Chapter 12.5, Problem 18E

(a)

To determine

To Calculate: The length of the arc s on the helix as a function of $t$ here, the function is $r\left(t\right)=\langle 2\cos t,2\sin t,t\rangle$.

(b)

To determine

To Calculate: The parametric equations of the curve $r\left(t\right)=\langle 2\cos t,2\sin t,t\rangle$ in terms of the arc length s.

(c)

To determine

To Calculate: The coordinates of the point on the helix $r\left(t\right)=\langle 2\cos t,2\sin t,t\rangle$ for arc lengths $s=\sqrt{5}$ and $s=4.$

(d)

To determine

To Prove: The relation $\Vert r^{\prime }\left(s\right)\Vert =1$