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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

Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured from the horizontal, and that the projectile is launched from the origin over a horizontal surfaceA projectile is fired at a speed of 880 m/sec at an angle of 42°. How long will it take to get 22 km downrange? Round your answer to the nearest whole number.