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