The essence of calculus

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 have in mind specifically.

Calculus has a lot of rules and formulas which

are often presented as things to be memorized.

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 3.

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 2 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.