Sketch the Derivative


 

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.

A.

B.

C.

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.


DRH 12/28/06   Last updated 1/15/07

Since 12/28/06