The average rate of change of a function $f$ on an interval $\left[a\,b\right]$ in its domain is given by the formula $r\=\frac{f\left(b\right)-f\left(a\right)}{b-a}$. Equivalently, this is the slope of the line connecting the points $\left(a\,f\left(a\right)\right)$ and $\left(b\,f\left(b\right)\right)$.

Relationship between Position and Velocity

The two plots below illustrate how well the average rate of change of a position function approximates the corresponding velocity function (assuming that the motion is along a straight line, to keep things manageable).

Click and drag on the Position graph and view the resulting information shown on the Position and Velocity graphs.

On the Position graph, the red point represents a position value. The green points, located some distance away from the red point, represent times at which position measurements have been taken. The slope of the secant line connecting the green points (the pink line) is the average rate of change of the position function over this time interval. This value is shown as the red point on the velocity graph. The vertical blue lines on the two graphs just indicate the same time value on each graph.

As you drag to the right in the position graph, the distance between the green points will increase. As you drag to the left, this distance will decrease. How well does the average rate of change approximate the actual velocity?

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