WEBVTT 00:00:14.980 --> 00:00:16.460 Hey everyone, Grant here. 00:00:16.820 --> 00:00:19.827 This is the first video in a series on the essence of calculus, 00:00:19.827 --> 00:00:23.599 and I'll be publishing the following videos once per day for the next 10 days. 00:00:24.300 --> 00:00:26.911 The goal here, as the name suggests, is to really get 00:00:26.911 --> 00:00:29.720 the heart of the subject out in one binge-watchable set. 00:00:30.320 --> 00:00:34.179 But with a topic that's as broad as calculus, there's a lot of things that can mean, 00:00:34.179 --> 00:00:36.200 so here's what I have in mind specifically. 00:00:36.939 --> 00:00:39.412 Calculus has a lot of rules and formulas which 00:00:39.412 --> 00:00:41.939 are often presented as things to be memorized. 00:00:42.479 --> 00:00:45.753 Lots of derivative formulas, the product rule, the chain rule, 00:00:45.753 --> 00:00:49.977 implicit differentiation, the fact that integrals and derivatives are opposite, 00:00:49.978 --> 00:00:52.460 Taylor series, just a lot of things like that. 00:00:52.960 --> 00:00:55.133 And my goal is for you to come away feeling like 00:00:55.133 --> 00:00:57.080 you could have invented calculus yourself. 00:00:57.640 --> 00:01:01.637 That is, cover all those core ideas, but in a way that makes clear where they 00:01:01.637 --> 00:01:06.000 actually come from, and what they really mean, using an all-around visual approach. 00:01:06.920 --> 00:01:10.427 Inventing math is no joke, and there is a difference between being 00:01:10.427 --> 00:01:14.040 told why something's true, and actually generating it from scratch. 00:01:14.680 --> 00:01:19.091 But at all points, I want you to think to yourself, if you were an early mathematician, 00:01:19.090 --> 00:01:22.031 pondering these ideas and drawing out the right diagrams, 00:01:22.031 --> 00:01:26.239 does it feel reasonable that you could have stumbled across these truths yourself? 00:01:26.819 --> 00:01:31.568 In this initial video, I want to show how you might stumble into the core ideas of 00:01:31.569 --> 00:01:35.566 calculus by thinking very deeply about one specific bit of geometry, 00:01:35.566 --> 00:01:36.840 the area of a circle. 00:01:37.780 --> 00:01:41.040 Maybe you know that this is pi times its radius squared, but why? 00:01:41.579 --> 00:01:44.459 Is there a nice way to think about where this formula comes from? 00:01:45.420 --> 00:01:49.326 Well, contemplating this problem and leaving yourself open to exploring the 00:01:49.325 --> 00:01:53.544 interesting thoughts that come about can actually lead you to a glimpse of three 00:01:53.545 --> 00:01:57.920 big ideas in calculus, integrals, derivatives, and the fact that they're opposites. 00:01:59.840 --> 00:02:04.840 But the story starts more simply, just you and a circle, let's say with radius 3. 00:02:05.700 --> 00:02:09.382 You're trying to figure out its area, and after going through a lot of 00:02:09.382 --> 00:02:13.485 paper trying different ways to chop up and rearrange the pieces of that area, 00:02:13.485 --> 00:02:16.852 many of which might lead to their own interesting observations, 00:02:16.852 --> 00:02:21.060 maybe you try out the idea of slicing up the circle into many concentric rings. 00:02:22.000 --> 00:02:25.806 This should seem promising because it respects the symmetry of the circle, 00:02:25.806 --> 00:02:29.459 and math has a tendency to reward you when you respect its symmetries. 00:02:30.360 --> 00:02:35.060 Let's take one of those rings, which has some inner radius r that's between 0 and 3. 