Chapter 12.2, Problem 24E

(a)

To determine

To calculate: The derivative $r\text{'}\left(t\right)$ for the function $r\left(t\right)=t^{3}i+\left(2t^2+3\right)j+\left(3t-5\right)k$.

(b)

To determine

To calculate: The second derivative $r\text{'}\text{'}\left(t\right)$ for the function $r\left(t\right)=t^{3}i+\left(2t^2+3\right)j+\left(3t-5\right)k$.

(c)

To determine

To calculate: The dot product $r\text{'}\left(t\right)\cdot r\text{'}\text{'}\left(t\right)$ for the function $r\left(t\right)=t^{3}i+\left(2t^2+3\right)j+\left(3t-5\right)k$.

(d)

To determine

To calculate: The cross product $r\text{'}\left(t\right)\times r\text{'}\text{'}\left(t\right)$ for the function $r\left(t\right)=t^{3}i+\left(2t^2+3\right)j+\left(3t-5\right)k$.