Slope of a line secant to a curve

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:03.060
So let's review
the idea of slope,

00:00:03.060 --> 00:00:05.930
which you might remember
from your algebra classes.

00:00:05.930 --> 00:00:09.470
The slope is just the
rate of change of a line.

00:00:09.470 --> 00:00:11.580
Or the rate of change
of y, with respect

00:00:11.580 --> 00:00:13.800
to x, as we go along a line.

00:00:13.800 --> 00:00:16.260
And you could also
view it as a measure

00:00:16.260 --> 00:00:17.820
of the inclination of a line.

00:00:17.820 --> 00:00:21.460
So the more incline the line
is, the more positive of a slope

00:00:21.460 --> 00:00:22.250
it would have.

00:00:22.250 --> 00:00:24.960
So this right over here,
this has a positive slope.

00:00:24.960 --> 00:00:27.210
It's increasing as x increases.

00:00:27.210 --> 00:00:29.460
And if this had an even
higher inclination like this,

00:00:29.460 --> 00:00:31.316
if it increased even
more as x increased,

00:00:31.316 --> 00:00:32.940
then it would even
have a higher slope.

00:00:32.940 --> 00:00:36.100
So this right over
here is some line.

00:00:36.100 --> 00:00:37.040
So that's some line.

00:00:37.040 --> 00:00:39.070
And just as a reminder,
we can figure out

00:00:39.070 --> 00:00:40.810
the slope between two points.

00:00:40.810 --> 00:00:42.490
Two points define a line.

00:00:42.490 --> 00:00:44.110
And between those
two points, we can

00:00:44.110 --> 00:00:48.750
find the rate of change
of y, with respect to x.

00:00:48.750 --> 00:00:50.250
So let's put two points on here.

00:00:50.250 --> 00:00:54.500
So let's say that this point
right over here, this x value,

00:00:54.500 --> 00:00:58.450
is x sub-- well, this
is pronounced x naught,

00:00:58.450 --> 00:01:04.890
or x sub 0 is just x naught--
and when x is x0 for this line,

00:01:04.890 --> 00:01:08.600
y is y0.

00:01:08.600 --> 00:01:12.590
So this is a point x0, comma y0.

00:01:12.590 --> 00:01:15.850
And let's say we have another
point all the way over here.

00:01:15.850 --> 00:01:19.330
And let's say that this x value,
this x value right over here

00:01:19.330 --> 00:01:20.890
is x sub 1.

00:01:20.890 --> 00:01:26.430
And the y value over
here is y sub 1.

00:01:26.430 --> 00:01:29.700
So this is the point
x sub 1, y sub 1.

00:01:29.700 --> 00:01:33.190
So just as a review,
the slope of this line,

00:01:33.190 --> 00:01:36.280
and a line by definition,
has a constant slope

00:01:36.280 --> 00:01:38.500
between any two
points that you pick.

00:01:38.500 --> 00:01:42.640
The slope of this line, which is
often denoted by the letter m,

00:01:42.640 --> 00:01:46.300
is your rate of change
of y with respect to x.

00:01:46.300 --> 00:01:47.800
Or another way of
thinking about it,

00:01:47.800 --> 00:01:52.980
for a given change in x,
how much are you changing y?

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Or a change in y divided
by a change in x.

00:01:55.550 --> 00:01:57.110
Just as a reminder,
this triangle,

00:01:57.110 --> 00:01:59.090
that's the Greek letter delta.

00:01:59.090 --> 00:02:02.140
It's shorthand for
change in whatever.

00:02:02.140 --> 00:02:04.400
So change in y over change in x.

00:02:04.400 --> 00:02:07.530
So let's think about what this
is going to be for this example

00:02:07.530 --> 00:02:08.990
right over here.

00:02:08.990 --> 00:02:10.889
Well, let's think about
change in x, first.

00:02:10.889 --> 00:02:14.200
So we are moving from x0 to x1.

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So our change in x.

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So this is our change
in x, right over here.

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We're starting at
x0 and going to x1.

00:02:21.690 --> 00:02:23.490
That is our change in x.

00:02:23.490 --> 00:02:25.550
I'll put that in
that pink color.

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That is our change in x.

00:02:27.350 --> 00:02:29.400
And what is it equal to?

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Well if we're ending
here and we started here,

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let's just do ending point
minus starting point.

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So it is x1 minus x0.

00:02:37.850 --> 00:02:39.687
And that way, doing
it this way, I

00:02:39.687 --> 00:02:41.770
would have made sure that
I have a positive value.

00:02:41.770 --> 00:02:45.250
I'm just assuming that
x1 is larger than x0.

00:02:45.250 --> 00:02:46.720
And what is my change in y?

00:02:46.720 --> 00:02:48.560
Well, once again,
ending point, ending

00:02:48.560 --> 00:02:51.030
y value minus starting y value.

00:02:51.030 --> 00:02:55.099
y1 minus y0.

00:02:55.099 --> 00:02:56.640
Now, you might be
saying hey, could I

00:02:56.640 --> 00:03:01.620
have done y0 minus y1
over x0 zero minus x1?

00:03:01.620 --> 00:03:02.500
Absolutely.

00:03:02.500 --> 00:03:03.810
You could have done that.

00:03:03.810 --> 00:03:05.580
Then you would have just
gotten the negative of each

00:03:05.580 --> 00:03:07.400
of these values in the
numerator and denominator,

00:03:07.400 --> 00:03:08.520
but they would
have canceled out.

00:03:08.520 --> 00:03:10.436
The important thing is
that you're consistent.

00:03:10.436 --> 00:03:13.480
If you're subtracting you're
starting value from your ending

00:03:13.480 --> 00:03:16.320
value in the numerator, you have
to subtract your starting value

00:03:16.320 --> 00:03:18.940
from your ending value in
the denominator as well.

00:03:18.940 --> 00:03:22.450
So this right here you probably
remember from algebra class.

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The definition of
slope is the rate

00:03:24.310 --> 00:03:27.800
of change of y
with respect to x.

00:03:27.800 --> 00:03:29.800
Or it's the rate of change
of our vertical axis,

00:03:29.800 --> 00:03:33.480
I should say, with respect
to our horizontal axis.

00:03:33.480 --> 00:03:36.100
Or change in y, or change
in our vertical axis

00:03:36.100 --> 00:03:38.350
over change in a
horizontal axis.

00:03:38.350 --> 00:03:41.150
Now I'm going to introduce
a little bit of a conundrum.

00:03:41.150 --> 00:03:43.812
So let me draw another
axis right over here.

00:03:43.812 --> 00:03:45.270
Scroll over a little
bit just so we

00:03:45.270 --> 00:03:48.150
have some space to work with.

00:03:48.150 --> 00:03:49.370
So that was for a line.

00:03:49.370 --> 00:03:52.090
And a line, by definition,
has a constant slope.

00:03:52.090 --> 00:03:55.960
If you calculate this between
any two points on the line,

00:03:55.960 --> 00:03:58.490
it's going to be
constant for that line.

00:03:58.490 --> 00:04:01.270
But what happens when we
start dealing with curves?

00:04:01.270 --> 00:04:06.580
' When we start dealing with
non-line or non-linear curves.

00:04:06.580 --> 00:04:10.140
So let's imagine a curve that
looks something like this.

00:04:14.390 --> 00:04:16.839
So what is the
rate of change of y

00:04:16.839 --> 00:04:19.959
with respect to x of this curve?

00:04:19.959 --> 00:04:21.750
Well, let's look at it
at different points.

00:04:21.750 --> 00:04:24.030
And we could at least
try to approximate what

00:04:24.030 --> 00:04:25.650
it might be in any moment.

00:04:25.650 --> 00:04:29.370
So let's say that this
is one point on a curve.

00:04:29.370 --> 00:04:33.956
Let's call that x1,
and then this is y1.

00:04:33.956 --> 00:04:36.080
And let's say that this is
another point on a curve

00:04:36.080 --> 00:04:39.790
right over here, x2.

00:04:39.790 --> 00:04:42.840
And let's call this y2.

00:04:42.840 --> 00:04:46.520
So this is a point x1, y1,
this is a point x2, y2.

00:04:46.520 --> 00:04:47.859
So we don't have the tools yet.

00:04:47.859 --> 00:04:49.650
And this is what's
exciting about calculus,

00:04:49.650 --> 00:04:52.490
we will soon have the
tools to figure out,

00:04:52.490 --> 00:04:55.060
what is the rate of
change of y with respect

00:04:55.060 --> 00:04:57.697
to x at exactly this point?

00:04:57.697 --> 00:04:59.030
But we don't have that tool yet.

00:04:59.030 --> 00:05:01.240
But using just the
tools from algebra,

00:05:01.240 --> 00:05:03.150
we could at least
start to think about,

00:05:03.150 --> 00:05:05.230
what is the average
rate of change

00:05:05.230 --> 00:05:08.380
over the interval from x1 to x2?

00:05:08.380 --> 00:05:10.140
Well, what's the
average rate of change?

00:05:10.140 --> 00:05:14.470
Well, that's just how
much did my y change--

00:05:14.470 --> 00:05:18.290
so that's my change in
y-- for this change in x.

00:05:20.950 --> 00:05:22.970
And so we would calculate
it the same way.

00:05:22.970 --> 00:05:27.050
y2 minus y1 over x2 minus x1.

00:05:27.050 --> 00:05:30.770
So our change in y
over this interval

00:05:30.770 --> 00:05:34.410
is equal to y2 minus
y1, and our change

00:05:34.410 --> 00:05:41.910
in x is going to be
equal to x2 minus x1.

00:05:41.910 --> 00:05:45.040
So just like that we
were able to figure out

00:05:45.040 --> 00:05:48.082
the rate of change
between these two points.

00:05:48.082 --> 00:05:49.540
Or another way of
thinking about it

00:05:49.540 --> 00:05:54.180
is, this is the average rate of
change for the curve between x

00:05:54.180 --> 00:05:56.600
equals x1, and x is equal to x2.

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This is the average
rate of change

00:05:58.130 --> 00:06:01.180
of y with respect to
x over this interval.

00:06:01.180 --> 00:06:04.310
But what if we have we
also figured out here?

00:06:04.310 --> 00:06:06.144
Well, we figured out
the slope of the line

00:06:06.144 --> 00:06:07.435
that connects these two points.

00:06:16.010 --> 00:06:17.950
And what we call a
line that intersects

00:06:17.950 --> 00:06:20.570
a curve in exactly two places?

00:06:20.570 --> 00:06:24.020
Well, we figured out, we
call that a secant line.

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So this right over
here is a secant line.

00:06:29.110 --> 00:06:32.480
So the big idea here is we're
extending the idea of slope.

00:06:32.480 --> 00:06:35.080
We said, OK, we already knew
how to find the slope of a line.

00:06:35.080 --> 00:06:36.860
A curve, we don't
have the tools yet,

00:06:36.860 --> 00:06:38.700
but calculus is about
to give it to us.

00:06:38.700 --> 00:06:40.519
But let's just use
our algebraic tools.

00:06:40.519 --> 00:06:42.310
We can at least figure
out the average rate

00:06:42.310 --> 00:06:46.600
of change of a curve, or a
function, over an interval.

00:06:46.600 --> 00:06:51.230
That is the same exact thing
as the slope of a secant line.

00:06:51.230 --> 00:06:52.980
Now just as a little
bit of foreshadowing,

00:06:52.980 --> 00:06:53.980
where is this all going?

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How will we eventually
get the tools, so

00:06:55.860 --> 00:06:58.540
that we can figure out the
instantaneous rate of change,

00:06:58.540 --> 00:07:00.060
not just the average?

00:07:00.060 --> 00:07:03.570
Well just imagine what happens
if this point right over here

00:07:03.570 --> 00:07:05.670
got closer and
closer to this point.

00:07:05.670 --> 00:07:08.200
Then the secant line is
going to better and better

00:07:08.200 --> 00:07:11.430
and better approximate
the instantaneous rate

00:07:11.430 --> 00:07:12.580
of change right over here.

00:07:12.580 --> 00:07:16.336
Or you could even think of it as
the slope of the tangent line.


Slope of a secant line example 1

WEBVTT
Kind: captions
Language: en

00:00:01.150 --> 00:00:04.690
A curve has the equation y
equals the natural log of x,

00:00:04.690 --> 00:00:08.980
and passes through the
points P equals e comma 1.

00:00:08.980 --> 00:00:12.780
And Q is equal to
x natural log of x.

00:00:12.780 --> 00:00:16.970
Write an expression in x that
gives the slope of the secant

00:00:16.970 --> 00:00:20.500
joining P and Q. So I think I'm
going to need my little scratch

00:00:20.500 --> 00:00:22.550
pad for this one
right over here.

00:00:22.550 --> 00:00:24.710
So this is the same
question over again.

00:00:24.710 --> 00:00:27.790
Now let's just try to visualize
this curve right over here.

00:00:27.790 --> 00:00:31.380
So let me draw my axes.

00:00:31.380 --> 00:00:37.760
So let's say this is my y-axis.

00:00:37.760 --> 00:00:41.600
This is my y-axis, and
then this is my x-axis.

00:00:41.600 --> 00:00:46.580
This is my x-axis
right over here.

00:00:46.580 --> 00:00:48.120
That's my x-axis.

00:00:48.120 --> 00:00:52.390
And natural log of x-- so let's
think about it a little bit.

00:00:52.390 --> 00:00:57.790
The natural log of 0, e to
what power is equal to 0?

00:00:57.790 --> 00:00:59.370
Well, that's going to be 1.

00:00:59.370 --> 00:01:03.710
So you're going to have
the point 1, 0 on this.

00:01:03.710 --> 00:01:06.140
So that's 1 right over there.

