Inferring limit from numerical data

WEBVTT
Kind: captions
Language: en

00:00:00.890 --> 00:00:02.930
Consider the table
with function values

00:00:02.930 --> 00:00:05.900
for f of x is equal
to 3 sine of 5x

00:00:05.900 --> 00:00:09.620
over sine 2x at x values near 0.

00:00:09.620 --> 00:00:11.170
From the table,
what does the limit

00:00:11.170 --> 00:00:15.240
as x approaches 0 of 3
sine of 5x over sine of 2x

00:00:15.240 --> 00:00:16.520
appear to be?

00:00:16.520 --> 00:00:19.310
So what they have is they're
approaching 0 from below here.

00:00:19.310 --> 00:00:25.060
So first negative 0.1,
and f of x is 7.239550.

00:00:25.060 --> 00:00:27.160
Then we get even a
little bit closer to 0.

00:00:27.160 --> 00:00:29.720
x is negative 0.01.

00:00:29.720 --> 00:00:33.140
Once again, we're approaching
0 from values less than 0.

00:00:33.140 --> 00:00:36.490
And now we've gone
up to 7.497375.

00:00:36.490 --> 00:00:42.289
We get even a little bit closer
to 0, and we get to 7.499974.

00:00:42.289 --> 00:00:44.080
Now let's see what
happens as we approach 0

00:00:44.080 --> 00:00:46.500
from above, from
values greater than 0.

00:00:46.500 --> 00:00:51.680
When x is 0.1, it's 7.239550,
actually the same as this value

00:00:51.680 --> 00:00:52.830
right over here.

00:00:52.830 --> 00:00:57.440
When it is 0.01, it's
7.497375, actually

00:00:57.440 --> 00:00:59.200
the same value as here.

00:00:59.200 --> 00:01:01.847
And once again,
when we're at 0.001,

00:01:01.847 --> 00:01:03.680
so we're approaching
from above, from values

00:01:03.680 --> 00:01:08.580
greater than 0, 7.499974.

00:01:08.580 --> 00:01:10.100
So this is an interesting thing.

00:01:10.100 --> 00:01:13.430
The limit in both cases
seems to be approaching,

00:01:13.430 --> 00:01:18.096
getting closer
and closer to 7.5.

00:01:18.096 --> 00:01:20.220
And so you could imagine
that this would be-- well,

00:01:20.220 --> 00:01:22.800
let's actually graph that just
to verify it for ourselves.

00:01:22.800 --> 00:01:26.150
So we could type in our answer
here and check our answer,

00:01:26.150 --> 00:01:27.130
and we got it right.

00:01:27.130 --> 00:01:29.030
But just to visualize
what that looks like,

00:01:29.030 --> 00:01:31.370
let's actually get our
graphing calculator.

00:01:31.370 --> 00:01:34.370
So I have the graph, just
to show you how I get here.

00:01:34.370 --> 00:01:38.610
So if I were to start
on the home screen,

00:01:38.610 --> 00:01:43.110
just press Graph,
define my function of x.

00:01:43.110 --> 00:01:59.180
And we could just write 3 sine
of 5x divided by sine of 2x.

00:01:59.180 --> 00:02:02.120
And then let me define my range.

00:02:02.120 --> 00:02:02.950
So, let's see.

00:02:02.950 --> 00:02:08.340
x, I want it to be-- yeah,
sure, I could make it around 0.

00:02:08.340 --> 00:02:10.229
And let's see, I'm
getting y values

00:02:10.229 --> 00:02:13.300
that go up to-- let's
see, around 7 or 8.

00:02:13.300 --> 00:02:16.620
So I could make my y, let me
do a little bit of negative y

00:02:16.620 --> 00:02:17.190
values.

00:02:17.190 --> 00:02:20.225
So let me start at
negative 1 for my y range,

00:02:20.225 --> 00:02:22.052
and go all the way
up to, let's say 8.

00:02:22.052 --> 00:02:23.510
Because this doesn't
look like it's

00:02:23.510 --> 00:02:25.560
getting above 8, or
at least in the range

00:02:25.560 --> 00:02:26.800
that we're talking about.

00:02:26.800 --> 00:02:29.287
And now let's graph this thing.

00:02:29.287 --> 00:02:30.370
So let's see what it does.

00:02:30.370 --> 00:02:34.930
OK, so the graph is doing
all sorts of neat things,

00:02:34.930 --> 00:02:37.500
but then as we get
closer and closer to 0,

00:02:37.500 --> 00:02:41.520
as x gets closer and closer to
0 from the negative direction,

00:02:41.520 --> 00:02:45.920
we see that the function
is approaching 7.5.

00:02:45.920 --> 00:02:49.150
Likewise, when x is
getting closer and closer

00:02:49.150 --> 00:02:51.740
to 0 from the
positive direction,

00:02:51.740 --> 00:02:54.820
the function seems to
be approaching 7.5.

00:02:54.820 --> 00:03:00.410
So you have this-- well, the
function isn't defined at 7.5,

00:03:00.410 --> 00:03:03.390
or it's pretty clear that when
you approach from both sides,

00:03:03.390 --> 00:03:05.920
you're getting there.


Unit vector notation

WEBVTT
Kind: captions
Language: en

00:00:00.474 --> 00:00:02.140
We've already seen
that you can visually

00:00:02.140 --> 00:00:06.420
represent a vector as an arrow,
where the length of the arrow

00:00:06.420 --> 00:00:10.700
is the magnitude of the vector
and the direction of the arrow

00:00:10.700 --> 00:00:12.650
is the direction of the vector.

00:00:12.650 --> 00:00:15.650
And if we want to represent
this mathematically,

00:00:15.650 --> 00:00:19.180
we could just think
about, well, starting

00:00:19.180 --> 00:00:22.720
from the tail of
the vector, how far

00:00:22.720 --> 00:00:27.120
away is the head of the vector
in the horizontal direction?

00:00:27.120 --> 00:00:29.300
And how far away is it in
the vertical direction?

00:00:29.300 --> 00:00:32.720
So for example, in the
horizontal direction,

00:00:32.720 --> 00:00:37.020
you would have to
go this distance.

00:00:37.020 --> 00:00:38.730
And then in the
vertical direction,

00:00:38.730 --> 00:00:42.265
you would have to
go this distance.

00:00:42.265 --> 00:00:44.260
Let me do that in
a different color.

00:00:44.260 --> 00:00:49.140
You would have to go this
distance right over here.

00:00:49.140 --> 00:00:52.700
And so let's just say
that this distance is 2

00:00:52.700 --> 00:00:55.040
and that this distance is 3.

00:00:55.040 --> 00:00:56.870
We could represent
this vector-- and let's

00:00:56.870 --> 00:01:00.070
call this vector v. We
could represent vector

00:01:00.070 --> 00:01:08.250
v as an ordered list or a
2-tuple of-- so we could say we

00:01:08.250 --> 00:01:11.040
move 2 in the
horizontal direction

00:01:11.040 --> 00:01:14.980
and 3 in the vertical direction.

00:01:14.980 --> 00:01:16.880
So you could represent
it like that.

00:01:16.880 --> 00:01:19.390
You could represent
vector v like this,

00:01:19.390 --> 00:01:25.742
where it is 2
comma 3, like that.

00:01:25.742 --> 00:01:27.450
And what I now want
to introduce you to--

00:01:27.450 --> 00:01:29.616
and we could come up with
other ways of representing

00:01:29.616 --> 00:01:32.360
this 2-tuple-- is
another notation.

00:01:32.360 --> 00:01:34.180
And this really
comes out of the idea

00:01:34.180 --> 00:01:37.550
of what it means to
add and scale vectors.

00:01:37.550 --> 00:01:39.150
And to do that,
we're going to define

00:01:39.150 --> 00:01:40.825
what we call unit vectors.

00:01:45.630 --> 00:01:47.230
And if we're in
two dimensions, we

00:01:47.230 --> 00:01:49.950
define a unit vector for
each of the dimensions

00:01:49.950 --> 00:01:50.912
we're operating in.

00:01:50.912 --> 00:01:53.370
If we're in three dimensions,
we would define a unit vector

00:01:53.370 --> 00:01:56.210
for each of the three dimensions
that we're operating in.

00:01:56.210 --> 00:01:57.640
And so let's do that.

00:01:57.640 --> 00:02:00.040
So let's define a unit vector i.

00:02:00.040 --> 00:02:04.826
And the way that we denote
that is the unit vector

00:02:04.826 --> 00:02:06.450
is, instead of putting
an arrow on top,

00:02:06.450 --> 00:02:09.320
we put this hat on top of it.

00:02:09.320 --> 00:02:11.620
So the unit vector
i, if we wanted

00:02:11.620 --> 00:02:14.060
to write it in this
notation right over here,

00:02:14.060 --> 00:02:18.220
we would say it only goes 1 unit
in the horizontal direction,

00:02:18.220 --> 00:02:21.080
and it doesn't go at all
in the vertical direction.

00:02:21.080 --> 00:02:22.665
So it would look
something like this.

00:02:26.842 --> 00:02:29.810
That is the unit vector i.

00:02:29.810 --> 00:02:31.920
And then we can define
another unit vector.

00:02:31.920 --> 00:02:33.740
And let's call
that unit vector--

00:02:33.740 --> 00:02:37.620
or it's typically
called j, which

00:02:37.620 --> 00:02:40.259
would go only in the
vertical direction and not

00:02:40.259 --> 00:02:41.425
in the horizontal direction.

00:02:41.425 --> 00:02:43.290
And not in the
horizontal direction,

00:02:43.290 --> 00:02:45.680
and it goes 1 unit in
the vertical direction.

00:02:45.680 --> 00:02:48.210
So this went 1 unit
in the horizontal.

00:02:48.210 --> 00:02:54.410
And now j is going to go
1 unit in the vertical.

00:02:54.410 --> 00:02:57.260
So j-- just like that.

00:02:57.260 --> 00:03:00.880
Now any vector, any
two dimensional vector,

00:03:00.880 --> 00:03:07.560
we can now represent as a sum of
scaled up versions of i and j.

00:03:07.560 --> 00:03:09.300
And you say, well,
how do we do that?

00:03:09.300 --> 00:03:13.470
Well, you could imagine
vector v right here

00:03:13.470 --> 00:03:15.820
is the sum of a vector
that moves purely

00:03:15.820 --> 00:03:18.930
in the horizontal direction
that has a length 2,

00:03:18.930 --> 00:03:21.740
and a vector that moves purely
in the vertical direction that

00:03:21.740 --> 00:03:23.320
has length 3.

00:03:23.320 --> 00:03:28.530
So we could say
that vector v-- let

00:03:28.530 --> 00:03:33.230
me do it in that
same blue color--

00:03:33.230 --> 00:03:36.690
is equal to-- so if we want
a vector that has length 2

00:03:36.690 --> 00:03:39.160
and it moves purely in
the horizontal direction,

00:03:39.160 --> 00:03:42.150
well, we could just scale
up the unit vector i.

00:03:42.150 --> 00:03:44.245
We could just
multiply 2 times i.

00:03:44.245 --> 00:03:49.385
So let's do that-- is equal
to 2 times our unit vector i.

00:03:49.385 --> 00:03:52.580
So 2i is going to
be this whole thing

00:03:52.580 --> 00:03:55.250
right over here or
this whole vector.

00:03:55.250 --> 00:03:56.800
Let me do it in
this yellow color.

00:03:56.800 --> 00:04:00.510
This vector right over
here, you could view as 2i.

00:04:00.510 --> 00:04:08.420
And then to that, we're going to
add 3 times j-- so plus 3 times

00:04:08.420 --> 00:04:08.920
j.

00:04:11.510 --> 00:04:12.740
Let me write it like this.

00:04:12.740 --> 00:04:15.890
Let me get that color.

00:04:15.890 --> 00:04:18.200
Once again, 3 times
j is going to be

00:04:18.200 --> 00:04:20.760
this vector right over here.

00:04:20.760 --> 00:04:25.700
And if you add this yellow
vector right over here

00:04:25.700 --> 00:04:29.150
to the magenta vector,
you're going to get-- notice,

00:04:29.150 --> 00:04:31.110
we're putting the tail
of the magenta vector

00:04:31.110 --> 00:04:32.760
at the head of
the yellow vector.

00:04:32.760 --> 00:04:34.410
And if you start at the
tail of the yellow vector

00:04:34.410 --> 00:04:36.830
and you go all the way to the
head of the magenta vector,

00:04:36.830 --> 00:04:42.340
you have now constructed
vector v. So vector v,

00:04:42.340 --> 00:04:46.220
you could represent it as a
column vector like this, 2 3.

00:04:46.220 --> 00:04:48.060
You could represent
it as 2 comma 3,

00:04:48.060 --> 00:04:51.440
or you could represent it as
2 times i with this little hat

00:04:51.440 --> 00:04:55.080
over it, plus 3 times j,
with this little hat over it.

00:04:55.080 --> 00:04:57.980
i is the unit vector in
the horizontal direction,

00:04:57.980 --> 00:04:59.570
in the positive
horizontal direction.

00:04:59.570 --> 00:05:00.455
If you want to go
the other way, you

00:05:00.455 --> 00:05:02.190
would multiply it by a negative.

00:05:02.190 --> 00:05:05.716
And j is the unit vector
in the vertical direction.

00:05:05.716 --> 00:05:07.340
As we'll see in future
videos, once you

00:05:07.340 --> 00:05:11.320
go to three dimensions,
you'll introduce a k.

00:05:11.320 --> 00:05:13.491
But it's very natural to
translate between these two

00:05:13.491 --> 00:05:13.990
things.

00:05:13.990 --> 00:05:16.840
Notice, 2, 3-- 2, 3.

