Velocity and Acceleration: Put the pedal to the metal

Now we need some everyday uses for the derivative. We need to tie it into your day-to-day life, make you feel as if it has some immediacy for you. We want to show how it can become more than just an acquaintance, it can be a real friend. But what makes someone your friend? Well, it helps if they do things for you. Send you cards on your birthday. Make you a special dinner once in a while. Derivatives will not do that kind of stuff for you. But they will do things that your other friends can't do, things that will convince you to open your heart. For instance, derivatives will tell you how fast you are going. Say you are driving down the road, and you look at the speedometer, and it reads 65 mph. That speedometer is telling you the instantaneous velocity of the car. It's the instantaneous velocity, because if you lean on the accelerator a little, you will speed up, and the velocity will change. The speedometer is telling you the current speed, not the average speed over the entire trip.

The distinction between constant velocity  and varying velocity is important. You know how people always say velocity = distance/time. That's certainly true if you are talking about the average velocity over the entire trip. But unless you have cruise control, you are probably going faster than the average velocity some of the time and slower than the average velocity  other parts of the time. Be careful to distinguish between the average velocity, given by

$$\frac{\text{total distance}}{\text{total time}}$$

and the instantaneous velocity, which is what we are going to talk about now.

What is the instantaneous velocity? We can think of it as the average velocity over a very short interval of time. So suppose f(t) tells us our position at time t, where we think of our position as a value along a line. (You can think of the line as corresponding to the road going from Walla Walla, WA to Pocatello, ID.) At a particular time t, we would like to find our instantaneous velocity v(t). Well, let $\Delta t$ be a very small interval of time. Then f(t) is where we are at time t and $f(t + \Delta t)$ is where we are at time $t+ \Delta t$. We have traveled a distance $f(t+ \Delta t)-f(t)$ over the time interval $\Delta t$. Therefore, our average velocity  over the interval of time $\Delta t$ is

$$\frac{ f(t+ \Delta t)-f(t)}{ \Delta t} .$$

Now, we don't want the average velocity over a little time interval, WE WANT TO KNOW THE VELOCITY RIGHT NOW, AT THIS VERY INSTANT. We just take the average velocities as the length of the time interval shrinks to nothing. So

$$v(t) = \lim _{\Delta t \to 0} \frac{ f(t + \Delta t) - f(t) }{\Delta t}.$$

But hey, you would have to be mighty hung over not immediately to say, ``Wait a minute, that's just the limit definition for the derivative. Well, I'll be hornswaggled. That velocity function is just the derivative of f(t)."

That's right. v(t) = f'(t), where f(t) is the position function.