# How to derive Hermite polynomial from a given data set?

The problem is asking to find a Hermite polynomial to predict the position of the car and its speed when t = 10s.
The Hermite polynomial formula is defined as: $$H_{2n+1}(x) = f[z_0] + \sum_{k=1}^{2n+1} f[z_0, \ldots, z_k](x - z_0)(x - z_1)\ldots (x - z_{k-1})$$ To evaluate this formula, I have to use divided difference formula. It's understandable when the derivative of $f(x)$ is given.

However, the data set given for this problem don't have any derivative:

So my guess was trying to find the formula for distance which is $s = v.t$ then derivative of $s$ with respect to $t$ becomes $v$. On the other hand, the solution from the back of the book is odd,

It makes sense only for $$75x$$ After that, I have no idea where do those numbers $0.222\ldots, 0.03111\ldots$ come from! Because if I choose $t$ to be variable, then my table would look like this:
$$\begin{array}{rrc} z & f(z) & f'(z) \\ \hline 0 & 0 & f'(z) = 75 \\ 0 & 0 & \\ 3 & 225 & f'(z) = 77 \\ 3 & 225 & \\ 5 & 383 & f'(z) = 80 \\ 5 & 383 & \\ 8 & 623 & f'(z) = 74 \\ 8 & 623 & \\ 13 & 993 & f'(z) = 72 \\ 13 & 993 & \end{array}$$

The first value I computed is way off comparing with the coefficient from the answer, so I guess I must interpret the data in a wrong way. Does anyone could shed me some light? Thank you.

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"However, the data set given for this problem don't have any derivative..." - of course it does. What is the derivative of distance with respect to time called? –  Ｊ. Ｍ. Nov 27 '11 at 9:52
You'll want to see this on how to do divided differences for computing Hermite interpolants. –  Ｊ. Ｍ. Nov 27 '11 at 11:28