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{{Math-h}}

==The Fundamental Theorem of Calculus==

==The Fundamental Theorem of Calculus==

The Fundamental Theorem of Calculus, first proven by [[James Gregory]], 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:

The Fundamental Theorem of Calculus, first proven by [[James Gregory]], 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.

*[[Differentiation]]—find the slope of a function's graph at a given point.

*[[integral|Integration]] -- find the area under a graph between two given limits.

*[[integral|Integration]]—find the area under a graph between two given limits.

==The Theorem==

==The Theorem==

<br />It then states that the definite integral of the function <math>f</math> from <math>a</math> to <math>b</math> is equal to <math>F</math> evaluated at <math>b</math>  minus <math>F</math> evaluated at <math>a</math>.

<br />It then states that the definite integral of the function <math>f</math> from <math>a</math> to <math>b</math> is equal to <math>F</math> evaluated at <math>b</math>  minus <math>F</math> evaluated at <math>a</math>.

==External Links ==

==External links ==

* [http://archives.math.utk.edu/visual.calculus/4/ftc.9/ Proof for the Fundamental Theorem of Calculus]

* [http://archives.math.utk.edu/visual.calculus/4/ftc.9/ Proof for the Fundamental Theorem of Calculus]