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

zero and let the right endpoint vary.

Are you able to find a function A(x)