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Thank you so much!!! 😊

Hi @cyanidepress thank you so much for your comment and for sharing! 😊 I am so happy to read it and I am so happy that your kid is learning calculus too! 😊 I am excited to create more calculus content to help support! 🥳 I hope you and your family have an amazing day/evening/night! 😊

Hi @aravindhvijayanandan3010 thank you so much for your comments on the channel! I am so happy to read them and I hope you have an amazing day/evening/night! 😊

Hi @anaa.9108 thank you SO much for your amazing and supportive comments! 😊 I am so happy to hear that you loved the video!!! 🥳 Your support and feedback mean so much to me. 🎉 I've got math content at all levels on my channel. If you let me know what kind of math you are interested in, I am always happy to recommend more videos to you and also create more videos similar to your interests!!! 🎁 Thank you so much for watching and I hope you have an amazing day/evening/night! 😊

Hi @Vishakh_Patel thank you so much for your comment and for sharing your insights! 😊Yes, that's a cool observation and somehow we have to combine the property that such a function would be discontinuous at every irrational number and (somehow) continuous at every rational number, which seems (simultaneously) hard though there are strange functions out there! I'll keep the suspense on this for the sequel video, but thinking about properties the irrational numbers satisfy that rational numbers don't is definitely super clever and insightful. 🥳 Thanks for sharing and I hope you have an amazing day/evening/night! 😊 (Also, I love the picture of the dog! 😊)

Hi @deltalima6703 thank you so much for sharing! 😊 Yes, that's an example and important to highlight! 😊 The conclusion of the video is that every function equal to its own derivative is a constant multiple of e^x, which technically also includes the zero function, since 0 = 0*e^x, but I agree it is important to mention this too! Thanks so much and I hope you have an amazing day/evening/night! 😊

You are not wrong. When I saw the thumnail the zero function immediately jumped to mind along with a periodic function, like a trig function. I couldnt think of exactly how to formulate such a function. A trig function would be out of phase with its derivative, obviously, so its not trivial to formulate one.

@@deltalima6703 Hi! Thank you so much for sharing your thoughts! 😊 Yes, a trig function is a great idea, and in fact these are closely related to the exponential function (e.g., due to Euler's formula). Also, for y = sin x or y = cos x, they do satisfy y'' = -y and y'''' = y (i.e., they equal their fourth derivative). Thanks again for sharing and I hope you have an amazing day/evening/night! 😊

Hi @koenth2359 thanks so much for your comment! 😊I think you are misreading the board since it does say f(x + y) = f(x)f(y). I think you might be seeing the second f I have written down in f(x)f(y) (on the right side of the equality) as a + since I didn't curl it at the top more, but please have a second look at it, and you'll see it's the equation f(x + y) = f(x)f(y). (I do agree that if the equation were f(x + y) = f(x) + f(y), we would have another interesting problem and in that case f(2) = 2a.) I hope you have an amazing day/evening/night! 😊

​@@MathMasterywithAmiteshYes, sorry, I misread it, thanks for your nice answer!

Hi @seyupo thanks so much for your comment and the discussion! 😊 Yes, the differentiability (or nondifferentiability) of the popcorn function is a very interesting question. It turns out that it is nowhere differentiable and I might do a video on this too in the future! However, here's a cool fact that is a bit easier (and fun) to prove and you might enjoy thinking about: the popcorn function has a local max at every rational number! 🥳 I hope you have an amazing day/evening/night! 😊

Hi @nbooth thank you so much for your comment! 😊 Yes, it's true that we basically discover exponential functions through this "exponent law" f(x + y) = f(x)f(y). From this perspective, it's nice to know (and have a rigorous proof) that you can't discover any other weird function that is not simply f(x) = a^x for some constant a > 0! I hope you have an amazing day/evening/night! 😊

Hi @PrabhakarKrishnamurthyprof thank you so much for your kind and supportive comment! 😊 I really appreciate the feedback. I hope you have an amazing day/evening/night! 😊

@@MathMasterywithAmitesh I am a senior citizen and lost studying mathematics as you teach now. Now I have regained what I have lost after 60 years! It is refreshing.

@@MathMasterywithAmitesh Best wishes and blessings to you.

@@PrabhakarKrishnamurthyprof Hi! Thank you so much for the kind words and supportive comments! I am so happy to read them and they mean a lot to me. 😊 I also wish you the best and I am extremely happy to hear that you are enjoying studying math and regained what you lost after a long time, that is so amazing and really makes me very happy! I wish you the absolute best and I hope you have an amazing day/evening/night! 😊

Hi @МаксимАндреев-щ7б thank you so much for your comment! 😊 It looks really interesting but I didn't quite understand what you meant at the beginning by "we can extend this hexagon in 2/√3 times". If you get the chance, could you please clarify the exact construction? I'd love to know because your estimate looks cool! 😊 I hope you have an amazing day/evening/night! 😊

