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Hello I am sorry for the inconvenient caused and I wish to apologise in advance for my language (I am not a native English speaker).

Here is an exercise I have been trying to do but I guess I am missing something out.

Wording: Let's denote $F$ a differentiable function $(F'= f)$.

Express $F(t)$ $\in $ $[0,T]$, in integral form depending on $f$ and $F(0)$.

Deduct that we can approximate the trajectory $ t -> F(t)$ using this scheme:

$F^n$ $(t_{i+1})$ $=$ $F^n(t_i)$ + $f(t_i)\Delta $$t_{i+1}$

$F^n(0) = F(0)$

Thank you very much for your help,

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    Are you familiar with [Fundamental theorem of calculus](https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus#Formal_statements) – 2017-01-05
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    Hi Zoran, a little bit unfortunately thank you for the link I am going to look inside – 2017-01-05

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What I have done so far is:

$F(t) = \int_0^t f(x) \ dx + F(0)$

Since $\int_0^t$ is the sum of small integrals I can approximate it by

$\Delta t_i$ but I do not know if there is a proof or theorem that can allow me to write F at the power n

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    I didn't even notice the $n$ in the superscript; it is, most likely, some kind of mistake. You yourself, in the question above, say you want to approximate the trajectory $t \mapsto F(t)$ not $t \mapsto F^n(t)$. – 2017-01-05