Preview Activity 1.6.1.
The position of a car driving along a straight road at time \(t\) in minutes is given by the function \(y = s(t)\) whose graph is provided. The car’s position function has units measured in thousands of feet. For instance, the point \((2,4)\) on the graph indicates that after 2 minutes, the car has traveled 4000 feet.
(a)
In everyday language, describe the behavior of the car over the provided time interval. In particular, you should carefully discuss what is happening on each of the time intervals \([0,1]\text{,}\) \([1,2]\text{,}\) \([2,3]\text{,}\) \([3,4]\text{,}\) and \([4,5]\text{,}\) plus provide commentary overall on what the car is doing on the interval \([0,12]\text{.}\)
(b)
On the lefthand axes provided, sketch a careful, accurate graph of \(y = s'(t)\text{.}\)
(c)
What is the meaning of the function \(y = s'(t)\) in the context of the given problem? What can we say about the car’s behavior when \(s'(t)\) is positive? when \(s'(t)\) is zero? when \(s'(t)\) is negative?
(d)
Rename the function you graphed in (b) to be called \(y = v(t)\text{.}\) Describe the behavior of \(v\) in words, using phrases like “\(v\) is increasing on the interval \(\ldots\)” and “\(v\) is constant on the interval \(\ldots\text{.}\)”
(e)
Sketch a graph of the function \(y = v'(t)\) on the righthand axes provide in (b). Write at least one sentence to explain how the behavior of \(v'(t)\) is connected to the graph of \(y=v(t)\text{.}\)
