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## Differentiating parametric curves

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# Vector valued function derivative example

## Video transcript

What I want to do in this
videos is to make to parametrizations of essentially
the same curve, but we're going to go along the curve
a different rates. And hopefully we'll be able to
use that to understand, or get a better intuition, behind what
exactly it means to take a derivative of a position
vector valued function. So let's say my first
parametrization, I have x of t is equal to t. And let's say that y of t
is equal to t squared. And this is true for t is
greater than or equal to 0, and less than or equal to 2. And if I want to write this
as a position vector valued function, let me write this. x1, call that y1, and let me
write my position vector valued function; I could say r1-- I'm
numbering them because I'm going to do a different version
of this exact same curve with a slightly different
parametrization --so r1 one of t, we could say is x1 of t
times i-- the unit vector i --so we'll just say t times i
plus-- this is just x of t right here, or x1 of t; I'm
numbering them because I'll later have an x2 t --plus
t squared times j. And if I wanted to graph this,
I'm going to be very careful graphing it because I really
want to understand what the derivative means here. Try to draw it
roughly to scale. So let's say that this is
one, two, three, four. Then let me draw my x-axis. That's good enough. And my x-axis, I want it to be
roughly to scale, one and two. And so at t equals 0, both my x
and y coordinates are at 0-- or this is just going to be the 0
vector, so this is where we are a t equals 0 --at t equals 1
this is going to be one times i-- we're going to be just
like that --plus 1 times j. 1 squared is j, so we're
going to be right there. And then at t is equal to 2,
we're going to be at 2i. So 2i-- you could imagine 2
times i would be this vector right there --2 times i plus
4-- 2 squared is 4 --4 times j, so plus 4 times j. If you add these two vectors
heads to tails, you're going to get a vector that's
end point is right there. The vector is going to
look something like this. So this is what, just to
make it clear what we're doing, that's r1 of 2. This is r1 of 0. This is r1 of 1. But the bottom line is the
path looks like this: it's a parabola. So the path will
look like that. Now that's in my first
parametrization of it. Actually, let me draw a
little bit more carefully. I want to get rid of this
arrows, just because I want it to be a nice clean drawing. So it's going to be a parabola. Let me get rid of that other
point, too, just because I didn't draw it exactly where
it needs to be; it needs to be right there. And my parabola, or part of
my parabola is going to look something like that. All right. Good enough. So this is the first
parametrization. Now I'm going to do this exact
same curve, but I'm going to do it slightly differently. So let's say I'll do it
in different colors. So x2 of t, let's it equals 2t. And y2 of t, let's say
it's equal to 2t squared. Or we could alternatively write
that, that's the same thing as 4t squared, just phrasing
both of these guys to the second power. And then let's say instead of
going from t equals 0 to 2, we're going to go from
t goes from 0 to 1. But we're going to see,
we're going to cover the exact same path. And our second position vector
valued function, r2 of t, is going to be equal to 2t times i
plus-- I could say 2t squared --4t squared times j. And if I were to graph this guy
right here, it would look like-- let me draw my axes
again; it's going to look the same, but it's I think useful
to draw it because I'm going to draw the derivatives and
all that on it later. One, two, three, four. One, two. And then let's see what happens
when t is equal to 0-- or r of 0; all these are going to be 0,
we're just going to have the zero vector; x and y are both
equal to 0 --when t is equal to 1/2 what are we
going to get here? 1/2 times 2 is 1. And then we're going to
get the point 1/2 squared is 1/4 times 4 is 1. So when t is equal 1/2 we're
going to be at the point 1, 1. And when the t is equal
to 1 we're going to be at the point 2, 4. So notice the curve is
exactly, the path we go is exactly the same. But before we even do the
derivatives, these two paths are identical. I want to think
about something. Let's pretend that our
parameter, t, really is time. And that tends to be the
most common, that's why they call it t. It doesn't have to be time,
but let's say it is time. So what's happening here? In the first parametrization
when we go from 0 to 2 seconds we cover this path. You can imagine after 1
second the dot moves here, then it moves there. You can imagine a dot moving
along this curve, and it takes two seconds to do so. In this situation we have a dot
moving along the same curve, but it's able to cover the same
curve in only one second; and half a second it gets here. It took this guy one
second to get here. In a one second, this guy's all
the way over here; this guy takes two seconds
