Firstly, this is a homework problem, so I would appreciate it if you might not just write the answer and rather, if I am wrong, provide suggestions only.
I am given a parametric curve with the components \begin{align} \begin{matrix} x=t^3-1,\\y=t^4+1, \\z=t, \end{matrix} \end{align} and I must find the tangent line (parametrically) to this curve at the point $\left(0,2,1\right)$. Since $t=1$ gives us this coordinate, I have found that \begin{align} \begin{matrix} x'\left(1\right)=3\left(1\right)^2=3,\\y'\left(1\right)=4\left(1\right)^3=4,\\z'\left(1\right)=1, \end{matrix} \end{align} and thus in using the point-slope form of a line in each of the components I have arrived at the parametric line \begin{align} \begin{matrix} x=3\left(t-1\right)=3t-3,\\y=4\left(t-1\right)+2=4t-2,\\z=1\left(t-1\right)+1=t. \end{matrix} \end{align} Am I correct? And additionally, I want it to be known that I realize it is not your job to do my homework. I'm just hoping to see where I may have gone wrong.