Defining a function at a point to make it continuous

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:04.960
The function, f of x is equal
to 6x squared plus 18x plus 12

00:00:04.960 --> 00:00:08.039
over x squared minus
4, is not defined

00:00:08.039 --> 00:00:11.010
at x is equal to
positive or negative 2.

00:00:11.010 --> 00:00:14.395
And we see why that is, if x is
equal to positive or negative 2

00:00:14.395 --> 00:00:17.400
then x squared is going
to be equal to positive 4,

00:00:17.400 --> 00:00:19.419
and 4 minus 4 is
0, and then we're

00:00:19.419 --> 00:00:20.960
going to have a 0
in the denominator.

00:00:20.960 --> 00:00:22.100
And that's not defined.

00:00:22.100 --> 00:00:24.600
We don't know what that happens
when you divide-- well we've

00:00:24.600 --> 00:00:27.210
never defined what happens
when you divide by 0.

00:00:27.210 --> 00:00:31.100
So they say, what value should
be assigned to f of negative 2

00:00:31.100 --> 00:00:34.766
to make f of x
continuous at that point?

00:00:34.766 --> 00:00:36.140
So to think about
that, let's try

00:00:36.140 --> 00:00:38.530
to actually simplify f of x.

00:00:38.530 --> 00:00:42.220
So f of x-- I'll just rewrite
it-- is equal to-- Actually let

00:00:42.220 --> 00:00:44.490
me just start simplifying
right from the get go.

00:00:44.490 --> 00:00:46.640
So in the numerator
I can factor out a 6

00:00:46.640 --> 00:00:48.480
out of every one of those terms.

00:00:48.480 --> 00:00:52.810
So it's 6 times
x squared plus 3x

00:00:52.810 --> 00:00:56.360
plus 2 over-- and
the denominator, this

00:00:56.360 --> 00:00:57.690
is a difference of squares.

00:00:57.690 --> 00:01:03.200
This is x plus 2
times x minus 2.

00:01:03.200 --> 00:01:05.340
And then we can factor
this expression up here.

00:01:05.340 --> 00:01:07.860
So this is going to
be equal to 6 times--

00:01:07.860 --> 00:01:09.530
let me do it a different color.

00:01:09.530 --> 00:01:12.120
So we think of two numbers
and if I take their product

00:01:12.120 --> 00:01:12.750
I get 2.

00:01:12.750 --> 00:01:14.530
If I take their sum I get 3.

00:01:14.530 --> 00:01:16.570
The most obvious one is 2 and 1.

00:01:16.570 --> 00:01:22.287
So this is 6 times x
plus 2 times x plus 1.

00:01:22.287 --> 00:01:23.870
When you take the
product there you'll

00:01:23.870 --> 00:01:28.010
get x squared plus 3x
plus 2, and then all

00:01:28.010 --> 00:01:33.000
of that over x plus
2 times x minus 2.

00:01:33.000 --> 00:01:40.120
Now, if we know that x
does not equal negative 2.

00:01:40.120 --> 00:01:43.210
Then we can divide both the
numerator and the denominator

00:01:43.210 --> 00:01:44.530
by x plus 2.

00:01:44.530 --> 00:01:46.630
The reason why I'm
making that constraint

00:01:46.630 --> 00:01:49.410
is that if x were to
be equal to negative 2

00:01:49.410 --> 00:01:51.695
then x plus 2 is going
to be equal to 0.

00:01:51.695 --> 00:01:53.070
And you won't be
able to do that.

00:01:53.070 --> 00:01:53.570
You can't.

00:01:53.570 --> 00:01:56.250
We don't know what it means
divide something by 0.

00:01:56.250 --> 00:01:57.760
So we could say
that this is going

00:01:57.760 --> 00:02:01.390
to be equal to-- so we
can divide the numerator

00:02:01.390 --> 00:02:04.950
and denominator by x plus 2 but
we have to assume that x is not

00:02:04.950 --> 00:02:06.290
equal to negative 2.

00:02:06.290 --> 00:02:13.980
So this is equal to 6 times--
we're going divided by x plus 2

00:02:13.980 --> 00:02:16.290
in the numerator, x plus
2 in the denominator--

00:02:16.290 --> 00:02:22.460
so it's going to be 6 times
x plus 1 over x minus 2.

00:02:25.520 --> 00:02:27.220
And we have to put
the constraint here

00:02:27.220 --> 00:02:28.428
because now we've changed it.

00:02:28.428 --> 00:02:30.600
Now this expression
over here is actually

00:02:30.600 --> 00:02:32.630
defined at x equals negative 2.

00:02:32.630 --> 00:02:34.970
But in order to be equivalent
to the original function

00:02:34.970 --> 00:02:37.370
we have to constrain it.

00:02:37.370 --> 00:02:45.041
So we will say for x
not equal to negative 2.

00:02:45.041 --> 00:02:47.290
And it's also obvious that
x can't be equal to 2 here.

00:02:47.290 --> 00:02:49.590
This one also isn't
defined at positive 2

00:02:49.590 --> 00:02:51.030
because you're dividing by 0.

00:02:51.030 --> 00:02:54.450
So you could say, for x does not
equal to positive or negative

00:02:54.450 --> 00:02:57.080
2 if you want to make
it very explicit.

00:02:57.080 --> 00:03:00.400
But they ask us, what could
we assign f of negative 2

00:03:00.400 --> 00:03:03.030
to make the function
continuous at the point?

00:03:03.030 --> 00:03:07.020
Well the function is completely
equivalent to this expression

00:03:07.020 --> 00:03:12.420
except that the function is not
defined at x equals negative 2.

00:03:12.420 --> 00:03:15.230
So that's why we have to
put that constraint here

00:03:15.230 --> 00:03:19.510
if we wanted this to be the same
thing as our original function.

00:03:19.510 --> 00:03:23.530
But if we wanted to
re-engineer the function so it

00:03:23.530 --> 00:03:26.320
is continuous at that
point then we just

00:03:26.320 --> 00:03:29.270
have to set f of x equal to
whatever this expression would

00:03:29.270 --> 00:03:31.710
have been when x is
equal to negative 2.

00:03:31.710 --> 00:03:33.460
So let's think about that.

00:03:33.460 --> 00:03:34.470
Let's think about that.

00:03:34.470 --> 00:03:43.400
So 6 times negative 2 plus
1 over negative 2 minus 2

00:03:43.400 --> 00:03:47.750
is equal to-- this is
6 times negative 1.

00:03:47.750 --> 00:03:54.650
So it's negative 6 over negative
4, which is equal to 3/2.

00:03:54.650 --> 00:04:06.210
So if we redefine f of x, if
we say f of x is equal to 6x

00:04:06.210 --> 00:04:13.960
squared plus 18x plus 12
over x squared minus 4.

00:04:13.960 --> 00:04:20.870
For x not equal
positive or negative

00:04:20.870 --> 00:04:28.630
2, and it's equal to 3/2
for x equals negative 2.

00:04:28.630 --> 00:04:31.740
Now this function is going
to be the exact same thing

00:04:31.740 --> 00:04:32.740
as this right over here.

00:04:32.740 --> 00:04:34.370
This f of x, this new one.

00:04:34.370 --> 00:04:36.380
This new definition--
this extended definition

00:04:36.380 --> 00:04:39.880
of our original one-- is now
equivalent to this expression,

00:04:39.880 --> 00:04:45.140
is equal to 6 times x
plus 1 over x minus 2.

00:04:45.140 --> 00:04:46.670
But just to answer
their question,

00:04:46.670 --> 00:04:48.910
what value should be
assigned to f of negative 2

00:04:48.910 --> 00:04:51.010
to make f of x
continuous to that point?

00:04:51.010 --> 00:04:55.855
Well f of x should be-- or f
of negative 2 should be 3/2.


