Review of Calculus Better Explained: A Guide to Lasting Intuition

Before I get into the magic that is this book, I need to give the author a quick introduction and let you know about his website. Kalid Azad runs the and it is one of my favorite on the internet. The knowledge on this website should be implemented in every math program across the globe.

Azad’s gift is taking the complicated concepts in mathematics and explaining them using simple analogy and real world examples. I’m an intelligent guy, above average in mathematics, and a physics major, but it wasn’t until I found Azad’s site that I truly understood the numerous concepts I learned by rote memorization and repeated use.

For example, I understand what imaginary numbers represent in the real world, the number “e” (and why it’s its own derivative and integral), my trigonometric knowledge is intuitive and I actually know what the fourier transform is accomplishing. The explanations on Azad’s website are why I bought his book Calculus Better Explained: A Guide to Lasting Intuition.

I thought I knew calculus. I took levels 1-3 and earned A’s in each class. After reading the first few pages of the book, I realized that, while I knew how to do calculus, I didn’t know what it was or why it was so important. Calling myself a physicist without understanding how calculus works is like calling yourself a mechanic because you drive a car and can change your brakes and tires.

“The goal is to get Calculus in hours—not months—and have those insights stay with you.” This is the promise Azad makes on the very first page and he lives up to it. The way most of us are taught calculus is “like learning to build a car before driving it.” Put another way, we learn in the most difficult and unintuitive way possible. It’s so difficult and obscure that we don’t even both getting to the good part of driving. Even if we do get to the driving part, we spent all that time learning to build the car instead of driving it that we suck at driving!

The main processes in calculus are derivation and integration. We learn the rules for this process, but many students do not understand the derivate beyond the power and chain rule or integration beyond “it’s the area underneath the curve.” These explanations are correct, but they do nothing for developing mathematical intuition and knowledge. Azad instead explains these processes as “X-Ray Vision” and “Time Lapse Vision.” This sets the stage for how the book builds your working intuition of calculus.

Azad builds on these concepts by going over the entire function of calculus in each chapter, but increasing the level of detail each time. He explains these concepts first via an intuitive approach, then he adds a natural description, and finally he adds the symbols that we are familiar with from class. By the time you see the symbol for taking the derivative or integral, you have a completely revised and intuitive understanding of what it’s asking and what it’s looking for.

This is not the type of book that will convert you into a math lover. However, if you feel even the smallest fire in your belly for advanced mathematics this book will morph it into an inferno. Many would-be STEM students give up not because of the science, but the math. People like Azad, people who write books like this, teach you the subject the way it’s meant to be learned. He does it without complicated explanations and uses analogies involving pizza and trees. I strongly recommend this book to anyone taking calculus or planning to take calculus. Pick it up, read it. Your inner mathematician will thank you.

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