Last time we looked at how to count the parts of a polyhedron, and a mention was made of Euler’s Formula (also called the Descartes-Euler Polyhedral Formula), which says that for any polyhedron, with V vertices, E edges, and F faces, V – E + F = 2. We should take a close look at that simple, yet amazing, …
More on Faces, Edges, and Vertices: The Euler Polyhedral Formula Read More »
Over the years, we have had many questions, often from young students, asking how to count the parts (faces, edges, vertices) of a polyhedron (cube, prism, pyramid, etc.). The task requires understanding of terms, visualization of three-dimensional objects, and organizing the parts for accurate counting — all important skills. How can we help with this?
(A new question of the week) A recent question asked about one of our explanations of the limit of x2 (which we have discussed at least five times). This led to a deeper examination of what was said; and as I have looked through this and other pages, I have realized that it would be …
(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 …
Using the epsilon-delta definition of a limit in calculus can be challenging. (That’s why, after using it for a few examples, we derive some easier techniques, and never use the definition directly unless we have to!) We’ll start with an overview of what the definition means, and then look at several examples of how it …
(A new question of the week) We have had a number of challenging questions about inequalities recently. I want to show one of those here, because it involved a useful discussion about how to prove them. A proof problem Here is the question: Q) Show that a^4 + b^4 >= (a^3)b + (b^3)a for all …
(An archive problem of the week) Having just discussed quartiles, I want to look at related issues concerning percentiles. There, I briefly mentioned different perspectives on the concept of quartile, and focused on differences in the details of the calculations; here I will focus mostly on the different perspectives, and then touch on variations in …
Some time ago I discussed various issues pertaining to the concept of median in statistics. The same issues, and more, affect the concept of quartile (the median being the second quartile), so much so that different statistical software packages produce many different answers for quartiles. I have seen this affect students, who are taught one …
(A new question of the week) A recent question about probability has ties to Venn diagrams, tables, and Bayes’ Theorem. Questions about answering multiple-choice questions are common; this one offers a twist that provided opportunity to discuss several important concepts. Here is the initial question, from August: On a multiple choice question, only one answer is …
(An archive question of the week) One of the discussions we looked at last time involved rolling three dice and getting at least one six. I didn’t go into detail on the calculation there; but I found another place where we discussed it at length. We’ll look at that here. A wrong way and a …