Fancy algebra to find a limit and make a function continuous

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:03.510
Let f be the function
given by f of x

00:00:03.510 --> 00:00:07.500
is equal to the square root of
x plus 4 minus 3 over x minus 5.

00:00:07.500 --> 00:00:12.640
If x does not equal 5, and
it's equal to c if x equals 5.

00:00:12.640 --> 00:00:15.630
Then say, if f is
continuous at x equals 5,

00:00:15.630 --> 00:00:17.630
what is the value of c?

00:00:17.630 --> 00:00:21.060
So if we know that f is
continuous at x equals 5,

00:00:21.060 --> 00:00:27.670
that means that the limit
as x approaches 5 of f of x

00:00:27.670 --> 00:00:30.390
is equal to f of 5.

00:00:30.390 --> 00:00:32.870
This is the definition
of continuity.

00:00:32.870 --> 00:00:34.590
And they tell us
that f of 5, when

00:00:34.590 --> 00:00:36.970
x equals 5, the value of
the function is equal to c.

00:00:36.970 --> 00:00:38.917
So this must be equal to c.

00:00:38.917 --> 00:00:40.500
So what we really
need to do is figure

00:00:40.500 --> 00:00:45.600
out what the limit of f of x
as x approaches 5 actually is.

00:00:45.600 --> 00:00:47.810
Now, if we just
try to substitute 5

00:00:47.810 --> 00:00:51.150
into the expression right
up here, in the numerator

00:00:51.150 --> 00:00:53.120
you have 5 plus 4 is 9.

00:00:53.120 --> 00:00:55.020
The square root of
that is positive 3,

00:00:55.020 --> 00:00:57.220
the principal root
is positive 3.

00:00:57.220 --> 00:00:58.550
3 minus 3 is 0.

00:00:58.550 --> 00:01:00.170
So you get a 0 in the numerator.

00:01:00.170 --> 00:01:02.520
And then you get 5 minus
5 in the denominator,

00:01:02.520 --> 00:01:03.940
so you get 0 in the denominator.

00:01:03.940 --> 00:01:07.507
So you get this
indeterminate form of 0/0.

00:01:07.507 --> 00:01:09.090
And in the future,
we will see that we

00:01:09.090 --> 00:01:12.540
do have a tool that allows
us, or gives us an option

00:01:12.540 --> 00:01:14.050
to attempt to find
the limits when

00:01:14.050 --> 00:01:15.400
we get this indeterminate form.

00:01:15.400 --> 00:01:17.080
It's called L'Hopital's rule.

00:01:17.080 --> 00:01:19.370
But we can actually tackle
this with a little bit

00:01:19.370 --> 00:01:20.920
of fancy algebra.

00:01:20.920 --> 00:01:22.840
And to do that, I'm
going to try to get

00:01:22.840 --> 00:01:25.990
this radical out
of the numerator.

00:01:25.990 --> 00:01:27.050
So let's rewrite it.

00:01:27.050 --> 00:01:36.090
So we have the square root of x
plus 4 minus 3 over x minus 5.

00:01:36.090 --> 00:01:39.622
And any time you see a radical
plus or minus something else,

00:01:39.622 --> 00:01:41.080
to get rid of the
radical, what you

00:01:41.080 --> 00:01:44.220
can do is multiply
by the radical--

00:01:44.220 --> 00:01:46.260
or, if you have a
radical minus 3,

00:01:46.260 --> 00:01:48.810
you multiply by
the radical plus 3.

00:01:48.810 --> 00:01:50.740
So in this situation,
you just multiply

00:01:50.740 --> 00:01:54.940
the numerator by
square root of x plus 4

00:01:54.940 --> 00:02:00.830
plus 3 over the square
root of x plus 4 plus 3.

00:02:00.830 --> 00:02:02.900
We obviously have to
multiply the numerator

00:02:02.900 --> 00:02:04.640
and the denominator
by the same thing

00:02:04.640 --> 00:02:08.729
so that we actually don't change
the value of the expression.

00:02:08.729 --> 00:02:10.560
If this right over
here had a plus 3,

00:02:10.560 --> 00:02:12.020
then we would do a minus 3 here.

00:02:12.020 --> 00:02:14.540
This is a technique that we
learn in algebra, or sometimes

00:02:14.540 --> 00:02:17.420
in pre-calculus class,
to rationalize usually

00:02:17.420 --> 00:02:20.300
denominators, but to rationalize
numerators or denominators.

00:02:20.300 --> 00:02:21.750
It's also a very
similar technique

00:02:21.750 --> 00:02:24.690
that we use often times to
get rid of complex numbers,

00:02:24.690 --> 00:02:27.319
usually in denominators.

00:02:27.319 --> 00:02:29.860
But if you multiply this out--
and I encourage you to do it--

00:02:29.860 --> 00:02:31.526
you notice this has
the pattern that you

00:02:31.526 --> 00:02:32.850
learned in algebra class.

00:02:32.850 --> 00:02:34.190
It's a difference of squares.

00:02:34.190 --> 00:02:37.500
Something minus something
times something plus something.

00:02:37.500 --> 00:02:41.460
So the first term is going to
be the first something squared.

00:02:41.460 --> 00:02:46.160
So square root of x plus
4 squared is x plus 4.

00:02:46.160 --> 00:02:49.530
And the second term is going
to be the second something,

00:02:49.530 --> 00:02:52.020
or you're going to subtract
the second something squared.

00:02:52.020 --> 00:02:55.940
So you're going to have
minus 3 squared, so minus 9.

00:02:55.940 --> 00:02:57.700
And in the denominator,
you're of course

00:02:57.700 --> 00:03:04.520
going to have x minus 5
times the square root of x

00:03:04.520 --> 00:03:07.930
plus 4 plus 3.

00:03:07.930 --> 00:03:11.270
And so this has-- I guess
you could say simplified to,

00:03:11.270 --> 00:03:13.010
although it's not
arguably any simpler.

00:03:13.010 --> 00:03:15.024
But at least we have
gotten our radical.

00:03:15.024 --> 00:03:16.690
We're really just
playing around with it

00:03:16.690 --> 00:03:20.020
algebraically to see if we
can then substitute x equals 5

00:03:20.020 --> 00:03:21.970
or if we can somehow
simplify it to figure out

00:03:21.970 --> 00:03:23.090
what the limit is.

00:03:23.090 --> 00:03:25.700
And when you simplify
the numerator up here,

00:03:25.700 --> 00:03:27.650
you get x plus 4 minus 9.

00:03:27.650 --> 00:03:35.570
Well, that's x minus 5 over x
minus 5 times the square root

00:03:35.570 --> 00:03:39.750
of x plus 4 plus 3.

00:03:39.750 --> 00:03:42.260
And now it pops out at
you, both the numerator

00:03:42.260 --> 00:03:46.910
and the denominator are
now divisible by x minus 5.

00:03:46.910 --> 00:03:49.235
So you can have a completely
identical expression

00:03:49.235 --> 00:03:50.860
if you say that this
is the same thing.

00:03:50.860 --> 00:03:53.670
You can divide the numerator
and the denominator by x minus 5

00:03:53.670 --> 00:03:55.920
if you assume x
does not equal 5.

00:03:55.920 --> 00:03:58.990
So this is going to
be the same thing as 1

00:03:58.990 --> 00:04:10.190
over square root of x plus 4
plus 3 for x does not equal 5.

00:04:10.190 --> 00:04:13.380
Which is fine, because in the
first part of this function

00:04:13.380 --> 00:04:16.480
definition, this is the
case for x does not equal 5.

00:04:16.480 --> 00:04:18.060
So we could actually
replace this--

00:04:18.060 --> 00:04:21.149
and this is a simpler
expression-- with 1

00:04:21.149 --> 00:04:26.500
over square root
of x plus 4 plus 3.

