Four functions, four corners, and one question: which one doesn’t belong? The trick is that there is no trick — every corner can be defended, so the game is never about the answer. It is about naming, precisely, the property your corner alone possesses — and in this course, the sharpest properties are about rates: not what a function is, but how it changes.
| A — | B — |
| C — | D — |
A defence for every corner
- A — the only one whose rate of change is itself a straight line: the slopes read , growing steadily forever. Also the only polynomial in the room.
- B — the only one that is its own rate of change: at every point, the slope equals the height. Also the only one with no zero — it never touches the axis at all.
- C — the only periodic one, and the only one whose slope changes sign forever: climbing, falling, climbing, falling, for the rest of time.
- D — the only one with a point where the rate of change has no value: zoom in on the corner at zero as long as you like and it never straightens into a line. The other three all do — a difference Zooming In hands your group a microscope for.
One variation
Four ways of writing, four corners: · the slope of the tangent to at · · . Three of the corners are the same number in different costumes — the fourth is an impostor, an average rate dressed convincingly as an instantaneous one, close enough to fool a glance (and for some curves, close enough to be exactly right). Finding it without computing anything is The Derivative earning its keep.
"It looks different" scores nothing
Precision is the whole game. Not “D is pointy” but “D has exactly one input where no tangent line exists, and everywhere else its slope is either or with nothing in between”. The slope-reading defences of corners A and B are the daily work of Derivative Rules — and the corner in D is the reason The Limit has to be stated carefully rather than waved at.