Change at an instant AB & BC

Over an interval $[a, b]$ the average rate of change of $f$ is the slope of the secant line joining the two endpoints:

$$\frac{f(b) - f(a)}{b - a}.$$

That number summarizes the whole interval and says nothing about any single moment inside it. A car covering 120 miles in 2 hours averaged 60 mph; it may have been stopped for a while and doing 90 elsewhere. The average is a fact about the trip, not about any instant of it.

Every quantity in that formula is a genuine difference. That is exactly why it fails at an instant: setting $b = a$ gives $\frac{0}{0}$, which is not a number.

The objection is real: over zero elapsed time the object moves zero distance, so $\frac{0}{0}$ is meaningless. The response is to never evaluate at the instant at all. Instead, watch what the average rate does as the interval shrinks toward the instant.

For $f(t) = t^2$ near $t = 2$, average rates over $[2, b]$ are:

1. $b = 3$: slope 5
2. $b = 2.1$: slope 4.1
3. $b = 2.01$: slope 4.01
4. $b = 2.001$: slope 4.001

The interval never becomes zero, and the slope is never $\frac{0}{0}$. But the numbers are unmistakably closing in on 4. The instantaneous rate at $t = 2$ is defined to be that limiting value. Calculus does not divide by zero; it sidesteps the need to.

Geometrically, each average rate is the slope of a secant line through $(a, f(a))$ and a second point on the curve. Slide the second point toward the first and the secants pivot, settling into a single line: the tangent at that point.

The tangent is not "a very short secant". A secant needs two distinct points, and the tangent has only one. It is the line the secants approach, which is a different kind of object arrived at by a limiting process. Writing the limit explicitly:

$$\text{instantaneous rate at } a = \lim_{b \to a}\frac{f(b) - f(a)}{b - a}.$$

That expression is the derivative, and Unit 2 is devoted to it. For now the point is only that the limit is what makes an instantaneous rate meaningful.

Because the value is inferred from a trend, how you sample matters. Three habits protect the conclusion:

1. Get genuinely close. Steps of 0.1 rarely settle anything. Push to 0.01 and 0.001 and watch the digits stabilize.
2. Approach from both sides. A trend from the right alone does not establish a two-sided limit. If the left side heads somewhere else, there is no limit at all.
3. Read the trend, not the endpoint. What the function does at the instant is a separate question, and in Topic 1.2 it turns out to be independent of the limit.

These are not fussiness. They are the difference between a limit and a guess, and each one has its own failure mode later in this unit.