Suppose we are concerned with the function
We can evaluate at natural numbers .
Indeed,

,
,
,
,

We are going to approximate by three polynomials, of degrees 1,
2, and 3. Let's call them , , and ,
respectively.
will be determined by the requirement that
will be determined by the requirement that
will be determined by the requirement that
Thus is simply the linear function whose graph passes through the
points and .
In fact,
.
is the quadratic function whose graph intersects the graph of
in the points , , and .
is the cubic function whose graph intersects the graph of
in the points , , , and .
The idea is illustrated in this Figure:
The graph of is shown in red. The graph of is
green,
that of is yellow, and that of is blue. The blue
graph is mostly covered up by the red graph, which indicates how
closely the cubic polynomial approximates the exponential
, particularly in the interval .