Shrink the Interval Until the Average Becomes an Instant

You'll learnto compute an average rate of change as a secant slope, and to get the instantaneous rate at a single point as the limit those averages close in on.

Two points on a curve give you a slope you can compute with nothing but arithmetic: change in height over change in time. That number is an average — a fact about the whole stretch between them, and about no single moment inside it. Now slide the second point in. The averages stop wandering and start crowding together, and the one number they crowd in on is the rate at the first point. This topic is that distinction, and the single quotient both sides of it are built from. It ends with one number at one named point; setting the point free is 2.2's job.

8 STEPS · 6 QUICK CHECKS · AVERAGE = A SECANT SLOPE ON AN INTERVAL · INSTANT = THE LIMIT OF THOSE AVERAGES · v1

average on [1, 4] = 75 m/s ⇄ rate at x = 1 = 90 m/s

Before you start

What you're looking at

The height of a ball, f(x) = 100x − 5x², x seconds after launch. Two points are marked, at x = 1 and x = 4, and the line through them is the secant whose slope we can measure.

The question

That secant slope is the average rate over the whole interval. What number is the rate at the single instant x = 1, and where does it come from?

Watch for

The right-hand point sliding in, the readout settling on 90, and the last step where 75 is offered as the rate at x = 1 and the line it draws cuts the curve twice.

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