Analyzing Accuracy

In most situations, it is not good enough merely to know what kind of model function is likely to give the best results. Usually it is important to have some idea of how accurate those results are as well.

Experiment #3

Since there is no way to discover what the true value is, one has to deduce what inaccuracies might have been introduced from the properties of the model function and the data.

Question 10

Using biological reasoning, argue that we might reasonably assume that the actual rate of CO2 production is monotonically decreasing from an hour past sunset until the next dawn.
The exploration page linked below shows a monotone decreasing function that passes through our sampled data for the time interval [13,24]. Superimpose both piecewise constant models and a piecewise linear model on the graph of the function. On the basis of this experiment, complete the following sentence. If it is not possible to complete the sentence for some model, explain why.
```
For ANY monotone decreasing function, the integral of the

 _________________ model always  _________(under, over)-estimates

the integral of the real function because ___________________.
       
```
How does the assertion change for each model if the underlying function that you are modeling is monotone increasing? Why?
On biological grounds, it might be safe to assume that the real CO2 rate function is concave up for a 2-3 hour period around sunrise. (Extra Credit: Give a biological explanation for this.) Repeat the experiment for this time interval, and complete the following sentence for any concave up function. Again, if it is not possible to complete the sentence for some models, explain why.
```
For concave up functions, the integral of the

_________________ model always  ________(under, over)-estimates

the integral of the real function because ___________________.
       
```
How does the assertion change for each model if the underlying function that you are modeling is concave down? Why?

Experiment #4

We are now in a position to formulate an estimate of how close the results we have obtained by numerical integration are to the actual CO2 concentration. Using what we discovered about over- and under-estimates for monotone and convex data, we will attempt to find upper and lower bounds for the actual net change in CO2 during the 24-hour period for which we have data.

For this experiment, use data set 1 again. Carefully record the intervals and model functions used as well as the results. You will need this information for Question 11.

Split the interval [0,23] into subintervals on which you assume that the underlying rate function is monotone. (There is no right answer, but choosing "large" intervals will make the subsequent analysis easier. For example, you may assume the underlying function is monotone on three intervals.)
For each interval on which the rate function is monotone, use the piecewise constant model (restricted to those intervals) to compute upper and lower bounds for the integral of the rate function.
Now split the interval [0,23] into subintervals on which you assume that the underlying rate function is concave. For each interval, choose models whose integrals give upper (and lower) bounds.

Question 11

At dawn, there were 2.600 mmol of CO2 per liter of water. Use your experimental findings to give upper and lower bounds for the actual CO2 concentration in the San Marcos River after 24 hours. Explain what you did to obtain your answer. Make sure that your final answer makes sense by checking that the values you obtained from Experiment #1 all lie within your estimates. (Hint: assume that the true function that generated the data is monotone and/or concave up/down on certain time intervals, then use the ideas from Experiments #3 and #4.)

Question 12

This is an open-ended question, so feel free to formulate conjectures provided you carefully explain how you attempted to answer it.

Of the values you recorded in Experiment #1, which would you give as the best approximation to the total change in CO2 for the river over this 24-hour period? Specify the model you prefer, and give reasons for your preferences. Based on your answer, do you think the river is healthy? Explain.

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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

for CO2 Rate Data Last modified: Mon Jan 8 13:05:20 1996