00:04:26.500 --> 00:04:28.667
And so now when we take the
limit as x approaches 5,

00:04:28.667 --> 00:04:30.541
we're going to get closer
and closer to five.

00:04:30.541 --> 00:04:32.800
We're going to get x values
closer and closer to 5,

00:04:32.800 --> 00:04:34.250
but not quite at 5.

00:04:34.250 --> 00:04:36.490
We can use this expression
right over here.

00:04:36.490 --> 00:04:39.310
So the limit of f of
x as x approaches 5

00:04:39.310 --> 00:04:42.923
is going to be the same
thing as the limit of 1

00:04:42.923 --> 00:04:49.690
over the square root of x plus
4 plus 3 as x approaches 5.

00:04:49.690 --> 00:04:52.220
And now we can
substitute a 5 in here.

00:04:52.220 --> 00:04:56.310
It's going to be
1 over 5 plus 4 is

00:04:56.310 --> 00:04:58.840
9, principal root of that is 3.

00:04:58.840 --> 00:05:01.030
3 plus 3 is 6.

00:05:01.030 --> 00:05:08.930
So if c is equal to 1/6, then
the limit of our function

00:05:08.930 --> 00:05:13.800
as x approaches 5 is going
to be equal to f of 5.

00:05:13.800 --> 00:05:17.340
And we are continuous
at x equals 5.

00:05:17.340 --> 00:05:19.331
So it's 1/6.


Limit and function defined at point of discontinuity

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:02.920
The graph of a function
f is shown below.

00:00:02.920 --> 00:00:08.290
If both the limit of f of x
as x approaches k and f of k

00:00:08.290 --> 00:00:12.280
exist, and f is not
continuous at k, then

00:00:12.280 --> 00:00:14.040
what is the value of k?

00:00:14.040 --> 00:00:16.950
So we have to find a k where
f is not continuous at k,

00:00:16.950 --> 00:00:18.990
but f of k is defined.

00:00:18.990 --> 00:00:23.330
And the limit as x approaches
k of f of x is also defined.

00:00:23.330 --> 00:00:26.000
So the easiest ones to
spot out just with our eyes

00:00:26.000 --> 00:00:29.420
might just to be to see
where f is not continuous,

00:00:29.420 --> 00:00:32.250
where f is not continuous.

00:00:32.250 --> 00:00:37.820
So you see here when x
is equal to negative 2,

00:00:37.820 --> 00:00:39.400
the function is not continuous.

00:00:39.400 --> 00:00:40.740
It jumps from up here.

00:00:40.740 --> 00:00:43.230
It looks like it's
approaching 3.

00:00:43.230 --> 00:00:44.790
And then it jumps
down to negative 3.

00:00:44.790 --> 00:00:46.670
So this is one of
our candidates.

00:00:46.670 --> 00:00:48.930
This is one of our
candidates for k.

00:00:48.930 --> 00:00:52.321
The other discontinuity
happens when x is equal to 3.

00:00:52.321 --> 00:00:54.570
Once again, we jump down
from-- looks like 4 and a 1/2

00:00:54.570 --> 00:00:56.120
all the way to negative 4.

00:00:56.120 --> 00:00:57.850
So that's another candidate.

00:00:57.850 --> 00:01:00.650
That's a point where
f is not continuous.

00:01:00.650 --> 00:01:05.620
And then we have when x is
equal to 8, we have this.

00:01:05.620 --> 00:01:07.870
As we approach this, it looks
like we're getting to 1.

00:01:07.870 --> 00:01:10.120
But then we jump
right at x equals 8.

00:01:10.120 --> 00:01:12.640
And then we continue
from 1 again.

00:01:12.640 --> 00:01:14.305
So this is our other candidate.

00:01:14.305 --> 00:01:15.680
So these are the
three candidates

00:01:15.680 --> 00:01:17.990
where the function
is not continuous.

00:01:17.990 --> 00:01:19.840
Now let's think about
which of these points,

00:01:19.840 --> 00:01:23.060
which of these x values,
does f of k exist.

00:01:23.060 --> 00:01:26.100
So if one of these is
k, does f of k exist?

00:01:26.100 --> 00:01:28.360
Well f of negative 2 exists.

00:01:28.360 --> 00:01:30.780
f of 3 exists, right over here.

00:01:30.780 --> 00:01:31.740
That's f of 3.

00:01:31.740 --> 00:01:38.290
This is f of negative 2.

00:01:38.290 --> 00:01:40.130
And f of 8, all exist.

00:01:40.130 --> 00:01:44.430
So all of these potential k's
meet this constraint-- f of k

00:01:44.430 --> 00:01:47.420
exists, and f is
not continuous at k.

00:01:47.420 --> 00:01:50.640
So that's true for x
equals 8, 3, or negative 2.

00:01:50.640 --> 00:01:52.930
Now let's look at
this first constraint.

00:01:52.930 --> 00:01:59.010
The limit of f of x as x
approaches k needs to exist.

00:01:59.010 --> 00:02:02.350
Well, if we tried to look at
x equals 2, the limit of f

00:02:02.350 --> 00:02:05.920
of x as x approaches
negative 2-- not 2,

00:02:05.920 --> 00:02:08.160
as x approaches
negative 2 here--

00:02:08.160 --> 00:02:11.230
the limit from the left,
the limit from values

00:02:11.230 --> 00:02:13.380
lower than negative 2, it
looks like our function

00:02:13.380 --> 00:02:15.545
is approaching something
a little higher.

00:02:15.545 --> 00:02:17.587
It looks like it's a
little higher than 3.

00:02:17.587 --> 00:02:19.170
And the limit from
the right, it looks

00:02:19.170 --> 00:02:22.030
like our function is
approaching negative 3.

00:02:22.030 --> 00:02:23.910
So this one, the
limit does not exist.

00:02:23.910 --> 00:02:26.750
You get a different limit from
the left and from the right.

00:02:26.750 --> 00:02:29.580
Same thing for x
equals positive 3.

00:02:29.580 --> 00:02:31.510
The limit from the
left seems like it's

00:02:31.510 --> 00:02:34.110
approaching 4 and 1/2, while
the limit from the right

00:02:34.110 --> 00:02:36.050
looks like it's
approaching negative 4.

00:02:36.050 --> 00:02:38.070
So this is also not a candidate.

00:02:38.070 --> 00:02:39.390
So we only have one left.

00:02:39.390 --> 00:02:41.970
So for this one, the
limit should exist.

00:02:41.970 --> 00:02:49.820
And we see the limit as
f of x as x approaches 8

00:02:49.820 --> 00:02:55.210
from the negative
direction, it looks

00:02:55.210 --> 00:02:59.020
like f of x is approaching 1.

00:02:59.020 --> 00:03:00.800
And it looks like,
as we approach 8

00:03:00.800 --> 00:03:08.560
from the positive
direction, the limit of f

00:03:08.560 --> 00:03:12.060
of x as x approaches 8 from
the positive direction.

00:03:12.060 --> 00:03:13.410
It's also equal to 1.

00:03:13.410 --> 00:03:15.350
So your left- and your
right-sided limits

00:03:15.350 --> 00:03:16.810
approach the same value.

00:03:16.810 --> 00:03:22.390
So the limit of f of x as x
approaches 8 is equal to 1.

00:03:22.390 --> 00:03:23.930
This limit exists.

00:03:23.930 --> 00:03:26.870
Now, the reason why the
function isn't continuous there

00:03:26.870 --> 00:03:29.680
is that the limit of
f of x as x approaches

00:03:29.680 --> 00:03:35.170
8, which is equal to 1, it does
not equal the value of f of 8.

