6524123 Essay

827 words - 3 pages

Concepts of Calculus Calculus was invented independently by two mathematicians, Issac Newton of England and Gottfried Wilhelm Leibniz of Germany in the 1680s. Leibniz published his research in the journal ¡¥Acta Eruditorum¡¦ in 1684 and Newton ¡¥s treatise was published in ¡¥Principia Mathematica¡¦ in 1687.Calculus was mainly developed to solve two types of problems ¡V the determination of tangents to curves, fig.1, and the calculation of area, fig. 2, especially for irregular shapes. Both of these problems are closely related to the rate of change of continuous function and the fundamental concept of calculus (also known as the ¡¥limit¡¦). The following shows what limit is and what are its applications.Fig.1 Given a function f(x) and a point P(x,y) on its graph, find an equation of the line tangent to the graph at P.Fig.2 Given a function f(x), find the area between the graph of f(x) and an interval [a,b] on the axis.In plane geometry, a line is called a ¡§tangent¡¨ to the circle if it meets the circle at precisely one point, fig.3a. However, this definition is not correct for all kinds of curves and irregular shapes. In fig.3b, the line meets the curve exactly once, yet it is not a tangent. Meanwhile in fig.3c, the line is a tangent, yet it meets the curve more than once.Fig.3 For tangent that applies to curves other than circles as in fig.4a, point P on curve in the xy-plane and Q is any point on the curve different from P, the line through P and Q is the ¡¥secant¡¦ line (the secant line is a line that intercepts a curve at at least two points). If Q moves along the curve toward P, fig.4b., the secant line will rotate toward a ¡¥limiting¡¦ position. The line occupying this limiting position is considered to be the tangent line at P.Fig.4 The area of the plane regions also leads to the concept of a ¡§limit¡¨ (the approximated area of a curve). Area of some plane regions can be calculated by subdividing them into finite number of rectangles or triangles, then adding the areas of its parts, fig.5. In calculus the area of an irregular shape can be approximated by inscribing rectangles of equal width under the curve and adding the areas of these rectangles, fig.6. If the...

Find Another Essay On 6524123

charant Creon as the Main Character of Antigone

1231 words - 5 pages Creon as the Main Character of Antigone   Throughout the Greek play Antigone by Sophocles, there exists a dispute as to who should receive the designation of main character. Antigone, the daughter of the cursed King Oedipus, as well as Creon, stately king of Thebes, both appear as the key figures in this historic play. I believe that Creon, king of Thebes, should be considered the main character in this work of Greek theater. Three

Free Macbeth Essays: Sleep and Sleeplessness

525 words - 2 pages The Sleep and Sleeplessness Motif in Macbeth We have consciences that function to tell us the difference between right and wrong. If we have clear consciences, we usually possess the ability to sleep. But when our consciences are full of guilt, we experience a state of sleeplessness. In Macbeth, Shakespeare uses the sleep and sleeplessness motif to represent Macbeth's and Lady Macbeth's consciences and the effect Macbeth's conscience has on

Life Outside of Life in Hawthorne’s Wakefield

898 words - 4 pages Life Outside of Life in Hawthorne’s Wakefield   Efficacy lies at the heart of human desires for immortality. Characters throughout literature and art are depicted as wanting to step aside and see what their world would be like without their individual contributions. The literary classic A Christmas Carol and the more recent, but ageless, film It’s Wonderful Life both use outside influences (three ghosts and Clarence the Angel

Essay on Identity in Song of Solomon

2172 words - 9 pages Searching for Identity in Song of Solomon         Abstract: Whether Africans really fly or just escape a monumental burden, perhaps only through death, is a decision Toni Morrison has apparently left to her readers. Never the less, no matter what you believe, within Song of Solomon, the suggestion is, that in order to "fly" you must go back to the beginning, back to your roots. You must learn the "art" from the old messages.   O

The Character of Oedipus in Oedipus and The Infernal Machine

904 words - 4 pages The Character of Oedipus in Oedipus and The Infernal Machine    The stories of Oedipus, as told through Seneca's Oedipus and Cocteau's The Infernal Machine, contain both similarites and differences. Both authors portray the character of Oedipus as being obstinate, ignorant, and inquisitive. Yet Seneca and Cocteau differ on their interpretation of the motives that propelled these characteristics of Oedipus. Seneca portrays Oedipus as a

