How I wish logistic growth was taught to me in Calc 2

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[@bloom945]

At this point you should just say whenever the video is NOT sponsored by Brilliant 😂

[@jimpim6454]

I honestly can't believe people still use brilliant the thing fell apart after they 'sunsetted' the community feature where people talked about daily problems. It leads me to the conclusion that nobody punting this website has ever actually used it. Tell me, does anyone actually use it anymore?

[@woody442]

That would be very cheap advertising for brilliant. Brilliant!

[@zachstar]

😂 fair

[@luisdmarinborgos9497]

Might fk around and get an annual suscriptions

[@tomkerruish2982]

​@zachstar Wasn't there a video you had to repost because the original sponsor didn't like it, and you had to find another?

[@supernovaxs9480]

It’s so weird seeing Zach go back into serious STEM mode from his second channel

[@mgm6723]

Zach is still able and willing to teach use, thanks bro

[@fredthechamp3475]

When you are getting paid to do it you get the motivation, I guess.

[@julianw1010]

able and willing I see what you did there

[@ChalkyWhiteChalkyWhite]

mate just make your own series' of videos courses your teaching is quality.

[@DMitsukirules]

I was waiting for the punchline and then remembered I originally subbed to you because you are an engineer who made math videos 😂

[@no-bk4zx]

The timing is impeccable. I was just taught population modelling this semester but never really understood exactly what the differential equation meant. I knew how to solve it and how to model the curves but not what it meant. Thanks a lot this was genuinely insightful.

[@anindyaguria6615]

I'm able AND willing to solve this differential eqn.

[@dank.]

Out of curiosity, is the phrasing of this comment derived from primis Richtoffen from cod zombies? (If you don't know what I mean I'm referring to, kzbin.info/www/bejne/b4evmmqKe8abl9Usi=xemX0mzff4HMqSOK at 10:30)

[@anindyaguria6615]

@@dank. No it's from Zach's second channel. Check out his airplane emergency exit skit, lol!

[@ekandrot]

I remember a few years ago on the news, everyone was explaining virus spread using exponential growth rather than logistic growth. At that time and still, I believe logistic is what should have been used and your video is a great explanation as to why and what the correct curves should have been.

[@christeanaz]

Looking back, the SEIR (Susceptible-Exposed-Infectious-Recovered) model would have been the best model. The logistical model would have oversimplifes the disease's progression and spread. The logistics model doesn't account for the incubation period (Exposed stage) or differentiate stages of infectiousness and recovery.

[@ekandrot]

@@christeanaz I agree that there were much better models over all, ones that account for feedback loops, etc. Exponential was the simplest model, but a simple swap to logistic would have been the same level of complexity for News audiences, and would have shown that there is a cap and a slowing down period - key features missing from the news (and likely what scared most people silly).

[@ersetzbar.]

Accuracy does not sell as good as drama

[@user-tt9uy5gg9o]

Actually, exponential growth is a very good approximation to logistic growth when you are less than 20% through the curve, and much easier to understand the short term consequences.

[@whannabi]

​@@user-tt9uy5gg9obut if you say that this is an exponential curve, someone's gonna think in the long term that this will keep behaving this way.

[@_mishi]

Gotta mention existence and uniqueness theorem there. For this exact differential equation its fine, but generally this rule can be used only if the existence and uniqueness is satisfied, otherwise further analysis is required

[@stokedfool]

Logistics growth explained with visuals. Great example involving the effects of hunting on deer populations.

[@northernlight1000]

Great video Zach! and yes often times in math, the higher level you learn certain fundamental topics a lot of the time people really stop looking for actual explicit solutions to things for the most part and focus much more on trying to find all the possible qualitative information they can about that certain problem

[@jamesjohn2537]

Thanks, for clarity. Visual interpretation of big idea logistics ODE telling us! Cheers and happy Easter

[@futuresmkt]

Great description!!!!

