1. **Problem Statement:** We are given the position function $s(t)$ of a car along a straight road and asked to identify which graph (A-F) best represents the derivative of the velocity function $v'(t)$, where $v(t) = s'(t)$ is the velocity. 2. **Understanding the Functions:** - $s(t)$ is the position function (in thousands of feet). - $v(t) = s'(t)$ is the velocity function (rate of change of position). - $v'(t)$ is the acceleration function (rate of change of velocity). 3. **Key Concepts:** - The derivative of position $s(t)$ is velocity $v(t)$. - The derivative of velocity $v(t)$ is acceleration $v'(t)$. - When $v'(t)$ is positive, velocity is increasing (car speeding up). - When $v'(t)$ is negative, velocity is decreasing (car slowing down). - The position graph $s(t)$ is stepwise increasing with flat sections and sharp rises, so velocity $v(t)$ is zero during flat sections and positive during sharp rises. 4. **Analyzing the Graphs:** - Since $s(t)$ has flat sections, $v(t)$ is zero there, so $v'(t)$ should be zero or near zero during those times. - Sharp rises in $s(t)$ correspond to positive velocity spikes. - The acceleration $v'(t)$ should show spikes at the points where velocity changes abruptly. 5. **Selecting the Best Graph:** - Graph B shows positive and negative spikes crossing zero frequently, matching the sudden changes in velocity. - Other graphs either show continuous oscillations or only positive values, which do not fit the stepwise nature of $s(t)$. 6. **Evaluating the Statements:** - A. False. $v'(t)$ negative means velocity is decreasing, not necessarily moving backwards. - B. False. The statement is self-contradictory. - C. True. Negative $v'(t)$ means slowing down. - D. False. $v'(t)$ is acceleration, not position. - E. True. Velocity can be nonzero when acceleration is zero. - F. True. $v'(t)$ represents acceleration. - G. False. Zero acceleration does not imply zero velocity. - H. False. Some statements are true. **Final answers:** - Best graph representing $v'(t)$ is **Graph B**. - True statements: **C, E, F**.