A new question of the week A simple question about the Basic Proportionality Theorem (which goes by several other names, including Thales’ Theorem) and its relationship to similar triangles leads to some helpful ideas about how to choose a suitable manipulation at each step in a proof, a skill central to good problem solving. In …
Last week we looked at some recent questions about limits, where we focused first on what limits are, in terms of graphs or tables, and then on finding them by algebraic simplification. This week, we’ll look at two old questions about a trigonometric limit that can’t be determined that way: sin(x)/x, as x approaches zero.
(A new question of the week) I am looking back at recent questions I’ve skipped because, though having useful content, the discussions were cut short. In the two cases we’ll see here, the student who asked a question never read the final answer, perhaps because it went to their spam folder, so the discussion was …
Limit Basics: Tables, Graphs, and Simplification Read More »
(A new question of the week) Counting ways to select teams can be simple, or quite complex. Here we’ll look at a few tricky examples.
A recent question (whose asker refused to cooperate by showing work, so that we were unable to help) reminded me that we haven’t yet shown a similar kind of problem that can be quite interesting: problems where we are given the value of one or several expressions in several variables, and are asked for the …
(An archive question of the week) In preparing the last couple posts, on recurrence relations, I ran across an answer to a much harder question, that illustrates what it can take to solve one that doesn’t fit the convenient forms. It’s linear, but the coefficients are not constant as they have been in all our …
A Challenging Homogeneous Second-Order Recurrence Read More »
Last week we looked at Ask Dr. Math questions about homogeneous linear recurrences; this time we’ll see some on simple (first-order) non-homogeneous recurrences, which will bring us back to the topic two weeks ago, when we looked at the examples of this type that a student had the most trouble with. This will be an …
Last week we looked at a recent question about recurrence relations, and I realized it needs a companion article to introduce these ideas. So here we will look at some answers from Ask Dr. Math about the simpler case, including general methods, why they work, and applications.
(A new question of the week) A recent question asked us to find errors in solving recurrence relations by the method of undetermined coefficients. We’ll see several things that can go wrong, and correct some misunderstandings.
Here is a short discussion of a common type of problem in trigonometry classes: finding a trig function of the sum or difference of two angles, given minimal information about them.