How do you answer these questions regarding the Taylor series method?

Question

(a) Approximate $f'(x_0)$ and $f''(x_0)$ using the values $x_0-h$, $x_0$ and $x_0 + \alpha h$ $(0 < \alpha)$ by applying the Taylor series method.

(b) Assuming $f(x)\in C^3$, evaluate the approximation error. What is the approximation order?

Answer 1

$f(x_0 + \delta)$ is approximately $f(x_0) + f'(x_0)\delta + \dfrac{f''(x_0)}{2}\delta^2$.

So:

$$\begin{align} f(x_0 + 0) & = f(x_0) + f'(x_0)0 + \dfrac{f''(x_0)}{2}0^2\\ f(x_0 - h) & = f(x_0) - f'(x_0)h + \dfrac{f''(x_0)}{2}h^2\\ f(x_0 + \alpha h) & = f(x_0) + f'(x_0)\alpha h + \dfrac{f''(x_0)}{2}\alpha^2h^2\\ \end{align}$$

rearranging:

$$ \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 1 & -h & \frac{h^2}{2} \\ 1 & \alpha h & \frac{\alpha^2 h^2}{2}\end{array}\right] \left[ \begin{array}{c} f(x_0) \\ f'(x_0) \\ f''(x_0) \end{array} \right] = \left[ \begin{array}{c} f(x_0)\\ f(x_0 - h)\\ f(x_0 + \alpha h)\end{array}\right] $$

Solve the linear system and you approximate $f'(x_0)$ and $f''(x_0)$

Answer 2

The (b) part can be solved with Peano form of the remainder:

If $\ T_n(x)$ is the n-th Taylor polynomial, then let $R_n = f(x) - T_n(x)$.

From Peano's theorm we know, that $lim_{x\to x_0}\frac{R_n(x)}{{(x-x_0)}^n} = 0$

As far as (a) goes, what can we assume about $f$? Do we know anything about it?