrew-johnson/

William Seward, Andrew Johnson's Secretary of State, purchased Alaska rather cheaply from Russia in 1867. Many thought that Seward, a holdover from the Lincoln administration, made a poor business deal, as few Americans knew anything about Alaska. They thought the land was barren and called it "Seward's Folly" and "Seward's Icebox." The land was not contiguous to the rest of the United States unlike other land acquisitions, and there was little chance that Americans would be willing to move there in 1867. There would also be some controversy with Britain about where British Columbia began and Alaska ended—this border would not be fully settled until the end of the nineteenth century. Seward realized that the ports of Alaska extended far down the west coast of Canada and these ports would be valuable to America's commercial goals in the Pacific. Seward also wanted to gain the rich timber and fishing rights in Alaska. Seward would not live long enough to see Alaska's true potential, as gold and oil would make the territory quite rich.  

Why was it possible for the Confederacy to ship cotton via European ships docked in Mexico?

The Confederacy adopted a policy known as "King Cotton" during the Civil War (1861-1865). The idea was to deny Europe, especially England and France, the cotton imports it needed in order to force them to aid the South. The Confederate leaders knew that French intervention was decisive in the Revolutionary War (1775-1783), and they believed that cotton was the weapon that would coerce European intervention again.
As part of the Confederacy, Texas had to follow the policy laid out by the government in Richmond. But the state's geography was unique because it bordered Mexico. Small Texas towns such as Brownsville had the ability to ship cotton from Matamoros, Mexico. The cotton was then put on foreign vessels and shipped to Europe in exchange for munitions. The business was lucrative and some individuals on both sides of the border became rich. This trade was not without its share of problems, however. For example, bandits often raided the wagons carrying cotton. But the acquisition of weaponry in this manner did help the South; some officials in the Lincoln administration wanted to invade Texas to stop this trade.
In the final analysis, King Cotton diplomacy was a failure. Britain resented the South's aggressive policy, and it would not go to war with the North over cotton. Texas's unique experience shipping cotton helped the South, and it was a decisive factor in the war.

Cotton was an important tool in the Civil War; European nations, particularly Britain, depended on American cotton, and withholding of it was a policy agreed upon as a way of trying to encourage foreign intervention in the war. However, cotton was also a means of garnering much-needed funds, so it was agreed that the Confederate state of Texas could use its border with Mexico to ship out cotton on the understanding that proceeds would be returned to the central Confederate government.
This cotton was shipped from Texas border towns all the way down the Rio Grande to the Mexican port town of Bagdad, where it could be placed upon European ships. The historic European trade partners remained desperate for US cotton, so it was not difficult to find traders willing to take this route.
https://tshaonline.org/handbook/online/articles/drw01

What does Macbeth's reaction to the news of his wife's death say about his state of mind?

Whatever else we might say about the Macbeths earlier in the play, there was no doubt that they were essentially a loving couple. By the end of the play, however, they seem to have "grown apart," as we might say today. Early in the play, Lady Macbeth is remorseless and cruel as she goads her husband into the murder of Duncan and urges him not to feel any guilt for doing so. By the end of the play, she herself has been overcome with guilt, as revealed in the first scene of Act V, when she attempts to wash imaginary blood from her hands while sleepwalking. In the meantime, Macbeth has become a bloody, murderous tyrant, without regard for human life. This development is underscored by this speech, in which he essentially expresses no grief whatsoever at his formerly beloved wife's death. Immediately before receiving the news, he says that after everything he has done, "direness...cannot once start me." He is immune, in short, to horror.
At the same time, Macbeth's speech reveals a sort of grim, existential resignation. Life, he says in reaction to her death, is essentially meaningless:

Life's but a walking shadow, a poor playerThat struts and frets his hour upon the stageAnd then is heard no more. It is a taleTold by an idiot, full of sound and fury,Signifying nothing.

This is perhaps the bleakest passage in all of Shakespeare's plays, and it is obviously Lady Macbeth's passing that has evoked this profound sense of pessimism in her husband. If everything is meaningless--just a march toward "dusty death," then what was the point of everything they have done to seize the throne? In any case, Macbeth's state of mind, already astonishingly bleak, does not improve when he receives the news that Birnam Wood (Malcolm's men concealing themselves with tree boughs) apparently advancing on his castle, thus fulfilling one half of the witches' prophecy.

What theme does the author of the History of Plymouth Plantation want to convey? How does this theme relate to God's providence?

