Find the curvature of the curve rt 3 sin 2tj 3 cos 2tj at

Question

1. Given a path

r(t) = e^(1/4)t sin ((1/2)t)i + e^(1/4)t cos ((1/2)t)j + e^(1/4)tk.

2. Find the velocity vector, acceleration vector and the speed of the curve

Let the acceleration vector be given by

a(t) = 6ti - 5j + 12t² k.

Find the velocity vector, and position vector, with the initial conditions r₀ = 3i + 4j and v₀ = 4j - 5k.

3. Find the arc length of

r(t) = 3 sin (2t)i + 3 cos (2t)j + 8tk.

between t = 0 and t = π

4. Find the curvature of the curve

r(t) = 3 sin (2t)j + 3 cos (2t)j at (0, 3).