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Let $f:[a\,..b]\rightarrow \mathbb R^2$ be continuosly differentiable on $[a\,..b]$. Let $[a\,..b]$ be subdivided into $\{t_0,...,t_N\}$, where $t_i=a+i\dfrac{(b-a)}{N}$. Let $|x$| be the length of $x$, i.e. distance of $x$ and $0$.

Wikipedia's page on arc length in its section on derivation of the $L(f)=\displaystyle \int_a^b|f(t)| \,dt$ formula, has the following quote:

[...] the definition of the derivative as a limit implies that there is a positive real function ${\displaystyle \delta (\epsilon )}$ of positive real ${\displaystyle \epsilon }$ such that ${\displaystyle \Delta t<\delta (\epsilon )}$ implies ${\displaystyle \left|{\bigg |}{\frac {f(t_{i})-f(t_{i-1})}{\Delta t}}{\bigg |}-{\Big |}f'(t_{i}){\Big |}\right|<\epsilon}$.

I don't know what's going on here - the definition of the derivative implies that for any $\epsilon$ there indeed exists $\delta$ such that for any $t:0<|t_i-t_{i-1}|=\Delta t<\delta$ it's true that $\bigg|\dfrac{f(t_i)-f(t_{i-1})}{\Delta t}-f'(t_{i-1})\bigg|<\epsilon$.

Why does Wikipedia use $f'(t_i)$ instead of $f'(t_{i-1})$? Why are the quantities inside the $|\dfrac{}{}|$ bars normed too, rather than left alone?

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Well, one thing to note is that $\big||a|-|b|\big|\leq|a-b|$.

The other is that using $f'(t_{i-1})$ is okay, since you could use $\Delta t=t_{i-1}-t_i$ and then

$$f(t_i+\Delta t)-f(t_i)=f(t_{i-1})-f(t_i)$$

Basically, what's important in the limit of $f'(x)$ is that $x$ remains in $\Delta t$.

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