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Proofs from Taylor polynomials

All proofs from Taylor polynomials.

[PROOF]

Write $Q$ in powers of $x-a$ :

Q(x)=c_0+c_1(x-a)+\cdots+c_n(x-a)^n.

Evaluating at $a$ gives $Q(a)=c_0$ . After differentiating once, evaluating at $a$ gives $Q'(a)=c_1$ and after differentiating $j$ times it gives

Q^{(j)}(a)=j!c_j.

The matching condition therefore forces

c_j=\frac{f^{(j)}(a)}{j!}

for every $0\le j\le n$ . These are exactly the coefficients of $P_n$ , so $Q=P_n$ .

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