00:01:06.140 --> 00:01:08.880
And then the natural log of
smaller and smaller numbers

00:01:08.880 --> 00:01:12.610
as we approach 0 is going to
get more and more negative,

00:01:12.610 --> 00:01:15.130
all the way going down
to negative infinity.

00:01:15.130 --> 00:01:19.175
So this curve is going to
look something like this.

00:01:19.175 --> 00:01:28.310
It's going to look
something like that.

00:01:28.310 --> 00:01:31.640
And we know it also has
the point e comma 1 on it.

00:01:31.640 --> 00:01:33.590
So it has a point,
the natural log of e.

00:01:33.590 --> 00:01:36.320
So if that's 1,
that'll be about 2.

00:01:36.320 --> 00:01:37.660
That'll be about 3.

00:01:37.660 --> 00:01:42.200
So that is e right over here,
roughly e, right over here.

00:01:42.200 --> 00:01:44.600
And that's the point e comma 1.

00:01:44.600 --> 00:01:47.240
So that's the point e
comma 1 right over there.

00:01:47.240 --> 00:01:49.280
And I'll just mark
that as P there

00:01:49.280 --> 00:01:50.420
to give us a label for it.

00:01:50.420 --> 00:01:51.920
So that is P.

00:01:51.920 --> 00:01:54.760
And we want to find the
secant line between P

00:01:54.760 --> 00:01:57.690
and an arbitrary
Q for any given x.

00:01:57.690 --> 00:02:02.340
The y value is going
to be natural log of x.

00:02:02.340 --> 00:02:06.300
So let's say that this
is our Q right over here.

00:02:06.300 --> 00:02:11.490
So let's say this right
over here is our-- actually,

00:02:11.490 --> 00:02:15.321
I want to make it clear that
it is a, I'll do it our here.

00:02:15.321 --> 00:02:16.570
I'm being a little indecisive.

00:02:16.570 --> 00:02:18.403
So let's say that is
our Q right over there.

00:02:18.403 --> 00:02:22.860
And that's the point x
natural log, natural log of x.

00:02:22.860 --> 00:02:25.590
And we want to find the slope
of the secant line joining

00:02:25.590 --> 00:02:26.440
these two points.

00:02:29.870 --> 00:02:32.880
So this, the slope
of this line, I

00:02:32.880 --> 00:02:35.010
want to try to make it so
it doesn't look tangent,

00:02:35.010 --> 00:02:36.390
so it's secant.

00:02:36.390 --> 00:02:38.430
So you see it cuts
through the curve.

00:02:38.430 --> 00:02:40.300
So that's the secant
line right over there.

00:02:40.300 --> 00:02:43.830
It intersects the
curve at P and at Q.

00:02:43.830 --> 00:02:46.580
So I want to find the
slope of that secant line.

00:02:46.580 --> 00:02:48.700
Well, to find the slope
of the secant line,

00:02:48.700 --> 00:02:50.630
I just need to find
the change in y

00:02:50.630 --> 00:02:54.040
and the change in x
between these two points.

00:02:54.040 --> 00:02:56.290
So what's the change in--
so let's be clear here.

00:02:56.290 --> 00:02:59.240
This is the point, this is
when x is equal to-- well,

00:02:59.240 --> 00:03:01.580
it's just a kind of arbitrary x.

00:03:01.580 --> 00:03:06.650
And this right over
here is the point lnx.

00:03:06.650 --> 00:03:08.250
So what is our change in x?

00:03:08.250 --> 00:03:11.340
Our change in x-- this
value right over here,

00:03:11.340 --> 00:03:14.740
our change in x-- is just
going to be x minus e

00:03:14.740 --> 00:03:17.760
is equal to x minus e.

00:03:17.760 --> 00:03:20.230
And what is our change in y?

00:03:20.230 --> 00:03:24.380
Our change in y is
going to be lnx is going

00:03:24.380 --> 00:03:28.640
to be the natural log
of x minus 1, minus 1.

00:03:28.640 --> 00:03:30.570
That's this distance
right over here.

00:03:30.570 --> 00:03:33.060
So the slope of
this line, the line

00:03:33.060 --> 00:03:36.460
that contains both of these
points, the slope-- I could

00:03:36.460 --> 00:03:38.910
write m for slope-- is
going to be our change

00:03:38.910 --> 00:03:44.270
in y over our change in x,
which is equal to lnx minus 1

00:03:44.270 --> 00:03:47.570
over x minus e.

00:03:47.570 --> 00:03:49.490
And now we can just
input that and make sure

00:03:49.490 --> 00:03:50.660
that we got that right.

00:03:50.660 --> 00:03:54.890
So let me try to remember it--
lnx minus 1 over x minus e.

00:03:54.890 --> 00:03:57.260
So let me input that.

00:03:57.260 --> 00:04:03.000
So we have the natural
log of x minus 1 over--

00:04:03.000 --> 00:04:06.380
and it interprets it
nicely for us right below,

00:04:06.380 --> 00:04:12.400
so we make sure that we're
doing over x minus e.

00:04:12.400 --> 00:04:16.040
And now let us check our answer.

00:04:16.040 --> 00:04:18.240
And we got it right.


Slope of a secant line example 2

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:03.280
The data for three points
on a smooth function f

00:00:03.280 --> 00:00:04.297
is given in the table.

00:00:04.297 --> 00:00:05.880
So let's actually
graph these, just so

00:00:05.880 --> 00:00:07.980
that we can visualize
it a little bit.

00:00:07.980 --> 00:00:10.590
So our vertical axis,
let's just call that y,

00:00:10.590 --> 00:00:12.480
is equal to f of x.

00:00:12.480 --> 00:00:15.310
And then we have
our horizontal axis.

00:00:15.310 --> 00:00:18.630
We have our horizontal
axis, this is our x-axis.

00:00:18.630 --> 00:00:20.570
And we're going to
skip some space here,

00:00:20.570 --> 00:00:24.460
just because we immediately go
to very high values of f of x.

00:00:24.460 --> 00:00:26.110
And I'll even skip
some space here,

00:00:26.110 --> 00:00:29.830
since we already go up to
6.5 all the way up to 7.5.

00:00:29.830 --> 00:00:32.639
So let me just show that I'm
skipping some space here.

00:00:32.639 --> 00:00:33.680
And we're going to start.

00:00:33.680 --> 00:00:40.620
Let's say this is
108, 108, 109, 110.

00:00:40.620 --> 00:00:46.160
And let's say that this right
over here is 6.5, that is 7,

00:00:46.160 --> 00:00:47.890
and then that is 7.5.

00:00:47.890 --> 00:00:49.890
And we can plot these points.

00:00:49.890 --> 00:00:54.950
6.5 comma 108.25, that'll
put us right over there.

00:00:54.950 --> 00:01:01.930
7 comma 109.45, that puts
us right around there.

00:01:01.930 --> 00:01:10.970
And then 7.5, 110.15 puts
us right around here.

00:01:10.970 --> 00:01:13.556
And these are just three
points on a smooth function f.

00:01:13.556 --> 00:01:15.430
So we can visualize what
that smooth function

00:01:15.430 --> 00:01:16.310
f might look like.

00:01:16.310 --> 00:01:20.540
It might look
something like this.

00:01:20.540 --> 00:01:24.380
So maybe it looks
something like this.

00:01:24.380 --> 00:01:27.494
So our smooth function
f might look something--

00:01:27.494 --> 00:01:29.160
who knows what it
looks like after this.

00:01:29.160 --> 00:01:31.370
So that's, this
right over here is,

00:01:31.370 --> 00:01:35.580
this is the graph of
y is equal to f of x,

00:01:35.580 --> 00:01:37.330
Now let's try to
answer their questions.

00:01:37.330 --> 00:01:38.970
What is the average
rate of change

00:01:38.970 --> 00:01:46.450
of f with respect to x as x
goes from 6.5 to 7, 7 to 7.5,

00:01:46.450 --> 00:01:48.160
and 6.5 to 7.5?

00:01:48.160 --> 00:01:49.720
So let's do them one at time.

00:01:49.720 --> 00:01:59.230
6.5 to 7, our change in x here,
our change in x is 7 minus 6.5,

00:01:59.230 --> 00:02:02.100
which is equal to 0.5.

00:02:02.100 --> 00:02:08.330
And our change in y here
is equal to-- let's see.

00:02:08.330 --> 00:02:15.710
We end up at 109.45 minus
108.25, which is 1.2.

00:02:15.710 --> 00:02:17.780
So our average rate of
change over this interval

00:02:17.780 --> 00:02:21.790
is our change in y over
our change in x, or 1.2

00:02:21.790 --> 00:02:23.912
divided by 0.5.

00:02:23.912 --> 00:02:24.870
Let me write this down.

00:02:24.870 --> 00:02:30.610
Our change in y over our change
in x is equal to 1.2 over 0.5,

00:02:30.610 --> 00:02:32.740
which is equal to 2.4.

00:02:32.740 --> 00:02:34.770
So this is equal to 2.4.

00:02:34.770 --> 00:02:36.850
Now let's do the next interval.

00:02:36.850 --> 00:02:43.000
Our change in x, once
again, is 7.5 minus 6.5.

00:02:43.000 --> 00:02:45.160
Change in x is still 0.5.

00:02:45.160 --> 00:02:48.940
That's hard to read, let me
write that a little bit neater.

00:02:48.940 --> 00:02:55.280
So our change in x is
still 0.5, 7 to 7.5.

00:02:55.280 --> 00:02:59.120
And our change in y, let's see.

00:02:59.120 --> 00:03:01.090
We end up at 110.15.

00:03:01.090 --> 00:03:03.810
We started at 109.45.

00:03:03.810 --> 00:03:07.320
So our change is 0.7.

00:03:07.320 --> 00:03:09.750
So it's 0.7.

00:03:09.750 --> 00:03:13.830
So our change in y over
our change in x is equal

00:03:13.830 --> 00:03:20.030
to 0.7/0.5, which
is equal to 1.4.

00:03:20.030 --> 00:03:22.075
And so you can see that
the slope of the secant

00:03:22.075 --> 00:03:23.450
line-- this
essentially the slope

00:03:23.450 --> 00:03:26.420
of the secant line
between these two points.

00:03:26.420 --> 00:03:28.610
So this one, this
one right over here--

00:03:28.610 --> 00:03:30.540
let me try my best to draw it.

00:03:30.540 --> 00:03:31.730
This one right over here.

00:03:31.730 --> 00:03:35.630
You can see it's steeper
than the second one,

00:03:35.630 --> 00:03:37.450
than this secant
line right over here,

00:03:37.450 --> 00:03:40.284
than this secant line
right over there.

00:03:40.284 --> 00:03:42.700
And now let's find the slope--
the average rate of change.

00:03:42.700 --> 00:03:44.180
I should say, or the
slope of the secant.

00:03:44.180 --> 00:03:45.790
The average rate of change
over this interval, which

00:03:45.790 --> 00:03:47.831
is the same thing as the
slope of the secant line

00:03:47.831 --> 00:03:49.980
between that point
and that point.

00:03:49.980 --> 00:03:54.022
So let's think
about our delta x.

00:03:54.022 --> 00:03:55.230
And maybe I'll do it up here.

00:03:55.230 --> 00:04:00.250
So our delta x, we're
going from 6.5 to 7.5.

00:04:00.250 --> 00:04:06.120
So our delta x is equal to
1 and what is our delta y?

00:04:06.120 --> 00:04:10.550
So our delta y, our change
in y, is equal to, let's see.

00:04:10.550 --> 00:04:12.240
We end up at 110.15.

00:04:12.240 --> 00:04:14.450
We started at 108.25.

00:04:14.450 --> 00:04:22.150
110.15 minus 108.25,
we increase by 1.9.

00:04:22.150 --> 00:04:27.560
So our change in y over change
in x is equal to 1.9 over 1,

00:04:27.560 --> 00:04:31.040
which is equal to 1.9.

00:04:31.040 --> 00:04:31.770
Fair enough.

00:04:31.770 --> 00:04:35.150
And that's the slope of this
secant line between this point

00:04:35.150 --> 00:04:36.520
and this point right over here.

00:04:36.520 --> 00:04:38.534
Slope of the secant line.

00:04:38.534 --> 00:04:39.950
It would look
something like that.

00:04:39.950 --> 00:04:43.234
You see that it's slightly less
steep than the magenta secant,

00:04:43.234 --> 00:04:45.400
and it's slightly more steep
than the orange secant.

00:04:45.400 --> 00:04:47.670
It's kind of in between the two.

00:04:47.670 --> 00:04:49.980
Now they ask us
this final question.

00:04:49.980 --> 00:04:53.060
Use the average rate of
change of f on the larger

00:04:53.060 --> 00:04:55.760
interval from here to here--
which we already figured out,

00:04:55.760 --> 00:04:59.270
that's 1.9-- as an approximation
for the slope of the line

00:04:59.270 --> 00:05:01.980
tangent to f at x equals 7.

00:05:01.980 --> 00:05:04.310
So we're trying to approximate
the slope of the line

00:05:04.310 --> 00:05:06.310
tangent to f at f equals 7.

00:05:06.310 --> 00:05:09.726
So the line tangent might
look something like this.

00:05:09.726 --> 00:05:11.100
And we see, at
least visually, it

00:05:11.100 --> 00:05:13.410
looks like that little
blue line we drew.

00:05:13.410 --> 00:05:15.702
This one right over here does
look like it's-- at least

00:05:15.702 --> 00:05:18.034
the way I've drawn it-- it
looks like it's a pretty good

00:05:18.034 --> 00:05:19.760
approximation for
the slope up there.

00:05:19.760 --> 00:05:22.930
So we'll use this as the
slope, as an approximation

00:05:22.930 --> 00:05:27.840
for the slope of the tangent
line to f at x equals 7.

00:05:27.840 --> 00:05:36.740
They say, write an equation for
the line tangent f at 709.45

00:05:36.740 --> 00:05:37.827
using point slope form.

