Some time ago I looked at questions about trisecting an angle by compass and straightedge, which entailed discussing the rules for such constructions. We left open another common question: Why are such constructions important, and why do we use those particular tools? This probably isn’t explained as often as it should be. Why does it …
The next few posts will look at a powerful technique for finding limits in calculus, called L’Hôpital’s Rule. Here, we’ll introduce what it is, and why it works. In the next post we’ll examine some harder cases.
We’ve looked at the basic concept of significant digits, then at how they interact with operations, which is one reason for defining them. This time I want to look a little closer at why they are defined as they are, which will involve considering some special cases.
Last time, we introduced what we mean by significant digits (or figures), and touched on why they are defined as they are. Here we will look at how significant digits and decimal places differ, and how they are affected by operations (primarily addition and multiplication). This is another aspect of why they are defined at …
Last time we looked at questions about how to shift, stretch, or flip a graph by changing the equation of a function. All our examples involved only a single transformation. Now we can look at cases where two or more transformations are combined. As we do this, we will develop a deeper understanding of how …
Combining Function Transformations: Order Matters Read More »
A common topic in algebra courses is how to transform functions and their graphs. In the series starting today, we’ll start with the basics of how and why a graph is moved or stretched, then combine transformations and look at various special cases and other transformations, ending up with graphing trigonometric functions.
Back in January, I discussed the issue of division by zero. There is a special case of that that causes even more trouble, in every field from arithmetic to calculus: zero divided by zero. I’ll look at several typical questions that we answered at different levels.
(An archive question of the week) Many calculus courses start out with a chapter on limits; or they may be introduced in a “precalculus” course. But too often the concept is not sufficiently motivated. What good are limits? Why did they have to be invented? Are they as simple as they seem? Why is an epsilon-delta …
(An archive question of the week) Last time, as part of our series on estimation, we looked at some numerical methods for solving equations approximately. I mentioned the Method of False Position, but when I looked for more detailed expositions in our archive, I realized that in a sense it is really two different things, …
(An archive question of the week) Last time I looked at reasons for learning to estimate. In searching for answers on that topic, I ran across a question that touches not just on reasons for estimation, but on other ways to check an answer, and on some of the specific ideas we will be looking …