This week we're going to talk about AC signal analysis. Now before we can do that we have to have certain methamtical tools at our disposal. Now we advertise this coursera course as requiring nothing more than high school algebra and trigonometry... But we do want to use a little bit of calculus and complex numbers and complex exponential notation, but we're going to develop that in the following lecture. So the this is certainly not intended to be a comprehensive review of all of that, but I think that it will give you enough background and there will be a few simple problems throughout That should set you up to be able to understand the basics of this and start to use this as a tool. Now the moving on to the content of once mathematical tools are in place then the actual. Content, what we want to get at at the, the end of this lecture is the idea of a phasor representation of an AC signal. And phasors are used in AC circuit analysis problems and it's a very convenient and easy way to represent time dependent signals. Now, before we get to that, we have to do a brief mathematical review of the idea of a derivative and look at a few simple derivatives that we may need throughout. And then in particular we want to talk about the simple harmonic oscillator. the physics of, of that and how one can set up the differential equation that describes simple harmonic oscillator and then solve it using the simple derivatives that we've learned about. And this also brings us to the very special number e which is is a base for an exponential function like e to the x is something that pops up over and over again and naturally comes out of several differential equations problem that one can set up in calculus. and then we're going to kind of switch gears a little bit and talk about complex numbers, and then at the end we're going to put all of this together, the complex numbers and exponential notation and e, and talk about complex exponentials and in particular Euler's formula. So if you're familiar with this, then you might fast forward through this. But for those of you had not encountered this before, hopefully this will set you up to be able to do AC circuit analysis and what follows. We're going to start by talking about the idea of a derivative. And in particular the deriviatve as the limit of the slope of a line joining two points on a curve. Now the, let's starat by just drawing any old curve f of x versus x. Now let's focus our attention on a certain point, we'll call that x0, and the function has a value f of x0 at that point. Now what we're trying to do is find the instantaneous rate of change of the of this curve, and what that's equivalent to is the slope of the tangent line at that point. Now to find an expression for that, we're going to start by considering another point on the curve, delta x. Further along the x axis than x zero. And the function has a value at that point. And what I want to do is concstruct a straight line that joins those two points and I want to write down an expression for the slope of that straight line. So, The slope of this line is just the, the rise over the run. So it's the change in y value divided by the change in x value. So the change in the y value is just f, at x0 plus delta x, minus f of x0, so that's the delta y. And then the Delta x is x0 plus Delta x minus x0, down here. Now if I cancel the x0 factors I'm left with this. It's just the change in f of x divided by Delta x. Now what I want to do is take the limit as delta goes to 0. So I'm going to move this point down the curve, and as I do that the line joining the two points is going to approach this tangent line. And the tangent line then will tell you How much the f of x is changing when I have a very small change in x. That's the instantaneous rate of change of this function evaluated at the point x to 0. So if I take the limit as delta x goes to 0 of this expression up here. That's the thing that we're going to define as the derivative. Of the function f of x, evaluated at x zero. And the shorthand notation for that is f prime, so f prime means the derivative of the function f. And that f prime is evaluated at x zero in this case. Okay, now let's look at a I made a little animation to help you visualize what's going on here. So here's our curve, we have the point f of x zero, and x zero on the curve. And we're trying to Find the slope of this tangent line at that point. And so we start off with the another point, delta x, a finite distance away on the x axis. And f, the function has a different value there, so it's easy to write the slope of that line. Now, as [COUGH], what we do is we take the limit as delta x goes to zero. So that means the point I started at is moving toward x0. And so when I construct the, the, the line joining those two points when delta x gets somewhat smaller. That line is starting to be more of an approximation of the tangent line. And as delta X goes to 0 that eventually becomes the same as that blue tangent line that I drew in the first place.