The essence of calculus
3Blue1Brown
youtube· 17:00standard
1.Introduction to the Essence of Calculus
0:00 / 1:46This video series aims to demystify calculus by exploring its core ideas visually, making them intuitive and understandable. The goal is to help viewers grasp the origins and meanings of fundamental calculus concepts, rather than just memorizing formulas.
- Calculus
- Visual Approach
- Core Ideas
What's inside this course
- 0:00
1. Introduction to the Essence of Calculus
This video series aims to demystify calculus by exploring its core ideas visually, making them intuitive and understandable. The goal is to help viewers grasp the origins and meanings of fundamental calculus concepts, rather than just memorizing formulas.
- 1:46
2. Area of a Circle: A Calculus Starting Point
The video begins by using the familiar problem of finding the area of a circle (πr²) to introduce core calculus concepts. By deeply contemplating this geometric problem, one can stumble upon the ideas of integrals, derivatives, and their inverse relationship.
- 3:18
3. Slicing a Circle into Concentric Rings
To find the area of a circle, the video proposes slicing it into many concentric rings. By approximating each ring as a thin rectangle with width 2πr (circumference) and thickness dr (a tiny difference in radius), we can calculate its approximate area.
- 5:08
4. Approximating Ring Area with Rectangles
Each unwrapped ring is approximated as a rectangle with area 2πr * dr. This approximation becomes more accurate as dr (the thickness of the ring) gets smaller. This method allows us to sum the areas of these thin rectangles to estimate the total area.
- 6:47
5. Visualizing Ring Areas as a Graph
The areas of the approximated rings (2πr * dr) can be visualized as thin rectangles stacked side-by-side under the graph of y = 2πr. Each rectangle has a thickness dr and a height corresponding to the circumference at that radius. This forms a triangular shape.
- 8:57
6. Deriving the Circle Area Formula
As dr approaches zero, the sum of the areas of the rectangles under the graph of 2πr becomes exactly the area of the triangle formed. This triangle has a base of R (the circle's radius) and a height of 2πR, leading directly to the formula πR² for the area of a circle.
- 10:40
7. The Essence of Integration
The process of summing many small, approximate quantities (like the ring areas) to find a precise total is the essence of integration. This technique is applicable to many problems in science and math, such as calculating distance from velocity, by reframing them as finding the area under a graph.
- 12:09
8. Introducing the Integral Function
The video then poses a new challenge: finding the area under a more complex curve, like x², between 0 and a variable x. This unknown function, which gives the area, is called an integral of x². This type of problem is common in math and science.
- 13:51
9. Discovering the Derivative
By considering how a tiny change in x (dx) affects the area under the curve (da), we find that the ratio da/dx is approximately equal to the height of the graph at that point (x²). This ratio, as dx approaches zero, is called the derivative, a measure of a function's sensitivity to input changes.
- 15:50
10. The Fundamental Theorem of Calculus
Derivatives are crucial for solving integral problems. The relationship where the derivative of an area function gives back the original function is known as the Fundamental Theorem of Calculus. This theorem establishes that integrals and derivatives are inverse operations, connecting these two major calculus ideas.
Every chapter ends with a checkpoint (quiz, flashcards, retell, diagram, or prediction) and the course closes with a final boss-fight. More courses →
Full transcript of “The essence of calculus”
— 243 segments
0:15Hey everyone, Grant here.
0:17This is the first video in a series on the essence of calculus,
0:20and I'll be publishing the following videos once per day for the next 10 days.
0:24The goal here, as the name suggests, is to really get
0:27the heart of the subject out in one binge-watchable set.
0:30But with a topic that's as broad as calculus, there's a lot of things that can mean,
0:34so here's what I have in mind specifically.
0:37Calculus has a lot of rules and formulas which
0:39are often presented as things to be memorized.
0:42Lots of derivative formulas, the product rule, the chain rule,
0:46implicit differentiation, the fact that integrals and derivatives are opposite,
0:50Taylor series, just a lot of things like that.
0:53And my goal is for you to come away feeling like
0:55you could have invented calculus yourself.
0:58That is, cover all those core ideas, but in a way that makes clear where they
1:02actually come from, and what they really mean, using an all-around visual approach.
1:07Inventing math is no joke, and there is a difference between being
1:10told why something's true, and actually generating it from scratch.
