What Are Derivatives and How Do They Work? Calculus in Context!

Подписаться 31 тыс.

Просмотров 11 тыс.

50% 1

Turns out there is not just one way to think about Derivatives but at least FIVE (for students in a first semester calculus course). And these different meanings each have different advantages in calculus. But one of them really takes the cake when it comes to real world problem solving with math. In this video we discuss these five conceptions, apply them to some pretty awesome real world contexts, and share a research study on how mathematicians, scientists, and engineers think about derivatives.
Link to Potato Gun Video: • My Potato Gun Teaches ...
Link to Article: journals.aps.org/prper/abstra...
It is important to note that the actual experiment is framed as a multivariable calculus problem, although it could be solved without multivariable calculus thinking or partial derivative thinking. We simplify the story to illustrate it's main point for our video here by just referring to the problem as derivatives.
Math The World is dedicated to bringing real world math problems into the classroom and answering the age old question “when will I ever use this?”
We use unique topics for algebra, trigonometry, calculus, and much more and go beyond context problems and use a technique called mathematical modeling to find solutions to real world questions and real world problems. These videos are great for students who plan to enter technical fields that require real world problem solving, and can be a great resource for teachers looking for ways to bring real world contexts into their classroom.
Instagram: / maththeworld
Twitter/X: / math_the_world
Facebook: / maththeworldproject
Email: MathTheWorld@byu.edu
Created by Doug Corey
Script: Doug Corey and Jennifer Canizales
Audio: Doug Corey
Animation: Jennifer Canizales
Music: Coma Media
© 2023 BYU

Опубликовано:

4 июн 2024

Ссылка:

Скачать:

Готовим ссылку...

Добавить в:

Мой плейлист

Посмотреть позже

Комментарии : 67

@susanafaciolince7755 23 дня назад

i don’t think the problem was that the mathematicians didn’t think of relating the derivative to the rations. it is more likely that they didn’t go to do the experiment because they tried to get the actual answer. maths is the only one of these fields where an approximate solution most of the time isn’t a real solution.

@MathTheWorld 23 дня назад

I actually think that it is more than that. Their attempt at creating a mathematical model would still just be an approximation (and they would of course know that). As George Box is credit with saying, "No model is correct, but some are useful" (or something close to that). So it isn't the issue of approximation. It might be an issue of trying to get a better approximation, but the mathematicians would know whatever numeric value they gave as an answer would still just be an approximation, and not be the "actual" answer.

@Magieschnuller 23 дня назад

@MathTheWorld I would assume it is about generalisation. Obviously a mathematical formula would still be "just a model", and won't give "true" solutions. But if you have a formula you can solve all sorts of similar arrangements of pulleys, masses, etc.. With the experimental approach you can't really do that, since you would need to gather the data again every time. In some cases however that might still be easier than coming up with a generalized formula. So when it comes to real world applications, one should definitely consider whether a generalized solution is worth the effort or not. And I would think that mathematicians tend to gravitate towards the generalized approach since that's kind of what mathematics is all about.

@BlueDog15391 23 дня назад

It seems like the problem is not with the mathematicians using the slope intuition, but with mathematicians trying to solve the puzzle presented to them instead of measuring some numbers and giving a numerical approximation as if it is an answer.

@pmnt_ 23 дня назад

to be honest, if the prompt is "find ∂x/∂Fₓ" and the answer they are looking for is just a single number, a difference quotient, I would be angry. It's like promting to "solve this Rubik's cube" and they measure the time until you start to peel the stickers off.

@Alfaomegabetagamma 23 дня назад

Exactly my thoughts!

@kgangadhar5389 22 дня назад

I thought the same. And I would still prefer the mathematicians' way. If their way is to generate a general solution, that will get the derivatives and help analyze many more things.

@MathTheWorld 22 дня назад

I was using the video to teach an idea in calculus, as well as showing how I was applying it, so I spent more time than what I actually did in practice. In practice I didn't even do the calculations, or even graph them. I just realized that the graphs would be something like that, and I did one experiment in a game to make sure my estimated areas were reasonable. One of the points that I was hoping people would take away is that the power of understanding calculus (and many other ideas about the world) are often not used how you learned them in school (in math you are always calculating/solving) but are powerful tools toe reason about the world.

