# What are the parametric equations for the tangent line at t=3 for the motion of a particle given by x(t)=4t^2+3, y(t)=3t^3?

Jun 9, 2018

$\boldsymbol{l} \left(\lambda\right) = \left(39 , 81\right) + \lambda \left(8 , 27\right)$

#### Explanation:

$\boldsymbol{r} \left(t\right) = \left(4 {t}^{2} + 3 , 3 {t}^{3}\right)$

$\boldsymbol{r} \left(3\right) = \left(39 , 81\right)$

$\boldsymbol{r} ' \left(t\right) = \left(8 t , 9 {t}^{2}\right)$

That is the tangent vector.

$\boldsymbol{r} ' \left(3\right) = \left(24 , 81\right)$

The tangent line is:

$\boldsymbol{l} \left(\lambda\right) = \boldsymbol{r} \left(3\right) + \lambda \boldsymbol{r} ' \left(3\right)$

$= \left(39 , 81\right) + \lambda \left(24 , 81\right)$

We can factor the direction vector a little:

$\boldsymbol{l} \left(\lambda\right) = \left(39 , 81\right) + \lambda \left(8 , 27\right)$