00:05:16.840 --> 00:05:19.550
And so with that, let's actually
do some vector operations

00:05:19.550 --> 00:05:21.470
using this notation.

00:05:21.470 --> 00:05:24.090
So let's say that I
define another vector.

00:05:24.090 --> 00:05:27.195
Let's say it is vector b.

00:05:31.090 --> 00:05:33.400
I'll just come up with
some numbers here.

00:05:33.400 --> 00:05:39.130
Vector b is equal to negative 1
times i-- times the unit vector

00:05:39.130 --> 00:05:48.000
i-- plus 4 times the unit vector
in the horizontal direction.

00:05:48.000 --> 00:05:50.320
So given these two
vector definitions,

00:05:50.320 --> 00:05:57.500
what would the would be the
vector v plus b be equal to?

00:05:57.500 --> 00:06:00.410
And I encourage you to pause
the video and think about it.

00:06:00.410 --> 00:06:02.000
Well once again,
we just literally

00:06:02.000 --> 00:06:04.404
have to add
corresponding components.

00:06:04.404 --> 00:06:05.820
We could say, OK,
well let's think

00:06:05.820 --> 00:06:07.760
about what we're doing in
the horizontal direction.

00:06:07.760 --> 00:06:09.830
We're going 2 in the
horizontal direction here,

00:06:09.830 --> 00:06:11.540
and now we're going negative 1.

00:06:11.540 --> 00:06:13.410
So our horizontal
component is going

00:06:13.410 --> 00:06:20.470
to be 2 plus negative
1-- 2 plus negative 1

00:06:20.470 --> 00:06:22.200
in the horizontal direction.

00:06:22.200 --> 00:06:24.690
And we're going to multiply
that times the unit vector i.

00:06:24.690 --> 00:06:26.710
And this, once
again, just goes back

00:06:26.710 --> 00:06:30.110
to adding the corresponding
components of the vector.

00:06:30.110 --> 00:06:36.715
And then we're going to have
plus 4, or plus 3 plus 4-- And

00:06:36.715 --> 00:06:47.410
let me write it that way--
times the unit vector

00:06:47.410 --> 00:06:50.430
j in the vertical direction.

00:06:50.430 --> 00:06:54.140
And so that's going
to give us-- I'll

00:06:54.140 --> 00:06:59.270
do this all in this one color--
2 plus negative 1 is 1i.

00:06:59.270 --> 00:07:01.120
And we could literally
write that just as i.

00:07:01.120 --> 00:07:02.120
Actually, let's do that.

00:07:02.120 --> 00:07:03.810
Let's just write that as i.

00:07:03.810 --> 00:07:05.940
But we got that from 2
plus negative 1 is 1.

00:07:05.940 --> 00:07:08.260
1 times the vector is just
going to be that vector,

00:07:08.260 --> 00:07:15.380
plus 3 plus 4 is 7-- 7j.

00:07:15.380 --> 00:07:19.130
And you see, this is exactly
how we saw vector addition

00:07:19.130 --> 00:07:22.160
in the past, is that we
could also represent vector

00:07:22.160 --> 00:07:23.430
b like this.

00:07:23.430 --> 00:07:28.805
We could represent it
like this-- negative 1, 4.

00:07:31.450 --> 00:07:33.050
And so if you were
to add v to b,

00:07:33.050 --> 00:07:35.520
you add the corresponding terms.

00:07:35.520 --> 00:07:39.270
So if we were to add
corresponding terms, looking

00:07:39.270 --> 00:07:41.440
at them as column
vectors, that is

00:07:41.440 --> 00:07:46.340
going to be equal to 2 plus
negative 1, which is 1.

00:07:46.340 --> 00:07:49.070
3 plus 4 is 7.

00:07:49.070 --> 00:07:52.982
So this is the exact same
representation as this.

00:07:52.982 --> 00:07:54.440
This is using unit
vector notation,

00:07:54.440 --> 00:07:58.190
and this is representing
it as a column vector.


Multiplying a vector by a scalar

WEBVTT
Kind: captions
Language: en

00:00:00.321 --> 00:00:10.075
Let's say that I have the vector a, and let's say that it's equal to (2,1)

00:00:10.075 --> 00:00:12.603
So we could draw it right over here,

00:00:12.603 --> 00:00:16.475
So it's equal to (2,1), so if we were to start at the origin

00:00:16.475 --> 00:00:20.974
and we would move 2 in the horizontal direction, and 1 in the vertical direction

00:00:20.974 --> 00:00:24.572
so we would end up right over here.

00:00:24.572 --> 00:00:27.408
Now what I want to do is think about, how we can define

00:00:27.408 --> 00:00:31.270
multiplying this vector by a scalar

00:00:31.270 --> 00:00:39.496
for example, if I were to say 3 times the vector a.

00:00:40.004 --> 00:00:47.780
which is the same thing as saying 3 times (2,1)

00:00:47.934 --> 00:00:50.937
So 3 is just a number. One way to think about a scalar quantity,

00:00:50.937 --> 00:00:56.176
it is just a number, versus a vector is giving you how much you're moving

00:00:56.176 --> 00:01:01.935
in the various directions. It's giving you both the magnitude and a direction.

00:01:01.935 --> 00:01:04.870
while this is just a plain number right over here.

00:01:04.870 --> 00:01:10.181
But how do we define multiplying 3 times this vector?

00:01:10.750 --> 00:01:13.075
Well one reasonable thing that might jump out at you is,

00:01:13.075 --> 00:01:16.483
why don't we just multiply the 3 times each of these components?

00:01:16.483 --> 00:01:21.005
So this could be equal to.. we have 2 and 1..

00:01:21.005 --> 00:01:24.011
And we're going to multiply each of these with 3.

00:01:24.011 --> 00:01:27.675
So 3 times 2 and 3 times 1.

00:01:27.675 --> 00:01:32.906
And then the resulting vector is still going to be a 2-dimensional vector.

00:01:32.906 --> 00:01:37.134
And it's going to be the 2-dimensional vector (6,3).

00:01:37.134 --> 00:01:40.735
Now I encourage you to get some graph paper out and to actually plot this vector,

00:01:40.735 --> 00:01:46.100
and think about how it relates to this vector right over here.

00:01:46.316 --> 00:01:47.739
So let me do that..

00:01:47.739 --> 00:01:50.741
So the vector (6,3), if we started at the origin..

00:01:50.741 --> 00:01:56.337
We would move 6 in the horizontal direction.. 1, 2, 3, 4, 5, 6..

00:01:56.337 --> 00:01:59.536
And 3 in the vertical.. 1, 2, 3..

00:01:59.536 --> 00:02:04.936
So it gets us right over there, so it would look like this.

00:02:05.536 --> 00:02:08.434
So what just happened to this vector?

00:02:08.604 --> 00:02:11.269
Well notice, one way to think about it is what has changed,

00:02:11.269 --> 00:02:14.405
and what has not changed about this vector?

00:02:14.405 --> 00:02:17.725
Well what's not changed is still pointing in the same direction.

00:02:17.725 --> 00:02:21.336
So this right over here has the same direction.

00:02:21.336 --> 00:02:24.669
Multiplying by the scalar, at least the way we defined it..

00:02:24.669 --> 00:02:28.539
did not change the direction that my vector is going in.

00:02:28.539 --> 00:02:32.072
Or at least in this case it didn't.. But it did change its magnitude.

00:02:32.072 --> 00:02:35.202
Its magnitude is now 3 times longer, which makes sense!

00:02:35.202 --> 00:02:36.935
Because we multiplied it by 3.

00:02:36.935 --> 00:02:39.601
One way to think about it is we scaled it up by 3.

00:02:39.601 --> 00:02:43.404
The scalar scaled up the vector. That might make sense.

00:02:43.404 --> 00:02:46.737
Or it might make an intuition of where that word scalar came from.

00:02:46.737 --> 00:02:52.484
The scalar, when you multiply it, it scales up a vector.

00:02:52.484 --> 00:02:57.145
It Increased its magnitude by 3 without changing its direction.

00:02:57.145 --> 00:02:59.149
Well let's do something interesting..

00:02:59.149 --> 00:03:04.501
Let's multiply our vector a by a negative number.

00:03:04.609 --> 00:03:08.148
Let's just multiply it by -1 for simplicity.

00:03:08.148 --> 00:03:13.351
So let's just multiply -1 times a.

00:03:13.536 --> 00:03:16.338
Well using the convention that we just came up with..

00:03:16.338 --> 00:03:20.148
We would multiply each of the components by -1.

00:03:20.148 --> 00:03:26.364
So 2 times -1 is -2, and 1 times -1 is -1.

00:03:26.564 --> 00:03:32.602
So now -1 times a is going to be (-2,-1)

00:03:32.602 --> 00:03:34.321
So if we started at the origin,

00:03:34.321 --> 00:03:39.737
we would move in the horizontal direction -2, and in the vertical -1

00:03:39.737 --> 00:03:47.859
So now what happened to the vector? When I did that?

00:03:48.229 --> 00:03:50.671
Well now it flipped its direction!

00:03:50.671 --> 00:03:54.070
Multiplying it by this -1, it flipped it's direction.

00:03:54.070 --> 00:03:56.333
Its magnitude actually has not changed,

00:03:56.333 --> 00:03:59.901
but its direction is now in the exact opposite direction.

00:03:59.901 --> 00:04:04.522
Which makes sense, that multiplying by a negative number would do that.

00:04:04.522 --> 00:04:08.274
In fact when we dealt with the traditional number line, that's what happened

00:04:08.274 --> 00:04:11.643
If you took 5 times -1, well now you're going in the other direction

00:04:11.643 --> 00:04:14.069
you're at -5, you're 5 to the left of zero.

00:04:14.069 --> 00:04:16.936
So it makes sense that this would flip its direction.

00:04:16.936 --> 00:04:24.394
So you could imagine, if you were to take something like -2 times your vector a,

00:04:24.394 --> 00:04:26.151
-2 times your vector a..

00:04:26.151 --> 00:04:29.474
And I encourage you to pause this video and try this on your own..

00:04:29.474 --> 00:04:33.940
What would this give? And what would be the resulting visualization of the vector?

00:04:33.940 --> 00:04:39.816
Well let's see, this would be equal to -2 times 2 is -4,

00:04:39.816 --> 00:04:43.738
-2 times 1 is -2, so this vector..

00:04:43.738 --> 00:04:46.669
if you were to start at the origin! remember you don't have to start at the origin..

00:04:46.669 --> 00:04:52.898
but if you were.. it would go 0, 1, 2, 3, 4.. 1, 2..

00:04:52.898 --> 00:04:56.742
It looks just like this..

00:04:57.358 --> 00:05:04.957
And so just to remind ourselves.. our original vector a looked like this..

00:05:05.404 --> 00:05:08.564
(2, 1) looks like this..

00:05:08.564 --> 00:05:15.934
And then when you multiply it by -2.. you get a vector that looks like this..

00:05:15.934 --> 00:05:17.068
Let me draw it like this..

00:05:17.068 --> 00:05:19.870
I'm purposely not having them all start at the origin,

00:05:19.870 --> 00:05:22.069
because they don't HAVE to all start at the origin..

00:05:22.069 --> 00:05:26.552
But you get a vector that looks like this..

00:05:27.737 --> 00:05:31.610
So what's the difference between a and -2 times a?

00:05:31.610 --> 00:05:33.973
well the negative flipped it over,

00:05:33.973 --> 00:05:37.526
and then the two flipped it over and now it has twice the magnitude

00:05:37.526 --> 00:05:41.526
but because of the negative it has twice the magnitude in the other direction.


Adding vectors

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:02.970
So I have two
2-dimensional vectors

00:00:02.970 --> 00:00:05.970
right over here,
vector a and vector b.

00:00:05.970 --> 00:00:08.575
And what I want to think
about is how can we define

00:00:08.575 --> 00:00:09.950
or what would be
a reasonable way

00:00:09.950 --> 00:00:15.137
to define the sum of
vector a plus vector b?

00:00:15.137 --> 00:00:16.970
Well, one thing that
might jump at your mind

00:00:16.970 --> 00:00:19.160
is, look, well, each of
these are two dimensional.

00:00:19.160 --> 00:00:20.410
They both have two components.

00:00:20.410 --> 00:00:23.220
Why don't we just add the
corresponding components?

00:00:23.220 --> 00:00:27.610
So for the sum, why don't
we make the first component

00:00:27.610 --> 00:00:30.130
of the sum just a sum of
the first two components

00:00:30.130 --> 00:00:31.110
of these two vectors.

00:00:31.110 --> 00:00:35.090
So why don't we just make
it 6 plus negative 4?

00:00:35.090 --> 00:00:38.660
Well, 6 plus negative
4 is equal to 2.

00:00:38.660 --> 00:00:41.010
And why don't we just
make the second component

00:00:41.010 --> 00:00:43.350
the sum of the two
second components?

00:00:43.350 --> 00:00:48.360
So negative 2 plus 4
is also equal to 2.

00:00:48.360 --> 00:00:50.540
So we start with two
2-dimensional vectors.

00:00:50.540 --> 00:00:52.560
You add them together,
you get another two

00:00:52.560 --> 00:00:53.600
2-dimensional vectors.

00:00:53.600 --> 00:00:56.630
If you think about it in terms
of real coordinate spaces,

00:00:56.630 --> 00:00:59.040
both of these are
members of R2--

00:00:59.040 --> 00:01:02.040
I'll write this down here just
so we get used to the notation.

00:01:02.040 --> 00:01:14.032
So vector a and vector b
are both members of R2,

00:01:14.032 --> 00:01:15.490
which is just
another way of saying

00:01:15.490 --> 00:01:16.980
that these are both two tuples.

00:01:16.980 --> 00:01:21.100
They are both two-dimensional
vectors right over here.

