Problem
A student collects five points that lie on a line and concludes “the law is linear everywhere”. Discuss why this is a risky extrapolation, looking at the diagram where the line and the real curve coincide only over the measured range.
Solution
The crux of the problem: interpolation versus extrapolation. Within the range where the data were collected (to the left of the “end of data” line), the five points lie on a line and the real law — even though in truth it is a curve — is locally well approximated by that line. Any smooth function, when observed over a sufficiently small interval, appears almost linear. Concluding that the points lie on a line over the measured range is therefore legitimate (interpolation).
Why extrapolating is risky. Claiming that “the law is linear everywhere” means extending the line beyond the measured range too (extrapolation). But there we have no data at all: as the diagram shows, beyond the “end of data” line the real curve (red) progressively departs from the line (blue). The collected data contain no information whatsoever about the system’s behaviour in that region. The agreement observed over the measured range guarantees nothing outside it.
Historical example: the ultraviolet catastrophe. The Rayleigh-Jeans law for black-body radiation magnificently reproduced the data at low frequencies, but extrapolated to high frequencies it diverged towards infinity, in conflict with experiment. It was precisely that failure of extrapolation that demanded a new idea, Planck’s quantisation of energy, and opened up quantum physics. Good local agreement is never proof of universal validity.
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Topics: Metodo sperimentale Skills: Lettura dei grafici · Linearizzazione dei dati