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# Vector-valued functions differentiation

AP.CALC:
CHA‑3 (EU)
,
CHA‑3.H (LO)
,
CHA‑3.H.1 (EK)

## Video transcript

in the last video we hopefully got ourselves a respectable understanding of how a vector-valued function works or even better a position vector-valued function that is in some ways a replacement for our traditional parametrizations to describe a curve and what I want to do in this video is just get a little bit of a gut sense of what it means to take of a derivative of a vector-valued function in this case it'll be with respect to our parameter T so let's let me let me draw some new stuff right here so let's say I have the vector valued function R of T R of T and this is no different than what I did in the last video X of T times unit vector I plus y of T times the unit vector J if we're dealing in three dimensions we'd add a Z of T times K but let's keep things relatively simple and let's say that this is this describes a curve and let's say the curve let's say we're dealing with between T is between a and B and this curve a little look something like let me do my best effort to draw the curve just draw some random curve here so let's say the curve looks something like that this is when T is equal to a so it's going to go in this direction this is when T is equal to B right here this is T is equal to a so this right here would be X of a this right here is Y of a and similarly this up here this is X of B and this over here is y of B now we saw in the last video that this the the the the endpoints of these position vectors are what's describing this curve so R of a we saw in the last video it describes that point right there I don't want to review that too much but what I want to do is think about what is the difference between two points so let's say that we take some random point here let's say some random T here let's call that R of T let's say laughter I'm going to a different point just cuz I would make it a little bit clearer so let's say switch colors let's say that that right there is our of some tea some particular tea right there that is our of tea it's going to be you know a plus something so that's some particular tea and let's say that we want to figure out and let's say we we increase tea by a little bit by H so let's say that R of T plus h well if we view the parameter T as time kind of move we've moved in forward in time by some amount so our little particle has moved a little bit and let's say that we're over here so that is that right there and yellow is R of T plus h just a slightly larger value for H now one question we might ask ourselves is how quickly is our changing with respect with respect to T so the first thing we might want to say well what's the difference between these two if I were to take if I were to take and I want to visualize it if I were to take R the position vector that that's the or that we get by evaluating R at T plus h and from that our subtract R of T R of T what do we get well you might want to review some of your vector algebra but we're essentially just going to get this vector we're going to be doing in a nice vibrant color we're going to get this vector right there that I'm doing in magenta so that magenta vector right there is let me do it that magenta one right there is the vector R of T plus h minus R minus R of T and it should make sense because when you add vectors you go heads to tails you could alternately write this as R of T R of T plus this character right here plus R of T plus h minus R of T it when you add two vectors you're adding let me make it very clear I'm adding this vector to this vector right here you put the tail of the second vector at the head of the first so this is the first vector and then I put the tail of the second there and then the sum of those two as we predicted should be equal to this last one it should be equal to r of t plus h and we see that is the case and even algebraically you would see that obviously this guy and that guy are going to cancel out so hopefully that satisfies you and I want to be clear this all of a sudden this isn't a position vector we're not necessary we're not saying that hey let's nail this guy's tail at the origin and use this guy to describe a unique position now all of a sudden he's it's just kind of a pure vector it's describing just a change between two other position vectors so these got this guy is right out here but this vector literally describes the change but let's say we care and and how would this look algebraically if we would expand it like that so this so this is going to be equal to this is going to be equal to what's R of T plus h that's the same thing as X of I want to move it over here this is the same thing as X of T plus h times the unit vector I plus y of T plus h times the unit vector J that's just that piece that piece right there is that piece - - so this piece so - I'll do it in the second line I could have done it out here but I'm running out of space minus X of T right rft is just X of T times I plus but I'll just distribute the minus sign so it's minus y of T times J Y actually let me write it this would be - let me write it this way Plus this so you realize that this is really just this guy right here I'm just evaluating at T so you have X of T and Y if T and then later we can distribute right if you distribute this minus sign you get a minus X 50 and a minus y of T and in vector addition and this might be you might need a little review on this if you haven't seen in a while you know that you can just add the corresponding component you can add the X components and you can add the Y components so this is going to be equal to let me rewrite it over here because I think I'm going to need some space later on so let me rewrite it over here so I have R of T plus h minus R of T is equal to now I'm just going to say I'm just going to group the X and the y components this is equal to the X components added together but this is a negative so we're going to subtract this guy from that guy so X of T plus h minus X of T and then all of that times our unit vector in the X direction and then we'll have plus y of T plus h minus y of T times the unit vector in the J direction I'm just rearranging things right now and this will tell us this will tell us what is our change between any two between any two R's for a given change in distance and our change in distance here is H between any two position vectors now what we what I set out at the beginning of this video I said well I want to figure out the change and we're going to think about the instantaneous change with respect to T so I want to see well how much did this change over a period of H over a period of H we could have instead of writing H we could