Why are the middle points the most `sensitiveʹ in the sensitivity experiments?

Variation, Δ, in the life-table parameters causes the proper increment in the intrinsic rate of population increase. While the general approximation theory states that the increment is exactly equal to Δ times the sensitivity coefficient taken at some intermediate point of Δ, particular observations revealed the approximations to be surprisingly accurate when taken at the middle point of Δ. We have found the mathematical reason for that via thorough insight into the Taylor-series expansion of the increment: while, in the remainder term, there may generally be summands of the first and second order of smallness, they both vanish at the middle point of the variation.