Okay folks. We were talking about doing calculus on parametric curves. What I mean by that is, for example, suppose you have a parametric curve. Consider the parametric curve whose parametric equation is x is some function of t and y sub function of t. And t is between a and B. And this traces out some curve, it doesn't matter what it looks like. Okay? Something like that. That's your curve C. Okay? So what are the things we want to do? We want to find slope of C and the concavity of C at some point P, let's say it's this. Okay? What's the slope? Slope is what's the slope of the tantent line and what's the concavity there? Of course, you can see in the picture, it seems that it's concave down. But that's a generic picture. How do we find these things? Parametric curves. All right? The idea is that for of the equals f, right? If your curve is on the form y equals x, okay? And you're at the point, then slow is dy by dx and concavity at d2y by dx squared. Okay? But how does this apply to curves which are given in this parametric form? All right? So the idea is that if your curve is in this parametric form, right, and you're at this point say, and you want to find the slope and the concavity at that point, okay? And the is given here by, in this form, x equals x, t, y equals y, t and t between a and B, okay? Now, I know the formula at the point. If the curve could be given in the form y equals x, but you can do that, okay? What we can do is x is given in terms, as a function of t. In principle, one can solve for t, like for example, x is t squared. You can solve for t in terms of x. One can solve for t in terms of x. Let for example say it was x equals t square, y is something doesn't matter. Then you could, in principle, solve for t plus or minus square root of x. Then plug it in, and you would get something like in terms of x, right? In principle, one can do this. It's not always easy. The equation is not always simple. Like that, if your curve is of the form x equals some function of t, y equals some function of t, then can write t in terms of x. And then y equals y becomes y equals Y of t of x. Okay? So therefore you dy by dt would be, sorry, dy by dx using the chain rule. Dy by dx would be dy by T. Times dt by d x, right? Which is d by dt divided by dx by dt, okay? So what it says is that if you want to find dy by dx by the parametric curve, just compute dy by dt and dx by dt, and you have by dx, You don't have to actually solve for in terms of x or anything like that, okay? So for parametric curves, dy dx is just dy by dt divided by D xt, which people write as for shorthand, y over x dot, dot means t, okay? Dt is not the same thing as D x. You have to make a distinction, okay? That's one formula which is going to be very handy whenever you want. By dx, you do dy by dt times dt by dx, like you do that and then you flip it over, okay? So that's taking care of y by dx. How do we find second derivative? Okay, same idea apply. Okay? I want d2y by dx squared, which means I want d by dx of dy by dx by dx has already been computed. Let's write this by dx has already been computed. Only problem is this is in terms of and by dx has been computed above, but it is in terms of t, correct. But I want to take the x derivative. What I'm going to do is compute the x derivative. I will use the chain rule again. It's going to be d by dt of dy by dx times dt by dx. Okay? Which is d by dt, dy by dx divided by dx by dt. So that's how you compute the second deliver. Okay, Let's apply this to an example. Find the slope and the concavity of the cur. X equal t cubed minus two. Y is t square minus t at the point corresponding to equals one. That is that t is one. It is one minus one. Okay. The shape of the curve is unimportant, but A looks like this. This is the x, y lane. The curve looks like this. That's a curve. C minus, sorry, it's wrong. Hydrotro, it goes through this point, it comes like this. Minus 10, which is this point. Okay? So I want to find the slow and the concavity at this point. Well, slope dy by dx, which is dy by dt divided by dx by dt. We can compute here dx by dt is three t squared minus two. Dy by dt is t minus one. Therefore, slope, which is y by tx, is three t squared minus two over to t minus one. Therefore, slope at t equals one, that is at minus 10, is equal to the value of this at t equals one. Which turns out to be 101, which is one. That's a slope at t equals one. Okay? Remember that we found by dx here, because I'm going to need for the next step, concavity, okay? Which is the sine of d2y, dx squared. Okay? So I want d2y by dx squared, which is D x of Y by dx by dx, is there, Okay? It's a function of t though. And I'm doing x here, so you change it to a t by chain rule. Bat of y bx times x, which is bad x divided by x t. This I already know, D x bt I computed before. I just need bat of dx, it's three square minus two siva x over two minus one. Okay? Now this is, you can apply the quotient rule. It's t minus one times 60, minus two times three t squared minus 22t minus one squared. Okay? Now, I'm not going to simplify it. In fact, you don't need to simplify it because we're only interested at t equals one. This is equal to one times six minus two times 11 squared, that's four. That's just this. Okay? A t equals one, dx by dt, which is three, T squared minus two is also one. Therefore, d by dx of y by dx. Okay? Sorry, this is yeah, d2y by this is using starar's. D by dt, dy by dx divided by dx by dt. This is at t equals one. At t equals one, and we figured out side, it's 401, which is four, okay? Greater than zero. So it's concave up. That t equals one. The second derivative is actually more than just concup. It actually tells you the curvature and all that stuff. I haven't drawn the figure correctly here. It doesn't look right. Okay, so this is how we do it. You want dy by dx, You go by dt divided by dx by dt. And you compute those from your parametric equation. Okay? And then you can find it at t equals one. So that's a slope. You want the second derivative, you want d by dx, dy by dx. You need to differentiate this quantity with respect to x, but it's all in t. You change it to d by dt times dt by dx. Or d by dt, dx by dt. Once you bring this in, the denominator flips. I compute the numerator here and the denominator we already know, put everything together. All right. That's how we find slopes. Slopes and curvature. That's part of it. Now we are other calculus we do with curves as find area. Okay, certainly. Let me just summarize that. Okay? For a aametric curve, x equals x t, y equals y t between a and B, y by dx is equal to y over x dot is by dt. Okay? And then if you want the second derivative, that's the same thing as x, sorry, you want to compute the second derivative we've shown. It's the same thing as bat divided by D x batt. All right, that's the things we do, okay? The second thing which we do is area enclosed by parametric curves. So normally we do this, all right? If you have x and you have y, okay? If you say curve, and then there's another curve, Px, and someone says find this area, or A to B, okay? And you want to find this area. This area is equal to integral A to B Y at the top, minus y at the bottom, D x, so, so it's just okay. But what about curves are enclosed by more general regions, not just piles, effects and les X in these vertical lines, right? Parametric curves can enclose more area, the special case general. Okay? It's the following proposition. So suppose you have, suppose you have some loop here. We'll see something more general in a second, okay? And this is the curve, and you want to find this area. So you want to find this area. Let's say we know the parametric equation of this curve C. When you have the parametric equation of the curve x equals x t, y equals y t. Let's be the loop. When you go from A to B, suppose it does the loop. Which, which represents the loop. Represents the loop. The area enclosed by C is the same form, it's y dx. Then the area of the region enclosed by integral y dx. Over when you go from A to B, this curve C will trace a path, right? It's a particle. If it goes around anticlockwise, then the area is this. If it covers clockwise and just the negative area clockwise. Okay. So you look at the parametric equation, whatever your thing is, and look at it going this says anticlockwise. Sorry, this is I want to do Yx. Yeah, sorry. This is clockwise. Just let me make sure I'm not making a mistake here. Yeah, that's correct, This is clockwise. Positive sign is with the clockwise. Let me this goes clockwise and negative, this goes anti clockwise. Just bill check. Yeah, correct. Okay, so you say, you know, how does this formula tie in with the earlier results Like is this captured in what we were doing before? Okay, well, I'll come back to that. Let's just do an example
Calculus of parametric curves, Part A Math 242 Fall 2023
From Rakesh Rakesh November 16, 2023
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