In the Wikipedia derivation they use the Mean Value Theorem (MVT) to approximate the distance between two points of a function $\vec{r}: \mathbb{R} \to \mathbb{R}^{n}$ by saying that $|\vec{r}(t_{i} + \Delta t) - \vec{r}(t_{i})| \approx|\vec{r}'(t_{i})|\Delta t$
I have two questions regarding this:
How can they use the MVT here for a vector valued function? I thought that there was no analog for the MVT of a vector valued function
Assuming, that you can use the MVT here why are we using the point $t_{i}$ as an input into $\vec{r}'$? I thought the MVT said that there is a point in the interval $c \in [t_{i},t_{i} + \Delta t]$ such that $|\vec{r}(t_{i} + \Delta t) - \vec{r}(t_{i})| = |\vec{r}'(c)|\Delta t$
(i.e. why is it $|\vec{r}'(t_{i})|$ and not $|\vec{r}'(c)|$?)
Is it just because we are assuming the interval $[t_{i},t_{i} + \Delta t]$ is small and thus $|\vec{r}'(t_{i})|\Delta t$ would be a good approximation to $|\vec{r}(t_{i} + \Delta t) - \vec{r}(t_{i})|$?