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.
Methods
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.
How to Evaluate Trig Functions (By Hand?)
In discussing the value of radians, we introduced the idea that trig functions are easier to evaluate that way. That raises the question, how do you find the value of a trigonometric function without a calculator, and how do calculators themselves do it? Let’s look into that.
Casting Out Nines: What and How
This old technique for checking arithmetic is both easy and hard to describe: easy to explain in advanced terms, but hard to explain in elementary terms. We’ll try to do it all here, but a fuller explanation of the “why” will come next week.
Monotonic Functions, Inequalities, and Optimization
Looking for a cluster of questions on similar topics, I found several from this year in which monotonic functions (functions that either always increase, or always decrease) provide shortcuts for various types of problems (optimization with or without calculus, and also algebraic inequalities). We’ll look at a few of these.
Prime Factorization of a Number (Advanced)
Last time we looked at basic methods for finding the prime factorization of a number. Here we will look at some special techniques for large numbers, demonstrating them for not-too-large numbers. This takes us a step beyond previous tests that told us whether a number was composite, without actually factoring them.
Prime Factorization of a Number (Basics)
I’ll close this series on prime numbers by looking at how to find the prime factorization of a number, starting with the most basic ideas applicable to relatively small numbers, and then (next week) looking at some advanced methods for larger numbers.
Prime Numbers: Checking for a Prime (Part 2)
Last time we saw how to test small or medium sized numbers to see if they are prime, including details on the elementary Trial Division method, and introduced the most popular test for larger numbers, the Fermat test. Here we’ll review Fermat, and then go beyond. This is not for the faint-hearted! (I myself am …
Prime Numbers: Checking for a Prime (Part 1)
Last time we looked at how to efficiently make a list of prime numbers. But if you want to check a single large number to see if it is a prime, you don’t want to have to make a list of all primes up to that number. That’s today’s subject, where we’ll start with Trial …
Prime Numbers: Making a List
We’ve looked at what prime numbers are, and how the concept extends (or doesn’t) to 0, 1, and negative integers. The next question many students have is, how can I make a list of prime numbers (or write a computer program to do so)? We’ll learn about the Sieve of Eratosthenes, and list all the …