00:03:35.170 --> 00:03:38.130
f of 8, we're seeing,
is equal to 7.

00:03:38.130 --> 00:03:40.380
So that's why it meets
the last constraint.

00:03:40.380 --> 00:03:42.520
The function is not
continuous there.

00:03:42.520 --> 00:03:43.420
The function exists.

00:03:43.420 --> 00:03:46.100
It's defined, f of
8 is equal to 7.

00:03:46.100 --> 00:03:47.620
And the limit exists.

00:03:47.620 --> 00:03:50.370
But the limit of f of
x as x approaches k

00:03:50.370 --> 00:03:53.460
is not the same thing,
or is not the same

00:03:53.460 --> 00:03:56.990
as the value of the function
evaluated at that point.

00:03:56.990 --> 00:04:00.960
And so x equals 8 meets
all of our constraints.

00:04:00.960 --> 00:04:05.060
So we could say k is equal to 8.


Overview of Chinese history 1911 - 1949

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:02.250
For those of you who
are just starting

00:00:02.250 --> 00:00:04.700
to learn about the
history of China

00:00:04.700 --> 00:00:07.000
in the first half
of the 20th century,

00:00:07.000 --> 00:00:09.750
it can be a little
bit confusing.

00:00:09.750 --> 00:00:13.040
So the goal of this video is
really to give you an overview,

00:00:13.040 --> 00:00:17.070
to give you a scaffold, of
the history of the first half

00:00:17.070 --> 00:00:19.980
of the 20th century in China.

00:00:19.980 --> 00:00:22.680
So as we go into
the early 1900s,

00:00:22.680 --> 00:00:27.390
you have the end of imperial
dynastic rule in China.

00:00:27.390 --> 00:00:28.380
This is a big deal.

00:00:28.380 --> 00:00:31.510
China has been ruled
by various dynasties

00:00:31.510 --> 00:00:34.670
for multiple thousands of years.

00:00:34.670 --> 00:00:38.340
But as you get into the
1900s, the dynastic rule,

00:00:38.340 --> 00:00:43.190
in particular the Qing Dynasty,
was getting weaker and weaker.

00:00:43.190 --> 00:00:45.670
It had suffered at the
hands of the Japanese

00:00:45.670 --> 00:00:50.310
during the first Sino-Japanese
War at the end of the 1800s.

00:00:50.310 --> 00:00:53.290
There was growing discontent
amongst the opposition

00:00:53.290 --> 00:00:55.400
that the dynasty,
that the emperors,

00:00:55.400 --> 00:00:57.440
were not modernizing
China enough.

00:00:57.440 --> 00:00:59.121
Remember, this is
the early 1900s.

00:00:59.121 --> 00:01:01.370
The rest of the world was
becoming a very, very modern

00:01:01.370 --> 00:01:02.110
place.

00:01:02.110 --> 00:01:06.380
China in the 1800s had suffered
at the hands of Western powers

00:01:06.380 --> 00:01:10.220
who were essentially exerting
their own imperial influence

00:01:10.220 --> 00:01:11.210
in China.

00:01:11.210 --> 00:01:14.480
Many people felt that this
was because China was not

00:01:14.480 --> 00:01:17.950
as modernized
economically, politically,

00:01:17.950 --> 00:01:20.480
technologically as
it needed to be.

00:01:20.480 --> 00:01:22.860
And so you fast-forward to 1911.

00:01:22.860 --> 00:01:26.220
You have what is known as
the Wuchang Uprising, which

00:01:26.220 --> 00:01:30.400
led to the overthrow
of the Qing dynasty.

00:01:30.400 --> 00:01:35.510
By 1912, a Republic of China
was established in Nanjing.

00:01:35.510 --> 00:01:38.470
So Nanjing right over here
was where it was established.

00:01:38.470 --> 00:01:43.550
Beijing was, of course, the
seat of dynastic rule in China.

00:01:43.550 --> 00:01:45.870
And the first
provisional president

00:01:45.870 --> 00:01:51.530
of the Republic of China was Dr.
Sun Yat-sen, right over here.

00:01:51.530 --> 00:01:53.700
And he actually did not
directly participate

00:01:53.700 --> 00:01:55.970
in this final
uprising that finally

00:01:55.970 --> 00:01:59.460
led to the overthrow
of the Qing dynasty.

00:01:59.460 --> 00:02:02.660
He was actually in Denver at
the time, Denver, Colorado.

00:02:02.660 --> 00:02:05.810
But he was a leading or one of
the leading figures in the run

00:02:05.810 --> 00:02:07.830
up to this uprising, one
of the leading figures

00:02:07.830 --> 00:02:10.449
who was providing
opposition and had tried

00:02:10.449 --> 00:02:13.940
multiple times to
overthrow the dynasty.

00:02:13.940 --> 00:02:17.090
Now along with Sun Yat-sen,
he was essentially in cahoots

00:02:17.090 --> 00:02:21.820
with Yuan Shikai, who was a
general in the old dynasty.

00:02:21.820 --> 00:02:24.270
And he has his own
fascinating history.

00:02:24.270 --> 00:02:27.600
And Sun Yat-sen struck
a deal with Yuan Shikai,

00:02:27.600 --> 00:02:30.250
who was very
politically ambitious.

00:02:30.250 --> 00:02:33.830
Yuan Shikai said, hey, if I
can get the emperor Puyi, who

00:02:33.830 --> 00:02:37.230
was the last emperor of China,
if I can get him to officially

00:02:37.230 --> 00:02:40.740
abdicate, I want to
become the president.

00:02:40.740 --> 00:02:43.320
So Sun Yat-sen agrees to this.

00:02:43.320 --> 00:02:48.050
So Yuan Shikai
becomes the president

00:02:48.050 --> 00:02:50.880
of the Republic of China.

00:02:50.880 --> 00:02:52.450
But that wasn't enough for him.

00:02:52.450 --> 00:02:56.030
He declares himself
emperor in 1915,

00:02:56.030 --> 00:02:58.390
which you could imagine did
not make many people happy

00:02:58.390 --> 00:03:01.370
because they were tired
of having emperors.

00:03:01.370 --> 00:03:06.990
And by 1916, he abdicates
and he passes away, actually.

00:03:06.990 --> 00:03:09.420
And this actually
begins a period

00:03:09.420 --> 00:03:13.510
of extremely fragmented
rule for China.

00:03:13.510 --> 00:03:16.440
Even under imperial rule,
the Chinese military

00:03:16.440 --> 00:03:18.240
was not one consolidated body.

00:03:18.240 --> 00:03:21.640
The military was controlled
by various warlords

00:03:21.640 --> 00:03:25.590
in various regions that all
had allegiance to the emperor.

00:03:25.590 --> 00:03:29.630
Once you have Yuan Shikai
abdicating and then dying

00:03:29.630 --> 00:03:31.960
in 1916, and even
prior to that, when

00:03:31.960 --> 00:03:33.980
he declared himself
emperor, people

00:03:33.980 --> 00:03:37.220
did not want to pledge
allegiance to Yuan Shikai.

00:03:37.220 --> 00:03:40.460
And so you had what is known
as the beginning of the Warlord

00:03:40.460 --> 00:03:42.130
Era in China.

00:03:42.130 --> 00:03:44.790
And this is a fragmented
period where you did not

00:03:44.790 --> 00:03:47.850
have any centralized leadership.

00:03:47.850 --> 00:03:51.310
This map over here shows
kind of the rough picture

00:03:51.310 --> 00:03:53.040
of what the Warlord
Era looked like.

00:03:53.040 --> 00:03:55.010
Each of these regions
were controlled

00:03:55.010 --> 00:03:57.390
by a different warlord
who was in charge

00:03:57.390 --> 00:03:59.850
of a different military.

