Essence of calculus, chapter 2. "The paradox of the derivative"
Added 2017-01-13 21:39:59 +0000 UTCHey folks!
Chapter 2 is ready for your viewing and reviewing. As always, I welcome any feedback you have, even though I might wait a month or two to go through it all and incorporate improvements. I'm especially curious to hear from the calculus teachers in the audience; the more anecdotes about a specific student's learning experience, the better.
There are some aspects of the sound that are slightly wonky, e.g. it seems like I hit the microphone once or twice while recording, but I figure I'll wait to do a re-recording until I've compiled feedback.
It might also be worth sharing a few scattered philosophies I have about this series:
A few people commented on Chapter 1 that it might be best to start with limits. I feel pretty strongly that it's better to precede any discussion of limits by showing why we care about the idea of "approaching" in the first place, and why it's worth formalizing. And we might as well motivate them with half of the reasons we're talking about them in the first place: Derivatives. I see no issue with describing the derivative using the word "approach" in an unformalized way.
That's not to say I couldn't be convinced with the right counterargument, but here's where I'm coming from: With a private student, I found that when I tried to start a series of calculus lessons with talk about limits, it came off as just one more algebra thing to know. And I found myself unable to answer the question "why should I care" satisfactorily without resorting to saying "just wait". When I switched gears to derivatives, things went much more smoothly, My current plan is cover the formalization of limits using epsilons in chapter 5, after the essentials of derivatives have been addressed.
Another thing I'm doing a bit differently is treating "dt", "dx", etc. somewhat literally, which is to say I often treat them rather *finitely*. I don't draw a distinction between "delta t" meaning a finite change and "dt" being reserved for something more infinitesimal. The thought here is that it's much better to have a concrete thought to hold when reading an expression involving differentials than it is to cast it off as mere notation, and to set the precedent from the get-go that it helps to first read a differential as an actual "tiny nudge".
Thanks, as always, for your support
-Grant
Comments
Just fixed, sorry about that!
3blue1brown
2019-07-22 19:01:06 +0000 UTCThe video is not available
Ahmed
2019-07-22 16:19:22 +0000 UTCLit'rally, you just made the Heisenberg uncertainty principle make so much sense. Thanks for unplugging my erroneous belief in instantaneous velocity.
Kathryn Schmiedicke
2017-03-15 06:28:19 +0000 UTCGreat video. I'm out of college and revisiting calculus on my own recently too. Loving this series. I'm not sure if this would be important to you, but I noticed that in the first video in this series, you downplayed the value of tangent lines as a visual representation of the derivative, yet in this video, you use this visualization in a way that makes it sound like you're happy with them. I understand that the example in this video is different, and I would agree that the tangent line's "meaning" is more intuitive here, but to some, this might appear as disingenuous? Just a thought. I can't wait to see the rest of this series!!! Thanks for being you!
ManMachine
2017-01-30 05:29:34 +0000 UTCHi Grant! Chapter 2 is looking beautiful. One minor suggestion - at exactly 4 minutes in, something along the lines of '==> 20 meters/second" would feel about as nice as a closed set of parantheses.)
Dillon Strichman
2017-01-28 05:16:38 +0000 UTCI agree with you that it is appropriate to wait to introduce the formal notion of a limit until a few examples of why we want to "approach" certain values is made clear
David Wych
2017-01-27 20:35:57 +0000 UTCGreat point! Thanks for the feedback. My only reason for those comments was in case any viewer was more familiar with how they'd seen things in a classroom and had a (potentially subconscious) dissonant feeling with my particular presentation choices.
3blue1brown
2017-01-22 01:27:25 +0000 UTCHere's the quote from which I derived my first criticism: Don't use adjectives which merely tell us how you want us to feel about the thing you are describing. I mean, instead of telling us a thing was "terrible," describe it so that we'll be terrified. Don't say it was "delightful"; make us say "delightful" when we've read the description. You see, all those words (horrifying, wonderful, hideous, exquisite) are only like saying to your readers, "Please will you do my job for me."
