The position vector r describes the path of an object moving in space. (a) Find the velocity vector, speed, and acceleration vector of the object. Position Vector: r(t) = ti + t²j + ½t²k Time: t = 4

Evaluate the definite integral.

$$∫_0^3 ǁti + t²jǁ dt$$

The position vector r describes the path of an object moving in space. (b) Evaluate the velocity vector and acceleration vector of the object at the given value of t. Position Vector: r(t) = 3ti + tj + ¼t²k Time: t = 2

The position vector r describes the path of an object moving in space. (a) Find the velocity vector, speed, and acceleration vector of the object. (b) Evaluate the velocity vector and acceleration vector of the object at the given value of t. Position Vector. $\mathbf{r}(t)=4 t \mathbf{i}+t^{3} \mathbf{j}-t \mathbf{k}$. TIme. t=1

r(t) = ti + 3tj + t²k, u(t) = 4ti + t²j + t³k Use the properties of the derivative to find the following. (a)r′(t) (b)d/dt [3r(t) - u(t)] (c)d/dt (5t)u(t) (d)d/dt [r(t)·u(t)] (e)d/dt [r(t)×u(t)] (f)d/dt r(2t)

Find the curvature K of the curve at the point P. r(t) = 3ti + 2t²j, P(-3, 2)

Find the curvature K of the curve. r(t) = ti + t²j + t²/2 k

The position vector r describes the path of an object moving in the xy-plane. (b) Evaluate the velocity vector and acceleration vector of the object at the given point. Position Vector: r(t) = ⟨e^-t, e^t⟩ Point: (1, 1)

The position vector r describes the path of an object moving in space. (b) Evaluate the velocity vector and acceleration vector of the object at the given value of t. Position Vector: r(t) = ⟨t, -tan t, e^t⟩ Time: t = 0

The position vector r describes the path of an object moving in space. (a) Find the velocity vector, speed, and acceleration vector of the object. Position Vector: r(t) = ⟨cos³ t, sin³ t, 3t⟩ Time: t = π

What is the formula for the rotational inertia of a thin spherical shell of inertia m and radius R about an axis tangent to any point on its surface?

The position vector r describes the path of an object moving in space. (b) Evaluate the velocity vector and acceleration vector of the object at the given value of t. Position Vector: r(t) = ti + t²j + ½t²k Time: t = 4

Find r(t) if r'(t)=2ti+3t^2j+t^1/2k and r(1)=i+j

The position vector r describes the path of an object moving in the xy-plane. Position Vector: r(t) = 2 cos ti + 2 sin tj Point: (√2, √2) (c) Sketch a graph of the path, and sketch the velocity and acceleration vectors at the given point.