Level 6: Speed at an instant

Change and motion · level 6 of 8 · rough age 13-14 (a guide for parents; anyone can start anywhere)

A speedometer shows your speed at this very instant, yet speed is distance divided by time, and an instant has no time in it. This level solves that old puzzle, the one that started calculus. You shrink the interval until the average speeds settle on one number, watch secant lines turn into a tangent, estimate rates from real data, discover that exponential growth has a rate proportional to its size, and find where change is fastest and where it stops.

Lesson	You will be able to
The problem of instant speed	Explain why \(0 \div 0\) is meaningless, and why instant speed is defined by where average speeds are heading
Shrinking the interval	Find an exact instant speed with tables and algebra, such as 19.6 m/s for the falling stone at 2 s
From secant to tangent	Find the tangent line at a point as the limit of secant lines, and see why a smooth curve looks straight close up
Estimating instantaneous rates from data and graphs	Draw a tangent by eye, and use forward, backward and central estimates from a table, knowing which to trust
Rates in exponential change	Show that exponential growth and decay have a rate proportional to the amount, and use it for bacteria, savings and cooling
Where is change fastest?	Find the steepest point of an S-curve, and the highest and lowest points of a curve where the rate is zero

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