@MideoKuze 22 дня назад

It works a little better as a parable. In the real world, mathematicians (and, I'll add, theoreticians in science) will usually reach first for some a priori explanation, and then see if it works in the real world, while empirical scientists will usually look for data, and try to fit an explanation afterwards, testing to see if it's predictive, then iterating. Both approaches emphasize different skills, and can lead to different sources of error. Theoreticians can become lost in the theoretical weeds, spending a lot of energy on systematic, complete explanations for data they can't get. Empiricists, on the other hand, can under-theorize, and convince themselves that a good fit for the data they have is a complete story, even if it doesn't say much about the data besides a cause-effect relationship (and even then, not always). It's worth noting here that both approaches can get you to the right answer. In this case, empiricists will make some tests to estimate the derivative near a sample of points, but fitting a curve to those estimates is still needed to predict the true answer, and while a finite number of iterations won't usually get you to an exact answer, unless getting good estimates generates some insight that lead you to the theoretically correct solution, it can get you arbitrarily close. On the other hand, if you start with a priori reasoning about a complex problem, you may find yourself searching through many possible models, in a way that is too constrained to quickly get you to the theoretically correct solution you're after. In any case, the insight is that probing the system you're trying to model often gives you constraints that make answering theoretical questions about it easier. The point here is to understand that in applied problems can be complex and impure in ways that require you to get off the page and into the world to reason about efficiently. I realize this is a long reply to a joke comment but I think it's worth adding anyway.

@bobpillow6792 23 дня назад

Resentful mathematician here :P Any mathematician worth 2 cents would know that a ratio of small changes is an approximation of the derivative. There is no way that a mathematician (or really anyone who works with math) would miss that. If anything, this video shows that the mathematicians tried to find an exact solution while everybody else went straight for the easy answer. The point of this video should be that thinking about mathematical concepts in different ways is extremely helpful. Approximations are extremely important and very practical. But from your video it seems that thinking about the theory is bad when you can just approximate, which I strongly disagree with. Video more clickbaity than educational. Salty mathematician rant over

@lih3391 23 дня назад

True, but most physical systems have way too many forces and factors to be analyzed quickly

@MathTheWorld 23 дня назад

I think I am going to make the same point as @lih3391, but creating a mathematical model of the device would still just be an approximation because the model is not exact. And, as you put it, any mathematician worth 2 cents knows that as well. So I think your argument is valid if you focus on increased accuracy, but not exactness. Your claim about the point of the video is something we emphasized in the video, but apparently not enough.

@thefatcat2790 22 дня назад

@@MathTheWorld Well, it's not all about exactness, it's about finding a formula that you know will describe it in better detail than a type of function guessed from the measurements

@JustinLe 22 дня назад

Telling a mathematician that their way of thinking is not practical for real world problems is honestly the biggest compliment you could give them. Their ego would soar to the moon.

@ablubulu 22 дня назад

I think that every mathematician that watch your video will feel a little bit offended. Most of us study derivatives in an analysis course and not in calculus, many universities dont even have calculus in their math program. And the different education we receive is what I believe to be the root of what we see in the experiement. If you ask me what a derivative is I'll answer that is a linear transformation, and this perception includes all the meanings you shared in the video and multiple more. And thats the thing, mathematicians are dealing all the time with generalizations. There is a simple question that was not made in the video. Does the derivative even exists? Both scientists and engineers found an answer but is it even correct? It cant be if I prove there is no answer. How do you even prove an answer exist and that if I find one with aproximations, it is correct? These are very simple question that most mathematicians are trained since their first analysis classes to think of. I'm not saying that this is what happened but a glimpse of the problem gave me ODE and PDE PTSD and I know many problems that can happen in these systems so I'm definitely in the team that would take a long time to give an answer, if any. But thinking of derivative just as slopes is very silly, and claiming that that's how mathematicians understand it is, no offense, to be ignorant about what we actually study.

@ablubulu 22 дня назад

The experiement itself is very interesting, and I understand the point you make in the video. But I felt like I had to defend myself a little bit.

