Last week we examined three probability problems that had problems. Looking further back, I find that Jonathan, who asked the first of those questions, asked a group of questions about rolling multiple dice in 2022. They provide some additional lessons about easy mistakes to make.
It’s been a while since we’ve looked at probability. Here, we’ll look at three questions that we received last year. In each case, we have to detect an error! They’re good examples of what can go wrong, and what to do when your answer appears to be wrong.
A recent question dealt with how to write the general solution to a trigonometric equation. I want to combine that with an older question that will set the stage for the issue. This topic was touched on in Trigonometric Equations: An Overview.
Here is an interesting question we got recently, that turns a common maximization problem (the open-top box) inside-out. What do you do when you’re given the answer and have to find the problem? We’ll hit a couple snags along the way that provide useful lessons in problem-solving.
A couple recent questions offered tricks for integrating rational functions, opportunistically modifying or working around the usual method of partial fractions. We have previously discussed this method in Partial Fractions: How and Why, and in Integration: Partial Fractions and Substitution, where we looked at other variations.
Some time ago we looked into the probability that a random set of sides (from, say, a broken stick) form a triangle. A recent question asked about the probability that a random triangle is acute (all angles acute) or obtuse (at least one angle obtuse), which led to more discussion of what it means for …
Last time, we considered the Chain Rule for derivatives. This time, we’ll look at the product and quotient rules, focusing on how to keep the formulas straight, and make them easier to apply. We’ll look primarily at the quotient rule to start with, and then examine the product rule at the end.
Having recently helped some students (in person) with the rules of differentiation, I’m reminded to do so here, starting with the chain rule. It is easy to make this topic look harder than it really is; the two main ways to state the rule are often confusing, and different approaches fit different problems. We’ll try …
I am always interested in problems that can be solved in different ways, particularly because this can give a student a chance to be creative, as well as learning from experience that you don’t have to do it “the teacher’s way”. Here we’ll use trigonometry, and two different ways to add lines to a figure …
Last week’s discussion reminded me of another question, from July, about a probability problem that was solved in a hard (but educational) way and an easy way. This instance is more extreme, and, due to its length, requires extreme editing in order to fit here.