I am wondering whether there is any general method of approximating the midpoint of a given curve, given the coordinates of the endpoints and the equation of the curve. I know the calculus method of finding it exactly, but I am looking for a purely algebraic method that works for all elementary functions.
Considering it, I have come up with 4 methods:
Finding the intersection of the vertical line down from the midpoint of the straight line connecting the two points, and the curve;
finding the intersection of the horizontal line down from the midpoint of the straight line connecting the two points, and the curve;
finding the intersection of the perpendicular line to the straight line connecting the two points and that passes through that line's midpoint, and the curve;
guessing and checking via numerical integration.
The first 2 are clearly very poor approximations, the 3rd only somewhat better, and the last, is of course undesirable as a guess and check method.
The definition I am using is based on arc length, and I just want the method to work for as many functions as possible, I don’t really care if they are elementary or not.