Difference between revisions of "Fundamental theorem of calculus"
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m (→The Fundamental Theorem of Calculus: deleted text to avoid repetition) |
m (→The Theorem: indented equations for clarity) |
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The first can be written as: | The first can be written as: | ||
Let the function <math>f</math> be continuous function defined on a closed interval <math>[a,b]</math>. Define <math>F(x)</math> as: | Let the function <math>f</math> be continuous function defined on a closed interval <math>[a,b]</math>. Define <math>F(x)</math> as: | ||
| − | + | :<math>F(x) = \int_a^x f(t)dt</math> | |
| − | <math>F(x) = \int_a^x f(t)dt</math> | ||
<br />It follows that: | <br />It follows that: | ||
| − | + | :<br />The first part states that if a function <math>F</math> is the antiderivative of a function <math>f</math>, then the derivative of <math>F</math> is <math>f</math>. In other words, antiderivative and derivative are opposite functions. | |
| − | <br />The first part states that if a function <math>F</math> is the antiderivative of a function <math>f</math>, then the derivative of <math>F</math> is <math>f</math>. In other words, antiderivative and derivative are opposite functions. | ||
===Part 2=== | ===Part 2=== | ||
If: | If: | ||
| − | + | :<math>f(x) = F'(x)</math> | |
<br />Then: | <br />Then: | ||
| − | + | :<br />The second part begins with what we know from part 1. | |
| − | <br />The second part begins with what we know from part 1. | ||
<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>. | ||
Revision as of 19:14, December 30, 2008
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:
- 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
Part 1
The first can be written as: Let the function be continuous function defined on a closed interval . Define as:
![{\displaystyle [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)
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 .
