Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. Select the correct answer.
$x = t ^ { 11 } , y = t ^ { 5 } , z = t ^ { 6 } ; ( 1,1,1 )$
A) $x = 11 t , y = 1 + 5 t , z = 1 + 6 t$
B) $x = 1 + 11 t , y = 1 + 5 t , z = 1 - 6 t$
C) $x = 1 + 11 t , y = 1 + 5 t , z = 1 + 6 t$
D) $x = 1 + 11 t , y = 1 - 5 t , z = 1 + 6 t$
E) $x = 1 - 11 t , y = 1 + 5 t , z = 1 + 6 t$

Question

Quizplus · Accepted Answer

To find the equation of the tangent line, we need to find the derivative of the parametric equations and evaluate them at the given point. The derivatives are:
$$\frac{dx}{dt}=11t^{10}, \frac{dy}{dt}=5t^4, \frac{dz}{dt}=6t^5$$
At the point (1,1,1), we have $\frac{dx}{dt}=11$, $\frac{dy}{dt}=5$, and $\frac{dz}{dt}=6$. Therefore, the parametric equations of the tangent line are:
$$x=1+11t, y=1+5t, z=1+6t$$
which is choice C.

Find Parametric Equations for the Tangent Line to the Curve x=t11,y=t5,z=t6;(1,1,1)x = t ^ { 11 } , y = t ^ { 5 } , z = t ^ { 6 } ; ( 1,1,1 )x=t11,y=t5,z=t6;(1,1,1)

Find Parametric Equations for the Tangent Line to the Curve $x = t ^ { 11 } , y = t ^ { 5 } , z = t ^ { 6 } ; ( 1,1,1 )$