[@iteerrex8166]

Love differential equations. They are so powerful and it applies to so much. 👍😁

[@mattias2576]

One of the most fun courses i had in University was non linear dynamics which was essentially doing analysis of differential equations which are non linear (hard to solve usually) in a qualitative way. There is so much information you can infer about solutions without ever solving anything This video reminded me a lot about how we approached things there

[@_mark_3814]

linear has a precise definition. Some non-linear difeqs are easy for instance separable ones.

[@maalikserebryakov]

Linear and non Linear are very artificial terms. The number of categories within non linear is massive. It is like categorising objects in the world as Chocolate and not chocolate. Childish

[@mattias2576]

@@maalikserebryakov well of course it is massive, and most definitions are artificial. But it still was usefull, the course was not so much about the individual dif eqs as it was about how to think in general when met with something you cannot solve analytically, and building knowledge about such systems in a way very similar to the video. In this case it was usefull to seperate into linear and non-linear since non-linear equations often are unsolvable analytically whilst linear equations are in general solvable The course name might have been childish (i dont think so), but the content was very good which is what is important

[@Impatient_Ape]

Very nice! Somewhere between taking the course decades ago and now, "DiffEQ" courses started to use visualization tools from non-linear dynamics analysis (like the flow fields you show here), which makes it WAYYYY easier to understand. They're even useful for *partial* differential equations. Without the visualization, you're just memorizing a bunch of procedures for each of dozens of cases, which doesn't motivate deeper understanding.

[@lethargogpeterson4083]

Love the shirt.

[@natanprzybylko7227]

Saw this video a couple weeks ago before I learned about logistic growth, now I'm coming back while I'm learning it lol

[@tobybartels8426]

This kind of thing is often covered in an introductory Differential Equations course, and you might even get the higher-order version of this (where a non-linear autonomous equation or system of equations usually can't be solved exactly, but you can still analyse its qualitative features). But we could certainly put this in Calculus 2, with maybe one extra day on the subject.

[@Dr.Rajkumz]

7:30 wow I was mind blown. Never thought of it like that.

[@surrealistidealist]

My cat jumped onto my lap and fell asleep, so I had to stop working through my Calc book. Good thing I had this video to watch! Btw, any favorite books on DiffEQ?

[@kennethhicks2113]

Your FUNNIEST vidy yet!

[@RFmath_]

Great video! Could you do one on hyperbolastic growth?

[@dcterr1]

Very good, educational video! It seems I've always placed too much importance on solving differential equations - perhaps because they're like puzzles in a way and fun to solve. But now I see that it's not inherently necessary to do so. Instead, what's needed is a good intuitive grasp of what the solutions look like. In any case, if you do want precise solutions, all you need to do is to run a computer simulation using numerical methods, lile Runge-Kutta.

[@happyeevee4955]

u should explain the lotka-volterra equations next

[@sonnyward9857]

Learned Logistic Growth today in pre-calc. After I took the test, and got a 98%, I wanted to relax and watch some KZbin. The first video I see is this. What are the odds?

[@void2258]

I never knew you did videos like this. I had only seen the comedy stuff.

[@JxH]

BEWARE !! In the real world, it should (sometimes) be possible to cross the equilibrium lines. For example, if the logistics equation includes an equilibrium line at 4.5, but (for example) human babies arrive in integer units. So the curve might presently be at 4, and then this example could 'tunnel' straight over to 5, leaping over the equilibrium line at 4.5. Imagine if the 'litter' was a larger number, such as six puppies or piglets. It could overshoot the line. What happens then depends on the rest of it. ...Sorry, I always see the exceptions. It's in my nature.

[@alejandroduque772]

In that case a new term would appear in the form of a time-dependent source term added to the logistic equation.

[@JxH]

@@alejandroduque772 Nope. You missed the point entirely. My original post above was quite clear, so I'll not try to restate it again.

