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Date: Thu, 6 May 2010 08:15:14 +0000 (UTC)
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"Roger Stafford" <ellieandrogerxyzzy@mindspring.com.invalid> wrote in message <hrt7e0$drs$1@fred.mathworks.com>...
>   Prepare yourself for a long-winded explanation. .......
- - - - - - - - -
  Please add this onto the previous discussion I gave.

  If y = f(x) is some continuous function with its first three derivatives also continuous over the interval xa < xb < xc, then by an extension of the famous Rolle's theorem of elementary calculus, there is a value xi (Greek letter) such that the difference between the expression above for an approximation of the derivative at xb and the actual derivative, f'(xb) is equal to

  1/6*f'''(xi)*(xc-xb)*(xb-xa)

where xi lies in xa < xi < xc and f'''(xi) is the third derivative value of f(x) there.  A remarkable fact!

  What this signifies is that if the function whose derivative you are attempting to approximate with the formula I gave you has a third derivative which remains very small - in other words is reasonably smooth - then you are guaranteed a close approximation.

Roger Stafford