Integration and the fundamental theorem of calculus | Chapter 8, Essence of calculus
Vložit
- čas přidán 8. 06. 2024
- Intuition for integrals, and why they are inverses of derivatives.
Help fund future projects: / 3blue1brown
An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: 3b1b.co/lessons/integration#th...
Check out the Art of Problem Solving: aops.com/3blue1brown
Timestamps:
0:00 - Car example
8:20 - Areas under graphs
11:18 - Fundamental theorem of calculus
16:20 - Recap
17:45 - Negative area
18:55 - Outro
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Vietnamese: ngvutuan2811
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with CZcams, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).
If you are new to this channel and want to see more, a good place to start is this playlist: 3b1b.co/recommended
Various social media stuffs:
Website: www.3blue1brown.com
Twitter: / 3blue1brown
Patreon: / 3blue1brown
Facebook: / 3blue1brown
Reddit: / 3blue1brown
How cruel it was to learn calculus for years in school and college without knowing the fundamentals. Glad the dark ages of pre internet era is gone. Thanks so much for this videos. You are a gift to mankind.
A sad tale indeed but we shall no longer be in the darkness but in light Thnx to the magicians of CZcams
indeed so thankful for the internet lol
Blessed be the internet for it's bestowing of knowledge...
Yea, i remember doing this in 7th grade. It wasn't so bad tho
And yet I throw away my valuable time in school when I can simply learn without pressure from books and the internet. But no, monopolies hold so much influence over the education system, changing the education system would be the equivalent of those monopolies loosing millions. I guess I have to force myself to "learn" in a monotonous curriculum that was designed by the school district in order for me to regurgitate information and throw it up on a test.
This channel, The Organic Chemistry Tutor, and Khan Academy are currently giving me the tools to master integral calculus... for free. What a wild time to be alive.
oversimplified reference?
Don't forget blackpenredpen!
It's awesome ^^
Definitely great channels
Professor Leonard also
Schools should just adopt this as an intro video and along with some worksheets the lectures can take two weeks off.
Haha, he is really really good.
lol absolutely how it should be.
In my Calculus cource it is actually recommended to watch this video series
then the lecturers don't improve as teachers. Usually they're there to get better at their job and make more money by having successful students with their own content
My school has already!
being in a car and only looking at the speedometer sounds incredibly dangerous
you think outside the box
Not if you are a passenger.
sigh... *rolls eyes* whatever dude
and then when u go backward
Well, if the speedometer shows 0 you should be alright. Assuming it isn't broken, of course.
Wish these videos existed when I was first learning calculus :P
yeah i never got where that area business came from. i just accepted it and memorised the rules. and got done with it. i knew the area was used to discover some integrals. like 1/x well if we used the 1/alpha rule we´d get 1/0*1 (as x to the zeroth power would be 1) and that is utter nonsense first it would be a constant no function second it´s literal physical pain for everyone with a bit (even my mini mini mini bit) of mathematical understanding. 1/0 that hurts. does that mean yo ucan´t integrate it? looks like it right? but i think Leibniz (might be wrong here though) found that the Area under the curve grew logarithmically. setting up the real butt saver S 1/x dx=lnx . but i always just accepted "okay the integral is magically bound to the Area under the curve".
My calculus class in the school is dull, i'm lucky to learn very important things here and Mathologer channel
Carlos Alberto López Reyes thanks for the mathologer tip :) didn't knew the channel
ikr!
That's because most math teachers aren't programmers who can utilize animation
I'm just a 12-year-old in 8th grade who likes math and this series so far has fascinated me and I love how these videos won't end without providing a full understanding of what's taught. This series is amazing.
Please keep learning math for as long as it fascinates you.
This kid is a rare one. I was not even fully conscious at 12. lol
I'm 15 and also learning it! Wanna chat?
@@user-mv4ix7jd8o Uh sure ig
@@Matthias27182 Lmao it only took a few weeks for me to get into chess and forget about math
Best 10 days in the life of CZcams!!!
Shubham Shinde agreed!!!!
Yes I agree.
Shubham Shinde hell yea
I agree to.
