Final answer:
To analyze particle motion, differentiate the position function to get velocity and twice for acceleration. The signs of these functions indicate the direction of movement and whether the particle is speeding up or slowing down. However, specific intervals for the given functions require the correct initial velocity function, which was not provided.
Step-by-step explanation:
The student's question pertains to the motion of a particle along a line, where the particle's position is described by a given function of time. To find the velocity and acceleration as functions of time, we differentiate the position function. Once we obtain the velocity v(t) by taking the first derivative of the position function s(t), we find the acceleration a(t) by differentiating the velocity function.
To determine the intervals where the particle is moving right or left, we analyze the sign of the velocity function. If v(t) > 0, the particle moves to the right; if v(t) < 0, it moves to the left. For speeding up and slowing down, the particle is speeding up when the velocity and acceleration have the same signs, and slowing down when they have opposite signs.
The acceleration as a function of time, as given by the equation, is a(t) = 18t - 18. To find the intervals of speeding up and slowing down, one would look at the intervals where the velocity and acceleration are both positive or both negative (speeding up) and where one is positive and the other is negative (slowing down). However, without the specific function for velocity that the student provided, the intervals cannot be conclusively determined in this answer.