1 00:00:14,980 --> 00:00:16,460 Hey everyone, Grant here. 2 00:00:16,820 --> 00:00:19,827 This is the first video in a series on the essence of calculus, 3 00:00:19,827 --> 00:00:23,599 and I'll be publishing the following videos once per day for the next 10 days. 4 00:00:24,300 --> 00:00:26,911 The goal here, as the name suggests, is to really get 5 00:00:26,911 --> 00:00:29,720 the heart of the subject out in one binge-watchable set. 6 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, 7 00:00:34,179 --> 00:00:36,200 so here's what I have in mind specifically. 8 00:00:36,939 --> 00:00:39,412 Calculus has a lot of rules and formulas which 9 00:00:39,412 --> 00:00:41,939 are often presented as things to be memorized. 10 00:00:42,479 --> 00:00:45,753 Lots of derivative formulas, the product rule, the chain rule, 11 00:00:45,753 --> 00:00:49,977 implicit differentiation, the fact that integrals and derivatives are opposite, 12 00:00:49,978 --> 00:00:52,460 Taylor series, just a lot of things like that. 13 00:00:52,960 --> 00:00:55,133 And my goal is for you to come away feeling like 14 00:00:55,133 --> 00:00:57,080 you could have invented calculus yourself. 15 00:00:57,640 --> 00:01:01,637 That is, cover all those core ideas, but in a way that makes clear where they 16 00:01:01,637 --> 00:01:06,000 actually come from, and what they really mean, using an all-around visual approach. 17 00:01:06,920 --> 00:01:10,427 Inventing math is no joke, and there is a difference between being 18 00:01:10,427 --> 00:01:14,040 told why something's true, and actually generating it from scratch. 19 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, 20 00:01:19,090 --> 00:01:22,031 pondering these ideas and drawing out the right diagrams, 21 00:01:22,031 --> 00:01:26,239 does it feel reasonable that you could have stumbled across these truths yourself? 22 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 23 00:01:31,569 --> 00:01:35,566 calculus by thinking very deeply about one specific bit of geometry, 24 00:01:35,566 --> 00:01:36,840 the area of a circle. 25 00:01:37,780 --> 00:01:41,040 Maybe you know that this is pi times its radius squared, but why? 26 00:01:41,579 --> 00:01:44,459 Is there a nice way to think about where this formula comes from? 27 00:01:45,420 --> 00:01:49,326 Well, contemplating this problem and leaving yourself open to exploring the 28 00:01:49,325 --> 00:01:53,544 interesting thoughts that come about can actually lead you to a glimpse of three 29 00:01:53,545 --> 00:01:57,920 big ideas in calculus, integrals, derivatives, and the fact that they're opposites. 30 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. 31 00:02:05,700 --> 00:02:09,382 You're trying to figure out its area, and after going through a lot of 32 00:02:09,382 --> 00:02:13,485 paper trying different ways to chop up and rearrange the pieces of that area, 33 00:02:13,485 --> 00:02:16,852 many of which might lead to their own interesting observations, 34 00:02:16,852 --> 00:02:21,060 maybe you try out the idea of slicing up the circle into many concentric rings. 35 00:02:22,000 --> 00:02:25,806 This should seem promising because it respects the symmetry of the circle, 36 00:02:25,806 --> 00:02:29,459 and math has a tendency to reward you when you respect its symmetries. 37 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. 38 00:02:36,219 --> 00:02:39,921 If we can find a nice expression for the area of each ring like this one, 39 00:02:39,921 --> 00:02:42,254 and if we have a nice way to add them all up, 40 00:02:42,254 --> 00:02:45,500 it might lead us to an understanding of the full circle's area. 41 00:02:46,419 --> 00:02:49,119 Maybe you start by imagining straightening out this ring. 42 00:02:50,800 --> 00:02:54,990 And you could try thinking through exactly what this new shape is and what its 43 00:02:54,990 --> 00:02:59,180 area should be, but for simplicity, let's just approximate it as a rectangle. 44 00:03:00,180 --> 00:03:03,861 The width of that rectangle is the circumference of the original ring, 45 00:03:03,861 --> 00:03:05,439 which is 2 pi times r, right? 46 00:03:05,860 --> 00:03:08,060 I mean, that's essentially the definition of pi. 47 00:03:08,680 --> 00:03:09,379 And its thickness? 48 00:03:10,199 --> 00:03:14,150 Well, that depends on how finely you chopped up the circle in the first place, 49 00:03:14,151 --> 00:03:15,620 which was kind of arbitrary. 