that if has a continuous second derivative, then the error is bounded by

That is, the approximation between two points on a given function gets worse with the second derivative of the function that is approximated. This is intuitively correct as well: the "curvier" the function is, the worse the approximations made with simple linear interpolation become.

History and applications

Linear interpolation has been used since antiquity for filling the gaps in tables. Suppose that one has a table listing the population of some

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