@glenm99 23 дня назад

A funny thing about the strongman scenario: bending the knees to bring the pillars inward also changes the angle of the line of force, counteracting the advantage to some extent (and at some point negating it entirely). In terms of applying minimal force to keep the pillars in place, an athlete should stand on their toes! (That said, that strategy creates torque which must be counteracted by the shoulders... and it's easier to pull the arms down than push them up.) I wonder if those competing in these events have thought this through in a systematic way, or if it's mostly an empirical process.

@mathbrotherc 23 дня назад

This is a great point. Calculating the force is definitely susceptible to an analytic analysis without needing empirical data. We are already planning on a few videos were we analyze a few of the events in the world strongman competition in more depth. I will make a note of this and see if we can get it in. (Oops, I see I am not on my Maththeworld account, but am the one that writes most of the scripts for the channel)

@AutumnReel4444 23 дня назад

Upon reflection, the framing of the mathematicians' perceived failure was perhaps presented not as well as it could have been.

@MathTheWorld 22 дня назад

Maybe we should change the thumbnail to be grades of A, A, and C. Would that be better?

@Luca_5425 22 дня назад

@@MathTheWorldyeah, I think so... in general, like one "salty mathematician" said in another comment, I don't think the way it was presented in the video was ideal, since it made mathematicians look like they're not even sure about what derivatives are (which isn't the case), but rather were trying to get to a general answer rather than geting a specific one... I do think changing the thumbnail will help, but the video itself sends the wrong message (or at least it did for me, before I read your comments/answers to people's comments)

@atlas4074 23 дня назад

One interpretation I think you missed that at least some mathematicians would recognise (though I'm not sure how many would think of it first) is as the best linear approximation to the function around a point. I typically see this emphasised when going over the mean value theorem or in the multivariable context where (total) derivatives are no longer numbers or vectors in the same space as the output of the functions and instead are linear transformations. Every analysis book I've seen touch multivariable derivatives always motivates it from this angle: f'(x) is the unique linear transformation A so that f(x+h) = Ah + f(x) + o(h) where o(h)/|h| goes to 0 as h goes to 0. Because the o term is small when h is small, f(x+h), the value of the function very close to x, is approximately Ah + f(x). The problems presented in the video moreso involve a 'small ratios' interpretation, so I doubt this would've helped, but it's an important interpretation in practice because tying things to linear algebra usually helps in complex problems. Nice video!

@MathTheWorld 23 дня назад

Thank you for the positive feedback. Your point makes me wish I would have emphasized in the video that we were talking about conceptions that students would learn about in first semester calculus. There are certainly others we could add to this list as students progress through higher, and different fields of, mathematics. I think a video about how derivatives are thought about at different levels and in different fields would be interesting. I'll have to do some leg work on that see if I can fit it to the theme of our channel. Thanks again for the comment.

@MideoKuze 22 дня назад

The mathematician salt is pretty amusing. As a theoretician, I get a lot of guff for working hard on abstract problems and not letting empirical constraints guide me enough, so I kind of sympathize with them, as someone who often tastes the disrespect of empiricists. But just as I trust my friends to bring me down to Earth when I need it, even if they are sometimes irritating about it, I think it's worth paying attention to the suggestion that living in theory-world is not always the best way to interact with a problem. Specifically, real world problems such as this often contain a mess of factors that can make finding a really tidy model and getting an exact solution for it quite hard, and the techniques for solving the pure problems theoreticians like to work with inefficient for finding a good predictor.

@samueldeandrade8535 23 дня назад

Where is the reference of the experiment? If it is true, I really want to know the "mathematicians" that participated ... they definitely do NOT represent mathematicians.

@jacobharmon6162 23 дня назад

My guess is math teachers, not mathematicians as in PhDs in a university/college math department

@samueldeandrade8535 23 дня назад

@@jacobharmon6162 "Finally, we interviewed two mathematicians who are both assistant professors and whose research is in mathematics education at the collegiate level. All of those we interviewed have doctoral degrees in their disciplinary areas and work in physics, engineering, or mathematics department".

@samueldeandrade8535 23 дня назад

@@jacobharmon6162 the whole article is just a joke. The objective was "Find dx/dF_x", the "d" actually dels, the symbol for partial derivative, having some machine with springs and flags. They wrote that on a board and expected the "experts" to make a table with some values and getting an approximation ... and called this "solving the problem". Hahahahaha. Just silly.

