Two things a worksheet reaches for that have no closed form: where a function crosses zero, and the area under it. Both take the name of a function, the way map and iterate do, and both do a fixed amount of work rather than testing a tolerance — so both give the same bits on every machine.
A root, by bisection. It needs a bracket and says so if it does not have one.
Dimensions travel through: a root of a length equation is a length.
root needs a bracket. roots needs only a window: it samples a fixed 200 intervals across it and bisects every sign change it finds. That is the question a worksheet usually has — where does this cross zero between here and here — when the answer may be one place, two, or none.
A wider window holds both zeros of the same parabola, and they come back in increasing order.
The slope of a function at a point, exactly: no step size, because every value carries how fast it is changing and every operation carries the chain rule.
The second derivative is where a curve turns, and it divides the dimension twice: a distance by a time twice over is an acceleration.
Which is what makes a peak findable: it is where the slope crosses zero, so the two numerical methods compose.
A definite integral, by Simpson's rule over a fixed number of panels. Exact up to a cubic, which covers the load shapes a beam calculation actually uses.
The resultant of a triangular distributed load, which is where this earns its place: the dimension falls out of f(x)·dx with no rule about integration.
Its line of action, from the first moment of the same load.
A system of equations linear in its unknowns, which is what statics is: the reactions of a body under load. f is a residual — what is left over when the unknowns are substituted — and the second argument names them by dimension.
The Euclidean norm, and the unit vector that comes of dividing by it.