 Hi, I'm Zor. Welcome to Indizor Education. Today I would like to talk about some kind of synergy between geometry and derivatives. Well, there is actually a subject called differential geometry, which is a very complex subject that's usually started in universities. But there is something like a beginning of that thing, which is relatively simple, and that's what I'm going to introduce to you today. This lecture is part of the advanced course of mathematics for teenagers and high school students presented on Unizor.com. I recommend you to watch this lecture from this website, because, number one, it has very detailed notes for each lecture, and number two, if you sign in, then you can actually take exams, for instance, and basically participate in the educational process, which is very helpful, I believe. Alright, so we are talking about certain geometry things. Well, so far we used to talk about functions and their derivatives. And the function, if it's represented by a graph, then derivative is basically the tangent of tangential line, of the angle between tangential line and positive direction of the x-axis. Now, this is a function, so it helps us to analyze smooth curves on the plane. Not all smooth curves of the plane are graphs of some smooth functions. For instance, this is also relatively smooth. Well, excuse my drawing, but I actually wanted to draw a smooth curve. But it's not a graph of any function, because for a single value of the argument x, we have more than one values of the function. So it's not a function in the traditional sense of the word. So how can we analyze any relatively smooth curve on a plane, including these ones? That's what this is all about. That's what this lecture is about. So first of all, we will talk about how to define a curve of that type on the plane without using something like this, this algebraic representation of the function. And here is what we will do, which is actually a very simple thing. It's called parametric representation of the curve. You see, every point on the curve has x and y coordinates, right? So I can always say that my x is some kind of a function of some parameter t, and my y is a function, as t belongs to some interval. Finite or infinite doesn't really matter. So for instance, if t is equal to zero, these two functions give you one point. Then as t goes, let's say, increases, then the next point will be actually part of this curve. And if my functions, this one and this one, are smooth functions, then our curve will always be smooth as well. I mean, that's kind of a non-rigorous definition of the smoothness. In our case, what we will assume is differentiability after whatever necessary level of derivative. The first derivative should exist, or first and derivatives should exist. It doesn't really matter, but some kind of a level of smoothness we assume with these two functions, which will ensure that this function is also smooth. Now we have basically exactly the same problem as we have with a regular function. What is a tangential line, and would be nice to have an equation of the tangential line. Now, so let's take any point, for instance this point, x0, y0, and have a tangential line. And what we want is this angle, which will give us basically enough information to come up with the equation of the tangential line. So that's the task, and that's what we are going to basically address right now. Alright, I don't need this primitive picture, we need something more basic, which would give us basically the same information. So this is part of our curve, and this is the point, let's say, x0, y0, where I would like to draw a tangential line, so this is x of t, y of t, any point. So my question is, how can I basically come up with the equation of the tangential line. Alright, first of all, if you have a point through which a line actually goes, then you can immediately have an equation of this line, which goes through the point x0, y0. That's kind of obvious, right? Because if x equals to x0 and y equals to y0, then we have the equation here, right? I mean, it obviously can be somehow expressed differently, like y is in the form y is equal to ax plus b. Yes, it can be done in this way, and you will substitute x0 and y0, and you're supposed to have equality, and then you will get this, where m is actually some kind of unknown slope of this particular line. So all we need actually is a slope. You see, since we have one point, basically we have how many unknowns we have in the equation of the line? Two, right? a and b. So we need some kind of two equations or two points through which line is going. x0, for instance, is one of them, and some kind of other point, if we know. Now we have two equations with two unknowns, a and b, and we can actually result if we know two points, x0, y0, and x1, y1. And then we can just calculate what's a and b. Now in this form, we have already resolved it once, and all we need to do is basically to find out m, which is the slope. Now, let's think about how can we get to a tangent. Well, let's just take any other point, x1, y1, and let's do a second, the second line. Now, obviously, if x1, y1 is getting closer and closer to x0, y0, or considering that x0 is equal to x of t0, and y0 is y of t0, and x1 is x of t1, and y1 is y of t1. So this is the position of the point when our parameter is equal to t0, and this is the position of the point where our parameter is equal to t1. That's how we get x0, x, y0 as one point and x1, y1 another point. If this point is getting closer to this one, or which is the same thing if t1 is converges to t0, then our second will be closer and closer to a tangential line, and the limit will be exactly tangential line. So if I would like to know the slope of the tangential line, I can actually get the slope of the second and have the limit as t1 goes to t0. So what is the slope of tangential line? Well, now we know we have two points. If we have two points, we can actually determine the slope very easily. So what is this? This is y1, y1 minus y0, y1 minus y0. And this is x1, this is x1, this is x0, this is y0 and this is y1. So it's y1 minus y0 divided by x1 minus x0, right? That's the