00:05:37.827 --> 00:05:39.410
So it's going to be
a line where we're

00:05:39.410 --> 00:05:42.090
going to use this as an
approximation for slope.

00:05:42.090 --> 00:05:44.930
And it's going to
contain this line.

00:05:44.930 --> 00:05:45.600
It's tangent.

00:05:45.600 --> 00:05:48.112
It touches to the
curve at this point.

00:05:48.112 --> 00:05:49.945
And so point slope form,
just as a reminder,

00:05:49.945 --> 00:05:54.040
it's just another way of saying
that every point on this line,

00:05:54.040 --> 00:05:56.400
every point on
this line, x comma

00:05:56.400 --> 00:05:59.660
y-- If we were to find
the change in x and change

00:05:59.660 --> 00:06:02.180
in y relative to this
point right over here,

00:06:02.180 --> 00:06:04.840
it's always going to
have a constant slope.

00:06:04.840 --> 00:06:07.320
So one way you could say is
if you take an arbitrary point

00:06:07.320 --> 00:06:12.510
x, y on this line, you
could say your change in y.

00:06:12.510 --> 00:06:21.960
So y minus 109.45
over your change

00:06:21.960 --> 00:06:26.160
in x, so we know that
the 0.7 is there.

00:06:26.160 --> 00:06:36.290
Or the 0.7 is-- when x is
equal to 7, f of x is 109.45.

00:06:36.290 --> 00:06:38.530
So we're referencing
the same point.

00:06:38.530 --> 00:06:41.900
When our change in x is x
minus 7, our change in y

00:06:41.900 --> 00:06:44.630
is y minus 109.45.

00:06:44.630 --> 00:06:47.020
And this is always going
to be constant for any x, y

00:06:47.020 --> 00:06:48.980
that I pick on this line there.

00:06:48.980 --> 00:06:50.970
And we're using this
as an approximation

00:06:50.970 --> 00:06:52.230
for what that's going to be.

00:06:52.230 --> 00:06:54.795
That's going to be equal to 1.9.

00:06:54.795 --> 00:06:56.670
So if you want to put
it in point slope form,

00:06:56.670 --> 00:06:59.850
you just multiply both
sides times x minus 7.

00:06:59.850 --> 00:07:10.460
And you get y minus 109.45 is
equal to 1.9 times x minus 7.

00:07:10.460 --> 00:07:11.620
And we are done.

00:07:11.620 --> 00:07:13.200
That is an approximation.

00:07:13.200 --> 00:07:14.570
We've approximated the slope.

00:07:14.570 --> 00:07:16.111
And if this was the
slope, this would

00:07:16.111 --> 00:07:20.360
be the equation of the tangent
line in point slope form.


Powers of fractions

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.130
Let's go through more
exponent examples.

00:00:03.130 --> 00:00:05.850
So to warm up, let's think
about taking a fraction

00:00:05.850 --> 00:00:06.640
to some power.

00:00:06.640 --> 00:00:10.220
So let's say I have
2/3, and I want

00:00:10.220 --> 00:00:13.680
to raise it to the
third power here.

00:00:13.680 --> 00:00:15.180
Now, we've already
learned there are

00:00:15.180 --> 00:00:17.040
two ways of thinking about this.

00:00:17.040 --> 00:00:19.792
One way is to say
let's take three 2/3's.

00:00:19.792 --> 00:00:26.751
So that's one 2/3, two
2/3's, and three 2/3's.

00:00:26.751 --> 00:00:29.236
So that's one, two, three, 2/3.

00:00:29.236 --> 00:00:31.224
And then we multiply them.

00:00:35.120 --> 00:00:37.670
And we will get-- let's
see, the numerator

00:00:37.670 --> 00:00:40.600
will be 2 times 2
times 2, which is 8.

00:00:40.600 --> 00:00:42.660
And the denominator's
going to be 3 times 3 times

00:00:42.660 --> 00:00:46.830
3 times 3, which is equal to 27.

00:00:46.830 --> 00:00:50.590
Now, the other way of viewing
this is you start with a 1,

00:00:50.590 --> 00:00:53.740
and you multiply it
by 2/3 three times.

00:00:53.740 --> 00:00:56.950
So you multiply by 2/3
once, twice, three times.

00:00:56.950 --> 00:00:59.820
You will get the exact
same result here.

00:00:59.820 --> 00:01:01.800
So let's do one more
example like that.

00:01:01.800 --> 00:01:05.529
So lets say I had 4/9,
and I want to square it.

00:01:05.529 --> 00:01:07.320
When I raise something
to the second power,

00:01:07.320 --> 00:01:08.680
people often say,
you're squaring it.

00:01:08.680 --> 00:01:10.471
Also, raising something
to the third power,

00:01:10.471 --> 00:01:12.580
people sometimes say,
you're cubing it.

00:01:12.580 --> 00:01:14.950
But let's raise 4/9
to the second power.

00:01:14.950 --> 00:01:15.917
Let's square it.

00:01:15.917 --> 00:01:17.500
And I encourage you
to pause the video

00:01:17.500 --> 00:01:18.910
and work this out yourself.

00:01:18.910 --> 00:01:21.180
Well, once again,
you could view this

00:01:21.180 --> 00:01:26.810
as taking two 4/9's
and multiplying them.

00:01:26.810 --> 00:01:29.380
Or you could view this
as starting with a 1,

00:01:29.380 --> 00:01:32.620
and multiplying it
by 4/9 two times.

00:01:32.620 --> 00:01:34.840
Either way, your
numerator is going

00:01:34.840 --> 00:01:37.432
to be 4 times 4, which is 16.

00:01:37.432 --> 00:01:38.890
And your denominator
is going to be

00:01:38.890 --> 00:01:45.110
9 times 9, which is equal to 81.


Patterns in raising 1 and -1 to different powers

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:04.520
Let's think about exponents
with ones and zeroes.

00:00:04.520 --> 00:00:07.330
So let's take the
number 1, and let's

00:00:07.330 --> 00:00:10.084
raise it to the eighth power.

00:00:10.084 --> 00:00:11.500
So we've already
seen that there's

00:00:11.500 --> 00:00:12.890
two ways of thinking about this.

00:00:12.890 --> 00:00:15.348
You could literally view this
as taking eight 1's, and then

00:00:15.348 --> 00:00:16.450
multiplying them together.

00:00:16.450 --> 00:00:17.158
So let's do that.

00:00:17.158 --> 00:00:21.557
So you have one, two, three,
four, five, six, seven,

00:00:21.557 --> 00:00:26.010
eight 1's, and then you're
going to multiply them together.

00:00:26.010 --> 00:00:29.966
And if you were to do that,
you would get well, 1 times 1

00:00:29.966 --> 00:00:31.340
is 1, times 1--
it doesn't matter

00:00:31.340 --> 00:00:33.560
how many times you
multiply 1 by 1.

00:00:33.560 --> 00:00:36.170
You are going to just get 1.

00:00:36.170 --> 00:00:38.204
You are just going to get 1.

00:00:38.204 --> 00:00:39.120
And you could imagine.

00:00:39.120 --> 00:00:40.300
I did it eight times.

00:00:40.300 --> 00:00:41.820
I multiplied eight 1's.

00:00:41.820 --> 00:00:44.070
But even if this was
80, or if this was 800,

00:00:44.070 --> 00:00:46.480
or if this was 8 million,
if I just multiplied 1--

00:00:46.480 --> 00:00:50.490
if I had 8 million 1's, and I
multiplied them all together,

00:00:50.490 --> 00:00:53.760
it would still be equal to 1.

00:00:53.760 --> 00:00:57.474
So 1 to any power is just
going to be equal to 1.

00:00:57.474 --> 00:00:59.640
And you might say, hey,
what about 1 to the 0 power?

00:01:02.790 --> 00:01:05.850
Well, we've already said
anything to 0 power,

00:01:05.850 --> 00:01:07.300
except for 0--
that's where we're

00:01:07.300 --> 00:01:09.450
going to-- it's
actually up for debate.

00:01:09.450 --> 00:01:13.269
But anything to the 0 power
is going to be equal to 1.

00:01:13.269 --> 00:01:14.810
And just as a little
bit of intuition

00:01:14.810 --> 00:01:16.290
here, you could
literally view this

00:01:16.290 --> 00:01:19.840
as our other definition
of exponentiation, which

00:01:19.840 --> 00:01:22.160
is you start with a
1, and this number

00:01:22.160 --> 00:01:24.820
says how many times you're
going to multiply that 1 times

00:01:24.820 --> 00:01:25.580
this number.

00:01:25.580 --> 00:01:30.021
So 1 times 1 zero times
is just going to be 1.

00:01:30.021 --> 00:01:32.020
And that was a little bit
clearer when we did it

00:01:32.020 --> 00:01:34.930
like this, where
we said 2 to the,

00:01:34.930 --> 00:01:39.710
let's say, fourth power
is equal to-- this

00:01:39.710 --> 00:01:42.190
was the other definition
of exponentiation we had,

00:01:42.190 --> 00:01:45.040
which is you start with a
1, and then you multiply it

00:01:45.040 --> 00:01:50.280
by 2 four times, so times
2, times 2, times 2,

00:01:50.280 --> 00:01:56.850
times 2, which is equal to--
let's see, this is equal to 16.

00:01:56.850 --> 00:01:59.350
So here if you start
with a 1 and then

00:01:59.350 --> 00:02:02.057
you multiply it by
1 zero times, you're

00:02:02.057 --> 00:02:03.890
still going to have
that 1 right over there.

00:02:03.890 --> 00:02:07.320
And that's why anything
that's not 0 to the 1 power

00:02:07.320 --> 00:02:09.360
is going to be equal to 1.

00:02:09.360 --> 00:02:12.070
Now let's try some other
interesting scenarios.

00:02:12.070 --> 00:02:14.370
Let's start try some
negative numbers.

00:02:14.370 --> 00:02:17.760
So let's take negative 1.

00:02:17.760 --> 00:02:22.950
And let's first raise
it to the 0 power.

00:02:22.950 --> 00:02:25.070
So once again,
this is just going,

00:02:25.070 --> 00:02:28.740
based on this definition,
this is starting with a 1

00:02:28.740 --> 00:02:30.899
and then multiplying it
by this number 0 times.

00:02:30.899 --> 00:02:32.440
Well, that means
we're just not going

00:02:32.440 --> 00:02:33.690
to multiply it by this number.

00:02:33.690 --> 00:02:35.620
So you're just going to get a 1.

00:02:35.620 --> 00:02:37.720
Let's try negative 1.

00:02:37.720 --> 00:02:41.310
Let's try negative 1
to the first power.

00:02:41.310 --> 00:02:43.800
Well, anything to the first
power, you could view this--

00:02:43.800 --> 00:02:45.890
and I like going with
this definition as opposed

00:02:45.890 --> 00:02:46.800
to this one right over here.

00:02:46.800 --> 00:02:48.290
If we were to make
them consistent,

00:02:48.290 --> 00:02:49.880
if you were to make
this definition

00:02:49.880 --> 00:02:52.830
consistent with this, you would
say hey, let's start with a 1,

00:02:52.830 --> 00:02:56.110
and then multiply
it by 1 eight times.

00:02:56.110 --> 00:02:58.480
And you're still going to
get a 1 right over here.

00:02:58.480 --> 00:03:00.070
But let's do this
with negative 1.

00:03:00.070 --> 00:03:02.270
So we're going to
start with a 1,

00:03:02.270 --> 00:03:06.410
and then we're going to multiply
it by negative 1 one time--

00:03:06.410 --> 00:03:07.950
times negative 1.

00:03:07.950 --> 00:03:11.660
And this is, of course, going
to be equal to negative 1.

00:03:11.660 --> 00:03:14.270
Now let's take
negative 1, and let's

00:03:14.270 --> 00:03:15.440
take it to the second power.

00:03:15.440 --> 00:03:16.979
We often say that
we are squaring it

00:03:16.979 --> 00:03:18.770
when we take something
to the second power.

00:03:18.770 --> 00:03:21.830
So negative 1 to the
second power-- well,

00:03:21.830 --> 00:03:23.650
we could start with a 1.

00:03:23.650 --> 00:03:26.620
We could start with a 1, and
then multiply it by negative 1

00:03:26.620 --> 00:03:32.630
two times-- multiply
it by negative 1 twice.

00:03:32.630 --> 00:03:34.327
And what's this
going to be equal to?

00:03:34.327 --> 00:03:35.910
And once again, by
our old definition,

00:03:35.910 --> 00:03:38.060
you could also just say,
hey, ignoring this one,

00:03:38.060 --> 00:03:39.935
because that's not going
to change the value,

00:03:39.935 --> 00:03:42.260
we took two negative 1's
and we're multiplying them.

00:03:42.260 --> 00:03:45.580
Well, negative 1
times negative 1 is 1.

00:03:45.580 --> 00:03:48.440
And I think you see
a pattern forming.

00:03:48.440 --> 00:03:54.780
Let's take negative
1 to the third power.

00:03:54.780 --> 00:03:56.840
What's this going
to be equal to?

00:03:56.840 --> 00:03:58.820
Well, by this definition,
you start with a 1,

00:03:58.820 --> 00:04:01.890
and then you multiply it
by negative 1 three times,

00:04:01.890 --> 00:04:06.945
so negative 1 times
negative 1 times negative 1.

00:04:06.945 --> 00:04:08.570
Or you could just
think of it as you're

00:04:08.570 --> 00:04:10.736
taking three negative 1's
and you're multiplying it,

00:04:10.736 --> 00:04:12.630
because this 1 doesn't
change the value.

00:04:12.630 --> 00:04:15.560
And this is going to be equal
to negative 1 times negative 1

00:04:15.560 --> 00:04:19.730
is positive 1, times
negative 1 is negative 1.