to go over here. So in this second
parametrization even though the path is the same, the curves
are the same, the dot is faster. I want you to keep that in
mind when we think about the derivatives of both
of these position vector valued functions. So just remember the dot is
moving faster for every second it's getting further along the
curve than here; that's why it only took them one second. Now let's look at the
derivatives of both of these guys. So the derivative here, so if I
were to write r prime, r1 prime of t-- let me do that in a
different color, actually, already used the orange; so let
me do it in the blue --r1 prime us t. So the is the derivative now. It's going to be, remember,
it's just the derivative of each of these times
the unit vectors. So the derivative of t with
respect to t, that's just 1. So it's 1 times i. I'll just write 1i plus-- I
didn't have to write the one there --plus the derivative
of t squared with respect to t is 2t plus 2t j. And let me take the
derivative over here. r2 prime of t. The derivative of 2t with
respect to t is 2, so 2i, plus the derivative
of 4t squared is 8t. 2 times 4, it is rt. Just like that. Now the question is, what do
their respective derivative vectors look like at
different points? So let's look at, I don't know,
let's see how fast they're moving when time is equal to 1. So let's take it at
a specific point. This is just the general
formula, but let's figure out what the derivative
is at a specific point. So let's take r1 when
time is equal to 1. And I want to take this
specific point on the curve, not the specific point in time. So this point on the curve
here is when time is equal to 1, you could say second. This point over here, which
is the exact corresponding point, is when time
is equal 1/2 second. So r1 of 1 is equal to--
we're taking the derivative there --is equal to 1i. It's not dependent on t at all. So it's 1i plus 2
times 1j, so plus 2j. So at this point the derivative
of our position vector function is going to be 1i plus 2j. So we can draw it like this. so
if we do 1i is like this: 1i. And then 2j. Just 2j is like that. So our derivative right there,
I'll do it in the same color that I wrote it in. It's in this green color; it's
going to look like this. And notice it looks like, at
least its direction is-- let me do it a little bit straighter
--its direction looks tangent to the curve; it's going in
the direction that my particle is moving. Remember my particle is moving
from here to there, so it's going in the direction. And I'm going to think about,
in a second, what this length of this to derivative
vector is. This right here, just to
be clear is, r1 prime. It's a vector, so it's telling
us the instantaneous change in our position vector with
respect to t, or time, when time is equal to 1 second. That's this thing right here. Now let's take the exact same
position here on our curve. But that's going to occur at a
different time for this guy. We already said it only takes
him, he's here at time is equal to 1/2 second. So let's take-- --I'll
do it in the same color --so here we have r2. We're going to evaluate it at
1/2 half because this is at time is equal 1/2 second. And this is going to be equal
to 2i-- this isn't dependent at all on time --so 2i
plus 8 times the time. So time right here is 1/2. So 8 times 1/2 is 4. So plus 4j. So what does this look like? The instantaneous
derivative here. Oh, and this is the derivative;
have to be very clear. So 2i-- let me draw some
more --so 2i maybe gets us about that far. Plus 4j will get us up
to right around there. Plus 4j is that factor. So when you add those two heads
to tails, you get this thing: you get something that-- let me
like --you get something that looks like that. I didn't draw it as neatly
as I would like to. But let's notice something:
both of these vectors are going in the exact same direction. They're both tangential to
the path, to our curve. But this vector is going, its
length, its magnitude, is much larger than this
vector's magnitude. And that makes sense because I
hinted at it when we first talked about these vector
valued position functions and their derivatives; is that the
length, you can kind of view it as the speed. The length is equal to the
speed if you imagine t being time and these parametrizations
are representing a dot moving along these curves. So in this case, the particle
only takes a second to go there, so at this point in its
path, it's moving much faster than this particle is. So if you think about it, this
vector right here, if you imagine this is a position
factor, this is velocity. Velocity is speed
plus the direction. Speed is just you know,
how fast are you going? Velocity is how fast you're
going in what direction? I'm going this fast-- and you
could calculate it using the Pythagorean Theorem, but I just
want to give you the intuition right here --I'm going that
fast in this direction. Here I'm going this fast;
I'm going even faster. That's my magnitude,
but I'm still going in the same direction. So hopefully you have a gut
feeling now of what the derivative of these position
vectors really are.