Solución

Apartado a

Primer paso: $\bold{r'}(t) = −sent\bold{i}+cost\bold{j}$

Segundo paso

\|\bold{r'}(t)\| = \sqrt{(-sent)^2+(cost)^2} = 1

Tercer paso:

\bold{T}(t) = \frac{\bold{r'}(t)}{\|\bold{r'}(t)\|} = \frac{−sent\bold{i}+cost\bold{j}}{1} = −sent\bold{i}+cost\bold{j}

Apartado b

Primer paso: $\bold{u'}(t) = (6t+2)\bold{i}−12t^2\bold{j}+6\bold{k}$

Segundo paso:

$\begin{aligned} \|\bold{u'}(t)\| &= \sqrt{(6t+2)^2+(-12t^2)^2+6^2}\\ &= \sqrt{144t^4+36t^2+24t+40}\\ &= 2\sqrt{36t^4+9t^2+6t+10} \end{aligned}$

Tercer paso

\begin{aligned} \frac{\bold{u'}(t)}{\|\bold{u'}(t)\|} &= \frac{(6t+2)\bold{i} −12t^2\bold{j} +6\bold{k}}{2\sqrt{36t^4+9t^2+6t+10}}\\ &= \frac{3t+1}{\sqrt{36t^4+9t^2+6t+10}}\bold{i}-\frac{6t^2}{\sqrt{36t^4+9t^2+6t+10}}\bold{j} + \frac{3}{\sqrt{36t^4+9t^2+6t+10}}\bold{k} \end{aligned}