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Sample Problem 2
Use a graph plotter to look at the graph of the function
f(x) = sin(tan(x))
What is happening at each of the "fuzzy bits"?
If the function is periodic, then why do each of these appear different?
Figure 5: Viewing f(x) = sin(tan(x)) using xFunctions 2.3 on the Macintosh
- Level 1: General information
Think of a function for a moment as an "input-output" machine, with numbers "going in" and new values "coming out."
The input numbers make up the DOMAIN, the output is the RANGE for the function.
- * Level 2: Specific information
In the case of a COMPOSITE function, the "input-output" process occurs more than once. For sin(tan(x)), values transformed by theTangent function are then passed through the Sine function to produce the values for the range.
- * Level 3: Suggest action
View the graph of the function Sin(tan(x)) and use your graph plotter to "zoom in" for a closer view of the "fuzzy bits."
- * Level 4: Suggest sequence
This is a case where the graph gives quite limited information. The table of values, however, may be used to examine points of interest in detail.
- * Level 5: Demonstrate
Take a value, such as 1.57, and study the table of values and graph.
Try to find a point where the output shifts from large negative values to large positive values (or vice versa).
So are the "fuzzy bits" a product of the function or of the computer?
This is an example of a mathematical situation ideally suited to the use of computer technology, and yet the very problem itself may be an artefact of the technology, rather than the mathematics itself. The nature of the computer as a "rational number machine" means that it plots graphs by joining many points - but these are always rational points! If something unusual happens at an irrational point (as is the case here) how can we be sure that the computer is not simply "stepping over" the point and missing the "real picture"?
Sample Problem 1. Sample Problem 3.
Last updated: 1st May, 1996
Stephen Arnold
crsma@cc.newcastle.edu.au
© 1996 The University of Newcastle
Challenge and Support Index
Courses | Software | Readings | Links | Comments? © 1996: The University of Newcastle: Faculty of Education