00:01:21.100 --> 00:01:23.720
Now, this might make sense just
looking at how we represented

00:01:23.720 --> 00:01:25.340
it, but how does
this actually make

00:01:25.340 --> 00:01:27.150
visual or conceptual sense?

00:01:27.150 --> 00:01:29.310
And to do that, let's
actually plot these vectors.

00:01:29.310 --> 00:01:32.100
Let's try to represent
these vectors in some way.

00:01:32.100 --> 00:01:33.880
Let's try to visualize them.

00:01:33.880 --> 00:01:37.550
So vector a, we could
visualize, this tells us

00:01:37.550 --> 00:01:39.640
how far this vector
moves in each

00:01:39.640 --> 00:01:42.070
of these directions--
horizontal direction

00:01:42.070 --> 00:01:44.020
and vertical direction.

00:01:44.020 --> 00:01:45.860
So if we put the,
I guess you could

00:01:45.860 --> 00:01:48.050
say the tail of the vector
at the origin-- remember,

00:01:48.050 --> 00:01:49.720
we don't have to put
the tail at the origin,

00:01:49.720 --> 00:01:52.261
but that might make it a little
bit easier for us to draw it.

00:01:52.261 --> 00:01:53.940
We'll go 6 in the
horizontal direction.

00:01:53.940 --> 00:01:56.620
1, 2, 3, 4, 5, 6.

00:01:56.620 --> 00:01:58.420
And then negative
2 in the vertical.

00:01:58.420 --> 00:01:59.980
So negative 2.

00:01:59.980 --> 00:02:05.050
So vector a could
look like this.

00:02:05.050 --> 00:02:09.139
Vector a looks like that.

00:02:09.139 --> 00:02:11.050
And once again,
the important thing

00:02:11.050 --> 00:02:12.750
is the magnitude
and the direction.

00:02:12.750 --> 00:02:15.970
The magnitude is represented
by the length of this vector.

00:02:15.970 --> 00:02:17.740
And the direction
is the direction

00:02:17.740 --> 00:02:19.030
that it is pointed in.

00:02:19.030 --> 00:02:21.810
And also just to emphasize,
I could have drawn vector a

00:02:21.810 --> 00:02:26.280
like that or I could
have put it over here.

00:02:26.280 --> 00:02:30.210
These are all
equivalent vectors.

00:02:30.210 --> 00:02:32.850
These are all equal to vector a.

00:02:32.850 --> 00:02:37.450
All I really care about is the
magnitude and the direction.

00:02:37.450 --> 00:02:40.100
So with that in mind,
let's also draw vector b.

00:02:40.100 --> 00:02:44.750
Vector b in the horizontal
direction goes negative 4-- 1,

00:02:44.750 --> 00:02:50.170
2, 3, 4, and in the vertical
direction goes 4-- 1, 2, 3, 4.

00:02:50.170 --> 00:02:52.560
So its tail if we
start at the origin,

00:02:52.560 --> 00:02:55.050
if its tail is at
the origin, its head

00:02:55.050 --> 00:02:59.105
would be at negative 4, 4.

00:02:59.105 --> 00:03:03.110
So let me draw that
just like that.

00:03:03.110 --> 00:03:07.260
So that right over
here is vector b.

00:03:07.260 --> 00:03:10.150
And once again, vector b
we could draw it like that

00:03:10.150 --> 00:03:15.240
or we could draw it-- let me
copy and let me paste it--

00:03:15.240 --> 00:03:17.710
so this would also be
another way to draw vector b.

00:03:21.620 --> 00:03:23.350
Once again, what I
really care about

00:03:23.350 --> 00:03:25.540
is its magnitude
and its direction.

00:03:25.540 --> 00:03:27.900
All of these green vectors
have the same magnitude.

00:03:27.900 --> 00:03:29.650
They all have the same
length and they all

00:03:29.650 --> 00:03:31.520
have the same direction.

00:03:31.520 --> 00:03:33.660
So how does the way that
I drew vector a and b

00:03:33.660 --> 00:03:36.440
gel with what its sum is?

00:03:36.440 --> 00:03:38.730
So let me draw
its sum like this.

00:03:38.730 --> 00:03:41.920
Let me draw its sum
in this blue color.

00:03:41.920 --> 00:03:44.610
So the sum based on this
definition we just used,

00:03:44.610 --> 00:03:47.060
the vector addition
would be 2, 2.

00:03:47.060 --> 00:03:48.100
So 2, 2.

00:03:48.100 --> 00:03:53.600
So it would look
something like this.

00:03:53.600 --> 00:03:55.650
So how does this make
sense that the sum,

00:03:55.650 --> 00:04:00.250
that this purple vector
plus this green vector

00:04:00.250 --> 00:04:02.464
is somehow going to be
equal to this blue vector?

00:04:02.464 --> 00:04:03.880
I encourage you
to pause the video

00:04:03.880 --> 00:04:07.300
and think about if
that even makes sense.

00:04:07.300 --> 00:04:10.600
Well, one way to think about
it is this first purple vector,

00:04:10.600 --> 00:04:12.290
it shifts us this much.

00:04:12.290 --> 00:04:14.621
It takes us from this
point to that point.

00:04:14.621 --> 00:04:16.870
And so if we were to add it,
let's start at this point

00:04:16.870 --> 00:04:18.990
and put the green
vector's tail right there

00:04:18.990 --> 00:04:22.270
and see where it
ends up putting us.

00:04:22.270 --> 00:04:25.320
So the green vector, we
already have a version.

00:04:25.320 --> 00:04:28.450
So once again, we
start the origin.

00:04:28.450 --> 00:04:30.240
Vector a takes us there.

00:04:30.240 --> 00:04:32.480
Now, let's start over
there with the green vector

00:04:32.480 --> 00:04:34.460
and see where green
vector takes us.

00:04:34.460 --> 00:04:35.720
And this makes sense.

00:04:35.720 --> 00:04:38.440
Vector a plus vector b.

00:04:38.440 --> 00:04:41.210
Put the tail of vector b
at the head of vector a.

00:04:41.210 --> 00:04:43.260
So if you were to
start at the origin,

00:04:43.260 --> 00:04:45.190
vector a takes you
there then if you

00:04:45.190 --> 00:04:48.520
add on what vector b takes you,
it takes you right over there.

00:04:48.520 --> 00:04:52.210
So relative to the
origin, how much did you--

00:04:52.210 --> 00:04:53.830
I guess you could say-- shift?

00:04:53.830 --> 00:04:55.420
And once again,
vectors don't only

00:04:55.420 --> 00:04:56.435
apply to things
like displacement.

00:04:56.435 --> 00:04:57.345
It can apply to velocity.

00:04:57.345 --> 00:04:58.920
It can apply to
actual acceleration.

00:04:58.920 --> 00:05:01.350
It can apply to a
whole series of things,

00:05:01.350 --> 00:05:03.500
but when you
visualize it this way,

00:05:03.500 --> 00:05:05.890
you see that it does
make complete sense.

00:05:05.890 --> 00:05:08.360
This blue vector,
the sum of the two,

00:05:08.360 --> 00:05:11.745
is what results where
you start with vector a.

00:05:11.745 --> 00:05:15.130
At that point right over there,
vector a takes you there,

00:05:15.130 --> 00:05:17.960
then you take vector b's
tail, start over there

00:05:17.960 --> 00:05:21.140
and it takes you to
the tip of the sum.

00:05:21.140 --> 00:05:23.570
Now, one question you
might be having is well,

00:05:23.570 --> 00:05:25.900
vector a plus vector
b is this, but what

00:05:25.900 --> 00:05:31.510
is vector b plus vector a?

00:05:31.510 --> 00:05:33.540
Does this still work?

00:05:33.540 --> 00:05:35.200
Well, based on the
definition we had

00:05:35.200 --> 00:05:37.010
where you add the
corresponding components,

00:05:37.010 --> 00:05:39.509
you're still going to
get the same sum vector.

00:05:39.509 --> 00:05:40.800
So it should come out the same.

00:05:40.800 --> 00:05:43.720
So this will just be
negative 4 plus 6 is 2.

00:05:43.720 --> 00:05:46.430
4 plus negative 2 is 2.

00:05:46.430 --> 00:05:48.420
But does that make visual sense?

00:05:48.420 --> 00:05:49.836
So if we start with vector b.

00:05:49.836 --> 00:05:51.460
So let's say you
start right over here.

00:05:51.460 --> 00:05:54.480
Vector b takes you
right over there.

00:05:54.480 --> 00:05:56.943
And then if you were
to go there and you

00:05:56.943 --> 00:05:59.960
were to start with vector
a-- so let's do that.

00:05:59.960 --> 00:06:02.760
So actually, let me make this
a little bit-- actually, let me

00:06:02.760 --> 00:06:06.240
start with a new vector b.

00:06:06.240 --> 00:06:11.230
So let's say that that's our
vector b right over there.

00:06:11.230 --> 00:06:13.066
And then-- actually,
let me give this

00:06:13.066 --> 00:06:15.230
a place where I'll have
some space to work with.

00:06:15.230 --> 00:06:18.450
So let's say that's my
vector b right over there.

00:06:18.450 --> 00:06:22.320
And then let me get a
copy of the vector a.

00:06:22.320 --> 00:06:23.832
That's a good one.

00:06:23.832 --> 00:06:26.630
So copy and let me paste it.

00:06:26.630 --> 00:06:31.170
So I could put vector a's
tail at the tip of vector b,

00:06:31.170 --> 00:06:35.590
and then it'll take
me right over there.

00:06:35.590 --> 00:06:42.680
So if I start right over
here, vector b takes me there.

00:06:42.680 --> 00:06:45.750
And now I'm adding to that
vector a, which starting here

00:06:45.750 --> 00:06:47.080
will take me there.

00:06:47.080 --> 00:06:52.520
And so from my original starting
position, I have gone this far.

00:06:52.520 --> 00:06:53.860
Now, what is this vector?

00:06:53.860 --> 00:06:57.360
Well, this is exactly
the vector 2, 2.

00:06:57.360 --> 00:06:58.860
Or another way of
thinking about it,

00:06:58.860 --> 00:07:02.630
this vector shifts you 2 in
the horizontal direction and 2

00:07:02.630 --> 00:07:04.410
in the vertical direction.

00:07:04.410 --> 00:07:06.920
So either way, you're going
to get the same result,

00:07:06.920 --> 00:07:09.770
and that should, hopefully,
make visual or conceptual sense

00:07:09.770 --> 00:07:11.460
as well.


Real coordinate spaces

WEBVTT
Kind: captions
Language: en

00:00:00.689 --> 00:00:02.230
When you get into
higher mathematics,

00:00:02.230 --> 00:00:03.870
you might see a
professor write something

00:00:03.870 --> 00:00:05.411
like this on a board,
where it's just

00:00:05.411 --> 00:00:08.730
this R with this extra
backbone right over here.

00:00:08.730 --> 00:00:10.790
And maybe they write R2.

00:00:10.790 --> 00:00:12.540
Or if you're looking
at it in a book,

00:00:12.540 --> 00:00:14.260
it might just be
a bolded capital

00:00:14.260 --> 00:00:17.790
R with a 2
superscript like this.

00:00:17.790 --> 00:00:19.490
And if you see this,
they're referring

00:00:19.490 --> 00:00:22.570
to the two-dimensional
real coordinate

00:00:22.570 --> 00:00:27.190
space, which sounds very fancy.

00:00:30.362 --> 00:00:31.820
But one way to
think about it, it's

00:00:31.820 --> 00:00:33.950
really just the
two-dimensional space

00:00:33.950 --> 00:00:37.400
that you're used to dealing
with in your coordinate plane.

00:00:37.400 --> 00:00:39.020
To go a little
bit more abstract,

00:00:39.020 --> 00:00:42.870
this isn't necessarily
this visual representation.

00:00:42.870 --> 00:00:45.470
This visual
representation is one way

00:00:45.470 --> 00:00:47.830
to think about this
real coordinate space.

00:00:47.830 --> 00:00:50.340
If we were to think about it
a little bit more abstractly,

00:00:50.340 --> 00:00:54.800
the real R2, the two-dimensional
real coordinate space--

00:00:54.800 --> 00:01:00.750
let me write this down-- and the
two-dimensional real coordinate

00:01:00.750 --> 00:01:01.250
space.

00:01:01.250 --> 00:01:03.090
And just to break
down the notation,

00:01:03.090 --> 00:01:05.420
the 2 tells us how many
dimensions we're dealing with,

00:01:05.420 --> 00:01:09.020
and then the R tells us this
is a real coordinate space.

00:01:09.020 --> 00:01:11.010
The two-dimensional
real coordinate space

00:01:11.010 --> 00:01:15.870
is all the possible
real-valued 2-tuples.

00:01:15.870 --> 00:01:17.950
Let me write that down.

00:01:17.950 --> 00:01:30.830
This is all possible
real-valued 2-tuples.

00:01:30.830 --> 00:01:33.030
So what is a 2-tuple?

00:01:33.030 --> 00:01:37.160
Well, a tuple is an
ordered list of numbers.

00:01:37.160 --> 00:01:38.910
And since we're talking
about real values,

00:01:38.910 --> 00:01:41.180
it's going to be ordered
list of real numbers,

00:01:41.180 --> 00:01:44.960
and a 2-tuple just says it's
an ordered list of 2 numbers.

00:01:44.960 --> 00:01:47.750
So this is an ordered list
of 2 real-valued numbers.

00:01:47.750 --> 00:01:49.540
Well, that's exactly
what we did here

00:01:49.540 --> 00:01:51.990
when we thought about a
two-dimensional vector.

00:01:51.990 --> 00:01:53.690
This right over
here is a 2-tuple,

00:01:53.690 --> 00:01:55.780
and this is a
real-valued 2-tuple.

00:01:55.780 --> 00:01:58.590
Neither of these have
any imaginary parts.

