Calculus of variations: big-O notation?

Question

I have a formula in my text-book $$y(x+C) = y(x) + \frac{dy}{dx}C + O(C^2)$$

Can someone explain this formula?

Answer 1

The formula results from the Taylor series expension:

$$y(x+C) = y(x) + \frac{dy}{dx}C + \frac{d^2y}{dx^2}\frac{C^2}{2} +...$$

Supposing $C$ small relatively to $x$ and other assumptions about the function $y(x)$

$$y(x+C) = y(x) + \frac{dy}{dx}C + O(C^2)$$

The symbol $O$(...) is a short way to express that the terms with $C$ at power $2$ and higher are considered as negligable. More precisely, see : http://mathworld.wolfram.com/LandauSymbols.html