Here is the graph of the position function $x(t) = 9.75 + 1.50t^3$ for the time interval from $t = 2.00$ s to $t = 3.00$ s.

Graph Details:

• The blue curve represents the position $x(t)$ as a function of time.
• The red points mark the positions at $t = 2.00$ s, $t = 2.50$ s, and $t = 3.00$ s.
• The green arrows indicate the instantaneous velocities at these points, calculated from the derivative $v(t) = 4.50t^2$.

Summary of Results:

• Average Velocity over the interval $t = 2.00$ s to $t = 3.00$ s is 28.5 cm/s (as given).
• Instantaneous Velocity:
  • At $t = 2.00$ s: $18.00$ cm/s
  • At $t = 2.50$ s: $28.12$ cm/s
  • At $t = 3.00$ s: $40.50$ cm/s

Would you like any more details or have any questions? Here are five related questions to explore further:

1. How would the graph change if the cubic coefficient were negative?
2. What is the significance of the slope of the position-time graph?
3. How do the instantaneous velocities compare to the average velocity?
4. What happens to the velocity as time increases beyond $t = 3.00$ s?
5. How would you find the exact time when the particle is at the midpoint between two given positions?

Tip: The slope of the position-time graph at any point represents the instantaneous velocity at that time.