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

Linear Algebra is Scary, Even for Future Mathematicians

The Calculus of Friendship: What a Teacher and a Student Learned about Live While Corresponding about Math (2009) is a delightful book by Steven Strogatz. Strogatz is Jacob Gould Schurman Professor of Applied Mathematics at Cornell University, and he writes beautifully. The book is about his decades-long correspondence with one of his high-school mathematics teachers, … Read more

What is Calculus?

What is calculus and what is it used for? Calculus includes an enormous number of ideas, methods, and applications, and this post is an attempt to provide an overview. Most of the interesting phenomena that are analyzed scientifically involve change. The flow of wind and water, the orbits of the planets, the path of a … 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 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

Tossing a Ball

Once upon a time I was playing catch with a small child. The child was just learning to catch a ball, and so I drew closer to her because she dropped the last one I threw. However,  the dear child was habitually teased by an older sibling for being a “baby” and she was therefore … Read more