 ## 字幕列表 影片播放

• [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

• 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

• 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

• 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