00:01:58.590 --> 00:02:00.290
So you have a 3 and a 4.

00:02:00.290 --> 00:02:01.200
Order matters.

00:02:01.200 --> 00:02:08.990
We view this as a different
2-tuple than, say, 4, 3.

00:02:08.990 --> 00:02:12.740
And even if we were trying
to represent them in our axes

00:02:12.740 --> 00:02:15.260
right over here,
this vector, 4, 3,

00:02:15.260 --> 00:02:18.130
would be 4 along
the horizontal axis,

00:02:18.130 --> 00:02:22.220
and then 3 along
the vertical axis.

00:02:22.220 --> 00:02:24.339
And so it would look
something like this.

00:02:24.339 --> 00:02:26.630
And remember, we don't have
to draw it just over there.

00:02:26.630 --> 00:02:28.840
We just care about its
magnitude and direction.

00:02:28.840 --> 00:02:32.230
We could draw it
right over here.

00:02:32.230 --> 00:02:38.489
This would also be 4, 3,
the column vector, 4 3.

00:02:38.489 --> 00:02:40.030
So when we're talking
about R2, we're

00:02:40.030 --> 00:02:43.610
talking about all of the
possible real-valued 2-tuples.

00:02:43.610 --> 00:02:47.660
So all the possible
vectors that you can have,

00:02:47.660 --> 00:02:50.380
where each of its components--
and the components

00:02:50.380 --> 00:02:52.190
are these numbers
right over here--

00:02:52.190 --> 00:02:56.000
where each of its components
are a real number.

00:02:56.000 --> 00:02:57.780
So you might have 3, 4.

00:02:57.780 --> 00:02:59.580
You could have
negative 3, negative 4.

00:02:59.580 --> 00:03:04.640
So that would be 1,
2, 3, 1, 2, 3, 4,

00:03:04.640 --> 00:03:06.567
might look something
like-- actually,

00:03:06.567 --> 00:03:08.400
I should make the scale
a little bit bigger,

00:03:08.400 --> 00:03:12.040
so it looks the
same-- 1, 2, 3, 4.

00:03:12.040 --> 00:03:16.060
So it might look
something like that.

00:03:16.060 --> 00:03:18.660
So that would be the
vector, negative 3--

00:03:18.660 --> 00:03:22.630
let me write a little bit
better than that-- negative 3,

00:03:22.630 --> 00:03:24.210
and negative 4.

00:03:24.210 --> 00:03:27.040
So if you were to take all
of the possible 2-tuples,

00:03:27.040 --> 00:03:30.960
including the vector 0,
0-- so it has no magnitude,

00:03:30.960 --> 00:03:34.240
and you could debate what its
direction is right over there--

00:03:34.240 --> 00:03:36.550
you take all of those
combined, and then

00:03:36.550 --> 00:03:39.690
you have created your
two-dimensional real coordinate

00:03:39.690 --> 00:03:40.300
space.

00:03:40.300 --> 00:03:43.440
And that is referred to as R2.

00:03:43.440 --> 00:03:44.830
Now, as you can
imagine, the fact

00:03:44.830 --> 00:03:46.246
that we wrote this
2 here-- we had

00:03:46.246 --> 00:03:48.680
to specify-- it's like, hey,
well, could I put a 3 there?

00:03:48.680 --> 00:03:51.470
And I would say, absolutely,
you could put a 3 there.

00:03:51.470 --> 00:03:54.920
So R3 would be the
three-dimensional real

00:03:54.920 --> 00:03:56.170
coordinate space.

00:03:56.170 --> 00:04:02.750
So 3D real coordinate space.

00:04:02.750 --> 00:04:04.840
And so you would
view this as all

00:04:04.840 --> 00:04:06.780
the possible
real-valued 3-tuples.

00:04:13.170 --> 00:04:20.436
So, for example, that
would be a member of R3.

00:04:20.436 --> 00:04:22.060
And let me actually
label these vectors

00:04:22.060 --> 00:04:23.476
just so we get in
the habit of it.

00:04:23.476 --> 00:04:26.570
So let's say we
call this vector x.

00:04:26.570 --> 00:04:31.210
Let's say we have a vector
b, that looks like this.

00:04:31.210 --> 00:04:34.970
Negative 1, 5, 3.

00:04:34.970 --> 00:04:38.200
Both of these would
be members of R3.

00:04:38.200 --> 00:04:41.090
And if you want to see some
fancy notation, a member

00:04:41.090 --> 00:04:47.510
of a set-- so this
is a member of R3--

00:04:47.510 --> 00:04:51.030
it is a real-valued 3-tuple.

00:04:51.030 --> 00:04:53.600
Now you say, well, what
would not be a member of R3?

00:04:53.600 --> 00:04:55.450
Well, this right over
here isn't a 3-tuple.

00:04:55.450 --> 00:04:57.260
This right over here
is a member of R2.

00:04:57.260 --> 00:04:59.260
Now, you might be able
to extend it in some way,

00:04:59.260 --> 00:05:01.470
add a zero or
something, but formally,

00:05:01.470 --> 00:05:03.500
this is not a 3-tuple.

00:05:03.500 --> 00:05:06.910
Another thing that would not
be a member of R3-- let's say

00:05:06.910 --> 00:05:11.130
someone wanted to make
some type of vector that

00:05:11.130 --> 00:05:12.810
had some imaginary parts in it.

00:05:12.810 --> 00:05:15.980
So let's say it had i, 0, 1.

00:05:15.980 --> 00:05:18.350
This is no longer real-valued.

00:05:18.350 --> 00:05:22.260
We have put an imaginary--
this number up here

00:05:22.260 --> 00:05:23.840
has an imaginary part.

00:05:23.840 --> 00:05:28.690
So this is no longer
a real-valued 3-tuple.

00:05:28.690 --> 00:05:30.380
And what's neat about
linear algebra is,

00:05:30.380 --> 00:05:31.546
we don't have to stop there.

00:05:31.546 --> 00:05:35.800
R3 we can visualize, we
can plot these things.

00:05:35.800 --> 00:05:37.342
In your previous
mathematical career,

00:05:37.342 --> 00:05:39.841
especially if you have some
type of a hologram or something,

00:05:39.841 --> 00:05:42.500
it's not hard to visualize
things in three dimensions.

00:05:42.500 --> 00:05:44.860
But what's neat is that we
can keep extending this.

00:05:44.860 --> 00:05:48.430
We can go into 4, 5, 6,
7, 20, 100 dimensions.

00:05:48.430 --> 00:05:51.120
And obviously there it becomes
much harder, if not impossible,

00:05:51.120 --> 00:05:52.360
to visualize it.

00:05:52.360 --> 00:05:55.450
But then we can at least
represent it mathematically

00:05:55.450 --> 00:05:57.879
with an n-tuple of vectors.

00:05:57.879 --> 00:06:00.170
And so if we were to talk
about a real coordinate space

00:06:00.170 --> 00:06:04.840
generally, you'll often
see the notation Rn,

00:06:04.840 --> 00:06:06.380
with n as a superscript.

00:06:06.380 --> 00:06:12.670
So this right over here is an
n-dimensional real coordinate

00:06:12.670 --> 00:06:14.220
space.


Vector intro for linear algebra

WEBVTT
Kind: captions
Language: en

00:00:00.830 --> 00:00:05.840
A vector is something that has
both magnitude and direction.

00:00:05.840 --> 00:00:09.620
Magnitude and direction.

00:00:09.620 --> 00:00:12.660
So let's think of an example
of what wouldn't and what

00:00:12.660 --> 00:00:14.380
would be a vector.

00:00:14.380 --> 00:00:15.990
So if someone tells
you that something

00:00:15.990 --> 00:00:20.340
is moving at 5 miles per hour,
this information by itself

00:00:20.340 --> 00:00:22.870
is not a vector quantity.

00:00:22.870 --> 00:00:26.010
It's only specifying
a magnitude.

00:00:26.010 --> 00:00:27.760
We don't know what
direction this thing is

00:00:27.760 --> 00:00:30.010
moving 5 miles per hour in.

00:00:30.010 --> 00:00:31.810
So this right over
here, which is often

00:00:31.810 --> 00:00:38.100
referred to as a speed, is not a
vector quantity just by itself.

00:00:38.100 --> 00:00:41.510
This is considered to
be a scalar quantity.

00:00:41.510 --> 00:00:43.520
If we want it to be a
vector, we would also

00:00:43.520 --> 00:00:45.970
have to specify the direction.

00:00:45.970 --> 00:00:48.070
So for example,
someone might say

00:00:48.070 --> 00:00:50.670
it's moving 5 miles
per hour east.

00:00:50.670 --> 00:00:54.490
So let's say it's moving
5 miles per hour due east.

00:00:54.490 --> 00:00:57.740
So now this combined 5
miles per are due east,

00:00:57.740 --> 00:00:59.470
this is a vector quantity.

00:00:59.470 --> 00:01:01.460
And now we wouldn't
call it speed anymore.

00:01:01.460 --> 00:01:04.110
We would call it velocity.

00:01:04.110 --> 00:01:06.940
So velocity is a vector.

00:01:06.940 --> 00:01:09.900
We're specifying the
magnitude, 5 miles per hour,

00:01:09.900 --> 00:01:12.580
and the direction east.

00:01:12.580 --> 00:01:15.650
But how can we actually
visualize this?

00:01:15.650 --> 00:01:18.241
So let's say we're
operating in two dimensions.

00:01:18.241 --> 00:01:19.740
And what's neat
about linear algebra

00:01:19.740 --> 00:01:21.420
is obviously a lot
of what applies

00:01:21.420 --> 00:01:23.420
in two dimensions
will extend to three.

00:01:23.420 --> 00:01:26.170
And then even four, five, six,
as made dimensions as we want.

00:01:26.170 --> 00:01:29.080
Our brains have trouble
visualizing beyond three.

00:01:29.080 --> 00:01:30.860
But what's neat is
we can mathematically

00:01:30.860 --> 00:01:33.820
deal with beyond three
using linear algebra.

00:01:33.820 --> 00:01:35.770
And we'll see that
in future videos.

00:01:35.770 --> 00:01:38.590
But let's just go back to
our straight traditional

00:01:38.590 --> 00:01:41.640
two-dimensional vector
right over here.

00:01:41.640 --> 00:01:43.270
So one way we
could represent it,

00:01:43.270 --> 00:01:46.020
as an arrow that
is 5 units long.

00:01:46.020 --> 00:01:50.530
We'll assume that each of our
units here is miles per hour.

00:01:50.530 --> 00:01:52.590
And that's pointed
to the right, where

00:01:52.590 --> 00:01:55.030
we'll say the right is east.

00:01:55.030 --> 00:01:58.220
So for example, I could start
an arrow right over here.

00:01:58.220 --> 00:01:59.670
And I could make its length 5.

00:01:59.670 --> 00:02:03.130
The length of the arrow
specifies the magnitude.

00:02:03.130 --> 00:02:08.020
So 1, 2, 3, 4, 5.

00:02:08.020 --> 00:02:09.610
And then the direction
that the arrow

00:02:09.610 --> 00:02:13.290
is pointed in specifies
it's direction.

00:02:13.290 --> 00:02:19.160
So this right over here could
represent visually this vector.

00:02:19.160 --> 00:02:21.980
If we say that the
horizontal axis is say east,

00:02:21.980 --> 00:02:24.310
or the positive horizontal
direction is moving

00:02:24.310 --> 00:02:26.630
in the east, this would be
west, that would be north,

00:02:26.630 --> 00:02:28.682
and then that would be south.

00:02:28.682 --> 00:02:30.890
Now, what's interesting
about vectors is that we only

00:02:30.890 --> 00:02:32.970
care about the magnitude
in the direction.

00:02:32.970 --> 00:02:36.240
We don't necessarily
not care where we start,

00:02:36.240 --> 00:02:39.060
where we place it when we think
about it visually like this.

00:02:39.060 --> 00:02:41.500
So for example, this would
be the exact same vector,

00:02:41.500 --> 00:02:44.500
or be equivalent vector to this.

00:02:44.500 --> 00:02:45.997
This vector has the same length.

00:02:45.997 --> 00:02:47.205
So it has the same magnitude.

00:02:47.205 --> 00:02:49.010
It has a length of 5.

00:02:49.010 --> 00:02:51.250
And its direction
is also due east.

00:02:51.250 --> 00:02:53.627
So these two vectors
are equivalent.

00:02:53.627 --> 00:02:56.210
Now one thing that you might say
is, well, that's fair enough.

00:02:56.210 --> 00:02:57.500
But how do we represent
it with a little bit

00:02:57.500 --> 00:02:58.930
more mathematical notation?

00:02:58.930 --> 00:03:00.554
So we don't have to
draw it every time.

00:03:00.554 --> 00:03:02.670
And we could start
performing operations on it.

00:03:02.670 --> 00:03:04.340
Well, the typical
way, one, if you

00:03:04.340 --> 00:03:06.440
want a variable to
represent a vector,

00:03:06.440 --> 00:03:08.780
is usually a lowercase letter.

00:03:08.780 --> 00:03:11.121
If you're publishing a
book, you can bold it.

00:03:11.121 --> 00:03:12.870
But when you're doing
it in your notebook,

00:03:12.870 --> 00:03:15.670
you would typically put a
little arrow on top of it.

00:03:15.670 --> 00:03:17.925
And there are several
ways that you could do it.

00:03:17.925 --> 00:03:20.050
You could literally say,
hey 5 miles per hour east.

00:03:20.050 --> 00:03:21.758
But that doesn't feel
like you can really

00:03:21.758 --> 00:03:23.620
operate on that easily.

