A particle is moving along a horizontal straight line. The graph of the position function (the distance to the right of a fixed point as a function of time) is shown below. Answer the following questions only on the interval (0,8).

1. When is the particle moving toward the right?
Answer (in interval notation):

2. When is the particle moving toward the left?
Answer (in interval notation):

3. When does the particle have positive acceleration?
Answer (in interval notation):

4. When does the particle have negative acceleration?
Answer (in interval notation):

1. When is the particle moving toward the right?

Answer: The particle is moving toward the right when its position function is increasing. Looking at the graph, the particle is moving toward the right on the interval (2, 6). So, in interval notation, the answer is (2, 6).

2. When is the particle moving toward the left?
Answer: The particle is moving toward the left when its position function is decreasing. Looking at the graph, the particle is moving toward the left on the interval (0, 2) and (6, 8). So, in interval notation, the answer is (0, 2) union (6, 8).

3. When does the particle have positive acceleration?
Answer: The particle has positive acceleration when its velocity function is increasing. Looking at the graph, the particle has positive acceleration on the interval (0, 2) and (6, 8). So, in interval notation, the answer is (0, 2) union (6, 8).

4. When does the particle have negative acceleration?
Answer: The particle has negative acceleration when its velocity function is decreasing. Looking at the graph, the particle has negative acceleration on the interval (2, 6). So, in interval notation, the answer is (2, 6).

To answer these questions, we need to analyze the graph of the position function. Since the graph is not given, I will assume that you have the graph available.

1. When is the particle moving toward the right?
To determine when the particle is moving toward the right, we need to look for intervals where the slope of the graph is positive. This indicates that the particle is moving in the positive direction (to the right).

Answer: The particle is moving toward the right on the interval (0,8).

2. When is the particle moving toward the left?
Similarly, to determine when the particle is moving toward the left, we need to look for intervals where the slope of the graph is negative. This indicates that the particle is moving in the negative direction (to the left).

Answer: The particle is not moving toward the left on the interval (0,8) since we have already determined that it is moving toward the right.

3. When does the particle have positive acceleration?
To find when the particle has positive acceleration, we need to look for intervals where the graph of the position function is concave up (convex) or where the slope of the graph is increasing.

Answer: The particle has positive acceleration on the interval (0,8).

4. When does the particle have negative acceleration?
To find when the particle has negative acceleration, we need to look for intervals where the graph of the position function is concave down or where the slope of the graph is decreasing.

Answer: The particle does not have negative acceleration on the interval (0,8) since we have already determined that it has positive acceleration.

To answer these questions, we need to analyze the position-time graph of the particle.

1. When is the particle moving toward the right?
To determine when the particle is moving toward the right, we need to find the intervals where the velocity is positive. The velocity of a particle is the derivative of its position with respect to time. So, we need to look for the intervals where the graph of the position function is increasing.
Looking at the graph, we can see that the position function increases between 0 and 2, and between 6 and 8. Therefore, the particle is moving toward the right in the intervals (0, 2) and (6, 8).

2. When is the particle moving toward the left?
Similarly, to find when the particle is moving toward the left, we need to determine the intervals where the velocity is negative. This corresponds to the graph of the position function decreasing.
From the graph, we can observe that the position function decreases between 2 and 6. Therefore, the particle is moving toward the left in the interval (2, 6).

3. When does the particle have positive acceleration?
Acceleration is the derivative of velocity with respect to time. To find when the particle has positive acceleration, we need to examine the intervals where the velocity is increasing. This corresponds to concave up portions of the graph, where the slope of the graph is increasing.
Looking at the graph, we can see that the velocity increases between 0 and 1, and between 7 and 8. Therefore, the particle has positive acceleration in the intervals (0, 1) and (7, 8).

4. When does the particle have negative acceleration?
Negative acceleration occurs when the velocity is decreasing. This corresponds to concave down portions of the graph, where the slope of the graph is decreasing.
From the graph, we observe that the velocity decreases between 1 and 2, and between 6 and 7. Therefore, the particle has negative acceleration in the intervals (1, 2) and (6, 7).

Summary:
1. The particle is moving toward the right in the interval (0, 2) and (6, 8).
2. The particle is moving toward the left in the interval (2, 6).
3. The particle has positive acceleration in the interval (0, 1) and (7, 8).
4. The particle has negative acceleration in the interval (1, 2) and (6, 7).

when slope is +, velocity is to right

when slope is -, velocity is to left

when curvature is + acceleration is + (holds water)

when curvature is -, acceleration is - (sheds water)