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Suppose that <math>f(x)</math> is a continuously differentiable function on | Suppose that <math>f(x)</math> is a continuously differentiable function on | ||
<math>[a,b]</math>. Let <math>N</math> be a positive integer and let | <math>[a,b]</math>. Let <math>N</math> be a positive integer and let | ||
| − | <math>M=\text{Max}\ |f'(x)|: a\le x\le b\}</math>. Define <math>R_N</math> | + | <math>M=\text{Max}\ \{ |f'(x)|: a\le x\le b\}</math>. Define <math>R_N</math> |
| − | to the the ''right endpoint'' Riemann Sum | + | to be the the ''right endpoint'' Riemann Sum |
<math>R_N = \sum_{n=1}^N f(a+n\Delta x)\Delta x</math> | <math>R_N = \sum_{n=1}^N f(a+n\Delta x)\Delta x</math> | ||
Latest revision as of 03:31, 16 September 2008
The basic estimate for the rectangle method
Suppose that $ f(x) $ is a continuously differentiable function on $ [a,b] $. Let $ N $ be a positive integer and let $ M=\text{Max}\ \{ |f'(x)|: a\le x\le b\} $. Define $ R_N $ to be the the right endpoint Riemann Sum
$ R_N = \sum_{n=1}^N f(a+n\Delta x)\Delta x $
where $ \Delta x = (b-a)/N $, and let
$ I=\int_a^b f(x)\ dx $.
We shall prove that the error, $ E=|R_N-I| $ satisfies the estimate,
$ E\le \frac{M(b-a)^2}{N} $.
