(A new question of the week) Average rate of change is a topic taught in pre-calculus and calculus courses, primarily as preparation for the derivative, though it has more immediate applications. A recent question asked about when the concept is valid, which I found interesting.
Last week, we looked at exactly what the mean is, referring specifically to the arithmetic mean, the one we first learn as the “average”. But just as we previously saw that there are several things called “average” (mean, median, mode), there are in fact several different kinds of “mean”. We’ll look here at the arithmetic, …
There are three different statistics that are commonly taught as “averages”, or “measures of central tendency”, of a set of numbers: mean, median, and mode. (There are others as well, which we will get to later.) What are they? How do they differ? How do you use them? We’ll look into questions like these as …
To close out this series on the definition of the derivative, I want to look at a few questions about alternative versions of the definition, primarily the “symmetric difference quotient”. We’ll see that this leads to a slightly different result, not always equivalent to the original, and we’ll observe some associated ways that calculators can …
To start a series of posts on differentiation (one of the basic concepts studied in calculus), I’d like to look at a number of answers we’ve given to the basic question, What is a derivative? This includes questions about the meaning of the concept and its definition, as well as examples. Something all these questions …
(An archive question of the week) We’ve looked at the scalar (dot) product and the vector (cross) product; but there is one answer in the Ask Dr. Math archives that was too long to fit in either post. Here we’ll see again where the two familiar products come from, while looking deeper into the math …
Last time, we looked at the scalar, or dot, product of vectors, focusing on proving the equivalence of two ways to define it. This time, we’ll look at the vector, or cross, product in the same way. The distinction between dot and cross product reflects the symbol used, u · v vs. u × v, …
Having covered the basics of defining and adding vectors, multiplying by scalars and finding unit vectors, it’s time to look at multiplying vectors together. What makes this entirely unlike working with numbers is that there are two ways (in fact, more than two!) to multiply two vectors. We’ll look at one of those today, the …
(A new question of the week) Since we looked at a question about economics last week, let’s examine another, which is very different, relating the supply and demand curves to the concept of variation or proportion. We are not economists, so we can’t go deeply into that subject, but it makes us think about some …
(A new question of the week) As we approach a new year, I want to start cleaning up a backlog of recent questions, and start posting more typical interactions, rather than waiting for the most momentous. Many of these will therefore be relatively short! This one goes back to last July, but it connects to …