According to the Separatists, the group to which William Bradford, the author of Of Plymouth Plantation, belonged, God controlled the universe, and everything that happened in the universe was His providence. 
The theme that Bradford wanted to convey is that the relocation of the Separatists from England—and then the Netherlands to the American colonies—was an expression of God's providence.  Bradford's sect believed that the Anglicans had not separated the church profoundly enough from Catholicism; they also could not align themselves with some other Protestants even though, just as they were, many were following Calvinist teachings.  They believed that their creation of a "city upon a hill" was what God wanted and that they would serve as a model of how a properly reformed theocracy functioned.
Bradford's many anecdotes about the sacrifices the Separatists made in their harrowing crossing aboard the Mayflower and the first winter in Massachusetts were colored with frequent references to God's providence. John Howland was miraculously saved after falling overboard because it "pleased God." When friendly Native Americans including Squanto assisted the Separatists, he attributed their generosity to God; in fact, Bradford calls Squanto "a special instrument of God."

One of the major themes of the History of Plymouth Plantation is, in fact, God's divine Providence. Throughout the book, Bradford interprets every event that occurs, both good and bad for the Pilgrims, as God's will, and connected to some divine purpose that was usually impossible for human beings to understand. On the voyage to Plymouth, for example, a very profane young man, who was given to blasphemy and insulting the pious Pilgrims, got very sick and died. Bradford reflects that this was surely God's way of chastening the people, reminding them of proper behavior for a Christian:

Thus his curses light on his own head, and it was an astonishment to all his fellows for they noted it to be the just hand of God upon him.

Bradford believed that whatever successes the Plymouth settlement experienced were the result of God's mercy and Providence, which would be extended to them only as long as they maintained their faith in God. Almost every event, good or bad, is prefaced by the phrase "it pleased God." "It pleased God," for example, "to visit them this year with an infectious fever," or to "send home a great quantity of beaver." Everything that happened to and around the Pilgrims portrayed by William Bradford was an example of God's will. So essentially, the main theme of the book is in fact God's providence.
http://eada.lib.umd.edu/text-entries/of-plymouth-plantation/

Create a chart or diagram to compare and contrast Martin Luther King Jr. and Malcolm X.

I will compare and contrast the two leaders—my advice would be to put this information in a Venn diagram, with the commonalities of the two leaders in the middle and their differences in the separate part of the circles.  Martin Luther King Jr. was a civil rights leader who organized boycotts and peaceful protests.  His position as an evangelical pastor made him popular with all but the most strident segregationists.  King was known for his speeches where he envisioned a world where black and white people could live together—this is most eloquently put in King's "I Have a Dream" speech.  
Malcolm X was a member of the Nation of Islam and he viewed separation of the two races as the most acceptable way to achieve justice.  Early in his speaking career, he argued that the white race would fail and that black people were superior to whites.  He accused leaders such as Martin Luther King as being too soft.  Malcolm X spoke to the anger that some African Americans felt having lived under segregation.  Malcolm X would eventually separate from the Nation of Islam and adopt a more moderate message, but he still argued for black political participation in order to achieve civil rights goals.  
Both leaders were assassinated—Malcolm X in 1965 by a member of the Nation of Islam, and King in 1968 by a segregationist.  Both men were controversial in that they were arguing to change the status quo.  Both men were also against the war in Vietnam. The Vietnam War claimed a higher percentage of black than white lives.  

What change should be expected in the velocity of a body to maintain the same kinetic energy, if its mass is increased sixteen times? How?

Hello!
Kinetic energy is that part of full energy which a body has due to its motion. The formula for kinetic energy is E_k = (m V^2)/2, where m is the mass and V is the speed (regardless of direction).
Usually a body remains the same during its motion, and the mass of the body also remains the same. In our problem, the mass of the body is supposed to increase 16 times, roughly speaking some other bodies will join our initial body.
In such a case, its kinetic energy becomes E'_k = ((16 m) V^2)/2 = 16 E_k. To compensate this change by a speed change, we have to reduce V^2  16 times, which means to reduce V  sqrt(16)=4 times.
This is the answer: body's speed must be reduced 4 times to maintain the same kinetic energy.

What is Plotinus saying in the eighth tractate about the souls decent into body?

