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    "segs": [ {
      "utf8": "Hey everyone, Grant here."
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    "segs": [ {
      "utf8": "This is the first video in a series on the essence of calculus,"
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    "segs": [ {
      "utf8": "and I'll be publishing the following videos once per day for the next 10 days."
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    "segs": [ {
      "utf8": "The goal here, as the name suggests, is to really get"
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    "segs": [ {
      "utf8": "the heart of the subject out in one binge-watchable set."
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  }, {
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    "segs": [ {
      "utf8": "But with a topic that's as broad as calculus, there's a lot of things that can mean,"
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  }, {
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    "segs": [ {
      "utf8": "so here's what I have in mind specifically."
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    "segs": [ {
      "utf8": "Calculus has a lot of rules and formulas which"
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    "dDurationMs": 2527,
    "segs": [ {
      "utf8": "are often presented as things to be memorized."
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    "segs": [ {
      "utf8": "Lots of derivative formulas, the product rule, the chain rule,"
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  }, {
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    "segs": [ {
      "utf8": "implicit differentiation, the fact that integrals and derivatives are opposite,"
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  }, {
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    "segs": [ {
      "utf8": "Taylor series, just a lot of things like that."
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  }, {
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    "segs": [ {
      "utf8": "And my goal is for you to come away feeling like"
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  }, {
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    "segs": [ {
      "utf8": "you could have invented calculus yourself."
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    "segs": [ {
      "utf8": "That is, cover all those core ideas, but in a way that makes clear where they"
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  }, {
    "tStartMs": 61638,
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    "segs": [ {
      "utf8": "actually come from, and what they really mean, using an all-around visual approach."
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  }, {
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    "segs": [ {
      "utf8": "Inventing math is no joke, and there is a difference between being"
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  }, {
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    "segs": [ {
      "utf8": "told why something's true, and actually generating it from scratch."
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    "segs": [ {
      "utf8": "But at all points, I want you to think to yourself, if you were an early mathematician,"
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    "segs": [ {
      "utf8": "pondering these ideas and drawing out the right diagrams,"
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    "segs": [ {
      "utf8": "does it feel reasonable that you could have stumbled across these truths yourself?"
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    "segs": [ {
      "utf8": "In this initial video, I want to show how you might stumble into the core ideas of"
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    "tStartMs": 91569,
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    "segs": [ {
      "utf8": "calculus by thinking very deeply about one specific bit of geometry,"
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  }, {
    "tStartMs": 95566,
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    "segs": [ {
      "utf8": "the area of a circle."
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    "segs": [ {
      "utf8": "Maybe you know that this is pi times its radius squared, but why?"
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    "segs": [ {
      "utf8": "Is there a nice way to think about where this formula comes from?"
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    "segs": [ {
      "utf8": "Well, contemplating this problem and leaving yourself open to exploring the"
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  }, {
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    "segs": [ {
      "utf8": "interesting thoughts that come about can actually lead you to a glimpse of three"
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  }, {
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    "segs": [ {
      "utf8": "big ideas in calculus, integrals, derivatives, and the fact that they're opposites."
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    "segs": [ {
      "utf8": "But the story starts more simply, just you and a circle, let's say with radius 3."
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    "segs": [ {
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    "segs": [ {
      "utf8": "paper trying different ways to chop up and rearrange the pieces of that area,"
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  }, {
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    "segs": [ {
      "utf8": "many of which might lead to their own interesting observations,"
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  }, {
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    "segs": [ {
      "utf8": "maybe you try out the idea of slicing up the circle into many concentric rings."
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    "dDurationMs": 3807,
    "segs": [ {
      "utf8": "This should seem promising because it respects the symmetry of the circle,"
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  }, {
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    "segs": [ {
      "utf8": "and math has a tendency to reward you when you respect its symmetries."
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    "segs": [ {
      "utf8": "Let's take one of those rings, which has some inner radius r that's between 0 and 3."
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    "segs": [ {
      "utf8": "If we can find a nice expression for the area of each ring like this one,"
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    "segs": [ {
      "utf8": "and if we have a nice way to add them all up,"
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    "segs": [ {
      "utf8": "it might lead us to an understanding of the full circle's area."