Okonkwo's Tragic Flaws in Chinua Achebe's Things Fall Apart

3121 words - 12 pages        An increasing amount of contemporary literature traces its origins back to the early works of Greece. For ages, humans have fascinated themselves with the impossible notion of perfection. Unrealistic expectations placed on those who were thought to be the noblest or most honorable individuals have repeatedly led to disappointment and frustration, either on the part of those particular individuals or those they influence. Classic

Sophocles' Antigone - Antigone Must Challenge Creon

889 words - 4 pages Antigone Must Challenge Creon in Antigone   In his "Funeral Oration" Pericles, Athens's leader in their war with other city-states, rallies the patriotism of his people by reminding them of the things they value. He encourages a sense of duty to Athens even to the point of self-sacrifice. He glorifies the free and democratic Athenian way of life and extravagantly praises those willing to die for it. In Antigone, Creon, Thebes's leader in

The Role of Women in Homer’s Iliad

796 words - 3 pages The Role of Women in Homer’s Iliad Homer’s Iliad is undoubtedly focused on its male characters: Achilles, primarily, but also Hector and Agamemnon. Nevertheless, it seems that the most crucial characters in the epic are female. Homer uses the characters of Thetis, Andromache, and Helen as a basis for comparison to the male characters. Homer wants his audience to see and understand the folly of his male characters in choosing war over peace

A Comparison of Butler's Life and Kindred

1915 words - 8 pages A Comparison of Butler's Life and Kindred   What lies in the mind of an author as he or she begins the long task of writing a fiction novel? This question can be answered if the author's life is studied and then compared to the work itself. Octavia E. Butler's life and her novel Kindred have remarkable comparisons. This essay will point out important events of Butler's life and how they link to the mentioned novel. Octavia Estelle

Pillars of Metaphorical Ambiguity in The Scarlet Letter

1439 words - 6 pages Pillars of Metaphorical Ambiguity in The Scarlet Letter Among the multiplicity of arcane elements hidden beneath the words in Hawthorne's "The Scarlet Letter", none is so apparent, yet strikingly subtle to the reader's perception and consumption of characterization than the allegorical play on words within the names of the characters.  Both the protagonist and her rival within the plot are blessed with conveniently appropriate, fitting

An Analysis of Robert Ji-Song Ku's Leda

2003 words - 8 pages An Analysis of Robert Ji-Song Ku's Leda       In Robert Ji-Song Ku's short story "Leda," the main character, Sorin, leads a life of imitation. He applies himself to his graduate studies in comparative literature a little too readily: he compares not just text to text; he also compares his life to text, to "works of literature" (Wong 281). If his life does not match that of at least one literary character on several levels of interpretation

Similar Essays

Reality And Illusion In Shakespeare's Hamlet Reality, Appearance And Deception

896 words - 4 pages Reality and Illusion in Hamlet   Shakespeare’s play, Hamlet, begins with the appearance of a ghost, an apparition, possibly a hallucination. Thus, from the beginning, Shakespeare presents the air of uncertainty, of the unnatural, which drives the action of the play and develops in the protagonist as a struggle to clarify what only seems to be absolute and what is actually reality. Hamlet's mind, therefore, becomes the central force of the

Sub Plots In Hamlet Essay

1118 words - 4 pages Sub-plots in Hamlet   There are many things that critics say make Hamlet a "Great Work," one of which is the way that Shakespeare masterfully incorporates so many sub-plots into the story, and ties them all into the main plot of Hamlet’s revenge of his father’s murder. By the end of Act I, not only is the main plot identified, but many other sub-plots are introduced. Among the sub-plots are trust in the Ghost of King Hamlet, Fortinbras, and

Hamlet As Victim And Hero Essay

1301 words - 5 pages Hamlet as Victim and Hero      Hamlet, Prince of Denmark, a Shakespearean tragedy, tells the story of Prince Hamlet, who gained the knowledge of a terrible incident that his kingdom had suffered. Claudius, the king of Denmark and Hamlet's uncle, had killed his own brother, the king, who was also the father of Hamlet, and married his brother's widow. Hamlet suffered these traumas to a severe degree, and his only relief was to defeat his

Essay On Light And Dark In Antigone

1188 words - 5 pages Use of Light and Dark in Antigone   The "Golden Age" of Greece is noted for its many contributions to the creative world, especially in its development of the play. These performances strived to emphasize Greek morals, and were produced principally for this purpose. Antigone, by Sophocles, is typical. The moral focused on in Antigone is the conflict between physis (nature) and nomos (law), with physis ultimately presiding over nomos