00:04:19.730 --> 00:04:20.910
So you see the pattern.

00:04:20.910 --> 00:04:22.240
Negative 1 to the 0 power is 1.

00:04:22.240 --> 00:04:24.420
Negative 1 to the first
power is negative 1.

00:04:24.420 --> 00:04:25.930
Then you multiply
it by negative 1,

00:04:25.930 --> 00:04:27.221
you're going to get positive 1.

00:04:27.221 --> 00:04:30.190
Then you multiply it by negative
1 again to get negative 1.

00:04:30.190 --> 00:04:31.840
And the pattern
you might be seeing

00:04:31.840 --> 00:04:38.240
is if you take negative
1 to an odd power you're

00:04:38.240 --> 00:04:40.100
going to get negative 1.

00:04:40.100 --> 00:04:42.130
And if you take it
to an even power,

00:04:42.130 --> 00:04:46.839
you're going to get 1 because
a negative times a negative is

00:04:46.839 --> 00:04:47.880
going to be the positive.

00:04:47.880 --> 00:04:50.270
And you're going to have an
even number of negatives,

00:04:50.270 --> 00:04:52.811
so that you're always going to
have negative times negatives.

00:04:52.811 --> 00:04:56.639
So this right over
here, this is even.

00:04:56.639 --> 00:04:57.930
Even is going to be positive 1.

00:04:57.930 --> 00:05:00.138
And then you could see that
if you went to negative 1

00:05:00.138 --> 00:05:01.480
to the fourth power.

00:05:01.480 --> 00:05:04.350
Negative 1 the fourth power?

00:05:04.350 --> 00:05:06.720
Well, you could start
with a 1 and then

00:05:06.720 --> 00:05:11.180
multiply it by negative 1 four
times, so a negative 1 times

00:05:11.180 --> 00:05:15.510
negative 1, times negative
1, times negative 1,

00:05:15.510 --> 00:05:18.450
which is just going to
be equal to positive 1.

00:05:18.450 --> 00:05:21.960
So if someone were to
ask you-- we already

00:05:21.960 --> 00:05:26.120
established that if someone were
to take 1 to the, I don't know,

00:05:26.120 --> 00:05:32.430
1 millionth power, this is
just going to be equal to 1.

00:05:35.820 --> 00:05:38.350
If someone told you
let's take negative 1

00:05:38.350 --> 00:05:44.210
and raise it to the 1 millionth
power, well, 1 million

00:05:44.210 --> 00:05:46.730
is an even number,
so this is still

00:05:46.730 --> 00:05:49.870
going to be equal to positive 1.

00:05:49.870 --> 00:05:58.600
But if you took negative
1 to the 999,999th power,

00:05:58.600 --> 00:06:00.380
this is an odd number.

00:06:00.380 --> 00:06:06.606
So this is going to be
equal to negative 1.


Introduction to fractions

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:06.180
- [Sal] What I want to do in this video is think about the idea of a fraction.

00:00:06.180 --> 00:00:13.280
And for the sake of this video as we just learn what a fraction is, you can really think of it as a part of a whole.

00:00:13.280 --> 00:00:16.660
Later on we'll think of it in even more ways.

00:00:16.660 --> 00:00:19.820
What do I mean by a part of a whole?

00:00:19.820 --> 00:00:23.660
So let's imagine that we have a pizza.

00:00:23.660 --> 00:00:26.900
I will do it in a pizza colored. . . so I'll do it an orange

00:00:26.900 --> 00:00:29.120
I guess that's roughly the color of a pizza.

00:00:29.120 --> 00:00:32.500
Pizza I guess is yellow and kind of red for the tomato sauce so yellow for the cheese.

00:00:32.500 --> 00:00:36.520
So let's say this was a perfect circle. I'm gonna try my best to draw a circle right over here.

00:00:36.520 --> 00:00:40.640
And let's say that I divided into four equal parts

00:00:40.640 --> 00:00:49.060
so just like that I've divided it into two equal parts and now I will divide it into four equal parts.

00:00:49.060 --> 00:00:55.440
So there's a total of four slices, four equal slices of this pizza.

00:00:55.440 --> 00:00:58.040
Four equal slices of this pizza.

00:00:58.040 --> 00:01:03.200
And let's say on three of the slices I have cheese only cheese.

00:01:03.200 --> 00:01:04.460
only

00:01:04.460 --> 00:01:06.320
only cheese.

00:01:06.320 --> 00:01:12.380
And lets say on this fourth slice, I have cheese and, I don't know, I have olives

00:01:12.380 --> 00:01:15.380
I have olives right over here.

00:01:15.380 --> 00:01:16.900
So we can ask ourselves,

00:01:16.900 --> 00:01:20.960
"What fraction of this pizza has olives on it?"

00:01:20.960 --> 00:01:27.760
Well, we have four total slices. They all are equal slices. We have four equal sections here

00:01:27.760 --> 00:01:34.780
and one of those equal sections has olives, so we could say one-fourth of this pizza

00:01:34.780 --> 00:01:37.940
one-fourth of the pizza

00:01:37.940 --> 00:01:39.120
the pizza,

00:01:39.120 --> 00:01:40.620
has,

00:01:40.620 --> 00:01:43.880
has olives.

00:01:43.880 --> 00:01:46.880
One out of the four. Now someone said,

00:01:46.880 --> 00:01:50.360
"Well what fraction of this pizza only has cheese?".

00:01:50.360 --> 00:01:54.300
Well this fraction has cheese, but it also has olives so we won't count that.

00:01:54.300 --> 00:02:00.080
Then we have one, two, three, three fractions three of these equal pieces have only cheese

00:02:00.080 --> 00:02:06.260
so we could say three out of a total of four pieces of. . .

00:02:06.260 --> 00:02:16.440
so three fourths of the pizza has only cheese.

00:02:16.440 --> 00:02:20.380
Let's do a couple more examples because this is probably one of the most important

00:02:20.380 --> 00:02:23.440
and most useful ideas that you can have in your brains.

00:02:23.440 --> 00:02:25.300
Let's do another, another few examples.

00:02:25.300 --> 00:02:27.700
So this is what we sliced up, a pizza.

00:02:27.700 --> 00:02:33.720
Let's imagine instead that we're looking at a, some animals of some. . .

00:02:33.720 --> 00:02:36.720
I'd say fruit. Fruit is easier for me to draw

00:02:36.720 --> 00:02:42.480
So let's say that I have, I have an orange

00:02:42.480 --> 00:02:44.720
so that's an orange right over there,

00:02:44.720 --> 00:02:52.800
and I have a Banana. So that's my best attempt at drawing a banana right over here.

00:02:52.800 --> 00:02:56.160
And I have a lemon.

00:02:56.160 --> 00:02:59.880
I have a lemon.

00:02:59.880 --> 00:03:05.920
Just like that. And let's say that I have an apple.

00:03:05.920 --> 00:03:08.360
I'll draw. . .

00:03:08.360 --> 00:03:10.600
Actually I'll draw an Apple.

00:03:10.600 --> 00:03:13.760
It's a little bit of a strange color for an apple, but we'll go with it.

00:03:13.760 --> 00:03:22.560
So that is my apple right over there, and actually let me draw one more, let me draw a grape.

00:03:22.560 --> 00:03:23.720
Or grapes.

00:03:23.720 --> 00:03:27.360
So I have a bunch of grapes right over here.

00:03:27.360 --> 00:03:30.400
And if I were to ask you, at least the way that it is drawn,

00:03:30.400 --> 00:03:35.960
"What fraction of my fruit that I have is. . ."

00:03:35.960 --> 00:03:37.880
Let's say,

00:03:37.880 --> 00:03:40.440
". . .is yellow?"

00:03:40.440 --> 00:03:44.200
"What fraction of my fruit as drawn is yellow?"

00:03:44.200 --> 00:03:50.720
Well, we could say, we have a total of one, two, three, four, five pieces of fruit

00:03:50.720 --> 00:03:53.560
So there's a total of five pieces of fruit.

00:03:53.560 --> 00:03:57.760
And then what fraction of those pieces of fruit are yellow?

00:03:57.760 --> 00:04:01.280
Well, the way I drew it, I have two yellow pieces right over here.

00:04:01.280 --> 00:04:08.680
So we'd say, "Two fifths of the fruit is. . ."

00:04:08.680 --> 00:04:11.720
and it's strange that I'm writing this in white so I'll write it in yellow

00:04:11.720 --> 00:04:14.800
". . .is yellow."

00:04:14.800 --> 00:04:16.920
Two fifths is yellow.

00:04:16.920 --> 00:04:21.520
Let me do one more example to just really make the point clear.

00:04:21.520 --> 00:04:28.640
So let me draw a candy bar, and this is all food-related. I guess it's getting close to dinnertime. I apologize.

00:04:28.640 --> 00:04:31.400
So let's say I have a candy bar. Just like that.

00:04:31.400 --> 00:04:33.160
And let me divide it into

00:04:33.160 --> 00:04:35.720
let me divide it into 5 equal pieces 1

00:04:35.720 --> 00:04:39.800
one, two, three, four

00:04:39.800 --> 00:04:41.200
that doesn't look too equal.

00:04:41.200 --> 00:04:42.480
So the whole point is to do it equal,

00:04:42.480 --> 00:04:45.080
one, two, three, four

00:04:45.080 --> 00:04:46.720
that looks pretty close.

00:04:46.720 --> 00:04:49.360
So let's assume that these are equal pieces.

00:04:49.360 --> 00:04:52.880
So my drawing isn't perfect but let's assume all of these pieces are equal

00:04:52.880 --> 00:04:56.040
and I'm hungry so let's say that I eat

00:04:56.040 --> 00:05:01.880
I eat this piece, and I eat this piece so they essentially go away.

00:05:01.880 --> 00:05:05.120
So what fraction of the pieces have I eaten?

00:05:05.120 --> 00:05:12.200
Well, I have a total of five pieces. There's a total of five equal sections of this candy bar I guess I could say.

00:05:12.200 --> 00:05:15.480
And I ate two of them, so two fifths

00:05:15.480 --> 00:05:20.520
two-fifths of this bar, so this is, I've taken them away

00:05:20.520 --> 00:05:26.800
two-fifths of the bar have been eaten.

00:05:26.800 --> 00:05:29.000
Have been eaten.

00:05:29.000 --> 00:05:33.680
So I'll leave you there, maybe I'll make another video that doesn't have as many food analogies,

00:05:33.680 --> 00:05:37.840
but this is a super useful concept and I encourage you to think about this as much as possible

00:05:37.840 --> 00:05:40.400
and then try the exercises on Khan Academy


Recognizing fractions exercise

WEBVTT
Kind: captions
Language: en

00:00:00.764 --> 00:00:04.253
- [Voiceover] What fraction
of the circle is shaded red?

00:00:04.253 --> 00:00:07.105
So let's see, this circle is divided into

00:00:07.105 --> 00:00:11.272
one, two, three, four, five,
six, seven, eight equal parts,

00:00:12.289 --> 00:00:15.887
and of those eight equal
parts seven are shaded red.

00:00:15.887 --> 00:00:20.054
So, we can say that seven
out of eight are shaded red,

00:00:20.999 --> 00:00:23.022
and this isn't how we
traditionally seen fraction

00:00:23.022 --> 00:00:23.883
when we've written them.

00:00:23.883 --> 00:00:26.439
When we've written them
we've written it like this.

00:00:26.439 --> 00:00:30.574
We have written seven
over eight are shaded red.

00:00:30.574 --> 00:00:32.089
And actually, let me do it in that color,

00:00:32.089 --> 00:00:33.618
oh, I don't have that exact,

00:00:33.618 --> 00:00:35.359
let me get closer to an actual red color.

00:00:35.359 --> 00:00:38.893
So, we could literally say seven out of,

00:00:38.893 --> 00:00:41.094
and then we need like a grey color was,

00:00:41.094 --> 00:00:42.015
was the non-shaded.

00:00:42.015 --> 00:00:46.302
Seven out of a total eight are shaded red.

00:00:46.302 --> 00:00:50.155
Are shaded red, that's how
we're used to seeing it,

00:00:50.155 --> 00:00:52.771
but sometimes, or usually
when we type it in

00:00:52.771 --> 00:00:55.082
there's no way to write
the seven over the eight,

00:00:55.082 --> 00:00:58.915
so instead, instead we
write seven over eight.

00:00:59.964 --> 00:01:03.800
Over eight are shaded red,
or seven out of eight.

00:01:03.800 --> 00:01:04.953
So, let's do that.

00:01:04.953 --> 00:01:06.551
Go back to what we we're doing.

00:01:06.551 --> 00:01:08.032
So, seven out of eight are shaded red.

00:01:08.032 --> 00:01:09.224
Let's check our answer.

00:01:09.224 --> 00:01:11.513
Let's do two more of these.

00:01:11.513 --> 00:01:13.636
What fraction of the bar is shaded red?

00:01:13.636 --> 00:01:16.454
So, the bar has one, two, three, four,

00:01:16.454 --> 00:01:19.938
five, six, seven, eight,
nine equal sections,

00:01:19.938 --> 00:01:22.253
and eight of them are shaded red.

00:01:22.253 --> 00:01:25.711
So, we can say that eight
out of nine are shaded red,

00:01:25.711 --> 00:01:28.666
or eight-ninths, or eight over nine.

00:01:28.666 --> 00:01:30.166
Let's do one more.

00:01:31.391 --> 00:01:33.729
What fraction of this
circle is shaded red?

00:01:33.729 --> 00:01:35.715
So, we have one, two, three, four,

00:01:35.715 --> 00:01:39.101
five, six, seven, eight,
nine equal sections.

00:01:39.101 --> 00:01:39.934
Did I count that right?

00:01:39.934 --> 00:01:40.767
Let me count again.

