Notes

The integral `int sin^3(3*x) dx` has to be determined. `int sin^3(3*x) dx` Let `y=3x` `dy/dx = 3` The given integral can be written as: `1/3int sin^3 y dy` = `1/3int sin y * sin^2 y dy` = `1/3int sin y * (1 - cos^2 y) dy` Let `z = cos` y, `dz/dy = -sin y` => `-(1/3)*int (1 - z^2) dz ` => `(-1/3)*(z - z^3/3)` As `z = cosy` => `-1/3(cos y - (cos^3y)/3)` And `y = 3x` => `-1/3(cos 3x - (cos^3(3x))/3)` The integral `int sin^3(3*x) dx = -1/3(cos 3x - (cos^3(3x))/3) + C`

To determine the convergence or divergence of the series `sum_(n=0)^oo (-1)^n/((2n+1)!)` , we may apply ratio test. In Ratio test , we determine the limit as: `lim_(n-gtoo)|a_(n+1)/a_n| = L` or `lim_(n-gtoo)|a_(n+1)*1/a_n| = L`  Then ,we follow the conditions: a) `L lt1` then the series converges absolutely . b) `Lgt1` then the series diverges . c)` L=1` or does not exist  then the test is inconclusive .The series may be divergent, conditionally convergent, or absolutely convergent. For the series `sum_(n=0)^oo (-1)^n/((2n+1)!)` , we have: `a_n=(-1)^n/((2n+1)!)` Then, `1/a_n=((2n+1)!)/(-1)^n` `a_(n+1)=(-1)^(n+1)/((2(n+1)+1)!)`             `=(-1)^(n+1)/((2n+2+1)!)`             `=(-1)^(n+1)/((2n+3)!)`             `=((-1)^n*(-1))/((2n+3)(2n+2)((2n+1)!))` Applying the Ratio test on the power series, we set-up the limit as: `lim_(n-gtoo) |((-1)^n*(-1))/((2n+3)(2n+2)((2n+1)!)) *((2n+1)!)/(-1)^n|` Cancel out common factors: `(-1)^n` and `(2n+1)!` . `lim_(n-gtoo) |(-1)/((2n+3)(2n+2))|` Evaluat...

In The Picture of Dorian Gray , Wilde uses a number of literary techniques to develop his characters and to enhance some of the novel's key themes. Allusions, for example, are widely used in Chapter One to reveal more about Dorian's appearance and character. He is likened to Adonis, a man of great beauty from Greek mythology, to demonstrate his visual appeal to the reader. He is also likened to Narcissus, another figure from Greek mythology, who fell in love with his own image and died because he could not stop looking at himself in the reflection of a pond. By including this allusion, Wilde also foreshadows Dorian's future and suggests that his self-love will develop into an unhealthy obsession. Wilde also uses a number of symbols in the novel. The portrait, for example, becomes a living representation of Dorian's soul which degrades as his levels of vice and immorality increase.  It also represents the unavoidable process of ageing which Dorian is so keen to prevent. ...

Throughout The Plague of Doves , Billy and Corwin Peace evolve in opposite ways. While Billy becomes a cruel tyrant, Corwin eventually grows from a troubled youth to a talented musician. Billy Peace As the younger brother of Maggie Peace, Billy starts out as a shy yet principled young man. When Maggie becomes pregnant with John Wildstrand's child, Billy overcomes his shy demeanor to demand that John treats his sister fairly. He is drawn into John's kidnapping scheme, and enlists in the military to avoid being arrested. War alters Billy's character, and he develops a fanatical perspective on religion that leads him to form his own cult. Billy changes from a shy youth to a charismatic religious leader, and he exerts absolute control over his congregation as well as his wife and children. He soon becomes abusive towards his family and dies at the hands of his own wife. Corwin Peace Corwin is the illegitimate son of Maggie Peace and John Wildstrand. He is also Billy Peace's...

Romeo is smitten with both Rosaline and Juliet, but his love for Rosaline is unrequited. Thus, we find him at the beginning of the play pining for Rosaline, who has decided to remain chaste. His friends urge him to get over Rosaline, but he only does so when he encounters Juliet. The Friar, to whom he has spoken at length about his love for Rosaline, is skeptical about his newly-professed love for Juliet. He understandably thinks that Romeo has impulsively given his love to another, which suggests that neither love was really sincere. It is hard to know whether his love for Rosaline was legitimate—it certainly seems so at the beginning of the play—but his love for Juliet certainly is. He is willing to die for her. I would argue Romeo loves both women passionately, but his love for Juliet seems more substantial.

First, what do you mean when you say "when he is precisely halfway between the planets as measured in the planet frame" and "the explosions are simultaneous in the frame of the planets"? The only option I see is: before the spaceship neared the first planet, observers on both planets synchronized their clocks (the speed of these clocks is the same because the planets are in rest relative to each other), measured the speed of the spaceship, and estimated when it would approach the first planet. Then they computed how much time it would take for a spaceship to travel half of the distance and wrote down that time. And, finally, they explode bombs (actually, emit light) when their clocks show this computed time. From the planets' point of view, the half of the flight will take `1/(0.6)` hours. Now let's look on this from an astronaut's point of view. He travels between planets with the same speed `0.6 c.` The distance between the planets for him is less than...

What Cassius says and does throughout the play paints a very negative picture of him. There is not much good one can say about him since he comes across as sly, scheming, manipulative, greedy, ruthless, corrupt and cowardly. In the end, one can understand Caesar's suspicion of him when he tells Antony, during the Lupercalian Festival, in Act 1, scene 2: Let me have men about me that are fat; Sleek-headed men and such as sleep o' nights: Yond Cassius has a lean and hungry look; He thinks too much: such men are dangerous. Caesar is accurate in his assessment. Cassius' 'lean and hungry look' portrays him as both a scavenger and a predator. He is hungry with ambition. The fact that 'he thinks too much' suggests that he is constantly plotting, which is exactly what Cassius does.  His scheming and manipulative nature is pertinently displayed when he draws the honourable Brutus into his web. He flatters him and says exactly what he knows Brutus wants to hear. He pr...