Subtitles section Play video Print subtitles [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