
We use limits to compute instantaneous velocity.

When we compute average velocity, we look at To obtain the (instantaneous) velocity, we want the change in time to “go to” zero. By this point we should know that “go to” is a buzz-word for a limit. The change in time is often given as the length of a time interval, and this length goes to zero.

The average velocity on the (time) interval $[a,b]$ is given by Here $s(t)$ denotes the position, at the time $t$, of an object moving along a line.

Let’s put all of this together by working an example.

In our previous example, we computed average velocity on several different intervals.

For example, the average velocity on the time interval $[2,t]$ is $v_{\text {av}}=8-16t$.

Note that the size or the length of that time interval is $t-2$.

If we let $t\to 2$, the size of the interval will go to $0$.

So, as $t$ approaches $2$, we are computing the average velocity on smaller and

smaller time intervals, and the limit of these average velocities should be called

the instantaneous velocity at $t=2$.

Limits will allow us to compute instantaneous velocity.

Let’s use the same setting as before.