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mLine 9:
Line 9: − <br />+ − − <br /><math>F'(x) = f(x)</math>+ − − <br /><math>f(x) = F'(x)</math>+ − <br /><math>\int_a^b f(x) = F(b) - F(a)</math>+ −
Fundamental theorem of calculus (view source)
Revision as of 19:14, December 30, 2008
, 19:14, December 30, 2008→The Theorem: indented equations for clarity
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>.