We know that
is at least three times differentiable, so we will use the second degree Maclaurin polynomial, which is
That is,
is approximated by
.
To estimate the error of our approximation, by the Lagrange Remainder Theorem we have
where
is in the interval
. Note that this expression is always positive for
in
. Furthermore, the numerator is decreasing on this interval and the denominator attains its minimum at
, which implies that this error is maximised at
. We conclude that the error
, as required.