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    "segs": [ {
      "utf8": "Maybe you start by imagining straightening out this ring."
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    "segs": [ {
      "utf8": "And you could try thinking through exactly what this new shape is and what its"
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    "segs": [ {
      "utf8": "area should be, but for simplicity, let's just approximate it as a rectangle."
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    "segs": [ {
      "utf8": "The width of that rectangle is the circumference of the original ring,"
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    "segs": [ {
      "utf8": "which is 2 pi times r, right?"
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    "segs": [ {
      "utf8": "I mean, that's essentially the definition of pi."
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    "segs": [ {
      "utf8": "And its thickness?"
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    "segs": [ {
      "utf8": "Well, that depends on how finely you chopped up the circle in the first place,"
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  }, {
    "tStartMs": 194151,
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    "segs": [ {
      "utf8": "which was kind of arbitrary."
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    "segs": [ {
      "utf8": "In the spirit of using what will come to be standard calculus notation,"
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    "segs": [ {
      "utf8": "let's call that thickness dr for a tiny difference in the radius from one ring to"
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    "segs": [ {
      "utf8": "the next."
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    "segs": [ {
      "utf8": "Maybe you think of it as something like 0.1."
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    "segs": [ {
      "utf8": "So approximating this unwrapped ring as a thin rectangle,"
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    "segs": [ {
      "utf8": "its area is 2 pi times r, the radius, times dr, the little thickness."
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    "segs": [ {
      "utf8": "And even though that's not perfect, for smaller and smaller choices of dr,"
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    "segs": [ {
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    "segs": [ {
      "utf8": "since the top and the bottom sides of this shape are going to get closer and closer to"
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    "segs": [ {
      "utf8": "So let's just move forward with this approximation,"
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    "segs": [ {
      "utf8": "keeping in the back of our minds that it's slightly wrong,"
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    "segs": [ {
      "utf8": "but it's going to become more accurate for smaller and smaller choices of dr."
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    "segs": [ {
      "utf8": "That is, if we slice up the circle into thinner and thinner rings."
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    "segs": [ {
      "utf8": "So just to sum up where we are, you've broken up the area of the circle into"
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    "segs": [ {
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    "segs": [ {
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    "segs": [ {
      "utf8": "spaced out by whatever the thickness is that you choose for dr, something like 0.1."
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    "segs": [ {
      "utf8": "And notice that the spacing between the values here corresponds to the"
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    "segs": [ {
      "utf8": "thickness dr of each ring, the difference in radius from one ring to the next."
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    "segs": [ {
      "utf8": "In fact, a nice way to think about the rectangles approximating each"
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    "segs": [ {
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    "segs": [ {
      "utf8": "Each one has a thickness dr, which is why they fit so snugly right there together,"
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    "segs": [ {
      "utf8": "and the height of any one of these rectangles sitting above some specific value of r,"
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    "segs": [ {
      "utf8": "like 0.6, is exactly 2 pi times that value."
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    "segs": [ {
      "utf8": "That's the circumference of the corresponding ring that this rectangle approximates."
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    "tStartMs": 309560,
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    "segs": [ {
      "utf8": "Pictures like this 2 pi r can get tall for the screen,"
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    "segs": [ {
      "utf8": "I mean 2 times pi times 3 is around 19, so let's just throw up a y axis that's"
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    "segs": [ {
      "utf8": "scaled a little differently so that we can actually fit all of these rectangles"
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      "utf8": "on the screen."
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    "segs": [ {
      "utf8": "A nice way to think about this setup is to draw the graph of 2 pi r,"
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    "segs": [ {
      "utf8": "Each of these rectangles extends up to the point where it just barely touches that graph."
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    "segs": [ {
      "utf8": "Again, we're being approximate here."
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    "segs": [ {
      "utf8": "But remember, that approximation, 2 pi r times dr,"
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    "segs": [ {
      "utf8": "And this has a very beautiful meaning when we're"
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      "utf8": "looking at the sum of the areas of all those rectangles."
