This Is the Calculus They Won't Teach You

“Infinity is mind numbingly weird. How is it even legal to use it in calculus?”

“After sitting through two years of AP Calculus, I still feel like I don’t know anything about it.”

“What do you mean that historically, integrals came before derivatives? Seriously, what’s the deal with calculus?”

There is an inspiring story hidden behind the formulas and word problems of single-variable calculus–a story that should be told in every introductory calculus course, but usually isn’t. Regardless of if you’ve taken a few calculus courses before or are just beginning your study of calculus, this video aims to give a taste of the insights that can be gained from learning about the history and philosophy behind the key ideas of this field of math… brought to you by a dog and some rough animations!

This video was part of #SoME2, a math video making contest created by @3blue1brown in the summer of 2022.


0:00 Chapter 1: Infinity
1:49 Chapter 2: The history of calculus (is actually really interesting I promise)
2:19 Chapter 2.1: Ancient Greek philosophers hated infinity but still did integration
7:43 Chapter 2.2: Algebra was actually kind of revolutionary
11:14 Chapter 2.3: I now pronounce you derivative and integral. You may kiss the bride!
19:53 Chapter 2.4: Yeah that’s cool and all but isn’t infinity like, evil or something
24:14 Chapter 3: Reflections: What if they teach calculus like this?



Thank you very much to Dr. David Bressoud for letting me interview him!

If you have any questions, thoughts, recommendations or you notice any mistakes, please do share!


CORRECTIONS LIST (periodically being updated):

3 : 47 I start a description of Archimedes’ circle proof here, but I want to note that the one I present is a modified version of it; it does contain a few “hand-wavy” mistakes near the end and may still be difficult to follow; that’s on me. My goal with showing this proof wasn’t to give a proper explanation of it, but to give a taste of what I thought is the spirit of rigorous reasoning. If you would like to see something that is closer to Archimedes’ argument, there’s a link in the further reading you can check out–or you could also just google “Archimedes circle proof”.

8 : 45 I made a typo… Viete lived from 1540-1603, not 1540-1693

14 : 35 This is embarrassing. Kepler’s 2nd law is true because angular momentum is constant, not because velocity is constant. Whoops!

A lot of comments have brought up Indian contributions to calculus that I left out, particularly the Kerala School and Madhava of Sangamagrama, where the first instances of derivation came about. A few reasons for why I missed this in my research could be because attributing the beginnings of calculus with Indian mathematicians seems like a relatively new historical narrative, and many sources in English likely have a bit of Eurocentric bias. Regardless, this goes to show how the narrative I shared does have its flaws, and similarly to the misleading impression of calculus I criticized modern introductory courses of spreading, this narrative is not the whole picture. If you have any sources you’d like me to add to the further reading, please let me know!

I may have suggested that Euler discovered the number e, but he did not–some attribute it to Napier, and many others before Euler used it. On that note, I probably misleadingly attributed a lot of other ideas I brought up to one or a few people, when they are much more nuanced; so take the briefly brought up associations in this video with a grain of salt. This video is intended to share a broad narrative instead of establish definitive history of calculus. Hopefully this can just act as an intro to the subject!


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