[Book Notes] Infinitesimals, Derivatives, and Beer – Full Frontal Calculus (Ch. 1)
"Where there is life, there is change; where there is change, there is calculus." — Seth Braver
When I decided to return to math and rebuild my foundations for AI/ML, I didn’t expect to enjoy a calculus book this much. Shocking, I know. But that’s exactly what happened with Full Frontal Calculus.
Can calculus feel intuitive? Even fun? From the first few pages? Okay, from page 8 to be exact. For me, the answer is yes.
Why This Book Clicked for Me
As part of my self-study, I’m posting short, progressive reviews of the books I’m working through. On today’s menu: Chapter 1 of Full Frontal Calculus by Seth Braver.
Before I stumbled on Full Frontal Calculus, I tried a few limit-based calculus books and textbooks, but none of them spoke to me. Luckily, there’s no shortage of calculus material these days, so it’s easy to shop around and try different sources.
Braver’s book grabbed me right away. The early focus on infinitesimals, the tight writing, and the emphasis on intuition won me over. I even caught myself smiling more than once. Rare for a math book.
Chapter 1 Highlights
Chapter 1 starts with infinitesimals: “an infinitely small number, smaller than any positive real number, yet greater than zero.” One early example shows how a circle, imagined as a polygon with infinitely many infinitesimal sides, leads to the familiar area formula πr². If your geometry or trig is rusty, don’t worry - it still makes sense. Braver then uses the same idea to show how curves appear straight on a small enough (infinitesimal) scale, which is the heart of differential calculus.
Things really clicked for me in the section titled A Gift From Leibniz: d-Notation. Braver’s explanation of dy/dx shows how it captures infinitesimal change in a way that just makes sense. It helped me understand why derivatives represent slopes and rates in a way I could explain to a 10-year-old. Working through the derivative of x² from first principles was also deeply satisfying.
Practically speaking, Chapter 1 covers:
what infinitesimals are
how they help us define rates of change
the geometric meaning of derivatives
the elegant dy/dx notation from Leibniz
why we ignore higher-order infinitesimals like (dx)² or du * dv
and a first-principles derivation of the derivative of x²
The chapter ends with two powerful tools: the power rule and linearity properties. These let you compute derivatives of polynomials using just basic mental math.
The writing is sharp and often funny, in a math kind of way. There’s even a cameo by the Sumerian beer goddess Ninkasi, who helps explain rate of change and derivatives using a vat of beer. It sounds quirky, but it works.
The book’s style, clarity, and focus on intuition made me want to keep going. That’s not something I’ve felt with many math books.
Final Thoughts and Tips
If you’re following along or just curious about studying calculus again, I recommend giving Chapter 1 a shot. It’s not always light reading, and the exercises are essential, but it might click for you like it did for me. Chapter 1 is available for free on the author’s site, so you can explore it before deciding whether to dive in.
If you do decide to dive into the book, here are a few tips to get the most out of it:
If you’re rusty on pre-calculus (I was), make sure you’ve got slope, rate of change, the point-slope formula, and the slope-intercept form down cold before the Rates of Change section on page 10. For that, Seth Braver’s other book Precalculus Made Difficult has excellent material on those topics. You can probably get through it in a day.
Read slowly, with a pen or pencil in hand. Write in the margins (get a paperback copy). It might feel painfully slow at times (pun intended), but it’s a recipe for deeper understanding.
The book includes answers to many exercises and is great for self-study. But the solutions are compact, so I recommend using Grok or ChatGPT to expand on them and deepen your understanding.
Once you’ve finished the chapter and exercises, check out the author’s YouTube videos that go along with the book. They’re criminally underrated and oddly hard to find. You might enjoy them as much as I do.
For topics that are hard to retain, try spaced repetition with active recall. Anki works great for that, or use whatever tool you prefer.
Chapter 1 sealed the deal. This is the calculus book I’m sticking with. Looking forward to seeing how Braver develops the ideas from here.
Dig deeper. More to come soon!
— Ruslan
P.S. I’m not affiliated with the author. I just really enjoy the book and wanted to share it.