50 00:03:16,340 --> 00:03:20,094 In the spirit of using what will come to be standard calculus notation, 51 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 52 00:03:24,431 --> 00:03:24,960 the next. 53 00:03:25,479 --> 00:03:27,879 Maybe you think of it as something like 0.1. 54 00:03:28,979 --> 00:03:32,848 So approximating this unwrapped ring as a thin rectangle, 55 00:03:32,848 --> 00:03:37,599 its area is 2 pi times r, the radius, times dr, the little thickness. 56 00:03:38,599 --> 00:03:42,436 And even though that's not perfect, for smaller and smaller choices of dr, 57 00:03:42,437 --> 00:03:46,481 this is actually going to be a better and better approximation for that area, 58 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 59 00:03:50,992 --> 00:03:52,599 being exactly the same length. 60 00:03:53,539 --> 00:03:55,932 So let's just move forward with this approximation, 61 00:03:55,932 --> 00:03:58,700 keeping in the back of our minds that it's slightly wrong, 62 00:03:58,700 --> 00:04:02,359 but it's going to become more accurate for smaller and smaller choices of dr. 63 00:04:03,219 --> 00:04:06,400 That is, if we slice up the circle into thinner and thinner rings. 64 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 65 00:04:12,383 --> 00:04:17,189 all of these rings, and you're approximating the area of each one of those as 66 00:04:17,189 --> 00:04:22,057 2 pi times its radius times dr, where the specific value for that inner radius 67 00:04:22,057 --> 00:04:26,803 ranges from 0 for the smallest ring up to just under 3 for the biggest ring, 68 00:04:26,803 --> 00:04:31,979 spaced out by whatever the thickness is that you choose for dr, something like 0.1. 69 00:04:33,139 --> 00:04:36,973 And notice that the spacing between the values here corresponds to the 70 00:04:36,973 --> 00:04:41,300 thickness dr of each ring, the difference in radius from one ring to the next. 71 00:04:42,259 --> 00:04:46,012 In fact, a nice way to think about the rectangles approximating each 72 00:04:46,012 --> 00:04:49,819 ring's area is to fit them all upright side by side along this axis. 73 00:04:50,660 --> 00:04:55,820 Each one has a thickness dr, which is why they fit so snugly right there together, 74 00:04:55,819 --> 00:05:01,230 and the height of any one of these rectangles sitting above some specific value of r, 75 00:05:01,230 --> 00:05:04,000 like 0.6, is exactly 2 pi times that value. 76 00:05:04,639 --> 00:05:08,959 That's the circumference of the corresponding ring that this rectangle approximates. 77 00:05:09,560 --> 00:05:12,548 Pictures like this 2 pi r can get tall for the screen, 78 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 79 00:05:16,922 --> 00:05:21,350 scaled a little differently so that we can actually fit all of these rectangles 80 00:05:21,350 --> 00:05:22,180 on the screen. 81 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, 82 00:05:26,940 --> 00:05:29,539 which is a straight line that has a slope 2 pi. 83 00:05:30,100 --> 00:05:34,800 Each of these rectangles extends up to the point where it just barely touches that graph. 84 00:05:36,000 --> 00:05:37,459 Again, we're being approximate here. 85 00:05:37,899 --> 00:05:40,013 Each of these rectangles only approximates the 86 00:05:40,014 --> 00:05:42,220 area of the corresponding ring from the circle. 87 00:05:42,939 --> 00:05:46,242 But remember, that approximation, 2 pi r times dr, 88 00:05:46,242 --> 00:05:50,800 gets less and less wrong as the size of dr gets smaller and smaller. 89 00:05:51,800 --> 00:05:53,966 And this has a very beautiful meaning when we're 90 00:05:53,966 --> 00:05:56,539 looking at the sum of the areas of all those rectangles. 91 00:05:57,079 --> 00:06:00,057 For smaller and smaller choices of dr, you might at first 92 00:06:00,057 --> 00:06:03,139 think that turns the problem into a monstrously large sum. 93 00:06:03,600 --> 00:06:05,560 I mean, there's many many rectangles to consider, 94 00:06:05,560 --> 00:06:08,399 and the decimal precision of each one of their areas is going to be an 95 00:06:08,399 --> 00:06:09,199 absolute nightmare. 96 00:06:10,060 --> 00:06:15,300 But notice, all of their areas in aggregate just looks like the area under a graph. 97 00:06:15,980 --> 00:06:19,365 And that portion under the graph is just a triangle, 98 00:06:19,365 --> 00:06:23,400 a triangle with a base of 3 and a height that's 2 pi times 3. 