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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!

What is the particle's velocity at time t = 12, its acceleration at time t = 2, its position at time t = 2, and at what time during the first 18 sec does s have the largest value?

Explore how to determine a particle's velocity, acceleration, and position using calculus. This problem uses derivatives to find velocity and acceleration, and integrals to calculate position. Perfect for college students studying calculus.

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