hey guys welcome back in this video i'm going to teach you everything you need to know about the parametric equations um for your ap calculus bc or just if you guys are taking a different course okay so in this video i'm gonna teach you the formulas for them and what you need to know but if you guys want the proofs you're gonna have to go to a different video for it okay so you're just going to basically have to accept the formulas in this video and if you want the proofs for them you can go to my other video so these are the um six items that you need to know for the bc exam first how to find the slope then how to find the equation of the line at a given time how to find the second derivative the speed the distance and the position so let's get to it how do we find dydx well it's actually very simple take this pair of parametric equations if you were going to take the derivative of y with respect to t then d y dt would be 2t plus e to the t dx dt would be derivative sine is cosine angle times 2t well check it out if you guys divide d y dt by dx dt the dt's cancel leaving you with d y d x so literally that's all you have to do you just have to say d y d x is equal to 2 t plus e to the t over two t cosine t squared now most likely they'll ask you at a specific time so if they said at time t equals let's say one then you would put a one in there and you can actually just leave your answer as two plus e over two cosine one okay now typically these are calculator problems um so that that means that they're either number one or number two so once again to find d y d x all you have to do is take the derivative of y with respect to t the derivative of x with respect to t and then just divide them because d y dt over dx dt the t's cancel each other out so that's all how do we find the equation of the line tangent to the curve at a specific time well any time you're trying to find the equation of the line first thing that you need is a point so what is y of 2 well y of 2 because remember this is t this is t so y of 2 is going to be 4 plus 2 which is 6. what is x of 2 x of 2 is going to be 6 minus 4 that's 2. so at time t equals 2 the particle is at position 2 6. now what is the derivative because we need the slope of the line tangent to the curve so we just found it in the last page so how do we do that d y d x is equal to d y d t over d x d t and that's very simple you just say the derivative of y is two t plus one over the derivative of x which is three you plug in remember this is a t so you have to plug in the t and that's going to be two times two four plus 1 is 5 and then you get 5 3 and then you just say y minus the y is equal to m times x minus x and then you could just leave your answer just like that if i'm going too fast slow it down and uh but it's pretty basic how do we find the second derivative so i'm actually going to show you why this is in a different video but for now i just want you to know how to get it so if you want the proof for this uh just watch my other video on uh proofs of parametric equation okay so how do we do this this is going to be a little bit different first thing to find the second derivative what you're going to do is you're going to take the derivative of the first one with respect to t now remember when you took the derivative of d y dx okay it was in terms of t 2t plus 1 over 3. it was before i did it here it was in terms of t so because it's in terms of t you're going to take the derivative of d y d x by itself so you're just going to take the regular derivative of the first derivative however and again i can show you this in the proof video you have to divide it by the derivative of the x okay so in this example here the derivative of this guy i'm going to do x prime of t is 2t the derivative of this person is 2. so therefore the first derivative is going to be y over the x one and so you get one over t now we know that this could be t to the negative 1. so what do we do to find the second derivative so d2y over dx squared is going to be the derivative of this which is negative 1 t to the negative 2 divided by the derivative of the x which is 2t okay i'm going to leave it like that but i just wanted to show you how to do it so you take the derivative of the derivative and then you divide it by the derivative of the x with respect to t okay so that's how you find the second derivative it's not frequently asked but it does uh pop up here and there here are my other two formulas and um that you guys definitely need to know for this it's the speed and the distance and it's actually really simple to see it it's like a vector right and it's like a squared plus b squared equals c squared so the speed formula is going to be the square root of the derivative squared of x plus the derivative squared of y okay so take the derivative of x and square it plus the derivative of y and square it and that's how you get the speed if you integrate this you get what we call a distance and an arc length once again i will prove this in the proof of parametric equation formula the next two items that we need to do is the position um and how to find the position at a certain time is rather simple they have to start off by telling you where the particle is at a certain time so then all you have to do is say okay i'm right here and they basically have to tell you where you need to end up okay so they can say at time t equals zero x is at five and y is at three where is uh the particle's position at time t equals one so you say at time t equals 0 i'm here plus the times 0 so a to b the integral of the x because you want to know where you are in the x direction so just x prime of t and then where you are right now for y of t so where you are plus how much you traveled so same exact thing and so with that being said let me just show you a quick example of a ap problem i can choose this one here the 2016 it says here is y of t says at time t the position of a particle moving in the x y plane is given by the parametric functions x of t plus y of t where dx dt is t squared plus sine 3t squared and the graph of y is shown above at time t equals zero x equals five at time t equals zero y is equal to one that's important because they always ask you for the position they typically ask you for either an x or y but in this case they said find the position of the particle at time t equals three all you have to do is say right now in the x so they want at time t equals three so all you have to do is say at time t equals zero i'm going to be at five plus how much i travel from zero to three of dx dt okay i need a calculator for this i'm not going to do it but that should give you x of 3 okay and now y of 3 is going to be y right now is at 1 and then you need from zero to three d y d t right and so from you're going to have to figure out the integrals it's basically area uh sorry that's gonna be from here so this should cancel this out so you can have a negative so that's negative 1 and then this is 1 by 2 and so 1 by 2 divided by 2 is a negative 1 also so together that makes negative two so you're gonna get one negative two um i may have messed up something there but basically you're going to have to integrate and find the area from zero to three um i don't i think that's gonna be probably one and a half i think um okay something like that so uh here we go x of three equals x of zero plus from zero to three of x prime of t and y of 3 i'm being silly right now i'm not thinking y of 3 this is the y function so y of 3 is just the value and you're going to estimate it at negative a half my apologies um i totally forgot that wasn't uh d y d t okay so that's one of them the slope is going to be very very simple to find the slope of the tangent line you're basically going to take this at time t equals three you're gonna take d y d x is equal to d y d t over d x d t and so the slope at three here is going to be positive um two one one half and then the slope here is gonna be at three it's gonna be like nine plus sine um 27 so basically you take the derivative of y derivative x and i'm just trying to get through the video quickly so i don't waste anyone's time the next one is the speed so the speed is going to be speed is equal to dx dt squared plus dydt squared and you will find that you would need a calculator at time t equals three which is fine and then the total distance is the integral of it so basically guys that's what we need to know for all of these problems position slope speed total distance and once again speed total distance it doesn't matter line tangent to the curve you're probably talking about the slope and once again all of these are position speed distance every single problem is exactly the same so i hope this was helpful and we will try more problems in different videos