Answer
The greatest magnitude of the average rate of change was in the period 2007-2009, when emigration from Ireland was increasing by $10,000$ people per year.
Work Step by Step
The average rate of change of $f$ over the interval $[a, b]$ is given by:
Average rate of change of $f\displaystyle \ \ =\frac{f(b)-f(a)}{b-a}$
The units of the average rate of change of $f$ are units of $f(x)$ per unit of $x$.
Taking 2-year intervals,
$\left[\begin{array}{llll}
a & b & [f(b)-f(a)]/(b-a) & \\
& & & \\
6 & 8 & (45-38)/2 & =3.5\\
7 & 9 & (60-40)/2 & =10\\
8 & 10 & (30-45)/2 & =-7.5\\
& & &
\end{array}\right]\quad $
thousand people per year
$a.$
The greatest magnitude of the average rate of change was in the period 2007-2009, when emigration from Ireland was increasing by $10,000$ people per year.
