I’ve introduced a few derivative formulas but a really important one that Ieft out was exponentials. So here, I want to talk about the derivatives of functions like Two to the x, seven to the x, and also to show why e to the x is arguably the most important of the exponentials. First of…

# Tag: three brown one blue

## But how does bitcoin actually work?

What does it mean to have a bitcoin? Many people have now heard of bitcoin, that’s it’s a fully digital currency, with no government to issue it and no banks needed to manage accounts and verify transactions. That no one actually knows who invented it. Yet many people don’t know the answer to this question,…

## Why is pi here? And why is it squared? A geometric answer to the Basel problem

I’m gonna guess that you have never had the experience of your heart rate increasing in excitement while you are imagining an infinitely large lake with lighthouses around it Well if you feel anything like I do about math, that is gonna change by the end of this video Take 1 plus 1/4 plus 1/9…

## Solving the heat equation | DE3

We last left off studying the heat equation in the one-dimensional case of a rod the question is how the temperature distribution along such a rod will tend to change over time and This gave us a nice first example for a partial differential equation It told us that the rate at which the temperature…

## But what is a Fourier series? From heat flow to circle drawings | DE4

Here, we look at the math behind an animation like this, what’s known as a “complex Fourier series”. Each little vector is rotating at some constant integer frequency, and when you add them all together, tip to tail, they draw out some shape over time. By tweaking the initial size and angle of each vector,…

## Derivative formulas through geometry | Essence of calculus, chapter 3

Now that we’ve seen what a derivative means, and what it has to do with rates of change. Our next step is to learn how to actually compute these guys, as in if I give you some kind of function with an explicit formula you’d want to be able to find what the formula for…

## Hilbert’s Curve: Is infinite math useful?

Let’s talk about space-filling curves. They are incredibly fun to animate and they also give a chance to address a certain philosophical question. Math often deals with infinite quantities, sometimes so intimately that the very substance of a result only actually makes sense in an infinite world. So the question is, how can these results…

## But why is a sphere’s surface area four times its shadow?

Some of you may have seen in school that the surface area of a sphere is 4pi*R^2, a suspiciously suggestive formula given that it’s an clean multiple of pi*R^2, the area of a circle with the same radius. But have you ever wondered why is this true? And I don’t just mean proving this 4pi*R^2…

## Cramer’s rule, explained geometrically | Essence of linear algebra, chapter 12

In a previous video, I’ve talked about linear systems of equations, and I sort of brushed aside the discussion of actually computing solutions to these systems. And while it’s true that number-crunching is something we typically leave to the computers, digging into some of these computational methods is a good litmus test for whether or…

## But what is a partial differential equation? | DE2

After seeing how we think about ordinary differential equations in chapter 1, we turn now to an example of a partial differential equation, the heat equation. To set things up, imagine you have some object like a piece of metal, and you know how the heat is distributed across it at one moment; what the…