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) = (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.  

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 1160 m/sec at an angle of 60°. How long will it take to get 29 km downrange? Round your answer to the nearest whole number.

Solve the initial value problem.Differential Equation: = (cos t)i + (4t3- 3)jInitial Condition: r(0) = -8j

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.

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

Find the domain of the vector-valued function.r(t) = ln (t – 1)i + j

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

Solve the initial value problem.Differential Equation: = (3t2 + 2t)i + (4t3+ 9)jInitial Condition: r(0) = 6j

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