Chapter 12, Problem 13PS

(a)

To determine

To Graph: The function $r\left(t\right)=\langle t\cos \pi t,t\sin \pi t\rangle$ where $0\le t\le 2$.

(b)

To determine

To calculate: The arc length of the function $r\left(t\right)=\langle t\cos \pi t,t\sin \pi t\rangle$ where $0\le t\le 2$.

(c)

To determine

To calculate: The curvature K as a function of t for the function $r\left(t\right)=\langle t\cos \pi t,t\sin \pi t\rangle$. Also, find the curvature at $t=0,t=1\text{ and }t=2$.

(d)

To determine

To Graph: The function of curvature $K=\frac{\pi \left({\pi }^2{t}^2+2\right)}{{\left({\pi }^2{t}^2+1\right)}^{\frac{3}{2}}}$.

(e)

To determine

To calculate: The value of $\underset{t\to \infty }{\lim }K$.

(f)

To determine

A conjecture about the graph of $r\left(t\right)=\langle t\cos \pi t,t\sin \pi t\rangle$ as $t\to \infty$ by the use of the result of part (e).