Sign Issues in Integration
Several recent questions involve things that go wrong with signs in integrating, and reveal some subtleties that are easily overlooked. We’ll also see some creative thinking!
Calculus
When Is an Improper Integral Not an Improper Integral?
Having looked at improper integrals last time, let’s look at some questions we’ve had involving integrals that either look improper but aren’t, or are improper but were missed, or that have other issues with their interval of integration.
Two-sided Improper Integrals: Can I Take Both Limits at Once?
We have a question about an improper integral, where one is strongly tempted to take a shortcut that makes it convergent, though the proper definition does not. Why can’t we do this? We’ll see something of the freedom mathematicians have in the matter of definitions, as well as why the standard definition has to be …
Two-sided Improper Integrals: Can I Take Both Limits at Once? Read More »
How to Question a Proof, and Find Answers
Last time, we looked a little more deeply than usual at an epsilon-delta proof of a simple function of one variable. Here, we have two questions about such proofs for a function of two variables, to illustrate how you can question aspects of such a proof and satisfy yourself that it is valid – or …
Epsilon and Delta Revisited
Some time ago we looked at the meaning of the definition of limits, and I included several links to additional discussions on the subject. Now I want to took at three of those, which fit together rather nicely. We’ll look deeply into the proof that the limit of \(x^2\), as x approaches 2, is 4, …
When Newton’s Method Fails – and Why It’s So Good When It Doesn’t!
Last time, we saw how Newton’s method works. Here, we’ll look at a question about why it might not work, which will lead to a deep examination of how iterative methods work in general, from which we will discover why Newton’s method is as good as it is. I have to say, as I read …
When Newton’s Method Fails – and Why It’s So Good When It Doesn’t! Read More »
Solving Equations with Newton’s Method
Last time we solved some of the equations connected with a segment of a circle using Newton’s Method. Let’s take a closer look at the method – how it works, why it works, and a few caveats.
Turning a Maximization Problem Inside-Out
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.
Pitfalls of Inverse Trig Functions
A couple recent questions involved errors made both by students and by the authors of their textbooks, involving trigonometric or inverse trigonometric functions. These offer some good lessons in pitfalls to be aware of.
Integration: Sometimes It Just Can’t Be Done!
Having looked at what it takes to work out an indefinite integral, using all our tools, we need to face something that isn’t explained often enough: Some integrals aren’t just difficult; they’re impossible! We’ll look at what we’ve said in several cases where this issue arose.