00:03:23.620 --> 00:03:26.640
The typical way is to specify,
if you're in two dimensions,

00:03:26.640 --> 00:03:29.180
to specify two
numbers that tell you

00:03:29.180 --> 00:03:33.770
how much is this vector moving
in each of these dimensions?

00:03:33.770 --> 00:03:35.564
So for example,
this one only moves

00:03:35.564 --> 00:03:36.730
in the horizontal dimension.

00:03:36.730 --> 00:03:39.640
And so we'll put our
horizontal dimension first.

00:03:39.640 --> 00:03:43.590
So you might call
this vector 5, 0.

00:03:43.590 --> 00:03:48.120
It's moving 5, positive 5
in the horizontal direction.

00:03:48.120 --> 00:03:51.320
And it's not moving at all
in the vertical direction.

00:03:51.320 --> 00:03:52.690
And the notation might change.

00:03:52.690 --> 00:03:55.315
You might also see notation, and
actually in the linear algebra

00:03:55.315 --> 00:03:57.110
context, it's more
typical to write it

00:03:57.110 --> 00:04:01.270
as a column vector
like this-- 5, 0.

00:04:01.270 --> 00:04:03.130
This once again,
the first coordinate

00:04:03.130 --> 00:04:07.090
represents how much we're moving
in the horizontal direction.

00:04:07.090 --> 00:04:09.870
And the second coordinate
represents how much

00:04:09.870 --> 00:04:15.309
are we moving in the
vertical direction.

00:04:15.309 --> 00:04:16.850
Now, this one isn't
that interesting.

00:04:16.850 --> 00:04:18.058
You could have other vectors.

00:04:18.058 --> 00:04:20.640
You could have a vector
that looks like this.

00:04:20.640 --> 00:04:24.950
Let's say it's moving 3 in
the horizontal direction.

00:04:24.950 --> 00:04:26.590
And positive 4.

00:04:26.590 --> 00:04:30.700
So 1, 2, 3, 4 in the
vertical direction.

00:04:30.700 --> 00:04:36.320
So it might look
something like this.

00:04:36.320 --> 00:04:39.400
So this could be another
vector right over here.

00:04:39.400 --> 00:04:42.420
Maybe we call this
vector, vector a.

00:04:42.420 --> 00:04:45.250
And once again, I want to
specify that is a vector.

00:04:45.250 --> 00:04:49.410
And you see here that if
you were to break it down,

00:04:49.410 --> 00:04:57.330
in the horizontal direction,
it's shifting three

00:04:57.330 --> 00:05:00.900
in the horizontal direction,
and it's shifting positive four

00:05:00.900 --> 00:05:02.580
in the vertical direction.

00:05:02.580 --> 00:05:04.650
And we get that by
literally thinking

00:05:04.650 --> 00:05:07.890
about how much we're moving
up and how much we're

00:05:07.890 --> 00:05:11.130
moving to the right when we
start at the end of the arrow

00:05:11.130 --> 00:05:12.840
and go to the front of it.

00:05:12.840 --> 00:05:19.650
So this vector might
be specified as 3, 4.

00:05:19.650 --> 00:05:20.792
3, 4.

00:05:20.792 --> 00:05:22.500
And you could use the
Pythagorean theorem

00:05:22.500 --> 00:05:24.940
to figure out the actual
length of this vector.

00:05:24.940 --> 00:05:27.870
And you'll see because this is
a 3, 4, 5 triangle, that this

00:05:27.870 --> 00:05:31.480
actually has a magnitude of 5.

00:05:31.480 --> 00:05:33.480
And as we study more
and more linear algebra,

00:05:33.480 --> 00:05:35.980
we're going to start extending
these to multiple dimensions.

00:05:35.980 --> 00:05:38.810
Obviously we can visualize
up to three dimensions.

00:05:38.810 --> 00:05:40.929
In four dimensions it
becomes more abstract.

00:05:40.929 --> 00:05:42.970
And that's why this type
of a notation is useful.

00:05:42.970 --> 00:05:47.100
Because it's very hard to draw
a 4, 5, or 20 dimensional arrow

00:05:47.100 --> 00:05:48.910
like this.


Introduction to proportional relationships

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:03.091
What I want to introduce you to
in this video is the notion of

00:00:03.091 --> 00:00:05.493
a proportional relationship.

00:00:05.493 --> 00:00:09.755
And a proportional relationship
between two variables

00:00:09.755 --> 00:00:12.252
is just a relationship
where the ratio

00:00:12.252 --> 00:00:13.572
between the two variables

00:00:13.572 --> 00:00:16.683
is always going to be the same thing.

00:00:16.683 --> 00:00:18.467
So let's look at an example of that.

00:00:18.467 --> 00:00:21.981
So let's just say that we want
to think about the relationship

00:00:21.981 --> 00:00:26.737
between x and y. And let's say that when

00:00:26.737 --> 00:00:35.234
x is one, y is three, and then
when x is two, y is six.

00:00:35.234 --> 00:00:40.055
And when x is nine, y is 27.

00:00:40.055 --> 00:00:44.339
Now this is a proportional relationship.
Why is that?

00:00:44.339 --> 00:00:48.194
Because the ratio between y and x
is always the same thing.

00:00:48.194 --> 00:00:51.297
And actually the ratio between y and x
or, you could say the ratio

00:00:51.297 --> 00:00:53.666
between x and y, is always the same thing.

00:00:53.666 --> 00:00:56.704
So, for example--
if we say the ratio y over x--

00:00:56.704 --> 00:01:00.544
this is always equal to--
it could be three over one,

00:01:00.544 --> 00:01:04.676
which is just three.
It could be six over two,

00:01:04.676 --> 00:01:06.846
which is also just three.

00:01:06.846 --> 00:01:11.545
It could be 27 over nine,
which is also just three.

00:01:11.545 --> 00:01:17.896
So you see that y over x is
always going to be equal to three,

00:01:17.896 --> 00:01:19.933
or at least in this table right over here.

00:01:19.933 --> 00:01:22.896
And so, or at least based on
the data points we have just seen.

00:01:22.896 --> 00:01:26.270
So based on this, it looks like that
we have a proportional relationship

00:01:26.270 --> 00:01:30.640
between y and x. So this one
right over here is proportional.

00:01:30.640 --> 00:01:34.555
So given that, what's an example of
relationships that are not proportional.

00:01:34.555 --> 00:01:37.367
Well those are fairly easy to construct.

00:01:37.367 --> 00:01:41.947
So let's say we had-- I'll do it with
two different variables.

00:01:41.947 --> 00:01:47.419
So let's say we have a and b.

00:01:47.419 --> 00:01:52.020
And let's say when a is one, b is three.

00:01:52.020 --> 00:01:56.118
And when a is two, b is six.

00:01:56.118 --> 00:02:02.289
And when a is 10, b is 35.

00:02:02.289 --> 00:02:04.492
So here-- you might say look, look

00:02:04.492 --> 00:02:06.305
when a is one, b is three so

00:02:06.305 --> 00:02:11.459
the ratio b to a-- you could say b to a--

00:02:11.459 --> 00:02:14.072
you could say well when
b is three, a is one.

00:02:14.072 --> 00:02:15.605
Or when a is one, b is three.

00:02:15.605 --> 00:02:18.227
So three to one. And that's also the case

00:02:18.227 --> 00:02:22.132
when b is six, a is two.
Or when a is two, b is six.

00:02:22.132 --> 00:02:25.048
So it's six to two. So these ratios
seem to be the same.

00:02:25.048 --> 00:02:27.161
They're both three.
But then all of sudden

00:02:27.161 --> 00:02:29.204
the ratio is different right over here.

00:02:29.204 --> 00:02:36.000
This is not equal to 35 over 10.

00:02:36.000 --> 00:02:38.612
So this is not a
proportional relationship.

00:02:38.612 --> 00:02:42.038
In order to be proportional
the ratio between the two variables

00:02:42.038 --> 00:02:46.741
always has to be the same.
So this right over here--

00:02:46.741 --> 00:02:49.699
This is not proportional.

00:02:49.699 --> 00:02:53.519
Not proportional.

00:02:53.519 --> 00:02:56.693
So the key in identifying
a proportional relationship

00:02:56.693 --> 00:02:59.784
is look at the different values
that the variables take on

00:02:59.784 --> 00:03:01.788
when one variable is one value,

00:03:01.788 --> 00:03:03.798
and then what is the
other variable become?

00:03:03.798 --> 00:03:05.608
And then take the ratio between them.

00:03:05.608 --> 00:03:08.062
Here we took the ratio y to x,
and you see y to x,

00:03:08.062 --> 00:03:12.290
or y divided by x-- the ratio of y to x
is always going to be the same here

00:03:12.290 --> 00:03:15.499
so this is proportional. And you could
actually gone the other way.

00:03:15.499 --> 00:03:18.893
You could have said, well
what's the ratio of x to y?

00:03:18.893 --> 00:03:22.797
Well over here it would be one to three,

00:03:22.797 --> 00:03:25.577
which is the same thing as two to six,

00:03:25.577 --> 00:03:29.588
which is the same thing as nine to 27.

00:03:29.588 --> 00:03:33.175
When you take this ratio--
if you say the ratio of x to y

00:03:33.175 --> 00:03:36.263
instead of y to x, you see that
it is always one third.

00:03:36.263 --> 00:03:39.489
But any way you look at it--
the ratio between these two variables--

00:03:39.489 --> 00:03:42.066
if you say y to x,
it's always going to be three.

00:03:42.066 --> 00:03:44.821
Or x to y is always going to be one third.

00:03:44.821 --> 00:03:49.126
So this is proportional
while this one is not.


Determining rates with fractions

WEBVTT
Kind: captions
Language: en

00:00:02.879 --> 00:00:06.389
It's a little daunting because it has fractions, but then when we work through it step-by-step

00:00:06.549 --> 00:00:10.589
Hopefully, it'll feel a little bit more intuitive so it says Calvin cleans

00:00:11.110 --> 00:00:17.670
three-fifths of his bathroom with one third of a bottle of cleaning solution at this rate

00:00:17.670 --> 00:00:22.799
What fraction of the bottle of cleaning Solution will Calvin used to clean his entire bathroom?

00:00:23.199 --> 00:00:27.149
And like always encourage you to pause the video and try to take a stab at this yourself

00:00:27.580 --> 00:00:29.160
So as [I] mentioned it's a little bit

00:00:29.160 --> 00:00:32.700
You know it's a three-fifths of his bathroom with one third of a bottle

00:00:33.070 --> 00:00:36.750
How do we [think] about this and what my brain does is I'd like to say well

00:00:37.059 --> 00:00:44.069
How would we like to answer the question what fraction of the bottle of cleaning solution will Calvin used to clean his entire bathroom?

00:00:44.440 --> 00:00:49.800
So we want to figure out is we want to figure out how many bottles so??

00:00:50.680 --> 00:00:52.300
bottles

00:00:52.300 --> 00:00:54.039
bottles

00:00:54.039 --> 00:00:56.039
Per let me write it this way

00:00:56.649 --> 00:01:01.198
So we want to figure out he uses a certain number up with that question mark

00:01:02.260 --> 00:01:04.180
bottles

00:01:04.180 --> 00:01:06.180
bottles per

00:01:06.220 --> 00:01:12.419
Bathroom if we knew this then we have the answer to the question if this would this might be I don't know [two-fifths]

00:01:12.670 --> 00:01:16.350
Bottles two-fifths of a bottle per bathroom it might be two

00:01:16.570 --> 00:01:20.790
Bottles per bathroom, but if we know whatever this question mark is then we know the answer

00:01:20.950 --> 00:01:25.950
How many bottles does he need to take to clean a bathroom or what fraction of a bottle?

00:01:25.950 --> 00:01:28.829
We don't know, they're hinting that it's a fraction of a bottle

00:01:28.829 --> 00:01:34.559
but how much of a bottle or how many bottles per bathroom? Or another way to think about this if we want to know if

00:01:34.560 --> 00:01:39.780
We wanted to express it as a as a rate more mathematically. We could say "question mark

00:01:40.720 --> 00:01:42.670
bottles

00:01:42.670 --> 00:01:44.909
bottles per bathroom"

00:01:46.030 --> 00:01:48.689
and the Reason why this is helpful

00:01:48.689 --> 00:01:52.529
It makes it clear that look we want to figure out we want to take our

00:01:52.990 --> 00:01:57.839
Units tell us that we want to divide the number of bottles or the fraction of a bottle it takes

00:01:58.000 --> 00:02:04.769
To clean the number of bathrooms or certain fraction of the bathrooms. And they tell us over here they tell us that

00:02:05.290 --> 00:02:08.129
he's able to take one third of a bottle

00:02:08.649 --> 00:02:11.899
He's able to take one third of a bottle so I could write it here "one third

00:02:13.020 --> 00:02:15.020
of a bottle to

00:02:15.670 --> 00:02:17.670
clean to clean

00:02:19.120 --> 00:02:21.120
three-fifths of a bathroom"

00:02:21.490 --> 00:02:23.490
three-fifths of a bathroom

00:02:23.770 --> 00:02:26.070
three-fifths of a bathroom

00:02:27.130 --> 00:02:30.540
So hopefully it's clear now. Why this was helpful? We say okay,

00:02:30.540 --> 00:02:36.929
We want to take how many bottles it takes to clean a certain number of bathrooms one third of a bottle take can clean

00:02:37.090 --> 00:02:40.950
three-fifths of a bathroom and makes it clear that we need to take the one-third and

00:02:41.020 --> 00:02:46.020
Divide it by three-fifths because then we're going to get bottles per bathroom. We're going to get the rate

00:02:46.020 --> 00:02:50.040
We're not going to get bathrooms per bottle, we're going to get bottles per bathroom

00:02:50.040 --> 00:02:52.889
Which is what we care about what fraction of the bottle will

00:02:52.890 --> 00:02:57.149
It will it take to clean entire bathroom to clean one entire bathroom?

