Newton's Forward Interpolation Formula

When to use it

Newton's forward formula estimates $y$ for an $x$-value near the BEGINNING of an equally-spaced table, using the first row's own forward differences.

The formula

With $p=\dfrac{x-x_0}{h}$ ($x_0$ the first tabulated point, $h$ the common spacing):

Worked reasoning

Using the table from the previous section ($x_0=0,y_0=1,\Delta y_0=2,\Delta^2y_0=2,\Delta^3y_0=0$), estimate $y$ at $x=0.5$: here $h=1$, so $p=\frac{0.5-0}{1}=0.5$.