Let's go through each question step-by-step.

1. Which vehicle has traveled farther at $t = 2$ seconds?

• At $t = 2$ seconds, compare the positions of $s = f(t)$ and $s = g(t)$.
• Observing the graph:
  • $s = f(t)$ is higher than $s = g(t)$ at $t = 2$.
• Answer: Vehicle $f$ has traveled farther at $t = 2$ seconds.

2. What is the approximate velocity of the $f$ vehicle at $t = 3$ seconds?

• The velocity is the slope (or derivative) of the position function $f(t)$ at $t = 3$.
• Estimate the slope by observing the tangent line at $t = 3$ for $f(t)$:
  • $f(t)$ seems to be increasing at about 2 meters per second at $t = 3$.
• Approximate Answer: 2 meters per second.

3. What is the approximate velocity of the $g$ vehicle at $t = 3$ seconds?

• The velocity is the slope (or derivative) of the position function $g(t)$ at $t = 3$.
• Estimate the slope by observing the tangent line at $t = 3$ for $g(t)$:
  • $g(t)$ appears to be increasing more steeply, approximately at 4 meters per second.
• Approximate Answer: 4 meters per second.

4. Which vehicle is traveling faster at $t = 4$ seconds?

• Look at the slopes of $f(t)$ and $g(t)$ at $t = 4$.
• The function $g(t)$ is much steeper than $f(t)$ at $t = 4$.
• Answer: Vehicle $g$ is traveling faster at $t = 4$ seconds.

Would you like further details on any of these steps or have other questions?

Follow-Up Questions

1. How can we accurately calculate the slope of a tangent line on a graph?
2. Why is the derivative of a position function equivalent to the velocity?
3. How would we determine acceleration from this graph?
4. What differences can we expect between position, velocity, and acceleration graphs?
5. How could we determine the average speed of each vehicle over a certain time interval?

Tip:

When estimating slopes on a graph, pick two points on the line close to your point of interest and calculate the rise over the run to approximate the slope.