1. ## Forward/Backward-Difference formula

Hi,
I'm learning some Numerical Analysis but this is a very calculus-like question so I'll put it here.

Forward/Backward - Difference formula,

Assumptions:

$f$ is continuous on $[a,b]$ and $f''(x)$ exists for all $x\in[a,b]$.
$x_0\in[a,b]$ and $x_1=x_0+h$ for $h$ such that $x_1\in[a,b]$.
$\xi$ is also in $[a,b]$.

$f'(x_0)=\frac{f(x_0+h)-f(x_0)}{h}-\frac{h}{2}f''(\xi).$

Since I do not have any information on f $''(\xi)$, the error in my approximation is bounded by $\frac{M|h|}{2}$ where $M$ is the maximum value of $f''(x)$ on my interval. How can I find the error bound without actually taking the second derivative of $f$?

Thanks.

2. Originally Posted by Mollier
Hi,
I'm learning some Numerical Analysis but this is a very calculus-like question so I'll put it here.

Forward/Backward - Difference formula,

Assumptions:

$f$ is continuous on $[a,b]$ and $f''(x)$ exists for all $x\in[a,b]$.
$x_0\in[a,b]$ and $x_1=x_0+h$ for $h$ such that $x_1\in[a,b]$.
$\xi$ is also in $[a,b]$.

$f'(x_0)=\frac{f(x_0+h)-f(x_0)}{h}-\frac{h}{2}f''(\xi).$

Since I do not have any information on f $''(\xi)$, the error in my approximation is bounded by $\frac{M|h|}{2}$ where $M$ is the maximum value of $f''(x)$ on my interval. How can I find the error bound without actually taking the second derivative of $f$?

Thanks.
Without knowing the function, and its second derivative I think you have done all that can be expected.

CB

3. Originally Posted by CaptainBlack
Without knowing the function, and its second derivative I think you have done all that can be expected.

CB
What if have a function but am unable to analytically find the second derivative?
Thanks