@MathTheWorld 23 дня назад

The purpose of the study was to study how different disciplines think about derivatives and they accomplished exactly that. The question was intentionally vague in order to really see how these groups of people intuitively think about derivatives. These Scientists and Engineers conceptualized derivatives in a more application based way, ratio of small changes. Something you can estimate for a single moment and don't have to generalize. While these Mathematicians thought about derivatives as slopes of tangent lines and generalized equations which are useful in other situations but you can't always find an equation to then "take the derivative of". Do the people chosen for this study represent all scientists, engineers, and mathematicians? Of course not. As we stated many mathematicians can and do think about derivatives as ratios of small changes. But the study highlights an important point when teaching derivatives. That limiting student's understanding of derivatives to just derivative rules and finding derivative equations is problematic when it comes to real world applications.

@mathbrotherc 23 дня назад

The article is referenced in the description.

@jon9103 22 дня назад

Items 1 and 2 are saying the same thing, 1. In plain English and 2. using an equation. The whole thing about units applies just as well to 2. its sometimes called dimensional analysis.

@algorithminc.8850 23 дня назад

Great video. Reminds me of thoughts related to practical people and theoretical people --- that it's best to improve both aspects of one's self - to be a good theoretician and actually do something in the real world. I seem to remember even Einstein talking about this. Theory helps practice, and practice helps theory. In any case, enjoying your channel. Thanks. Cheers

@MathTheWorld 22 дня назад

Thank you for the comment and the positive vibes. Your comment reminds me of the philosophy of the new Universty of Austin (I believe it is called). Where the first two years the focus is on being a great thinker, and the last two year the focus is on being a great doer (applying your thinking skills to help solve an important real-world problem).

@algorithminc.8850 22 дня назад

@@MathTheWorld Thanks. Sounds like a great school.

@PubicGore 23 дня назад

The article does not say mathematicians could not solve a particular problem. It says they were reluctant to approximate solutions with a partial derivative machine.

@MathTheWorld 23 дня назад

I actually think that it is more than that. Their attempt at creating a mathematical model would still just be an approximation (and they would of course know that). so isn't the issue of approximation. It might be an issue of trying to get a better approximation, but they would know whatever numeric value they gave would still just be an approximation.

@chixenlegjo 23 дня назад

@@MathTheWorldI think it depends on why they wanted to figure it out. If it was to predict the behavior of a similar system, then yeah, an approximation would work, but that’s not what a mathematician does. They work with purely theoretical models that usually don’t have real-world interpretations. The mathematicians were asked about a function, and so they wanted to give a function.

@jcorey333 23 дня назад

This was a great video! You only got one thing "wronge" at 8:08, lol. This is a really helpful way of thinking about math, thanks!

@MathTheWorld 23 дня назад

I thought we actually did that on purpose (since it is the wrong way to spell wrong).

@yeehawmcfly1794 22 дня назад

Is is possible for you to tell me how you came up with that equation at 9:06 ? I didn't quite understand lol

@yeehawmcfly1794 22 дня назад

I understood that arctan(d/132) provides the angle at which the pillar leans but I do not understand how the other stuff gives you a force or why you used 3100 oof

@remjoleea5560 18 дней назад

I’m coming up on the end of my second calc quarter and you hit a problem I’ve been thinking of on the head!! Sure, if given a formula for whatever situation I can solve for what they’re asking for, but I’ve been wondering how they get the formulas in the first place! It’s most likely just made up problems written for a textbook but it’s always seemed super unlikely to me that in a real world math situation you would stop to find a formula for exactly how whatever it is you’re describing works with exact coefficients / etc. It’s probably possible in a way I just haven’t seen, or easier in some situations than others, but it’s still been on my mind.

@ExzaktVid 23 дня назад

Waiting for the next Arbys video, I loved the previous two.

@MathTheWorld 23 дня назад

I think it is 3 weeks away. Sorry for the wait. We are working around some summer vacations.