Just to be clear, this matter is discussed by Plotinus in the fourth Ennead, suitably entitled "On the soul's descent into the body." Heavily influenced by Plato, Plotinus sees the soul as essentially trapped inside the body. As a spiritual entity, it has more reality for Plotinus than the merely physical body. This particular section of the book is important because it illustrates Plotinus's general system of metaphysics, which is based on a hierarchy of being.
At the top of this ontological hierarchy (ontological means relating to the nature of being) sits the perfect, indivisible One. All lesser forms of being such as intellect, bodies, plants, animals, and minerals are further down, ultimately emanating from the One. A good way of understanding Plotinus's philosophical system is to try to see it as a kind of pyramid with the One at the very top and various types of matter, such as bodies, right at the bottom.
For Plotinus, the soul is superior to the body because it partakes of the eternal. The body, on the other hand, is restricted to this world; it is susceptible to decay and corruption, and it will eventually die. Plotinus describes the body as "the house of poverty" for the soul. Yet, the fact that the soul is housed in the body does not necessarily have to be a bad thing. The soul is unique in that it can look two ways, as it were. Although it has descended into the body, it can still focus on the higher realities; it can still partake of the eternal. All too often, however, the body disturbs the soul, attacking it with hunger, pleasures, desires, sorrows, and all other kinds of distraction. These divert the soul's attention from reconnecting to the world of higher things.
The soul is caught between the higher and lower stages of Plotinus's hierarchy of being; the soul has something from above and something from below. It was by focusing on the things of this world, the lower stage of reality, that the soul originally became individualized and first came to reside within the body. However, if the soul chooses to turn its attention to higher things, to the eternal truths, then it can no longer be damaged by its temporary home in an imperfect body. It can ascend, by way of a mystical experience, back to where it came from: the One, from which the soul, as with every other being, originated.

What is a summary of chapters 90–92?

Chapters 90–92 of Wonder deal with the aftermath of Daisy's death and Olivia's role in the school play. In chapter 90, Auggie ponders life in heaven and if the family's dog, Daisy, is there. He thinks that in heaven, his face won't matter, because Daisy never cared about his face and just loved him for being him.
In Chapter 91, the family goes to see the school play, Our Town. They are surprised when Via steps on stage as Emily, since it was supposed to be Miranda. While Miranda believes she worked hard on the role of Emily, no one in her family is able to come. When she sees that Via's whole family is in the audience, she pretends to be sick and gives the role to Via.
In chapter 92, everyone congratulates Via and Justin for their role in the play. Auggie gets lost in the backstage crowd and begins to panic, but Miranda rescues him. Miranda has always seen Auggie as her little brother, to the point that she developed a lie about him actually being her brother at camp that summer. The act of saving Auggie gives her the opportunity to protect him like a big sister would.

Why is prejudice an integral part of Maycomb ?

The story is set in the fictional small town of Maycomb, Alabama during the mid-1930s, which was a time when Jim Crow laws were enforced throughout the South. Jim Crow laws were state and local laws that enforced racial segregation, which marginalized and discriminated against African American citizens. African American citizens were forbidden from using or occupying white-only public facilities throughout the South, and they were considered second-class citizens. These organized discriminatory state laws and institutions perpetuated and encouraged racial discrimination, which is an integral part of Maycomb's culture throughout the novel. The conflict throughout the story centers around the trial of an innocent African American man who is accused of assaulting and raping a white woman. Despite Atticus's valiant defense, Tom Robinson becomes the unfortunate victim of racial injustice. Harper Lee chose to set the story in a Southern town that enforced Jim Crow laws in order to emphasize the significance of race that would develop the conflict throughout the story. Both Jem and Scout lose their childhood innocence after they witness racial injustice firsthand and begin to acknowledge the fact that the majority of Maycomb's citizens are racist.

Who do Napoleon's dogs symbolize in Animal Farm?

Napoleon's cadre of privileged guard dogs represent the police in a totalitarian state. They are often likened to Stalin's secret police, but they are not secret: they are an open form of terror. It's important to note that while Orwell was critiquing Stalinist Russia, he was also critiquing any form of government based on terror. He had Nazi Germany on his mind as well as the Soviet Union while writing this work.
In the broadest sense, the dogs symbolize that Napoleon's dictatorial regime is based on violence and terror. They represent that violence. Through show trials, in which the dogs viciously kill animals who "confess" to crimes against the state, the other animals are frightened into silence. Just the presence of the dogs—or a warning growl—can be enough to keep the other animals from speaking out. Like the police in a totalitarian state, the dogs are given special privileges not afforded the rest of the animals.

At the beginning of the story, Orwell writes that Napoleon's main focus is on educating the youth, which differs from Snowball's egalitarian political agenda. Napoleon ends up training nine puppies in the loft of the barn and turns them into extremely loyal, ruthless killers. While Snowball is giving a speech in front of the animals, Napoleon calls on his nine ferocious dogs, and they chase Snowball off the farm. Napoleon then becomes the leader of Animal Farm, and the nine dogs surround him wherever he travels, acting as his personal bodyguards. Napoleon also uses the dogs to intimidate and punish the other animals who disagree with his political decisions. Napoleon's nine ferocious dogs symbolically represent The People's Commissariat for Internal Affairs, abbreviated NKVD, which was Stalin's secret police force. The NKDV was responsible for carrying out The Great Purge, which occurred from 1936 to 1938 and was notorious for its brutality during Stalin's reign.

