#### The Essence of Calculus, Chapter 1

[INTRO MUSIC] Hey everyone, Grant here. This is the first video in a series on the essence of calculus. and I’ll be publishing the following videos once per day for the next 10 days. The goal here, as the name suggests, is to really get the heart of the subject out in one binge watchable set. But with a topic that’s as broad as calculus. there’s a lot of things that can mean. So, here’s what I’ve in my mind specifically. Calculus has a lot of rules and formulas which are often presented as things to be memorised. Lots of derivative formulas, the product rule, the chain rule, implicit differentiation, the fact that integrals and derivatives are opposite, Taylor series; just a lot of things like that. And my goal is for you to come away feeling like you could have invented calculus yourself. That is, cover all those core ideas, but in a way that makes clear where they actually come from and what they really mean using an all-around visual approach. Inventing math is no joke, and there is a difference between being told why something’s true and actually generating it from scratch. But at all points I want you to think to yourself if you were an early mathematician, pondering these ideas and drawing out the right diagrams, does it feel reasonable that you could have stumbled across these truths yourself? In this initial video, I want to show how you might stumble into the core ideas of calculus by thinking very deeply about one specific bit of geometry: the area of a circle. Maybe you know that this is pi times its radius squared, but why? Is there a nice way to think about where this formula comes from? Well, contemplating this problem and leaving yourself open to exploring the interesting thoughts that come about can actually lead you to a glimpse of three big ideas in calculus: integrals, derivatives, and the fact that they’re opposites. But the story starts more simply—just you and a circle; let’s say with radius three. You’re trying to figure out its area, and after going through a lot of paper trying different ways to chop up and rearrange the pieces of that area, many of which might lead to their own interesting observations, maybe you try out the idea of slicing up the circle into many concentric rings. This should seem promising because it respects the symmetry of the circle, and math has a tendency to reward you when you respect its symmetries. Let’s take one of those rings which has some inner radius R that’s between 0 and 3. If we can find a nice expression for the area of each ring like this one, and if we have a nice way to add them all up, it might lead us to an understanding of the full circle’s area. Maybe you start by imagining straightening out this ring. And you could try thinking through exactly what this new shape is and what its area should be, but for simplicity let’s just approximate it as a rectangle. The width of that rectangle is the circumference of the original ring, which is two pi times R. Right? I mean that’s essentially the definition of pi; and its thickness? Well that depends on how finely you chopped up the circle in the first place, which was kind of arbitrary. In the spirit of using what will come to be standard calculus notation, let’s call that thickness dr, for a tiny difference in the radius from one ring to the next. Maybe you think of it as something like 0.1. So, approximating this unwrapped ring as a thin rectangle, its area is 2 pi times R, the radius, times dr, the little thickness. And even though that’s not perfect, for smaller and smaller choices of dr, this is actually going to be a better and better approximation for that area, since the top and the bottom sides of this shape are going to get closer and closer to being exactly the same length. So let’s just move forward with this approximation, keeping in the back of our minds that it’s slightly wrong, but it’s going to become more accurate for smaller and smaller choices of dr. That is, if we slice up the circle into thinner and thinner rings. So just to sum up where we are, you’ve broken up the area of the circle into all of these rings, and you’re approximating the area of each one of those as two pi times its radius times dr, where the specific value for that inner radius ranges from zero, for the smallest ring, up to just under three, for the biggest ring, spaced out by whatever the thickness is that you choose for dr—something like 0.1. And notice that the spacing between the values here corresponds to the thickness dr of each ring, the difference in radius from one ring to the next. In fact, a nice way to think about the rectangles approximating each ring’s area is to fit them all up-right side by side along this axis. Each one has a thickness dr, which is why they fit so snugly right there together, and the height of any one of these rectangles sitting above some specific value of R—like 0.6—is exactly 2 pi times that value. That’s the circumference of the corresponding ring that this rectangle approximates. Pictures like this two pi R can actually get kind of tall for the screen. I mean 2 times pi times 3 is around 19, so let’s just throw up a y-axis that’s scaled a little differently, so that we can actually fit all of these rectangles on the screen. A nice way to think about this setup is to draw the graph of two pi r which is a straight line that has a slope two pi. Each of these rectangles extends up to the point where it just barely touches that graph. Again we’re being approximate here. Each of these rectangles only approximates the area of the corresponding ring from the circle, but remember, that approximation, 2 pi R times dr, gets less and less wrong as the size of dr gets smaller and smaller. And this has a very beautiful meaning when we’re looking at the sum of the areas of all those rectangles. For smaller and smaller choices of dr, you might at first think that that turns the problem into a monstrously large sum. I mean there’s many many rectangles to consider and the decimal precision of each one of their areas is going to be an absolute nightmare! But notice; all of their areas in aggregate just looks like the area under a graph, and that portion under the graph is just a triangle. A triangle with a base of 3 and a height that’s 2 pi times 3. So its area, 1/2 base times height, works out to be exactly pi times 3 squared; or if the radius of our original circle was some other value capital R that area comes out to be pi times R squared, and that’s the formula for the area of a circle! It doesn’t matter who you are or what you typically think of math that right there is a beautiful argument. But if you want to think like a mathematician here, you don’t just care about finding the answer; you care about developing general problem-solving tools and techniques. So take a moment to meditate on what exactly just happened and why it worked, cause the way that we transitioned from something approximate to something precise is actually pretty subtle, and it cuts deep to what calculus is all about. You had this problem that can be approximated with the sum of many small numbers, each of which looked like 2 pi R times dr for values of R ranging between 0 and 3. Remember, the small number dr here represents our choice for the thickness of each ring—for example 0.1. And there are two important things to note here. First of all, not only is dr a factor in the quantities we’re adding up—2 pi R times dr—it also gives the spacing between the different values of R. And secondly, the smaller our choice for dr the better the approximation. Adding all of those numbers could be seen in a different pretty clever way as adding the areas of many thin rectangles sitting underneath a graph. The graph of the function 2 pi R in this case. Then—and this is key—by considering smaller and smaller choices for dr corresponding to better and better approximations of the original problem, this sum, thought of as the aggregate area of those