POSSIBLE SOLUTIONS

1. A bicyclist travels .5 mile. Obtain a data set consisting of (at least) 25 (time in sec, distance in ft) pairs of observations. (Or you can use the data set below.)

There are several ways to obtain the data. We include here a dummy data set on {time (sec), position (ft)).

Input := 

data = {{0, 0}, {6., 7.}, {12., 28.}, {18., 61.}, 
{24., 102.}, {30., 148.}, {36., 195.},  {42., 241.}, 
{48., 285.}, {54., 327.}, {60., 370.}, {66., 419.}, 
{72., 479.}, {78., 557.},  {84., 658.}, {90., 787.}, 
{96., 947.}, {102., 1137.}, {108., 1353.}, {114., 1590.}, 
 {120., 1835.}, {126., 2075.}, {132., 2295.}, 
 {138., 2476.}, {144., 2599.}, {150., 2644.}};

Alternatively, the students may be asked to collect the data, and given specific instructions about how to do it: divide a quarter mile track into sections (not necessarily of equal length), and then write down the time at which the cyclist passes each mark, using a stopwatch. A third option is to simply ask the students to collect some minimum number of (time, distance) pairs, and accept the likelihood that the students may have to collect data more than once before obtaining a useful data set. The data will be more interesting if the cyclist varies his/her speed considerably. See Teacher Notes for more ideas.

The intent of the next two questions is to have the students `discover' the relationship between slope and velocity (speed), and also to see the distance function become more smooth as more data points are included. This will help them to answer `smoothly curving' in response to the fourth question. It might be worthwhile to point out that the curve would pass through all of our data points.

2. Plot every 8th data point. What was the average speed of the bicycle in each interval? Connect the data points with line segments. What is the slope of each segment? - The slope of each line segment is the average slope of the bicyclist in each interval.

3. Plot every 4th, every other, and every data point, and answer the same questions. The slope of each line segment is the average slope of the bicyclist in each interval.

4. Let the line segments that connect the data points be our `distance function'. What would our distance function look like if we took a data point every foot? every inch? if we had an electronic recording device that took observations continuously?

It would look something very much like our last plot for every data point, getting even smoother.

5. What happens to the length of the line segments as we take measurements more frequently? What does the slope of the line segments tell us? If we have continuous observations, what slope (the slope of what line) would give us the same information?

The fifth question asks the students to apply their budding intuitive understanding of limits to the slope. Especially if the students are working in groups, they are likely to `discover' the tangent line slope as velocity without additional prodding.

6. Estimate when the velocity ( in ft/sec and mi/hr) is greatest in this one-half mile cycling trip.

We use the plot of velocity based on every data point.

Input := 

ListPlot[vdataE,
	PlotStyle  -> { PointSize[.02] },
	PlotJoined -> True,
	AxesLabel  -> {"time [sec]", "vel [ft/sec]"}]

Output =

-Graphics-

From the plot of velocity based on every data point, it would appear that the point (120, 40.5) is the maximum on the velocity vs. time plot. Therefore it looks like the maximum velocity is 40.5 ft/sec which is approximately 27. 6 mi/hr.

Input := 

40.5 ft/sec 3600 sec/hr /(5280 ft/mi)

Output =

27.6136 mi
----------
    hr

The next two questions concern data collection.

7. How would you decide how many observations to take?

The goal is to have the student say something along the lines of `Choose the interval so that there is not a large amount of change from one point to the next' or `so that a plot does not look very jagged.' By this time, it is not unreasonable to expect that some students will answer `so that the change in slope between adjacent segments is small.'

8. In some research or industrial settings, observations may be very expensive or dangerous to obtain. Crash testing of automobiles and `weatherbeater' airplanes sent to obtain data about hurricanes are two examples. In these situations, additional factors play a role in determining the number of observations to obtain. Discuss some appropriate issues using these examples and/or some of your own.

This question raises some ethical questions in data collection, and is meant to stimulate thought and discussion.