Find parametric equations for the tangent line to the curve with parametric equations $x = 3 t$, $y = 5 t ^ { 2 } , z = 4 t ^ { 3 }$ at the point with $t = 1$.
A) $x = 1 + 3 t , y = 1 + 10 t , z = 1 + 12 t$
B) $x = 3 + 3 t , y = 5 + 10 t , z = 4 + 12 t$
C) $x = 1 + 3 t , y = 1 + 5 t , z = 1 + 4 t$
D) $x = 3 + 3 t , y = 5 + 5 t , z = 4 + 4 t$

Question

Quizplus · Accepted Answer

To find the tangent line at the point with $t = 1$, we need to find the values of $x$, $y$, and $z$ at $t = 1$ and the corresponding derivatives.

At $t = 1$, we have $x = 3$, $y = 5$, and $z = 4$.

To find the derivatives, we differentiate each component of the parametric equations with respect to $t$:

$$\frac { d x } { d t } = 3,\quad \frac { d y } { d t } = 10 t,\quad \text{and}\quad \frac { d z } { d t } = 12 t ^ { 2 }. $$

At $t = 1$, we have $\frac { d x } { d t }=3,\frac { d y } { d t }=10$, and $\frac { d z } { d t }=12$.

So the parametric equation of the tangent line at $t=1$ is 
$$x=3+3t,\quad y=5+10t,\quad z = 4+12t.$$

Therefore, the best choice is B.

Find Parametric Equations for the Tangent Line to the Curve x=3tx = 3 tx=3t

Find Parametric Equations for the Tangent Line to the Curve $x = 3 t$