Activity 5.
Consider a moving object whose position function is given by \(s(t) = t^2\text{,}\) where \(s\) is measured in meters and \(t\) is measured in minutes.
(a)
Determine the most simplified expression for the average velocity of the object on the interval \([3, 3+h]\text{,}\) where \(h \gt 0\text{.}\)
Hint.
\(s(3+h) = (3+h)^2\text{.}\)
Answer.
\(6 + h\)
Solution.
Observe that \(AV_{[3, 3+h]} = \frac{s(3+h)-s(3)}{h} = \frac{(3+h)^2 - 3^2}{h} = \frac{9 + 6h + h^2 - 9}{h} = \frac{6h + h^2}{h} = \frac{h(6 + h)}{h} = 6 + h\text{.}\)
(b)
Determine the average velocity of the object on the interval \([3,3.2]\text{.}\) Include units on your answer.
Hint.
Recall that \(AV_{[a,b]} = \frac{s(b)-s(a)}{b-a}\text{.}\)
Answer.
\(6.2\) meters/min.
Solution.
Using the expression just found in (a) with \(h = 0.2\text{,}\) \(AV_{[3,3.2]} = 6 + 0.2 = 6.2\) meters/min.
(c)
Determine the instantaneous velocity of the object when \(t = 3\text{.}\) Include units on your answer.
Hint.
Consider \(\lim_{h \to 0} \frac{s(3+h)-s(3)}{h}\) and use your work in (a).
Answer.
\(6\) meters per minute.
Solution.
Taking the limit of average velocity and using our work from (a), we find that
\begin{equation*}
IV_{t = 3} = \lim_{h \to 0} AV_{[3, 3+h]} = \lim_{h \to 0} 6+h = 6\text{,}
\end{equation*}
so the instantaneous velocity of the object when \(t = 3\) is 6 meters per minute.
