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  • [INTRO MUSIC]

  • Hey everyone, Grant here.This is the first video in the series of 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 the 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, product rule, chain rule, implicit diffrentiation

  • 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 & 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 circles

  • 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 it's area is 2 [pi]

  • times R the radius times dr are 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

  • zer for the smallest ring up to just

  • under three, for the biggest ring spaced

  • out by whatever the thicknesses that you

  • choose for dr are 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

  • rings 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

  • at 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*[pi]*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 it's 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 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

  • because 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 have 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 & 3.

  • Remember the small numbered 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-on 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