Answer
$\displaystyle \langle\frac{6}{7},\ \displaystyle \frac{2}{7},\ \displaystyle \frac{3}{7}\rangle$
Work Step by Step
Unit tangent vector at t:$\quad \mathrm{T}(t)= \displaystyle \frac{\mathrm{r}^{\prime}(t)}{|\mathrm{r}^{\prime}(t)|}$
Parametric equations: $\mathrm{r}(t):\quad\left\{\begin{array}{ll}
x=t^{3}+3t & \\
y=t^{2}+1 & \\
z=3t+4 &
\end{array}\right.$
Differentiate $(\displaystyle \frac{d}{dt})$ each component function.
$\mathrm{r}^{\prime}(t):\quad\left\{\begin{array}{ll}
x=3t^{2}+3 & \\ \\
y=2t\\ & \\
z=3 &
\end{array}\right. $
$\mathrm{r}^{\prime}(1)=\langle 3+3, 2(1),\ 3\rangle=\langle 6,2,3\rangle$
$|\mathrm{r}^{\prime}(1)|=\sqrt{6^{2}+2^{2}+3^{2}}=7$
$\displaystyle \mathrm{T}(1)=\frac{1}{|\mathrm{r}^{\prime}(1)|}\mathrm{r}^{\prime}(1)=\frac{1}{7}\langle 6,2,3\rangle=\langle\frac{6}{7},\ \displaystyle \frac{2}{7},\ \displaystyle \frac{3}{7}\rangle$.
