The essence of calculus

0:14 Hey everyone, Grant here.

0:16 This is the first video in a series on the essence of calculus,

0:19 and I'll be publishing the following videos once per day for the next 10 days.

0:24 The goal here, as the name suggests,

0:25 is to really get the heart of the subject out in one binge-watchable set.

0:30 But with a topic that's as broad as calculus,

0:32 there's a lot of things that can mean,

0:34 so here's what I have in mind specifically.

0:36 Calculus has a lot of rules and formulas

0:39 which are often presented as things to be memorized.

0:42 Lots of derivative formulas, the product rule,

0:44 the chain rule, implicit differentiation,

0:46 the fact that integrals and derivatives are opposite,

0:49 Taylor series, just a lot of things like that.

0:52 And my goal is for you to come

0:54 away feeling like you could have invented calculus yourself.

0:57 That is, cover all those core ideas, but in a way that makes clear where

1:01 they actually come from, and what they really mean,

1:04 using an all-around visual approach.

1:06 Inventing math is no joke,

1:08 and there is a difference between being told why something's true,

1:11 and actually generating it from scratch.

1:14 But at all points, I want you to think to yourself,

1:17 if you were an early mathematician,

1:19 pondering these ideas and drawing out the right diagrams,

1:22 does it feel reasonable that you

1:23 could have stumbled across these truths yourself?

1:26 In this initial video, I want to show how you might stumble into the core ideas

1:31 of calculus by thinking very deeply about one specific bit of geometry,

1:35 the area of a circle.

1:37 Maybe you know that this is pi times its radius squared, but why?

1:41 Is there a nice way to think about where this formula comes from?

1:45 Well, contemplating this problem and leaving yourself

1:47 open to exploring the interesting thoughts that come

1:50 about can actually lead you to a glimpse of three big ideas in calculus,

1:55 integrals, derivatives, and the fact that they're opposites.

1:59 But the story starts more simply,

2:01 just you and a circle, let's say with radius 3.

2:05 You're trying to figure out its area,

2:07 and after going through a lot of paper trying different

2:10 ways to chop up and rearrange the pieces of that area,

2:13 many of which might lead to their own interesting observations,

2:16 maybe you try out the idea of slicing up the circle into many concentric rings.

2:22 This should seem promising because it respects the symmetry of the circle,

2:25 and math has a tendency to reward you when you respect its symmetries.

2:30 Let's take one of those rings,

2:32 which has some inner radius r that's between 0 and 3.

2:36 If we can find a nice expression for the area of each ring like this one,

2:39 and if we have a nice way to add them all up,

2:42 it might lead us to an understanding of the full circle's area.

2:46 Maybe you start by imagining straightening out this ring.

2:50 And you could try thinking through exactly what

2:53 this new shape is and what its area should be,

2:55 but for simplicity, let's just approximate it as a rectangle.

3:00 The width of that rectangle is the circumference of the original ring,

3:03 which is 2 pi times r, right?

3:05 I mean, that's essentially the definition of pi.

3:08 And its thickness?

3:10 Well, that depends on how finely you chopped up the circle in the first place,

3:14 which was kind of arbitrary.

3:16 In the spirit of using what will come to be standard calculus notation,

3:20 let's call that thickness dr for a tiny difference

3:22 in the radius from one ring to the next.

3:25 Maybe you think of it as something like 0.1.

3:28 So approximating this unwrapped ring as a thin rectangle,

3:32 its area is 2 pi times r, the radius, times dr, the little thickness.

3:38 And even though that's not perfect, for smaller and smaller choices of dr,

3:42 this is actually going to be a better and better approximation for that area,

3:46 since the top and the bottom sides of this shape are

3:49 going to get closer and closer to being exactly the same length.

3:53 So let's just move forward with this approximation,

3:55 keeping in the back of our minds that it's slightly wrong,

3:58 but it's going to become more accurate for smaller and smaller choices of dr.

4:03 That is, if we slice up the circle into thinner and thinner rings.

4:07 So just to sum up where we are,

4:09 you've broken up the area of the circle into all of these rings,

4:13 and you're approximating the area of each one

4:16 of those as 2 pi times its radius times dr,

4:19 where the specific value for that inner radius ranges from 0

4:22 for the smallest ring up to just under 3 for the biggest ring,

4:26 spaced out by whatever the thickness is

4:29 that you choose for dr, something like 0.1.

4:33 And notice that the spacing between the values

4:35 here corresponds to the thickness dr of each ring,

4:38 the difference in radius from one ring to the next.

4:42 In fact, a nice way to think about the rectangles approximating each ring's

4:46 area is to fit them all upright side by side along this axis.

4:50 Each one has a thickness dr,

4:52 which is why they fit so snugly right there together,

4:55 and the height of any one of these rectangles

4:58 sitting above some specific value of r,

5:01 like 0.6, is exactly 2 pi times that value.

5:04 That's the circumference of the corresponding

5:06 ring that this rectangle approximates.

5:09 Pictures like this 2 pi r can get tall for the screen,

5:12 I mean 2 times pi times 3 is around 19,

5:14 so let's just throw up a y axis that's scaled a little differently

5:18 so that we can actually fit all of these rectangles on the screen.