Redstone Jazz
2017-01-21 02:14:01 +0000 UTCLet me add, your work is indeed some of the best I've ever seen for explaining Math well. There was more good than bad in this last calculus video. I suppose only the bad seemed worth pointing out :)
Redstone Jazz
2017-01-21 02:12:36 +0000 UTCIt seems your enthusiasm for teaching Math has been expressing itself in a couple ways you may not be aware of. I may be jumping the gun on pointing these out. Maybe these are intentional. Firstly, you tell the viewers how they ought to feel about the Material you're presenting them with. It was done again 13:08 of this video. It seems the desired presentation would naturally yield an intuition for the Beauty of Math. Secondly, you seem to make right-teaching of Math part of the video Material. Often instead of merely showing the viewer the Material you've prepared, you also tell them why it's presented in the particular way you've chosen. (For some examples of these meta-comments: 0:28, 4:42, 9:37) It's not bad to teach about Teaching. But - if I may cast my vote - I enjoyed your older videos where my only role was the Student. Adding your thoughts on presentation choice distracts me from the Math. I'm happy to support your videos regardless, but as for suggestions; why not make videos where you teach about Teaching? Your thoughts already seem to display themselves when you teach about Math. Why not give them room to breathe?
Redstone Jazz
2017-01-21 02:02:15 +0000 UTCI love the fading out of the insignificant part of the equation at 12:30ish. It's a really good visual way of getting the point across.
Scott Ramsay
2017-01-20 13:15:17 +0000 UTCChris
2017-01-19 20:38:31 +0000 UTCM V
2017-01-16 19:52:13 +0000 UTCCallum
2017-01-16 19:41:42 +0000 UTCNoam Ta Shma
2017-01-16 16:35:39 +0000 UTCMarko Nikic
2017-01-16 14:39:35 +0000 UTCDavid Rysdam
2017-01-15 14:01:50 +0000 UTC3blue1brown
2017-01-15 05:41:10 +0000 UTCMyles Buckley
2017-01-15 05:26:21 +0000 UTCGuido Gambardella
2017-01-15 00:31:51 +0000 UTCGuido Gambardella
2017-01-15 00:06:09 +0000 UTC3blue1brown
2017-01-14 23:15:26 +0000 UTC3blue1brown
2017-01-14 23:13:57 +0000 UTC3blue1brown
2017-01-14 23:10:00 +0000 UTC3blue1brown
2017-01-14 23:08:29 +0000 UTCTodd Nagel
2017-01-14 22:21:25 +0000 UTC3blue1brown
2017-01-14 18:36:17 +0000 UTCJavier Almeida
2017-01-14 17:26:13 +0000 UTCBenjamin Grossmann
2017-01-14 14:21:09 +0000 UTCJavier Almeida
2017-01-14 13:53:37 +0000 UTCKerigorrical
2017-01-14 07:30:46 +0000 UTCGuido Gambardella
2017-01-14 06:37:12 +0000 UTCJosh B.
2017-01-14 06:36:59 +0000 UTCJosh B.
2017-01-14 06:33:38 +0000 UTCGuido Gambardella
2017-01-14 06:16:21 +0000 UTCLouis Ng
2017-01-14 05:47:11 +0000 UTC
2017-01-14 03:46:26 +0000 UTC
3blue1brown
2017-01-14 01:54:00 +0000 UTC3blue1brown
2017-01-14 01:50:02 +0000 UTCJake Palmer
2017-01-14 01:44:07 +0000 UTCBenjamin Grossmann
2017-01-14 01:31:01 +0000 UTCBenjamin Grossmann
2017-01-14 01:15:52 +0000 UTCBenjamin Grossmann
2017-01-14 01:12:09 +0000 UTCBenjamin Grossmann
2017-01-14 01:07:32 +0000 UTCjason black
2017-01-14 00:28:24 +0000 UTCDebbie Newhouse
2017-01-13 23:41:36 +0000 UTCLuop90
2017-01-13 23:23:06 +0000 UTCJake Palmer
2017-01-13 23:18:57 +0000 UTCMajed Samad
2017-01-13 22:45:43 +0000 UTCJacob Mirra
2017-01-13 22:27:46 +0000 UTCJacob Mirra
2017-01-13 22:22:39 +0000 UTCJacob Mirra
2017-01-13 22:20:39 +0000 UTCLouis Ng
2017-01-13 22:15:18 +0000 UTC
