Chapter 15.2, Problem 57ES

Suppose that a curve C in the $xy-\text{plane}$ is smoothly parametrized by $\text{r}(t)=x(t)\text{i}+y(t)\text{j}(a\le t\le b)$ In each part, part refer to the notation used in the derivation of Formula (9).

(a) Let m and M denote the respective minimum and maximum values of $‖\text{r'}(t)‖\text{on [a,b]}\text{. Prove that 0}\le m(\text{max}\Delta t_k)\le \max \Delta s_k\le M(\max \Delta t_k)$

(b) Use part (a) to prove that max $\Delta s_k\to 0.$ if and only if max $\Delta t_k\to 0.$