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Let $f$ be a function defined on the interval $[x_0 −h, x_0, x_0 +h]$, and $f \in C^3[x_0 −h, x_0 +h]$.

Let $h$ be the Lagrange interpolation polynomial of $f$ at the nodes $x = x_0 − h$, $x_0$, $x_0 + h$.

If we approximate the derivative $f'(x_0)$ by $h'(x_0)$, why is it true that the above approximation is exact if $f$ is a polynomial of degree less than or equal to $2$?

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  • $\begingroup$ How many parabolas can one draw through three points? $\endgroup$ – user147263 Nov 9 '15 at 2:54
  • $\begingroup$ two? (if you mean there is only one line passing through them) $\endgroup$ – lulu Nov 9 '15 at 3:12
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    $\begingroup$ Only one. Because if you had two quadratic polynomials attaining the same values in three points, their difference would have three roots. Use this to conclude that $h\equiv f$. $\endgroup$ – user147263 Nov 9 '15 at 3:13

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