00:02:36.219 --> 00:02:39.921 If we can find a nice expression for the area of each ring like this one, 00:02:39.921 --> 00:02:42.254 and if we have a nice way to add them all up, 00:02:42.254 --> 00:02:45.500 it might lead us to an understanding of the full circle's area. 00:02:46.419 --> 00:02:49.119 Maybe you start by imagining straightening out this ring. 00:02:50.800 --> 00:02:54.990 And you could try thinking through exactly what this new shape is and what its 00:02:54.990 --> 00:02:59.180 area should be, but for simplicity, let's just approximate it as a rectangle. 00:03:00.180 --> 00:03:03.861 The width of that rectangle is the circumference of the original ring, 00:03:03.861 --> 00:03:05.439 which is 2 pi times r, right? 00:03:05.860 --> 00:03:08.060 I mean, that's essentially the definition of pi. 00:03:08.680 --> 00:03:09.379 And its thickness? 00:03:10.199 --> 00:03:14.150 Well, that depends on how finely you chopped up the circle in the first place, 00:03:14.151 --> 00:03:15.620 which was kind of arbitrary. 00:03:16.340 --> 00:03:20.094 In the spirit of using what will come to be standard calculus notation, 00:03:20.094 --> 00:03:24.431 let's call that thickness dr for a tiny difference in the radius from one ring to 00:03:24.431 --> 00:03:24.960 the next. 00:03:25.479 --> 00:03:27.879 Maybe you think of it as something like 0.1. 00:03:28.979 --> 00:03:32.848 So approximating this unwrapped ring as a thin rectangle, 00:03:32.848 --> 00:03:37.599 its area is 2 pi times r, the radius, times dr, the little thickness. 00:03:38.599 --> 00:03:42.436 And even though that's not perfect, for smaller and smaller choices of dr, 00:03:42.437 --> 00:03:46.481 this is actually going to be a better and better approximation for that area, 00:03:46.480 --> 00:03:50.992 since the top and the bottom sides of this shape are going to get closer and closer to 00:03:50.992 --> 00:03:52.599 being exactly the same length. 00:03:53.539 --> 00:03:55.932 So let's just move forward with this approximation, 00:03:55.932 --> 00:03:58.700 keeping in the back of our minds that it's slightly wrong, 00:03:58.700 --> 00:04:02.359 but it's going to become more accurate for smaller and smaller choices of dr. 00:04:03.219 --> 00:04:06.400 That is, if we slice up the circle into thinner and thinner rings. 00:04:07.699 --> 00:04:12.382 So just to sum up where we are, you've broken up the area of the circle into 00:04:12.383 --> 00:04:17.189 all of these rings, and you're approximating the area of each one of those as 00:04:17.189 --> 00:04:22.057 2 pi times its radius times dr, where the specific value for that inner radius 00:04:22.057 --> 00:04:26.803 ranges from 0 for the smallest ring up to just under 3 for the biggest ring, 00:04:26.803 --> 00:04:31.979 spaced out by whatever the thickness is that you choose for dr, something like 0.1. 00:04:33.139 --> 00:04:36.973 And notice that the spacing between the values here corresponds to the 00:04:36.973 --> 00:04:41.300 thickness dr of each ring, the difference in radius from one ring to the next. 00:04:42.259 --> 00:04:46.012 In fact, a nice way to think about the rectangles approximating each 00:04:46.012 --> 00:04:49.819 ring's area is to fit them all upright side by side along this axis. 00:04:50.660 --> 00:04:55.820 Each one has a thickness dr, which is why they fit so snugly right there together, 00:04:55.819 --> 00:05:01.230 and the height of any one of these rectangles sitting above some specific value of r, 00:05:01.230 --> 00:05:04.000 like 0.6, is exactly 2 pi times that value. 00:05:04.639 --> 00:05:08.959 That's the circumference of the corresponding ring that this rectangle approximates. 00:05:09.560 --> 00:05:12.548 Pictures like this 2 pi r can get tall for the screen, 00:05:12.548 --> 00:05:16.921 I mean 2 times pi times 3 is around 19, so let's just throw up a y axis that's 00:05:16.922 --> 00:05:21.350 scaled a little differently so that we can actually fit all of these rectangles 00:05:21.350 --> 00:05:22.180 on the screen. 