Let Ct sin 3t cos 3t 2t32a Find parametric equations of the

Let C(t)= (sin 3t, cos 3t, 2t^(3/2)).(a) Find parametric equations of the tangent line to the curve at (0,1,0)

(b) find the arc length of the curve between (0,1,0) and (sin 3, cos 3, 2)

Solution

The direction vector of the tangent line will be the derivative with respect to t at the given point.
curve = {sin 3t, cos 3t, 2t^(3/2)}
d(curve)/dt = {3cos 3t, -3sin 3t, 3t^(1/2)}

The given point is at x=t=0, so the parametric equation for the line (with parameter \"k\") is
line = (given point) + k*(direction vector at t=0)
line = {0, 1, 0} + k*{3, 0, 3}
.. line = {3k, 1, 3k}