99 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. 100 00:06:31,360 --> 00:06:35,134 Or if the radius of our original circle was some other value, 101 00:06:35,134 --> 00:06:38,660 capital R, that area comes out to be pi times r squared. 102 00:06:39,379 --> 00:06:41,459 And that's the formula for the area of a circle. 103 00:06:42,319 --> 00:06:45,411 It doesn't matter who you are or what you typically think of math, 104 00:06:45,411 --> 00:06:47,379 that right there is a beautiful argument. 105 00:06:50,180 --> 00:06:52,817 But if you want to think like a mathematician here, 106 00:06:52,817 --> 00:06:55,195 you don't just care about finding the answer, 107 00:06:55,196 --> 00:06:58,920 you care about developing general problem-solving tools and techniques. 108 00:06:59,680 --> 00:07:03,766 So take a moment to meditate on what exactly just happened and why it worked, 109 00:07:03,766 --> 00:07:07,587 because the way we transitioned from something approximate to something 110 00:07:07,586 --> 00:07:11,779 precise is actually pretty subtle and cuts deep to what calculus is all about. 111 00:07:13,819 --> 00:07:18,908 You had this problem that could be approximated with the sum of many small numbers, 112 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. 113 00:07:26,600 --> 00:07:31,947 Remember, the small number dr here represents our choice for the thickness of each ring, 114 00:07:31,947 --> 00:07:32,980 for example 0.1. 115 00:07:33,519 --> 00:07:35,639 And there are two important things to note here. 116 00:07:36,079 --> 00:07:40,639 First of all, not only is dr a factor in the quantities we're adding up, 117 00:07:40,639 --> 00:07:45,579 2 pi r times dr, it also gives the spacing between the different values of r. 118 00:07:46,240 --> 00:07:50,519 And secondly, the smaller our choice for dr, the better the approximation. 119 00:07:52,199 --> 00:07:55,202 Adding all of those numbers could be seen in a different, 120 00:07:55,202 --> 00:07:58,468 pretty clever way as adding the areas of many thin rectangles 121 00:07:58,468 --> 00:08:02,420 sitting underneath a graph, the graph of the function 2 pi r in this case. 122 00:08:02,939 --> 00:08:07,425 Then, and this is key, by considering smaller and smaller choices for dr, 123 00:08:07,425 --> 00:08:12,587 corresponding to better and better approximations of the original problem, the sum, 124 00:08:12,588 --> 00:08:15,906 thought of as the aggregate area of those rectangles, 125 00:08:15,906 --> 00:08:18,180 approaches the area under the graph. 126 00:08:19,000 --> 00:08:23,476 And because of that, you can conclude that the answer to the original question, 127 00:08:23,476 --> 00:08:28,519 in full unapproximated precision, is exactly the same as the area underneath this graph. 128 00:08:30,860 --> 00:08:35,260 A lot of other hard problems in math and science can be broken down and 129 00:08:35,260 --> 00:08:38,360 approximated as the sum of many small quantities, 130 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. 131 00:08:44,759 --> 00:08:48,769 In a case like that, you might range through many different points in time, 132 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, 133 00:08:53,260 --> 00:08:57,858 which would give the corresponding little bit of distance traveled during that little 134 00:08:57,859 --> 00:08:58,180 time. 135 00:08:59,259 --> 00:09:03,155 I'll talk through the details of examples like this later in the series, 136 00:09:03,155 --> 00:09:07,377 but at a high level many of these types of problems turn out to be equivalent 137 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. 138 00:09:13,200 --> 00:09:16,147 This happens whenever the quantities you're adding up, 139 00:09:16,147 --> 00:09:19,038 the one whose sum approximates the original problem, 140 00:09:19,038 --> 00:09:23,240 can be thought of as the areas of many thin rectangles sitting side by side. 141 00:09:24,639 --> 00:09:29,779 If finer and finer approximations of the original problem correspond to thinner and 142 00:09:29,779 --> 00:09:35,105 thinner rings, then the original problem is equivalent to finding the area under some 143 00:09:35,105 --> 00:09:35,539 graph. 