00:05:23.259 --> 00:05:26.940 A nice way to think about this setup is to draw the graph of 2 pi r, 00:05:26.940 --> 00:05:29.539 which is a straight line that has a slope 2 pi. 00:05:30.100 --> 00:05:34.800 Each of these rectangles extends up to the point where it just barely touches that graph. 00:05:36.000 --> 00:05:37.459 Again, we're being approximate here. 00:05:37.899 --> 00:05:40.013 Each of these rectangles only approximates the 00:05:40.014 --> 00:05:42.220 area of the corresponding ring from the circle. 00:05:42.939 --> 00:05:46.242 But remember, that approximation, 2 pi r times dr, 00:05:46.242 --> 00:05:50.800 gets less and less wrong as the size of dr gets smaller and smaller. 00:05:51.800 --> 00:05:53.966 And this has a very beautiful meaning when we're 00:05:53.966 --> 00:05:56.539 looking at the sum of the areas of all those rectangles. 00:05:57.079 --> 00:06:00.057 For smaller and smaller choices of dr, you might at first 00:06:00.057 --> 00:06:03.139 think that turns the problem into a monstrously large sum. 00:06:03.600 --> 00:06:05.560 I mean, there's many many rectangles to consider, 00:06:05.560 --> 00:06:08.399 and the decimal precision of each one of their areas is going to be an 00:06:08.399 --> 00:06:09.199 absolute nightmare. 00:06:10.060 --> 00:06:15.300 But notice, all of their areas in aggregate just looks like the area under a graph. 00:06:15.980 --> 00:06:19.365 And that portion under the graph is just a triangle, 00:06:19.365 --> 00:06:23.400 a triangle with a base of 3 and a height that's 2 pi times 3. 00:06:24.139 --> 00:06:30.500 So its area, 1 half base times height, works out to be exactly pi times 3 squared. 00:06:31.360 --> 00:06:35.134 Or if the radius of our original circle was some other value, 00:06:35.134 --> 00:06:38.660 capital R, that area comes out to be pi times r squared. 00:06:39.379 --> 00:06:41.459 And that's the formula for the area of a circle. 00:06:42.319 --> 00:06:45.411 It doesn't matter who you are or what you typically think of math, 00:06:45.411 --> 00:06:47.379 that right there is a beautiful argument. 00:06:50.180 --> 00:06:52.817 But if you want to think like a mathematician here, 00:06:52.817 --> 00:06:55.195 you don't just care about finding the answer, 00:06:55.196 --> 00:06:58.920 you care about developing general problem-solving tools and techniques. 00:06:59.680 --> 00:07:03.766 So take a moment to meditate on what exactly just happened and why it worked, 00:07:03.766 --> 00:07:07.587 because the way we transitioned from something approximate to something 00:07:07.586 --> 00:07:11.779 precise is actually pretty subtle and cuts deep to what calculus is all about. 00:07:13.819 --> 00:07:18.908 You had this problem that could be approximated with the sum of many small numbers, 00:07:18.908 --> 00:07:24.060 each of which looked like 2 pi r times dr, for values of r ranging between 0 and 3. 00:07:26.600 --> 00:07:31.947 Remember, the small number dr here represents our choice for the thickness of each ring, 00:07:31.947 --> 00:07:32.980 for example 0.1. 00:07:33.519 --> 00:07:35.639 And there are two important things to note here. 00:07:36.079 --> 00:07:40.639 First of all, not only is dr a factor in the quantities we're adding up, 00:07:40.639 --> 00:07:45.579 2 pi r times dr, it also gives the spacing between the different values of r. 00:07:46.240 --> 00:07:50.519 And secondly, the smaller our choice for dr, the better the approximation. 00:07:52.199 --> 00:07:55.202 Adding all of those numbers could be seen in a different, 00:07:55.202 --> 00:07:58.468 pretty clever way as adding the areas of many thin rectangles 00:07:58.468 --> 00:08:02.420 sitting underneath a graph, the graph of the function 2 pi r in this case. 00:08:02.939 --> 00:08:07.425 Then, and this is key, by considering smaller and smaller choices for dr, 00:08:07.425 --> 00:08:12.587 corresponding to better and better approximations of the original problem, the sum, 00:08:12.588 --> 00:08:15.906 thought of as the aggregate area of those rectangles, 00:08:15.906 --> 00:08:18.180 approaches the area under the graph. 