By David Singmaster
ISBN-10: 0894900439
ISBN-13: 9780894900433
ISBN-10: 0894900579
ISBN-13: 9780894900570
Notes on Rubik's 'Magic dice'
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Additional resources for Notes on Rubik's Magic Cube
Example text
A good model simplifies reality enough to permit mathematical calculations but is accurate enough to provide valuable conclusions. It is important to realize the limitations of the model. In the end, Mother Nature has the final say. There are many different types of functions that can be used to model relationships observed in the real world. In what follows, we discuss the behavior and graphs Unless otherwise noted, all content on this page is © Cengage Learning. Copyright 2012 Cengage Learning.
These same basic functions are often used to model real-world phenomena, so we begin with a discussion of mathematical modeling. We also review briefly how to transform these functions by shifting, stretching, and reflecting their graphs as well as how to combine pairs of functions by the standard arithmetic operations and by composition. MATHEMATICAL MODELING A mathematical model is a mathematical description (often by means of a function or an equation) of a real-world phenomenon such as the size of a population, the demand for a product, the speed of a falling object, the concentration of a product in a chemical reaction, the life expectancy of a person at birth, or the cost of emission reductions.
H͑x͒ and h͑Ϫx͒ Ϫh͑x͒, we conclude that h is neither even nor ■ (c) The graphs of the functions in Example 7 are shown in Figure 16. Notice that the graph of h is symmetric neither about the y-axis nor about the origin. FIGURE 16 INCREASING AND DECREASING FUNCTIONS y B The graph shown in Figure 17 rises from A to B, falls from B to C, and rises again from C to D. The function f is said to be increasing on the interval ͓a, b͔, decreasing on ͓b, c͔, and increasing again on ͓c, d͔. Notice that if x 1 and x 2 are any two numbers between a and b with x 1 Ͻ x 2, then f ͑x 1 ͒ Ͻ f ͑x 2 ͒.
Notes on Rubik's Magic Cube by David Singmaster
by Thomas
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