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In the last video, we hopefully got ourselves a respectable

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understanding of how a vector-valued function works,

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or even better, a position vector-valued function, that

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is, in some ways, a replacement for traditional

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parameterization to describe a curve.

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And what I want to do in this video is just get a little bit

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of gut sense of what it means to take of a derivative of

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a vector-valued function.

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In this case, it'll be with respect to our parameter t.

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So let me draw some new stuff right here.

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So let's say I have the vector-valued function r of t,

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and this is no different than what I did in last video. x of

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t times unit vector i plus y of t times the unit factor j.

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If we were doing it in 3 dimensions, we'd add a z of t

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times k, but let's keep things relatively simple, and let's

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say that this describes a curve, and let's say the curve

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we're dealing with, t is between a and b, and this curve

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will look something like, let me do my best effort to draw

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the curve, I'll just draw some random curve here, so

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let's say the curve looks something like that.

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This is when t is equal to a, so it's going to

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go in this direction.

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This is when t is equal to b right here, this is t is equal

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to a, so this right here would be x of a, this right here is y

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of a, and similarly, this up here, this is x of b, and

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this over here is y of b.

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Now, we saw in the last video that the endpoints of these

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position vectors are what's describing this curve.

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So r of a we saw in the last video, it describes

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that point right there.

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I don't want to review that too much.

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But what I want to do is think about, what is the

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difference between 2 points?

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So let's say that we take some random point here.

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Let's say, some random t here.

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Let's call that r of t.

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Well actually, I'm going to do a different point, just because

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I want to make it a little bit clearer.

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So let's say-- I'm going to switch colors-- let's say

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that that right there is r of some t.

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Some particular t, right there.

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That is r of t.

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It's going to be, you know, a plus something.

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So that's some a particular t.

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And let's say that we want to figure out, and let's say we

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increase t by a little bit.

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By h.

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So let's say that r of t plus h, well, if we view the

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parameter t as time, we've moved in forward in time by

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some amount, so our little particle has moved

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a little bit.

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And let's say that we're over here.

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So that is that right there, in yellow, is r of t plus h.

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Just a slightly larger value for h.

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Now, one question we might ask ourselves, is how quickly is f

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changing with respect to t?

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So the first thing we might want to say, well, what's the

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difference between these two?

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If I were to take, and I want to visualize it.

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If I were to take r, the position vector, that we get by

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evaluating r at t plus h, and from that, I would

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subtract r of t.

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What do we get?

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Well, you might want to review some of your vector algebra

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but we're essentially just going to get this vector.

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Let me do it in a nice, vibrant color.

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We're going to get this vector right there, that

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I'm doing in magenta.

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So that magenta vector right there is, let me do it, that

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magenta one right there, is the vector r of t

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plus h minus r of t.

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And it should make sense, because when you add vectors,

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you go heads to tails.

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You could alternatively write this as r of t plus this

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character right here, plus r of t plus h minus r of t.

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When you add two vectors, you're adding, let me make

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it very clear, I'm adding this vector to this

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vector right here.

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You put the tail of the second vector at the

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head of the first.

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So this is the first vector, and I put the tail of the

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second there, and then the sum of those two, as we predicted,

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should be equal to this last one.

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It should be equal to r of t plus h.

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And we see that is the case, and algebraically, you would

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see that obviously this guy and that guy are

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going to cancel out.

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So hopefully that satisfies you.

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And I want to be clear.

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This, all of a sudden, this isn't a position vector.

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We're not saying that hey, let's nail this guy's tail at

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the origin and use this guy to describe a unique position.

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Now all of a sudden he's, it's just kind of a pure vector.

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It's describing just a change between two

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other position vectors.

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So this guy is right out here.

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But this vector literally describes the change.

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But say we care, and how would this look algebraically if we

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were to expand it like that?

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So this is going to be equal to, what's r of t plus h?

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That's the same thing as x of, let me do it over here.

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This is the same thing as x of t plus h times the unit vector

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i plus y of t plus h times the unit vector j, that's just that

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piece, that piece right there is that piece, minus this

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piece, so minus, I'll do it in the second line, I could have

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done it out here, but I'm running out of space.

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Minus x of t, right r of t is just x of t times i, plus, but

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I'll just distribute the minus sign, so it's minus

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y of t times j.

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Actually let me write it, this would be minus, let me

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write this way, plus this.

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So you realize that this is really just this

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guy right here.

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I'm just evaluating at t.

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So you have x of t and y of t, and then later we

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can distribute, right?

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If you distribute this minus sign, you get a minus x

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of t and a minus y of t.

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And in vector addition, you might need a little review on

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this if you haven't seen it in a while, you know that you can

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just add the corresponding components.

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You can add the x-components and you can add

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the y-components.

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So this is going to be equal to, let me rewrite it over

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here, because I think I'm going to need some space later on.

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So let me rewrite it over here.

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So I have r of t plus h minus r of t is equal to, and I'm just

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going to group the x- and the y-components, this is equal to

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the x-components added together, but this is a

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negative, so we're going to subtract this guy

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from that guy.

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So x of t plus h minus x of t, and then all of that times our

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unit vector in the x-direction, and then we'll have plus y of t

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plus h minus y of t times a unit vector the j-direction,

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I'm just rearranging things right now, and this will tell

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us what is our change between any 2 r's for given

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change in distance.

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And our change in distance here is h between any

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2 position vectors.

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Now, what I set out at the beginning of this video, I

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said, well, I wanted to figure out the change, and we're going

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to think about the instantaneous change

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with respect to t.

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So I want to see, well, how much did this change

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over a period of h?

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Instead of writing h we could have written delta t, it

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would've been the same thing.

