# Category Archives: Mathematics

## How does the value of an inverse cosine function change when the unit of its argument changes?

How does the value of differ from ?   This is an interesting question, especially because the question of units in trigonometric and inverse trigonometric functions is rarely discussed in mathematics textbooks.   If you’re interested, think this problem through … Continue reading

## A Neat Trick For Determining The Integrals Of exp(x) cos x And exp(x) sin x

The standard method (typically found in first-year calculus textbooks) for determining the integrals and is to integrate by parts twice. If you haven’t seen the standard method, I’ll show you how to do the first one; the second one is … Continue reading

Posted in Calculus, Mathematics | | 11 Comments

## A Number Riddle, Updated With Solution, And Some Comments On Iterative Playgrounds

Two weeks ago I posted a puzzle sent to me by my nephew Matthew: 1 is 3, 3 is 5, 5 is 4, and 4 is cosmic. Why is 4 cosmic? What happens to the other numbers? As I mentioned … Continue reading

## How Much Mathematics Should A Student Memorize? Part 6, Derivatives Of Exponential And Logarithmic Functions

In teaching calculus many, many times over the years, I strove to present to my students my approach to learning and mastering the subject. Part of this approach can be summarized by the slogan memorize the minimum As a teacher, … Continue reading

## How Much Mathematics Should A Student Memorize? Part 5, The Multiplication Table

There has been a war in the mathematics education world for the past few decades about whether students should master basic skills, or whether they should use calculators or software for basic skills to save time and energy for higher-level … Continue reading

## Reviews of “Proofiness,” by Charles Seife

I recently finished reading Proofiness, written by Charles Seife, science writer and journalism professor at the Arthur L. Carter Journalism Institute at New York University, and it’s excellent. There are a number of glowing reviews out there (see Stephen Strogatz … Continue reading

## A Question From The USSR Olympiad Problem Book; Updated With Solution

A few days ago I posted on a problem (Problem 4(a) on Page 7) from the very nice The USSR Olympiad Problem Book, by D.O. Shklarsky, N.N. Chentzov, and I.M. Yaglom: We are given 80 coins of the same denomination; … Continue reading