\[ f(x)\approx L(x)=f(a)+f'(a)(x-a)\]
\[\text{FEIL:}\quad f(x)= L(x)+\frac12 f''(s)(x-a)^2\quad \text{for \(s\) mellom \(x\) og \(a\)}\]
\[f(x)\approx P_n(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}2(x-a)^2+\dots + \frac{f^{(n)}(a)}{n!}(x-a)^n\]
\(P_1(x)=x\) \(P_3(x)=x-\frac16 x^3\) \(P_5(x)=x-\frac16 x^3+\frac{1}{5!}x^5\) |
\[f(x)=P_n(x)+E_n(x)\]
der feilen
\[E_n(x)=\frac{f^{(n+1)}(s)}{(n+1)!}(x-a)^{n+1}\quad
\text{for en \(s\) mellom \(x\) og \(a\)}\]