# First order purturbation of an abstract chaotic map

I am going through the lecture notes of David Gross on dynamical systems.

In Section 1.1.3 on page 10, the first equation in the section is given below.

$$f(x_0 + t) = f(x_0) + f'(x_0) t + \mathcal{O(t^2)}$$

Could anyone help me to understand the intuition behind this particular equation? I assume $$f'$$ is the first derivative. Also, why the last term is $$t^2$$ and not some other function of $$t$$?

You can also write this as $$f(x_0+t)-f(x_0)=t\int_0^1f'(x_0+st)\,ds=f'(x_0)t+t^2\int_0^1(1-s)f''(x_0+st)\,ds$$ which is the Taylor expansion with integral remainder term.
As long as $$f$$ is twice continuously differentiable, the last integral is a continuous function in $$x_0$$ and $$t$$, so that indeed the term is $$O(t^2)$$.
we have, $$f'(x_0) = \frac{f(x_0 + t)-f(x_0)}{t} + \frac{O(t^2)}{t}$$
Note : we can also replace $$O(t^2)$$ by $$o(t)$$
• thanks for the reply. I am familiar with the big O notation. My question is why it is $t^2$? Sep 3 '20 at 6:20