Hi @МаксимАндреев-щ7б thank you so much for your detailed comment and for sharing that summary of the math! 😊 I hope you have an amazing day/evening/night! 😊

Hi @MorgKev thank you so much for your comment! 😊 Yes, while doing the video I somehow forgot to make the -6 a 0 at that step. However, it's so nice you also spotted that and I really appreciate your comment! I hope you have an amazing day/evening/night! 😊

Hi @manuelgonzales2570 thank you so much for your kind and supportive comments! 😊 I really appreciate them and they mean so much to me! 😊 I hope you have an amazing day/evening/night! 😊

Hi @kennethkho7165 thank you so much for your kind and supportive comment! 😊 I really appreciate the positive feedback and it means a lot to me. I am looking forward to making more videos that you will enjoy watching! 🥳 I hope you have an amazing day/evening/night! 😊

Hi @hasanmertsoycan2736 thanks so much for sharing that ODE technique and for your kind and supportive comment! 😊I really appreciate it and it means a lot to me. 😊 I really like your method because it applies more widely to lots of ODE problems (the separation of variables technique) but I personally prefer the method in the video for this specific problem because there are two steps in the separation of variables technique that someone might ask for justification about which at least lengthens the explanation: (1) How do we know 1/y makes sense, i.e., that y is nowhere zero (a priori, just using y' = y)? Although it is possible to justify this, I think it takes some extra effort. (I guess that is one advantage of the method in the video for this problem, because when we take the ratio f(x)/e^x, we don't need to know f(x) ≠ 0 everywhere a priori or even address that point.) (2) I guess some beginning students might object to the separation of dx and dy and argue dy/dx is not really a fraction - again, this can be argued/fixed by writing it differently and saying y' = y => y'/y = 1 and then integrate both sides with respect to x and use u-substitution (with u = y). So it's just syntax about the way we write it and we aren't really separating dy and dx, but it still at least needs addressing. Anyway, that's not to say that I don't think your method is better because of being more broadly applicable, but rather to point out the advantages of the method in the video for this specific problem. I am glad you mentioned this technique so some other people may learn from it. Thank you so much for sharing and I hope you have an amazing day/evening/night! 😊

He literally just explained exactly this in the video...

​@@ginopagano7293yes, but I'm not from a math background and I had to search a little to how his statements imply this. I feel explicitly stating this is helpful to few people. Thanks for helping me clarify my comment with a justification.

Hi @MridulGupta94 thank you so much for your comment and for your kind words! 😊 I am always happy to discuss/clarify points in my video! Yes, that's the key intuition as you say. 😊 I thought I discussed this starting at 5:00 and ending with the title card at 5:34 stating this explicitly (before starting the proof), which may be what @ginopagano7293 referred to. (I also reiterated this point after the proof from 12:40 - 12:54.) However, if you felt this segment 5:00 - 5:36 (and 12:40 - 12:54) wasn't explicit, I'm happy to hear why so I can keep it in mind for future videos. 😊 (A minor point (just for anyone reading this in general) is that technically the denominators of rational numbers approaching an irrational number don't have to keep strictly increasing at every step but do have to go to ∞ (or - ∞, if negative). For example, if we had an irrational number x = 1/2 + π/10^8, we could start off with a sequence {1/5, 1/2, ...} that converges to x, and 1/2 is closer to x than 1/5 even though its denominator is smaller. In particular, the denominators could still jump around over time, but in the limit, the denominators still must -> ∞ which is the key point.) I'm so happy you watched the video and it's so cool that you are interested even though you're not from a math background! 😊 I also really appreciate you sharing your comment, which may help emphasize this point. I hope you have an amazing day/evening/night! 😊

@@MathMasterywithAmitesh Hi Amitesh, I was watching the video half asleep, out of interest (as I'm not from math background) so I missed it. Thanks for pointing out the parts I missed. I still think you could have concluded towards the end as to what it means to be extremely explicit. The following chain of connections towards the end of the video would have been nice: denominator of rational gets larger => q gets larger => 1/q get smaller => this sequence approaches 0 (value of f at irrationals) from the rationals. I feel this translates the math definition of continuity at irrational i written at the bottom right of the board @12:40 to English and nicely connects it to the value of f at rationals close to i. I only see that |1/q|<1/N. But I don't see lim f(p/q) = 0. :)

@@MridulGupta94 Hi! Thanks so much for the feedback! 😊 I understand what you are saying and I really appreciate that you were interested in this topic and video even though you aren't from a math background! The chain of implications you wrote is super clear and I agree that writing it down explicitly would have added to the video. But based on your comments and explanations, I actually think you understood the math and the video quite deeply beyond any extra explanations I could have added 😊 (although I get it might have taken some extra effort, if it came from your own thinking/effort it's probably more rewarding). Thank you so much for taking the time to share this feedback, it's really helpful for me to get your perspective - I appreciate it and I'll keep it in mind for future videos! I hope you have an amazing day/evening/night! 😊

Must be a good element, given how much charisma and intellect he is casually exuding. Your education on the other hand, seems to have failed you.

