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.
Calculus
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.
Integration: It Takes a Whole Toolbox
Individual techniques of integration, as discussed in the last two posts, don’t represent the reality of the process, any more than demonstrating how to use a hammer or a screwdriver shows how to do carpentry. Let’s look at two questions we’ve had about challenging integrals that require a combination of methods. We’ll be using substitution, …
Integration: Choosing a Substitution to Try
Having looked at two basic techniques of integration, let’s start putting things together. How do you approach an integral without knowing what method to use? We’ll focus on substitution here, which is also called “change of variables”.
Figuring Out Integration by Parts
Having looked at some issues in integration, let’s look at some old questions about integration by parts.
Integrating Rational Functions: Beyond Partial Fractions
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.
Using Symmetry to Find a Definite Integral
(A new question of the week) Definite integrals can sometimes be solved by finding an antiderivative; but when that is either difficult or impossible, there may be special tricks available. Here we’ll lead a student gradually to a solution using symmetry; and then we’ll look at an earlier problem that used essentially the same trick …
Where Do Logarithms Come From?
Having answered many questions recently about logarithms, I realized we haven’t yet covered the basics of that topic. Here we’ll introduce the concept by way of its history, and subsequently we’ll explore how they work.