rectangles, approaches the area under the graph; and because of that, you can conclude that the answer to the original question in full un-approximated precision is exactly the same as the area underneath this graph. A lot of other hard problems in math and science can be broken down and approximated as the sum of many small quantities. Things like figuring out how far a car has traveled based on its velocity at each point in time. In a case like that you might range through many different points in time and at each one multiply the velocity at that time times a tiny change in time, dt, which would give the corresponding little bit of distance traveled during that little time. I’ll talk through the details of examples like this later in the series, but at a high level many of these types of problems turn out to be equivalent to finding the area under some graph. In much the same way that our circle problem did. This happens whenever the quantities that you’re adding up, the one whose sum approximates the original problem, can be thought of as the areas of many thin rectangles sitting side-by-side like this. If finer and finer approximations of the original problem correspond to thinner and thinner rings, then the original problem is going to be equivalent to finding the area under some graph. Again, this is an idea we’ll see in more detail later in the series, so don’t worry if it’s not 100% clear right now. The point now is that you, as the mathematician having just solved a problem by reframing it as the area under a graph, might start thinking about how to find the areas under other graphs. I mean we were lucky in the circle problem that the relevant area turned out to be a triangle. But imagine instead something like a parabola, the graph of x squared. What’s the area underneath that curve say between the values of x equals zero and x equals 3? Well, it’s hard to think about, right? And let me reframe that question in a slightly different way: we’ll fix that left endpoint in place at zero and let the right endpoint vary. Are you able to find a function A(x) that gives you the area under this parabola between 0 and x? A function A(x) like this is called an integral of x-squared. Calculus holds within it the tools to figure out what an integral like this is, but right now it’s just a mystery function to us. We know it gives the area under the graph of x squared between some fixed left point and some variable right point, but we don’t know what it is. And again, the reason we care about this kind of question is not just for the sake of asking hard geometry questions; it’s because many practical problems that can be approximated by adding up a large number of small things can be reframed as a question about an area under a certain graph. And I’ll tell you right now that finding this area this integral function, is genuinely hard and whenever you come across a genuinely hard question in math a good policy is to not try too hard to get at the answer directly, since usually you just end up banging your head against a wall. Instead, play around with the idea, with no particular goal in mind. Spend some time building up familiarity with the interplay between the function defining the graph, in this case x squared, and the function giving the area. In that playful spirit if you’re lucky here’s something that you might notice When you slightly increase x by some tiny nudge dx look at the resulting change in area represented with this sliver that I’m going to call dA for a tiny difference in area. That sliver can be pretty well approximated with a rectangle one whose height is x squared and whose width is dx, and the smaller the size of that nudge dx the more that sliver actually looks like a rectangle. Now this gives us an interesting way to think about how A(x) is related to x-squared. A change to the output of A, this little dA, is about equal to x squared, where X is whatever input you started at, times dx, the little nudge to the input that caused A to change. Or rearranged dA divided by dx, the ratio of a tiny change in A to the tiny change in x that caused it, is approximately whatever x squared is at that point, and that’s an approximation that should get better and better for smaller and smaller choices of dx. In other words, we don’t know what A(x) is; that remains a mystery, but we do know a property that this mystery function must have. When you look at two nearby points for example 3 & 3.001 consider the change to the output of A between those two points—the difference between the mystery function evaluated at 3.001 and evaluated at 3. That change divided by the difference in the input values, which in this case is 0.001, should be about equal to the value of x squared for the starting input—in this case 3 squared. And this relationship between tiny changes to the mystery function and the values of x-squared itself is true at all inputs not just 3. That doesn’t immediately tell us how to find A(x), but it provides a very strong clue that we can work with. and there’s nothing special about the graph x squared here. Any function defined as the area under some graph has this property that dA divided by dx—a slight nudge to the output of A divided by a slight nudge to the input that caused it—is about equal to the height of the graph at that point. Again, that’s an approximation that gets better and better for smaller choices of dx. And here, we’re stumbling into another big idea from calculus: “Derivatives”. This ratio dA divided by dx is called the derivative of A, or more technically the derivative is whatever this ratio approaches as dx gets smaller and smaller. Although, I dive much more deeply into the idea of a derivative in the next video, but loosely speaking it’s a measure of how sensitive a function is to small changes in its input. You’ll see as the series goes on that there are many many ways that you can visualize a derivative depending on what function you’re looking at and how you think about tiny nudges to its output. And we care about derivatives because they help us solve problems, and in our little exploration here, we already have a slight glimpse of one way that they’re used. They are the key to solving integral questions, problems that require finding the area under a curve. Once you gain enough familiarity with computing derivatives, you’ll be able to look at a situation like this one where you don’t know what a function is but you do know that its derivative should be x squared and from that reverse engineer what the function must be. And this back and forth between integrals and derivatives where the derivative of a function for the area under a graph gives you back the function defining the graph itself is called the “Fundamental theorem of calculus”. It ties together the two big ideas of integrals and derivatives, and it shows how, in some sense, each one is an inverse of the other. All of this is only a high-level view: just a peek at some of the core ideas that emerge in calculus, and what follows in the series are the details for derivatives and integrals and more. At all points I want you to feel that you could have invented calculus yourself. That if you drew the right pictures and played with each idea in just the right way, these formulas and rules and constructs that are presented could have just as easily popped out naturally from your own explorations, and before you go it would feel wrong not to give the people who supported this series on Patreon a well-deserved thanks both for their financial backing as well as for the suggestions they gave while the series was being developed. You see supporters got early access to the videos as I made them, and they’ll continue to get early access for future essence of type series and as a thanks to the community I keep ads off of new videos for their first month. I’m still astounded that I can spend time working on videos like these, and in a very direct way you are the one to thank for that.

