[Axonymous]

Go to wolframalpha.com and ask for:

"derivative of integral of (sin(pi*x)-(a*x))^2 from -1 to 1 with respect to a"

Set the result equal to 0 and solve for a. This gives you the line that is the "best" approximation. (I believe.)

It is not the answer to question 4. It is the result that our simulation method hopefully gets close to.

What's interesting to me is how far from this value for a our simulation result is.