#### To determine

**To find:** When the particle is speeding up and slowing down.

#### Answer

The particle speeds up on the interval (0, 1).

The particle slows down on the interval (1, 2).

#### Explanation

The given figure represents the velocity function of the particle.

It is identified from the graph that the velocity is positive on (0, 2) and is negative on (2, 3).

Also, observe that the graph increases on the interval (0, 1) and decreases on (1, 3). Thus, the acceleration is positive in the interval (0, 1) and negative in the interval (1, 3).

Recall the fact that the particle speeds up when the velocity and acceleration have the same sign and slows down when the velocity and acceleration have the opposite sign.

Therefore, it can be concluded that the particle speeds up on the interval (0, 1) as *v* and *a* have the same sign and slows down on the interval (1, 2) as *v* and *a* have the opposite sign.

#### To determine

**To find:** When the particle is speeding up and slowing down.

#### Answer

The particle speeds up on the intervals (1, 2) and (3, 4).

The particle slows down on the intervals (0, 1) and (2, 3).

#### Explanation

The given figure represents the velocity function of the particle.

It is identified from the graph that the velocity is positive on (0, 3) and is negative on (3, 4).

Also, observe that the graph is increasing on the interval (1, 2) and decreasing on (0, 1) and (2, 4). Thus, the acceleration is positive the interval (1, 2) and negative on the intervals (0, 1) and (2, 4).

Recall the fact that the particle speeds up when the velocity and acceleration have the same sign and slows down when the velocity and acceleration have the opposite sign.

Therefore, it can be concluded that the particle speeds up on the interval (1, 2) and (3, 4) as *v* and *a* have the same sign and slows down on the intervals (0, 1) and (2, 3) as *v* and *a* have the opposite sign.