3blue1brownThe better way to understand Taylor's polynomial expansion
Taylor's polynomial expansion is a core part of high-school level calculus. However, I was never satisfied with the way it was taught to me, as the motivation for it seemed to come out of nowhere. In this video, I show how Taylor's polynomial, an explicit formula for the error of the polynomial approximation, and a generalized version of Taylor's polynomial with multiple centres, are all the result of just applying the fundamental theorem of calculus over and over again.
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Comments
2.5
1. Lack of motivating examples
2. Overuse of manim
3. Excessive formulas
4. Well-structured
5.2
I have the feeling that this would have been better as a written paper and not as a video. Long passages of whiten text do not do well in the video format. And even the few charts and graphs you have could have just as well been diagrams and not animations.
however, the idea is unique and does show new and interesting ways of approaching math.
9
I had never seen this derivation before, it is just beautiful and so intuitive. Thanks for making this, fantastic entry!
8.2
Well-paced, voice & video timing match very well.
8.1
Good novel derivation of Taylor expansion.
However, I don't think the discussion of the error is too relevant, and you can just cut it out, because people who will watch the video already knows Taylor expansion (and the error stuff). In other words, those parts are not as novel, and cutting them off would yield a much better viewing experience.
5.4
I suggest that more emphasis / time should be taken for the last part, with the different cis. Indeed the remark is good and not frequently presented in math courses, whereas the fist part of the video is somewhat less original.
8.1
Excellent video. In the section about the radius of convergence, it would be nice to mention the Cauchy-Hadamard theorem by name after deriving it. It would also be nice to mention the term “analytic function”, and the fact that even if a Taylor series converges on some interval, it still might not equal the original function there (ie. non-analytic smooth functions exist).
6.9
I genuinely learned a lot here, extremely good explanation and clear pacing. You linked so much together so easily.
7.1
I like this video a lot. It is a very intuitive way of thinking about Taylor series approximations. The animation and narration are very good and clear.
4.5
Between the 30-second mark and the 9:30 mark, there is only formula after formula. If there is a way to provide some visualization to break that up, that would help. With integrals, there should be some example curve you can show to ground the reader.
6.8
Provided a much nicer intuition