Putting aside the brilliant content in your videos, the little pi character with its different expressions is so so cute! It adds so much to the explanations and the visual of your videos!

7:40

The revelation at 6'50" in this video is awesome. Thx so much for sharing.

How can i feel so relaxed and so enlightened at the same time?

as someone else put it, this channel is ART. it has a fantasy feel to it that even people who don't love maths can undestand. the thng i hated about my regular school and college maths was always the fact that it just gave solutions to the problems. never the answers. i always had to find the answers myself and being a procrastinator, i seldom did. but this channel, this is true mathematics.

AMAZING

Bealtilful

But how do you derive the circumference of circle 2(pie)r?? You are using that here without showing how it was born??

You are the best. I love calculus and your videos made me understand the 'why' in calculus, so that formulas now have a meaning instead of just being sry, static formulas to memorize. Higher mathematics has such an intrinsic beauty! And your videos help people see it. Thank you.

Thank you lord blue

That's fascinating. You are a kind of genius. We need more videos like that.

Starting calculus II this coming semester, but it's been three years since my Calculus I class. Always love your videos, so I'm gonna watch this series over the summer to prepare!

Thank you.

한국인없나?

i payed this at 0.75x speed. it makes it easier to take in…great content…

this is nice

I'm starting my first mathematics module at university in October and the thing I was dreading was calculus. I'm so happy you made these videos, they're really easy to watch and you lay it out so that even an easily distracted idiot like myself can follow. Great job!

وأخيرا، وجدت قناة تشرح الرياضيات كما أتخيلها تماما.

ربط الأرقام بالواقع، واستعمال المخيلة.

شكرا جزيلا.

that's great, thank you…

from Algeria

Wow this video watches like entertainment but I actually learned something. Wish there were videos like this back when I was in school. We only had terrible notepad tutorials, and unregisteredhypercam2 with loud techno music.