@kruksog 22 дня назад

Numerical approximation is not a solition for a mathermatician. Hence why they "didnt" solve it. Bad clickbait video

@simeonsurfer5868 22 дня назад

hum, i've studied both physics and mathematics, and I can say 1 thing: The slope interpretation is almost never used in mathematics. The limit is used as definition, the transformation rule is used to calculate, and the ratio of small changes is used as interpretation, but the slope is just bad for 2 reason: not precise enough, and it cannot be used with 2 variable function in a lot of case. "But the slope is how it's teach" you could say, and you will be right, but that's because with one variable it's true and because it's a lot easier to teach! you can see what it's mean, and have a feel of how it works, that's what is esential in this. So, your video is not true, or should I say not for the reason you tell, it's true that mathematician would have a harder time doing real case, but this is for one major reason: approximation. In math, if you do an approximation, you found an upper bound and a lower bound and this make it possible for you to you to say that it's about right while in physics you mesure, you say that by observation, you can say that the rule should be like this, and then you use your conjecture to find a result and test if it work, you don't need to proove that this conjecture is true by pure logic but need to test your conjecture in real case to test if it's about right in the case of the experiment, wich need much less work but is good in real world case.

@alexandersanchez9138 20 дней назад

This is not your best video. I have two points of feedback: 1) The subtext of this video is that scientists and engineers are more capable of solving real-world problems than mathematicians because their basic conception of the derivative is better suited for real-world scenarios. If you deny this, then you are simply rejecting reality (“…and substituting your own” ~Adam Savage, Mythbusters). However, there is a simpler explanation of the outcome of the experiment that has nothing to do with derivatives: the task was simply closer to the training the scientists and engineers had and nothing like the training the mathematicians had. If, instead, they had all been handed a spreadsheet with fixed observational data instead of an apparatus, I wager that the mathematicians would have outperformed the scientists and engineers at obtaining estimates for the derivatives. That means the salient variable determining the performance was likely the task, not the differences in the ways the various groups thought about derivatives. 2) You are equivocating the purely geometric interpretation of “slope of the tangent line” with the mathematician’s algebro-geometric “slope of the tangent line/linear approximation”. I challenge you to find an actual freshman calculus course which, ALONGSIDE any mention of “tangent lines” or other purely geometric language, fails to state something to the effect f(x+dx)~f(x)+f’(x)*dx, or “changes in f are locally well-approximated as proportional to changes in x according to the derivative”. It is then immediate to compute approximations to f’ by experimentally collecting observations of f in response to various choices of x and dx. There is simply no inadequacy in the mathematician’s concept of the derivative-only, perhaps, that it assumes a competent user who has understood the correspondence between geometry and algebra (perhaps, lamentably, a strong assumption for many students failed by their prior education to convey this powerful and important idea). P.S. If I had taken this test, I would have smoked the scientists and engineers. ;)

@jon9103 22 дня назад

Frankly most people who studied calculus probably became scientists or engineers.

@nikolairubinskii6450 22 дня назад

That's some dim mathematicians...

@Vectorized_mind 16 дней назад

Scientists and Engineers are ALWAYS sloppy and cutting corners in their low tier dumbed down version of mathematics,and Mathematicians always have to come and clean up the disgusting displays of mathematics put together by Scientists and Engineers with no appreciation for precise mathematical definitions,theorems and proofs.

@user-pf6jl6mu1n 21 день назад

According to engeniers, pi=e=3 and g=9m/s^2. Yeahhh...

@aaab6054 23 дня назад

Your third definition ignores multivariable calculus.

@MathTheWorld 23 дня назад

The context of the video was on the meanings of derivatives that students could/would first encounter, so first semester calculus. I am sorry we didn't make that clear. There are other meanings that we could add (or change some of our meanings) at different levels, and within different fields, of mathematics. Commenter @samueldeandrade8535 raises some of these in another comment.

@samueldeandrade8535 23 дня назад

7:23 I am sorry, but this video ... is unacceptable.

@zilord3264 22 дня назад

I watch the video very early and i immediately knew that butthurt mathematicians would show up 😂. Great video btw.

@Vectorized_mind 16 дней назад

Real Mathematicians know this is DUMB,Scientists and Engineers are sloppy and not as rigorous and sophisticated as Mathematicians when it comes to problem solving,and this is evident in this clickbaity video.