144 00:09:36,600 --> 00:09:40,423 Again, this is an idea we'll see in more detail later in the series, 145 00:09:40,423 --> 00:09:43,179 so don't worry if it's not 100% clear right now. 146 00:09:43,779 --> 00:09:47,150 The point now is that you, as the mathematician having just 147 00:09:47,150 --> 00:09:50,577 solved a problem by reframing it as the area under a graph, 148 00:09:50,577 --> 00:09:54,519 might start thinking about how to find the areas under other graphs. 149 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, 150 00:10:00,317 --> 00:10:03,759 but imagine instead something like a parabola, the graph of x2. 151 00:10:04,759 --> 00:10:08,013 What's the area underneath that curve, say between 152 00:10:08,013 --> 00:10:10,680 the values of x equals 0 and x equals 3? 153 00:10:12,080 --> 00:10:14,759 Well, it's hard to think about, right? 154 00:10:15,220 --> 00:10:18,019 And let me reframe that question in a slightly different way. 155 00:10:18,019 --> 00:10:23,059 We'll fix that left endpoint in place at 0, and let the right endpoint vary. 156 00:10:26,860 --> 00:10:30,556 Are you able to find a function, a of x, that gives 157 00:10:30,556 --> 00:10:34,180 you the area under this parabola between 0 and x? 158 00:10:35,620 --> 00:10:39,580 A function a of x like this is called an integral of x2. 159 00:10:40,500 --> 00:10:44,602 Calculus holds within it the tools to figure out what an integral like this is, 160 00:10:44,602 --> 00:10:47,199 but right now it's just a mystery function to us. 161 00:10:47,500 --> 00:10:51,317 We know it gives the area under the graph of x2 between some fixed left 162 00:10:51,317 --> 00:10:54,919 point and some variable right point, but we don't know what it is. 163 00:10:55,659 --> 00:10:59,848 And again, the reason we care about this kind of question is not just for 164 00:10:59,849 --> 00:11:03,980 the sake of asking hard geometry questions, it's because many practical 165 00:11:03,980 --> 00:11:08,053 problems that can be approximated by adding up a large number of small 166 00:11:08,053 --> 00:11:12,299 things can be reframed as a question about an area under a certain graph. 167 00:11:13,419 --> 00:11:17,371 I'll tell you right now that finding this area, this integral function, 168 00:11:17,371 --> 00:11:21,991 is genuinely hard, and whenever you come across a genuinely hard question in math, 169 00:11:21,991 --> 00:11:25,776 a good policy is to not try too hard to get at the answer directly, 170 00:11:25,777 --> 00:11:29,340 since usually you just end up banging your head against a wall. 171 00:11:30,080 --> 00:11:33,780 Instead, play around with the idea, with no particular goal in mind. 172 00:11:34,340 --> 00:11:38,564 Spend some time building up familiarity with the interplay between the function 173 00:11:38,563 --> 00:11:42,360 defining the graph, in this case x2, and the function giving the area. 174 00:11:44,090 --> 00:11:48,019 In that playful spirit, if you're lucky, here's something you might notice. 175 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, 176 00:11:54,778 --> 00:12:00,419 represented with this sliver I'm going to call da for a tiny difference in area. 177 00:12:01,379 --> 00:12:05,506 That sliver can be pretty well approximated with a rectangle, 178 00:12:05,506 --> 00:12:08,620 one whose height is x2 and whose width is dx. 179 00:12:09,659 --> 00:12:12,227 And the smaller the size of that nudge dx, the 180 00:12:12,227 --> 00:12:15,019 more that sliver actually looks like a rectangle. 181 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. 182 00:12:22,000 --> 00:12:26,577 A change to the output of a, this little da, is about equal to x2, 183 00:12:26,577 --> 00:12:30,184 where x is whatever input you started at, times dx, 184 00:12:30,184 --> 00:12:34,000 the little nudge to the input that caused a to change. 185 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 186 00:12:40,316 --> 00:12:45,780 change in x that caused it, is approximately whatever x2 is at that point. 187 00:12:46,559 --> 00:12:48,759 And that's an approximation that should get better 188 00:12:48,759 --> 00:12:50,960 and better for smaller and smaller choices of dx. 189 00:12:52,100 --> 00:12:55,639 In other words, we don't know what a of x is, that remains a mystery. 190 00:12:56,080 --> 00:12:59,500 But we do know a property that this mystery function must have. 191 00:13:00,159 --> 00:13:04,971 When you look at two nearby points, for example 3 and 3.001, 192 00:13:04,971 --> 00:13:10,184 consider the change to the output of a between those two points, 193 00:13:10,184 --> 00:13:16,119 the difference between the mystery function evaluated at 3.001 and 3.001. 