00:02:57.490 --> 00:03:01.169
So now we just have to take one-third and divide it by three-fifths

00:03:01.170 --> 00:03:06.570
So this is going to be equal to this is going to be equal to you can write it this way

00:03:07.750 --> 00:03:09.750
one third divided by

00:03:10.930 --> 00:03:12.760
three-fifths

00:03:12.760 --> 00:03:18.359
Divided by three fifths and the units are going to be bottles per bathroom. I'll write it like that

00:03:19.780 --> 00:03:27.450
Bottles or since if we know that that's going to be a fraction. We could just say bottle per bathroom bottle

00:03:29.920 --> 00:03:34.079
Bathroom and so this is going to be the same thing as well

00:03:34.080 --> 00:03:37.020
we're going to have one third divided by three fifths is the same thing as

00:03:37.330 --> 00:03:42.510
One third times the reciprocal of three fifths, so one-third times five thirds

00:03:43.810 --> 00:03:46.860
one third times five thirds

00:03:48.370 --> 00:03:52.890
Bottle per bathroom, so let me write that down bottle per

00:03:55.750 --> 00:03:57.010
bathroom

00:03:57.010 --> 00:03:59.700
Bottle per bathroom and what is this going to be? Well,

00:03:59.700 --> 00:04:05.819
We multiply the numerators 1 times 5 is going to give us 5 and the denominator three times three is ninths

00:04:06.130 --> 00:04:10.499
There's going to be five ninths of a bottle of a bottle per

00:04:11.290 --> 00:04:13.290
bathroom

00:04:14.380 --> 00:04:18.059
Bathroom I wrote these a little bit further apart than I would want to let me write that a little bit closer together

00:04:18.730 --> 00:04:21.960
five ninths of a bottle

00:04:23.130 --> 00:04:25.130
per bathroom

00:04:25.490 --> 00:04:27.639
So just as a reminder of what we did here

00:04:27.639 --> 00:04:29.440
It could see him a little bit daunting

00:04:29.440 --> 00:04:34.809
But we said okay look we want how much, what fraction of the bottle to clean his entire bathroom

00:04:34.810 --> 00:04:38.500
So we care about bottles per bathroom or since we're talking about a fraction

00:04:38.500 --> 00:04:44.079
I guess we could just say bottle some fraction of a bottle per bathroom so it's bottle divided by

00:04:44.870 --> 00:04:47.079
Bathrooms that's going to give us the correct rate

00:04:47.180 --> 00:04:49.209
So it took one third of a bottle to clean

00:04:49.400 --> 00:04:52.570
Three-fifths of a bathroom so you divide one third by three-fifths

00:04:52.669 --> 00:04:57.879
You get five ninths of a bottle per bathroom, so this question mark right over here that is

00:04:58.520 --> 00:05:04.230
five ninths, it takes five ninths of a bottle of cleaning solution to clean his entire bathroom


How to estimate the average rate of change of a modeling function from a graph

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.600
Teresa went skydiving.

00:00:02.000 --> 00:00:10.000
The graph below discribes Teresa's height, measured in meters, as a function of time, measured in seconds.

00:00:10.000 --> 00:00:15.140
So let's look at this graph over here. It's actually quite large, so let me zoom out a little bit.

00:00:15.700 --> 00:00:24.700
And we can see that at time zero her height is seven hundred meters and then as time increases, as we move to the right her height is decreasing.

00:00:26.380 --> 00:00:32.300
And her height is decreasing at faster and faster rates as we move to the right so her- her-

00:00:32.480 --> 00:00:38.000
the rate of decline of her height is quite steep as we approach ten seconds after she jumps

00:00:38.400 --> 00:00:42.760
And then we can see all of a sudden, then we can see all of a sudden her rate of decline slows down

00:00:42.760 --> 00:00:48.040
She's still declining as -as we move forward in time, but she's declining at a slower rate.

00:00:48.760 --> 00:00:54.760
And so we can say she's declining at a --- or she is --- her height is changing at a less negative rate

00:00:54.780 --> 00:00:59.060
It's -It's quite a negative rate right over here --- seems roughly a fairly negative rate

00:00:59.460 --> 00:01:02.560
but then it becomes a less negative rate right over here

00:01:02.560 --> 00:01:04.200
her height is changing at a less negative rate

00:01:04.200 --> 00:01:08.600
and it makes sense that this is when she deploys the parachute, so after 10 seconds

00:01:08.600 --> 00:01:10.060
she deploys the parachute

00:01:10.060 --> 00:01:13.440
so she jumps at 0 seconds, 10 seconds she deploys the parachute.

00:01:14.320 --> 00:01:16.400
Alright so let's see what they're asking --- actually asking us.

00:01:16.400 --> 00:01:16.900
They say

00:01:16.900 --> 00:01:18.860
Complete the following sentence

00:01:18.860 --> 00:01:24.740
Between 3 seconds and 8 seconds after Teresa jumped, her height decreased, on average, by approximately

00:01:24.920 --> 00:01:26.940
blank meters per second.

00:01:26.980 --> 00:01:33.320
So between 3 seconds, and 8 seconds, so at 3 seconds --- so time is 3 right over there

00:01:33.320 --> 00:01:35.320
So, Let's see what H(3) is.

00:01:35.660 --> 00:01:39.659
what is her height at 3 seconds and I'm just looking
at a graph so I'm going to have to

00:01:39.659 --> 00:01:47.340
Ballpark it so at 3 seconds at 3 seconds
her height looks pretty close it's

00:01:47.340 --> 00:01:52.649
pretty close we just have to approximate
it so her height at the height after 3

00:01:52.649 --> 00:01:57.960
seconds it looks like it's about halfway
between 600 and 700 so it looks like

00:01:57.960 --> 00:02:06.060
it's about 650 meters and then we care
between 3 seconds and 8 seconds so our

00:02:06.060 --> 00:02:12.830
height at 8 seconds let's look at that
at 8 seconds let's see this looks about

00:02:12.830 --> 00:02:25.220
halfway it looks about halfway between
350 between 350 and 400 so I'll say her

00:02:25.220 --> 00:02:29.000
height at 8 seconds actually since I'm
approximating it let me put a little

00:02:29.000 --> 00:02:33.350
squiggly equal sign here her height
after 8 seconds looks like it's

00:02:33.350 --> 00:02:43.280
approximately 300 and looks
approximately 375 meters 375 meters so

00:02:43.280 --> 00:02:47.840
what is her average rate of change her
height decreased on average by

00:02:47.840 --> 00:02:50.990
approximately so we what we want to do
is we forgot to figure out the average

00:02:50.990 --> 00:02:55.340
rate of change which you could view as
the slope of the line that connects

00:02:55.340 --> 00:02:59.360
these two points the slope of this line
is going to be her average rate of

00:02:59.360 --> 00:03:03.080
change so let's think about that her
average rate of change her change in

00:03:03.080 --> 00:03:08.990
height over the change in time for that
interval well her change in height after

00:03:08.990 --> 00:03:15.260
at 8 seconds she is at let me write it
this way her height at 8 seconds - her

00:03:15.260 --> 00:03:19.190
height at 3 seconds so this is going to
be her change in height and the change

00:03:19.190 --> 00:03:24.380
in time is 8 seconds she finishes at 8
seconds - where she started or the

00:03:24.380 --> 00:03:30.350
interval that we care about starting at
3 seconds and so H of 8 we already said

00:03:30.350 --> 00:03:38.620
this is approximately 375 H of 3 this is
650 and then of course 8 minus 3 is

00:03:38.630 --> 00:03:41.930
going to be equal to 5
I just want emphasize this is just her

00:03:41.930 --> 00:03:46.400
average rate of change for approximate
average rate of change over this

00:03:46.400 --> 00:03:51.769
interval as we as we go from 3 to 8
seconds our height goes from H of 3 to H

00:03:51.769 --> 00:03:58.670
of 8 so this is going to be let's see
375 - 650

00:03:58.670 --> 00:04:06.380
let's see if it was 375 - 675 it would
be negative 300 and so this is going to

00:04:06.380 --> 00:04:10.640
be but then we're not subtracting 675
we're subtracting 650 so it's going to

00:04:10.640 --> 00:04:18.830
be 25 it's going to be 25 more so this
is going to be negative 275 over 5 let

00:04:18.830 --> 00:04:22.640
me make sure I did that math right let
me make sure I did that math right so

00:04:22.640 --> 00:04:29.449
it's 375 - 650 is
negative 275 does that make sense

00:04:29.449 --> 00:04:34.639
let's see 275 plus 375 would be 2 650
yeah that is right all right so let's

00:04:34.639 --> 00:04:39.590
just figure out what this is so 5 I'll
just figure out what 5 goes into 275 and

00:04:39.590 --> 00:04:45.440
then we can remember the negative 5 goes
into 27 5 times 5 times 5 is 25

00:04:45.440 --> 00:04:51.110
subtract we get a to bring down to 5 5
goes into 25 five times and then we're

00:04:51.110 --> 00:04:57.880
not going to have a remainder so this is
going to be equal to negative 55

00:04:57.880 --> 00:05:02.750
negative 55 and the unit's our height is
given in meters so this part up here

00:05:02.750 --> 00:05:07.669
this is in meters up here meters per
second so her change in height or

00:05:07.669 --> 00:05:13.070
average change in height over or the
average rate of change of height over

00:05:13.070 --> 00:05:18.889
this five seconds over this five seconds
is negative 55 meters per second one way

00:05:18.889 --> 00:05:23.020
to think about it the slope right over
here the slope is equal to negative

00:05:23.020 --> 00:05:29.780
55 now it might be tempting to just
write negative 55 right over here but

00:05:29.780 --> 00:05:33.289
let's just think about whether that
would be right between 3 seconds and 8

00:05:33.289 --> 00:05:38.660
seconds after Teresa jumped her height
decreased on average by approximately

00:05:38.660 --> 00:05:42.409
negative 55 meters per second and
decrease is important because they're

00:05:42.409 --> 00:05:45.289
already saying that it's decreased when
this negative is telling us that we're

00:05:45.289 --> 00:05:50.360
decreasing we're decreasing at a rate of
55 meters per second our average rate of

00:05:50.360 --> 00:05:55.000
change is a decrease our average rate of
high a change of height over time is a

00:05:55.000 --> 00:05:59.870
decrease of 55 meters per second well
the negative is already saying the

00:05:59.870 --> 00:06:03.680
decrease so we're not decreasing at a
negative rate of change we would be

00:06:03.680 --> 00:06:09.620
decreasing at 55 meters per second so
let me just write it 55 meters per

00:06:09.620 --> 00:06:15.139
second if they asked if they asked her
height if we asked her height if we if

00:06:15.139 --> 00:06:23.900
they asked the average change of height
average rate of change let me write this

00:06:23.900 --> 00:06:36.349
the average rate of change of change of
H over or let me say of H from 3 seconds

00:06:36.349 --> 00:06:39.620
3 seconds
to 8 seconds

00:06:39.620 --> 00:06:44.570
well now this would be negative 55
meters per second but when they're

00:06:44.570 --> 00:06:47.390
saying that her height decrease that's
re taking the negative into

00:06:47.390 --> 00:06:49.970
consideration they're saying it's
definitely decreasing that's what the

00:06:49.970 --> 00:06:54.110
negatives already telling us it's
decreasing by a rate of 55 meters per

00:06:54.110 --> 00:06:56.466
second hopefully that makes some sense


How to represent the average rate of change of a modeling function as an expression

WEBVTT
Kind: captions
Language: en

00:00:00.020 --> 00:00:05.580
Karina drained a bucket of water. Let W(t)
denote the amount of water W

00:00:05.580 --> 00:00:09.940
(measured in milliliters) that remained
in the bucket after t seconds.

00:00:10.140 --> 00:00:12.620
Which equation best represents
the following statement?

00:00:12.920 --> 00:00:20.800
Over the first 25 seconds,
the amount of water remaining

00:00:20.960 --> 00:00:25.900
decreased by an average of four milliliters

00:00:25.960 --> 00:00:28.240
each second.

00:00:28.800 --> 00:00:32.200
Let me rewrite this, let me paraphrase
this and then maybe

00:00:32.560 --> 00:00:34.740
we'll be able to think about the math a little bit.

00:00:34.900 --> 00:00:38.820
They're saying that the average

00:00:39.300 --> 00:00:44.400
rate of change

00:00:44.800 --> 00:00:51.940
of the amount of remaining water, so I could say the average rate of change of W, with respect to time,

00:00:52.440 --> 00:00:59.820
over first 25 seconds

00:01:01.200 --> 00:01:04.360
is equal to

00:01:04.940 --> 00:01:07.940
a decrease.  The water decreased

00:01:08.300 --> 00:01:12.480
by four milliliters each second. So if we're decreasing,

00:01:12.560 --> 00:01:17.140
if our rate of change, if W is going down, our rate of change is going to be negative.

00:01:17.360 --> 00:01:21.640
Every second that goes by, W is going to go down by some amount, so it's going to be

00:01:21.640 --> 00:01:24.700
negative 4 milliliters,

00:01:24.720 --> 00:01:26.280
I'll just write that as mL,

00:01:26.520 --> 00:01:33.420
negative 4 milliliters per second.

00:01:33.420 --> 00:01:37.060
Now can we write this in a more "mathy" way?

00:01:37.060 --> 00:01:40.460
"The average rate of change of W
over the first twenty five seconds?"

00:01:41.780 --> 00:01:46.360
The rate of change of W is going to be
our change in W over our change in time.

00:01:46.580 --> 00:01:51.360
It's our change in W over the first 25 seconds, divided by the change of time over the first 25 seconds

00:01:51.360 --> 00:01:52.740
which is just 25 seconds.

