Understanding Instantaneous Speed and Derivatives

Telling a cop your average speed was under the limit will not get you out of a ticket.

That’s because your average speed is just the total distance you traveled divided by the total time.

It smooths out all the moments you were speeding and slowing down. The cop's radar

gun, however, clocks your instantaneous speed—how fast you're going in that one

single moment. And that's where calculus comes in. To find that instantaneous speed,

we take the average speed over an infinitely tiny slice of time. This idea is called the derivative,

and it’s the heart of calculus.