NOTE: No good demo collection would be
complete without a demonstration involving the definition of
derivative. While neither the ideas nor the approach are novel,
this demo can be used by any introductory course in calculus. The
software codes allow an automated investigation of approximation of the
tangent line by a sequence of secant lines.
Objective: The
definition of the derivative at a point x = a involves the limiting
behavior (as h approaches zero) of the slopes of the secant lines
passing through through (a,f(a)) and (a+h,f(a+h)),. This demo
provides a visualization of the secant lines as they approach the
tangent line.
Level:
This demo is appropriate for any course in which derivatives are
introduced.
Prerequisites: Students should be familiar
with limits. In addition, familiarity with the terms "secant line"
and "tangent line" to a graph is recommended.
Platform: Various (see software
downloads below). The MATLAB version of the demo displays a
table of the numerical values of the slopes of the secant lines and the
slope of the tangent at (a,f(a)). Thus students may see a
numerical approach to the limiting value of the slope as well as a
graphical approach to the tangent line. While the main emphasis of
this demo is on both a numerical and graphical approach to the slope,
only the MATLAB utility gives the numerical approach simultaneously as
the secant lines are generated. Please note that Maple VI and
Mathematica, do not support the simultaneous display of the table
of slopes as illustrated in the demo page animations. Rather, an
array of slopes is displayed separately from the graph. The
Mathcad animation displays the values of h (as h->0) and the slopes
as they change.
Instructor's
Notes: A secant line passing through a point C on a graph
is a line that passes through C and (at least) one other point Q on the
graph.
The slope of the tangent line at a point
C can be approximated using a secant line passing through C and a second
point Q on the graph, as illustrated in Figure 1.
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Figure
1. |
If we allow Q to approach C along the
graph of f, the limit of the slopes of the secants (if the limit exists)
approaches the slope of the tangent line. Figure 2 and the
animation at the beginning of this demo illustrate a right-hand approach
to C.
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Figure 2. Right-hand
Approach to C. |
Figure 3 illustrates a left-hand
approach. We leave out the labels for point Q in this
animation.
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Figure 3. Left-hand Approach
to C. |
By choosing two points, one on either
side of C, we can approach C from the left and right (Figure
4).
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Figure 4. Approach from
both sides of C. |
Using computer software to generate the
graphics greatly enhances the instructor's ability to provide many more
illustrative examples than hand sketches or textbook pictures. The
MATLAB animations allow students to see both a geometric and numerical
approach to the limiting tangent line.
Software Routines for
Download
MATLAB R11: A very useful
utility, secline.m,
was developed by Dr. David R.
Hill. The user may choose from 4 built-in demos OR may input
his or her own function and plotting interval. The user decides
whether to select the point C by mouse input OR by giving the value of x
used to compute the coordinates of C. When the graph is displayed,
the user selects (by mouse input) a point to the left, right, or two
points on either side of C. The secant lines are generated as the
selected point(s) approach C. In addition to the moving secant
lines, a table showing the slopes of each secant line and finally the
slope of the tangent line is also given. Thus, students may see a
graphical and numerical approach of the slopes of the secant lines to
the slope of the tangent line.
An illustrative sample of the animations
produced by secline.m can be seen below.
Routines for alternate platforms are
given below. Please note that Maple VI and Mathematica do
not support the simultaneous display of a table of slopes. Rather,
an array of slopes is displayed separately from the
graph. The Mathcad animation displays the values of h (as
h==>0) and the slopes as they change.
MAPLE VI:
defofderiv.mws
Mathematica 4:
defofderiv.nb
Mathcad
8: defofderiv_mcd8.mcd
Credits:
This demo was submitted by
Dr.
Lila F. Roberts
Mathematics and Computer Science
Department
Georgia College and State University
Milledgeville, GA
31061
and is included in Demos with
Positive Impact with her permission.
