The Essence of Calculus, Chapter 1

The Essence of Calculus, Chapter 1


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

100 comments

  1. Calculus was required for my major and I had failed it twice! I ended up dropping out and joined to army to take a break and after a year I’m trying to learn it on my own before I give it a 3rd try. Videos like these make it so simple and break it down into ways my professors never could, I actually have hope now that I’ll be able to pass it. Thank you!

  2. I am very curious as to why everybody here seems to think that this presentation is so great. I have seen stuff like this in the late 1970s on educational tv. That's how I learned calculus all by myself before it ever became a topic in high school.

  3. தெய்வமே தெய்வமே .
    நன்றி சொல்வேன் தெய்வமே
    தேடினேன் தேடினேன் .
    கண்டு கொண்டேன் உண்மையை
    கண்டு கொண்டேன் உண்மையை 🙏🙏🙏

  4. Besides all his other knowledge, abilities, and powers, just think of the moment-to-moment computing power it took and takes for God to create and to sustain the entire created universe. And people say "prove to me that God exists"? We haven't a scintilla of the mind power required to prove any such thing, and that's why salvation has to be simply a matter of faith, and is why those unwilling to seriously humble themselves and ask for that God-given faith will never come to believe in and know God. They are relying on a degree of empiricism they haven't personally the capacity to imagine, much less understand.

  5. Watching these videos can save souls and help billions to avoid aversion to math. They should just show those in schools.

  6. I’m a junior in high school taking pre calculus. I am VERY confused. I don’t have a reason to know this stuff yet for all we are learning about is mostly algebra and some basic geometry. But I still feel the urge to understand this

  7. "At all points I want you to feel that you could have invented calculus yourself." Newton wants to know your location~

  8. Your channel is endlessly stimulating! I'm finding that calculus is NOT like riding a bike; if you don't use it, you will lose it. Thanks for reminding me how beautiful calc is after 30 years!

  9. Bruh I’m 14 and it’s like one in the morning and I have algebra at 8 but here I am watching calculus for absolutely no reason. YouTube is a very large platform and I could be watching LITERALLY ANYTHING ELSE right now but no. Calculus. I hate my algebra class. So why do u care about calculus? Well, truthfully I do not. Considering that I need sleep to be active in my classes later, I thought I needed something boring to put me to sleep. Here I am. Calculus. I know that you will NEVER read this but Mr. Arnold, why must you constantly make love to your calculator in class? He’s always molesting his Texas Instruments Ti-84 graphing calculator plus silver edition. Make sure that you have stat diagnostics on. Use alpha y= to get fractions. Why am I doing this. Your not taking algebra. Your all taking calculus. Honest to god who invented this garbage. When I work at McDonald’s for the rest of my life I won’t have to use calculus to determine how a Big Mac is set up. When I sell high quality vacuum bags to Fahim Gupta from the irs, I won’t have to use calculus to steal his credit card information. I’ve wasted too much of my time typing this mongaloid comment. Personally, calculus looks really boring. I can’t believe this dude makes videos about the worst topic of any subject known to human kind. I’m gonna go. Peace out. Bologna sandwich. Calculus is bullshit.

  10. Not to be negative but I learned calculus in college in the late eighties and it made perfect sense to me, it still does. This kinda confuses me and I know what calculus is, lol. I don't know but maybe it's just how I learned it.

  11. This is the problem with the education system. We are wasting money having to hire people when we can just have thousands of students watch videos of masters of their craft explain concepts.

  12. I love music and dancing but have no rhythm now I’ve a high foot arch
    Love maths but can’t work shit out
    Definitely cursed

  13. I wish school we were taught the deep stuff, the background, so that we could be able to easily invent our own as well, that's real education and your channel follows that

  14. Hey. I am a Mechanical Engineering masters student. Its Saturday night, and I wasn't planning on doing anything serious. However, I had fun binge watching your videos! this is how amazing you are at teaching. You should know that. You should be told that everyday.

  15. I haven't looked at a calculus (let alone any other math textbook), since I graduated from high school 14 years ago. This is a beautiful reintroduction to the subject and I thank you!

  16. 9:06 Here in the graph they assumed the curve to be touching the corners of the left half of the rectangles leaving some extra space in between and cutting the right half rectangles leaving some area of rectangles out of the curve. Did they do so because parts of rectangles left out would make up for the extra space left and hence the area under the curve would be exactly correct? And is it a proof that calculus is exactly correct?

  17. I literally tried to derive a formula for the area of a circle by taking the circumference and rolling out. I'm pissed because I didn't keep going and instead tried to predict the area by taking the rolled out circumference and turning it into a square. Still don't know why it doesn't work and why I need pie. Damnit brain!

  18. As a person who always got told off for asking why/how a formula works, your videos really help answer many of these questions!😊

    Describing it in animation shows how deeply you truly understand these topics, it truly is mind blowing after so many years of just "following the formula". Thanks for introducing us to the world of calculus, in the most simple, proper and eloquent way possible, this, changed me.

  19. School kids of this generation are very lucky to have sources such as this. We literally wasted our life in an egoistic academic system.

  20. I think the ads in the video work well as they give one a bit of repose. That’s important when developing cognitive skills.

  21. I tried watching this before I started calculus 1 and was so disinterested. Now that I'm at the end of calc 1, I can't get enough of this. It's amazing to see what's going on.

  22. Although not an explanation that satisfy my need for a set theory and fundamentals of mathematics perspective on the subject this is one of the best tutorials I've found on the subject, a gourgeous one when you reach a level of abstraction and cognitive awareness as to observe mathematical beauty. I'll download this, it's a very nice guide to dive deeper into calculus and develop actual skills for reckoning and general use in everyday life.

  23. 1:20 Woah!!! I never thought of doing the PRODUCT RULE by drawing an AREA DIAGRAM. Makes it all so clear! I have derived the product rule algebraically many times, to remind myself how it works. But seeing it done on an area diagram makes it a thousand times clearer than proving it algebraically ever did.

  24. "Mathematics has a tendency to reward you when you respect its symmetries". Damn!! That's so true. Next time I find myself stuck, I'll remind myself: RESPECT THE SYMMETRIES. Respect the symmetries and the symmetries will respect you. "This is how you use the force, Luke". (or something) 🙂

  25. Thank you! Just thank you for this beautiful series. It helps me a lot with my studies (informatics). I think this is the third time I am watching this.

  26. Could someone please clarify for me? As you decrease the value of dr, 3B1B says the answer becomes more accurate. But how does that make sense? Wouldn't the left over space in between the rectangles and the slope not just remain the same value – decreasing in size, but increasing in quantity?

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