Part 1: Local Motion and Small Errors

Calculus starts with a simple question: what does it mean to know your speed at one exact moment?

Imagine riding a bike down a straight road. Your position at time $t$ is $x(t)$, and your speed at time $t$ is $v(t)$.

Suppose that at time $t = 10$ seconds, your speed is 5 meters per second. If that speed stayed constant for a short time, then after $h$ more seconds you would travel $5h$ meters, so

$$x(10+h) = x(10) + 5h.$$

That is the constant-speed picture.

Real motion is usually not that simple. Maybe right after $t = 10$ you start pedaling harder, so your speed begins to increase. Then the straight-line formula is still a good first guess, but it is no longer exact. A more honest statement is

$$x(10+h) = x(10) + 5h + \text{(extra distance)}.$$

Instead of trying to compute that extra distance exactly, we will only track how large it is.

For example,

$$\pi = 3.14 + A(0.005)$$

means that $\pi$ differs from 3.14 by at most 0.005. We are not saying exactly what the difference is. We are only saying that it is small.

Now go back to the bike. If $h$ is small, then your speed does not have much time to change, so the extra distance should be much smaller than the main term $5h$. If $h$ is small enough, and the motion is not changing too wildly over that short interval, it is reasonable to write, a bit imprecisely,

$$x(10+h) = x(10) + 5h + A(h^2).$$

This is not meant to be a fully precise statement yet. It is a good short-time model, and it captures the picture we want.

The term $5h$ is the main change in position, and the $A(h^2)$ term is the leftover error. When $h$ is small, that error is much smaller than the main term. For example, if $h = 0.01$, then $h^2 = 0.0001$. So for a first approximation we can ignore the error and get

$$x(10+h) \approx x(10) + 5h,$$

which is the equation of a line in the variable $h$. In other words, if we zoom in to a short enough time scale, the motion starts to look linear.

This suggests a useful way to think about speed. We say, still a bit imprecisely for now, that the speed at time $t$ is the number $v(t)$ for which

$$x(t+h) = x(t) + v(t)\,h + A(h^2)$$

holds when $h$ is small. So speed is the slope of the best short-time line. In the next note, we will see how this matches the more familiar idea of average speed.