What is an intuitive explanation for the fact that the remainder of a Taylor polynomial has this equivalency? Why is z necessarily between x and c? Why does such a value for z exist in the first place?
2026-03-30 06:44:14.1774853054
Intuition behind Taylor's Theorem
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For n = 0, it's applying the mean value theorem to f(x).
For n = 1, it's applying the mean value theorem to
f(c) + f'(c)(x - c).
And so to n. So it's basically an elaborate MVT.