00:03:59.850 --> 00:04:02.440
When this was going on
during the Warlord Era,

00:04:02.440 --> 00:04:05.680
especially as we go back
to the early '20s, in 1921

00:04:05.680 --> 00:04:09.120
in particular, Sun
Yat-sen hasn't given up.

00:04:09.120 --> 00:04:12.340
He goes to the
south in Guangzhou

00:04:12.340 --> 00:04:16.510
and sets up, essentially,
a revolutionary government,

00:04:16.510 --> 00:04:19.820
essentially a desire
from there to try

00:04:19.820 --> 00:04:23.810
to consolidate power in
China again and reestablish

00:04:23.810 --> 00:04:25.970
the Republic of China.

00:04:25.970 --> 00:04:27.540
So he goes there.

00:04:27.540 --> 00:04:32.440
But unfortunately he passes
away in 1925 from cancer.

00:04:32.440 --> 00:04:38.790
And the hands or the power of
the movement that he started,

00:04:38.790 --> 00:04:42.710
which is now being referred
to as the Kuomintang-- Let me

00:04:42.710 --> 00:04:43.550
write that down.

00:04:53.790 --> 00:04:56.150
Essentially, the
power there passes on

00:04:56.150 --> 00:05:00.042
to Generalissimo
Chiang Kai-shek.

00:05:00.042 --> 00:05:01.500
And Chiang Kai-shek,
the reason why

00:05:01.500 --> 00:05:03.166
we say the power
essentially goes to him

00:05:03.166 --> 00:05:05.530
is because he was in
control of the major part

00:05:05.530 --> 00:05:09.540
of the military forces
of the Kuomintang.

00:05:09.540 --> 00:05:13.040
And this is essentially the
very nascent early stages

00:05:13.040 --> 00:05:15.050
of what would essentially
be the Chinese Civil

00:05:15.050 --> 00:05:22.890
War because in the period from
1921 until Sun Yat-sen's death,

00:05:22.890 --> 00:05:24.990
you actually had a
lot of collaboration

00:05:24.990 --> 00:05:27.590
between the Chinese
nationalists, the Kuomintang,

00:05:27.590 --> 00:05:31.540
and the Soviet Union, and
the Chinese Communist Party.

00:05:31.540 --> 00:05:33.910
They were trying to
collaborate in order

00:05:33.910 --> 00:05:36.810
to think about how
China would unify.

00:05:36.810 --> 00:05:41.830
But then once Sun Yat-sen dies
and the power of the Kuomintang

00:05:41.830 --> 00:05:44.890
essentially goes into the
hands of Generalissimo Chiang

00:05:44.890 --> 00:05:48.080
Kai-shek, he starts
to consolidate power.

00:05:48.080 --> 00:05:50.070
And right from the
get-go, he doesn't

00:05:50.070 --> 00:05:51.810
antagonize the communists.

00:05:51.810 --> 00:05:56.010
But by 1927, he's
starting to consolidate,

00:05:56.010 --> 00:06:01.840
he's starting to merge
these various factions

00:06:01.840 --> 00:06:02.970
in the rest of China.

00:06:02.970 --> 00:06:04.900
So he's able to
consolidate power.

00:06:04.900 --> 00:06:08.470
But he also starts to
go after the communists.

00:06:08.470 --> 00:06:11.976
So Chiang Kai-shek,
by '27, also starts

00:06:11.976 --> 00:06:13.100
to go after the communists.

00:06:13.100 --> 00:06:14.683
And the communists
are saying, hey, we

00:06:14.683 --> 00:06:18.110
are the ones that really
represent the spirit of what

00:06:18.110 --> 00:06:21.375
Sun Yat-sen represented,
while the Kuomintang

00:06:21.375 --> 00:06:23.750
under the leadership of Chiang
Kai-shek said, no, no, no.

00:06:23.750 --> 00:06:28.060
We represent what Sun Yat-sen
represented when he first

00:06:28.060 --> 00:06:31.330
established the
Republic of China.

00:06:31.330 --> 00:06:36.531
And so in 1927, you have the
beginning of the Chinese Civil

00:06:36.531 --> 00:06:37.030
War.

00:06:37.030 --> 00:06:39.650
This is when the Kuomintang,
as part of its efforts

00:06:39.650 --> 00:06:41.550
to consolidate
power, not only tries

00:06:41.550 --> 00:06:43.350
to consolidate power
of the warlords,

00:06:43.350 --> 00:06:46.670
but also goes after
the Communist Party.

00:06:46.670 --> 00:06:48.990
Now while all of
this is happening,

00:06:48.990 --> 00:06:53.210
as we get into the early
1930s, Japan once again

00:06:53.210 --> 00:06:56.470
is trying to exert its
imperial, its military,

00:06:56.470 --> 00:06:58.860
might on the Chinese mainland.

00:06:58.860 --> 00:07:00.970
They had already
captured Formosa,

00:07:00.970 --> 00:07:03.400
which is now known
as Taiwan, and Korea

00:07:03.400 --> 00:07:07.340
during the first Sino-Japanese
War at the end of the 1800s.

00:07:07.340 --> 00:07:16.250
And then in 1931, the Japanese
start to encroach on Manchuria.

00:07:16.250 --> 00:07:19.860
And this would essentially
become a multi-year occupation

00:07:19.860 --> 00:07:23.960
and infiltration of
Japan into China.

00:07:23.960 --> 00:07:26.790
And this continues all
the way until 1937,

00:07:26.790 --> 00:07:29.350
when it becomes an
official all-out war

00:07:29.350 --> 00:07:32.870
between the Japanese
and the Chinese.

00:07:32.870 --> 00:07:35.130
And I have a map
here that shows kind

00:07:35.130 --> 00:07:40.050
of the maximum Japanese
control over this period.

00:07:40.050 --> 00:07:43.250
And so in east Asia between
the Chinese and the Japanese,

00:07:43.250 --> 00:07:49.440
World War II was really just
part of the Sino-Japanese War.

00:07:49.440 --> 00:07:51.780
The Japanese had
already encroached

00:07:51.780 --> 00:07:56.310
on the mainland of China
well before World War

00:07:56.310 --> 00:07:58.790
II had officially begun.

00:07:58.790 --> 00:08:01.020
Now while all this
is happening, Japan

00:08:01.020 --> 00:08:04.640
is encroaching into Manchuria,
in 1934, you have to remember,

00:08:04.640 --> 00:08:08.370
the Kuomintang, the Nationalist
Party under Chiang Kai-shek

00:08:08.370 --> 00:08:11.160
is going after the communists.

00:08:11.160 --> 00:08:16.560
And in 1934, he almost
has them, or he does.

00:08:16.560 --> 00:08:20.510
The communists are
nearly defeated.

00:08:20.510 --> 00:08:22.810
They're surrounded by
the Nationalist Party.

00:08:22.810 --> 00:08:25.590
And this becomes what
is a fairly famous event

00:08:25.590 --> 00:08:28.790
in Chinese history, the
famous Long March, where

00:08:28.790 --> 00:08:31.410
the Chinese Communist
Party, their military,

00:08:31.410 --> 00:08:34.820
is marched through
extremely tough terrain

00:08:34.820 --> 00:08:38.330
all the way to the
northwest of China.

00:08:38.330 --> 00:08:42.490
So this right over here is
a map of the Long March.

00:08:42.490 --> 00:08:46.060
The Chinese Communist Party
seemed to be on the ropes

00:08:46.060 --> 00:08:48.370
here in 1934.

00:08:48.370 --> 00:08:50.580
And it was during
this Long March

00:08:50.580 --> 00:08:55.670
that Mao Zedong really started
to exert and show leadership.

