Formula help needed algebraic formula for intra-interval rate of acceleration

[seasnake]

This isn't for a class, just for my own personal use as I am not a student, but I need an equation in algebraic form that allows me to calculate the average acceleration between spreadsheet intervals given that the intra-interval rate of acceleration is to be treated as a constant, and the speed at the beginning of the interval is known and the speed at the end of the interval is also known.

T = time
Speed at T = 4 is 6
Speed at T = 5 is 4
Speed at T = 6 is 7
Speed at T = 7 is 7

I am searching for an algebraic formula (non-derivitative, non-trig) that gives the average rate of intra-interval acceleration if the rate of intra-interval acceleration were constant between T = 5 and T = 6. Speed at T = 4 and speed at T =7 should have no impact to this intra-interval multiplier.

Said another way, I have a spreadsheet and I have formulated it so that the average rate of interval acceleration between T = 5 and T = 6 is (Speed at T = 5 + Speed at T = 6)/2, but I know that this is incorrect as the intra-interval rate should rise exponentially between T = 5 and T = 6, so instead of multiplying by 0.5 I should be multiplying by a different fraction that is exponentially related.

 

[CompuChip]

Acceleration is change of speed per time step, not average velocity during the time step.
So the acceleration between time T and T' is
[(speed at T') - (speed at T)] / (T' - T)

 

[HallsofIvy]

Since this is not "Linear and Abstract Algebra", I am moving it to "General Math".