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The position function of an object is the function that models where the particle is located at time t, which means the function will be given in terms of the variable t. Since velocity is the derivative of position, and acceleration is the derivative of velocity, we can differentiate the position function in order to find the velocity and acceleration functions.

Once we have all three of the these functions, we can figure out all kinds of things about the behavior of the particle. In this video we’ll:

– Find velocity at time t.

– Find velocity after 2 seconds and after 4 seconds.

– Say when the particle is at rest by setting the velocity function equal to 0 and solving for t.

– Find out when the particle is moving forward by identifying where the velocity function is positive.

– Find the distance traveled by the particle during the first 5 seconds.

– Find the acceleration at time t by calculating the second derivative of the position function.

– Figure out the acceleration after 4 seconds.

– Calculate when the particle is speeding up and when it’s slowing down by finding out where the velocity function is positive and negative, where the acceleration function is positive and negative, and then comparing the points in time at which both velocity and acceleration are positive, and where both velocity and acceleration are negative. When they are both positive or both negative (have the same sign), the particle will be speeding up. Whenever they have opposite signs, it means that the particle is slowing down.

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