Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $x = t ^ { 11 } , y = t ^ { 3 } , z = t ^ { 6 } ; ( 4,4,4 )$
A) $x = 4 - 11 t , y = 4 + 3 t , z = 4 + 6 t$
B) $x = 4 + 11 t , y = 4 + 3 t , z = 4 + 6 t$
C) $x = 4 + 11 t , y = 4 + 3 t , z = 4 - 6 t$
D) $x = 11 t , y = 4 + 3 t , z = 4 + 6 t$
E) $x = 4 + 11 t , y = 4 - 3 t , z = 4 + 6 t$

Question

Quizplus · Accepted Answer

To find the tangent line, first find the derivative of each component with respect to $t$: $dx/dt = 11t^{10}$, $dy/dt = 3t^2$, $dz/dt = 6t^5$. At the point (4,4,4), solve $t^{11} = 4$, $t^3 = 4$, $t^6 = 4$ to find $t = 2^{1/3}$. The direction vector is $(11 \cdot 2^{10/3}, 3 \cdot 2^{2/3}, 6 \cdot 2^{5/3})$. The tangent line is $x = 4 + 11t$, $y = 4 + 3t$, $z = 4 + 6t$.

Find Parametric Equations for the Tangent Line to the Curve x=t11,y=t3,z=t6;(4,4,4)x = t ^ { 11 } , y = t ^ { 3 } , z = t ^ { 6 } ; ( 4,4,4 )x=t11,y=t3,z=t6;(4,4,4)

Find Parametric Equations for the Tangent Line to the Curve $x = t ^ { 11 } , y = t ^ { 3 } , z = t ^ { 6 } ; ( 4,4,4 )$