This Is the fondamental theorem of calculus, but explained in a really effective way
@understandingmathandphysic9438
Жыл бұрын
You are absolutely right! I've taken more analysis after since making this video, so I know it isn't very rigorous. But it is really intuitive and helped me out a ton.
@lovishnahar1807
8 ай бұрын
@@understandingmathandphysic9438and guess what sir , im from india going to make college entrance very soon and also know the same meaning from very long and even asked my school teacher way back (end up being debate he didnt believed ), and asked same on our infamous discord server n similar situation there and finally i see u, very thankfull to u now i know i wasnt wrong at all
@hablo_papøl
8 ай бұрын
I have another way of understanding it intuitively. Given that the indefinite integral of a function is the function's area function, the derivative of the function's indefinite integral is the function that tells you the rate at which the area under the curve of the function changes. Intuitively, the rate at which the area under a curve changes is equal to the height of the curve at the point we are evaluating, which is just the value of the function at that point, so the rate of change (the derivative) of the area function (anti-derivative) is the function itself, why differentiation and integration are inverse operations.
@990user
7 ай бұрын
thats good. thats real good. ima tryna understand why they are inverse operators and that helped me. thanks bro. i now understand why the derivative of the integral is the function, but i have yet to understand why the integral of the derivative is the function. halfway there...
@hablo_papøl
7 ай бұрын
@@990user I am very glad to hear. That is exactly what I wanted when writing the comment. I remember being irked for a long time on getting an intuitive understanding of why this is true. One night, right before I lay in my bed, I found this. It was and still is beautiful and a beautifully satisfactory connection. But now you bring this up, the inverse, the integral of the derivative. Why haven't I thought of this? Perhaps this too will bug me for a while. this will be interesting...
@990user
7 ай бұрын
@@hablo_papøl cool man :) let me know when you find out, so I can find out too ;)
@juliavixen176
7 ай бұрын
@@990userThe "non-standard" way that I understand it is that integration is the addition of an infinite number of Infitesimals, and a derivative is the subtraction of an infinite number of Infitesimals (from the previous and next real numbers). Like, if you have a list of the square of every integer: 1, 4, 9, 16, 25, 36, etc. And you subtract each number from the next one: (4-1), (9-4), (16-9), (25-16), (36-25), etc. You get 3, 5, 7, 9, 11, etc. Then subtract each pair of numbers in that list: (5-3), (7-5), (9-7), (11-9), etc. and you get 2, 2, 2, 2, etc. Because the derivative of x² is 2x The derivative is like that, but instead of the numbers in your sequence being "one" apart from each other, they are "epsilon" apart from each other (being consecutive (uncountably infinite) real numbers)
@aryanjoshi3342
Жыл бұрын
This proof is great. I've seen many intuitions for the integral and antiderivative being the same, but none of them actually clicked except this one
@BabySeahorse420
9 ай бұрын
I think my immense interest in math only started this year because of statistics and calculus, but mainly calculus. Seeing drastically different ideas connect so seamlessly is amazing
@me_hanics
7 ай бұрын
You need to try combinatorics. I have never found more interest in anything than strategizing in games, coming up with optimization algorithms and so on.. I think it's also the "future", methods in combinatorics are usually not that "general" (except linear programming), compared to linear algebra but this specificity, complexity makes things much more effective in some practices
@enasalareef5961
5 ай бұрын
Just wait until you get to linear algebra. I thought the same thing too when I started calculus. But the journey really begins AFTER taking differential equations and linear algebra. (*_*)
@mpm165
2 ай бұрын
once I saw the definition of the derivative pop up it all made sense. what a beautiful proof
@ingenuity23
2 жыл бұрын
that is one of the most beautiful proofs ive seen. loved it!
@understandingmathandphysic9438
Жыл бұрын
This video has gotten far more views than I ever expected it to, thank you! I have just recently released a video about the calculus 3 second derivative test. I would greatly appreciate it if you gave it a watch and let me know what you think.
@prembhullar850
Жыл бұрын
Today you have earned a subscriber. Great video , i am a student learning calculus and i also many times wonder the reasoning behind some of the concepts in calculus . It's very easy to remember a concept when you know why it is done
@understandingmathandphysic9438
Жыл бұрын
@@prembhullar850 amen! That’s exactly why I like math. Once I have an understanding for a concept I feel amazing and have more confidence applying that knowledge. Please do recommend concepts or problems you would like me to cover. I’m looking to vamp up and make lots of new videos.
@CliffSedge-nu5fv
6 ай бұрын
The geometric/ non-geometric way I've always thought about it (not a proof, really) is that integral as sum of areas does the operations of multiplication followed by addition, and derivative as slope does subtraction followed by division. Those are inverse processes.
@Viki13
5 ай бұрын
I also had the same idea. Obviously not a proof but at least an elementary intuition
@WolfgangFeist
8 ай бұрын
If the function is not monotonically increasing in [x; x+h] use min an max in stead of the end-point-values :-). Otherwise: quite good presented. (we intuitivly assume that f is continuous; but I agree, for practical purposes we can do that).
