Objective:
To provide instructors with interactive examples for the classroom or student
assignments involving
functions for which students are asked to sketch the derivative.Level:
First term calculus.
Prerequisites:
The derivative as a function, the connection between the derivative and the
tangent line to y = f(x), and graphing the derivative using properties
of the original function y = f(x) including intervals over which f(x)
increases, intervals over which f(x) decreases, and points at which the
function has a horizontal tangent.
Platform:
Included are Excel worksheets that will work on both a PC and a MAC, and
animations in both Flash and QuickTime.
Instructor's
Notes: In many text books after the concept of a
derivative has been discussed in terms of limits, tangent lines, and the rate
of change there is a discussion involving the derivative function from a
graphical point of view. A part of such a discussion involves the slope of the
tangent line to the curve and the sign of the derivative. For example:
- If the slope of the tangent line is positive, then
f ' is positive.
- If the slope of the tangent line is negative, then
f ' is negative.
- If the tangent line is horizontal, then f ' is
zero.
The discussions involve such behavior at points and
over intervals. This leads to statements like the following:
If f ' > 0 on an interval, then we say f
is increasing over that interval.
If f' < 0 on an interval, then we say f
is decreasing over the interval.
In addition, it is noted that the magnitude of the
derivative effects the rate at which the function increases or decreases
over and interval. If f ' is large and positive, then f
increases rapidly, while if f ' is a large negative value, then f
decreases rapidly.
With such basic connections between the slope of the
tangent line (alias the value of the derivative) and the geometric behavior
of the graph of y = f(x) we can establish enough information to
produce a reasonable sketch of the derivative function f ' (x). We
use the intrinsic properties of the graph of y = f(x) to produce a
sketch of f ' (x) that usually does not have a precise scale, but
does reflect the qualitative aspect of the graph of the derivative curve.
Here we do not assume that techniques for computing
derivative formulas have been discussed. In fact, our examples provide only
the graph of the function y = f(x). Also in our examples we do not
consider one-sided differentiability concepts.
Given the graph of a function y = f(x) a standard
approach is to identity intervals over which the graph is increasing, other
intervals over which it is decreasing, and points at which the tangent line
is horizontal. Next use this information to identify behavior of the graph
of f ' (x), and then provide a sketch based on the information
recorded. We illustrate this approach in Example 1.
Example 1. Sketch a graph of the f '(x)
for the function shown in Figure 1.
Figure 1. |
In Figure 2 we identity features of the graph of y
= f(x) and connect them to properties of the graph of f '(x).
Figure 2. |
In Figure 3 we use the information
derived from Figure 2 about f '(x) to sketch a graph that has the
approximate shape of a graph of f ' (x), but not necessarily with the
proper scale.
Figure
3. |
The turning point of the curve shown in
Figure 3 may not be in the correct position, but such a point must occur
based on the properties of the original curve.
■
In the introductory animation at the top
of this page and the Excel worksheets
included in this demo the original curve is sketched and simultaneously
three choices for an approximate graph of the derivative are sketched. The
user is to select the correct choice for the (approximate) derivative curve,
but must perform their own analysis like that illustrated in Example 1. For
instance, as Figure 1 would be sketched the three choices appearing below
would be generated.
Example 2. In Animation #1
that follows we develop an easy method for determining the behavior of the
derivative of function by generating a sketch of the function itself. As the curve is sketched by plotting points the tangent line to
the curve is also displayed and moves along the curve. By recording the sign
of the slope of the tangent line as points are plotted a pattern of the
behavior of the graph of its derivative is constructed. (The Flash files
below are smaller and load faster.) To get the Flash player click
here; to get the QuickTime player click
here.
Start Animation #1, which includes
audio, by clicking one of
the following:
Flash file
QuickTime file
After you view and understand the idea in
Animation #1 use Animation #2.
At the end of this animation decide which figure is a good approximation the
derivative of the function.
Start Animation #2 by clicking one of
the following: Flash file
QuickTime file
■
We have developed a gallery of 10 Excel
routines that sketch a function y = f(x) and simultaneously generate
three choices for possible approximations to the graph of the derivative
f '(x). The user is asked to determine which of the three graphs is a
good approximation to the graph of the derivative based on properties of the
function f(x). These demos can be used by instructors as part of
classroom demonstrations or can be part of an assignment for students for
sketching a derivative approximation.
Click here to go to the
Sketching the Derivative
Gallery.
Credits:
This demo, the Excel files, and the animations were developed by
Dr.
David R. Hill
Department of Mathematics
Temple University
and is included in Demos
with Positive Impact with his permission.