Since the series started, everyday I wake up that's the first thing I do. It's so good to watch it
I've been watching you for ages and JUST realised that this "group of pi monsters" is made up of 3 Blue and 1 Brown pi monsters....
Just realized that from reading this comment.
you are not the last
@@andyjiang9094 neither you
ur right
I never noticed that
Friday ... check
Little or no work to do ... check
3blue1brown uploads a video ... check
Today is a good day!
Faculty of Khan same, except for the work to do :(
neglecting to do my geometry homework now to watch a calculus video
I should be studying analytic geometry, conic sections, to be exact... but screw it, right?! i shouldn't be doing it..
Oliver Hees But Geometry is also interesting... that is if the subject is taught as pure mathematics.
I'm not sure about the day ... check
"I should not be wasting time on writing this comment" amount of work left to do ... check
What was the last one, ok I have to go. ... check
Someday will be a good day!
Grant is the Richard Feynman of CZcams
Totally
Couldn't agree more. He's an exceptional teacher!
never heared a more appropriate comparison before
Exactly!
If you mean that you think you understand everything when following along but half an hour afterwards you haven't got a clue about anything, then yeah. Just like Feyman.
0:00 intro
0:55 distance from velocity
2:27 area under a graph
4:08 approximating and refining
6:29 writing an expression: the integral of v(t)
8:30 how does this help?
9:34 the area as derivative
11:17 the antiderivative
14:57 the fundamental theorem of calculus
16:18 recap
17:46 signed area
18:55 outro & sponsor
Hey, I see lots of comments wishing they had this in their school days and, know what? I'm in my school days!!!!! I'm taking a lot of advantage of this series, THANK YOU!!!
Happy Mathin'
I’m so lucky that my calc professor seems to have the same love you do for teaching why things work the way they do, not just teaching how to do problems. His classes are always as engaging to me as your videos. I honestly can’t wait for calc 2 in fall with him.
seriously man... A good teacher can make you love the subject even if you never knew about it before
That professor of yours is a man of great value!
I think you should give him a gift if you can.. i think he deserves it!
School: You gotta learn this.
Me: Why?
School: You gotta learn this.
Me: But what is it good for?
School: You gotta learn this.
A few years later in university ...
Me: Geee, I wonder how I can calculate the area under that curve so I can get the consumer surplus from non-linear demand and supply functions ...
Are you at the AUT Uni? Btw what you say is relatable.
you could argue with a highschooler that they should learn it because building up the capacity to think in abstract of students is overall good for society but what's the point man, teens are morons and won't really care
I’m actually in algebra 2 right now but taught myself precalc in one day and am now getting into calc. So far I’ve done a lot of anti derivatives, derivatives, and integrals. Pretty cool stuff I just came here to be given a picture and practice whatever problems he put on the screen. Pretty good examples honestly
@MrComrade I'ts actually opposite. Everyone ABOVE 25 y.o. is a complete moron and thus needs to be eliminated.
@@tomasito6417 SAME. LITERALLY. I’m doing all of precalc today while starting to do calc in algebra II
I'm currently studying mechanical engineering and working as a tutor. I have to say, this series is absolutely amazing! I can only imagine how easier these concepts would have been to grasp if it were to watch your series when I first started learning calculus. I might just incorporate some of your notions in my tutoring sessions, if you don't mind!
M studying mathematics in college at the moment.. This videos n its execution has been helping me with the gaps in my knowledge
why is your profile pic euler
@@squibble311 why not>>>???
@@ostapigor1607 Gauss is better
@@rahimeozsoy4244 dude sssshhh you wanna start a war in here?!
Next up is a different perspective on why the area under one graph is related to the slope of another. Full playlist at 3b1b.co/calculus
Please do a vid on stochastic integration
I second that! Since the video where the derivative of x^2 was done using a sqare with the tiny (dx)^2 part vanishing, i had wanted to see something on quadratic variation. :D
3Blue1Brown Thumps up for stochastics. It's like mating calculus with a massive application for it. Beautiful!
Is there a way you can explain concavity and the second derivative in an intuitive sense?