@koalamistico
2 ай бұрын
Amazing proof. Obviously it needs to be a little bit more rigorous (hypothesis and negative cases), but I think it's one of the best ways to introduce the concept of derivatives and integrals being the inverse operation of each other (when valid).
@integrationmaster
Жыл бұрын
Thank you so so much for making me understand something that I've been stuck for long time :))
@understandingmathandphysic9438
Жыл бұрын
I was hung up on this connection for a long time as well! Glad I could help.
@Spacexioms
9 ай бұрын
This how I was taught FTC. Much more intuitive and actually makes learning differential forms much more easily
@nope_sup_yup
2 жыл бұрын
This video should be at trending. Thanks sir
@pableraspfgpfg468
Жыл бұрын
Really good explanation. For the first time I clearly understand this concept and avoid to repeat "because of the Fundamental Theorem of Calculus".
@jordanrutledge7943
5 ай бұрын
this is fantastic, I wonder if there is something equally intuitive going the other direction, from the graph of the derivative, proving it’s anti derivative takes us back to the function itself
@Viki13
5 ай бұрын
My explanation is that the slope is a division and the area is a multiplication so they're inverse operations
@AmmoBops
Жыл бұрын
This is awesome, I honestly kindve made my own assumptions as to why, but I was wrong 😂, thank you for the actual proof as to why and it makes a lot more sense now. It never really made too much sense when the teacher just kept repeating “the integral is the area under a curve” I was like “but how”
@understandingmathandphysic9438
Жыл бұрын
I’m happy I could play my role in passing along this awesome relation between two very crucial operators!
@mr.slytherin8290
Жыл бұрын
OMG! I have watched a lot of videos about this and I didnt take the idea behind it! You deserve more views for sure! Amazing video, amazing explanation about the topic and passed your thought in a clear way! Thanks for your work, I subscribed !!!
@understandingmathandphysic9438
Жыл бұрын
Thank you for your kind comment and support. The idea in this video really helped me understand how integration and differentiation are related visually. I’m glad it was able to do the same for you!
@Noobatsecs69
Жыл бұрын
I watched many other videos on KZitem about the fundamental theorem of calculus, and none of them made sense like you did. The geometric interpretation makes a lot of sense, thank you so much for the video.
@vikraal6974
2 ай бұрын
I loved this proof by Herbert Gross from M.I.T. Let f(x) be some curve and A(x) be area under curve in interval [ x, x+dx ] f(x) dx < A(x) < f(x+dx) dx Then d [ f(x) dx < A(x) < f(x+dx) dx ] / dx = f(x) < A'(x) < f(x+0) A'(x) = f(x)
@lincolnuland5443
3 ай бұрын
excellent explanation. you're a great teacher.
@bigheaddanny3377
7 ай бұрын
Very beautiful proof, makes the intuition click.
@niftkislay
5 ай бұрын
This is such a good way to explain this concept. Thank you so much.
@lovishnahar1807
8 ай бұрын
keep exploring things like this sir , its remarkable
@warisulimam3440
7 ай бұрын
Thank you for the wonderful explanation. I was already familiar with this proof already but watching your video raised a question in my mind. What you proved here is that the derivative of the integral of a function is the function itself. But to prove p(x) and q(x) are inverses, we have to prove both p(q(x)) = x as well as q(p(x)) = x. So what I'm asking is: how to show that the integral of the derivative of a function is just the function too?
@doraemon402
5 ай бұрын
That's cool but it only applies to functions that are strictly increasing in that x to x+h interval. It can easily be generalized, you just have to change the language to maximums and minimums instead of simply stating f(x)< f(x+h) . Aside from that, it's a good explanation.
@mhm6421
10 ай бұрын
That's the best and simplest proof of this I've seen so far
@arittroarindomsporsho4446
Ай бұрын
Loved it.
@mmmmmmok5292
9 ай бұрын
this is actually amazing, thanks so much
@CryzUwU
5 ай бұрын
Brilliant.
@marvinlee7657
8 ай бұрын
Great !!!!!!!!!!!!!!!!!!!!!!!!! Hope this video can last forever !!!!!!!
@chharlessweeney
5 ай бұрын
Well done ... thanks.
@simonettarinaldi5468
Жыл бұрын
Your Channel Is very good I Wish I found It before my calc exam!
@cyberjack1406
8 ай бұрын
Such a beautiful proof.
@ninadee9753
8 ай бұрын
THANK YOU!!!
@nayeem7359
10 ай бұрын
Thank you
@sumtensor
15 күн бұрын
This might be due to my physicist brain, but to me it has always seemed obvious from the fact that a derivative in terms of infinitesimals, you divide by dt, and in an integral you multiply by dt.
@timothycardinal5934
Жыл бұрын
Super job, thanks!!!