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    "segs": [ {
      "utf8": "For smaller and smaller choices of dr, you might at first"
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      "utf8": "think that turns the problem into a monstrously large sum."
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      "utf8": "I mean, there's many many rectangles to consider,"
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      "utf8": "and the decimal precision of each one of their areas is going to be an"
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  }, {
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    "segs": [ {
      "utf8": "absolute nightmare."
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    "segs": [ {
      "utf8": "But notice, all of their areas in aggregate just looks like the area under a graph."
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    "segs": [ {
      "utf8": "And that portion under the graph is just a triangle,"
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    "segs": [ {
      "utf8": "a triangle with a base of 3 and a height that's 2 pi times 3."
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    "segs": [ {
      "utf8": "So its area, 1 half base times height, works out to be exactly pi times 3 squared."
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    "segs": [ {
      "utf8": "Or if the radius of our original circle was some other value,"
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    "segs": [ {
      "utf8": "capital R, that area comes out to be pi times r squared."
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  }, {
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    "segs": [ {
      "utf8": "And that's the formula for the area of a circle."
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    "segs": [ {
      "utf8": "It doesn't matter who you are or what you typically think of math,"
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    "segs": [ {
      "utf8": "that right there is a beautiful argument."
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      "utf8": "you care about developing general problem-solving tools and techniques."
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  }, {
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    "segs": [ {
      "utf8": "So take a moment to meditate on what exactly just happened and why it worked,"
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    "segs": [ {
      "utf8": "because the way we transitioned from something approximate to something"
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      "utf8": "precise is actually pretty subtle and cuts deep to what calculus is all about."
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    "segs": [ {
      "utf8": "You had this problem that could be approximated with the sum of many small numbers,"
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    "segs": [ {
      "utf8": "each of which looked like 2 pi r times dr, for values of r ranging between 0 and 3."
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    "segs": [ {
      "utf8": "Remember, the small number dr here represents our choice for the thickness of each ring,"
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    "segs": [ {
      "utf8": "for example 0.1."
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    "segs": [ {
      "utf8": "And there are two important things to note here."
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    "segs": [ {
      "utf8": "First of all, not only is dr a factor in the quantities we're adding up,"
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    "segs": [ {
      "utf8": "2 pi r times dr, it also gives the spacing between the different values of r."
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    "segs": [ {
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      "utf8": "Adding all of those numbers could be seen in a different,"
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      "utf8": "pretty clever way as adding the areas of many thin rectangles"
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      "utf8": "sitting underneath a graph, the graph of the function 2 pi r in this case."
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    "segs": [ {
      "utf8": "Then, and this is key, by considering smaller and smaller choices for dr,"
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    "segs": [ {
      "utf8": "corresponding to better and better approximations of the original problem, the sum,"
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    "segs": [ {
      "utf8": "And because of that, you can conclude that the answer to the original question,"
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    "segs": [ {
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      "utf8": "A lot of other hard problems in math and science can be broken down and"
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      "utf8": "approximated as the sum of many small quantities,"
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    "segs": [ {
      "utf8": "like figuring out how far a car has traveled based on its velocity at each point in time."
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    "segs": [ {
      "utf8": "In a case like that, you might range through many different points in time,"
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    "segs": [ {
      "utf8": "and at each one multiply the velocity at that time times a tiny change in time, dt,"
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    "segs": [ {
      "utf8": "which would give the corresponding little bit of distance traveled during that little"
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    "segs": [ {
      "utf8": "I'll talk through the details of examples like this later in the series,"
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    "segs": [ {
      "utf8": "but at a high level many of these types of problems turn out to be equivalent"
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    "tStartMs": 547378,
    "dDurationMs": 4762,
    "segs": [ {
      "utf8": "to finding the area under some graph, in much the same way that our circle problem did."
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    "segs": [ {
      "utf8": "This happens whenever the quantities you're adding up,"
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      "utf8": "the one whose sum approximates the original problem,"
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    "tStartMs": 559038,
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    "segs": [ {
      "utf8": "can be thought of as the areas of many thin rectangles sitting side by side."
