We want to take a minute and talk about this number e, and what's so special about it. Now, e is just a number. It's 2.718. And here it is to 40 or so decimal places. And it's just a number that arises in certain types of problems, that, you run into. Such as, like, compounding interest. Or in certain types of calculus problems, and I'll give you a simple example of that now. Now, it has the special property that the derivative or the slope of e to the x is equal to the value of e to the x. Now, it turns out, if I think about any base, a to the x power. So, let's just let a be any number and I asked the question, what is the slope of this curve and what is the value of that function? There's only one value of a where the slope is always equal to the value of the function. And that value is e. So that's what makes e special. Now let's illustrate where this comes from. Let's look at a very simple problem here. The population of a group of rabbits or any species. So let's say we have a colony of these rabbits and that each pair of rabbits has two offspring per year, which is probably an underestimate. But I want to know, how does the population grow over time? And we're going to make the simplifying assumption that rabbits never die, they just keep making new rabbits, so we don't have to account for that. So to set this up, let's say that y, as a function of time, is going to be the number of rabbits at time t. And y, at t equals 0, the number of rabbits I start with at t equals 0, is 2. So that's the initial pair of rabbits. Now, the statement of the problem says that, the rate of change of the number of rabbits in the colony is equal to the number of rabbits that I currently have. Now, there's a one here and not a two because it takes two rabbits to produce two more rabbit. So each rabbit produces one more rabbit, per year. Now, the solution of this, the function that solves this equation. The slope is equal to, the, the slope of this function is equal to the value of the function itself, where the function itself we know is e to the t. And I could multiply by some constant a, and it would be carried along on both sides. So it's perfectly fine to multiply by this constant. So this is the general solution of this differential equation. Now, I have to make this general solution satisfy my particular initial condition, which is at t equals 0, I have two rabbits. So if I plug t equals 0 in here, e to the 0, any number to the 0 power, is unity. And so that means that a has to be 2. So the solution that we're looking for for the number of rabbits as a function of time t is 2 times e to the t. And if I plot that, I start off initially with two rabbits. And over time, the number of rabbits increases and over the course of ten years, I have close to 50,000 rabbits. So this is an example of exponential increase of a population. And so, the whole point of this is that this number e comes about in certain types of problems. This is a very naturally occurring problem. Another example is like radioactive decay. Where the number of decaying atoms at any, or decaying nuclei at any particular time is proportional to the number of nuclei that I have at that time. And so, that would give me a decreasing exponential function. but e pops up all the time from certain naturally occurring problems. Let's take a minute now and look at the derivatives of sine and cosine. So I'm going to start by plotting just one cycle of sin wave here. And we said before that the derivative of sine of t is cosine t. So that says that the slope of the tangent line to every point on the sine curve is equal to the cosine at that point. So here's a plot of one cycle of a cosine wave, and let's just take a look at what the, the slope of the tangent lines would be. So here, at t equals 0, if I draw the tangent line, it is a positive slope, and it turns out to be maximum at this point. As I go up the hill, the slope starts to decrease. And when I get to the top of the hill, the slope is 0, so it's a horizontal tangent line up here. Now, the function starts heading down the hill, and the slope of the tangent line is negative. And it becomes maximum, its maximum negative valueaAt the crossing point. And then it starts to level out again, and goes back to 0. So the slope here is 0. And now the slope with the tangent line is positive. And it reaches it's maximum value here at the crossing point. Now, if I take the derivative of cosine. That is minus sine of t. So let's do the same kind of construction. And say, okay here on the cosine curve at t equals 0. The slope is 0, the slope at the tangent line is 0. So that's right there. And now, I'm heading down the hill, the slope is negative. So, the and it becomes maximum negative at the crossing point. So here's the crossing point. And as I keep going down into the valley of cosine curve, the slope is always negative, but it's coming back towards 0. So the slope of the tangent line at the bottom of the valley is zero. And now, the slope turns positive and I'm heading up the hill and the slope becomes maximum at the crossing point. And then after I go through the crossing point, it starts to level off and the slope goes back to 0. Now, the last if I take the derivative of this minus sign t. Then that's just minus cosign t because if I just multiply, I'm just multiplying both sides of this equation by negative 1. So that derivative is then minus cosine t. Now, if I take the derivative of minus cosine t. That's just like this equation, multiplied by negative 1. And so that's going to be plus sine t. And so that brings me back to where I started. So sine and cosine have this interesting property that if I take the derivative four times, I'm back to the original function again.