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In the context of motion along a coordinate line, the position function, often denoted as x(t), represents the location of an object at any given time t. Think of it as a snapshot of where something is at a specific moment. It’s the starting point for understanding the object's motion.

For example, the exercise provided uses the position function x(t) = 5t - t^3, where t represents time. By setting t to any value, we can find out where the object will be at that time. Specifically, at time t_0 = 3, we substitute the value into the function to get the object's position: x(3) = 5(3) - (3)^3 = 15 - 27 = 6. So at t = 3 seconds, the object is at the position 6 on the coordinate line.

Moving beyond position, velocity in calculus represents how fast an object's position is changing over time. It is the first derivative of the position function with respect to time, and it provides a sense of direction (forward or backward) and speed of movement.

In calculus, finding the velocity function involves differentiating the position function. From our exercise, differentiating x(t) = 5t - t^3 gives us the velocity function v(t) = 5 - 3t^2. This tells us how the velocity of the object changes as time progresses. If we want to find the velocity at time t_0 = 3, we simply substitute 3 into v(t), yielding v(3) = 5 - 3(3)^2 = -22. A negative sign here indicates that the object is moving in the opposite direction of the positive coordinate axis.

Acceleration in calculus is all about the change in velocity over time. Just like velocity is the first derivative of position, acceleration is the first derivative of velocity or, equivalently, the second derivative of position.

When it comes to our exercise, we obtain the acceleration by differentiating the velocity function v(t) = 5 - 3t^2 to get a(t) = -6t. This acceleration function depicts how the object's velocity changes at different times. Substituting time t_0 = 3 into the acceleration function a(t), we find out that the acceleration at that moment is a(3) = -6(3) = -18. A negative value for acceleration suggests that the object is slowing down in the direction of the positive coordinate axis or speeding up in the opposite direction.

Students often confuse speed with velocity, but they're not the same in physics. Velocity is a vector quantity, meaning it has both magnitude and direction, while speed is a scalar—it only has magnitude. In simple terms, speed tells us how fast an object is moving regardless of its moving direction.

In the exercise, the velocity at t_0 = 3 is -22 units per second, which includes the direction (indicated by the negative sign). However, when we talk about speed, we are interested in the absolute value of this number, without care for direction. So, we take the absolute value of the velocity to find the speed at t_0 = 3, which is |v(3)| = |-22| = 22 units per second. This tells us how fast the object is moving, but not its direction of motion.