Why, according to the second clown, is Ophelia really being given a Christian burial?

It is generally accepted that Ophelia has committed suicide. At that time, suicide was considered a sin, and a very serious one at that. Those who killed themselves were therefore denied burial in consecrated ground.
In act 5, scene 1 of Hamlet, a couple of grave-digging clowns are preparing the recently departed Ophelia's final resting place. They engage in all manner of wordplay and witty badinage to lighten the oppressive mood hanging over the scene. The gravediggers argue whether Ophelia will be given a proper Christian burial, being as how she's committed suicide. One of them comes up with a novel explanation of Ophelia's death, one that can be construed in such a way as to make it seem that she didn't intend to kill herself after all. He says that Ophelia didn't drown herself; the water drowned her:

If the man go to this water and drown himself, it is, will he nill he, he goes. Mark you that. But if the water come to him and drown him, he drowns not himself. Argal, he that is not guilty of his own death shortens not his own life.

In other words, according to the gravedigger, Ophelia didn't go to the water; the water came to her. As such, she didn't really commit suicide, and so can be given a good Christian burial.

How can I write a story titled "An Unforgettable Journey"? (It must be a creative writing essay with 300 words.)

It's not clear from the assignment whether your task is to write a non-fiction or fiction piece. Either way, you can interpret the idea of a journey in many ways. For example, a journey can be a trip to a physical location, or it can also be a metaphorical journey. Examples of a metaphorical journey include finding out something new about yourself, or about a loved one, or seeing a place where you've been before in a new way.
As you prepare for this assignment, perhaps go over old photographs and think about journeys, both physical and metaphorical, that you've made. You can think about trips you've taken, friends you've gotten to know, or ways you've gotten to know yourself better. When writing, think about starting with a hook that grabs the reader, and try to show the reader the changes you went through on your journey by writing dialogue and description rather than by simply telling the reader what you experienced. 

How does the past (childhood & nostalgia) affect Nathan Zuckerman in American Pastoral?

In American Pastoral,Nathan Zuckerman has created and cherished a series of romantic myths about Seymour "the Swede" Levov, the brother of his closest friend in school. When he encounters the Swede early in the novel, he feels that something has gone wrong—that the immense promise of the boy who was considered practically a demigod in high school has been compromised or destroyed (even though the Swede has remarried and now evidently leads a normal, happy life).
What Zuckerman knows of the Swede, up to this point, seems to confirm what everyone had expected of Seymour. He was a champion athlete, breaking all the school records in New Jersey. After returning from the service, he married, entered his father's business, and became the owner upon his father's retirement. He then moved out to the suburbs, to Morris County, far from his Jewish working-class roots in old Newark. Yet as the whole story unfolds, Zuckerman finds himself witnessing a Greek tragedy. Perhaps the story would not be so shocking if Zuckerman hadn't created such a deeply mythologized view of the Swede, and of his Newark childhood overall. For Philip Roth, in his Zuckerman persona and elsewhere, old Newark is an Arcadia, a prelapsarian world. It shakes Zuckerman that Seymour Levov, the one person who emerges from this Eden as a hero without a single flaw, is destroyed by circumstances beyond control. It's a testament to the inability of even the most virtuous and gifted person to escape tragedy.
The rebellion of Seymour's daughter Merry, though a shock in the context of the Swede's mythic stature, should not be surprising, of course. The late 1960s and early 70s were a time of vast social upheaval in the US. That Merry first becomes a "terrorist" and then a Jain, withdrawing from reality into a purgatory-like fantasy world, can be partly seen as a result of the Swede's own tragic flaw. Zuckerman's fixation on Seymour and his story stems from Zuckerman's own idealization of the childhood setting he reimagines over and over (as Roth himself does). Yet the Swede seemingly has not grasped, until the final deterioration of his daughter, that the course of his own life may have been wrong in some way. In his case, the marriage to a woman from a different background has backfired. The Swede and his wife create their own fantasy world in the rural setting of Old Rimrock. In spite of Seymour's devotion to the effective management of his father's glove business, by living on a farm he's trying to escape reality—the reality ironically embodied in Newark, where one identified strongly with one's ethnic and religious group and where one's Jewish identity was a prime factor in the direction of one's life. The Swede attempts to create for himself a "new" Eden in the countryside of Morris County, but the result is a catastrophe for him and for his family.