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Velocity is a key concept in calculus, particularly when it comes to analyzing the motion of particles. It tells us the rate at which a particle's position changes over time. To find the velocity of a particle, we differentiate the position function, in this case, \( s = f(t) = t^3 - 12t^2 + 36t \), with respect to time \( t \). This provides the velocity function:

• \( v(t) = \frac{ds}{dt} = 3t^2 - 24t + 36 \).

The velocity function helps us determine how fast the particle is moving at any given second.
To find the velocity at a specific time, such as 3 seconds, substitute the time into the velocity function: \( v(3) = 3(3)^2 - 24(3) + 36 = -9 \text{ ft/s} \).
The negative sign indicates the particle is moving in the opposite direction.The moments when the velocity is zero (at rest) are points where the particle stops momentarily. This occurs when solving \( 3t^2 - 24t + 36 = 0 \), leading to \( t = 2 \) and \( t = 6 \) seconds. These concepts emphasize how velocity provides insights into the dynamics of motion.

Acceleration measures how quickly the velocity of a particle changes with time. It is the derivative of the velocity function. In calculus, finding the acceleration involves differentiating the velocity function. For our example:

• The velocity function is \( v(t) = 3t^2 - 24t + 36 \)
• The acceleration function is found by differentiating: \( a(t) = \frac{dv}{dt} = 6t - 24 \).

This function tells us how the speed of the particle is increasing or decreasing.
To find the acceleration at a certain time, such as 3 seconds, replace \( t \) in \( a(t) = 6t - 24 \) with 3: \( a(3) = 6(3) - 24 = -6 \text{ ft/s}^2 \).
A negative acceleration means that the particle is decelerating. Knowing acceleration is crucial since it provides deeper understanding about how the particle's velocity changes, whether it's speeding up or slowing down.

Motion in calculus refers to the analysis of how a particle moves along a path over time. By breaking it into parts, we explore its position, velocity, and acceleration. Understanding the complete motion involves looking at each aspect:

• The position function \( f(t) = t^3 - 12t^2 + 36t \) shows where the particle is in space at any moment.
• The velocity function \( v(t) \) indicates how quickly the position changes.
• The acceleration \( a(t) \) describes how the velocity itself is changing.

To picture the total distance traveled by the particle in the first 8 seconds, check the position at specific intervals and calculate the path taken:

• Start at \( s(0) = 0 \), move to \( s(2) = 16 \), then \( s(6) = 0 \), and finish at \( s(8) = 64 \),
• Total distance is the sum of the absolute changes, equaling 96 ft.

Understanding motion helps us visualize not just where a particle is, but how fast it moves and its changing pace,providing a comprehensive view of movement dynamics.