00:01:55.860 --> 00:02:02.640
Our change in W is going to be our

00:02:02.660 --> 00:02:05.520
finishing amount remaining in the bucket,

00:02:05.520 --> 00:02:06.780
so W(25).

00:02:06.780 --> 00:02:11.460
That's how much we have at the end of this interval that we care about, how much water is remaining,

00:02:11.760 --> 00:02:17.640
minus how much water we started off with,

00:02:17.660 --> 00:02:19.800
divided by how much time goes by.

00:02:20.000 --> 00:02:21.480
And we could say "Hey,

00:02:21.720 --> 00:02:25.340
you know, we finished it at the 25th second, we started at the 0th second,

00:02:25.520 --> 00:02:30.400
or 25 minus 0 is just going to be 25."

00:02:30.800 --> 00:02:37.260
This expression I just wrote is the average rate of change of W over the first 25 seconds.

00:02:37.360 --> 00:02:38.480
Notice the way I wrote it.

00:02:39.440 --> 00:02:41.620
When I write it like this it might be a little bit clearer.

00:02:41.800 --> 00:02:45.560
This is our ending W minus our initial W,

00:02:45.800 --> 00:02:48.520
and this is our ending time minus our initial time.

00:02:48.880 --> 00:02:51.800
This last part just simplifies to 25.

00:02:52.140 --> 00:02:56.300
And they tell us that this is going to be
negative 4 milliliters per second.

00:02:56.800 --> 00:02:59.940
This is going to be equal to negative 4.

00:03:00.120 --> 00:03:02.720
And the units up here in the numerator,
this would be in milliliters,

00:03:02.720 --> 00:03:03.860
and down here would be in seconds.

00:03:03.860 --> 00:03:07.100
So it makes sense that this would end up
being in milliliters per second.

00:03:07.440 --> 00:03:09.280
But anyway, which of these choices have that?

00:03:09.640 --> 00:03:11.280
I have one more choice down here.

00:03:12.060 --> 00:03:18.360
This one over here looks exactly like what I just wrote.

00:03:18.740 --> 00:03:21.500
Now, a tempting one might be this one up here

00:03:21.500 --> 00:03:24.920
and the only difference between this one and this one

00:03:24.920 --> 00:03:27.080
is that we have a positive 4 over here.

00:03:27.500 --> 00:03:29.360
But keep in mind what this would imply.

00:03:29.620 --> 00:03:33.600
If W(25) minus W(0),

00:03:33.900 --> 00:03:38.500
in order for this to be positive (because we're dividing by a positive 25),

00:03:38.860 --> 00:03:42.760
then this would have to be positive.

00:03:43.260 --> 00:03:48.700
In order for this to be positive, that would mean that we have more water remaining after 25 seconds

00:03:48.700 --> 00:03:51.480
than we do after 0 seconds,
because in order for this to be positive

00:03:51.480 --> 00:03:53.420
this one has to be larger,

00:03:53.620 --> 00:03:57.020
which means that somehow the bucket
is filling up with water, not draining.

00:03:57.200 --> 00:04:01.820
But we know that the water is decreasing
by an average of 4 milliliters each second.

00:04:02.040 --> 00:04:06.820
So if we're decreasing, this value over here
needs to be equal to negative.

00:04:07.020 --> 00:04:10.980
You have to have a lower value after 25 seconds than you do initally.

00:04:10.980 --> 00:04:14.300
So that minus that needs to be a negative value.

00:04:14.380 --> 00:04:18.260
If you have a negative value up here, and you divide by a positive value,

00:04:18.260 --> 00:04:21.000
you should get a negative value.

00:04:21.120 --> 00:04:22.720
It also makes conceptual sense.

00:04:22.880 --> 00:04:26.900
The water is decreasing, the rate of change of water with respect to time

00:04:27.000 --> 00:04:30.980
should be negative, because the amount of water is decreasing.


Format changes to the new SAT

WEBVTT
Kind: captions
Language: en

00:00:00.175 --> 00:00:01.587
- [Voiceover] As you might already know

00:00:01.587 --> 00:00:03.253
or you're finding out through this video,

00:00:03.253 --> 00:00:07.489
a new SAT, a new SAT is coming.

00:00:07.489 --> 00:00:10.689
And the first administration
of that new SAT

00:00:10.689 --> 00:00:15.058
is going to be on March 2016.

00:00:15.058 --> 00:00:16.744
So the first question
you're probably asking,

00:00:16.744 --> 00:00:19.020
"well, does this apply to me?"

00:00:19.020 --> 00:00:20.747
And a simple rule of thumb is if

00:00:20.747 --> 00:00:22.423
you're going to need to apply to college

00:00:22.423 --> 00:00:25.623
before March 2016, and
so that's the majority

00:00:25.623 --> 00:00:29.717
of the folks who are
applying in the fall of 2015,

00:00:29.717 --> 00:00:31.159
well then the new SAT is probably

00:00:31.159 --> 00:00:33.851
going to come a little
bit too late for you.

00:00:33.851 --> 00:00:35.720
But if you're applying
for college after that,

00:00:35.720 --> 00:00:37.752
so the fall of 2016,

00:00:37.752 --> 00:00:40.526
you could probably use either SAT,

00:00:40.526 --> 00:00:42.953
the new SAT or the current SAT.

00:00:42.953 --> 00:00:45.585
And for sure if you're
applying for college

00:00:45.585 --> 00:00:48.175
in the fall of 2017, 2018,

00:00:48.175 --> 00:00:50.887
you're probably going
to focus on the new SAT.

00:00:50.887 --> 00:00:53.011
So applying, let me write this,

00:00:53.011 --> 00:00:55.388
so this most likely will apply

00:00:55.388 --> 00:01:00.388
to folks applying for college

00:01:02.283 --> 00:01:07.283
fall 2016 and after.

00:01:10.495 --> 00:01:12.160
So with that out of the way,

00:01:12.160 --> 00:01:13.653
what's the whole point?

00:01:13.653 --> 00:01:16.223
What is different about the new SAT?

00:01:16.223 --> 00:01:18.326
And I'll do another video
where I'll go into the details

00:01:18.326 --> 00:01:19.789
and I look at some of the sample items,

00:01:19.789 --> 00:01:21.353
and we think about how
they're different from

00:01:21.353 --> 00:01:24.553
the current SAT, but at a very high level,

00:01:24.553 --> 00:01:26.758
the goal is to really just be more aligned

00:01:26.758 --> 00:01:28.790
with what you learn in school.

00:01:28.790 --> 00:01:33.790
More aligned, more aligned with school.

00:01:34.692 --> 00:01:35.992
And the things that you need to know

00:01:35.992 --> 00:01:37.120
to be college ready.

00:01:37.120 --> 00:01:38.887
So, this even includes the vocabulary.

00:01:38.887 --> 00:01:42.282
The old SAT or you could
say the current SAT

00:01:42.282 --> 00:01:44.353
was famous for some words

00:01:44.353 --> 00:01:45.420
that you might not use in

00:01:45.420 --> 00:01:46.659
your everyday conversations,

00:01:46.659 --> 00:01:49.950
or you might not use in, even in college

00:01:49.950 --> 00:01:52.754
while the new SAT is going
to be more aligned with that.

00:01:52.754 --> 00:01:54.247
It still helps to know vocabulary,

00:01:54.247 --> 00:01:55.456
but they're going to be vocabulary words

00:01:55.456 --> 00:01:57.620
that you'll find using a lot in college

00:01:57.620 --> 00:01:59.652
or in your later life.

00:01:59.652 --> 00:02:00.759
So, it's going to be more aligned

00:02:00.759 --> 00:02:01.988
with what you learn in school,

00:02:01.988 --> 00:02:06.418
and at a very general level,
it's going to be simpler.

00:02:06.418 --> 00:02:07.952
And it's going to be simpler in terms of

00:02:07.952 --> 00:02:09.323
the number of sections,

00:02:09.323 --> 00:02:10.522
and we'll talk about that a little bit.

00:02:10.522 --> 00:02:11.619
And you're going to have more time

00:02:11.619 --> 00:02:13.752
for a question in general.

00:02:13.752 --> 00:02:15.185
It's also going to be simpler in terms

00:02:15.185 --> 00:02:17.054
of how you think about taking the test.

00:02:17.054 --> 00:02:19.025
The old or the current SAT,

00:02:19.025 --> 00:02:20.091
you have to think about,

00:02:20.091 --> 00:02:21.260
ok if I guess on this,

00:02:21.260 --> 00:02:22.651
there's going to be a
penalty for guessing.

00:02:22.651 --> 00:02:25.353
While on the new SAT, while on the new SAT

00:02:25.353 --> 00:02:26.613
there's no penalty for guessing.

00:02:26.613 --> 00:02:27.853
So if you can answer it, answer it.

00:02:27.853 --> 00:02:30.321
If you can rule out some choices,

00:02:30.321 --> 00:02:31.053
still answer it.

00:02:31.053 --> 00:02:32.872
You should just always
try to answer questions,

00:02:32.872 --> 00:02:34.120
so you don't have to really think about

00:02:34.120 --> 00:02:35.157
the gamesmanship of it,

00:02:35.157 --> 00:02:38.255
you just answer, answer the questions.

00:02:38.255 --> 00:02:39.525
So the next thing you're
probably thinking about-

00:02:39.525 --> 00:02:40.653
well, what are the sections,

00:02:40.653 --> 00:02:42.827
and how is it actually going to be,

00:02:42.827 --> 00:02:45.488
how is it actually going to be scored?

00:02:45.488 --> 00:02:47.693
So, what I've pre-drawn right over here,

00:02:47.693 --> 00:02:48.546
and I hope you can read it,

00:02:48.546 --> 00:02:49.681
the writing is a little bit small,

00:02:49.681 --> 00:02:54.681
is I'm comparing the
current/old SAT up here,

00:02:55.017 --> 00:02:58.054
so this line right over here is the,

00:02:58.054 --> 00:02:59.893
depending on your point of view,

00:02:59.893 --> 00:03:02.656
the current/old SAT.

00:03:02.656 --> 00:03:06.821
The one that is "sun-setting",
going into 2016.

00:03:06.821 --> 00:03:10.123
Your last chance to take
this is January 2016.

00:03:10.123 --> 00:03:12.520
And then this one is the new one,

00:03:12.520 --> 00:03:13.922
so this right over here,

00:03:13.922 --> 00:03:15.853
that is the new SAT.

00:03:15.853 --> 00:03:18.484
That's the one that we are talking about.

00:03:18.484 --> 00:03:21.724
So, big picture, you see
that the current SAT,

00:03:21.724 --> 00:03:23.573
you have three math sections.

00:03:23.573 --> 00:03:26.692
Two of them are 25 minutes,
one of them is 20 minutes.

00:03:26.692 --> 00:03:28.460
And it's not necessarily going
to be given in this order.

00:03:28.460 --> 00:03:30.918
In fact, it's likely
to be all shuffled up.

00:03:30.918 --> 00:03:34.717
While on the new SAT, you
have two math sections.

00:03:34.717 --> 00:03:36.983
One of them's 25 minutes
and one is longer.

00:03:36.983 --> 00:03:39.090
It's 55 minutes, but they only have

00:03:39.090 --> 00:03:40.122
those two sections,

00:03:40.122 --> 00:03:42.387
and as you can see here,
with the longer section,

00:03:42.387 --> 00:03:44.586
you're allowed to use a calculator.

00:03:44.586 --> 00:03:45.892
And so you'll probably see questions

00:03:45.892 --> 00:03:48.350
that may be helpful to use a calculator.

00:03:48.350 --> 00:03:49.387
And then the other one,

00:03:49.387 --> 00:03:50.880
you don't use a calculator.

00:03:50.880 --> 00:03:53.054
And you see that the time is,

00:03:53.054 --> 00:03:54.344
let's see you're comparing,

00:03:54.344 --> 00:03:59.344
this is 70 minutes and 54 questions

00:03:59.850 --> 00:04:02.918
versus 80 minutes and 58 questions.

00:04:02.918 --> 00:04:04.821
So on average, you have a little bit,

00:04:04.821 --> 00:04:07.855
you have more time per question.

00:04:07.855 --> 00:04:09.450
And now we can look here in the reading,

00:04:09.450 --> 00:04:11.451
again you have three sections of reading.

00:04:11.451 --> 00:04:13.788
Two 25 minute sections,
one 20 minute sections.

00:04:13.788 --> 00:04:17.719
Now you have one 65
minute reading sections.

00:04:17.719 --> 00:04:19.213
And once again, on average,

00:04:19.213 --> 00:04:21.519
you have more time per question

00:04:21.519 --> 00:04:23.926
in the new reading sections.

00:04:23.926 --> 00:04:25.054
And I just want to make it very clear,

00:04:25.054 --> 00:04:27.421
it's not necessarily going to
be administered in this order.

00:04:27.421 --> 00:04:29.422
I'm just showing you so
we can kind of compare.

00:04:29.422 --> 00:04:32.023
Reading to reading or math to math.

00:04:32.023 --> 00:04:34.989
Then in the current SAT, you have,

00:04:34.989 --> 00:04:37.854
the one that you would take
if you took it in 2015,

00:04:37.854 --> 00:04:40.018
you have two writing sections

00:04:40.018 --> 00:04:41.084
or two, I guess you could say,

00:04:41.084 --> 00:04:42.821
two non-essay writing sections.

00:04:42.821 --> 00:04:44.721
One is 25 minutes, one is 10 minutes.

00:04:44.721 --> 00:04:46.590
And then you also have an essay section

00:04:46.590 --> 00:04:49.821
that helps form part
of your writing score.

00:04:49.821 --> 00:04:51.893
While in the new SAT, you have just a

00:04:51.893 --> 00:04:55.621
35 minute writing section and
then the essay is optional.