[@aniruddhvasishta8334]

This is true; differential equations are just abstractions of discrete systems into continuous ones so this will happen sometimes. However, the model is only meant to be an approximation to the real world so it won't be exactly how it behaves. For example, the solution to the differential equation will take on noninteger values, but obviously a real population will not. However, the model will be a good approximation (say, if you round the decimals to whole numbers). This isn't a flaw per se, but it's still a good point to bring up.

[@alejandroduque772]

@@JxH I get what you mean and you are right. What I am trying to say if that in the scenario you are talking about, you would be working with a different equation than the one presented in this video.

[@sonnyward9857]

Kind of want to see what 4.5 babies would look like.

[@Stacee-jx1yz]

1) Calculus Foundations: Contradictory: Newtonian Fluxional Calculus dx/dt = lim(Δx/Δt) as Δt->0 This expresses the derivative using the limiting ratio of finite differences Δx/Δt as Δt shrinks towards 0. However, the limit concept contains logical contradictions when extended to the infinitesimal scale. Non-Contradictory: Leibnizian Infinitesimal Calculus dx = ɛ, where ɛ is an infinitesimal dx/dt = ɛ/dt Leibniz treated the differentials dx, dt as infinite "inassignable" infinitesimal increments ɛ, rather than limits of finite ratios - thus avoiding the paradoxes of vanishing quantities. 2) Foundations of Mathematics Contradictory Paradoxes: - Russell's Paradox, Burali-Forti Paradox - Banach-Tarski "Pea Paradox" - Other Set-Theoretic Pathologies Non-Contradictory Possibilities: Algebraic Homotopy ∞-Toposes a ≃ b ⇐⇒ ∃n, Path[a,b] in ∞Grpd(n) U: ∞Töpoi → ∞Grpds (univalent universes) Reconceiving mathematical foundations as homotopy toposes structured by identifications in ∞-groupoids could resolve contradictions in an intrinsically coherent theory of "motive-like" objects/relations. 3) The Unification of Physics Contradictory Barriers: - Clash between quantum/relativistic geometric premises - Infinities and non-renormalizability issues - Lack of quantum theory of gravity and spacetime microphysics Non-Contradictory Possibilities: Algebraic Quantum Gravity Rμν = k [ Tμν - (1/2)gμνT ] (monadic-valued sources) Tμν = Σab Γab,μν (relational algebras) Γab,μν = f(ma, ra, qa, ...) (catalytic charged mnds) Treating gravity/spacetime as collective phenomena emerging from catalytic combinatorial charge relation algebras Γab,μν between pluralistic relativistic monadic elements could unite QM/QFT/GR description. 4) Formal Limitations and Undecidability Contradictory Results: - Halting Problem for Turing Machines - Gödel's Incompleteness Theorems - Chaitin's Computational Irreducibility Non-Contradictory Possibilities: Infinitary Realizability Logics |A> = Pi0 |ti> (truth of A by realizability over infinitesimal paths) ∀A, |A>∨|¬A> ∈ Lölc (constructively locally omniscient completeness) Representing computability/provability over infinitary realizability monads rather than recursive arithmetic metatheories could circumvent diagonalization paradoxes. 5) Computational Complexity Contradictory: Halting Problem for Turing Machines There is no general algorithm to decide if an arbitrary program will halt or run forever on a given input. This leads to the unsolvable Turing degree at the heart of computational complexity theory. Non-Contradictory: Infinitary Lambda Calculus λx.t ≝ {x→a | a ∈ monadic realizability domain of t} Representing computations via the interaction of infinitesimal monads and non-standard realizers allows non-Church/Turing computational models avoiding the halting problem paradox.

[@km4168]

It would have been nice if you had spent another minute or two at 7:30 to expand this to Lyapunov stability. This intuition for a stable equilibrium is exactly the second condition for a lyapunov function. Nonlinear control (stability) relies on transforming (finding a lyapunov) your system (differential equation) into such a form that you can conclude stabillity similarly to what you show in your video.

