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Act out the following story Daniel McLaury found on MathOverflow:
During the Russian revolution, there is a story of a mathematician (I've heard Igor Tamm may be the one) who was mistaken by rebels to be a communist spy. He was promptly captured by a local gang and interrogated. When he said that he is a mathematician, the gang leader asked him to back up his claim by deriving the formula for the Taylor Remainder Theorem. He was warned that if he failed, he would be shot on the spot. After some sweating the mathematician finally derived the result. The gang leader was satisfied with the proof and let him go.
It turns out there are several possible forms of this "remainder". The most intuitive is the integral form of the error. There are also "mean value" forms (derived from the mean value theorem in various ways): one by Lagrange and one by Cauchy. Take your pick! :)