When finding the length of a curve using calculus, the first step is to determine whether $f(x)$ has a continuous derivative on the curve (i.e. on $[a,b]$). But how do we determine this?
My understanding is that you need to find the second derivative, i.e. $f''(x)$. If $f''(x)$ exists, this implies that $f'(x)$ is a continuous derivative along the curve of $f(x)$. Is that accurate?
I apologize if this is an obvious question. I just feel that my textbook/online resources give much more convoluted answers than this, when the solution is pretty simple. My concern is that I may be overlooking something.
Okay, it looks like my idea is wrong. Consider $y = 2x + 1$ on $[1,5]$, for example. This function has a continuous first derivative ($y' =2$) but does not have a meaningful second derivative ($y'' = 0$).