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

• With the current data, determine your best guess for the rate of soot production at 3:00 pm. What assumptions did you use in order to come up with that guess?

• Suppose the EPA worker comes back with an electronic device that measures the factory's rate of soot production at 20 minute intervals, beginning at 8:00 am.
• Will this additional data make it easier to know the amount of soot that the factory is producing? Why or why not?

• If you have access to this new data, explain how you could approximate the rate of soot production at 3:00, at 3:10, and at 3:15.

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.
• Plot the piecewise constant model on the domain [0,10] using the graph provided.

• Graphically represent the area under the graph of the PC model as a union of rectangles.

• Using geometric techniques, find the exact integral of the piecewise constant model on this domain.

• According to this model, how many kilograms of soot did the factory produce during ten hours of operation?

• One advantage to this model is that the integral is relatively easy to calculate. What disadvantages are there to this model? How realistic is the assumption that the rate of soot production remains constant between data points?

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.
• Plot the piecewise linear model on the domain [0,10] using the graph provided. You will need to make an additional assumption in order to graph the PL model on the interval [9,10]. Explicitly state what assumption you used.

• Graphically represent the area under the graph of the PC model as a union of trapezoids.

• Using geometric techniques, find the exact integral of the piecewise linear model on this domain. (Hint: recall the area of a trapezoid.)

• According to this model, how many kilograms of soot did the factory produce during ten hours of operation?

• What advantages and disadvantages are there to this model? Do you think that amount of soot predicted by this model is more or less valid than the amount predicted by the PC model? Why?

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.
• For the new data, you can still model the pollution-rate function as piecewise constant or piecewise linear. Do you expect the differences between the PC and PL models based on this new data set (with more data points) to be more or less pronounced than for the original data (the data set with four data points)?

• Suppose now the EPA worker gains access to a device that measures the factory's rate of soot production at 1 minute intervals. Show that the integral of the PL and PC models will be nearly identical. Construct an argument and sketch a diagram to indicate why the claim is true. (Hint: Look at the area under the PL model that is not under the PC model.)

Next:More Complicated Models
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