Chapter 12.2, Problem 25E

(a)

To determine

To calculate: The derivative $r\text{'}\left(t\right)$ for the function $r\left(t\right)=\langle \cos t+t\sin t,\sin t-t\cos t,t\rangle$.

(b)

To determine

To calculate: The second derivative $r\text{'}\text{'}\left(t\right)$ for the function $r\left(t\right)=\langle \cos t+t\sin t,\sin t-t\cos t,t\rangle$

(c)

To determine

To calculate: The dot product $r\text{'}\left(t\right)\cdot r\text{'}\text{'}\left(t\right)$ for the function $r\left(t\right)=\langle \cos t+t\sin t,\sin t-t\cos t,t\rangle$.

(d)

To determine

To calculate: The cross product $r\text{'}\left(t\right)\times r\text{'}\text{'}\left(t\right)$ for the function $r\left(t\right)=\langle \cos t+t\sin t,\sin t-t\cos t,t\rangle$.