Activity 4.1.2.
Suppose that a person is walking in such a way that their velocity varies slightly according to the information given in the following table and graph.
| \(t\) | \(v(t)\) |
| \(0.00\) | \(1.500\) |
| \(0.25\) | \(1.789\) |
| \(0.50\) | \(1.938\) |
| \(0.75\) | \(1.992\) |
| \(1.00\) | \(2.000\) |
| \(1.25\) | \(2.008\) |
| \(1.50\) | \(2.063\) |
| \(1.75\) | \(2.211\) |
| \(2.00\) | \(2.500\) |
(a)
Using the grid, graph, and given data appropriately, estimate the distance traveled by the walker during the two hour interval from \(t = 0\) to \(t = 2\text{.}\) You should use time intervals of width \(\Delta t = 0.5\text{,}\) choosing a way to use the function consistently to determine the height of each rectangle in order to approximate distance traveled.
(b)
How could you get a better approximation of the distance traveled on \([0,2]\text{?}\) Explain, and then find this new estimate.
(c)
Now suppose that you know that \(v\) is given by \(v(t) = 0.5t^3-1.5t^2+1.5t+1.5\text{.}\) Remember that \(v\) is the derivative of the walker’s position function, \(s\text{.}\) Find a formula for \(s\) so that \(s' = v\text{.}\)
(d)
Based on your work in (c), what is the value of \(s(2) - s(0)\text{?}\) What is the meaning of this quantity?
