# Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x=3 ln(t), y=4t^{frac{1}{2}}, z=t^{3}, (0, 4, 1)

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point.
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Step 1 Given: ......(1) $y=4{t}^{1/2}$ ......(2) $z={t}^{3}$ ......(3) For the specified points, Substitute 1 for z in equation 3. $z={t}^{3}$
$1={t}^{3}$
$t=1$ The function r(t) of these equations is, Substitute the values in the above equation. Step 2 Derivative of the vector r(t) is, Substitute 1 for t in the above vector. Step 3 The parametric equations for the tangent line are,

Substitute the values in the above equation. $X=3t$

Thus, the parametric equations for the tangent line are