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75.41 A “flat” function with some interesting properties and an application

The purpose of this note is to highlight some important properties of a function which plays a key role in the study of mathematical physics as well as in some aerodynamic applications. Although such material is normally beyond the scope of an "A" level student, the particular function in question has some interesting properties which are accessible to sixth formers, and gives practice in certain techniques of the calculus, including limits. Furthermore, the function occurs in the study of reaction rates in physical chemistry, also studied at "A" level. Since it is important, if possible, to discuss topics in pure mathematics, e.g. calculus, that are motivated by, or connected with an application, we provide a few brief details on this topic at the end. For the purposes of a classroom discussion, the ideas as presented here can be taken in reverse, with the application providing the motivation. Consider the function