00:01:40.767 --> 00:01:42.487
One, two, three, four, five,

00:01:42.487 --> 00:01:45.417
six, seven, eight, nine equal sections,

00:01:45.417 --> 00:01:48.218
and of those one, two, three, four,

00:01:48.218 --> 00:01:50.633
five, six, seven are shaded red.

00:01:50.633 --> 00:01:54.800
So, seven out of nine, or
seven-ninths are shaded red.


Raising a number to the 0 and 1st power

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:02.020
What I want to do
in this video is

00:00:02.020 --> 00:00:04.780
think about exponents in a
slightly different way that

00:00:04.780 --> 00:00:06.940
will be useful for
different contexts

00:00:06.940 --> 00:00:09.210
and also go through
a lot more examples.

00:00:09.210 --> 00:00:11.370
So in the last video, we
saw that taking something

00:00:11.370 --> 00:00:14.630
to an exponent means multiplying
that number that many times.

00:00:14.630 --> 00:00:17.780
So if I had the
number negative 2

00:00:17.780 --> 00:00:20.740
and I want to raise
it to the third power,

00:00:20.740 --> 00:00:26.230
this literally means
taking three negative 2's,

00:00:26.230 --> 00:00:30.100
so negative 2, negative
2, and negative 2,

00:00:30.100 --> 00:00:31.360
and then multiplying them.

00:00:34.350 --> 00:00:35.834
So what's this going to be?

00:00:35.834 --> 00:00:36.500
Well, let's see.

00:00:36.500 --> 00:00:40.310
Negative 2 times
negative 2 is positive 4,

00:00:40.310 --> 00:00:45.800
and then positive 4 times
negative 2 is negative 8.

00:00:45.800 --> 00:00:51.030
So this would be
equal to negative 8.

00:00:51.030 --> 00:00:53.480
Now, another way of
thinking about exponents,

00:00:53.480 --> 00:00:55.772
instead of saying you're just
taking three negative 2's

00:00:55.772 --> 00:00:58.313
and multiplying them, and this
is a completely reasonable way

00:00:58.313 --> 00:00:59.980
of viewing it, you
could also view it

00:00:59.980 --> 00:01:02.770
as this is a number
of times you're

00:01:02.770 --> 00:01:05.880
going to multiply
this number times 1.

00:01:05.880 --> 00:01:07.450
So you could
completely view this

00:01:07.450 --> 00:01:10.930
as being equal to-- so you're
going to start with a 1,

00:01:10.930 --> 00:01:15.210
and you're going to multiply 1
times negative 2 three times.

00:01:15.210 --> 00:01:22.080
So this is times negative
2 times negative 2

00:01:22.080 --> 00:01:25.350
times negative 2.

00:01:25.350 --> 00:01:27.012
So clearly these
are the same number.

00:01:27.012 --> 00:01:29.470
Here we just took this, and
we're just multiplying it by 1,

00:01:29.470 --> 00:01:33.120
so you're still going
to get negative 8.

00:01:33.120 --> 00:01:35.640
And this might be a
slightly more useful idea

00:01:35.640 --> 00:01:38.410
to get an intuition for
exponents, especially when you

00:01:38.410 --> 00:01:42.040
start taking things
to the 1 or 0 power.

00:01:42.040 --> 00:01:44.250
So let's think about
that a little bit.

00:01:44.250 --> 00:01:49.940
What is positive
2 to the-- based

00:01:49.940 --> 00:01:54.770
on this definition-- to the
0 power going to be equal to?

00:01:58.330 --> 00:01:59.470
Well, we just said.

00:01:59.470 --> 00:02:02.740
This says how many times are
going to multiply 1 times

00:02:02.740 --> 00:02:03.920
this number?

00:02:03.920 --> 00:02:08.250
So this literally says,
I'm going to take a 1,

00:02:08.250 --> 00:02:11.179
and I'm going to
multiply by 2 zero times.

00:02:11.179 --> 00:02:13.280
Well, if I want to multiply
it by 2 zero times,

00:02:13.280 --> 00:02:15.160
that means I'm just
left with the 1.

00:02:15.160 --> 00:02:18.440
So 2 to the zero power is
going to be equal to 1.

00:02:18.440 --> 00:02:21.590
And, actually, any non-zero
number to the 0 power

00:02:21.590 --> 00:02:23.551
is 1 by that same rationale.

00:02:23.551 --> 00:02:25.300
And I'll make another
video that will also

00:02:25.300 --> 00:02:27.050
give a little bit more
intuition on there.

00:02:27.050 --> 00:02:29.430
That might seem very
counterintuitive,

00:02:29.430 --> 00:02:31.330
but it's based on
one way of thinking

00:02:31.330 --> 00:02:33.812
about it is thinking
of an exponent as this.

00:02:33.812 --> 00:02:35.270
And this will also
make sense if we

00:02:35.270 --> 00:02:37.315
start thinking of what
2 to the first power is.

00:02:41.900 --> 00:02:44.830
So let's go to this definition
we just gave of the exponent.

00:02:44.830 --> 00:02:51.840
We always start with a 1, and we
multiply it by the 2 one time.

00:02:51.840 --> 00:02:53.440
So 2 is going to
be 1-- we're only

00:02:53.440 --> 00:02:54.690
going to multiply it by the 2.

00:02:54.690 --> 00:02:56.390
I'll use this for
multiplication.

00:02:56.390 --> 00:02:57.700
I'll use the dot.

00:02:57.700 --> 00:03:01.160
We're only going to
multiply it by 2 one time.

00:03:01.160 --> 00:03:03.700
So 1 times 2, well,
that's clearly

00:03:03.700 --> 00:03:06.380
just going to be equal to 2.

00:03:06.380 --> 00:03:08.530
And any number to
the first power

00:03:08.530 --> 00:03:11.379
is just going to be
equal to that number.

00:03:11.379 --> 00:03:13.670
And then we can go from there,
and you will, of course,

00:03:13.670 --> 00:03:14.710
see the pattern.

00:03:14.710 --> 00:03:21.370
If we say what 2 squared is,
well, based on this definition,

00:03:21.370 --> 00:03:27.180
we start with a 1, and we
multiply it by 2 two times.

00:03:27.180 --> 00:03:34.630
So times 2 times 2 is
going to be equal to 4.

00:03:34.630 --> 00:03:35.880
And we've seen this before.

00:03:35.880 --> 00:03:43.710
You go to 2 to the third,
you start with the 1,

00:03:43.710 --> 00:03:47.150
and then multiply
it by 2 three times.

00:03:47.150 --> 00:03:51.050
So times 2 times 2 times 2.

00:03:51.050 --> 00:03:54.780
This is going to
give us positive 8.

00:03:54.780 --> 00:03:56.940
And you probably
see a pattern here.

00:03:56.940 --> 00:04:00.200
Every time we multiply by 2--
or every time, I should say,

00:04:00.200 --> 00:04:04.420
we raise 2 to one more power,
we are multiplying by 2.

00:04:04.420 --> 00:04:07.920
Notice this, to go from
2 to the 0 to 2 to the 1,

00:04:07.920 --> 00:04:10.965
we multiplied by 2.

00:04:10.965 --> 00:04:13.630
I'll use a little x for the
multiplication symbol now,

00:04:13.630 --> 00:04:14.970
a little cross.

00:04:14.970 --> 00:04:17.440
And then to go from 2
to the first power to 2

00:04:17.440 --> 00:04:21.329
to the second power, we multiply
by 2 and multiply by 2 again.

00:04:21.329 --> 00:04:23.620
And that makes complete sense
because this is literally

00:04:23.620 --> 00:04:26.640
telling us how many times are
we going to take this number

00:04:26.640 --> 00:04:29.400
and-- how many times are we
going take 1 and multiply it

00:04:29.400 --> 00:04:30.534
by this number?

00:04:30.534 --> 00:04:33.200
And so when you go from 2 to the
second power to 2 to the third,

00:04:33.200 --> 00:04:37.580
you're multiplying
by 2 one more time.

00:04:37.580 --> 00:04:40.740
And this is another intuition
of why something to the 0 power

00:04:40.740 --> 00:04:41.899
is equal to 1.

00:04:41.899 --> 00:04:43.440
If you were to go
backwards, if, say,

00:04:43.440 --> 00:04:45.410
we didn't know what
2 to the 0 power

00:04:45.410 --> 00:04:46.420
is and we were just
trying to figure out

00:04:46.420 --> 00:04:47.965
what would make
sense, well, when

00:04:47.965 --> 00:04:50.390
we go from 2 to the third
power to 2 to the second,

00:04:50.390 --> 00:04:51.630
we'd be dividing by 2.

00:04:51.630 --> 00:04:53.170
We're going from 9 to 4.

00:04:53.170 --> 00:04:56.650
Then we'd divide by 2 again to
go from 2 to the second to 2

00:04:56.650 --> 00:04:57.519
to the first.

00:04:57.519 --> 00:04:59.060
And then it seems
like we should just

00:04:59.060 --> 00:05:01.830
divide by 2 again from
going from 2 to the first

00:05:01.830 --> 00:05:02.760
to 2 to the 0.

00:05:02.760 --> 00:05:05.550
And that would give us 1.


Introduction to exponents

WEBVTT
Kind: captions
Language: en

00:00:00.739 --> 00:00:02.574
You already know that we can view

00:00:02.574 --> 00:00:04.768
multiplication as repeated addition.

00:00:04.768 --> 00:00:08.894
So, if we had 2 times 3 (2 × 3), we could literally

00:00:08.894 --> 00:00:11.828
view this as 3 2's being added together.

00:00:11.828 --> 00:00:16.645
So it could be 2 + 2 + 2.

00:00:16.645 --> 00:00:20.270
Notice this is [COUNTING: 1, 2] 3 2's.

00:00:20.270 --> 00:00:24.736
And when you add those 2's together, you get 6.

00:00:24.736 --> 00:00:26.376
What we're going to introduce you to

00:00:26.376 --> 00:00:30.659
in this video is the idea of repeated multiplication –

00:00:30.659 --> 00:00:32.601
a new operation that really can be

00:00:32.601 --> 00:00:34.336
viewed as repeated multiplication.

00:00:34.336 --> 00:00:39.544
And that's the operation of taking an 'exponent.'

00:00:39.544 --> 00:00:41.675
And it sounds very fancy.

00:00:41.675 --> 00:00:42.958
But we'll see with a few examples

00:00:42.958 --> 00:00:44.659
that it's not too bad.

00:00:44.659 --> 00:00:46.635
So now, let's take the idea of

00:00:46.635 --> 00:00:50.104
2 to the 3rd power (2^3) –

00:00:50.104 --> 00:00:51.097
which is how we would say this.

00:00:51.097 --> 00:00:52.045
(So let me write this down

00:00:52.045 --> 00:00:53.713
in the appropriate colors.)

00:00:53.713 --> 00:00:57.537
So 2 to the 3rd power. (2^3.)

00:00:57.537 --> 00:00:59.026
So you might be tempted to say,

00:00:59.026 --> 00:01:01.491
"Hey, maybe this is 2 × 3, which would be 6."

00:01:01.491 --> 00:01:02.553
But remember, I just said

00:01:02.553 --> 00:01:04.527
this is repeated multiplication.

00:01:04.527 --> 00:01:07.196
So if I have 2 to the 3rd power, (2^3),

00:01:07.196 --> 00:01:11.065
this literally means multiplying 3 2's together.

00:01:11.065 --> 00:01:15.013
So this would be equal to,

00:01:15.013 --> 00:01:18.510
not 2 + 2 + 2, but 2 × ...

00:01:18.510 --> 00:01:20.538
(And I’ll use a little dot to signify multiplication.)

00:01:20.538 --> 00:01:24.276
... 2 × 2 × 2.

00:01:24.276 --> 00:01:27.202
Well, what's 2 × 2 × 2?

00:01:27.202 --> 00:01:29.230
Well that is equal to 8.

00:01:29.230 --> 00:01:31.365
(2 × 2 × 2 = 8.)

00:01:31.365 --> 00:01:35.268
So 2 to the 3rd power is equal to 8. (2^3 = 8.)

00:01:35.268 --> 00:01:37.813
Let's try a few more examples here.

00:01:37.813 --> 00:01:45.598
What is 3 to the 2nd power (3^2)

00:01:45.598 --> 00:01:47.183
going to be equal to?

00:01:47.183 --> 00:01:48.970
And I'll let you think about that for a second.

00:01:48.970 --> 00:01:51.454
I encourage you to pause the video.

00:01:51.454 --> 00:01:52.829
So let's think it through.

00:01:52.829 --> 00:01:58.084
This literally means multiplying 2 3's.

00:01:58.084 --> 00:02:00.573
So let's multiply 3 –

00:02:00.573 --> 00:02:02.810
(Let me do that in yellow.)

00:02:02.810 --> 00:02:07.093
Let's multiply 3 × 3.

00:02:07.093 --> 00:02:11.197
So this is going to be equal to 9.

00:02:11.197 --> 00:02:12.979
Let’s do a few more examples.

00:02:12.979 --> 00:02:19.938
What is, say, 5 to the – let's say –

00:02:19.938 --> 00:02:23.421
5 to the 4th power (5^4)?

00:02:23.421 --> 00:02:24.212
And what you'll see here is

00:02:24.212 --> 00:02:25.378
this number is going to get

00:02:25.378 --> 00:02:27.635
large very, very, very fast.

00:02:27.635 --> 00:02:30.090
So 5 to the 4th power (5^4) is going to be

00:02:30.090 --> 00:02:33.399
equal to multiplying 4 5's together.

00:02:33.399 --> 00:02:38.649
So 5^4 = 5 × 5 × 5 × 5.

00:02:38.649 --> 00:02:40.822
Notice, we have [COUNTING: 1, 2, 3] 4 5's.

