(A new question of the week) It’s surprising how many questions we get that end up being about problems that are poorly worded or simply wrong. But these can be as illuminating as good problems, by showing ways to catch the error. This is one simple in itself, but will lead us into a common …
Last time we looked at various ways to find tangent lines to a parabola without using calculus. Another standard calculus task is to find the maximum or minimum of a function; this is commonly done in the case of a parabola (quadratic function) using algebra, but can it be done with a cubic function? Yes, …
I always like solving advanced problems with basic methods. For example, many problems that we usually think of as “algebra problems” can be solved by creative thinking without algebra; and some “calculus problems” can be solved using only algebra or geometry. Using simple tools for a big job requires more thought than using “the right …
(A new question of the week) Sometimes a problem that looks complicated turns out to have a simple answer. And sometimes that simple answer turns out to be too simple for its own good. Today’s problem is an example of this.
(An archive question of the week) The indeterminate nature of 0/0, which we looked at last time, is an essential part of the derivative (in calculus): every derivative that exists is a limit of that form! So it is a good idea to think about how these ideas relate.
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.
(A new question of the week) I want to look at a question that came in recently that is, in one sense, very simple, but at the same time is quite challenging. It was given to a 12-year-old whose father asked us about it, and requires some skill in thinking about non-routine problems.
(An archive question of the week) While gathering combinatorics questions, there were several that stood out. This one will serve well to summarize the topic, showing multiple methods for counting, and contrasting other kinds of problems.
We have been looking at ways to count possibilities (combinatorics), including a couple ways to model a problem using blanks to fill in. Today, we’ll consider a special model called Stars and Bars, which can be particularly useful in certain problems, and yields a couple useful formulas. (I only remember the method, not the formulas.)
(A new question of the week) We have been looking at some combinatorics questions, both easy and challenging. Some questions have come to us in recent weeks that can illustrate how to think your way through relatively difficult problems, including catching errors and interpreting a textbook’s solutions. We’ll see yet again that there are usually …
Permutations and Combinations: Undercounts and Overcounts Read More »