[@micayahritchie7158]

Wait you still make math videos

[@attila0323]

A reminder that this is the same guy who makes video series about Washington and Lincoln time traveling, and other funny sketches.

[@ScrotN]

I just had this exact question a while ago. If an organism live in certain conditions without evolving to fit it's surrounding. How fast will the species die out? I didn't know differential equation, so I just used f(x)= 1/(x+a) +b; a,b in R; a [not]= -x to tinker around. But yeah I should've known this equation

[@xdzzz0]

6:35 - * Bill Gates drooling at the mouth *

[@vigyanumtube9154]

I just realized that he makes sketchs too

[@lazartomic5800]

I was worried that Zach only filmed comedy.

[@GarthWhoever]

Your brain is really awesome btw

[@citizencj3389]

Taylor series is the best tool learned in calculus 2 for engineering. It isnt even funny how often i use taylor series.

[@pendragon7600]

I mean you do learn this stuff in any intro ODEs course

[@YashTiwari-13]

Alright less goo!

[@eguineldo]

I understand why they might not talk about logistic growth in calc 2 in depth as there's a couple other important topics that are better suited for an ODE course.

[@_abdul]

I love to see people saying "Oh...he's the sketch guy" on math videos like this, There used to be comments like "Oh... he's that Math guy" when he started the himself channel. We've come a long way.

[@Meatloaf_TV]

I have heard calc 2 is pretty hard and im taking it ts summer for 6 weeks im starting to get nervous

[@trademarked2476]

It’s not bad, if you have a good understanding of the idea of a limit and the derivative, the integration is not too bad at all. For summation I would recommend just watching some numberphile videos on Xeno’s paradox or some infinite sums to see some super basic properties before you learn in class. For differential equations, I would recommend just a lot of studying, the process of solving them is quite easy to memorize but the actual understanding it quite hard. But very much easy to get a high grade on with practice. Also get super familiar with all the tests, they really aren’t hard individually it’s just the sheer amount of them.

[@Meatloaf_TV]

@@trademarked2476 thx

[@ersetzbar.]

Can you quantum tunnel the 0 line?

[@jormoria]

Lotka-Volterra?

[@o_s-24]

Calc TWO?? We only did improper integrals and power series in calc 2

[@trademarked2476]

You may have covered a different curriculum, my university has like 12 different calc 2 courses and they’re all drastically different but covering the same super basic common topics. Probably just has to do with your major.

[@ishupogs4432]

Bro how tf does George Washington know so much math

[@xxak47xx18]

This is the first video ive seen of Zach Star or Zach Star Himself where he is serious and the whole time ive been at edge waiting for him to turn something sarcastic and its just not happening fuck fuck fuck this must be worse than gooning

[@hammerbg5816]

i cant stop thinking god is going to call him any second now.

[@devilsadvocate3364]

Honestly, if you could add in humorous examples, i feel like that would be both easy to engage with, as well as help with my small-brain understand a bit better 😅

[@billylee5624]

Those hunters man, hm...when will they ever learn. Had they taken and stayed at least for the intro of Differential equations on population growth and predator prey models, they would've learnt a thing or too.

[@fegolem]

the sky is pickles

[@Yotubez]

third

[@drmsanford]

You’re just a quick step from phase lines and bifurcation diagrams. I’d throw in that the solutions don’t cross because of the theorem of existence and uniqueness.

[@passengerplanetearth]

Nearly tricked me into learning something, nice try dude!

[@JYF921]

wait, this is NOT a comedy channel? XD

[@wrathofainz]

6:35 Israel _loves_ this equation.

[@maalikserebryakov]

Hamas would love that equation too, but too bad they’re too dumb to understand it. 🫡.

[@owdeezstrauz]

Your sense of humor has drifted far from my ability to understand. If this is the new comedy I just won't understand what the world has become.

[@isojapo4219]

bald peanuts