00:04:55.621 --> 00:04:57.084
This little green thing right over here,

00:04:57.084 --> 00:04:58.354
this is the experimental section

00:04:58.354 --> 00:04:59.817
in the current SAT,

00:04:59.817 --> 00:05:01.656
and there might be experimental sections

00:05:01.656 --> 00:05:04.856
in the new SAT as well.

00:05:04.856 --> 00:05:07.290
So this is kind of a comparison of things,

00:05:07.290 --> 00:05:09.092
and kind of the big takeaways are

00:05:09.092 --> 00:05:10.585
you have fewer sections here,

00:05:10.585 --> 00:05:13.318
more time on average per question,

00:05:13.318 --> 00:05:15.752
and then the essay is optional.

00:05:15.752 --> 00:05:18.824
So, how is all of this going to be scored?

00:05:18.824 --> 00:05:23.824
Well, in the current SAT,
let me write it over here,

00:05:25.620 --> 00:05:29.084
so let me write the current SAT,

00:05:29.084 --> 00:05:30.821
or you can view it as the old SAT

00:05:30.821 --> 00:05:32.091
if you want to view it that way.

00:05:32.091 --> 00:05:34.021
Current/old SAT and I want to compare it

00:05:34.021 --> 00:05:38.368
to the new SAT

00:05:38.368 --> 00:05:43.368
In the current one, math is scored

00:05:44.678 --> 00:05:48.193
on a scale from 200 to 800.

00:05:48.193 --> 00:05:50.753
And that's still going to
be the case in the new SAT.

00:05:50.753 --> 00:05:55.751
Math is going to be scored
on a scale from 200 to 800.

00:05:55.751 --> 00:06:00.751
Now on the current SAT,
you have a reading score

00:06:01.592 --> 00:06:04.486
based on these three
sections right over here

00:06:04.486 --> 00:06:08.388
that's going to be scored from 200 to 800.

00:06:08.388 --> 00:06:11.121
Now in the new SAT, reading and writing

00:06:11.121 --> 00:06:12.757
is going to be combined.

00:06:12.757 --> 00:06:17.757
So reading plus writing is going

00:06:22.052 --> 00:06:26.491
to give you a 200 to 800 points of score.

00:06:26.491 --> 00:06:27.253
So this is going to be-

00:06:27.253 --> 00:06:29.021
I don't know whether to
write it in blue or orange

00:06:29.021 --> 00:06:30.220
so maybe I write it in both

00:06:30.220 --> 00:06:34.354
or let me, I'll write it in another color.

00:06:34.354 --> 00:06:37.849
200 to 800 points for this.

00:06:37.849 --> 00:06:40.534
And in the current or old SAT,

00:06:40.534 --> 00:06:45.534
you have writing/essay
as a separate section.

00:06:45.927 --> 00:06:50.927
Writing/essay and this also
gave you 200 to 800 points.

00:06:54.917 --> 00:06:57.284
While in the new SAT, you just have

00:06:57.284 --> 00:06:58.859
the essay section is separate from

00:06:58.859 --> 00:07:00.951
the writing in terms of score.

00:07:00.951 --> 00:07:04.294
So you just have the essay,

00:07:04.294 --> 00:07:05.756
and once again, this is optional.

00:07:05.756 --> 00:07:07.392
This is an optional thing.

00:07:07.392 --> 00:07:09.850
You have the essay which is optional,

00:07:09.850 --> 00:07:10.887
I'll write it again even though

00:07:10.887 --> 00:07:12.654
I wrote it down here as well.

00:07:12.654 --> 00:07:14.219
It's optional and that's
going to be scored

00:07:14.219 --> 00:07:19.219
from 2 to 4, sorry 2 to 8 points.

00:07:19.757 --> 00:07:22.051
So, what are the big takeaways here?

00:07:22.051 --> 00:07:23.148
Well in the current SAT,

00:07:23.148 --> 00:07:24.418
when people give you their SAT score,

00:07:24.418 --> 00:07:25.322
they're going to give you a score,

00:07:25.322 --> 00:07:26.460
if you add these up,

00:07:26.460 --> 00:07:27.750
the lowest possible score's going to be

00:07:27.750 --> 00:07:30.655
200 plus 200 plus 200
which is going to be 600.

00:07:30.655 --> 00:07:32.819
Then the highest one is 2400.

00:07:32.819 --> 00:07:34.617
So it's on a 2400 point scale.

00:07:34.617 --> 00:07:36.283
That's the current/old SAT.

00:07:36.283 --> 00:07:37.756
Well the new one, well people are

00:07:37.756 --> 00:07:38.793
likely to give their scores,

00:07:38.793 --> 00:07:41.424
the lowest possible score is 200 plus 200

00:07:41.424 --> 00:07:42.856
which is going to be 400,

00:07:42.856 --> 00:07:45.822
and the highest one is going to 1600.

00:07:45.822 --> 00:07:48.352
800 in math, 800 in reading plus writing,

00:07:48.352 --> 00:07:50.120
and then people might have also

00:07:50.120 --> 00:07:54.620
an optional essay score
that is as high as 8 points.

00:07:54.620 --> 00:07:55.920
So, those are the big takeaways.

00:07:55.920 --> 00:07:57.251
Hopefully that helps you a little bit.

00:07:57.251 --> 00:07:59.557
And in a future video, I'll
look at the actual items

00:07:59.557 --> 00:08:02.219
and think about how things have changed.


Interpreting linear functions — Harder example

WEBVTT
Kind: captions
Language: en

00:00:00.341 --> 00:00:02.640
- [Voiceover] Alice fills
up the gas tank of her car

00:00:02.640 --> 00:00:04.648
before going for a long drive.

00:00:04.648 --> 00:00:08.084
The equation below, and
that should be above.

00:00:08.084 --> 00:00:12.313
The equation above models
the amount of gas, g,

00:00:12.313 --> 00:00:16.970
in gallons in Alice's car
when she has driven m miles.

00:00:16.970 --> 00:00:19.561
What is the meaning of 32 in the equation?

00:00:19.561 --> 00:00:20.440
All right, let's look up here.

00:00:20.440 --> 00:00:24.661
This is the gallons that
she has left in her car,

00:00:24.661 --> 00:00:25.845
and what's going on here?

00:00:25.845 --> 00:00:27.696
So it makes sense that,
look, when m is zero,

00:00:27.696 --> 00:00:29.619
when she has driven
nothing, she's going to have

00:00:29.619 --> 00:00:31.715
15 gallons in her car.

00:00:31.715 --> 00:00:33.845
And then as you drive more and more,

00:00:33.845 --> 00:00:36.978
you're going to subtract more
and more gallons from her car.

00:00:36.978 --> 00:00:39.742
So this is, this term right over here,

00:00:39.742 --> 00:00:42.597
this is the number of gallons,

00:00:42.597 --> 00:00:45.453
this is the number of gallons used up.

00:00:45.453 --> 00:00:46.743
So why are you dividing?

00:00:46.743 --> 00:00:51.306
If you have m miles divided
by 32, how do you get gallons?

00:00:51.306 --> 00:00:54.450
What do the units on the 32 need to be?

00:00:54.450 --> 00:00:55.843
And so, let's just think about this.

00:00:55.843 --> 00:00:59.689
If I have miles, so this is miles up here,

00:00:59.689 --> 00:01:02.971
and I'm going to be dividing
them by some mystery units,

00:01:02.971 --> 00:01:06.633
some mystery units, I need to get gallons.

00:01:06.633 --> 00:01:09.249
I need to get, I need to get gallons,

00:01:09.249 --> 00:01:11.767
'cause this is going to be
the number of gallons used up.

00:01:11.767 --> 00:01:13.702
So what are the mystery units going to be?

00:01:13.702 --> 00:01:16.914
Well, we can treat these
units the way we would treat

00:01:16.914 --> 00:01:19.373
algebraic, algebraic variables.

00:01:19.373 --> 00:01:22.466
So we could solve for units here.

00:01:22.466 --> 00:01:24.792
So we could take the
reciprocal of both sides,

00:01:24.792 --> 00:01:27.508
and we could say the
units, the mystery units

00:01:27.508 --> 00:01:31.399
divided by miles is going
to be one over gallons,

00:01:31.399 --> 00:01:33.818
or I guess you could say per gallon.

00:01:33.818 --> 00:01:35.449
It's going to be one over gallons,

00:01:35.449 --> 00:01:38.515
and then multiply both sides times miles.

00:01:38.515 --> 00:01:40.351
Multiply both sides times miles.

00:01:40.351 --> 00:01:43.401
You're going to get the mystery units.

00:01:43.401 --> 00:01:45.213
The mystery units are going to be,

00:01:45.213 --> 00:01:47.351
if you multiply both sides by miles,

00:01:47.351 --> 00:01:50.541
and I don't want to get
confused with this over here.

00:01:50.541 --> 00:01:54.191
So we multiply both sides by miles,

00:01:55.041 --> 00:01:56.938
both sides by miles.

00:01:57.828 --> 00:02:00.587
Your mystery units are going to be miles,

00:02:01.377 --> 00:02:03.894
miles per gallon.

00:02:04.814 --> 00:02:07.414
So this right over here,
the units over here,

00:02:07.414 --> 00:02:10.456
in order for this thing
to come out to miles,

00:02:10.456 --> 00:02:12.596
in order to come out to gallons,

00:02:12.596 --> 00:02:14.252
in the numerator you have m miles

00:02:14.252 --> 00:02:18.142
divided by 32 miles per gallon,

00:02:18.142 --> 00:02:20.219
miles per gallon.

00:02:20.219 --> 00:02:22.623
This is telling you how much mileage

00:02:22.623 --> 00:02:26.965
you get per gallon for her
car, how much she is getting.

00:02:26.965 --> 00:02:28.299
So let's look at the choices.

00:02:28.299 --> 00:02:32.873
Alice uses 32 gallons of gas per mile.

00:02:32.873 --> 00:02:34.153
Now we have to be very careful.

00:02:34.153 --> 00:02:37.514
It's miles per gallon, not gas per mile.

00:02:38.214 --> 00:02:40.691
Alice's tank can hold 32 gallons of gas.

00:02:40.691 --> 00:02:45.195
No, Alice's tank can
hold 15 gallons of gas.

00:02:45.195 --> 00:02:48.081
She fills up her tank, and
before she's driven anything,

00:02:48.081 --> 00:02:49.672
we see here, when m is zero,

00:02:49.672 --> 00:02:51.559
she's going to have 15 gallons in her car.

00:02:51.559 --> 00:02:53.015
So that's not right.

00:02:53.015 --> 00:02:56.271
Alice can drive 32 miles on a tank of gas.

00:02:56.271 --> 00:03:00.932
That's right. The units here
are 32 miles per gallon.

00:03:00.932 --> 00:03:02.510
So that's that one.

00:03:02.510 --> 00:03:07.444
Alice's car can drive
32 miles to the gallon.

00:03:07.444 --> 00:03:11.530
Oh, whoa, that was, that
was, that was a close one.

00:03:11.530 --> 00:03:13.749
This isn't 32 miles per gallon.

00:03:13.749 --> 00:03:15.564
This is 32 miles on a tank of gas.

00:03:15.564 --> 00:03:17.841
This would be 32 miles on over 15 gallons.

00:03:17.841 --> 00:03:19.252
No, no, that's not right.

00:03:19.252 --> 00:03:21.050
Well, that was a close one.

00:03:21.050 --> 00:03:24.009
Alice's car can drive 32 miles per gallon.

00:03:24.009 --> 00:03:28.852
That's right, 32 miles,
32 miles per gallon.

00:03:28.852 --> 00:03:31.478
And if this unit business that I just did,

00:03:31.478 --> 00:03:35.116
you find it confusing, one
other way to think about it,

00:03:35.116 --> 00:03:37.083
I mean, you could have ruled,
you could have ruled out

00:03:37.083 --> 00:03:38.963
something like this, because
when you look at this,

00:03:38.963 --> 00:03:42.942
you can see that okay, her tank holds 32,

00:03:44.012 --> 00:03:46.569
her tank holds 15 gallons.

00:03:46.569 --> 00:03:50.147
You could have tried, you
could have tried this one,

00:03:50.147 --> 00:03:52.609
and you would have seen that
the units didn't work out.

00:03:52.609 --> 00:03:55.691
If you have 15 gallons,

00:03:56.301 --> 00:04:00.911
15 gallons minus m miles

00:04:00.911 --> 00:04:05.002
divided by 32, 32 gallons per mile,

00:04:05.002 --> 00:04:09.541
32 gallons per mile,

00:04:09.541 --> 00:04:11.170
what would the units end up being here,

00:04:11.170 --> 00:04:12.335
especially right over here?

00:04:12.335 --> 00:04:15.640
You would have miles
divided by gallons per mile.

00:04:15.640 --> 00:04:19.873
This is going to be, this is
going to be minus m times,

00:04:19.873 --> 00:04:23.040
I guess let me write
it this way, m over 32,

00:04:23.040 --> 00:04:27.585
miles times miles per gallon,

00:04:27.585 --> 00:04:30.848
miles per, miles per gallon,

00:04:30.848 --> 00:04:32.679
and you get miles squared per gallon.

00:04:32.679 --> 00:04:35.930
You'd get these crazy units up here.

00:04:35.930 --> 00:04:37.778
So, hopefully it makes sense

00:04:37.778 --> 00:04:39.866
that we would pick this last choice.

00:04:39.866 --> 00:04:41.952
And this one almost got me
when they said tank of gas.

00:04:41.952 --> 00:04:43.499
I was thinking, my brain
was thinking gallon.

00:04:43.499 --> 00:04:46.685
I was so already
programmed for the answer.

00:04:46.685 --> 00:04:49.503
But anyway, yes, the last choice

00:04:49.503 --> 00:04:51.192
is what we want to go with.