5:23 A nice way to think about this setup is to draw the graph of 2 pi r,

5:26 which is a straight line that has a slope 2 pi.

5:30 Each of these rectangles extends up to the point

5:32 where it just barely touches that graph.

5:36 Again, we're being approximate here.

5:37 Each of these rectangles only approximates the area

5:40 of the corresponding ring from the circle.

5:42 But remember, that approximation, 2 pi r times dr,

5:46 gets less and less wrong as the size of dr gets smaller and smaller.

5:51 And this has a very beautiful meaning when we're looking

5:54 at the sum of the areas of all those rectangles.

5:57 For smaller and smaller choices of dr,

5:58 you might at first think that turns the problem into a monstrously large sum.

6:03 I mean, there's many many rectangles to consider,

6:05 and the decimal precision of each one

6:06 of their areas is going to be an absolute nightmare.

6:10 But notice, all of their areas in aggregate

6:12 just looks like the area under a graph.

6:15 And that portion under the graph is just a triangle,

6:19 a triangle with a base of 3 and a height that's 2 pi times 3.

6:24 So its area, 1 half base times height,

6:27 works out to be exactly pi times 3 squared.

6:31 Or if the radius of our original circle was some other value,

6:35 capital R, that area comes out to be pi times r squared.

6:39 And that's the formula for the area of a circle.

6:42 It doesn't matter who you are or what you typically think of math,

6:45 that right there is a beautiful argument.

6:50 But if you want to think like a mathematician here,

6:52 you don't just care about finding the answer,

6:55 you care about developing general problem-solving tools and techniques.

6:59 So take a moment to meditate on what exactly just happened and why it worked,

7:03 because the way we transitioned

7:05 from something approximate to something precise is

7:08 actually pretty subtle and cuts deep to what calculus is all about.

7:13 You had this problem that could be

7:16 approximated with the sum of many small numbers,

7:18 each of which looked like 2 pi r times dr,

7:21 for values of r ranging between 0 and 3.

7:26 Remember, the small number dr here represents our choice

7:29 for the thickness of each ring, for example 0.1.

7:33 And there are two important things to note here.

7:36 First of all, not only is dr a factor in the quantities we're adding up,

7:40 2 pi r times dr, it also gives the spacing between the different values of r.

7:46 And secondly, the smaller our choice for dr, the better the approximation.

7:52 Adding all of those numbers could be seen in a different,

7:55 pretty clever way as adding the areas

7:57 of many thin rectangles sitting underneath a graph,

7:59 the graph of the function 2 pi r in this case.

8:02 Then, and this is key, by considering smaller and smaller choices for dr,

8:07 corresponding to better and better approximations of the original problem,

8:11 the sum, thought of as the aggregate area of those rectangles,

8:15 approaches the area under the graph.

8:19 And because of that, you can conclude that the answer to the original question,

8:23 in full unapproximated precision,

8:24 is exactly the same as the area underneath this graph.

8:30 A lot of other hard problems in math and science can

8:34 be broken down and approximated as the sum of many small quantities,

8:38 like figuring out how far a car has traveled

8:41 based on its velocity at each point in time.

8:44 In a case like that, you might range through many different points in time,

8:48 and at each one multiply the velocity at that time times a tiny change in time,

8:52 dt, which would give the corresponding little

8:55 bit of distance traveled during that little time.

8:59 I'll talk through the details of examples like this later in the series,

9:03 but at a high level many of these types of problems

9:06 turn out to be equivalent to finding the area under some graph,

9:09 in much the same way that our circle problem did.

9:13 This happens whenever the quantities you're adding up,

9:16 the one whose sum approximates the original problem,

9:19 can be thought of as the areas of many thin rectangles sitting side by side.

9:24 If finer and finer approximations of the original

9:27 problem correspond to thinner and thinner rings,

9:30 then the original problem is equivalent to finding the area under some graph.

9:36 Again, this is an idea we'll see in more detail later in the series,

9:40 so don't worry if it's not 100% clear right now.

9:43 The point now is that you, as the mathematician having just solved a problem

9:48 by reframing it as the area under a graph,

9:50 might start thinking about how to find the areas under other graphs.

9:55 We were lucky in the circle problem

9:57 that the relevant area turned out to be a triangle,

10:00 but imagine instead something like a parabola, the graph of x2.

10:04 What's the area underneath that curve,

10:07 say between the values of x equals 0 and x equals 3?

10:12 Well, it's hard to think about, right?

10:15 And let me reframe that question in a slightly different way.

10:18 We'll fix that left endpoint in place at 0, and let the right endpoint vary.

10:26 Are you able to find a function, a of x,

10:29 that gives you the area under this parabola between 0 and x?

10:35 A function a of x like this is called an integral of x2.

10:40 Calculus holds within it the tools to figure out what an integral like this is,

10:44 but right now it's just a mystery function to us.

10:47 We know it gives the area under the graph of x2

10:50 between some fixed left point and some variable right point,

10:52 but we don't know what it is.

