Hyperbolic functions are functions that form a sort of parallel universe to the trigonometric functions. They are typically introduced as an aside while teaching calculus – because they happen to be useful there, and also because they can’t really be understood without calculus. As a result, we can easily misunderstand them, forget about them, and …
The start of a new year seems like a good time to take a look at the big picture. A question from September raises a big topic: What is mathematics, in general? What does all math – arithmetic, algebra, geometry, trigonometry, calculus, and beyond – have in common?
A recent question led me to find a series of old questions, each answering it at a different level. All of them are worth presenting.
A couple recent questions asked what constitutes “standard form” for a quadratic equation; that will lead us to some older questions about “standard form” for a linear equation. We’ll see that “standard” isn’t quite as standard as you might think.
Having looked at improper integrals last time, let’s look at some questions we’ve had involving integrals that either look improper but aren’t, or are improper but were missed, or that have other issues with their interval of integration.
We have a question about an improper integral, where one is strongly tempted to take a shortcut that makes it convergent, though the proper definition does not. Why can’t we do this? We’ll see something of the freedom mathematicians have in the matter of definitions, as well as why the standard definition has to be …
Two-sided Improper Integrals: Can I Take Both Limits at Once? Read More »
A recent question reminded me I hadn’t yet written about the complexity surrounding the definition of ratio (and related terms, like rate and fraction). Here are four questions about the words.
A recent question asked about an interesting locus, which led me to realize we haven’t talked about that topic in general. Here we’ll look at what a locus is, using three simple examples, and then dig into a question about the wording.
We’ve looked at the basics of logs and how they work; now we have some questions testing the limits of the definition. We’ll focus on the inverse idea of exponential functions with a negative base, looking at this from several perspectives.
A recent question about two interpretations of the range of a data set in statistics leads us into some older questions and some mysteries. Is “range” defined as the interval containing the data, or the difference between largest and smallest values, or 1 more than that? Yes! All three are used, and are useful.