Ghosts of Departed Quantities

Calculus was developed by many workers, and their incremental progress was independently systematized by Newton and Leibnitz in the late 1600s. At that time the concept of limit had not been devised yet, and even the concept of a function was still in development, and there was not yet a precise definition of a function. … Read more

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 $\int e^x \cos x \, {\rm d}x$ and $\int e^x \sin x \, {\rm d}x$ 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 similar. … Read more

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, I took it as part of my responsibility to help students identify the essential core … Read more

How Much Mathematics Should a Student Memorize? Part 2, Integral Calculus

My basic attitude towards memorization in mathematics education is to memorize the absolute minimum, but memorize that minimum perfectly. Part of a mathematics teacher’s job, in my view, is to guide students to understand what this “minimum” is, and then encourage them to memorize it, helping them to find effective means for memorization. Effective means, … Read more

On the fundamental theorem of calculus

One day a graduate student submitted some writing to me, in which she was explaining rates of change at the high school level. She made an interesting statement: The slope of a secant line joining two points $(a, f(a))$ and $(b, f(b))$ on the graph of a differentiable function $f$ is the average of the … Read more