00:02:40.822 --> 00:02:42.110
And we are multiplying them.

00:02:42.110 --> 00:02:42.971
We are not adding them.

00:02:42.971 --> 00:02:44.416
This is not 5 × 4.

00:02:44.416 --> 00:02:45.668
This is not 20.

00:02:45.668 --> 00:02:47.962
This is 5 × 5 × 5 × 5.

00:02:47.962 --> 00:02:49.658
So what is this going to be?

00:02:49.658 --> 00:02:52.413
Well 5 × 5 is 25. (5 × 5 = 25.)

00:02:52.413 --> 00:02:56.573
25 × 5 is 125. (25 × 5 = 125.)

00:02:56.573 --> 00:03:01.531
125 × 5 is 625. (125 × 5 = 625.)


Estimating the line of best fit exercise

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:03.230
Find the line of
best fit, or mark

00:00:03.230 --> 00:00:05.010
that there is no
linear correlation.

00:00:05.010 --> 00:00:07.190
So let's see, we have
a bunch of data points,

00:00:07.190 --> 00:00:09.170
and we want to find a
line that at least shows

00:00:09.170 --> 00:00:10.889
the trend in the data.

00:00:10.889 --> 00:00:12.430
And this one seems
a little difficult

00:00:12.430 --> 00:00:14.513
because if we ignore these
three points down here,

00:00:14.513 --> 00:00:17.400
maybe we could do a line that
looks something like this.

00:00:17.400 --> 00:00:21.070
It seems like it kind of
approximates this trend,

00:00:21.070 --> 00:00:23.284
although this doesn't
seem like a great trend.

00:00:23.284 --> 00:00:25.200
And if we ignored these
two points right over,

00:00:25.200 --> 00:00:29.190
we could do something like
maybe something like that.

00:00:29.190 --> 00:00:30.940
But we can't just
ignore points like that.

00:00:30.940 --> 00:00:32.440
So I would say that
there's actually

00:00:32.440 --> 00:00:33.760
no good line of best fit here.

00:00:33.760 --> 00:00:35.300
So let me check my answer.

00:00:35.300 --> 00:00:38.200
Let's try a couple
more of these.

00:00:38.200 --> 00:00:41.150
Find the line of best-- well,
this feels very similar.

00:00:41.150 --> 00:00:44.004
It really feels like there's
no-- I mean, I could do that,

00:00:44.004 --> 00:00:45.420
but I'm ignoring
these two points.

00:00:45.420 --> 00:00:47.915
I could do something
like that, then

00:00:47.915 --> 00:00:49.610
I'd be ignoring these points.

00:00:49.610 --> 00:00:53.080
So I'd also say no good
best fit line exists.

00:00:53.080 --> 00:00:54.800
So let's try one more.

00:00:54.800 --> 00:00:57.439
So here it looks like there's
very clearly this trend.

00:00:57.439 --> 00:00:59.730
And I could try to fit it a
little bit better than it's

00:00:59.730 --> 00:01:01.060
fit right now.

00:01:01.060 --> 00:01:04.879
So it feels like something
like that fits this trend line

00:01:04.879 --> 00:01:06.414
quite well.

00:01:06.414 --> 00:01:08.580
I could maybe drop this
down a little bit, something

00:01:08.580 --> 00:01:09.079
like that.

00:01:09.079 --> 00:01:11.160
Let's check my answer.

00:01:11.160 --> 00:01:12.950
A good best fit line exists.

00:01:12.950 --> 00:01:14.380
Let me check my answer.

00:01:14.380 --> 00:01:16.370
We got it right.


Figuring out how much pizza is left

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:03.806
Brandon ate 5 slices
of apple-- of pie.

00:00:03.806 --> 00:00:05.180
I'm just assuming
it's apple pie.

00:00:05.180 --> 00:00:06.410
They didn't tell me that.

00:00:06.410 --> 00:00:08.940
Gabriela ate 3 slices.

00:00:08.940 --> 00:00:10.670
If there were
originally 9 slices,

00:00:10.670 --> 00:00:12.950
what fraction of
the pie was eaten?

00:00:12.950 --> 00:00:16.750
So let me see if I can
draw this thing out.

00:00:16.750 --> 00:00:19.160
So let me draw the pie.

00:00:19.160 --> 00:00:22.460
I will draw the pie
in a yellowish color.

00:00:22.460 --> 00:00:24.870
So let me try my best.

00:00:24.870 --> 00:00:27.620
So let's see how good
I am at drawing a pie.

00:00:27.620 --> 00:00:31.200
So I'll just draw it from
the top view as a circle.

00:00:31.200 --> 00:00:32.330
And there're 9 slices.

00:00:32.330 --> 00:00:34.050
I think it's a
reasonable assumption

00:00:34.050 --> 00:00:36.920
to say that they're
9 equal slices.

00:00:36.920 --> 00:00:39.970
So we have 9 equal
slices of pie.

00:00:39.970 --> 00:00:45.870
And I'll just make sure they're
initially 9 equal slices.

00:00:45.870 --> 00:00:47.350
What fraction of
the pie was eaten?

00:00:47.350 --> 00:00:50.030
So let's first divide
this into 9 sections.

00:00:50.030 --> 00:00:53.000
So one way I could do that, I
could divide it into 3 sections

00:00:53.000 --> 00:00:56.980
first, so it looks
like a peace symbol.

00:00:56.980 --> 00:00:59.450
It actually looks more
like the Mercedes emblem.

00:00:59.450 --> 00:01:01.940
So I'll draw it into
3 sections first.

00:01:01.940 --> 00:01:04.670
Then I'll do each of those into
3 sections, and I'll have 9.

00:01:04.670 --> 00:01:11.500
So let's see, I'll draw
like that and like that.

00:01:11.500 --> 00:01:15.280
Keep in mind, I'm trying to
make these as equal as possible.

00:01:15.280 --> 00:01:17.470
So bear with me if they
don't look 100% equal,

00:01:17.470 --> 00:01:18.780
but I'm trying.

00:01:18.780 --> 00:01:20.660
I am trying my best.

00:01:20.660 --> 00:01:26.910
So 9 equal slices-- so that
looks pretty respectable.

00:01:26.910 --> 00:01:31.750
So here's our pie that
initially had 9 equal slices.

00:01:31.750 --> 00:01:35.460
Now, they tell us that
Brandon ate 5 slices of pie.

00:01:35.460 --> 00:01:38.540
So Brandon eats-- he seems
like a hungry young man--

00:01:38.540 --> 00:01:46.850
so he eats 1, 2, 3, 4, 5.

00:01:46.850 --> 00:01:48.981
You could say that he
ate 5/9 of the pie.

00:01:48.981 --> 00:01:49.730
But that's not it.

00:01:49.730 --> 00:01:50.280
That's not what they're asking.

00:01:50.280 --> 00:01:52.488
It's saying total, not just
how much did Brandon eat,

00:01:52.488 --> 00:01:57.230
but how much was eaten total
between Brandon and Gabriela.

00:01:57.230 --> 00:02:00.100
And they tell us
Gabriela ate 3 slices.

00:02:00.100 --> 00:02:02.220
So she ate 1-- sounds
like they didn't really

00:02:02.220 --> 00:02:08.030
eat dinner-- 1, 2, 3.

00:02:08.030 --> 00:02:10.419
So now let's answer
their question.

00:02:10.419 --> 00:02:12.430
What fraction of
the pie was eaten?

00:02:12.430 --> 00:02:17.680
Well, we know that there was a
total of 9 equal slices of pie.

00:02:17.680 --> 00:02:19.180
What fraction was eaten?

00:02:19.180 --> 00:02:26.095
Well, as we see, 1,
2, 3, 4, 5, 6, 7, 8.

00:02:26.095 --> 00:02:30.490
8/9 of the pie was eaten.

00:02:30.490 --> 00:02:32.100
So let's actually type that in.

00:02:32.100 --> 00:02:33.725
And then we will do
it right over here.

00:02:33.725 --> 00:02:36.040
We'd say 8, and we use
that little slash symbol

00:02:36.040 --> 00:02:37.290
on your computer like this.

00:02:37.290 --> 00:02:45.180
8 over 9 or 8 divided by 9
or 8/9 of the pie was eaten.

00:02:45.180 --> 00:02:47.690
Let's do a couple more of these.

00:02:47.690 --> 00:02:49.720
Ishaan ate 2 slices of pizza.

00:02:49.720 --> 00:02:51.830
Omar ate 3 slices.

00:02:51.830 --> 00:02:53.760
If there were
originally 8 slices,

00:02:53.760 --> 00:02:57.800
what fraction of the
pie is remaining?

00:02:57.800 --> 00:02:58.880
So this is interesting.

00:02:58.880 --> 00:03:01.230
Actually, let me
copy and paste this

00:03:01.230 --> 00:03:06.590
so that I can do it on my
little notepad right over here.

00:03:06.590 --> 00:03:10.980
So let's do it right over--
trying to find some space.

00:03:10.980 --> 00:03:12.255
There you go.

00:03:12.255 --> 00:03:15.070
Let me put that in here.

00:03:15.070 --> 00:03:17.240
So the same thing,
we have-- well,

00:03:17.240 --> 00:03:19.510
this is a pizza now, not a pie.

00:03:19.510 --> 00:03:21.480
And so pizza I
will draw in brown.

00:03:21.480 --> 00:03:26.930
It has 8, and we can assume
it's initially 8 equal slices.

00:03:26.930 --> 00:03:30.440
So it's actually a little bit
easier to draw 8 equal slices,

00:03:30.440 --> 00:03:32.940
since 8 is an even number.

00:03:32.940 --> 00:03:34.190
So let's see.

00:03:34.190 --> 00:03:37.960
That is my best attempt
at drawing a circle.

00:03:37.960 --> 00:03:40.845
And let's see, first I can
divide the pizza into 2 slices,

00:03:40.845 --> 00:03:45.040
then I can divide
it into 4 slices.

00:03:45.040 --> 00:03:47.920
And I'm going to try to make
them look as equal as possible.

00:03:47.920 --> 00:03:50.455
And now I'm going to--
with two more cuts,

00:03:50.455 --> 00:03:55.620
I should be able to get 8 so--
and one more just like this.

00:03:55.620 --> 00:04:00.610
So there you have it, a pizza
that has 8 equal slices.

00:04:00.610 --> 00:04:04.470
Now, they tell us Ishaan
ate 2 slices of pizza.

00:04:04.470 --> 00:04:09.580
So he eats 1, 2 slices of pizza.

00:04:09.580 --> 00:04:11.830
Omar ate 3 slices.

00:04:11.830 --> 00:04:13.310
Omar ate 3.

00:04:13.310 --> 00:04:21.836
So he eats 1, 2, 1,
2, 3 slices of pizza.

00:04:21.836 --> 00:04:23.460
Now, you might
immediately say, oh, OK,

00:04:23.460 --> 00:04:27.250
the answer to this
must be 1, 2, 3, 4, 5.

00:04:27.250 --> 00:04:30.300
It must be that they
ate 5 slices of pizza

00:04:30.300 --> 00:04:33.240
over a total of 8
slices of pizza.

00:04:33.240 --> 00:04:35.620
So they ate 5/8 of the pizza.

00:04:35.620 --> 00:04:38.510
You would be right in saying
that they ate 5/8 of the pizza,

00:04:38.510 --> 00:04:42.450
because they ate 1, 2, 3, 4,
5 out of a total of 8 pieces.

00:04:42.450 --> 00:04:45.360
But that's not what
this question is asking.

00:04:45.360 --> 00:04:50.370
They are asking, what fraction
of the pizza is remaining?

00:04:50.370 --> 00:04:53.030
So what's left over
after Ishaan and Omar

00:04:53.030 --> 00:04:54.720
had their go at the pizza?

00:04:54.720 --> 00:04:59.520
Well, what's remaining
is 1, 2, 3 slices.

00:04:59.520 --> 00:05:05.790
So what's remaining is 3 out
of the original 8 equal parts

00:05:05.790 --> 00:05:08.420
or 3/8 of the
pizza is remaining.

00:05:08.420 --> 00:05:10.270
So let's input that.

00:05:10.270 --> 00:05:11.890
So we could go right here.

00:05:11.890 --> 00:05:15.870
And we would say 3 over
8, 3/8, is remaining.

00:05:15.870 --> 00:05:19.300
So let's check our answer,
and we got it right.


Plotting basic fractions on the number line

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:02.090
We've already seen
that fractions

00:00:02.090 --> 00:00:04.581
could be used to
represent parts of wholes.

00:00:04.581 --> 00:00:06.580
Now we're going to extend
that idea a little bit

00:00:06.580 --> 00:00:09.200
to think about them as actual
numbers on the number line.

00:00:09.200 --> 00:00:12.380
So let me draw a number
line right over here.

00:00:12.380 --> 00:00:14.100
So that's my number line.

00:00:14.100 --> 00:00:18.260
Let's say that this
is 0, this is 1,

00:00:18.260 --> 00:00:21.360
and let's say that this is 2.

00:00:21.360 --> 00:00:29.655
And let's say I wanted to
represent the fraction 3/4.

00:00:34.340 --> 00:00:37.700
So one way to think about it
is let's go between 0 and 1.

00:00:37.700 --> 00:00:39.920
So this fraction,
the numerator here

00:00:39.920 --> 00:00:41.180
is less than the denominator.

00:00:41.180 --> 00:00:43.580
So the value of this is
going to be less than 1.

00:00:43.580 --> 00:00:45.927
So this is going to be
someplace between 0 and 1.

00:00:45.927 --> 00:00:47.385
And one way we
could think about it

00:00:47.385 --> 00:00:50.770
is divide this section
into 4 equal spaces.

00:00:50.770 --> 00:00:53.380
So that would be dividing
it into 2 equal spaces.