Thank you

If you've watched this for a class there is an inherent error in the method which should be considered. Using straight or curved lines with a defined dR will provide inexact results. The trapezium rule is a technique for approximating definite integrals, there are other averaging methods that can get closer using "mathematical nudging" to adjust for dr. The problem with the trapezoidal rule, it will short an area when the curve is concave downwards, likewise, it will overestimate when the curve is concave up, optimally it may find exact areas when the curve is a straight, or when a function is linear.

Three rules to google for:

Trapezoid Rule (the average of the left hand and right hand)

Riemann Sums (more accurate approximations), and Simpson's Rule.

3b1b pointed approximation out at 3:44 but it seems that isn't getting a lot of traction after reading some of the comments. Cheers!

This stuff use to look so alien to me but I’ve been focusing on math now for a month and this stuff just clicks when you see it now.

The basic concept of calculus is any function can be approximated to a line segment in the vicinity of any value of x. This holds true provided the departure from x is very small. For the thus produced line segment the gradient can be found easily by evaluating the change in y devided by the change in x. The gradient function that is derived in this way is known as the derivative function.

Integration, on the other hand, is the reverse process of differentiation. This means, from a given derivative function, the original undifferentiated function is found. Some functions have no analytical integral.

The area under a curve in two dimentions is an example evaluation of integration. An integral function is evaluated for a given interval. This kind of integration often results in a plain numerical answer, provided the original function does not have parameters.

A yet another area of calculus is differential calculus. This deals with the expression and solution of equations involving differentials like dY/dx and higher order differentials.

Your videos are genius

man you are awesome, this is the way that i always want it to learn math, like understand everything that im doing on paper and not just "learn the formula", math builds upon itself, and it is core you should be able to intuitively get to the result by thinking a little a bit, and by having "magical formulas" that do the work for you, that is just impossible

Nice video. Nice to see you did proper preparation for class.

Just try to write straightly, it is not that difficult.

¿+? = 🙂

Los subtítulos en español están bastante mal hechos,mejor no los useis porque causan confusión debido a la mala traducción,con un nivel medio de inglés se entienden sin problema los subtítulos en eng.

Indonesia sub please

I dont understand english too much🙏

@3blue1brown

How come the width of the small rectangle formed by breaking the circle into many concentric circles is 2*pi*r?

at 9.21, example of car and distance travelled by it in a given time, how does the distance depend on the thickness of the path!! Just wondering. I mean technically, it needs a relook as the example begins.

JFC!…. SLOW DOWN, I CAN'T ABSORB, AND ASSIMILATE ALL OF THESE IDEAS, WITHOUT TAKING TIME TO THINK EACH ONE THRU, CAREFULLY!!! THIS GETS TO BE LIKE DRINKING FROM A FIRE HOSE!!!! WTF??!!!…..[GREAT GRAPHICS!!!!] THANKS!!!

Bruh dude took me like an extra 30 minutes of just thinking before I understood the first part with the circle and I wasn't even half way through the video

But how to draw the line ? Can you put the point exactly on 3.14……………………………………………………………… or any multiplied value of it ? How do you multiply number which does not have end ? Precisely not again approximately ?

I hate mathematics in high school 20 years ago and my scores are bad, but now I recognised mathematics is a beautiful game, and now I’m here to learn calculus

Lucky to me and this generation, we can learn something new by ourselves on YouTube

I loved math back in my HS days I said to myself "I will do math in college."

But when I got to college, I was sucked into computer programming.

Looking back, Math is relevant (either as 15 or 50), but my programming (Pascal, Basic, Cobol, Lisp, ASM….) were deemed obsolete immediately after college (there is always a new programming language out there after mastering one)….

I lost him at dr

in 2:50 when you cut out one of the concentric rings why does it turn out to be a trapezoid??(to me it seems to be a rectangle)

the area works out ok on a unit circle radius 1 , area = pi.. but then don't u have a hypotenuse the same as a leg.. so radius would be zero.. area of a straight line is zero..

i lost it when you went from circle to straight lines, I-cant understand wgat follows if you say ignore this part.

Honesytly someone explain pls?

When I 13 years old I actually had a maths teacher draw this on a blackboard good on you Br. Donald. Seeing this video I cannot understand why all maths is not taught like this now?

Very repetitive.

Binge-watching calculus. :thumbsup:

The essence of topology!!!

Your presentation style is exceptional. Thanks for making these gorgeous works on math.

omg, this explanation is so good. I needed this 10 years back.

Then what do you do when u have an uneven curve? The equation is not given.. So how to find out the equation for any given curve regardless of the shape of it?

5:04 actually is the area of the ring, which coincides whit the perimeter.

Excellent job man!

Yo notch supported him on patreon?

