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Math`e^(lnx)=4` To solve, apply the logarithm rule `e^(lnm) = m` . So the left side simplifies to: `x=4` Therefore, the solution of the given equation is x=4.

Math`f(x)=sec(x)` Take note that a function is strictly monotonic on a given interval if it is entirely increasing on that interval or entirely decreasing on that interval. To determine if f(x) is...

MathWe are asked to verify that the function f(x)=cos(x) defined on the interval `[0,pi) ` has an inverse function by determining that the function is strictly monotonic on the interval using the...

MathWe are asked to determine if the function `y=cot(x) "on" (0,pi) ` has an inverse function by finding if the function is strictly monotonic on the interval using the derivative. `y'=csc^2(x) ` . On...

Math`f(x)=4/x^2` `f(x)=4x^(2)` `f'(x)=8x^(3)=0` There are no critical values for x in the domain (0, `oo).` If you choose any x value in the domain `(0,oo),` `f'(x)=8x3<0` therefore the...

MathWe are asked to show that `f(x)=x+2,[2,oo) ` has an inverse by showing that the function is monotonic on the interval using the derivative: By definition, ` f(x)=x+2={[x+2,x+2...

MathGiven `f(x)=(x4)^2, [4,oo) ` we are asked to show that the function has an inverse since it is monotonic on the given interval: `f'(x)=2(x4) ` and `2(x4)>=0 forall x>=4 ` so the function...

MathWe are asked to determine if the function `y=cos((3x)/2) ` has an inverse function by finding if the function is strictly monotonic on its entire domain using the derivative. The domain is all real...

Math`f(x)=ln(x3)` Take note that a function is strictly monotonic if it is increasing on its entire domain or decreasing on its entire domain. For our function, `f(x)=ln(x3)` to determine if it is...

MathWe are asked to determine if the function `y=x^5+2x^3 ` has an inverse function by finding if the function is strictly monotonic on its entire domain using the derivative. The domain for...

MathWe are asked to determine if `y=x^4/42x^2 ` has an inverse function by finding if the function is strictly monotonic on its entire domain using the derivative. The domain for a polynomial is all...

Math`f(x)=x^36x^2+12x` `f'(x)=3x^212x+12=0` `3(x^24x+4)=0` `3(x2)(x2)=0` `x=2` A critical x value is `x=2.` The function increases in the interval `(oo, 2).` The function increases in the...

Math`f(x)=2xx^2` `f'(x)=12x=0` `12x=0` `2x=1` `x=1/2` A critical value for x is `x=1/2.` In the interval `(oo, 1/2)` the function increases. In the interval `(1/2,oo)` the function...

HistoryThe Civil Rights Movement of the 1960s was largely aimed at exposing the injustices AfricanAmericans faced in society, while the social justice movements of the 1970s concentrated on agitating for...

HistoryOne event from the 1850s that heightened tensions between the North and the South was the raid by the abolitionist John Brown on the federal armory at Harpers Ferry, which was then in the state of...

ReferenceAlthough slavery has ended and white supremacy is no longer encouraged in the United States, the truth is that the history of white supremacy still has a strong lingering impact on the US and...

Native American LiteratureThe writer is apprehensive and a little afraid of how he will be received when he first arrives in Woodruff. This is because the writer has presumably been away for a while. He hasn't seen his...

Patrick HenryPatrick Henry wanted to raise a militia to prevent the British from taking advantage of them. In this impassioned speech, Henry’s point was that they could not stand back and let the British walk...

A Christmas CarolHow do the Cratchits react to their Christmas feast, and what does their celebration show Scrooge...The Cratchit family is grateful for their feast even though it is meager, and Scrooge realizes that you do not need much to be happy as long as you have people you love. The Cratchit family reminds...

ArabyThe most obvious character who is not named in James Joyce's "Araby" is Mangan's sister. Throughout the story, she is known as "Mangan's sister," not as a named character despite the fact that the...

René Descartes"Je pense donc je suis" by Descartes can be translated into English as "I think, therefore I am" and into Latin as "cogito ergo sum." Descartes published this phrase in his Discourse on the Method...

FrindleWhen the novel begins, Nick is a discontented teen looking for ways to cause minor trouble in his Language Arts class. He finds a way by calling a pen “frindle,” never imagining how far this...

A White HeronMotifs are recurring symbols which support the themes or main ideas in any story. In A White Heron, the young sportsman's gun is a recurring motif. To Sylvia, the gun represents strange emotions...

Public HealthBased on your background in public health, here are some headings I would recommend for use in promotional materials regarding the Rotavirus: 1) What the Rotavirus is and how it affects the health...

