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Fundamental theorem of calculus

2007-09-05T15:28:15Z

111111NUMBERS: lol, okay, stop now

:''This page is over '''Calculus'''. Calculus is theory, '''not a fact'''. Information presented in this article should be critically interpreted and taken with an open mind. There are true challenges to this debated theory including the [[God dun maths]] theory''.
'''The Fundamental Theorem of Calculus''' is the rather remarkable result that the two fundamental operations of [[calculus]] are just inverses of each other. Those two operations are performed on [[functions]] from the [[real numbers]] to the real numbers, and are most easily visualized when the functions are expressed in terms of graphs. The operations are:
*Differentiation -- find the slope of a function's graph at a given point.
*Integration -- find the area under a graph between two given limits.
The Fundamental Theorem of Calculus says that the two operations are inverses -- to find the area under the graph of f(x) between a and b, find the function g(x) whose derivative is f(x) (that is, find the ''antiderivative'' of f.) The area under the graph of f is just g(b)-g(a).

The antiderivative of a function is often called the ''indefinite integral''. (Indefinite because the limits a and b haven't been specified.) So, for example, the derivative of <math>\frac{x^3}{3}+7</math> is <math>x^2</math>. From this it follows that the antiderivative of
<math>x^2</math> could be <math>\frac{x^3}{3}+7</math>. But note that the "7" in that formula was a red herring. Adding any constant to a function doesn't change its derivative, so the antiderivative of <math>x^2</math> could have any constant added to it. This arbitrary constant is usually written '''C''' and is called the "constant of integration. The indefinite integral could be written:

:<math>\int x^2\ \mathrm{d}x = \frac{x^3}{3} + C\,</math>,

The Fundamental Theorem of Calculus says that the area under the graph of <math>x^2</math> between a and b is the difference in the values of <math>\frac{x^3}{3}+C</math> between a and b. Note that the constant of integration cancels out.

This kind of integral is called a ''definite integral'', written with the limits:

:<math>\int_a^b x^2\ \mathrm{d}x = \frac{b^3}{3} - \frac{a^3}{3}\,</math>,

The above is a simplified "intuitive" treatment of calculus and of this theorem. The actual "rigorous" proof, "rigorous" definitions of derivative and integral, and statement of the conditions under which the theorem is true, are beyond the scope of this article.

[[Category:Mathematics]]

Internet

2007-09-05T15:27:52Z

111111NUMBERS: Replacing page with 'The '''Internet''' is a series of tubes.'

The '''Internet''' is a series of tubes.

Animal kingdom

2007-09-05T15:27:17Z

111111NUMBERS: 3 REVERT RULE IN EFFECT, ADMIN SHOULD LOCK PAGE TO PREVENT CENSORSHIP

{{merge with|Animalia}}
:''This is a page over [[life]]. Life is a theory, '''not a fact'''. Please consider the following information with an open mind. There are cirtical alternatives, including the well known '''God dun it''' theory.''
The Kingdom Animalia is a division of life that includes all the organisms that are classified as animals. They tend to have a multi-cellular structure. They consume other organic tissue to feed themselves. They are responsive to outside stimulus. They are motile durring some stage of life. Unlike plants, they do not have a cell wall.

The Animal Kingdom has the largest number of organisms yet recorded. It includes sponges, worms, insects, birds, mammals, crustations, as well as humans.

==See also==
*[[Species]]

[[Category:Biology]]

Form

2007-09-05T15:26:42Z

111111NUMBERS:

'''Form''' is the overall structural organization of a music composition and the interrelationships of music events within the overall structure. Also, it has something to do with physical objects.

[[Category:Music]]

Animal kingdom

2007-09-05T15:25:44Z

111111NUMBERS: rv 2

Fundamental theorem of calculus

2007-09-05T15:25:24Z

111111NUMBERS: NO U

Animal kingdom

2007-09-05T15:24:55Z

111111NUMBERS: rv censorship

Belle

2007-09-05T15:24:33Z

111111NUMBERS:

Belle--a screw tug completed at Philadelphia early in 1864--operated briefly at Boston before she was purchased there by the Union Navy on 2 June 1864. Commissioned in the next day or so, the tug headed south on 8 June to join the North Atlantic Blockading Squadron.

The [[Norfolk Navy Yard]] refitted Belle to serve as a torpedo boat so that she might help defend larger Union warships in the North Carolina Sounds against attacks by the Confederate ironclad ram Albemarle that, the previous spring, had emerged from the Roanoke River and defeated a Union flotilla. Ready by late July, Belle and three other tugs were towed to Hatteras Inlet by the sidewheel steamer Nansemond, entered the sounds, and proceeded under their own power to the mouth of the Roanoke. There, Belle performed picket duty, ready to sound the alarm should Albemarle reappear and then to attempt to torpedo the Southern ram. She continued this duty until Lt. William B. Cushing sank Albemarle in a daring attack on the night of 27 -28 October 1864. oh snaps

This victory cleared the way for a Union effort to retake Plymouth, N.C., on the last day of October 1864. Belle, by then commanded by Acting Master James G. Green, took part in the operation lashed to the port and unengaged side of Otsego so that she might keep her partner in motion should that double-ender’s engines be disabled. The Northern warships entered the Roanoke from Middle River above Plymouth and headed downstream toward the Confederate defensive works. About half an hour before noon, they opened fire; and, after almost an hour of bitter fighting, the Southern troops withdrew leaving the Union forces in control of the area.

Following that action, Belle remained in the sounds until after the collapse of the Confederacy the following spring. On 10 April 1865, the day after General Robert E. Lee surrendered at Appomattox, she had the satisfaction of accompanying the raised Albemarle to Hatteras Inlet as her former foe began her voyage to Norfolk.

Late in May 1865, Belle left North Carolina waters and headed for New York where she was sold at auction on 12 July 1865 to Cozzens and Co. Re-documented on 9 September 1865, the ship operated in private hands until abandoned in 1891.<ref>Dictionary of American Naval Fighting Ships [http://www.history.navy.mil/danfs/b4-list.htm]</ref>

==References==
<references/>

[[Category:Ships]]