00:08:19.000 --> 00:08:23.476 And because of that, you can conclude that the answer to the original question, 00:08:23.476 --> 00:08:28.519 in full unapproximated precision, is exactly the same as the area underneath this graph. 00:08:30.860 --> 00:08:35.260 A lot of other hard problems in math and science can be broken down and 00:08:35.260 --> 00:08:38.360 approximated as the sum of many small quantities, 00:08:38.360 --> 00:08:43.939 like figuring out how far a car has traveled based on its velocity at each point in time. 00:08:44.759 --> 00:08:48.769 In a case like that, you might range through many different points in time, 00:08:48.769 --> 00:08:53.260 and at each one multiply the velocity at that time times a tiny change in time, dt, 00:08:53.260 --> 00:08:57.858 which would give the corresponding little bit of distance traveled during that little 00:08:57.859 --> 00:08:58.180 time. 00:08:59.259 --> 00:09:03.155 I'll talk through the details of examples like this later in the series, 00:09:03.155 --> 00:09:07.377 but at a high level many of these types of problems turn out to be equivalent 00:09:07.378 --> 00:09:12.139 to finding the area under some graph, in much the same way that our circle problem did. 00:09:13.200 --> 00:09:16.147 This happens whenever the quantities you're adding up, 00:09:16.147 --> 00:09:19.038 the one whose sum approximates the original problem, 00:09:19.038 --> 00:09:23.240 can be thought of as the areas of many thin rectangles sitting side by side. 00:09:24.639 --> 00:09:29.779 If finer and finer approximations of the original problem correspond to thinner and 00:09:29.779 --> 00:09:35.105 thinner rings, then the original problem is equivalent to finding the area under some 00:09:35.105 --> 00:09:35.539 graph. 00:09:36.600 --> 00:09:40.423 Again, this is an idea we'll see in more detail later in the series, 00:09:40.423 --> 00:09:43.179 so don't worry if it's not 100% clear right now. 00:09:43.779 --> 00:09:47.150 The point now is that you, as the mathematician having just 00:09:47.150 --> 00:09:50.577 solved a problem by reframing it as the area under a graph, 00:09:50.577 --> 00:09:54.519 might start thinking about how to find the areas under other graphs. 00:09:55.639 --> 00:10:00.317 We were lucky in the circle problem that the relevant area turned out to be a triangle, 00:10:00.317 --> 00:10:03.759 but imagine instead something like a parabola, the graph of x2. 00:10:04.759 --> 00:10:08.013 What's the area underneath that curve, say between 00:10:08.013 --> 00:10:10.680 the values of x equals 0 and x equals 3? 00:10:12.080 --> 00:10:14.759 Well, it's hard to think about, right? 00:10:15.220 --> 00:10:18.019 And let me reframe that question in a slightly different way. 00:10:18.019 --> 00:10:23.059 We'll fix that left endpoint in place at 0, and let the right endpoint vary. 00:10:26.860 --> 00:10:30.556 Are you able to find a function, a of x, that gives 00:10:30.556 --> 00:10:34.180 you the area under this parabola between 0 and x? 00:10:35.620 --> 00:10:39.580 A function a of x like this is called an integral of x2. 00:10:40.500 --> 00:10:44.602 Calculus holds within it the tools to figure out what an integral like this is, 00:10:44.602 --> 00:10:47.199 but right now it's just a mystery function to us. 00:10:47.500 --> 00:10:51.317 We know it gives the area under the graph of x2 between some fixed left 00:10:51.317 --> 00:10:54.919 point and some variable right point, but we don't know what it is. 00:10:55.659 --> 00:10:59.848 And again, the reason we care about this kind of question is not just for 00:10:59.849 --> 00:11:03.980 the sake of asking hard geometry questions, it's because many practical 00:11:03.980 --> 00:11:08.053 problems that can be approximated by adding up a large number of small 00:11:08.053 --> 00:11:12.299 things can be reframed as a question about an area under a certain graph. 