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So I want to divide this by h.

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So I want to say, look.

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My vectors changed this much, but I want to say it's

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over a period of h.

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And this is analogous to when we do slope.

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We say rise over run, over delta y, or change in

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y, over change in x.

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This is kind of the change in our function per change in x.

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Let's just divide everything, or I shouldn't say change

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x, per change in t.

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So here, our change in t is h, right?

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The difference between t plus h and t is just going to be h.

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And so we're going to divide everything by h.

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When you multiply a vector by some scale, or divide it by

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some scale, or you're just the taking each of its components

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and multiplying or dividing by that scalar, and we

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get that right there.

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So this, for any finite difference right here, h,

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this'll tell us how much our vector changes per h.

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But if we want to find the instantaneous change, right,

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just like what we did when we first learned differential

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calculus, we said, ok.

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This is kind of analogous to a slope.

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This would be good, this would work out well for us, if the

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path under question looked something like this.

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If it was a linear path.

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If our path looked something like this.

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We could just calculate this, and we'll essentially have the

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average change in our position vectors, so you could imagine,

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2 position vectors, that's one of them.

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Well, actually, they'd all be parallel.

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Well, the position vectors, they don't have to be parallel.

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They could be like that.

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And then, this would just describe the change between

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these 2 per h, or how quickly are the position vectors

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changing per our change in our parameter, right?

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This is, the h, you could also consider, is kind of a delta t.

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Sometimes people find the h simpler, or sometimes

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they find the delta t.

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But anyway, I'm concerned with the instantaneous.

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We're dealing with curves, we're dealing with calculus.

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This would have been OK if we were just in an

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algebraic, linear world.

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So what do we do?

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Well maybe, we can just take the limit as h approaches 0.

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Let me scroll this over.

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So let's just take the limit, let me do this in a nice

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vibrant color, let's take, I'm running out of colors, the

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limit as h approaches 0 of both sides of this.

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So here, too, I'm going to take the limit as h approaches 0,

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and here, too, I'm going to take the limit as

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h approaches 0.

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So I just want to say, well, what happens, how much do I

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change per a change in my parameter t, but what's kind of

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the instantaneous change, as the difference gets smaller

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and smaller and smaller?

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This is exactly what we first learned when we learned about

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instantaneous slope, or instantaneous velocity, or

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slope of a tangent line.

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Well, this thing looks a little bit undefined to me, right now.

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We haven't defined limits for vector-valued functions, we

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haven't defined derivatives for vector-valued functions.

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But lucky for us, all of this stuff here looks

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pretty familiar.

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This is actually the definition of our derivative.

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And these are scalar-valued functions right here.

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They're multiplied by vectors, in order for us to get

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vector-valued functions.

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But this right here, by definition, this is the

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derivative, this is x prime of t.

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Or this is dx dt.

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This right here is y prime of t, or we could

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write that as dy dt.

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So all of a sudden we can define, we can say, and I'm

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being a little hand-wavy here, but I want to give you the

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intuition, more than anything.

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We can say that the derivative, we can call this expression

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right here, as the derivative of my vector-valued function r

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with respect to t, or we could call it dr dt, notice I keep

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the vector signs there.

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This is its derivative, and all it's going to be equal to, r

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prime of t, is going to be equal to, well, this is just

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the derivative of x with respect to t, is equal to x

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prime of t times the x-unit vector, the horizontal unit

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vector, plus y prime of t, times the y-unit vector, times

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j, the unit vector in the horizontal direction.

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That's a pretty nice and simple outcome.

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But the hard thing may be to a kind of visualize

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what it represents.

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So if we think about what happens, let me draw a big

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graph, just to get the visualization in a healthy way.

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So let's say my curve looks something like this.

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That's my curve.

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And let's say that this is, we want to figure out the

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instantaneous change at this point right here.

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So that is r of t.

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And then if we take r of t plus h, we saw this already, you

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know, t plus h might be something like right there.

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So this is r of t plus h.

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Right now, the difference between these two, and this is

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just the numerator when you take the difference, or how

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fast we're changing from this vector to that vector in terms

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of t, and it's hard to visualize here.

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And I'm going to do a whole video so we can think about

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the magnitudes here.

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That might be some vector.

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Well, the difference between these two is

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just going to be that.

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But then when you divide it by h, it's going to be a larger

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vector, right, if we assume that h is a small number.

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Let's say h is less than one.

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We're going to get a larger vector, right?

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But this is kind of the average change over this time.

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But as h gets smaller and smaller and smaller, this r

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prime of t is going to, its direction is going to be

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tangential to the curve.

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And I think you can visualize that, right?

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As these two guys get closer and closer and closer, the dr's

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get smaller, so the change, the dr, the difference between the

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two, the delta r's, get smaller and smaller, you can imagine if

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h was even smaller, if it was right here.

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Then all of a sudden, the difference between those two

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vectors is getting smaller.

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And it's getting more and more tangential to the curve.

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But then we're also dividing by a smaller h, so the actual

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derivative, as the limit of h approaches 0, it might be you

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know, maybe it's even a bigger number there.

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And actually, the magnitude of this vector, it's a

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little hard to visualize.

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It's going to be dependent our parameterization for the curve.

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it's not dependent on the shape of the curve.

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The direction of this vector is dependent on the shape of this

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curve, and the direction, so the direction, this will

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be tangent to the curve.

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Or you could imagine that this vector is on the tangent

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line to the curve.

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The magnitude of it is a little bit hard to understand.

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I'll try to give you a little bit of intuition on

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that in the next video.

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But this is what I want you to understand right now, because

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we're going to be able to use this in the future, when we do

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the line integral over vector-valued functions.

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