A claim goes up — “at the top of a smooth hill, the derivative is zero” — and everyone must file it under always, sometimes, or never. “Sometimes” is not a shrug: it obliges you to produce a case where the claim holds, one where it fails, and the exact boundary between them.
How we play
- Classify silently first. Gut verdicts welcome; they get audited.
- “Always” and “never” demand an argument covering every case.
- “Sometimes” demands an example, a counter-example, and the boundary.
The hilltop claim, argued
- ” peaks at the origin, and the tangent there is flat. Not never.”
- “Could a smooth curve peak on a slant? Walk toward the top: the secants arriving from the left rise, the secants leaving to the right fall. The tangent is squeezed between rising and falling — it has nowhere to go but flat.”
- “‘Smooth’ is doing real work in that argument. turned upside down peaks at a corner, and a corner has no slope at all — no tangent to be flat. The claim survives only because ‘smooth’ rules the corner out.”
- “So: always, for smooth hills. And the converse is the sometimes everyone should carry out the door: where the derivative is zero, is there a hilltop? flattens at the origin and keeps right on climbing. The boundary is whether the derivative changes sign — flat is a candidate, not a verdict.”
One variation
Claims from the vector weeks: ” equals ” — sometimes, and the boundary is sharper than it looks. Swapping the order of a cross product flips the answer’s direction, so the two sides disagree whenever the answer is a genuine arrow — and agree only when the answer is the zero vector, which happens exactly when the two vectors are parallel. A product that cares about order is new territory, and The Cross Product is where the room gets to argue about why.
"Sometimes" is where the mathematics is
The boundary of a claim is its content. The course before this one ended on a claim it could not settle: as the interval shrinks, the average rate of change settles on a single value. This course opens by settling it — sometimes, and the boundary runs exactly through the corners and jumps where no tangent exists. Drawing that line precisely is the job The Limit was invented for, and The Derivative lives on the “always” side of it.