Definition of Derivative--A Graphical and Numerical Approach

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.

 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.

 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.

 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).

 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.

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

Credits:  This demo was submitted by

Dr. Lila F. Roberts
College of Information & Mathematical Sciences
Clayton State University
Morrow, GA 30260

and is included in Demos with Positive Impact with her permission.

LFR 2/701        Last updated  9/15/2010  (DRH)
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