Let's go through each part of the question based on the graph and provided data. Here's a detailed breakdown:

a. What is the particle's velocity at time t = 12?

The velocity $v(t)$ of the particle is given by the value of the function $f(t)$, which is shown in the graph. At $t = 12$, the graph shows that $f(12) = 0$. Therefore, the velocity at time $t = 12$ is: $\boxed{0 \text{ m/sec}}$

b. Is the acceleration of the particle at time t = 2 positive or negative?

The acceleration $a(t)$ is the derivative of the velocity function, or the slope of $f(t)$ at $t = 2$. Looking at the graph, the slope at $t = 2$ is positive since the curve is increasing at that point. Therefore, the acceleration is: $\boxed{\text{positive}}$

c. What is the particle's position at time t = 2?

The position $s(t)$ is given by the integral of $f(x)$ from $0$ to $t$, that is: $s(t) = \int_0^t f(x) \, dx$ At $t = 2$, the position corresponds to the area under the curve from $0$ to $2$. Based on the graph, this area is approximately 1 square unit (as the value of $f(t)$ is about 1 in that region). Thus, the position at $t = 2$ is: $\boxed{1 \text{ m}}$

d. At what time during the first 18 sec does s have the largest value?

The position $s(t)$ is the accumulated area under the curve, and the largest value of $s(t)$ will occur when the area under $f(t)$ reaches its maximum before any negative contribution occurs (i.e., before the curve dips below the x-axis).

From the graph, the area under the curve keeps increasing until about $t = 10$, where $f(t)$ starts to decrease and cross the x-axis. Therefore, $s(t)$ has the largest value at $t = 10$: $\boxed{t = 10}$

Would you like more details or have any questions on this?

Here are some related questions for deeper exploration:

1. How would you compute the particle’s position at $t = 4$?
2. What is the velocity at $t = 10$ based on the graph?
3. Can you determine the total displacement of the particle by $t = 18$?
4. What is the net distance traveled by the particle between $t = 0$ and $t = 18$?
5. If $f(t)$ represents the velocity function, how can you find when the particle changes direction?

Tip: Always remember that when a velocity graph crosses the x-axis, the particle either stops or changes direction!