- 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?

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.

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.

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