It will also be awesome if you made a series about probability
I have watched this video 4 years ago, while I was in 9th grade and now I am studing math at university and watching it again. What a journey
Wow, you studied this in 9th grade at s hool or you were just interested in math?
@@exposingreality6391 I was just interested in mathematics. I knew some people who where way older than me and they carried me along.
@@Caspar__ im 13 lol
Great explanation. I am an aeronautic engineer that loves calculus but until now, it had been quite difficult for me to deeply understand why the integral is the "antiderivative" of a function.
Thank you for your great work!
I still don't understand why integral is the opposite of derivative, could you please explain what you understood?
Thank you
@@rikitobruece2386 I hope I can aid to your understanding, although it's been 2 weeks since you posted.
You can imagine the integral to be the "sum" of all the tiny strips that an area under the curve has been broken down into. The caveat is that you must remember that the integral is not just the sum of all area strips, it is the sum of the strips as these strips approach zero width. That's why integral is different than the ordinary sum.
Why is it opposite to the derivative?
Let's say you have the information on the distance travelled by a car with respect to time,and you want to find the velocity of the car during that time interval. You would use the derivative of that distance versus time function.
Now reverse the situation. Suppose it so happens that you have absolutely no data about the distance your car has travelled, but you just have access to the speedometer(i.e. velocity). You intend to find the distance travelled by your car through this data of velocity versus time. Earlier, you had distance versus time and you found velocity by taking the derivative. Now, when you have velocity versus time, and you want to find the distance, you would take the "integral" of the velocity data.
So in these two situations, you seek to find the derivative of a given function, and in the next situation, you already have the "derivative"(in the form of velocity data from speedometer), and you need to find the function which has this derivative(distance). This is called integral.
@@happysoul5031 Thank you so much for taking out your time. I have understood it well!!
@@lollel1490 Basically that's what I think will happen. I suck at calculus though so don't blame me if I'm wrong 😂😂😂. I used to hate calculus until i watched 3Blue1Brown videos now it's getting interesting.
I love that the gradient of the area rectangles animated the idea of smoothness. When there are 8 large rectangles, each rectangle's color is noticeably different from the color of the adjacent rectangle. As the rectangles get smaller (approaching the area under the curve) the colors approach a smooth gradient as well.
It seems that beautiful and subtle tricks are your forté, and this one is no exception. Thank you for your very helpful videos!
Nice spot. Should be developed.
On a more serious note, these videos are very well done. It's great to see the fundamentals (huehuehue) of mathematics being explained in such an intuitive manner. I think I could use your videos to help my own content, which is mostly geared towards higher-level undergrad/graduate science/math. If I incorporate animations and an intuitive angle, I imagine the explanations I give could get even better. Thank you for making these lessons!
- Faculty of Khan
This series is a perfect supplement to the Khan Academy curriculum or any school curriculum, which just teaches the "rules" of calculus.
Faculty of Khan
huehuehue? BR?
Dylan Wedel the gig is up bud we all know it's yours
David Valero ah fuck, you got me :'(
this is THE MOST succinct explanation i have ever heard for integrals - as a teacher myself , I tip my hat to you, very well done!
When I was in undergrad, educational videos on CZcams were really in their nascency. Having access to this, Khan Academy, and organic chemistry tutor would have been a game changer.
It’s wonderful to see that future generations will have access to such an efficient form of learning.
Been studying calculus for a semester, without any motivation. Now, I found joy in doing it, and I really enjoy your videos. Inspires me to do more work, and it drives me to try to be more creative with it. Thanks!
Man, the way you teach intuition in math is so beautiful. I wish we had a 3Blue1Brown in every subject.
I’m a first year Math student and I have to say, your channel is the reason I love math so much. It’s part of the reason I decided to study it. It hurts when I hear people complain about math when they’ve only been taught how to do things instead of why you do it that way. However, the hope that they might discover your videos keeps my head held high. Thank you for these “Essence of” videos. They help me to understand the ideas behind what I’m learning. I’m grateful that my profs do go into some detail as to why things are the way that they are. But when they don’t, you’re there to give me the intuition behind the math. I could never thank you enough for these videos.
