This is interesting Alan,
You use calculus in a way that I have not seen it used before (or I don't remember anyway)
I have just enough of knowlege to attempt to understand
2x^2 -2x +5
tangent function : 4x-2
Average = sum of all tangents(numerator makes sense) over number of tangents (okay I am having trouble with the denominator)
Maybe I just need to think it through. Can you help me get my mind around it? 