1:15But at all points, I want you to think to yourself, if you were an early mathematician,
1:19pondering these ideas and drawing out the right diagrams,
1:22does it feel reasonable that you could have stumbled across these truths yourself?
1:27In this initial video, I want to show how you might stumble into the core ideas of
1:32calculus by thinking very deeply about one specific bit of geometry,
1:36the area of a circle.
1:38Maybe you know that this is pi times its radius squared, but why?
1:42Is there a nice way to think about where this formula comes from?
1:45Well, contemplating this problem and leaving yourself open to exploring the
1:49interesting thoughts that come about can actually lead you to a glimpse of three
1:54big ideas in calculus, integrals, derivatives, and the fact that they're opposites.
2:00But the story starts more simply, just you and a circle, let's say with radius 3.
2:06You're trying to figure out its area, and after going through a lot of
2:09paper trying different ways to chop up and rearrange the pieces of that area,
2:13many of which might lead to their own interesting observations,
2:17maybe you try out the idea of slicing up the circle into many concentric rings.
2:22This should seem promising because it respects the symmetry of the circle,
2:26and math has a tendency to reward you when you respect its symmetries.
2:30Let's take one of those rings, which has some inner radius r that's between 0 and 3.
2:36If we can find a nice expression for the area of each ring like this one,
2:40and if we have a nice way to add them all up,
2:42it might lead us to an understanding of the full circle's area.
2:46Maybe you start by imagining straightening out this ring.
2:51And you could try thinking through exactly what this new shape is and what its
2:55area should be, but for simplicity, let's just approximate it as a rectangle.
3:00The width of that rectangle is the circumference of the original ring,
3:04which is 2 pi times r, right?
3:06I mean, that's essentially the definition of pi.
3:09And its thickness?
3:10Well, that depends on how finely you chopped up the circle in the first place,
3:14which was kind of arbitrary.
3:16In the spirit of using what will come to be standard calculus notation,
3:20let's call that thickness dr for a tiny difference in the radius from one ring to
3:24the next.
3:25Maybe you think of it as something like 0.1.
3:29So approximating this unwrapped ring as a thin rectangle,
3:33its area is 2 pi times r, the radius, times dr, the little thickness.
3:39And even though that's not perfect, for smaller and smaller choices of dr,
3:42this is actually going to be a better and better approximation for that area,
3:46since the top and the bottom sides of this shape are going to get closer and closer to
3:51being exactly the same length.
3:54So let's just move forward with this approximation,
3:56keeping in the back of our minds that it's slightly wrong,
3:59but it's going to become more accurate for smaller and smaller choices of dr.
4:03That is, if we slice up the circle into thinner and thinner rings.
4:08So just to sum up where we are, you've broken up the area of the circle into
4:12all of these rings, and you're approximating the area of each one of those as
4:172 pi times its radius times dr, where the specific value for that inner radius
4:22ranges from 0 for the smallest ring up to just under 3 for the biggest ring,
4:27spaced out by whatever the thickness is that you choose for dr, something like 0.1.
4:33And notice that the spacing between the values here corresponds to the
4:37thickness dr of each ring, the difference in radius from one ring to the next.
4:42In fact, a nice way to think about the rectangles approximating each
4:46ring's area is to fit them all upright side by side along this axis.
4:51Each one has a thickness dr, which is why they fit so snugly right there together,
4:56and the height of any one of these rectangles sitting above some specific value of r,
5:01like 0.6, is exactly 2 pi times that value.
5:05That's the circumference of the corresponding ring that this rectangle approximates.
5:10Pictures like this 2 pi r can get tall for the screen,
5:13I mean 2 times pi times 3 is around 19, so let's just throw up a y axis that's
5:17scaled a little differently so that we can actually fit all of these rectangles
5:21on the screen.
5:23A nice way to think about this setup is to draw the graph of 2 pi r,
5:27which is a straight line that has a slope 2 pi.
5:30Each of these rectangles extends up to the point where it just barely touches that graph.
5:36Again, we're being approximate here.
5:38Each of these rectangles only approximates the
5:40area of the corresponding ring from the circle.
5:43But remember, that approximation, 2 pi r times dr,
5:46gets less and less wrong as the size of dr gets smaller and smaller.
5:52And this has a very beautiful meaning when we're
5:54looking at the sum of the areas of all those rectangles.
5:57For smaller and smaller choices of dr, you might at first
6:00think that turns the problem into a monstrously large sum.
