Suppose that we perform an experiment that involves randomly throwing darts at a dartboard. We then record where they land. We would like to determine the probability of a dart landing in a given region of the dartboard.
Fibonacci, Lucas, and the ubiquitous Catalan numbers are delightful sources for experimentation, exploration, and conjecture (Askey 2005; Koshy 2001, 2006, 2008). Pell numbers are another similarly rich source. In this article, I will demonstrate some patterns associated with Pell numbers and then will show how they can be extracted from Pascal and Pascal–like triangles. I will provide a geometric interpretation of Pell numbers and conclude by citing a few opportunities for further exploration.
Whether teaching calculus or simply living life, we often find ourselves confronted with functions that we would like to optimize. For example, suppose that we have a function that tells us the amount in millions of dollars that we will earn as profit by producing x units of widgets. Suppose, for example, that our profit function is given by f (x) = -9x3 + 71x2 - 91x
Students who have grown up with computers and calculators may take these tools' capabilities for granted, but I find something magical about entering arbitrary values and computing transcendental functions such as the sine and cosine with the press of a button. Although the calculator operates mysteriously, students generally trust technology implicitly. However, beginning trigonometry students can compute the sine and cosine of any angle to any desired degree of precision using only simple geometry and a calculator with a square root key.
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