slope of the tangential line. Now, what happens if you start getting closer, t1 getting closer to t0? Well, this will be infinitesimal and this will be infinitesimal, so we don't really know yet what will be the ratio. But here is what we can actually say about this. y of t1 minus y of t0, right? That's what y1 minus divided by x of t1 minus x of t0 equals to y of t1 minus y of t0 divided by t1 minus t0 divided, this is another division, x of t1, let's put it explicitly, minus x of t0 divided by t1 minus t0. Obviously, this is equal to this. I just divide it and multiply it by t1 minus t0, right? Because this is a division, so it's actually inverted. So x1 minus x0 goes to the denominator, t1 minus t0 to the numerator, and t1 minus t0 is cancelling out. Now, as t1 goes to t0, what is this? Well, this is basically the ratio which was used to define a derivative of function y by its argument t at point t0. So, when we are talking about the limit of this, this piece will go to y derivative at point t0, or let me just write it differently, more, I would say, derivative by t at point t equals to t0 of function y of t. That's what it is, right? That's what this converges to when t1 goes to t0. And this one is obviously converging to a derivative at point t is equal to t0 of function x of t. So, this is the limit. So, this is exactly that m in this equation, okay? So, what we have done, we basically expressed the tangent of the angle which is formed between the tangential line at point x0, y0, and the positive direction of the x-axis through derivatives of each coordinate function x of t and y of t. So, if our curve is given to us by these two functions of some parameter t, then differentiation by this parameter t at any point gives me the slope of the tangential line. And considering I know which point we are talking about x0 and y0, which are x and y at point t0, we have the equation of our tangential line. So, very simply, just the ratio of two derivatives, all right? And I think it would be nice if we can actually do some kind of example to illustrate this. So, as our example, I will consider a very simple parameterized curve, a unit circle, unit circle with a radius one. And now I would like this point, which is at 45 angle, to be the point where I would like to draw a tangential line. So, this is an angle. Now, obviously, this is 45, then this is also 45, so this is 135 degrees, right? So, we can always find the tangential line using this. But I would like to use, instead of this trigonometry, I would like to use whatever I was just talking about, how to determine my equation for this line using this parameterized methodology. Okay, well, not that I will escape trigonometry because it's obviously about angles. So, what is my parameterized description of the circle? So, if I have any point, then what is its x-coordinate and y-coordinate? Well, it depends on what the parameter is, right? So, the parameter can be an angle, right? It doesn't matter what the parameter is. So, for instance, I assume parameter is an angle. I can as well have a parameter as the length of this particular part of the circle from this point to whatever my point is. Yes, I can. But in my case, I think it's more convenient to take the angle as a parameter. Now, if angle is a parameter, then, and it's basically the definition of the trigonometric functions, my x is cosine of this angle. Let's use t. So, angle is t. And y of t is sine of t, right? This is y. This is x. This is t. This is cosine. This is sine. This is basically definitions of trigonometric functions, right? On the unit circle. So, basically, I have my circle defined using these two functions. I cannot define it using function like y is equal to f at x because it's not a graph of any function. It has two halves. Each half probably can be, I mean, obviously can be like square root. y is equal to square root of whatever, 1 minus x square. Or this piece, y is equal to minus this square root. But if I would like to describe the whole curve, this is the only way how I can do it. Using the parameters. All right. So, I have this. Now, in particular, I'm interested at point t0 which is equal to 45 degrees. Or pi over 4. Let's use regions. So, t belongs to 0 to pi. When t is from 0 to 2 pi, I have exactly one circle. So, my t0, the angle, the parameter, if you wish, where I would like to have my tangential line is pi over 4. Okay. So, what I need to do is I have to take the derivative of the y function. y of t is equal to cosine of t. Right? Now, derivative of x is minus, from cosine, it's minus sine of t. And I have to put t is equal to t0 which is pi over 4. Which actually brings me to y at t0 is equal to cosine of pi over 4 is square root of 2. Divided by 2. And minus sine x at t0 derivative is minus square root of 2 over 2. Now, their ratio is minus 1. So, my equation is 1 minus y0 which is at pi over 4. It's equal to square root of 2 by 2 is equal to m which is minus 1 times x minus square root of 2 over 2. So, that's the equation. Or if you wish we can regroup it, it would be y is equal to minus x. This is minus its plus and this plus plus square root of 2. So, that's my equation of the tangential line. That's it. So, basically the whole mechanism of parameterized curve is very important. Because using this mechanism you can describe smooth but any kind of a curve on the plane. Not necessarily the curve which is a graph of some nice and smooth function, right? So, this is the way how you can define for instance something like this, a spiral, right? So, no matter how your curve is defined as long as it is defined using some parameterized expressions for x and y, then you can take their derivatives at certain point and the ratio of derivative of y over derivative of x will give you the slope and using an equation similar to this where y0 and x0, the points where you would like actually to have your tangential line there are known. So, that gives you the equation of tangential line which is very important in some cases. All right? Okay, I do recommend you to read whatever the notes are for this lecture on Unizor.com. And basically that's it. Thank you very much and good luck.