00:08:55.670 --> 00:08:57.580
The leadership during
this Long March,

00:08:57.580 --> 00:09:00.080
during this retreat to
the northwest of China,

00:09:00.080 --> 00:09:02.950
is really what allowed Mao
Zedong to eventually take

00:09:02.950 --> 00:09:06.180
control of the Chinese
Communist Party.

00:09:06.180 --> 00:09:10.140
Now as we fast forward, we know
that the Sino-Japanese War--

00:09:10.140 --> 00:09:12.700
you could view this as one
theater, eventually, of World

00:09:12.700 --> 00:09:16.360
War II-- eventually the
US goes in on the side

00:09:16.360 --> 00:09:19.200
of the Allies against
Japan after Pearl Harbor.

00:09:19.200 --> 00:09:25.210
And then in 1945,
you have the attacks

00:09:25.210 --> 00:09:27.900
on Hiroshima and Nagasaki
with atomic weapons,

00:09:27.900 --> 00:09:30.640
which essentially ends
the Pacific theater.

00:09:30.640 --> 00:09:36.240
It's defeat for Japan, and
Japan has lost World War II.

00:09:36.240 --> 00:09:38.530
And at this point,
full-scale civil war

00:09:38.530 --> 00:09:40.720
between the two parties
break out again.

00:09:40.720 --> 00:09:45.720
The Civil War started in 1927,
and then it kept continuing.

00:09:45.720 --> 00:09:48.780
But then once there was a common
enemy in Japan that was clearly

00:09:48.780 --> 00:09:53.140
aggressively trying to take
over more and more of China's

00:09:53.140 --> 00:09:56.750
people, resources, exert
its imperial influence,

00:09:56.750 --> 00:09:59.370
then you had the two parties
kind of go into a low-grade war

00:09:59.370 --> 00:10:01.420
and say, hey, we need
to fight these Japanese.

00:10:01.420 --> 00:10:04.880
But once World War
II ended in 1945,

00:10:04.880 --> 00:10:06.670
once the Japanese were
defeated, then you

00:10:06.670 --> 00:10:08.530
had full-scale
civil war break out

00:10:08.530 --> 00:10:10.100
again between the
Chinese Communist

00:10:10.100 --> 00:10:12.260
Party and the Kuomintang.

00:10:12.260 --> 00:10:14.850
And this is probably one of the
biggest comebacks in history.

00:10:14.850 --> 00:10:18.892
This was the Chinese Communist
Party that in 1934 and 1935

00:10:18.892 --> 00:10:20.350
looked like they
were on the ropes.

00:10:20.350 --> 00:10:23.964
They were forced into,
essentially, retreat.

00:10:23.964 --> 00:10:25.130
They were able to come back.

00:10:25.130 --> 00:10:28.480
And in 1949-- and
there's a lot of theories

00:10:28.480 --> 00:10:31.086
as to why they were
able to pull this off.

00:10:31.086 --> 00:10:33.800
That they were able to get
much more of the support

00:10:33.800 --> 00:10:35.090
from the rural population.

00:10:35.090 --> 00:10:37.460
They were more savvy about
getting support generally

00:10:37.460 --> 00:10:38.360
than the Kuomintang.

00:10:38.360 --> 00:10:41.070
But we could talk about
that in a future video.

00:10:41.070 --> 00:10:44.290
But by 1949, they
were able to defeat

00:10:44.290 --> 00:10:46.770
Chiang Kai-shek
and the Kuomintang,

00:10:46.770 --> 00:10:53.770
force the Kuomintang
to retreat to Taiwan,

00:10:53.770 --> 00:10:56.570
establish government in Taiwan.

00:10:56.570 --> 00:10:58.760
And ever since then, you
had the establishment

00:10:58.760 --> 00:11:02.440
by the Chinese
Communist Party in 1949

00:11:02.440 --> 00:11:06.144
of the People's
Republic of China.


FICO scores and credit bureaus

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:02.010
- As long as human beings

00:00:02.010 --> 00:00:04.050
have been lending things
to other human beings,

00:00:04.050 --> 00:00:05.970
there's always been the question

00:00:05.970 --> 00:00:09.302
of how likely is that person
that I'm lending that thing to

00:00:09.302 --> 00:00:11.400
going to give it back?

00:00:11.400 --> 00:00:15.630
So there's always been this
question of credit worthiness.

00:00:15.630 --> 00:00:17.760
Is this person a good credit risk?

00:00:17.760 --> 00:00:22.690
If I lend them money, how
likely are they going to be

00:00:22.690 --> 00:00:25.010
to pay me back?

00:00:25.010 --> 00:00:26.730
Now people have always found ways

00:00:26.730 --> 00:00:28.160
to come up with a solution here.

00:00:28.160 --> 00:00:30.400
How much money do they have?

00:00:30.400 --> 00:00:32.410
What kind of income do they have?

00:00:32.410 --> 00:00:35.687
But in 1956, two gentlemen said,

00:00:35.687 --> 00:00:40.687
"Hey, I think we can apply
the ideas of mathematics,

00:00:40.812 --> 00:00:42.907
"the ideas of statistics,

00:00:42.907 --> 00:00:44.787
"to actually become a
little bit more scientific

00:00:44.787 --> 00:00:47.320
"about how credit worthy someone is."

00:00:47.320 --> 00:00:51.030
and so these two gentlemen
started what is known

00:00:51.030 --> 00:00:54.369
as the Fair Isaac Corporation.

00:00:54.369 --> 00:00:56.680
Fair Isaac Corporation.

00:00:56.680 --> 00:00:58.030
And when I first heard the name,

00:00:58.030 --> 00:01:01.080
I assumed it was started by
a particularly fair Isaac,

00:01:01.080 --> 00:01:03.800
or at least an Isaac who
thought that he was fair,

00:01:03.800 --> 00:01:05.870
so much so that he named it fair Isaac,

00:01:05.870 --> 00:01:08.780
but it's actually named
after the two founders,

00:01:08.780 --> 00:01:12.513
Bill Fair, and Earl Isaac.

00:01:13.510 --> 00:01:16.570
Actually the first one
was Fair, comma, Isaac,

00:01:16.570 --> 00:01:19.700
and Corporation, then changed
to Fair Isaac Corporation.

00:01:19.700 --> 00:01:21.743
And now it's often known as FICO.

00:01:23.120 --> 00:01:24.850
Often known as FICO.

00:01:24.850 --> 00:01:26.160
And what they came up with,

00:01:26.160 --> 00:01:27.430
Bill Fair was an engineer,

00:01:27.430 --> 00:01:29.410
Earl Isaac was a mathematician,

00:01:29.410 --> 00:01:34.410
is an algorithm for computing
how credit worthy someone is.

00:01:34.490 --> 00:01:36.110
So they came up with an algorithm,

00:01:36.110 --> 00:01:39.000
and the algorithm has changed since 1956,

00:01:39.000 --> 00:01:41.360
because clearly, technology has changed,

00:01:41.360 --> 00:01:42.960
and our understanding of statistics,

00:01:42.960 --> 00:01:44.080
and even the world itself

00:01:44.080 --> 00:01:46.610
that we're trying to
understand has changed.

00:01:46.610 --> 00:01:49.485
But they started focusing
on the problem of,

00:01:49.485 --> 00:01:52.115
given some inputs, can I give a score

00:01:52.115 --> 00:01:56.370
that can give you a sense of
someone's credit worthiness.

00:01:56.370 --> 00:01:57.662
So they would take inputs,

00:01:57.662 --> 00:02:02.540
they would take inputs like
total debt, total debt,

00:02:02.540 --> 00:02:04.930
and actually not just total
debt, but available debt.

00:02:04.930 --> 00:02:06.110
Total debt to how much credit

00:02:06.110 --> 00:02:07.950
someone could get access to.

