Difference between revisions of "Fundamental theorem of calculus"
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==The Fundamental Theorem of Calculus== | ==The Fundamental Theorem of Calculus== | ||
| − | The Fundamental Theorem of Calculus | + | The Fundamental Theorem of Calculus is the 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]] | + | *[[Differentiation]]—find the slope of a function's graph at a given point. |
| − | *[[integral|Integration]] | + | *[[integral|Integration]]—find the area under a graph between two given limits. |
==The Theorem== | ==The Theorem== | ||
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<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 | + | ==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] | ||
Revision as of 04:52, July 15, 2018
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This article/section deals with mathematical concepts appropriate for late high school or early college. |
Contents
The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus is the 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 Theorem
There are two parts to the Fundamental Theorem of Calculus[1]
Part 1
The first can be written as:
Let the function 
be continuous function defined on a closed interval 
. Define 
as:
It follows that:
The first part states that if a function
is the antiderivative of a function , then the derivative of is . In other words, antiderivative and derivative are opposite functions.
Part 2
If:
Then:
The second part begins with what we know from part 1.
It then states that the definite integral of the function from 
to 
is equal to evaluated at minus evaluated at .