00:11:13.419 --> 00:11:17.371 I'll tell you right now that finding this area, this integral function, 00:11:17.371 --> 00:11:21.991 is genuinely hard, and whenever you come across a genuinely hard question in math, 00:11:21.991 --> 00:11:25.776 a good policy is to not try too hard to get at the answer directly, 00:11:25.777 --> 00:11:29.340 since usually you just end up banging your head against a wall. 00:11:30.080 --> 00:11:33.780 Instead, play around with the idea, with no particular goal in mind. 00:11:34.340 --> 00:11:38.564 Spend some time building up familiarity with the interplay between the function 00:11:38.563 --> 00:11:42.360 defining the graph, in this case x2, and the function giving the area. 00:11:44.090 --> 00:11:48.019 In that playful spirit, if you're lucky, here's something you might notice. 00:11:48.580 --> 00:11:54.778 When you slightly increase x by some tiny nudge dx, look at the resulting change in area, 00:11:54.778 --> 00:12:00.419 represented with this sliver I'm going to call da for a tiny difference in area. 00:12:01.379 --> 00:12:05.506 That sliver can be pretty well approximated with a rectangle, 00:12:05.506 --> 00:12:08.620 one whose height is x2 and whose width is dx. 00:12:09.659 --> 00:12:12.227 And the smaller the size of that nudge dx, the 00:12:12.227 --> 00:12:15.019 more that sliver actually looks like a rectangle. 00:12:16.799 --> 00:12:21.079 This gives us an interesting way to think about how a of x is related to x2. 00:12:22.000 --> 00:12:26.577 A change to the output of a, this little da, is about equal to x2, 00:12:26.577 --> 00:12:30.184 where x is whatever input you started at, times dx, 00:12:30.184 --> 00:12:34.000 the little nudge to the input that caused a to change. 00:12:34.779 --> 00:12:40.315 Or rearranged, da divided by dx, the ratio of a tiny change in a to the tiny 00:12:40.316 --> 00:12:45.780 change in x that caused it, is approximately whatever x2 is at that point. 00:12:46.559 --> 00:12:48.759 And that's an approximation that should get better 00:12:48.759 --> 00:12:50.960 and better for smaller and smaller choices of dx. 00:12:52.100 --> 00:12:55.639 In other words, we don't know what a of x is, that remains a mystery. 00:12:56.080 --> 00:12:59.500 But we do know a property that this mystery function must have. 00:13:00.159 --> 00:13:04.971 When you look at two nearby points, for example 3 and 3.001, 00:13:04.971 --> 00:13:10.184 consider the change to the output of a between those two points, 00:13:10.184 --> 00:13:16.119 the difference between the mystery function evaluated at 3.001 and 3.001. 00:13:16.120 --> 00:13:22.076 That change, divided by the difference in the input values, which in this case is 0.001, 00:13:22.076 --> 00:13:28.100 should be about equal to the value of x2 for the starting input, in this case 3 squared. 00:13:30.200 --> 00:13:34.416 And this relationship between tiny changes to the mystery function 00:13:34.416 --> 00:13:38.440 and the values of x2 itself is true at all inputs, not just 3. 00:13:39.419 --> 00:13:41.972 That doesn't immediately tell us how to find a of x, 00:13:41.972 --> 00:13:44.819 but it provides a very strong clue that we can work with. 00:13:46.259 --> 00:13:48.740 And there's nothing special about the graph x2 here. 00:13:49.279 --> 00:13:53.646 Any function defined as the area under some graph has this property, 00:13:53.647 --> 00:13:58.592 that da divided by dx, a slight nudge to the output of a divided by a slight 00:13:58.591 --> 00:14:03.728 nudge to the input that caused it, is about equal to the height of the graph at 00:14:03.729 --> 00:14:04.500 that point. 