Accumulating Rates of Change
The fundamental theorem of calculus tells us that if we know the rate
of change of some quantity, then adding up (or integrating) the
rate of change over some interval will give the total change in that
quantity over the same interval. For example, if a car is moving
along a straight line and we know the speed of the car as a function
of time, it is possible to determine the total change in the car's
position over some time interval. But what if we don't know a
formula for the car's velocity, but we only have measured its
velocity at certain instants of time? Is it possible to "integrate"
this discrete data in order to estimate the change in the car's
position? If so, how?
In this lab we learn to model functions that produce experimental
data. By integrating the model, we approximate the (true) integral of
the underlying (unknown) function. First, we integrate pre-collected
data concerning the rate at which carbon-dioide is produced in an
aquatic environment. Then we create, collect, and analyze data
concerning the relationship between velocity and position.
This lab is long. But some parts can be done independently of others.
After completing the first three sections below, you can move on to
the section on CO2 concentrations, or directly to the section on
Next:A Thought Experiment
A portion of this lab is based on a problem appearing in
the Harvard Consortium Calculus book, Hughes-Hallet, et al,
1994, p. 174
Support for the Curriculum Initiative Project at the
University of Minnesota
has been provided by a grant from the
National Science Foundation
(DUE 9456095) and by the Geometry Center.
The Geometry Center Calculus Development Team
Last modified: Fri Jan 5 11:19:51 1996