In general, you can say things are worn out.
What do you expect? I've been using them for the past two years. They're worn out!
For many objects, there are alternative words, but they're less general:
For your car example, I'd say dilapidated. This word works best for larger, more complex objects. A house or car can be dilapidated, but the phrases ?dilapidated keys and ?dilapidated headphones sound weird to me. You can also say a car is broken down, but only if it doesn't actually move.
For keys, I agree that rusty works, but only if they're actually rusty. If they're worn down, I'd say they're worn.
Sometimes you don't need a special word. I might complain about my broken old headphones. You could also say they're in a state of disrepair, but it doesn't sound very conversational to me.
? This symbol means "This phrase is of dubious acceptability. I suggest you avoid this phrase."
This is quite technical mathematical language, the vertex, or apex is the point at which the curvature is a minimum or a maximum. For a vertical parbola, the vertex occurs at the maximum point. For a car driver, the apex of a turn is the point at which the steering wheel is turned the most.
However, it is more common to speak of
- a "stationary point" (zero gradient, applies to 1D as well higher dimensions),
- a "turning point" (a stationary point that isn't an inflexion) or
- a "(local) maximum/minimum" (which will be at a stationary point, if the curve is differentiable)
"Turning point" is the most useful term for the top or bottom of a curve.
Find the turning points of the curve y = x³ - x, and for each point, determine if it is a maximum or minimum.
Best Answer
"disc" means "thin, flat, circular object" and is mostly applied to 3D objects. "Circle" is for 2D-ish circular things.
In math, "disc" has its own meaning; the meaning you mentioned, as well as the normal meaning.
Names of shapes cover both definitions you provide ("a line in the shape of X" and "filled X"), except circle (and in non-rigorous talk, they get conflated too).