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.