The question asks to identify which graph (Graph 1 or Graph 2) represents velocity and which graph represents distance, providing a hint to consider the derivative of the function.

1. Graph of Distance: The graph representing distance is typically smooth and increasing, as distance accumulates over time. The slope of the distance graph at any point represents the velocity.

2. Graph of Velocity: The graph representing velocity corresponds to the rate of change of distance, i.e., the derivative of the distance function. If velocity is positive, distance is increasing; if velocity is zero, distance is constant; and if velocity is negative, distance is decreasing.

Now, let's analyze the graphs:

• Graph 1 shows a curve that goes up and down, with peaks and troughs. This suggests it might represent velocity, as velocity can be positive (moving forward), negative (moving backward), or zero (at rest).
• Graph 2 shows a smooth increasing curve that mirrors what you'd expect for a distance graph, where distance accumulates over time.

Conclusion:

• Graph 1 represents velocity.
• Graph 2 represents distance.

Would you like more detailed reasoning or explanation of the derivative's role? Here are five related questions to help further your understanding:

1. What is the significance of the slope in a distance vs. time graph?
2. How can you identify points where the velocity is zero on a graph?
3. What does it mean when the velocity graph crosses the x-axis?
4. How does a negative velocity affect the distance graph?
5. Can acceleration be interpreted from these graphs? How?

Tip: Understanding the relationship between a function and its derivative is key to interpreting graphs of motion.