The average rate of change of the area of the circle with respect to its radius can be calculated by dividing the change in area by the change in radius. It is very similar to an average speed (which is rate of change of position with respect to time), a concept we are all familiar with. If we travel 150 miles in 3 hours, our average speed was 50 mph (i.e. 150 miles / 3 hours). We weren't necessarily going 50 mph the whole time, and likely weren't, but our overall speed across the entire 3 hours averaged out to that. And we covered the exact same distance as we would have if our speed had been exactly 50 mph for the entire 3 hours.
Applying those ideas to area of a circle looks like this:
A(ii) r = 2 -> r = 2.5
Area of circle when r = 2: A = π⋅22 = 4π ≈ 12.56
Area of circle when r = 2.5: A = π⋅2.52 = 6.25π ≈ 19.625
avg r o c of area w respect to r: (19.625 - 12.56) / (2.5 - 2) = 14.13
We interpret this answer to mean that over this interval (r going from 2 to 2.5) the area increased at a rate of 14.13 sq. units for every 1 unit increase in the radius.
