(A new question of the week) It is not uncommon for students to ask about why they get different answers using different methods. Usually the answer is that the answers are really equivalent. This time, the answers really are different! This was partly the result of being taught an incomplete technique, omitting important cautions. And …
(A new question of the week) Sometimes we have lots of quick questions and a number of long discussions, neither of which seems suitable for a post. This time I’ve chosen to combine two distantly related questions, one recent and one from several months ago, both involving tangent lines to functions.
(A new question of the week) It is not unusual for mathematicians to define a concept in multiple ways, which can be proved to be equivalent. One definition may lead to a theorem, which another presentation uses as the definition, from which the original definition can be proved as a theorem. Here, in yet another …
(A new question of the week) We received a couple different questions recently about solving differential equations by separation of variables, and why the method is valid. We’ll start with a direct question about it, and then look at an attempt at an alternate perspective using differentials.
Two recent questions (that came to us within two hours) dealt with apparent contradictions in integration. The first seems to give a result of zero that is clearly wrong; the second seems to give two different results for the same integral.
(A new question of the week) A good way to develop a sense of what limits are and how they work comes from working with visual representations of them, in the form of graphs. In particular when the functions are defined by graphs rather than by equations, we have a lot more flexibility in creating …
(A new question of the week) Average rate of change is a topic taught in pre-calculus and calculus courses, primarily as preparation for the derivative, though it has more immediate applications. A recent question asked about when the concept is valid, which I found interesting.
In response to a recent request for information about implicit differentiation (hi, Brian!), let’s take a look at that topic. It happens to be distantly related to Friday’s topic, which was about implicitly defined curves. We’ll start with a thorough explanation, and then look at several specific examples, capping it off with a weird one.
(A new question of the week) An interesting question we received in mid-January concerned two implicit derivative problems with an unusual feature: the derivative we are seeking disappears! How do you track down such elusive quarry? Each case is a little different.
(A new question of the week) Limits can be challenging. They can be even more challenging when they require L’Hôpital’s rule or more advanced methods (Maclaurin series), and then are turned inside-out by asking not for the limit itself, but for parameters that will result in a specified limit, or what values of the limit …