have written delta T it would have been the same thing so I want to divide this by H so I want to say look I changed I my vector has changed this much but I want to say it's over a period of age and this is analogous to when we do slope we say rise over run over Delta Y over change in Y over change in X this is kind of the change in our function per change in X so let's just divide everything or I shouldn't say change an expert' change in T so here our change in T is H right the difference between t plus h and t is just going to be H and so we're going to divide everything by H when you multiply a vector by some scale or divide it by some scalar you're just taking each of its components and multiplying or dividing by that scalar and we get that right there so this for any finite difference right here H this will tell us how much our vector changes per H but if we want to find the instantaneous change just like what we did when we first learned different differential calculus we said okay this is kind of analogous to a slope this would be good this would work out well for us if the path if the path looks something like if the path under question looks something like this if it was a if it was a linear path if our path looks something like this we could just calculate this and we'll essentially have the average change in our position vectors so you could imagine two position vectors that's one of them well actually they'd all be parallel well they don't always well the position vectors they don't have to be parallel but well I won't they don't have to be parallel they could be like that and then this would just describe the change between these two per H or how quickly are the position vectors changing per hour change in our parameter right this is this the H you could also consider is kind of a delta T sometimes people find the H more a simpler sometimes they find the Delta T but anyway I am concerned with the instantaneous we're dealing with curves we're dealing with calculus this would have been okay if we were just in an algebraic linear world so what do we do well maybe we can just take the limit as H approaches zero so let me scroll this over so let's just take the limit let's just let me do this in a nice vibrant color let's take I'm running out of colors the limit as H approaches zero of both sides of this so here too I'm going to take the limit as H approaches zero and here too I'm gonna take the limit as H approaches zero so I just want to say well what happens how much do I change per a change in my parameter T if but what's kind of the instantaneous change is the difference gets smaller and smaller and smaller this is exactly what we first learned when we learned about instantaneous slope or instantaneous velocity or slope of a tangent line well this thing looks a little bit undefined to me right now we haven't defined limits for vector valued functions we haven't defined derivatives for vector valued functions but lucky for us all of this stuff here looks pretty familiar this is just this is actually the definition of our derivative and these are scalar valued functions right here they're multiplied by vectors in order for us to get vector valued functions but this right here by definition this is the derivative this is X prime of T or this is DX DT this right here is Y prime of T or we could write that as dy DT so all of a sudden we can define we can say and I'm being a little hand wavy here but I want to give you the intuition more than anything we can say that the derivative we can call this expression right here as the derivative of my vector valued function R with respect to T or we could call it dr/dt notice I keep the vector signs there this is this derivative and all its going to be equal to all it's going to be equal to R prime of T is going to be equal to well this is just the derivative of X with respect to T is equal to X prime of T times the X unit vector the horizontal unit vector plus y prime of T Y prime of T times the Y unit vector times J the unit vector in the horizontal direction that's a pretty nice and simple outcome but the harder thing may be is to kind of visualize what it represents so if we think about what happens let me draw a big a big graph just to get the visualization going in a healthy way so let's say my curve looks something like this that's my curve and let's say that this is we want to figure out the instantaneous change at this point right here so that is R of T and then if we take R of T plus h we saw this already you know T plus h might be something like right there so this is R of T plus h now right now the difference between these two and this is just the numerator when you take the difference or how fast we're changing how fast we're changing from this vector to that vector in terms of T and it's hard to visualize here and I'm going to do a whole video so we can think about the magnitude here that might be some that vector well the difference between these two is just going to be that but then when you divide it by H it's going to be a larger vector right if we assume H is a smaller is a small number let's say H is less than 1 we're going to get a lot right but this is kind of the average change over this time but as H gets smaller and smaller and smaller as H gets smaller and smaller and smaller this R prime of T is going to its direction is going to be tangential to the curve and I think you can visualize that right is these two guys get closer and closer and closer and closer the DRS gets smaller so the change the dr the difference between the two the delta R is get smaller and smaller you could imagine if H was even smaller if it was right here then all of a sudden the difference between those two vectors is getting smaller and it's getting more and more tangential to the curve it's getting more and more tangential to the curve but then we're also dividing by a smaller H so the actual derivative as the limit as H approaches zero it might be you know maybe it's even a bigger number there and actually the magnitude of this vector it's a little hard to visualize it's going to be dependent on a parametrizations for the curve it's not dependent on the shape of the curve the direction of this vector is dependent on the shape of this curve and the direction so the direction this will be tangent tangent to the curve or you could imagine that it this vector is on the tangent line to the curve the the magnitude of it is a little bit harder to understand I'll try to give you a little bit of intuition on that in the next video but this is what I want you to understand right now because we're going to be able to use this in the future when we when we do the line integral over vector valued functions
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