### Problem StatementThe function \( s(t) \) describes the motion of a particle along a line.\[ s(t) = t^3 - 11t^2 + 35t - 250 \]1. **(a)** Find the velocity function \( v(t) \) of the particle at any time \( t \geq 0 \). - \( v(t) = \underline{\hspace{4cm}} \)2. **(b)** Identify the time interval(s) on which the particle is moving in a positive direction. (Enter your answer using interval notation.) - \(\underline{\hspace{8cm}}\)3. **(c)** Identify the time interval(s) on which the particle is moving in a negative direction. (Enter your answer using interval notation.) - \(\underline{\hspace{8cm}}\)4. **(d)** Identify the time(s) at which the particle changes direction. (Enter your answers as a comma-separated list.) - \( t = \underline{\hspace{6cm}} \)### ExplanationIn this exercise, we are given a function that represents the motion of a particle along a line, and we are required to:1. **Calculate the Velocity Function**: Derive the velocity function from the given position function \( s(t) \).2. **Determine Positive Movement Intervals**: Identify time intervals where the particle moves in a positive direction.3. **Determine Negative Movement Intervals**: Identify time intervals where the particle moves in a negative direction.4. **Identify Points of Direction Change**: Find the times at which the particle changes its direction.The solution involves using calculus to derive the velocity function \( v(t) = s'(t) \), analyzing the sign of \( v(t) \) to identify intervals of positive and negative motion, and determining when changes in direction occur by finding where \( v(t) = 0 \) and changes sign.

Name: ### Problem StatementThe function \( s(t) \) describes the motion of
Uploaded: 2024-03-05T11:56:34.000Z
Channel: Bartleby
Description: ### Problem StatementThe function \( s(t) \) describes the motion of a particle along a line.\[ s(t) = t^3 - 11t^2 + 35t - 250 \]1. **(a)** Find the velocity function \( v(t) \) of the particle at any time \( t \geq 0 \). - \( v(t) = \underline{\h