Change at an Instant, Without Dividing by Zero

You'll learnwhy an average rate is a statement about an interval and never about a moment inside it, why setting the two endpoints equal destroys the calculation rather than finishing it, and how shrinking the interval produces a definite number that the chords approach and never reach.

A speedometer reads 60 at a single instant, and yet in an instant the car covers no distance and no time passes. Every average rate you can actually compute needs two distinct moments, so the one number the speedometer is reporting is the one number the formula refuses to produce. Calculus does not answer this by dividing by zero. It answers it by never evaluating at the instant at all, and asking instead what the averages over shrinking intervals are heading toward.

8 STEPS · 6 QUICK CHECKS · APPROACH, NEVER ARRIVE · v1

an average rate needs two distinct points ⇒ the rate at one instant is the limit of those averages

Before you start

What you're looking at

One graph of a position function, with a fixed point marked at t = 2. A second point sits further along the curve, and a straight line is drawn through the two of them. That line's slope is the average rate over the interval between them.

The question

The average needs two distinct moments. So what number, if any, describes the rate at the single moment t = 2 — and where would it come from?

Watch for

The second point slides in and stops short every time. It is never allowed to arrive, and that restriction is what keeps every number on screen a genuine quotient rather than zero over zero.

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