# Note - polynomial models

Those who studied calculus in school recall that first and second derivatives can be used to highlight changes of direction for data modeled by polynomial equations. For the data used in the figures relating age to finish time, third-degree polynomial models fit the data very well (R2>0.97 in all cases).

On the graph on the previous page, after the local minimum at point B, finish times will increase at a faster and faster rate with age. But not all third-degree polynomials have distinct local maxima and minima. Rather they have a more "flattened" curvature such as is shown in the accompanying figure, where the graph continues upward throughout the range of age values, but the curvature changes from concave (concave downward) to convex (concave upward) at a certain point (point A in the graph). This point is known as the "inflection point," and it is the point beyond which finish times increase at a faster and faster rate with age. In the discussions of finish times with age on this website, both points of local minima and inflection are used, depending on the shape of the fitted polynomial, to highlight that critical age beyond which finish times cannot reasonably be expected to improve.