Choosing a Good Model Function

We would like to know the "best" answer to the question "What is the net change in CO2 concentration over a 24 hour time period?" Before we can we can get very far with this, we need to identify some of the relevant issues.

Experiment 1

Use each of the five models to approximately determine how much CO2 accumulated in the San Marcos River during a 24-hour period using "Data Set 1." This data set gives rates of carbon-dioxide during a time interval [0,24]. To do this, select a Model Function (such as "Piecewise Linear" and "Cubic Spline") and press the button labelled "Integrate the model function." The integral of the model over the selected interval of integration will be display under the graph of the model function.

Record your answers in the table you were given.

Question 7

One way of determining whether a particular model is valid for a given set of data is to compare the properties of the model functions with the properties of the system being modeled. Argue whether you think CO2 production by plants and animals in the river is a continuous function of time. Is it differentiable? Which models have these properties?

Question 8

Another important issue is whether the way the data was collected will react in some strange way with a given kind of model function.

Experiment 2

We know from the definition of the integral as a limit of approximating sums that if we collect enough data, at least the piecewise constant models will get closer and closer to the true value of the integral (provided the function we are modeling is continuous). However, because of the time and expense of collecting and processing data, researchers usually prefer to use as few data points as possible to get good results.

Integrate each of the five models a second time using data set 2. Data set 2 consists of the CO2 rate measurements taken every other hour during the day. Again record your answers in the table you were given. This data will be used in the next question.

Question 9

Based on your observations:
Next:Analyzing Accuracy
Previous:Introduction to Models of CO2 Concentrations
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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: Mon Jan 8 13:04:47 1996