Hans Georg Schaathun
September 2016
Approksimér \(f(x)\) rundt punktet \(x_0\)
$$P_1(x) = L(x) = f(x_0) + f'(x_0)(x-x_0) $$
$$\epsilon_1 = f(x) - L(x) = \frac{f''(s)}{2}(x-x_0)^2\quad s\in(x_0,x)$$
$$P_2(x) = f(x_0) + f'(x_0)(x-x_0) + \frac{f''(x_0)}{2}(x-x_0)^2$$
Approksimér \(f(x)\) rundt punktet \(x_0\)
$$P_1(x) = f(x_0) + f'(x_0)(x-x_0) $$
$$P_2(x) = f(x_0) + \frac{f'(x_0)}{1}(x-x_0) + \frac{f''(x_0)}{1\cdot2}(x-a)^2$$
$$\epsilon_2 = \frac{f'''(s)}{1\cdot2\cdot3}(x-a)^3\quad s\in(x_0,x)$$
$$ \begin{align} \begin{split} P_3(x) = f(x_0) &+ \frac{f'(x_0)}{1}(x-x_0) \\& + \frac{f''(x_0)}{1\cdot2}(x-x_0)^2 \\& + \frac{f'''(x_0)}{1\cdot2\cdot 3}(x-x_0)^3 \end{split} \end{align} $$
$$ \begin{align} \begin{split} P_n(x) = f(x_0) & + \frac{f'(x_0)}{1!}(x-x_0) \\& + \frac{f''(x_0)}{2!}(x-x_0)^2 \\& + \frac{f'''(x_0)}{3!}(x-x_0)^3 \\& + \cdots + \frac{f^{(n)}(x_0)}{n!}(x-x_0)^n \end{split} \end{align} $$
$$x_0 = 0$$
$$ \begin{align} \begin{split} P_n(x) = f(0) & + \frac{f'(0)}{1!}x \\& + \frac{f''(0)}{2!}x^2 \\& + \frac{f'''(0)}{3!}x^3 \\& + \cdots + \frac{f^{(n)}(0)}{n!}x^n \end{split} \end{align} $$
$$y = e^{-x}$$