Last time we looked at some false proofs, which are often used to help students understand what does and does not constitute a valid proof, and in particular, to remind them to be careful in algebraic proofs, looking for issues like division by zero and taking square roots. This time, we’ll look at two similar …
One of the ways to show the need for careful proof, which I discussed last time, is the existence of apparent proofs of false “facts”. These are also useful for training us to think carefully. In this and the next couple posts, I will look at a few categories of false proofs, focusing on what …
(A new question of the week) A recent question from a student working beyond what he has learned led to an interesting discussion of alternative methods for solving a minimization problem, both with and without calculus.
We have looked at how to calculate, apply, and undo a percent increase or decrease. Here we will look at some special terms used in business for percent increases, which have been a source of many questions over the years.
Having discussed how to calculate the percent change between two numbers, and how to apply such a change to one number to get a new number, we need to look at what may be one of the most common types of questions we get: reversing a percent change (increase or decrease) to find the original …
A question we got in March asked about working with percent increases. As I replied to this rather common topic, referring to an archived answer, I was reminded that this would is a very common type of question, and would be a good topic to cover in the blog. So that will be the topic …
(An archive question of the week) In collecting questions and answers about 0.999… for the last post, there were two that were too long to include, but that dig more deeply into issues that some of the standard answers tend to gloss over. So here, I want to look at those two answers, both of …
Having looked at two common questions in probability that are often challenged, let’s turn to the realm of numbers. Non-terminating decimals are inherently problematic, and one particular example causes difficulty for many, even after they fully accept the mathematics of it. Our FAQ page on this topic, at 0.9999… = 1, is very brief, and …
I’ll close out our look at transformations of functions with some trigonometric graphs. These are the best example of combined transformations, and involve some special tricks as well. We’ll start with an early question that gives an overview of the process, then focus in on important details.
Having looked at all the usual transformations of a function and its graph, there are two more situations I want to look at. The first of these are the horizontal and vertical transformations introduced by absolute values (with a bonus thrown in: square roots); then next time we’ll conclude this series with a look at …