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 Derivative chains. (Posted on 2009-03-29)
For any polynomial function f(x) if you know f(a), f'(a), f"(a), f"'(a)... for some value of a you can reconstruct the function. This is true even if the polynomial has an infinite number of terms.

(f' is the first derivative of f, f" is the second derivative etc.)

Define s(a) to be the sequence of f(a), f'(a), f"(a), f"'(a), ...

For each of the following, find the function:

(1) s(1) = 19, 23, 32, 18, 0, 0, 0, 0, ...

(2) s(0) = 1, 2, 4, 8, 16, 32, ...

(3) s(1) = 1, -1, 1/2, -1/6, 1/24, -1/120, 1/720, ...

(4) s(0) = 0, 1, 0, -1, 0, 1, 0, -1, ...

(5) s(0) = ln(2), ln(2ln(2)), ln(2ln(2ln(2) )), ...

 No Solution Yet Submitted by Jer No Rating

 Subject Author Date 4 of 5 solved Brian Smith 2016-06-25 22:19:38 re: Keep trying Harry 2009-04-03 00:06:20 Keep trying Jer 2009-04-02 13:35:30 re(2): Partially done but help needed (Spoiler) Daniel 2009-03-31 21:03:02 re: Partially done but help needed (Spoiler) JayDeeKay 2009-03-30 13:30:09 Partially done but help needed (Spoiler) Harry 2009-03-29 20:26:42

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