Haiku for Calculus:
Unfathomable
beauty in infinite curves
Elusive, yet plain
I'm in my final year of highschool taking calculus right now and these videos have proven invaluable to my understanding. my favourite part about math is when it finally clicks, and your videos are making it click. i'm at a point in my eductation where I can't just coast through because the material is actually becoming challenging, so you are like a bridge between what my school can teach me and what I want to know. i cannot thank you enough
This is the most beautiful series I have ever seen in my life.
This is my favourite series as of now.
FIRST! and also, this the THE BEST CALCULUS SERIES ON CZcams for beginners!!
Not only for beginners. For those like me that have already learned calculus but want a better understanding find this incredibly helpful
*For anyone
well, I have done calculus 4 years ago during my undergraduate years! And overall I am good at Calculus. Still this video was amazing for me as there were many new things I didnt knew he explained :)
@@geekyprogrammer4831 can you give an example what you learned here that you didn't know yet?
@@squeakybunny2776 I'd suppose the abstract intuition that you might realize while watching Grant do his bid. As a grad student, I sometimes just come back here just to witness the fundamentals being explained by him as it incites a cloud of imagination in my mind and lets me rethink the problem from an entirely different standpoint. Most of the time, it's more like I'll just be overthinking without considering the concrete fundamentals.
I cannot describe how much I love this series. I watched it once when I started learning calculus and enjoyed it a fair bit. But now that I can really appreciate the subtleness that it shows and the nuance it gives, it's even better!
I've just had a double lesson in economics... This is exactly the right thing to calm you down.
J&M Productions Right? Economics is a branch of Mathematics in the same way Astrology is a branch of Science.
loooooool !!! :DDD
J&M Productions Maths Bachelor here. I can only applaud.
+
+Noel Goetowski
well, at liest economics counts right
This series is really outstanding. Clear, interesting, accessible. Thank you so much for making these videos and for making them available to us!
What a fantastic video! I was just getting into integration in school and finding it tricky, but this video made it become so much more intuitive. Thank you so much for making these wonderful videos!
Holy moley! An interrobang! Your visualizations are absolutely superb, and you've got a great radio voice. And your videos are fantastic in laying bare the underlying structure, simplicity, and beauty of mathematical reasoning. Thanks!
as an undergrad student who is struggling in calc, thanks so very much for these videos. they're very very good!
I believe that you are the number one math youtube
Yep he's quite cognizant
Hey, I just wanted to say thanks for this video series. Awesome job!
From what I've seen, the main requests for another "Essence of" series are real analysis, complex analysis, group theory/abstract algebra, and topology. Any of these series from you Grant would be a gift from God. Thank you so much for these videos.
The Flagged Dragon also probability
Genius to use car velocity as the f(x) to demonstrate the principle of integration, making the understanding of the subject very intuitive and precise. Again, 3B1B is the best animated math video channel that makes the learning of math a breeze and very enjoyable. Soooo unique! Thank you very much !!
Thank you very much for the quality content and your exquisitely explained concepts using dynamic visualizations much needed for the intuitive grasp of such difficult subjects, especially as they so dryly spat at us at the university course! Keep the good work coming!
Holy frick look at those animations at 15:42, how the rectangles shift dimensions and how the area forms a wave, so pleasing...
Yeah, his animations are wonderful. I just would like to know which tool did he use to make those animations?
@@falikousoumaoro9831Manim.
You are one of the main reasons why I loved math in the first place. Thank you so much for showing me this beautiful subject that not many people I know enjoy it.
This and the last video are godsends. I've never had a teacher explain L'Hopital's Rule and Integration so clear and concisely as he does. Thank you for everything
Thanks Grant for such wonderful video series. I truly started falling in love with maths again!
9:37 - 11:25 my favorite part, this will be super helpful for beginners of calculus to understand the whole idea. Good video.
I don't know how many times I have watched the series but I finally understood what calculus is about. I would have never understood the theorum of calculus if I tried to learn from the textbook without animation. Thank you very much!!
never stop teaching math, you have a gift, I have Known calculus for over two years but this series truly made me understand so much more. Amazing Work, keep it up
0:18 to 0:31 this is what people should realise about math! So well said and relatable!!
