#### Velocity

How fast is the point moving? Using calculus, its velocity, , is defined as the derivative of with respect to time:

 (8.1)

Using numerical computations, is approximately equal to , in which denotes a small change in time and

 (8.2)

In other words, is the change in from the start to the end of the time change. The velocity can be used to estimate the change in over as

 (8.3)

The approximation quality improves as becomes smaller and itself varies less during the time from to .

We can write to indicate that velocity may change over time. The position can be calculated for any time from the velocity using integration as8.1

 (8.4)

which assumes that was known at the starting time . If is constant for all time, represented as , then . The integral in (8.4) accounts for being allowed to vary.

Steven M LaValle 2020-01-06