$\textbf{Build the whole divided difference table in one go, instead of computing b0, b1, b2 one at a time.}$
The data is $x: 2.0, 2.5, 3.0$ and $y: 3.0, 3.5, 5.0$.
$\textbf{Zeroth order column, the raw y values.}$
$f[x_0] = 3.0$, $f[x_1] = 3.5$, $f[x_2] = 5.0$.
$\textbf{First order column, the slope between consecutive points.}$
$f[x_0,x_1] = \dfrac{3.5 - 3.0}{2.5 - 2.0} = 1.0$
$f[x_1,x_2] = \dfrac{5.0 - 3.5}{3.0 - 2.5} = 3.0$
These two numbers are the slope over each small interval, and $f[x_0,x_1] = 1.0$ already matches the given $b_1 = 1$.
$\textbf{Second order column, the change of the slope.}$
$f[x_0,x_1,x_2] = \dfrac{f[x_1,x_2] - f[x_0,x_1]}{x_2 - x_0} = \dfrac{3.0 - 1.0}{3.0 - 2.0} = 2.0$
This last entry of the table is exactly $b_2$, since Newton's form uses $b_2 = f[x_0,x_1,x_2]$.
$\textbf{Final answer.}$
Reading straight off the divided difference table, $b_2 = 2$, which is option (A).
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