Preview Activity 7.1.1.
The position of a moving object is given by the function \(s(t)\text{,}\) where \(s\) is measured in feet and \(t\) in seconds. We determine that the velocity is \(v(t) = 4t + 1\) feet per second.
(a)
How much does the position change over the time interval \([0,4]\text{?}\)
(b)
Does this give you enough information to determine \(s(4)\text{,}\) the position at time \(t=4\text{?}\) If so, what is \(s(4)\text{?}\) If not, what additional information would you need to know to determine \(s(4)\text{?}\)
(c)
Suppose you are told that the object’s initial position \(s(0) = 7\text{.}\) Determine \(s(2)\text{,}\) the object’s position 2 seconds later.
(d)
If you are told instead that the object’s initial position is \(s(0) = 3\text{,}\) what is \(s(2)\text{?}\)
(e)
If we only know the velocity \(v(t)=4t+1\text{,}\) is it possible that the object’s position at all times is \(s(t) = 2t^2 + t - 4\text{?}\) Explain how you know.
(f)
Are there other possibilities for \(s(t)\text{?}\) If so, what are they?
(g)
If, in addition to knowing the velocity function is \(v(t) = 4t+1\text{,}\) we know the initial position \(s(0)\text{,}\) how many possibilities are there for \(s(t)\text{?}\)