@aliassaf5244
9 ай бұрын
very helpful, thank you man .❤
@someone-ol8wc
2 жыл бұрын
Amazing video! TYSM
@understandingmathandphysic9438
2 жыл бұрын
You are too kind! I really love understanding math on a deeper level than school usually teaches it at. I have improved my video qualities slightly more recently by using a digital writing board, but I have a long way to go. I'm excited to put more time into learning editing as well as using coding examples and animations for help visualizing ideas. Your encouragement means a lot to me.
@evannixon6402
9 ай бұрын
fantastic explanation!
@stevenwilson5556
3 ай бұрын
awesome
@MrVideoTronix
2 жыл бұрын
Very good explanation. Great video! 👍
@satyamraj2039
9 ай бұрын
great explanation
@abdullohyakubov9854
6 ай бұрын
but why is there an equal sign in the inequality? the Area is clearly less than the area of the rectangle at the right endpoint, and similarly the Area is clearl bigger than the area of the rectangle at the left endpoint. If there is not equal sign, then we cannot conclude that derivative of the Area function is actually the given function
@understandingmathandphysic9438
6 ай бұрын
Imagine the function is constant over the small interval x to x+h. That is the case in which they are equal. It wasn’t stated explicitly.
@UberHummus
Жыл бұрын
Nice I really dig this!
@understandingmathandphysic9438
Жыл бұрын
Thanks Hector! I just graduated… between school and work I wasn’t able to make videos much the last few years. I’ve spent the last few weeks planning and animating for a video deriving the 2nd derivative test learned in calc 3 (for 3D surfaces). Im exited to start making higher quality videos! I appreciate your encouragement.
@cesarzzzc
Жыл бұрын
This is great!! Keep it up!!!
@jeremiahjohnson1121
Жыл бұрын
Beautiful! Thanks mate!
@phillee3507
11 ай бұрын
Just a question, why are you taking the limit of h going to zero at 5:37?
@understandingmathandphysic9438
11 ай бұрын
It causes the left and right hand sides of the inequality to be equal. Also, the central term is the difference quotient, so taking its limit as h goes to zero will give us a derivative. It’s hard to answer your question. It’s like asking if 2x=8 why do we divide by 2? We do it because it gives us the solution we are looking for. That’s the real answer, I’m taking that limit because it leads to the solution I’m looking for.
@muhammadkumaylabbas8513
5 ай бұрын
Cool!
@toiletry
2 жыл бұрын
awesome explanation
@ElianAPinheiro
Жыл бұрын
this is awesome!! very good
@PedroSotelo188
Жыл бұрын
excelent proof
@markfowler673
Жыл бұрын
(That, and groups....)
@johnpro2847
5 ай бұрын
I'm getting there..
@markfowler673
Жыл бұрын
I can't see it on my cellphone -- Upgrade your graphics? Anyway, thanks for the great idea(s).
@understandingmathandphysic9438
Жыл бұрын
Thanks for the feedback, this was very early on and I didn’t have great equipment (still don’t).
@brendawilliams8062
8 ай бұрын
I hope he gets his equipment. It is nice presentation and the method of explaining it with hand drawn methods is invaluable
@RutajitChakrabarti
5 ай бұрын
Bro this guy needs to be a prof at University level.
@understandingmathandphysic9438
5 ай бұрын
Since making this video I have finished my physics degree and am now teaching introductory physics labs. My goal is to be a math/physics full professor :) Thank you for your support!
@pritamdodeja
2 жыл бұрын
Nice video!
@abdullahraja8496
Ай бұрын
Bro i think problem is people do not know the intitutive def of area
@aaronnorman9755
Жыл бұрын
Pretty damn cool
@TheWesternPrince
Жыл бұрын
cool proof 👍
@OmarAdnan-hb9yw
Жыл бұрын
nicee
@yisahak
2 жыл бұрын
Place proof chain rule
@monoman4083
Жыл бұрын
no balls here. good video..
@mrimmpossible
6 ай бұрын
Im sorry. Maybe I am dumb. But your video doesn't answer why Integration is the inverse of differentiation
@erzi3800
2 жыл бұрын
That works only if f is increasing
@understandingmathandphysic9438
2 жыл бұрын
True, but it would just be a matter of flipping the left and right sides of the inequality and the result will be the same. Thank you for pointing this out.
@erzi3800
2 жыл бұрын
@@understandingmathandphysic9438 but what if the function is not monotonic?
@understandingmathandphysic9438
2 жыл бұрын
Monotonicity is of no relevance to this proof. There are three cases, the case where h*f(x+h)>A(x+h)-A(x)>h*f(x) {The positive slope case), the case where h*f(x+h)=A(x+h)-A(x)=h*f(x) {zero slope case}, and lastly the case where h*f(x+h)
@understandingmathandphysic9438
2 жыл бұрын
@@erzi3800 Plus the arbitrary function I drew isn't monotonic, the local region from x to x+h is all that matters for this proof.
@understandingmathandphysic9438
2 жыл бұрын
Also, about your first comment, we should clarify that when we say increasing or decreasing, we are talking locally within the region between x and x+h.
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