8:40 I've never actually seen anyone use an interrobang
Mitch Kovacs they're used semi commonly in chess notation!
‽th
Even in chess annotation, the permutations '!?' and '?!' are usually used with slightly different meanings (with !? representing an interesting move, which is probably good but difficult to analyze, and ?! representing a dubious move, which is probably bad but difficult to refute). So the interrobang still gets stranded.
To me they're just visually confusing as most people are not trained to read them quickly. It's a lot less confusing to type something like "!?" in my opinion.
How have you not seen it yet ‽
(...I just have interrobang on my keyboard. ⸘What is wrong with me‽)
This video is with no doubt the most precious calculus material I could find on CZcams and amongst many books as well... I admire your work, Grant!
holy thank you man! i never knew i needed that "negative area" part more than i do now
You treat math simple, sweet and calm as math deserves to be treated. Thanks for sharing this sense with us.
It feels so good to finally be able to explain WHY the anti-derivative method of calculating integrals actually works and how it's connected to the sum definition.
Thank you so much 3b1b!
Ol of elasticity maximum of viscosity is a good evening sir and I am in detail please give me a CZcams channel for JEE ADVANCED and the logarithm step by step solution for JEE ADVANCED and the reaction is defined as a basic physics is defined as a basic physics question and answer show please explain me this paragraph from NCERT solutions videos practice videos and more exceptions in the first equation of the day of the day of the day of the day of the year ahead I will send it is about paramagnetic molecules of viscosity of viscosity of a person in a factory 🏭🏭🏭🏭🏭🏭🏭 to be a part but a lot is a good and I will not get the same dipole by the other words the same to you and you are you can be derived by the other day and I will not get a CZcams video 1 se to a new one of my life if the unit is a supercooled liquid by the fact and a half of a lake is the most important and a little more than the other unit is the after the same to you and you can I am a Huzaifa and the reaction of a person in a factory as well and I will not get a good and a half filled by the other unit is the most important hai is a good evening and I will not get a CZcams video of Physics Wallah by step solution containing the after school and the logarithm of a lake was absent today or the same dipole by the fact which are frequently in a circle what we can I call karna to a thread such as significance and applications for a long term and a little calculus of the following is the most common and a half filled nor the parents are spending by step solution of your JEE and a little more exceptions to be like to be a good evening and the reaction of a person who reads all this time is a supercooled of a lake is equal by the other unit of the day of the day of the day of the day of the
These videos are truly amazing, I mean it! You explain it not only in a very natural manner but beautifully and elegantly!
I love you, Grant, I would come here completely terrified of these concepts and your voice and animations would just instantly soothe me. You're my math pacifier. Thank you.
You should do a series on differential equations! Maybe even differential geometry if you're like trying to make our brains mushy (in a good way)
you prophet
Thank you a lot for this series, it was beautifully done and very useful. When can we expect the Essence of probability series?
This was such a surprisingly intuitive explanation for derivatives and integrals being opposites. Good job, man. Good job.
This is arguable the best math explanation video I have ever watched. Just insanely good explanation.
The most beautiful thing I have ever seen in my life and probably the best 20 minutes experience honestly in my entire life till the age of 16
Do update your comment, in due course.
@@tim40gabby25😁
Why is this so intuitive man? I'm in grade 9 and I'm studying additional maths for my IGCSE boards. I looked everywhere for explanations of differentiation and integration and this just explained everything so intuitively. Thank you so so much ❤️😊
same man. But the thing is my grade 9 just starts in a few days. AD MATH IS TOO EASY FOR ME. IM also in igcse lol.
It's so satisfactory to hear "fundamental theorem of calculus" after you've understood all the basics
Thank you so much for explaining everything so clear and detailed. The graphics are amazing, representing the concepts. Makes it so much easier to form an understanding. These videos just re-taught me two semesters of long forgotten Calculus, lol. Thanks a lot for your work!
