A skydiver steps out of a plane, and the distance she has fallen after seconds is metres. The question this whole course exists to answer, asked on its very first day: how fast is she falling at exactly ?
The task
Speed is distance over time — but at exactly , no time passes and no distance falls. So sneak up on it: compute her average speed from to . From to . To . To . Then sneak up from the other side: from to , and from . Draw each average as a line through the graph’s point at — what are the lines doing as the interval shrinks? Your group owes the room one defensible number — the speed at the instant — and the argument for why that number and no other. Then push: does the same sneaking-up work at ? Would it work if distances followed instead? And the question with teeth: if computing the instant head-on divides zero by zero, is “speed right now” even a real thing — or just a useful fiction?
Facilitation notes — for the teacher
Some students met a parade like this in MHF4U’s closing unit; this course begins by refusing to let it go. Insist on the table before the picture — the numerical march makes the limit feel inevitable before anyone names it. The productive fight is over whether the value the march points at is the answer or merely never reached — let both camps argue it out. Groups converge near m/s; the quick extension — convert it to km/h — lands a hair over , which makes the number feel real. Fast groups: find the instants at , , and , conjecture the pattern, and ask how a roadside camera could ever prove a speed at an instant — a question that returns wearing a badge in The Speed Camera.
What mathematics tends to surface
The averages parade in an orderly way, each closer to the last, and both parades — left and right — point at the same value like two compass needles agreeing. Geometrically, the lines through the point are secants, and they tilt toward one limiting line that touches the curve only there: the tangent. The speed at the instant is that tangent’s slope, reached by approach rather than by division. The Limit gives the settling a name.
Where it leads
Everywhere. This unit sharpens today’s manoeuvre into the definition in The Derivative; The Speed Camera puts it in front of a review board; and every unit after — the toolbox, the applications, even the vectors that close the course — keeps asking today’s question in new clothing.
The answer is not on this page
No worked solution appears here. The number — and the argument that forces it — belongs to your group at the boards.
Curriculum connection
A1.1
describe examples of real-world applications of rates of change, represented in a variety of ways (e.g., in words, numerically, graphically, algebraically)
Link to original
A1.3
make connections, with or without graphing technology, between an approximate value of the instantaneous rate of change at a given point on the graph of a smooth function and average rates of change over intervals containing the point (i.e., by using secants through the given point on a smooth curve to approach the tangent at that point, and determining the slopes of the approaching secants to approximate the slope of the tangent)
Link to original