Exercise 20

In this exercise, we will make the unit speed parameterization for the curve parameterized by $\vr(t)=\langle \frac{3}{2} t^2,2t^2,\frac{5}{3}t^3\rangle$ for $t\in[0,a]$.

1. Calculate the speed for the parameterization $\vr(t)=\langle \frac{3}{2} t^2,2t^2,\frac{5}{3}t^3\rangle$. Factor out as large a power of $t$ out of the square root as possible.

2. Use your speed calculation from above to compute the arc length on our curve $C$ as $t$ goes from 0 to $a$. Your answer should be in terms of $a$.

3. Your previous step should have created a function $s(a)$ which outputs the arclength traveled on $C$ for $t$ from 0 to $a$. Find the inverse function, $a(s)$, that outputs the time past 0 that is needed to travel $s$ along the curve.

4. Verify that the composition of $\vr(a(s))$ is a parameterization of $C$ with unit speed.