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    "tStartMs": 564640,
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    "segs": [ {
      "utf8": "If finer and finer approximations of the original problem correspond to thinner and"
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    "segs": [ {
      "utf8": "thinner rings, then the original problem is equivalent to finding the area under some"
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    "segs": [ {
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    "segs": [ {
      "utf8": "Again, this is an idea we'll see in more detail later in the series,"
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    "tStartMs": 580424,
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    "segs": [ {
      "utf8": "so don't worry if it's not 100% clear right now."
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    "segs": [ {
      "utf8": "The point now is that you, as the mathematician having just"
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    "segs": [ {
      "utf8": "solved a problem by reframing it as the area under a graph,"
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      "utf8": "might start thinking about how to find the areas under other graphs."
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    "segs": [ {
      "utf8": "We were lucky in the circle problem that the relevant area turned out to be a triangle,"
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    "segs": [ {
      "utf8": "but imagine instead something like a parabola, the graph of x2."
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    "segs": [ {
      "utf8": "What's the area underneath that curve, say between"
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    "segs": [ {
      "utf8": "Well, it's hard to think about, right?"
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    "segs": [ {
      "utf8": "And let me reframe that question in a slightly different way."
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    "segs": [ {
      "utf8": "We'll fix that left endpoint in place at 0, and let the right endpoint vary."
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    "segs": [ {
      "utf8": "Are you able to find a function, a of x, that gives"
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      "utf8": "you the area under this parabola between 0 and x?"
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    "segs": [ {
      "utf8": "A function a of x like this is called an integral of x2."
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    "segs": [ {
      "utf8": "Calculus holds within it the tools to figure out what an integral like this is,"
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    "segs": [ {
      "utf8": "but right now it's just a mystery function to us."
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    "segs": [ {
      "utf8": "We know it gives the area under the graph of x2 between some fixed left"
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    "segs": [ {
      "utf8": "point and some variable right point, but we don't know what it is."
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    "segs": [ {
      "utf8": "And again, the reason we care about this kind of question is not just for"
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    "segs": [ {
      "utf8": "the sake of asking hard geometry questions, it's because many practical"
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    "dDurationMs": 4074,
    "segs": [ {
      "utf8": "problems that can be approximated by adding up a large number of small"
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  }, {
    "tStartMs": 668054,
    "dDurationMs": 4246,
    "segs": [ {
      "utf8": "things can be reframed as a question about an area under a certain graph."
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    "tStartMs": 673420,
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    "segs": [ {
      "utf8": "I'll tell you right now that finding this area, this integral function,"
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    "segs": [ {
      "utf8": "is genuinely hard, and whenever you come across a genuinely hard question in math,"
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    "segs": [ {
      "utf8": "a good policy is to not try too hard to get at the answer directly,"
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    "segs": [ {
      "utf8": "since usually you just end up banging your head against a wall."
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    "segs": [ {
      "utf8": "Instead, play around with the idea, with no particular goal in mind."
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    "tStartMs": 694340,
    "dDurationMs": 4224,
    "segs": [ {
      "utf8": "Spend some time building up familiarity with the interplay between the function"
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    "segs": [ {
      "utf8": "defining the graph, in this case x2, and the function giving the area."
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    "segs": [ {
      "utf8": "In that playful spirit, if you're lucky, here's something you might notice."
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    "segs": [ {
      "utf8": "When you slightly increase x by some tiny nudge dx, look at the resulting change in area,"
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    "tStartMs": 714779,
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    "segs": [ {
      "utf8": "represented with this sliver I'm going to call da for a tiny difference in area."
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    "tStartMs": 721380,
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    "segs": [ {
      "utf8": "That sliver can be pretty well approximated with a rectangle,"
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      "utf8": "one whose height is x2 and whose width is dx."
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    "segs": [ {
      "utf8": "And the smaller the size of that nudge dx, the"
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    "segs": [ {
      "utf8": "more that sliver actually looks like a rectangle."
