Activity 5.6.3.
A car traveling along a straight road is braking and its velocity is measured at several different points in time, as given in the following table. Assume that \(v\) is continuous, always decreasing, and always concave down, as is suggested by the data.
| seconds, \(t\) | Velocity in ft/sec, \(v(t)\) |
| \(0\) | \(100\) |
| \(0.3\) | \(99\) |
| \(0.6\) | \(96\) |
| \(0.9\) | \(90\) |
| \(1.2\) | \(80\) |
| \(1.5\) | \(50\) |
| \(1.8\) | \(0\) |
(a)
Plot the given data on the set of axes provided in the figure, with time on the horizontal axis and the velocity on the vertical axis.
(b)
What definite integral will give you the exact distance the car traveled on \([0,1.8]\text{?}\)
(c)
Estimate the total distance traveled on \([0,1.8]\) by computing \(L_3\text{,}\) \(R_3\text{,}\) and \(T_3\text{.}\) Which of these under-estimates the true distance traveled?
(d)
Estimate the total distance traveled on \([0,1.8]\) by computing \(M_3\text{.}\) Is this an over- or under-estimate? Why?
(e)
Using your results from (c) and (d), improve your estimate further by using Simpson’s Rule.
(f)
What is your best estimate of the average velocity of the car on \([0,1.8]\text{?}\) Why? What are the units on this quantity?
