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Find the principal unit normal vector N for the curve r(t).r(t) = (4 + t)i + (1 + ln(cos t))k, -π/2 < t < π/2

Find the curvature of the curve r(t).r(t) = (8 + 7 cos 4t) i – (10 + 7 sin 4t)j + 5k

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 + (6t3)j

Find the length of the curve with the given vector equation. r(t) = (1 + 4t)i + (1 + 3t)j + (2 – 2t)k, -1 ≤ t ≤ 0 Type your answer in the given answer box.  You may need to use the equation editor found in the editing menu.  If the equation editor is not showing, click on the three vertical dots to expand the menu.  

Find the length of the curve with the given vector equation. r(t) = (1 + 5t)i + (1 + 6t)j + (3 – 3t)k, -1 ≤ t ≤ 0 Type your answer in the given answer box.  You may need to use the equation editor found in the editing menu.  If the equation editor is not showing, click on the three vertical dots to expand the menu.  

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