00:02:07.950 --> 00:02:09.930
Things like payment history,

00:02:09.930 --> 00:02:12.900
so have they paid all their debts on time?

00:02:12.900 --> 00:02:15.290
Payment history.

00:02:15.290 --> 00:02:19.090
And other things, like
length of credit history,

00:02:19.090 --> 00:02:20.700
and the types of credit.

00:02:20.700 --> 00:02:25.700
So length, length, and types of credit.

00:02:28.210 --> 00:02:30.080
And other things like how many people

00:02:30.080 --> 00:02:32.080
are inquiring about this person's credit,

00:02:32.080 --> 00:02:32.940
and things like that.

00:02:32.940 --> 00:02:34.780
And then they came up with a score.

00:02:34.780 --> 00:02:37.010
And today, the modern FICO score,

00:02:37.010 --> 00:02:39.150
and the FICO really is the, I would say,

00:02:39.150 --> 00:02:40.790
the dominant type of credit score.

00:02:40.790 --> 00:02:43.140
There are other types of algorithms

00:02:43.140 --> 00:02:43.973
that people use,

00:02:43.973 --> 00:02:46.070
but FICO is still the predominant one.

00:02:46.070 --> 00:02:49.573
It will give you a score
that is between 300 and 850.

00:02:51.810 --> 00:02:56.310
So 300 at the low end,
850 at the high end.

00:02:56.310 --> 00:02:58.860
And the general view is, the general view,

00:02:58.860 --> 00:03:00.660
and I actually talked to
some folks at some banks

00:03:00.660 --> 00:03:02.601
to get their sense of things,

00:03:02.601 --> 00:03:07.601
that anything between
780 and 850 is excellent.

00:03:09.140 --> 00:03:10.600
You have excellent credit,

00:03:10.600 --> 00:03:12.872
you are amongst the best credit risk.

00:03:12.872 --> 00:03:14.870
You are viewed to be very likely

00:03:14.870 --> 00:03:18.190
to pay off whatever loan
you're about to get.

00:03:18.190 --> 00:03:21.760
Anything between 720 and 780

00:03:21.760 --> 00:03:24.170
is considered still above average.

00:03:24.170 --> 00:03:25.970
It's still considered very good.

00:03:25.970 --> 00:03:29.860
And anything between 650, 650.

00:03:29.860 --> 00:03:31.530
We'll do that a different color.

00:03:31.530 --> 00:03:35.050
Anything between 650 and 720

00:03:35.050 --> 00:03:38.090
is considered kind of middle,
or average, or just okay.

00:03:38.090 --> 00:03:42.300
And anything less than 650 is not so good,

00:03:42.300 --> 00:03:44.150
often referred to as sub-prime.

00:03:44.150 --> 00:03:46.180
Prime would be an excellent credit risk,

00:03:46.180 --> 00:03:48.350
or a good credit risk,
or an okay credit risk.

00:03:48.350 --> 00:03:51.360
Sub-prime means okay, this
person's a little bit riskier

00:03:51.360 --> 00:03:52.780
to lend money to.

00:03:52.780 --> 00:03:57.460
So this is not so good.

00:03:57.460 --> 00:04:00.350
Now, the question is,
who actually collects

00:04:00.350 --> 00:04:01.520
all of this information

00:04:01.520 --> 00:04:04.520
in order to apply the FICO algorithm?

00:04:04.520 --> 00:04:07.791
And this is where credit
bureaus come into the picture.

00:04:07.791 --> 00:04:12.020
So FICO, and it's a large
multi-billion dollar company now,

00:04:12.020 --> 00:04:13.680
that does many, many, many things,

00:04:13.680 --> 00:04:18.350
but it's most famous for
devising the FICO algorithm.

00:04:18.350 --> 00:04:20.830
But the way that the credit
scores tend to be calculated

00:04:20.830 --> 00:04:22.920
is by the credit bureaus.

00:04:22.920 --> 00:04:25.043
So the credit bureaus are the folks

00:04:25.043 --> 00:04:30.043
that will actually collect the data,

00:04:30.150 --> 00:04:31.050
collect the data.

00:04:31.050 --> 00:04:33.510
Your total debt, payment
history, and other things,

00:04:33.510 --> 00:04:36.160
apply the FICO algorithm,

00:04:36.160 --> 00:04:39.300
and then give you your
actual credit score.

00:04:39.300 --> 00:04:41.377
And the three major credit bureaus

00:04:41.377 --> 00:04:46.377
are TransUnion, TransUnion.

00:04:46.750 --> 00:04:51.750
Experian, Experian, and Equifax.

00:04:52.780 --> 00:04:54.990
So, if you were about
to lend someone money,

00:04:54.990 --> 00:04:56.750
you could go to one of
these credit bureaus,

00:04:56.750 --> 00:04:59.290
and you could do an inquiry
into their credit score.

00:04:59.290 --> 00:05:01.110
And actually, you,
yourself, as an individual

00:05:01.110 --> 00:05:03.900
can go and inquire about your credit score

00:05:03.900 --> 00:05:05.830
with each of these bureaus.

00:05:05.830 --> 00:05:10.100
And they offer different
types of algorithms,

00:05:10.100 --> 00:05:12.730
but the FICO one is the most used one.

00:05:12.730 --> 00:05:14.040
So you could get a FICO score

00:05:14.040 --> 00:05:15.910
from each of these credit unions.

00:05:15.910 --> 00:05:17.350
And the score might be
a little bit different,

00:05:17.350 --> 00:05:20.780
because they all might have
slightly different data here.

00:05:20.780 --> 00:05:22.350
But the way that it would generally work,

00:05:22.350 --> 00:05:23.700
is if you were to go to your bank

00:05:23.700 --> 00:05:24.740
and ask for a mortgage,

00:05:24.740 --> 00:05:27.600
or go to a credit card company,
and ask for a credit card,

00:05:27.600 --> 00:05:29.100
or get a car loan,

00:05:29.100 --> 00:05:30.780
that entity, so let's say that

00:05:30.780 --> 00:05:32.730
this is the bank right over here,

00:05:32.730 --> 00:05:35.121
that might be interested
in lending you money,

00:05:35.121 --> 00:05:36.671
they are going to go,

00:05:36.671 --> 00:05:40.260
they are going to go to one
of these credit bureaus.

00:05:40.260 --> 00:05:42.090
They're gonna go to one
of these credit bureaus,

00:05:42.090 --> 00:05:44.820
and say, "Hey, give me this
person's credit score."

00:05:44.820 --> 00:05:45.720
And so that credit bureau's

00:05:45.720 --> 00:05:48.787
gonna look at all the data
that it has on that person,

00:05:48.787 --> 00:05:50.970
put it into the FICO algorithm,

00:05:50.970 --> 00:05:53.070
if the bank wants the FICO credit score,

00:05:53.070 --> 00:05:55.120
and give a score right over here.

00:05:55.120 --> 00:05:57.580
And if you have excellent credit,

00:05:57.580 --> 00:05:58.760
then you might get a low interest rate,

00:05:58.760 --> 00:06:00.757
or you might definitely get the loan.

00:06:00.757 --> 00:06:04.740
If you have a good credit
score, you'll get the loan,

00:06:04.740 --> 00:06:06.650
but it'll be a slightly
higher interest rate.

00:06:06.650 --> 00:06:07.760
But if you have really bad credit,

00:06:07.760 --> 00:06:10.298
your probably not going
to get the loan at all.

00:06:10.298 --> 00:06:12.090
Now the last question
you're probably asking

00:06:12.090 --> 00:06:15.577
is well, how do these credit
bureaus get the information?

00:06:15.577 --> 00:06:17.860
How do they know how
much total debt I have,

00:06:17.860 --> 00:06:19.180
and my payment history,

00:06:19.180 --> 00:06:21.150
and the lengths and types
of credit that I have,

00:06:21.150 --> 00:06:23.280
and the number of inquiries that come in.

