Chapter 12.2, Problem 35E

(a)

To determine

To calculate: The function $r\text{'}\left(t\right)$ of the provided function $r\left(t\right)=ti+3tj+t^{2}k$.

(b)

To determine

To calculate: The expression $\frac{d}{dt}\left[3r\left(t\right)-u\left(t\right)\right]$ for the provided functions $r\left(t\right)=ti+3tj+t^{2}k\text{ and }u\left(t\right)=4ti+t^{2}j+t^{3}k$.

(c)

To determine

To calculate: The expression $\frac{d}{dt}\left[\left(5t\right)u\left(t\right)\right]$ for the provided function $u\left(t\right)=4ti+t^{2}j+t^{3}k$

(d)

To determine

To calculate: The expression $\frac{d}{dt}\left[r\left(t\right)\cdot u\left(t\right)\right]$ for the provided functions $r\left(t\right)=ti+3tj+t^{2}k\text{ and }u\left(t\right)=4ti+t^{2}j+t^{3}k$.

(e)

To determine

To calculate: The expression $\frac{d}{dt}\left[r\left(t\right)\times u\left(t\right)\right]$ for the provided functions $r\left(t\right)=ti+3tj+t^{2}k\text{ and }u\left(t\right)=4ti+t^{2}j+t^{3}k$.

(f)

To determine

To calculate: The expression $\frac{d}{dt}\left[r\left(2t\right)\right]$ for the provided function $r\left(t\right)=ti+3tj+t^{2}k$.