The Devil's HighwayIn any game, there are always two opposing sides; this is certainly the case in the area of illegal immigration. In the book, Urrea discusses how the Border Patrol agents (La Migra) often find...

Fahrenheit 451When the story starts, Montag is much like everyone else in his society. He is a fireman, and he doesn’t question his bookless life. It doesn’t occur to him to be introspective until he meets...

The PrinceIn The Prince and Discourses on Livy, Machiavelli exhibits the characteristics of humanism in many ways. First, both works are full of allusions to classical politics, especially to the Roman...

To AutumnThe theme of "To Autumn" is the transitory quality of nature. In one of the marvelous letters written by Keats to friends, he explained that he composed "To Autumn" because Somehow a stubble plain...

ScienceThe laws of thermodynamics are as follows. The first law is that the increase in energy in a system is the sum of energy added to the system. This means that a system will not change energy unless...

Ode to a NightingaleIn the poem, the speaker claims to be "half in love" with death. He thinks that death will release him from his mental torment. Indeed, he's portrayed death in a soft light in "many a mused rhyme"...

War Is KindThe poet Stephen Crane is being intentionally ironic. He does not really believe that war is kind but that it is a horrible and senseless thing. The summary in the eNotes study guide for the poem...

ReferenceChildren of preschool age (23) often begin talking about what they dreamt about, though these reports usually involve static pictures rather than episodes with characters or emotions. Children...

Oedipus

Romeo and JulietI would like to say that people no longer fight for no reason or let their tempers get the best of them, or are ruled by affiliation and association loyalties more than logic, but this is not the...

P. D. JamesThe author of the novel upon which the 2006 film Children of Men is based is P. D. James. She was an English writer born in 1920 who died in 2014. Her novel's full title was The Children of Men and...

MacbethYour general themes are appearance versus reality and good versus evil. Frankenstein and Macbeth approach these themes quite differently, so you might want to structure your essay to compare and...

October SkyOctober Sky (originally titled Rocket Boys) a story about a boy who, despite growing up in a small mining town, develops an interest in rocketry. Sonny, the main character, founds the Bear Creek...

Guns, Germs, and SteelGuns, Germs, and Steel by Jared Diamond was intended to answer a question asked to Diamond by his friend Yali, a native of Papua New Guinea. Yali asked: Why is it that you white people developed...

Macbeth

HamletThese questions have been debated for centuries. They are called "The Hamlet Problem." I think the best answer was given by Samuel Taylor Coleridge, who said Hamlet procrastinates because he thinks...

Salvage the BonesIn short, Esch describes her love for Manny in a strange way, as "certainty" and "as deep as a tree's roots." This reveals that Esch's character is both passionate and loyal. It is the way that...

To Kill a MockingbirdIn Chapter 26, Scout is confused about Miss Gates' lack of consistency: while she claims to despise Hitler's horrific abuse of the Jews, she welcomes extremely biased treatment of the black people...

Sonnet 30Uncharacteristically, the turn in thought does not occur in Sonnet XXX until the final couplet. Perhaps in keeping with the "remembrance of things past," the lines of this sonnet return again and...

Global WarmingOn the level of media coverage of global climate change, I agree with Bill Nye's position that the media should focus on reliable information by scientific experts, serving to educate its audience...

The Duchess of MalfiFirst, we should think about the definition of a "revenge tragedy." The term was first used by the American Shakespeare scholar Ashley H. Thorndike (1871–1933) in a 1902 article "The Relations of...

Tales of a Fourth Grade NothingIn the book, Fudge is Peter's toddler brother. He's mischievous and often gets into trouble at home. In Chapter 10, we discover Fudge has swallowed Peter's pet turtle, Dribble. Frantic with fear...

The Great GatsbyThe Don McLean song "American Pie" might be a fitting song to describe Gatsby's death. For example, at the beginning of the song, the lyrics are as follows: "And I knew if I had my chance / That I...

MayflowerThe Pilgrims established peace with the Wampanoags around Cape Cod after they arrived in 1620. Their migration was followed by other migrations that brought increased English settlement to New...

Paradise of the BlindSome of the most significant recurrent symbols in Paradise of the Blind are blindness, sickness, water and river. Another is sky; it is tied closely to the river symbol. Being fluid symbols, like...

SeedfolksWhat does Florence mean by the word "Seedfolks"? Why do you think Fleischman chose this title for...Florence tells the story of how her greatgrandparents walked all the way from Louisana to Colorado in 1859. They were both freed slaves who settled alongside the Gunnison River. She comments that...