# Defining Change in an interval

I don't know an appropriate topic for this. Here is my question though.

How does one define change in something? For example, regarding Riemann sums one defines change in the width of the rectangles to be $$\Delta x_i = x_i - x_{i-1}~~~,~~~i=1,2,\ldots, n$$

Would I be wrong in defining change as $$\Delta x_i = x_{i+1} - x_i~~?$$ In other words, if I have an interval $[x_0, x_1]$, would I write $\Delta x_0 = x_1 - x_0$ or $\Delta x_1 = x_1 - x_0$ and why?

• math isn't an opinion, but notation sure is. – Exodd Jun 10 '15 at 16:01
• Both are fine to use. – muaddib Jun 10 '15 at 16:04

For example, a common definition for an approximation to the derivative could be $f'(x) = (f(x+h) - f(x))/h = (f(x_{i+1})-f(x_i))/h$. This is known as a 'forward difference' and has first order accuracy in the size of your interval $h$. So if you decrease the size of your $h$, the error in your derivative approximation will correspondingly decrease by that amount.
On the other hand if we define the approximation to the derivative be the 'centered difference' : $f'(x) = (f(x+h) - f(x-h))/(2h) = (f(x_{i+1})-f(x_{i-1}))/(2h)$ we can show that this results in $O(h^2)$ accuracy - i.e. if your $h$ is twice as small, the error in your derivative approximation will be 4 times less.
• I think it's $2h$ in the central difference – Exodd Jun 10 '15 at 16:14