10:55 And again, the reason we care about this kind of question

10:58 is not just for the sake of asking hard geometry questions,

11:02 it's because many practical problems that can

11:04 be approximated by adding up a large number

11:07 of small things can be reframed as a question

11:10 about an area under a certain graph.

11:13 I'll tell you right now that finding this area, this integral function,

11:17 is genuinely hard, and whenever you come

11:19 across a genuinely hard question in math,

11:21 a good policy is to not try too hard to get at the answer directly,

11:25 since usually you just end up banging your head against a wall.

11:30 Instead, play around with the idea, with no particular goal in mind.

11:34 Spend some time building up familiarity

11:36 with the interplay between the function defining the graph,

11:39 in this case x2, and the function giving the area.

11:44 In that playful spirit, if you're lucky, here's something you might notice.

11:48 When you slightly increase x by some tiny nudge dx,

11:52 look at the resulting change in area,

11:54 represented with this sliver I'm going to call da for a tiny difference in area.

12:01 That sliver can be pretty well approximated with a rectangle,

12:05 one whose height is x2 and whose width is dx.

12:09 And the smaller the size of that nudge dx,

12:11 the more that sliver actually looks like a rectangle.

12:16 This gives us an interesting way to think about how a of x is related to x2.

12:22 A change to the output of a, this little da, is about equal to x2,

12:26 where x is whatever input you started at, times dx,

12:30 the little nudge to the input that caused a to change.

12:34 Or rearranged, da divided by dx,

12:36 the ratio of a tiny change in a to the tiny change in x that caused it,

12:42 is approximately whatever x2 is at that point.

12:46 And that's an approximation that should get better

12:48 and better for smaller and smaller choices of dx.

12:52 In other words, we don't know what a of x is, that remains a mystery.

12:56 But we do know a property that this mystery function must have.

13:00 When you look at two nearby points, for example 3 and 3.001,

13:04 consider the change to the output of a between those two points,

13:10 the difference between the mystery function evaluated at 3.001 and 3.001.

13:16 That change, divided by the difference in the input values,

13:19 which in this case is 0.001,

13:22 should be about equal to the value of x2 for the starting input,

13:26 in this case 3 squared.

13:30 And this relationship between tiny changes

13:32 to the mystery function and the values

13:35 of x2 itself is true at all inputs, not just 3.

13:39 That doesn't immediately tell us how to find a of x,

13:41 but it provides a very strong clue that we can work with.

13:46 And there's nothing special about the graph x2 here.

13:49 Any function defined as the area under some graph has this property,

13:53 that da divided by dx, a slight nudge to the output of a divided

13:57 by a slight nudge to the input that caused it,

14:00 is about equal to the height of the graph at that point.

14:06 Again, that's an approximation that gets better

14:08 and better for smaller choices of dx.

14:11 And here, we're stumbling into another big idea from calculus, derivatives.

14:17 This ratio da divided by dx is called the derivative of a, or more technically,

14:22 the derivative is whatever this ratio approaches as dx gets smaller and smaller.

14:28 I'll dive much more deeply into the idea of a derivative in the next video,

14:32 but loosely speaking it's a measure of how sensitive

14:34 a function is to small changes in its input.

14:37 You'll see as the series goes on that there

14:40 are many ways you can visualize a derivative,

14:42 depending on what function you're looking at and how

14:44 you think about tiny nudges to its output.

14:48 We care about derivatives because they help us solve problems,

14:52 and in our little exploration here,

14:54 we already have a glimpse of one way they're used.

14:57 They are the key to solving integral questions,

15:00 problems that require finding the area under a curve.

15:04 Once you gain enough familiarity with computing derivatives,

15:07 you'll be able to look at a situation like

15:10 this one where you don't know what a function is,

15:12 but you do know that its derivative should be x2,

15:15 and from that reverse engineer what the function must be.

15:20 This back and forth between integrals and derivatives,

15:23 where the derivative of a function for the area under

15:26 a graph gives you back the function defining the graph itself,

15:30 is called the fundamental theorem of calculus.

15:34 It ties together the two big ideas of integrals and derivatives,

15:38 and shows how each one is an inverse of the other.

15:44 All of this is only a high-level view,

15:46 just a peek at some of the core ideas that emerge in calculus.

15:50 And what follows in this series are the details,

15:52 for derivatives and integrals and more.

15:54 At all points, I want you to feel

15:57 that you could have invented calculus yourself,

15:59 that if you drew the right pictures and played

16:01 with each idea in just the right way,

16:03 these formulas and rules and constructs that are presented could

16:06 have just as easily popped out naturally from your own explorations.

16:12 And before you go, it would feel wrong not to give

16:15 the people who supported this series on Patreon a well-deserved thanks,

16:18 both for their financial backing as well as for the suggestions

16:21 they gave while the series was being developed.

16:24 You see, supporters got early access to the videos as I made them,

16:27 and they'll continue to get early access for future essence-of type series.

16:32 And as a thanks to the community,

16:33 I keep ads off of new videos for their first month.

16:37 I'm still astounded that I can spend time working

16:39 on videos like these, and in a very direct way,

16:41 you are the one to thank for that.