00:00:53.380 --> 00:00:56.790
So if I divide those again,
now I've divided the space.

00:00:56.790 --> 00:00:58.420
Let me draw that a
little bit neater.

00:00:58.420 --> 00:01:01.960
Now I'm dividing the
space between 0 and 1

00:01:01.960 --> 00:01:04.170
into 4 equal spaces.

00:01:04.170 --> 00:01:10.000
Now, 3/4 means that I've gone 3
out of the 4 spaces towards 1.

00:01:10.000 --> 00:01:12.370
So I've gone 1, 2, 3.

00:01:12.370 --> 00:01:17.305
So this point right over here
on the number line, this is 3/4.

00:01:17.305 --> 00:01:18.930
This point right over
here, what do you

00:01:18.930 --> 00:01:21.280
think this is going to be?

00:01:21.280 --> 00:01:24.890
Well, we've moved 1, 2
of the spaces towards 4,

00:01:24.890 --> 00:01:27.860
so this is 2/4.

00:01:27.860 --> 00:01:30.719
And this space right over
here, or this mark right

00:01:30.719 --> 00:01:33.010
over here on the number line,
what is that going to be?

00:01:33.010 --> 00:01:36.510
Well, we've moved 1 of
the 4 spaces towards 1.

00:01:36.510 --> 00:01:38.810
So this is 1/4.

00:01:38.810 --> 00:01:40.170
What about this one?

00:01:40.170 --> 00:01:45.390
Well, you could call this 0/4,
which is the same thing as 0.

00:01:45.390 --> 00:01:48.140
So you normally wouldn't
call that just 0/4.

00:01:48.140 --> 00:01:50.250
And now we have another
interesting question.

00:01:50.250 --> 00:01:52.364
What could you also
call this space?

00:01:52.364 --> 00:01:54.530
Well, we already know based
on how we already marked

00:01:54.530 --> 00:01:57.260
the number line that this is 1.

00:01:57.260 --> 00:02:00.201
But using the 4's
in the denominator,

00:02:00.201 --> 00:02:01.200
we could say, hey, look.

00:02:01.200 --> 00:02:04.900
We've moved 1, 2, 3,
4 out of 4 spaces.

00:02:04.900 --> 00:02:08.889
We could call that 4/4/

00:02:08.889 --> 00:02:09.979
And that also makes sense.

00:02:09.979 --> 00:02:11.730
So let me make this clear.

00:02:11.730 --> 00:02:18.240
4 parts out of 4 total
parts equals 1 whole.

00:02:21.200 --> 00:02:23.700
But this also makes sense
from what we know of division.

00:02:23.700 --> 00:02:27.300
This little fraction
symbol that we keep using,

00:02:27.300 --> 00:02:29.050
this is the same
thing as divided by.

00:02:29.050 --> 00:02:31.210
This is 4 divided by 4.

00:02:31.210 --> 00:02:34.320
And we know that any
number divided by itself--

00:02:34.320 --> 00:02:35.180
except for 0.

00:02:35.180 --> 00:02:37.050
0 is a little bit strange.

00:02:37.050 --> 00:02:41.680
But any non-zero number divided
by itself is going to be 1.

00:02:41.680 --> 00:02:43.730
So this could be viewed as 4/4.

00:02:43.730 --> 00:02:47.290
So let's do a few exercises
with our newfound knowledge

00:02:47.290 --> 00:02:49.560
of fractions on the number line.

00:02:49.560 --> 00:02:52.090
So let me get a
little exercise going.

00:02:52.090 --> 00:02:58.610
So here we have move the orange
dot to 3/6 on the number line.

00:02:58.610 --> 00:03:00.820
So we already see that
they've divided up

00:03:00.820 --> 00:03:05.210
the space between 0 and
1 into 6 equal sections.

00:03:05.210 --> 00:03:08.440
1, 2, 3, 4, 5, 6.

00:03:08.440 --> 00:03:10.940
So we need to move it to
3 of those 4 sections.

00:03:10.940 --> 00:03:13.370
1, 2, 3.

00:03:13.370 --> 00:03:14.720
Let's check our answer.

00:03:14.720 --> 00:03:17.460
Let's do a couple more of these.

00:03:17.460 --> 00:03:19.820
Move the orange dot to
2/4 on the number line.

00:03:19.820 --> 00:03:22.750
So we have 1, 2, 3,
4 equal sections.

00:03:22.750 --> 00:03:25.900
We need to move it
to 1, 2 of them.

00:03:25.900 --> 00:03:28.310
If they didn't divide it
into 4 equal sections for us,

00:03:28.310 --> 00:03:31.060
then it wouldn't have been
so easy to just count 2.

00:03:31.060 --> 00:03:33.900
The number of sections has to be
the number in the denominator.

00:03:33.900 --> 00:03:36.233
And then the number in the
numerator's how many of those

00:03:36.233 --> 00:03:37.907
we move the dot to.

00:03:37.907 --> 00:03:38.990
Let's do one more of this.

00:03:38.990 --> 00:03:40.900
This is a lot of fun.

00:03:40.900 --> 00:03:42.850
Move the orange dot to
1/4 on the number line.

00:03:42.850 --> 00:03:45.099
Well, we already did that
when we were drawing it out.

00:03:45.099 --> 00:03:48.090
There's 1, 2, 3,
4 equal sections.

00:03:48.090 --> 00:03:52.070
And we are going to move to 1/4.


Decimals: choosing a number to represent place value

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:07.640
Choose the right number of
each place value to make 8,129.

00:00:07.640 --> 00:00:12.150
And we could say this
is 123 thousandths.

00:00:12.150 --> 00:00:17.520
Or we could say it is 1 tenth,
2 hundredths, and 3 thousandths.

00:00:17.520 --> 00:00:19.190
So let's just go place by place.

00:00:19.190 --> 00:00:20.399
So first we have 8.

00:00:20.399 --> 00:00:22.440
This 8 right over here is
in the thousands place.

00:00:22.440 --> 00:00:24.182
It represents 8 thousands.

00:00:24.182 --> 00:00:25.640
So we're going to
have 8 thousands.

00:00:25.640 --> 00:00:31.120
So let's select 1, 2, 3,
4, 5, 6, 7, 8 thousands.

00:00:31.120 --> 00:00:32.920
Then we have one 100.

00:00:32.920 --> 00:00:35.800
So we literally just have
only one 100 right over there.

00:00:35.800 --> 00:00:36.930
Then we have 2 tens.

00:00:36.930 --> 00:00:37.960
This 20 is 2 tens.

00:00:37.960 --> 00:00:39.670
So we're going to need 2 tens.

00:00:39.670 --> 00:00:42.290
Then this 9, well,
that's just 9 ones.

00:00:42.290 --> 00:00:47.900
So it's 1, 2, 3,
4, 5, 6, 7, 8, 9.

00:00:47.900 --> 00:00:50.250
And now we're going
into the tenths place.

00:00:50.250 --> 00:00:51.740
This is only 1 tenth.

00:00:51.740 --> 00:00:53.680
So 1 tenth right like that.

00:00:53.680 --> 00:00:56.040
Now this is 2 hundredths.

00:00:56.040 --> 00:00:57.810
So we're going to put
2 hundredths there.

00:00:57.810 --> 00:01:02.220
Then finally, 3 thousandths--
1, 2, 3 thousandths.

00:01:02.220 --> 00:01:03.230
And we check our answer.

00:01:03.230 --> 00:01:05.467
And we got it right.


Constructing solutions to systems of equations

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:03.500
Fill in the blanks to form
a system of linear equations

00:00:03.500 --> 00:00:08.579
in the variables x and y with
infinitely many solutions.

00:00:08.579 --> 00:00:10.620
So you're going to have
infinitely many solutions

00:00:10.620 --> 00:00:12.510
if essentially both
of these equations

00:00:12.510 --> 00:00:14.450
are describing the same line.

00:00:14.450 --> 00:00:17.200
If they're both essentially
the same equation,

00:00:17.200 --> 00:00:18.520
they are the same constraint.

00:00:18.520 --> 00:00:20.660
And we can graphically
imagine that.

00:00:20.660 --> 00:00:25.790
Let's say that that's our
y-axis, and this is our x-axis.

00:00:25.790 --> 00:00:28.460
And we could even try to
graph this right over here.

00:00:28.460 --> 00:00:31.380
If we were to put this
into slope-intercept form,

00:00:31.380 --> 00:00:35.310
you would have 5y is
equal to 2 minus 4x.

00:00:35.310 --> 00:00:38.670
And then if we divide both sides
by 5, you get y is equal to--

00:00:38.670 --> 00:00:43.900
and I'll swap these--
negative 4/5 x plus 2/5.

00:00:43.900 --> 00:00:47.790
So this line up here is going
to look something like this.

00:00:47.790 --> 00:00:51.260
It's y-intercept is at 2/5.

00:00:51.260 --> 00:00:53.410
And it's going to have
a negative 4/5 slope.

00:00:53.410 --> 00:00:55.160
So it's going to look
something like this.

00:00:59.320 --> 00:01:01.420
So that's what the
line looks like.

00:01:01.420 --> 00:01:03.680
So we're going to get an
infinite number of solutions

00:01:03.680 --> 00:01:04.810
for the system.

00:01:04.810 --> 00:01:07.020
If the second line,
if when we graph it,

00:01:07.020 --> 00:01:09.050
is essentially the
exact same line,

00:01:09.050 --> 00:01:11.760
it overlaps at
every x and y that

00:01:11.760 --> 00:01:13.960
satisfy the first equation.

00:01:13.960 --> 00:01:16.460
Now, the easiest way to think
about what these blanks should

00:01:16.460 --> 00:01:22.600
be is, well, how do
I fill in the blanks

00:01:22.600 --> 00:01:25.420
here so it is really just a
direct algebraic manipulation

00:01:25.420 --> 00:01:27.140
of this first equation?

00:01:27.140 --> 00:01:28.810
And we have a clue here.

00:01:28.810 --> 00:01:32.490
On the right-hand side of the
first equation, we have a 2.

00:01:32.490 --> 00:01:35.620
On the right-hand side of the
second equation, we have a 4.

00:01:35.620 --> 00:01:38.200
So if we want a direct
algebraic manipulation, clearly

00:01:38.200 --> 00:01:41.095
on the right-hand side,
you have multiplied by 2.

00:01:41.095 --> 00:01:42.970
And if you want this to
be the same equation,

00:01:42.970 --> 00:01:44.720
you have to do
that to both sides.

00:01:44.720 --> 00:01:48.130
So let's multiply
both sides by 2.

00:01:48.130 --> 00:01:51.740
So 4x times 2 is 8x.

00:01:51.740 --> 00:01:54.450
5y times 2 is 10y.

00:01:54.450 --> 00:01:56.100
So this equation
and this equation

00:01:56.100 --> 00:01:57.890
are the exact same constraint.

00:01:57.890 --> 00:02:00.150
They represent the
exact same line.

00:02:00.150 --> 00:02:03.770
They have an infinite
number of solutions.

00:02:03.770 --> 00:02:06.520
Let's do another example.

00:02:06.520 --> 00:02:10.729
Which of the following
choices of a will result

00:02:10.729 --> 00:02:14.790
in a system of linear
equations with no solutions?

00:02:14.790 --> 00:02:17.840
So you're going to have
no solutions as if you

00:02:17.840 --> 00:02:19.970
take the same
combination of x and y's,

00:02:19.970 --> 00:02:22.222
but you get to a
different number.

00:02:22.222 --> 00:02:23.680
Or another way of
thinking about it

00:02:23.680 --> 00:02:25.263
is if you plotted
this equation and it

00:02:25.263 --> 00:02:29.140
had the exact same slope, but
it had a different y-intercept.

00:02:29.140 --> 00:02:30.800
So let's think
about it both ways.

00:02:30.800 --> 00:02:35.280
So first, let's try to
algebraically manipulate

00:02:35.280 --> 00:02:38.450
the second equation so that
the left-hand side looks

00:02:38.450 --> 00:02:40.744
exactly the same as the
left-hand side up here.

00:02:40.744 --> 00:02:42.160
And then we just
have to make sure

00:02:42.160 --> 00:02:44.640
that the right-hand
side is different.

00:02:44.640 --> 00:02:47.240
And then you will
have no solutions.

00:02:47.240 --> 00:02:49.500
So let's think about
that a little bit.

00:02:49.500 --> 00:02:50.710
So it looks pretty close.

00:02:50.710 --> 00:02:53.220
If we multiply the left-hand
side by a negative 1,

00:02:53.220 --> 00:02:55.690
it's going to look just like
the left-hand side over here.

00:02:55.690 --> 00:02:56.440
So let's do that.

00:02:56.440 --> 00:02:59.960
Let's multiply the left-hand
and the right-hand side

00:02:59.960 --> 00:03:00.780
by negative 1.

00:03:00.780 --> 00:03:02.370
So it's essentially
the same equation.

00:03:02.370 --> 00:03:05.700
So negative 6x times
negative 1 is positive 6x.

00:03:05.700 --> 00:03:09.090
7y times negative 1 is minus 7y.

00:03:09.090 --> 00:03:12.100
And a times negative
1 is negative a.

00:03:12.100 --> 00:03:14.990
So notice, now on the
left-hand side we have the same

00:03:14.990 --> 00:03:16.820
combinations of x's and y's.

00:03:16.820 --> 00:03:18.740
We have 6 x's minus 7 y's.

00:03:22.040 --> 00:03:27.065
If on the right-hand side these
two things were to be equal,

00:03:27.065 --> 00:03:28.940
then we'd have the
scenario that we just saw.

00:03:28.940 --> 00:03:30.740
We would have the
exact same line.

00:03:30.740 --> 00:03:33.837
If negative a was equal to 4,
we would have the same line.