Hi @dominiquelarchey-wendling5829 thank you so much for sharing this amazing comment and detailed explanation! 🥳 I think that's a super cool way of looking at it. I really like your argument for why f(x) > 0 always (which extends the argument in the video where I prove f(x) ≥ 0 always in the Lemma), and then solving it with Cauchy's functional equation (which is another really nice uniqueness situation). Thank you so much for sharing this, I really enjoyed reading it, and I hope you have an amazing day/evening/night! 😊

@@MathMasterywithAmitesh Well basically what you do is to reproduce Cauchy's proof, up to an exponential. The proof I propose relies on Cauchy's proof. I did not describe it but it is a real classic to be found all over the place, eg th-cam.com/video/AkQYZubfUJE/w-d-xo.html Btw, you just need to assume that your f is continuous at 1.

@@dominiquelarchey-wendling5829 Hi! Thanks so much for your comment! 😊 (I think there may be a TH-cam bug because I wasn't notified of your reply, although I just saw it on my studio, and it also doesn't seem to be publicly visible on this page but anyway ...) Yes, I have seen Cauchy's proof before and I really like it, and as you say you can think of it as equivalent to the proof in this video. Thanks so much also for pointing out that only continuity at a single point is necessary due to the homogeneity of the equation - that is something I should have mentioned in this video, but I guess now people can read it in this thread too thanks to your comment! 😊 I hope you have an amazing day/evening/night! 😊

Hi @ethan_nas thank you so much for your very kind and supportive comment! 😊 I really appreciate it and it means so much to me! I definitely hope more people will see this video. 🥳 I hope you have an amazing day/evening/night! 😊

Hi @cyanidepress thank you so much for your comment and for sharing your approach! 😊 Yes, that's a cool way of looking at it! 🥳 As you say, it follows from that trig identity sin(x) = cos(π/2 - x). Indeed, if y = sin(x), then arcsin(y) = x (by definition of arcsin) and arccos(y) = π/2 - x (by the identity and the definition of arccos). In particular, arcsin(y) + arccos(y) = x + (π/2 - x) = π/2 for all y. An alternate approach is: since we know arcsin(x) + arccos(x) is a constant by differentiation, to figure out the constant we just need to plug in one value of x. We can (for example) plug in x = 0 and obtain arcsin(0) + arccos(0) = 0 + π/2 = π/2. (We could have also used x = 1/2, x = 1/√2, x = √3/2, x = 1, since we readily know the inputs that give these outputs for both sin and cos.) However, I think the identity sin(x) = cos(π/2 - x) that you mention is indeed the best way to solve this problem since it doesn't use differentiation and allows you to immediately why arcsin(x) + arccos(x) = π/2. 😊 Thanks for your support of the videos and I hope you have an amazing day/evening/night! 😊

Hi @RUDRARAKESHKUMARGOHIL thanks so much for your kind and supportive comment! 😊 I really appreciate it and it means a lot to me. The math you mentioned is super interesting and I would love to do some videos on it! The main challenge for me right now is balancing different audiences, levels and topics (ranging from intro math (beginner algebra), precalculus, calculus, to proof-based math and beyond) but over time I'd love to create such videos too! I've definitely got several exciting real analysis proof videos coming up and would love to get to topology soon too! 🥳 I got some ideas after reading your comment. I hope you have an amazing day/evening/night! 😊

Waiting / exciting for that videos...❤

Hi @chrysleague thank you so much for your kind and supportive comment! 😊 Yes, the point you raised is very interesting! I think there are many variations of Euclid's proof and some variations present it as "Let p_1, ..., p_k be the first k primes" which works with the same argument. (I think another popular version is a proof by contradiction: assume p_1, ..., p_k are all the primes, and then arrive at a contradiction with the same reasoning.) I think Euclid's original argument was just start with any list of primes and produce another one, and I agree it is cool that this slightly more general strategy also works! 😊 I hope you have an amazing day/evening/night! 😊

Hi @AndrewLappeman-w4d thank you so much for your comment! 😊 I think that's a great video idea and I would love to do a video on that. I've been trying to explain in some videos the thought process of proofs in addition to presenting the proof (I have another video proving √2 is irrational th-cam.com/video/y6zPPFWnEjA/w-d-xo.htmlsi=k5zaH0IiHpkFm_f6 where I explain the thought process more, give some tips, and walk through the steps). I'm also happy to hear if you want to share one or two specific proofs you are getting stuck on, and perhaps I can think about doing a related video as well. I hope you have an amazing day/evening/night! 🥳

Thank you so much!!! 😊

Hi @nidadursunoglu6663 thanks so much for your comment and sharing your thoughts! 😊 I completely agree, proofs themselves don't always lead to insight about why something is true. I often find that unless I think about something myself and try to prove it myself, I don't really feel like I fully understand the statement. I hope you have an amazing day/evening/night! 😊