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Find the indicated derivative for the given vector-valued function .Find r ‘ (t) for r(t) = (t4)i – (csc t)j + 4k.

Find the indicated derivative for the given vector-valued function .Find r ‘ (t) for r(t) = (t6)i – (csc t)j + 10k.

Find the length of the curve with the given vector equation.r(t) = 4ti + j + k; -6≤ t ≤ 1

Find the curvature of the curve r(t).r(t) = (5 + 5 cos 10t) i – (7 + 5 sin 10t)j + 3k

The position vector of a particle is r(t). Find the requested vector.The velocity at t = for r(t) = 5sec2(t)i – 9tan(t)j + 6t2k

Compute r”(t).r(t) = (3 ln(6t))i + (9t3)j

If r(t) is the position vector of a particle in the plane at time t, find the indicated vector.Find the acceleration vector.r(t) = (8 ln(3t))i + (2t3)j

The position vector of a particle is r(t). Find the requested vector.The velocity at t = for r(t) = 4sec2(t)i – 5tan(t)j + 7t2k

Find the required limit or indicate that it does not exist.

Find the domain of the vector-valued function.r(t) = i + j + ln k