The position of the particle is described by the function $x(t) = 4.00 t^3 - 1.00 t + 3.00$ (in meters), where $t$ is in seconds. We need to calculate the particle's:

(a) Position at $t = 3.00 \, \text{s}$
(b) Velocity at $t = 3.00 \, \text{s}$
(c) Acceleration at $t = 3.00 \, \text{s}$

Step 1: Position at $t = 3.00 \, \text{s}$

The position function is: $x(t) = 4.00 t^3 - 1.00 t + 3.00$ Substitute $t = 3.00$ into the equation.

Step 2: Velocity at $t = 3.00 \, \text{s}$

The velocity is the derivative of the position function with respect to time: $v(t) = \frac{dx(t)}{dt} = \frac{d}{dt} \left( 4.00 t^3 - 1.00 t + 3.00 \right)$ We'll compute this derivative and evaluate it at $t = 3.00 \, \text{s}$.

Step 3: Acceleration at $t = 3.00 \, \text{s}$

The acceleration is the derivative of the velocity, or the second derivative of the position function: $a(t) = \frac{dv(t)}{dt} = \frac{d^2 x(t)}{dt^2}$ We'll compute this second