3blue1brownWhy Calculus Only Works In Radians
Why Calculus Only Works In Radians
In this video, we discuss why Calculus has a bias towards radians. You see, the derivative of sin to be cos only works when theta is in radians!
Corrections
0:44 This is in fact a valid identity! Thank you to @billwindsor4224 who pointed this out to me. My point in showing this graphic was that e^ipi = -1 where pi is in degrees is false, so the thumbnail is still correct. But when we have the degree symbol in the argument this identity becomes true because "180°=π" (https://math.stackexchange.com/questions/1368049/eulers-identity-in-degrees)
3:52 The right hand side should be limx_a f(x) + limx_a g(x)
4:36 The limx_0 is the first step in the proof
4:37 The limit for sinx/x is 1, and the order of multiplication should be switched
For the Geometry Diagram, the whole circular sector involving theta is B
Attributions:
Calculus Image - https://wordsmithofbengal.wordpress.com/2021/08/02/dr-philos-the-creative-fantasy-of-differential-and-integral-calculus/
@3blue1brown Video - https://youtu.be/3d6DsjIBzJ4?si=EPTsw8CJDSsbkBYR (also inspiration)
Some Latex From - http://www.deepnlp.org/blog/series-formulas-latex
LaTeX Generated By - https://latexeditor.lagrida.com/
Animations are done with CapCut
Diagrams and some animations were made with Desmos
Chapters:
0:00 - Intro
1:00 - Proof of Euler's Identity
2:08 - Analysis of McLaurin Series
2:54 - The Derivative of sinx
3:10 - The Definition of the Derivative
3:33 - Back to sinx
4:40 - sinx/x
5:25 - The Radian and Geometry
6:08 - Conclusion
6:46 - Or is It?
Analytics
Comments
5.2
I was confused at the beginning (first minute), but I liked the second part. Some calculations are a bit too fast near the end, but it was clear overall.
6.4
Hey, you are a natural.
The topic wasn't that interesting, yet your explanation skills are quite good, keep going
3.1
The video is nicely put together but it really doesn't prove what the title says. It should be renamed as "Why calculus is simpler when you use radians instead of degrees"
7
I do appreciate it when a math video does attempt to keep the viewers attention.
3.9
Good motivation at the start. (Although I personally have never really thought of the theta on the LHS of the identity as an angle; to me we're just exponentiating a number.)
As for the rest of the content: although it's perfectly good mathematics to dig all the way down to the geometric proof that sin(x)/x -> 1 as x -> 0, I feel that it's pedagogically overkill. I think it would be just as satisfying, and probably more memorable, to just draw graphs of sin(x rad) and sin(x deg) and look at the slopes of their tangent lines at x=0. Of course you can justify the picture with the chain rule if you like.
More superficial criticisms: the calculations from 3:40 onwards are hard to follow because of somewhat choppy animation. (I certainly couldn't do better myself--but somehow the Manim experts always seem to make these things flow really well.) I also found it hard to follow in real time what was going on with the picture at 4:50--partly because a lot of information was being thrown at me at once, and partly because the B in the picture is written so small. And finally, although I was okay with most of your notational shorthand (like "Calculus <=> rad"), writing "θ ∈ °" at 5:45 was really a bridge too far for me. The symbol ∈ should really only be used for elements of a set.
5.7
Well edited, motivated, scripted, and narrated. Marks off for novelty but overall great.