Consider a particle moving along the x -axis where x(t) is...

Consider a particle moving along the $x$ -axis where $x(t)$ is the position of the particle at time $t, x^{\prime}(t)$ is its velocity, and $\int_{a}^{b}\left|x^{\prime}(t)\right| d t$ is the distance the particle travels in the interval of time. Repeat Exercise 97 for the position function given by $x(t)=(t-1)(t-3)^{2}, 0 \leq t \leq 5$.

Consider a particle moving along the $x$ -axis where $x(t)$ is the position of the particle at time $t, x^{\prime}(t)$ is its velocity, and $\int_{a}^{b}\left|x^{\prime}(t)\right| d t$ is the distance the particle travels in the interval of time.
Repeat Exercise 97 for the position function given by $x(t)=(t-1)(t-3)^{2}, 0 \leq t \leq 5$.

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Key Concepts

Position Function

The position function describes the location of a particle along a line at any given time. It gives the coordinates of the particle and is often represented as a function of time, which determines the path the particle follows.

Velocity Function

The velocity function is the derivative of the position function with respect to time. It indicates both the speed and the direction of the particle's motion, and its sign reveals whether the particle is moving forward or backward along the x-axis.

Critical Points

Critical points are the values of time when the velocity function is equal to zero or changes sign. These points are important because they mark transitions in the direction of motion and help in segmenting the timeline for calculating quantities like the total distance traveled.

Integration of Absolute Value of Velocity

The total distance traveled by a particle is found by integrating the absolute value of its velocity function over the time interval of interest. This integration accounts for all movement, regardless of direction, by summing the magnitudes of displacement over various intervals.

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