6:04I mean, there's many many rectangles to consider,
6:06and the decimal precision of each one of their areas is going to be an
6:08absolute nightmare.
6:10But notice, all of their areas in aggregate just looks like the area under a graph.
6:16And that portion under the graph is just a triangle,
6:19a triangle with a base of 3 and a height that's 2 pi times 3.
6:24So its area, 1 half base times height, works out to be exactly pi times 3 squared.
6:31Or if the radius of our original circle was some other value,
6:35capital R, that area comes out to be pi times r squared.
6:39And that's the formula for the area of a circle.
6:42It doesn't matter who you are or what you typically think of math,
6:45that right there is a beautiful argument.
6:50But if you want to think like a mathematician here,
6:53you don't just care about finding the answer,
6:55you care about developing general problem-solving tools and techniques.
7:00So take a moment to meditate on what exactly just happened and why it worked,
7:04because the way we transitioned from something approximate to something
7:08precise is actually pretty subtle and cuts deep to what calculus is all about.
7:14You had this problem that could be approximated with the sum of many small numbers,
7:19each of which looked like 2 pi r times dr, for values of r ranging between 0 and 3.
7:27Remember, the small number dr here represents our choice for the thickness of each ring,
7:32for example 0.1.
7:34And there are two important things to note here.
7:36First of all, not only is dr a factor in the quantities we're adding up,
7:412 pi r times dr, it also gives the spacing between the different values of r.
7:46And secondly, the smaller our choice for dr, the better the approximation.
7:52Adding all of those numbers could be seen in a different,
7:55pretty clever way as adding the areas of many thin rectangles
7:58sitting underneath a graph, the graph of the function 2 pi r in this case.
8:03Then, and this is key, by considering smaller and smaller choices for dr,
8:07corresponding to better and better approximations of the original problem, the sum,
8:13thought of as the aggregate area of those rectangles,
8:16approaches the area under the graph.
8:19And because of that, you can conclude that the answer to the original question,
8:23in full unapproximated precision, is exactly the same as the area underneath this graph.
8:31A lot of other hard problems in math and science can be broken down and
8:35approximated as the sum of many small quantities,
8:38like figuring out how far a car has traveled based on its velocity at each point in time.
8:45In a case like that, you might range through many different points in time,
8:49and at each one multiply the velocity at that time times a tiny change in time, dt,
8:53which would give the corresponding little bit of distance traveled during that little
8:58time.
8:59I'll talk through the details of examples like this later in the series,
9:03but at a high level many of these types of problems turn out to be equivalent
9:07to finding the area under some graph, in much the same way that our circle problem did.
9:13This happens whenever the quantities you're adding up,
9:16the one whose sum approximates the original problem,
9:19can be thought of as the areas of many thin rectangles sitting side by side.
9:25If finer and finer approximations of the original problem correspond to thinner and
9:30thinner rings, then the original problem is equivalent to finding the area under some
9:35graph.
9:37Again, this is an idea we'll see in more detail later in the series,
9:40so don't worry if it's not 100% clear right now.
9:44The point now is that you, as the mathematician having just
9:47solved a problem by reframing it as the area under a graph,
9:51might start thinking about how to find the areas under other graphs.
9:56We were lucky in the circle problem that the relevant area turned out to be a triangle,
10:00but imagine instead something like a parabola, the graph of x2.
10:05What's the area underneath that curve, say between
10:08the values of x equals 0 and x equals 3?
10:12Well, it's hard to think about, right?
10:15And let me reframe that question in a slightly different way.
10:18We'll fix that left endpoint in place at 0, and let the right endpoint vary.
10:27Are you able to find a function, a of x, that gives
10:31you the area under this parabola between 0 and x?
10:36A function a of x like this is called an integral of x2.
10:41Calculus holds within it the tools to figure out what an integral like this is,
10:45but right now it's just a mystery function to us.
10:48We know it gives the area under the graph of x2 between some fixed left
10:51point and some variable right point, but we don't know what it is.
10:56And again, the reason we care about this kind of question is not just for
11:00the sake of asking hard geometry questions, it's because many practical
11:04problems that can be approximated by adding up a large number of small
11:08things can be reframed as a question about an area under a certain graph.
11:13I'll tell you right now that finding this area, this integral function,
11:17is genuinely hard, and whenever you come across a genuinely hard question in math,
11:22a good policy is to not try too hard to get at the answer directly,
11:26since usually you just end up banging your head against a wall.