When you compute a definite integral - an integral between two values - the constants cancel out, so in this case we can ignore it (and could have ignored it even if we started at _t_ = 2). However, it becomes important when taking the indefinite integral, so I'm glad you mentioned it.
Can't thank you enough for your videos. This is one of my favorite channels on youtube! :)
One of the best videos i have ever seen for integration in whole youtube. The concept clarity from fundamental is great. The whole playlist is so fucking good with top quality content.
I’ve been almost obsessed with these videos recently. Got imtroduced from Stand up maths (which I also thoroughly enjoy!) but these videos are just so absolutely wonderful! I can listen to them as background or as I go to sleep because the content is so smooth and calming (well-written script, calm voice, pleasant enthusiasm) all so wonderful! Or I can focus on it intently to really absorb (relearn) theses deep mathematical topics.
Absolutely wonderful! I very much appreciate and take advantage of all the work that has gone into planning, preparing, and making this amazing content.
Thank you so much!
Fundamental theorem of calculus: The connection between derivatives and integration.
I am in love with math now. coz of u. ty
I've never seen such a great visualization of this before, thank you so much!
It truly feels very like a privilege to have access to the videos you made... I am very thankful for them! Keep up the good work!
22 people didn't pass the calculus exam.
112 now
@Ardian Maliqaj ±1
114
123
128
25 people are dirty nonstandard analyst,infinitesimal loving peasants
lol
I needed to subscribe because this series is bringing me back to high school. I took AP calc and in college in my high school senior year, then took the college placement test and was placed into calc 1. they gave me the option to instead receive 6 trig credits to satisfy my college math requirement. I did. I neglected my math for the rest of my college life because I thought "who needs math when you are going for a fine art degree?". I was so wrong back then. thank you for reigniting my curiosity of math after 20+ years. The abstract nature and creativity required for math really goes hand in hand with artistic creativity.
I have my analysis exam in 1 month and i'm so glad you're posting these videos. Nice job, keep up the good work!
Another clever way to obtain the total distance travelled is to multiply the average value of the velocity by the total time travelled. This still works when using curves!
Won't it depend on the degree of the curve?
Nex Ninja
No it doesn’t, that’s what’s so neat about it! Such an elegant solution to a apparently complex problem. First you’ve got to find your range for which you want to compute your area, then you invert that range and then multiply it by the integral of the range in question. This will give you the average value of velocity.
Yes, that is true, but to find the average value of the velocity of the function you need to integrate, so you might as well just integrate.
sir ,please make vedio on laplace transforms
And Fourier transforms
coming in chapter 10, if im not mistaken
I'm pretty sure chapter 10 is just Taylor series. Getting into Laplace transforms requires quite a bit of work building up intuitions about differential equations, frequency domain analysis, and complex numbers.
Nope, Taylor series.
To me, Fourier transform have priority over Laplace transform
Very well explained and easy to understand. Great job on making such a difficult topic seem so easy!
This explanation is absolutely amazing! It is quintessential to understand the simplified proof, before getting into more rigorous proving. I watch the videos before going to lecture, it is a perfect combination with course materials!!
There's seriously nobody else on CZcams who even gets CLOSE to your ability to teach.
I'm only 2 minutes in and this is *very* helpful! Thank you very much for this! :)
Christopher Okhravi you’re good tutor too !!
Such a simple and accurate way of simplifying the theorem of calculus! Thank you!
This is just pure awesome. Never before have I understood why you subtract the applied lower bound from the applied upper bound. Thank you SO much!
third time watching, good job! a 13 year old understands intergrals!
Lol you're a dumbfuck. If you don't understand calculus when you're in your mom's womb, you can't be successful in life.
@@bigfly1391 Well, I don't have enough humor then. (I deleted my previous comment now)
Now look at my teachers...i think they really don't know these things and due to such teachers students don't understand maths and science...thats why students hate...thank you very much for this knowledge...i just cant memorise formulae without understanding..i just can't....thank you very much...🙏🏻🙏🏻🙏🏻🙏🏻🙏🏻🙏🏻🙏🏻
Thank you very much.😍
You are fulfilling my keen desire to know the backend of these concepts, which i could not do during 11th and 12th in the school.
