Originally Posted by **roger** I have a test early next week on using the Taylor series.
Does the attached derivation look correct and is the error term correct?
Thank you all for your help so far. |

What you have done (correctly) is to

**verify** the given equation - it might be fun to try to

**derive** it from scratch. I made a table with 1st and 2nd differences, including backward extrapolation to (t-tau)

Code:

t-tau |**Extrapolate ** | | |
| |**Extrapolate** | |
t |f(t) | |**f(t+2tau)-2f(t+tau)+f(tau)**|
| |f(t+tau)-f(t) | |**Error Term**
t+tau |f(t)+..+..+.. | |f(t+2tau)-2f(t+tau)+f(tau)|
| |f(t+2tau)-f(t-tau)| |
t+2tau|f(t)+2..+4..+8| | |

Since we are neglecting the 3rd difference, it is the Error Term. Thus the 2nd difference is assumed to be constant, allowing you to calculate the 1st difference between t and t-tau. Then set

f'(t) = [sum of 1st differences]/(2 tau) = [f(t+tau) - f(t-tau)]/(2 tau)
That should produce the desired formula.

Looking at the "explicit expression" for the error term -- when you take 4(A) - (B), isn't it -(

**2**/3)tau^3 f'''(t) ? So the final expression would be (1/3)tau^2)|f'''(t)| ?

If you try to complete the derivation as outlined above, you can also extrapolate the error term back to see how it affects f'(t).

[BTW - for

**math** help you can also try the forums at

www.FreeMathHelp.com - they have a working LaTeX over there that makes it easier to "type" math.]