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

A spring gun at ground level fires a tennis ball at an angle of 37°. The ball lands 22 m away. What was the ball’s initial speed? Round your answer to the nearest tenth.

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

If r(t) is the position vector of a particle in the plane at time t, find the indicated vector.Find the velocity vector.r(t) = (3t2 + 15)i + 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

Solve the initial value problem.Differential Equation: = (sec2t)i + (4t3- 3)jInitial Condition: r(0) = -6j

A spring gun at ground level fires a tennis ball at an angle of 37°. The ball lands 18 m away. What was the ball’s initial speed? Round your answer to the nearest tenth.

Find the principal unit normal vector N for the curve r(t).r(t) = (1 + t)i + (6 + ln(cos t))k, -π/2 < t < π/2

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

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.