00:14:06.200 --> 00:14:10.360 Again, that's an approximation that gets better and better for smaller choices of dx. 00:14:11.639 --> 00:14:16.039 And here, we're stumbling into another big idea from calculus, derivatives. 00:14:17.100 --> 00:14:22.106 This ratio da divided by dx is called the derivative of a, or more technically, 00:14:22.106 --> 00:14:27.240 the derivative is whatever this ratio approaches as dx gets smaller and smaller. 00:14:28.179 --> 00:14:32.037 I'll dive much more deeply into the idea of a derivative in the next video, 00:14:32.038 --> 00:14:36.514 but loosely speaking it's a measure of how sensitive a function is to small changes in 00:14:36.514 --> 00:14:37.080 its input. 00:14:37.940 --> 00:14:42.174 You'll see as the series goes on that there are many ways you can visualize a derivative, 00:14:42.173 --> 00:14:45.169 depending on what function you're looking at and how you think 00:14:45.169 --> 00:14:46.740 about tiny nudges to its output. 00:14:48.600 --> 00:14:52.153 We care about derivatives because they help us solve problems, 00:14:52.153 --> 00:14:57.139 and in our little exploration here, we already have a glimpse of one way they're used. 00:14:57.840 --> 00:15:00.437 They are the key to solving integral questions, 00:15:00.437 --> 00:15:03.419 problems that require finding the area under a curve. 00:15:04.360 --> 00:15:07.669 Once you gain enough familiarity with computing derivatives, 00:15:07.669 --> 00:15:12.580 you'll be able to look at a situation like this one where you don't know what a function 00:15:12.581 --> 00:15:15.560 is, but you do know that its derivative should be x2, 00:15:15.559 --> 00:15:18.759 and from that reverse engineer what the function must be. 00:15:20.700 --> 00:15:23.826 This back and forth between integrals and derivatives, 00:15:23.826 --> 00:15:27.994 where the derivative of a function for the area under a graph gives you 00:15:27.994 --> 00:15:32.104 back the function defining the graph itself, is called the fundamental 00:15:32.104 --> 00:15:33.320 theorem of calculus. 00:15:34.220 --> 00:15:38.750 It ties together the two big ideas of integrals and derivatives, 00:15:38.750 --> 00:15:42.360 and shows how each one is an inverse of the other. 00:15:44.799 --> 00:15:47.304 All of this is only a high-level view, just a peek 00:15:47.304 --> 00:15:49.859 at some of the core ideas that emerge in calculus. 00:15:50.500 --> 00:15:54.419 And what follows in this series are the details, for derivatives and integrals and more. 00:15:54.980 --> 00:15:59.105 At all points, I want you to feel that you could have invented calculus yourself, 00:15:59.105 --> 00:16:03.434 that if you drew the right pictures and played with each idea in just the right way, 00:16:03.434 --> 00:16:07.254 these formulas and rules and constructs that are presented could have just 00:16:07.254 --> 00:16:10.259 as easily popped out naturally from your own explorations. 00:16:12.379 --> 00:16:16.304 And before you go, it would feel wrong not to give the people who supported this 00:16:16.304 --> 00:16:20.130 series on Patreon a well-deserved thanks, both for their financial backing as 00:16:20.130 --> 00:16:23.860 well as for the suggestions they gave while the series was being developed. 00:16:24.700 --> 00:16:27.888 You see, supporters got early access to the videos as I made them, 00:16:27.888 --> 00:16:31.560 and they'll continue to get early access for future essence-of type series. 00:16:32.139 --> 00:16:36.240 And as a thanks to the community, I keep ads off of new videos for their first month. 00:16:37.019 --> 00:16:40.461 I'm still astounded that I can spend time working on videos like these, 00:16:40.461 --> 00:16:43.419 and in a very direct way, you are the one to thank for that.