Part 3: From A to o(h)

In Parts 1 and 2, we wrote formulas like

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

That was a good way to track the leftover error, but it was intentionally a little loose. Now we want the precise version of the same idea.

One reason the A notation is useful is that you can calculate with it. For example,

$$\sin x = A(1),$$

because sine always stays between $-1$ and $1$. You can also manipulate it algebraically.

$$\begin{aligned} \bigl(3.14 + A(0.005)\bigr)\bigl(1 + A(0.01)\bigr) &= 3.14 + A(0.005) + A(0.0314) + A(0.00005) \\ &= 3.14 + A(0.03645) \\ &= 3.14 + A(0.04). \end{aligned}$$

Another example is

$$\begin{aligned} \bigl(1 + A(t)\bigr)^2 &= 1 + 2A(t) + A(t)^2 \\ &= 1 + 2A(t) + A(t^2) \\ &= 1 + 2A(t) + A(t) \qquad \text{if } t = A(1) \\ &= 1 + 3A(t). \end{aligned}$$

These examples all do the same job. The $A$ notation lets us carry an unnamed error term through algebra while still keeping track of its size.

Let's go back to

$$x(t+h) = x(t) + v(t)h + \text{error}.$$

The precise way to say that the error is much smaller than the main linear term is to write the error as $h\varepsilon(h)$, where $\varepsilon(h) \to 0$ as $h \to 0$. We do not need to know exactly what $\varepsilon(h)$ is. We only need to know that it becomes small as $h$ becomes small. So the exact local statement of speed is

$$x(t+h) = x(t) + v(t)h + h\varepsilon(h), \qquad \varepsilon(h) \to 0.$$

We abbreviate $h\varepsilon(h)$ using o notation, so the same statement becomes

$$x(t+h) = x(t) + v(t)h + o(h).$$

This is the precise version of the informal $A(h^2)$ language from Part 1. The $A$ notation gives an absolute bound on the error. The o notation gives a relative bound: once you divide the error by $h$, what is left goes to 0. In other words, the error is negligible compared with $h$ itself.

Part 4 will use three simple rules for o notation over and over, so it is worth stating them clearly.

So the old $A(h^2)$ language was a good way to bound errors, while o notation gives the exact local statement needed for proofs. In the next note, we will use that exact form to prove the power rule.