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    "segs": [ {
      "utf8": "This gives us an interesting way to think about how a of x is related to x2."
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    "segs": [ {
      "utf8": "A change to the output of a, this little da, is about equal to x2,"
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    "segs": [ {
      "utf8": "where x is whatever input you started at, times dx,"
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      "utf8": "the little nudge to the input that caused a to change."
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    "segs": [ {
      "utf8": "Or rearranged, da divided by dx, the ratio of a tiny change in a to the tiny"
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    "segs": [ {
      "utf8": "change in x that caused it, is approximately whatever x2 is at that point."
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    "segs": [ {
      "utf8": "And that's an approximation that should get better"
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      "utf8": "and better for smaller and smaller choices of dx."
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    "segs": [ {
      "utf8": "In other words, we don't know what a of x is, that remains a mystery."
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    "segs": [ {
      "utf8": "But we do know a property that this mystery function must have."
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    "segs": [ {
      "utf8": "When you look at two nearby points, for example 3 and 3.001,"
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      "utf8": "consider the change to the output of a between those two points,"
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    "segs": [ {
      "utf8": "the difference between the mystery function evaluated at 3.001 and 3.001."
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    "segs": [ {
      "utf8": "That change, divided by the difference in the input values, which in this case is 0.001,"
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      "utf8": "should be about equal to the value of x2 for the starting input, in this case 3 squared."
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    "segs": [ {
      "utf8": "And this relationship between tiny changes to the mystery function"
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    "segs": [ {
      "utf8": "and the values of x2 itself is true at all inputs, not just 3."
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    "segs": [ {
      "utf8": "That doesn't immediately tell us how to find a of x,"
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      "utf8": "but it provides a very strong clue that we can work with."
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      "utf8": "And there's nothing special about the graph x2 here."
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      "utf8": "Any function defined as the area under some graph has this property,"
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    "segs": [ {
      "utf8": "that da divided by dx, a slight nudge to the output of a divided by a slight"
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    "segs": [ {
      "utf8": "nudge to the input that caused it, is about equal to the height of the graph at"
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    "segs": [ {
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    "segs": [ {
      "utf8": "Again, that's an approximation that gets better and better for smaller choices of dx."
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    "segs": [ {
      "utf8": "And here, we're stumbling into another big idea from calculus, derivatives."
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    "segs": [ {
      "utf8": "This ratio da divided by dx is called the derivative of a, or more technically,"
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    "segs": [ {
      "utf8": "the derivative is whatever this ratio approaches as dx gets smaller and smaller."
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    "segs": [ {
      "utf8": "I'll dive much more deeply into the idea of a derivative in the next video,"
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    "segs": [ {
      "utf8": "but loosely speaking it's a measure of how sensitive a function is to small changes in"
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    "segs": [ {
      "utf8": "You'll see as the series goes on that there are many ways you can visualize a derivative,"
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    "segs": [ {
      "utf8": "depending on what function you're looking at and how you think"
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      "utf8": "and in our little exploration here, we already have a glimpse of one way they're used."
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    "segs": [ {
      "utf8": "Once you gain enough familiarity with computing derivatives,"
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      "utf8": "you'll be able to look at a situation like this one where you don't know what a function"
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    "segs": [ {
      "utf8": "and from that reverse engineer what the function must be."
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    "segs": [ {
      "utf8": "where the derivative of a function for the area under a graph gives you"
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      "utf8": "back the function defining the graph itself, is called the fundamental"
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    "segs": [ {
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    "segs": [ {
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    "segs": [ {
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    "segs": [ {
      "utf8": "these formulas and rules and constructs that are presented could have just"
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    "segs": [ {
      "utf8": "And before you go, it would feel wrong not to give the people who supported this"
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  }, {
    "tStartMs": 976305,
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    "segs": [ {
      "utf8": "series on Patreon a well-deserved thanks, both for their financial backing as"
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    "segs": [ {
      "utf8": "well as for the suggestions they gave while the series was being developed."
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    "segs": [ {
      "utf8": "You see, supporters got early access to the videos as I made them,"
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    "segs": [ {
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