It is very teasing to cram things and very beautiful and soothing to understand them.
I am so grateful for your contribution to the understanding of math! Being a math student, I am almost dependent on your clever and insightful videos. Thanks!
Grant, you genuinely deserve the Nobel Peace Prize
can you make a video on integration using the u substitution method
Wow. This video is 6 years old. These videos just have such a timeless feeling to them
This is such a great explanation I cannot stress this enough 4 mins in and I have a better grasp of calculus than from watching at least an hour of tutorials.
If you think that's weird seeing distance as area, you can analyse the units:
m/s × s/1 = (m.s)/s
The "s" cancels
m
There you go, velocity by time gives distance
Hi, i must admit the video has been amazing, but i don't understand, maybe the easiest thing: why v(t)= t(8-t) ?
I had just studied cinematic in physic this year (i'm 16) and actually, I can't come up with where it comes from
Thanks for the help
(And sorry if I've made any mistake while writing)
v(t) = t(8-t) is just an example
It could have been anything. In practice, you'd get the form for velocity by plotting position against time and grinding out the differences until you got something constant
The velocity function, v(t), is the derivative of the position function, s(t). He’s using this as an example
Just to clarify: by v(t)= t(8-t) is an example. It means that t(8-t) is just a function to describe the curve in the graph. v(t) can be v(t) = t^2 + 8t+8 or v(t) = t^4 + 8t^2+t but the curve would change accordingly. So this velocity function is a function and not a formula of velocity(v=s/t).
Indeed, that simple "t(8-t)" had me stuck in the mud until could figure a workable explanation. Namely, 3blue1brown needed the simplest function to demo the anti-derivative, do he chose a perfectlly smooth start-accell-decell-stop function that could EASILY be anti-derived from t(8-t) [or "8t-t^2"] to "4t^2 - t^3/3" . The simplest derivative he chose to start with was t(8-1) where "8" is the 8 seconds shown on the x-axis. Now, the distance travelled at each time unit 1 thru 8 EASILY computes using t(8-t) to 7,12,15,16,15,12,7, 0 meters. Hope this helps.
These videos are so relaxing to watch. I could genuinely watch these like movies with my cereal in the morning and enjoy myself. The animations are always so mesmerizing. If you guys want so see some even more mesmerizing videos, watch the ones in matrix algebra.
at 7:02, not only it a factor in each quantity that we are adding up it also indicates the spacing between each sample step. wow that is at the height of Grant's explanation. Really appreciate it.
Was assuming that since you covered techniques of differentiating like product rule, chain rule, implicit, exponential and logarithmic differentiation, you'd do the same for integration, giving intuition for how substitution, integration by parts etc work.
Ah well, they might not really be part of the 'essence' of calculus...
I wanted to include those topics, and I intend to revisit this playlist at some point and add three more integral videos, but unfortunately for the date I was hoping to publish, I did not have time to do integrals the full justice they deserve. But that is not to imply that integrals lie outside the "essence".
3Blue1Brown would be really great cause derivativing functions is most of the time easy if you know the special derivatives and the rules. i seriously don't have much mathematical knowledge but i can derivate (is that the correct verb? nonenglish speaker here) most functions with a glance. but integration even simple polynomials require thinking and for more difficult problems i need to have my old math lectures ready and use them like "manuals" because i just don't get things like partial integration or substitution in my head
Metalhammer1993 I think the verb is 'to differentiate', but the noun is 'derivative' - logic at its finest.
aye man, thanks. it´s really a bit tricky if you´re a non native speaker
Metalhammer1993 Yeah I know. You're welcome!
I can't be the only one who has a crush on this guy‽
Vruksha Nikam No u aren't
Hey it's me your this guy
About to start Calculus II and I only wish that I had stumbled through these videos in Calculus 1. These animations are so beautifully done that I recreate and screenshot these for notes. Doing this helped me finally understand related rates questions more intuitively which were my biggest struggle in Calculus I.
Thank you so much. I’m pregaming and studying all that you have provided.
this is the best teaching (explanation or proving) of integral to me at the moment, easy to understand with the idea derivative comes from!