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Theorems from Taylor polynomials

All theorems from Taylor polynomials.

[THEOREM]

Suppose $Q$ is a polynomial of degree at most $n$ satisfying

Q^{(j)}(a)=f^{(j)}(a),\qquad 0\le j\le n.

Then $Q=P_n$ , where $P_n$ is the Taylor polynomial of $f$ about $a$ .

How accurate this model is away from the centre is measured by the remainder in Taylor's theorem.

Part of the thread Local approximation, in 7 places.

Replace a hard object by a simple model that matches it near a point, and bound the error.

DFTaylor polynomialone variable: match derivatives at a centreCAL 3 / Here