00:03:33.837 --> 00:03:34.920
So let me write that down.

00:03:34.920 --> 00:03:38.085
If negative a is
equal to 4, then we

00:03:38.085 --> 00:03:41.085
are dealing with the same
line/equation/constraint.

00:03:46.270 --> 00:03:48.920
And like the
previous scenario, we

00:03:48.920 --> 00:03:50.790
would have an infinite
number of solutions.

00:03:55.130 --> 00:04:00.252
On the other hand, if
negative a is not equal to 4,

00:04:00.252 --> 00:04:01.960
then there's no way
that there's anything

00:04:01.960 --> 00:04:03.140
that will satisfy both of them.

00:04:03.140 --> 00:04:04.485
Here you're saying
that whatever your x is

00:04:04.485 --> 00:04:06.735
and whatever your y is, take
6 of the x's and subtract

00:04:06.735 --> 00:04:08.170
7 of the y's, and you get 4.

00:04:08.170 --> 00:04:10.250
If here you take 6 of
the x's and subtract 7

00:04:10.250 --> 00:04:11.440
of the y's and you get
a different number,

00:04:11.440 --> 00:04:12.898
then there's never
going to be an x

00:04:12.898 --> 00:04:15.610
and a y that satisfy
both of them.

00:04:15.610 --> 00:04:18.290
So if negative 8
is not equal to 4,

00:04:18.290 --> 00:04:22.180
then you're going to
have no solutions.

00:04:22.180 --> 00:04:26.900
Or another way of saying this is
if a does not equal negative 4,

00:04:26.900 --> 00:04:29.321
obviously, if a is
equal to negative 4,

00:04:29.321 --> 00:04:31.070
then negative a is
going to be equal to 4.

00:04:31.070 --> 00:04:33.560
So as long as a is not
equal to negative 4,

00:04:33.560 --> 00:04:37.940
you're going to
have no solutions.

00:04:37.940 --> 00:04:44.100
So a can be any number
except for negative 4.

00:04:44.100 --> 00:04:45.730
Now, another way
to think about this

00:04:45.730 --> 00:04:48.720
is to put them both in
slope-intercept form.

00:04:48.720 --> 00:04:50.900
And you'll see that they'll
have the same slope.

00:04:50.900 --> 00:04:51.880
And then you would
just want to have

00:04:51.880 --> 00:04:53.450
them have different
y-intercepts.

00:04:53.450 --> 00:04:56.630
And they're going to be parallel
lines that don't overlap.

00:04:56.630 --> 00:05:01.850
So this top one right over
here, if we subtract 6x

00:05:01.850 --> 00:05:04.070
from both sides,
you get negative 7y

00:05:04.070 --> 00:05:08.150
is equal to negative 6x plus 4.

00:05:08.150 --> 00:05:10.510
Divide both sides by negative 7.

00:05:10.510 --> 00:05:15.800
You get y is equal
to 6/7 x minus 4/7.

00:05:15.800 --> 00:05:18.220
So that's this first one
right up here written

00:05:18.220 --> 00:05:19.721
in slope-intercept form.

00:05:19.721 --> 00:05:21.220
Now, the second
one, I'll just start

00:05:21.220 --> 00:05:22.690
with exactly what they gave us.

00:05:22.690 --> 00:05:25.110
Let me add 6x to both sides.

00:05:25.110 --> 00:05:26.710
So this is the second one.

00:05:26.710 --> 00:05:34.580
So if I add 6x to both sides,
I get 7y is equal to 6x plus a.

00:05:34.580 --> 00:05:36.520
And then if I divide
both sides by 7,

00:05:36.520 --> 00:05:43.110
I get y is equal to
6/7 x plus a over 7.

00:05:43.110 --> 00:05:47.360
So notice in slope-intercept
form, our first equation

00:05:47.360 --> 00:05:48.170
looks like that.

00:05:48.170 --> 00:05:51.280
And in terms of a, our second
equation looks like that.

00:05:51.280 --> 00:05:56.570
So they definitely already
have the same slope.

00:05:56.570 --> 00:05:58.720
They have the exact same slope.

00:05:58.720 --> 00:06:03.662
If a is equal to
negative 4, then these

00:06:03.662 --> 00:06:05.370
are going to be the
exact same equations.

00:06:05.370 --> 00:06:07.411
They're going to have the
exact same y-intercept.

00:06:07.411 --> 00:06:09.940
You're going to have an
infinite number of solutions.

00:06:09.940 --> 00:06:12.820
On the other hand, if a is
anything other than negative 4,

00:06:12.820 --> 00:06:15.180
you're going to have a
different y-intercept.

00:06:15.180 --> 00:06:19.609
And these two things are
just going to be parallel.


Implied volatility

WEBVTT
Kind: captions
Language: en

00:00:00.874 --> 00:00:03.095
Voiceover: In the last video,
we already got an overview

00:00:03.095 --> 00:00:05.842
that if you give me a stock price,

00:00:05.842 --> 00:00:08.143
and an exercise price,

00:00:08.143 --> 00:00:11.603
and a risk-free interest rate,

00:00:11.603 --> 00:00:14.428
and a time to expiration

00:00:14.428 --> 00:00:19.025
and the volatility or
the standard deviation

00:00:19.025 --> 00:00:21.021
of the log returns,

00:00:21.021 --> 00:00:26.970
if you give me these six things,

00:00:26.970 --> 00:00:36.457
I can put these into the
Black-Scholes Formula,

00:00:36.457 --> 00:00:38.858
so Black-Scholes Formula,

00:00:38.858 --> 00:00:44.345
and I will output for
you the appropriate price

00:00:44.345 --> 00:00:48.370
for this European call option.

00:00:48.370 --> 00:00:50.607
So it sounds all very straightforward,

00:00:50.607 --> 00:00:52.205
and some of this is straightforward.

00:00:52.205 --> 00:00:55.129
The stock price is easy to look up.

00:00:55.129 --> 00:00:56.284
The exercise price,

00:00:56.284 --> 00:00:57.263
well, that's part of the contract.

00:00:57.263 --> 00:00:58.260
You know that.

00:00:58.260 --> 00:00:59.763
The risk-free interest rate,

00:00:59.763 --> 00:01:01.007
there are good proxies for it,

00:01:01.007 --> 00:01:02.639
money market funds,

00:01:04.070 --> 00:01:06.054
there's government debt, things like that,

00:01:06.054 --> 00:01:07.731
so that's pretty easy to figure out,

00:01:07.731 --> 00:01:09.527
or at least approximate.

00:01:09.527 --> 00:01:11.601
The time to expiration,

00:01:11.601 --> 00:01:12.339
well you know that,

00:01:12.339 --> 00:01:13.670
you know what today's date is.

00:01:13.670 --> 00:01:15.205
You know when this
thing's going to expire,

00:01:15.205 --> 00:01:17.211
so that's pretty straightforward.

00:01:17.211 --> 00:01:19.735
Now let's think a little
bit about volatility,

00:01:19.735 --> 00:01:22.556
so how do you actually
measure the standard deviation

00:01:22.556 --> 00:01:23.858
of log returns.

00:01:23.858 --> 00:01:26.215
Now, one of the assumption
about Black-Scholes Formula

00:01:26.215 --> 00:01:27.459
is that this is a constant thing.

00:01:27.459 --> 00:01:31.739
This is just some intrinsic
propety of this security.

00:01:31.739 --> 00:01:33.925
Well, the only way that
you can at least attempt

00:01:33.925 --> 00:01:36.857
to estimate it is by
looking at the history

00:01:36.857 --> 00:01:39.741
of the standard deviation of log returns.

00:01:39.741 --> 00:01:41.636
The way that people
would normally do it is

00:01:41.636 --> 00:01:44.208
they'll say, "Okay, what
has historically been

00:01:44.208 --> 00:01:46.967
"the standard deviation of log
returns over some time period

00:01:46.967 --> 00:01:49.801
"where that security has not
changed in some dramatic way?"

00:01:49.801 --> 00:01:51.730
And then use that as the input,

00:01:51.730 --> 00:01:54.292
and then they would
come up with some price.

00:01:54.292 --> 00:01:57.142
Well, that's all interesting,
but it's very important to know

00:01:57.142 --> 00:01:58.797
that this is an estimate.

00:01:58.797 --> 00:02:00.135
This right over here is an estimate.

00:02:00.135 --> 00:02:03.325
There's no way of us actually
knowing the actual intrinsic

00:02:03.325 --> 00:02:05.489
and it's even up for
grabs whether there is,

00:02:05.489 --> 00:02:07.256
whether you can even as assume

00:02:07.256 --> 00:02:09.863
that there's some constant
intrinsic property

00:02:09.863 --> 00:02:12.334
as this volatility that's
going to be constant

00:02:12.334 --> 00:02:13.793
over the life of this option.

00:02:13.793 --> 00:02:16.563
So this is just an estimate.
It's important to know that.

00:02:16.563 --> 00:02:19.989
But what is interesting is that
these things are being traded.

00:02:19.989 --> 00:02:23.126
These call options are being
traded all of the time,

00:02:23.126 --> 00:02:26.062
and so you could actually look
up the price of this call option.

00:02:26.062 --> 00:02:29.323
You could look up a call
option with this stock price,

00:02:29.323 --> 00:02:30.397
this exercise price.

00:02:30.397 --> 00:02:31.948
You know what these two things are,

00:02:31.948 --> 00:02:35.700
and you could say, "Hey, look.
This traded for $3 just now."

00:02:35.700 --> 00:02:38.985
So you actually can
figure out what this is,

00:02:38.985 --> 00:02:41.727
which raises a very, very
interesting question.

00:02:41.727 --> 00:02:43.674
If you know exactly what this is

00:02:43.674 --> 00:02:46.945
because you know what the
market is pricing this at,

00:02:46.945 --> 00:02:47.947
so let me write this.

00:02:47.947 --> 00:02:51.700
You know what the market
believes this price should be,

00:02:51.700 --> 00:02:55.017
so the market belief,

00:02:55.017 --> 00:02:57.217
and it's based on their
actual transcations,

00:02:57.217 --> 00:02:58.665
so it's based on transactions.

00:02:58.665 --> 00:03:01.665
This is what the market is
saying the correct price is.

00:03:01.665 --> 00:03:03.725
You can figure that out,
you can just look that up.

00:03:03.725 --> 00:03:05.687
You can figure out all
of this other stuff.

00:03:05.687 --> 00:03:09.020
Can you then take this
output plus all of these

00:03:09.020 --> 00:03:12.351
to work backwards through
Black-Scholes to figure out

00:03:12.351 --> 00:03:15.101
what the market is guessing about this,

00:03:15.101 --> 00:03:18.189
or what the market is
estimating about that.

00:03:18.189 --> 00:03:20.724
The answer is yes.

00:03:20.724 --> 00:03:23.615
This is where this whole idea

00:03:23.615 --> 00:03:26.681
about when people talk
about what is the volatility

00:03:26.681 --> 00:03:29.487
in the market, or even where
are carton volatility rates,

00:03:29.487 --> 00:03:32.541
or even what does the market
expect volatility to be?

00:03:32.541 --> 00:03:35.771
How do we know what the market
expects volatility to be?

00:03:35.771 --> 00:03:39.294
Well, we can look at what
markets are trading options at.

00:03:39.294 --> 00:03:41.289
We could look at all of
this other information

00:03:41.289 --> 00:03:43.263
that would be inputted into
Black-Scholes equation,

00:03:43.263 --> 00:03:44.877
and we can say, "Hey,
look. Based on the fact

00:03:44.877 --> 00:03:47.755
"that the market is
paying $5 for this option,

00:03:47.755 --> 00:03:49.132
"and all of these other variables,

00:03:49.147 --> 00:03:52.215
"they must assume that the
standard deviation of log returns

00:03:52.215 --> 00:03:54.539
"for this security is now this."

00:03:54.539 --> 00:03:56.972
Now, let's say that
things get really scary.

00:03:56.972 --> 00:03:59.495
The market becomes a
lot dicier and choppier.

00:03:59.495 --> 00:04:01.676
Well then, people are gonna
pay more for this option.

00:04:01.676 --> 00:04:05.607
It's gonna drive the
implied volatility up.

00:04:05.607 --> 00:04:08.578
So when you hear people talk
about implied volatility,

00:04:08.578 --> 00:04:10.542
or implied vol, and there are even people

00:04:10.542 --> 00:04:15.850
who will actually trade
on implied volatility,

00:04:16.345 --> 00:04:18.260
This is what they're talking about.

00:04:18.260 --> 00:04:21.003
They're saying, "Look. Options
are trading all the time."

00:04:21.003 --> 00:04:23.934
Can we use that price, the market belief

00:04:23.934 --> 00:04:25.996
of what those prices should be,

00:04:25.996 --> 00:04:28.868
and then work backwards
through Black-Scholes

00:04:28.868 --> 00:04:30.994
to figure out, because we
know these are all facts.

00:04:30.994 --> 00:04:32.748
We can look these things up,

00:04:32.748 --> 00:04:35.494
but based on what the market
is trading these options at,

00:04:35.494 --> 00:04:38.931
can we figure out what
the implied volatility,

00:04:38.931 --> 00:04:42.599
what the implied market
belief about volatility

00:04:42.599 --> 00:04:44.001
for that security is,

00:04:44.001 --> 00:04:45.421
and then we can actually aggregate it

00:04:45.421 --> 00:04:46.861
across many, many securities,

00:04:46.861 --> 00:04:48.673
and come up with an implied volatility

00:04:48.673 --> 00:04:51.743
for given markets at a time.

00:04:51.743 --> 00:04:54.001
So it's a very, very,
very interesting idea,

00:04:54.001 --> 00:04:56.528
but in some levels, it's
kind of a basic one.

00:04:56.528 --> 00:04:59.420
You're just working backwards
through Black-Scholes.