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: mathematics
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,…
How many ways can you arrange a deck of cards? – Yannay Khaikin
Pick a card, any card. Actually, just pick up all of them and take a look. This standard 52-card deck has been used for centuries. Everyday, thousands just like it are shuffled in casinos all over the world, the order rearranged each time. And yet, every time you pick up a well-shuffled deck like this…
Curl – Grad, Div and Curl (3/3)
Both grad and div involve finding fields using partial derivatives. We’ll look at yet another useful field. Once again it involves partial derivatives. Water can flow in many different and often complex ways. Let’s look at a relatively simple case. What’s the velocity field that describes the flow on the surface of a river? We…
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…