11:30Instead, play around with the idea, with no particular goal in mind.
11:34Spend some time building up familiarity with the interplay between the function
11:39defining the graph, in this case x2, and the function giving the area.
11:44In that playful spirit, if you're lucky, here's something you might notice.
11:49When you slightly increase x by some tiny nudge dx, look at the resulting change in area,
11:55represented with this sliver I'm going to call da for a tiny difference in area.
12:01That sliver can be pretty well approximated with a rectangle,
12:06one whose height is x2 and whose width is dx.
12:10And the smaller the size of that nudge dx, the
12:12more that sliver actually looks like a rectangle.
12:17This gives us an interesting way to think about how a of x is related to x2.
12:22A change to the output of a, this little da, is about equal to x2,
12:27where x is whatever input you started at, times dx,
12:30the little nudge to the input that caused a to change.
12:35Or rearranged, da divided by dx, the ratio of a tiny change in a to the tiny
12:40change in x that caused it, is approximately whatever x2 is at that point.
12:47And that's an approximation that should get better
12:49and better for smaller and smaller choices of dx.
12:52In other words, we don't know what a of x is, that remains a mystery.
12:56But we do know a property that this mystery function must have.
13:00When you look at two nearby points, for example 3 and 3.001,
13:05consider the change to the output of a between those two points,
13:10the difference between the mystery function evaluated at 3.001 and 3.001.
13:16That change, divided by the difference in the input values, which in this case is 0.001,
13:22should be about equal to the value of x2 for the starting input, in this case 3 squared.
13:30And this relationship between tiny changes to the mystery function
13:34and the values of x2 itself is true at all inputs, not just 3.
13:39That doesn't immediately tell us how to find a of x,
13:42but it provides a very strong clue that we can work with.
13:46And there's nothing special about the graph x2 here.
13:49Any function defined as the area under some graph has this property,
13:54that da divided by dx, a slight nudge to the output of a divided by a slight
13:59nudge to the input that caused it, is about equal to the height of the graph at
14:04that point.
14:06Again, that's an approximation that gets better and better for smaller choices of dx.
14:12And here, we're stumbling into another big idea from calculus, derivatives.
14:17This ratio da divided by dx is called the derivative of a, or more technically,
14:22the derivative is whatever this ratio approaches as dx gets smaller and smaller.
14:28I'll dive much more deeply into the idea of a derivative in the next video,
14:32but loosely speaking it's a measure of how sensitive a function is to small changes in
14:37its input.
14:38You'll see as the series goes on that there are many ways you can visualize a derivative,
14:42depending on what function you're looking at and how you think
14:45about tiny nudges to its output.
14:49We care about derivatives because they help us solve problems,
14:52and in our little exploration here, we already have a glimpse of one way they're used.
14:58They are the key to solving integral questions,
15:00problems that require finding the area under a curve.
15:04Once you gain enough familiarity with computing derivatives,
15:08you'll be able to look at a situation like this one where you don't know what a function
15:13is, but you do know that its derivative should be x2,
15:16and from that reverse engineer what the function must be.
15:21This back and forth between integrals and derivatives,
15:24where the derivative of a function for the area under a graph gives you
15:28back the function defining the graph itself, is called the fundamental
15:32theorem of calculus.
15:34It ties together the two big ideas of integrals and derivatives,
15:39and shows how each one is an inverse of the other.
15:45All of this is only a high-level view, just a peek
15:47at some of the core ideas that emerge in calculus.
15:51And what follows in this series are the details, for derivatives and integrals and more.
15:55At all points, I want you to feel that you could have invented calculus yourself,
15:59that if you drew the right pictures and played with each idea in just the right way,
16:03these formulas and rules and constructs that are presented could have just
16:07as easily popped out naturally from your own explorations.
16:12And before you go, it would feel wrong not to give the people who supported this
16:16series on Patreon a well-deserved thanks, both for their financial backing as
16:20well as for the suggestions they gave while the series was being developed.
16:25You see, supporters got early access to the videos as I made them,
16:28and they'll continue to get early access for future essence-of type series.
16:32And as a thanks to the community, I keep ads off of new videos for their first month.
16:37I'm still astounded that I can spend time working on videos like these,
16:40and in a very direct way, you are the one to thank for that.