194 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, 195 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. 196 00:13:30,200 --> 00:13:34,416 And this relationship between tiny changes to the mystery function 197 00:13:34,416 --> 00:13:38,440 and the values of x2 itself is true at all inputs, not just 3. 198 00:13:39,419 --> 00:13:41,972 That doesn't immediately tell us how to find a of x, 199 00:13:41,972 --> 00:13:44,819 but it provides a very strong clue that we can work with. 200 00:13:46,259 --> 00:13:48,740 And there's nothing special about the graph x2 here. 201 00:13:49,279 --> 00:13:53,646 Any function defined as the area under some graph has this property, 202 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 203 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 204 00:14:03,729 --> 00:14:04,500 that point. 205 00:14:06,200 --> 00:14:10,360 Again, that's an approximation that gets better and better for smaller choices of dx. 206 00:14:11,639 --> 00:14:16,039 And here, we're stumbling into another big idea from calculus, derivatives. 207 00:14:17,100 --> 00:14:22,106 This ratio da divided by dx is called the derivative of a, or more technically, 208 00:14:22,106 --> 00:14:27,240 the derivative is whatever this ratio approaches as dx gets smaller and smaller. 209 00:14:28,179 --> 00:14:32,037 I'll dive much more deeply into the idea of a derivative in the next video, 210 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 211 00:14:36,514 --> 00:14:37,080 its input. 212 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, 213 00:14:42,173 --> 00:14:45,169 depending on what function you're looking at and how you think 214 00:14:45,169 --> 00:14:46,740 about tiny nudges to its output. 215 00:14:48,600 --> 00:14:52,153 We care about derivatives because they help us solve problems, 216 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. 217 00:14:57,840 --> 00:15:00,437 They are the key to solving integral questions, 218 00:15:00,437 --> 00:15:03,419 problems that require finding the area under a curve. 219 00:15:04,360 --> 00:15:07,669 Once you gain enough familiarity with computing derivatives, 220 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 221 00:15:12,581 --> 00:15:15,560 is, but you do know that its derivative should be x2, 222 00:15:15,559 --> 00:15:18,759 and from that reverse engineer what the function must be. 223 00:15:20,700 --> 00:15:23,826 This back and forth between integrals and derivatives, 224 00:15:23,826 --> 00:15:27,994 where the derivative of a function for the area under a graph gives you 225 00:15:27,994 --> 00:15:32,104 back the function defining the graph itself, is called the fundamental 226 00:15:32,104 --> 00:15:33,320 theorem of calculus. 227 00:15:34,220 --> 00:15:38,750 It ties together the two big ideas of integrals and derivatives, 228 00:15:38,750 --> 00:15:42,360 and shows how each one is an inverse of the other. 229 00:15:44,799 --> 00:15:47,304 All of this is only a high-level view, just a peek 230 00:15:47,304 --> 00:15:49,859 at some of the core ideas that emerge in calculus. 231 00:15:50,500 --> 00:15:54,419 And what follows in this series are the details, for derivatives and integrals and more. 232 00:15:54,980 --> 00:15:59,105 At all points, I want you to feel that you could have invented calculus yourself, 233 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, 234 00:16:03,434 --> 00:16:07,254 these formulas and rules and constructs that are presented could have just 235 00:16:07,254 --> 00:16:10,259 as easily popped out naturally from your own explorations. 236 00:16:12,379 --> 00:16:16,304 And before you go, it would feel wrong not to give the people who supported this 237 00:16:16,304 --> 00:16:20,130 series on Patreon a well-deserved thanks, both for their financial backing as 238 00:16:20,130 --> 00:16:23,860 well as for the suggestions they gave while the series was being developed. 239 00:16:24,700 --> 00:16:27,888 You see, supporters got early access to the videos as I made them, 240 00:16:27,888 --> 00:16:31,560 and they'll continue to get early access for future essence-of type series. 241 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. 242 00:16:37,019 --> 00:16:40,461 I'm still astounded that I can spend time working on videos like these, 243 00:16:40,461 --> 00:16:43,419 and in a very direct way, you are the one to thank for that.