As to this beautiful derivation of the Taylor's Theorem, wouldn't it break the equality when we add the term $f(0)$ to the right side of $f(x) = \int_{0}^{x} f'(t)dt$ ?
2026-04-09 17:27:01.1775755621
Show the correctness of this derivation to the Taylor's Theorem?
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No. The reason is as follows: we have $$ \int_{0}^{x} f'(t) dt = \left [ f(t) \right ] ^ x _ 0 = f(x) - f(0)$$ (not just $f(x)$). Thus: $$f(x)=f(0)+ \int_{0}^{x} f'(t) dt$$