Would you kind enough to inform me from where you get the name 3 blue 1brown

Thanks in advance

The essence of calculus is to avoid this intuitive chopping and approximation and to inject precision. These approaches are unrigourous and often lead to fallacious results.The essence of calculus is mastering the epsilon-delta definition.

I came here after waiting a video about integral on Crush Corse, I’m telling you that this video is AMAZING!!!

Tq sir it is very helpful I like ur animations 😻 from karnataka

7:15 it's not precise. The presence of pi prevents it. The value of any circumference perimeter cannot be known. It can be approximated to one's needs, but never known for sure. A circumference does not have the property of length. At least, it's not numerically representable.

What about a video on the area of a rectangle?

I'm sorry, but can we please punch some numbers in these videos, I'm freaking out because I'm not sure if I got this correctly. So if I have a wave or any shape on a graph an I want to find (dA) I take the Height = x^2 x dx, right so If let's say my Height in y was 3, that would mean 3^2 which = 9, now my chosen value for dx is 0.25 along the x, when I input the values into the formula I get 2.25? what does this number constitute to? it can't be a square because it has no x-, & y-coordinate? Please can anyone help?

Hellooo, anyone? please help me understand : (

Love this video!!!! Sooooo wish I had it 20 years ago!!! Goes to show that MOST teachers have no clue.

I believe the slope is 2π.

Slope=rise/run

Slope=2πr/r

Slope=2π

In my school age I hate maths because for me any thing I want to learn with meaning and concept but unlucky , my maths teachers did teach in this form and now I love maths.thank u

After this I gained 3000 iq

Es un honor pertenecer a este canal. Realmente estoy agradecido por el desarrollo de los temas de cálculo. Espero mi suscripción sea necesario para mostrar mi gratitud por lo aprendido

Circles are actually triangles confirmed

isaac newton invented calculus on a whim and anyone who doesn't understand calculus should feel bad

You're my idol. I thought calculus was all about bunch of equations and rules. It's what I'm thought.

Hello sir I think it's good if u make a probability series.

I could not imagine a better way for learning and loving Math <3 Thank u So much for that!

Thank you for this series

Would you suggest to watch this series before learning schooly calculus or after learning schooly maths?

Simply amazing explanation

🐸🙏thank you so much👍

Very well explained man

OMG I FOUND IT! I FOUND IT! HURRAY! FINALLY!

Beautiful.

15:20

Omgggggg this was dope as fuccccccc*

****your first minute SUCKS, way to overwhelm people and make then totally tune you out ahead of key points

why do so many math teachers suck ass?

As an Engineer, i wished this kind of education and explanations existed when i was in college.

Today's generation is so lucky, they have almost limitless access to free education, free information, free teaching etc.

The most CLASSIC prestigious essence of explanation a mind can ever have

"Whenever you come across a genuinely hard question in math, a good policy is to not try too hard to get at the answer directly since usually just end up banging your head against a wall. Instead play around with the idea."

I wish someone had said this to me in college… I would have had a much easier time with calculus.

Ur channel is beyond amazing. Love from China ❤❤

난 이미 미분을 했었다

Thanks for the class. It is amazing

This is stunningly beautiful. You are a beautiful human being.

Now I understand what I learned 61 years ago. Thank you.

This guy is good but his voice is extraordinarily irritating

And this is exactly why I waited on going to college. Instead of struggling in a calc class to understand and possibly failing I can watch this for free and actually understand the concept to better apply it

Thanks for this insightful lecture. Now I am beginning to understand and love Calculus.

I'm a starter at math I only know basic math, any tips/advice or anything?

Can anyone explain why hypotenuse = 2*pi*r ? At 8:16

thanks!

I was born dumb. Math, the perfect language, is the 8th wonder of the world.

so beautiful

❤❤❤love and respect

@3Blue1Brown: Thank you for this.

This is the third time I watched this video.I don't understand a thing about calculus in highschool and I never thought I would ever be able to. And then I found this channel! This is the best visualization I've ever seen in my life. I get to understand more and more about the concept each time I watch and all of this just for free! Just want to give a big thanks to you. Your content is ridiculously good.

I hate math, dafuq am I doing here

I just finished watching this video, and wow. It's the most entertaining math video I have ever watched. I've never been a math person, and I never really enjoyed learning about the beauty and intricacies of math. This video, however, has changed me. I have never before felt this interested in learning math. This video is so well made, and I'm so glad I was able to see it. You have done a truly amazing job, and have managed to do what I thought was impossible, getting me interested in math. I sincerely thank you for making this video.

Very helpful for exam prepare. Thank you for your help from Al Habib ahmed Manipur 🇮🇳India

Thank you!