Teaching and Learning Theorems

Theorems are carefully worded statements about mathematical facts that have been proved to be true. Important (and some not so important) ideas in calculus and all of mathematics are summarized as theorems. When you come across a theorem you need to understand it; the author of your textbook would not have included it and the…

The Calculus of Inverses

Today we will consider computing the derivative of the inverse of a function. This is pretty standard and is in all the textbooks. The usual suspects are the inverse trigonometric functions. So let’s start with  and then rewrite this as . Differentiating this gives Since we would like this in terms of x we can…

The Range of the Inverse

The last two post discussed inverse functions and some concerns about them. We continue that today be considering that fact that sometimes the inverse of a function is not a function, and what can be done in that case. Since the square of both 3 and –3 is 9. Which number should you get when…

Writing Inverses

In my last post I identified two “problems” related to inverses. The first of these is that there may be no string of operations, no algebra or arithmetic, that tells us how to evaluate the inverse function. For simple functions you can find the inverse function by switching the x and the y and then…

Inverses

The next few posts will concern functions and their inverses. Today we will just get into the basics which, hopefully, students know from their work prior to calculus. Of course they will have forgotten some of this and claim they never learned it. (Which may be correct, but one hopes they were taught it.) This…