Monotonic Functions, Inequalities, and Optimization

Looking for a cluster of questions on similar topics, I found several from this year in which monotonic functions (functions that either always increase, or always decrease) provide shortcuts for various types of problems (optimization with or without calculus, and also algebraic inequalities). We’ll look at a few of these.

Implied Multiplication 2: Is There a Standard?

This is part 2 of a series of extracts from discussions we have had on whether multiplication implied by juxtaposition is to be done before division (which I call IMF, for Implied Multiplication First). Some people write to us claiming that there is one official correct answer. Are they right?

Anything to the Zero Power: Why 1?

We’ve been looking at oddities of zero. Because “nothing” behaves differently than “something”, operations with it can be surprising. Although students learn that \(x^0=1\) for any non-zero number x, they often wonder, why?? I’ve selected a few out of at least a dozen such questions in our archive.

Comparing Logarithms With Different Bases

Logarithms are not hard to work with when only one base is involved (as in most real-life problems); but they can be challenging when each log has a different base. Here, we’ll look at a few problems in which we have to compare logarithms with different bases, showing various strategies.

Is Zero Really a Number?

A recent comment on the site raised questions about zero, beyond what we have discussed in the past about division by zero. Here we’ll look at basic questions about whether zero is actually a number at all, and then about multiplication by zero, which confuses a lot of people.

Exponential Growth: Surprisingly Flexible

Two recent questions from the same student involve exponential functions: We can express different kinds of growth all using one base, called e; or we can use different bases (and ignore horizontal scaling transformations). And we can use different transformation to obtain the same graph. This relates to some important properties of exponential functions.