3.7 Section 7.1 Numerical differentiation and Richardson extrapolation

Some points to know

1. If a function $f(x)$ is known at $n$ points, and we also know that the function is a polynomial of at most $n-1$ degree, then we can find the polynomial exactly by solving $n$ equations and finding the $c_0,c_1,\cdots ,c_n$ coefficients. Hence no need to do numerical differentiation, we can do analytical differentiation.

2. Remember this for Taylor: $f(x+h)=f(x)+h{f}^{\prime }(x)+\frac{h^2}{2}{f}^{\prime \prime }\left(\xi \right)$ for this to be valid, $f(x),f^{\prime }\left(x\right)$ have to be continuous in the CLOSED interval between $x$ and $h$ while $f^{\prime \prime }(x)$ need to exist on the OPEN interval.

$f(\sqrt{2}+h)-\frac{df}{dx}@\left(\sqrt{2}\right)$

$f\left(\sqrt{2}\right)$

$f^{\prime }\left(\sqrt{2}\right)=\frac{f(\sqrt{2}+h)-f\left(\sqrt{2}\right)}{h}$

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