Models of Experimental Data

Let's think about the problem of integrating the experimental data gathered by our hypothetical EPA worker. The first thing is that we have to accept the fact that we do not know the underlying function that, for each instant of time, gives the rate at which soot is being produced at that instant. There is no possible way to know the underlying "pollution-rate function"; the best we can hope for is that our data gives us a good idea of this function. Note that the "pollution-rate function" is a rate; to obtain the amount of pollution, we need to integrate the pollution function.


Question 1


Since we don't have a formula for the "pollution-rate function", let's try to approximate the integral of this function based on our understanding of the graphical meaning of the integral and what little information we do have. The EPA worker measured the pollution-rate function at four instants of time. We will rewrite the data in a sightly different form and plot the new data:
    Hours        Measured rate of soot
    after 8:00   production (kg/hour)
    ------      -----------------------
      0                2
      2                3
      5                4
      9                1

Figure 2: A plot of experimental data for the "pollution-rate function": time versus rate of soot production.


There are two especially simple assumptions that we can use to "fill in" the missing portions of the graph of the pollution-rate function: the assumption that the pollution-rate function is constant when there are no data points, or the assumption that the pollution-rate function is linear between data points.

Question 2

Assume that the pollution-rate function is constant whenever no data is present. In other words, assume that the rate of soot production is exactly 2 kg/hour for every instant of time between 8:00 and 10:00, it is exactly 3 kg/hour from 10:00 until 1:00, and so on. The function that we get by this process is called the piecewise constant model (or PC model) for the pollution-rate function.

Question 3

Assume that the pollution-rate function is linear whenever no data is present. In other words, assume that the rate of soot production increases steadily from between 8:00 and 10:00, increases at a different steady rate between 10:00 and 1:00, and then decreases steadily from 1:00 until 5:00. Thus, for example, we will assume that the rate of soot production at 9:00 was 2.5 kg/h. The function that we get by this process is called the piecewise linear model (or PL model) for the pollution-rate function.

Question 4

Once again, suppose the EPA worker installs an electronic device that measures the factory's rate of soot production at 20 minute intervals, beginning at 8:00 am.
Next:More Complicated Models
Previous:A Thought Experiment
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The Geometry Center Calculus Development Team

A portion of this lab is based on a problem appearing in the Harvard Consortium Calculus book, Hughes-Hallet, et al, 1994, p. 174

Last modified: Fri Jan 5 09:51:05 1996