Activity 1.2.4.
For the moving object whose position \(s\) at time \(t\) is given by the graph provided, answer each of the following questions. Assume that \(s\) is measured in feet and \(t\) is measured in seconds.
The function \(y = s(t) = t + 0.5\sin(\pi t)\) is graphed on the interval \(-1.5 \lt t \lt 5.5\text{.}\) The vertical scale is the same as the horizontal scale. The graph rises and falls periodically, doing so above and below the line \(y = t\text{.}\)
(a)
Use the graph to estimate the average velocity of the object on each of the following intervals: \([0.5,1]\text{,}\) \([1.5,2.5]\text{,}\) \([0,5]\text{.}\) Draw each line whose slope represents the average velocity you seek.
(b)
How could you use average velocities or slopes of lines to estimate the instantaneous velocity of the object at a fixed time?
(c)
Use the graph to estimate the instantaneous velocity of the object when \(t = 2\text{.}\) Should this instantaneous velocity at \(t = 2\) be greater or less than the average velocity on \([1.5,2.5]\) that you computed in (a)? Why?
