Bottles and Divers
(Rates of Change)
High School Math Project
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Mathematically Speaking
The NCTM Standards give direction on the content of high school mathematics,
indicating that the underpinnings of calculus be included. The standards
recommend that maximum and minimum points of a graph, the limiting process,
the area under a curve, the rate of change, and the slope of a tangent line be examined from both a graphical and a numerical perspective. This lesson is an
informal exploration of rates of change and the derivative.
The derivative and related developments represent one of the most important
advances in the history of mathematics. The seventeenth century discoveries of
analytical geometry by Descartes and infinitesimal calculus by Newton and Leibnitz
are the beginning of modern mathematics. The concepts of limits and the derivative
are key ideas in calculus.
Some students do not have the opportunity to explore the concepts involved with
the derivative in an informal way before they begin the formal study of calculus. As
a result, these students may rely on memorizing formulas and rules rather than
clearly understanding the concepts involved. Explorations such as those in this
lesson allow students to investigate and explore the important mathematics
without the formality that they will encounter later. These explorations and the
applications on which the investigations are based help give students a solid
mathematical foundation.
In many calculus texts, the derivative of the function y = f(x) is defined as  which can also be written as  How does this differ from
 or 
? In the video lesson, the students calculate an approximation for the derivative
because they are actually using a particular value for Dx. Help your students realize
that there are several ways that an approximation may be calculated. They can
determine the slope of the line through points on either side of the given point,
through the point and a point slightly above the given point, or through the point
and a point slightly below the given point. Also help students to understand that
they can get a closer approximation to the instantaneous rate of change by using
smaller values for Dx. As students as they calculate an approximation for the derivative, help them realize
that the more they zoom in to a particular area of a graph, the more that particular
part of the graph will approximate a straight line. Have the students use the zoom
feature of their calculator to examine the graphical implications of calculating the
rate of change over a very small period as an approximation of the derivative or the
instantaneous rate of change at a particular point.
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