Microworld: Magnify
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Author:
Dan
Kalman
Two
of the most fundamental concepts in calculus are the limit and the derivative.
Each of these concepts can be visualized graphically, in terms of zooming
in on a point on the graph of a function. Here, zooming in refers to repeatedly
magnifying a part of the graph, so that it can be examined on finer and finer
scales. Think of looking at the graph with ever more powerful microscopes.
Both
the idea of limit and the idea of derivative concern the properties of the
graph under indefinitely high magnifications. For a limit, we imagine that
the graph is missing a point for one particular x. By zooming in, it may
become clear that there is a single obvious position for the missing point.
If so, at that position the y coordinate is what we call the limit. It allows
us to infer a natural y value for a missing point, based on nearby points.
For a derivative, we again imagine zooming in at a single point of the graph.
This time, we want to know whether the graph will appear to become a straight
line under sufficiently powerful magnifications. If so, the slope of the
straight line is what we call the derivative. It gives us a sense of the
slope of a curve at a single point. This computer activity will allow you
to experience this zooming in process for each concept.
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