00:06:23.280 --> 00:06:25.630
Well, in exchange for lenders

00:06:25.630 --> 00:06:27.390
being allowed to make these inquiries

00:06:27.390 --> 00:06:30.270
and they usually pay the credit bureaus

00:06:30.270 --> 00:06:31.580
to get the information,

00:06:31.580 --> 00:06:34.010
they also agree to share this information.

00:06:34.010 --> 00:06:37.170
So let's say this bank
does give you the loan,

00:06:37.170 --> 00:06:38.623
they will tell the credit bureaus,

00:06:38.623 --> 00:06:41.081
they agree to tell the credit bureaus

00:06:41.081 --> 00:06:44.020
that they gave you that loan,

00:06:44.020 --> 00:06:45.615
and they will also tell the credit bureau

00:06:45.615 --> 00:06:46.920
about your payment history.

00:06:46.920 --> 00:06:48.060
If you pay them all on time,

00:06:48.060 --> 00:06:49.560
the credit bureaus will know about that,

00:06:49.560 --> 00:06:51.690
and they'll be able to
input into your FICO score.

00:06:51.690 --> 00:06:53.030
But if you don't pay it on time,

00:06:53.030 --> 00:06:54.400
they will also know about that,

00:06:54.400 --> 00:06:56.643
and that will also affect your FICO score.


Proof: Sum of measures of angles in a triangle are 180

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:03.760
I've drawn an arbitrary
triangle right over here.

00:00:03.760 --> 00:00:07.010
And I've labeled the measures
of the interior angles.

00:00:07.010 --> 00:00:08.780
The measure of this angle is x.

00:00:08.780 --> 00:00:09.580
This one's y.

00:00:09.580 --> 00:00:10.750
This one is z.

00:00:10.750 --> 00:00:12.830
And what I want to
prove is that the sum

00:00:12.830 --> 00:00:15.960
of the measures of the interior
angles of a triangle, that x

00:00:15.960 --> 00:00:20.410
plus y plus z is
equal to 180 degrees.

00:00:20.410 --> 00:00:22.245
And the way that
I'm going to do it

00:00:22.245 --> 00:00:25.050
is using our knowledge
of parallel lines,

00:00:25.050 --> 00:00:26.660
or transversals
of parallel lines,

00:00:26.660 --> 00:00:28.600
and corresponding angles.

00:00:28.600 --> 00:00:30.407
And to do that,
I'm going to extend

00:00:30.407 --> 00:00:32.740
each of these sides of the
triangle, which right now are

00:00:32.740 --> 00:00:35.370
line segments, but
extend them into lines.

00:00:35.370 --> 00:00:39.000
So this side down
here, if I keep

00:00:39.000 --> 00:00:44.320
going on and on forever
in the same directions,

00:00:44.320 --> 00:00:49.220
then now all of a sudden
I have an orange line.

00:00:49.220 --> 00:00:51.750
And what I want to do is
construct another line

00:00:51.750 --> 00:00:54.140
that is parallel to
the orange line that

00:00:54.140 --> 00:00:58.070
goes through this vertex of
the triangle right over here.

00:00:58.070 --> 00:00:59.860
And I can always do that.

00:00:59.860 --> 00:01:01.390
I could just start
from this point,

00:01:01.390 --> 00:01:03.520
and go in the same
direction as this line,

00:01:03.520 --> 00:01:04.760
and I will never intersect.

00:01:04.760 --> 00:01:09.210
I'm not getting any closer or
further away from that line.

00:01:09.210 --> 00:01:13.710
So I'm never going to
intersect that line.

00:01:13.710 --> 00:01:16.900
So these two lines right
over here are parallel.

00:01:16.900 --> 00:01:19.780
This is parallel to that.

00:01:19.780 --> 00:01:21.280
Now I'm going to
go to the other two

00:01:21.280 --> 00:01:25.660
sides of my original triangle
and extend them into lines.

00:01:25.660 --> 00:01:27.860
So I'm going to extend
this one into a line.

00:01:27.860 --> 00:01:30.504
So, do that as neatly as I can.

00:01:30.504 --> 00:01:32.170
So I'm going to extend
that into a line.

00:01:32.170 --> 00:01:36.390
And you see that this is clearly
a transversal of these two

00:01:36.390 --> 00:01:38.370
parallel lines.

00:01:38.370 --> 00:01:42.670
Now if we have a transversal
here of two parallel lines,

00:01:42.670 --> 00:01:44.930
then we must have some
corresponding angles.

00:01:44.930 --> 00:01:47.260
And we see that
this angle is formed

00:01:47.260 --> 00:01:50.550
when the transversal intersects
the bottom orange line.

00:01:50.550 --> 00:01:52.160
Well what's the
corresponding angle

00:01:52.160 --> 00:01:55.000
when the transversal
intersects this top blue line?

00:01:55.000 --> 00:01:57.360
What's the angle on the top
right of the intersection?

00:01:57.360 --> 00:02:02.470
Angle on the top right of the
intersection must also be x.

00:02:02.470 --> 00:02:04.240
The other thing that
pops out at you,

00:02:04.240 --> 00:02:06.710
is there's another
vertical angle with x,

00:02:06.710 --> 00:02:08.600
another angle that
must be equivalent.

00:02:08.600 --> 00:02:10.400
On the opposite side
of this intersection,

00:02:10.400 --> 00:02:12.740
you have this angle
right over here.

00:02:12.740 --> 00:02:14.010
These two angles are vertical.

00:02:14.010 --> 00:02:16.200
So if this has measure
x, then this one

00:02:16.200 --> 00:02:18.424
must have measure x as well.

00:02:18.424 --> 00:02:20.840
Let's do the same thing with
the last side of the triangle

00:02:20.840 --> 00:02:23.160
that we have not
extended into a line yet.

00:02:23.160 --> 00:02:24.370
So let's do that.

00:02:24.370 --> 00:02:26.380
So if we take this one.

00:02:26.380 --> 00:02:28.010
So we just keep going.

00:02:28.010 --> 00:02:29.060
So it becomes a line.

00:02:29.060 --> 00:02:31.710
So now it becomes a transversal
of the two parallel lines

00:02:31.710 --> 00:02:34.220
just like the magenta line did.

00:02:34.220 --> 00:02:37.340
And we say, hey look this
angle y right over here,

00:02:37.340 --> 00:02:39.990
this angle is formed from the
intersection of the transversal

00:02:39.990 --> 00:02:41.580
on the bottom parallel line.

00:02:41.580 --> 00:02:43.270
What angle to
correspond to up here?

00:02:43.270 --> 00:02:45.940
Well this is kind of on the
left side of the intersection.

00:02:45.940 --> 00:02:48.060
It corresponds to this
angle right over here,

00:02:48.060 --> 00:02:50.100
where the green line,
the green transversal

00:02:50.100 --> 00:02:52.810
intersects the
blue parallel line.

00:02:52.810 --> 00:02:54.380
Well what angle
is vertical to it?

00:02:54.380 --> 00:02:55.630
Well, this angle.

00:02:55.630 --> 00:02:59.990
So this is going to
have measure y as well.

00:02:59.990 --> 00:03:02.890
So now we're really at the
home stretch of our proof

00:03:02.890 --> 00:03:05.787
because we will see that
the measure-- we have

00:03:05.787 --> 00:03:06.870
this angle and this angle.

00:03:06.870 --> 00:03:08.340
This has measure angle x.

00:03:08.340 --> 00:03:10.000
This has measure z.

00:03:10.000 --> 00:03:11.880
They're both adjacent angles.

00:03:11.880 --> 00:03:15.560
If we take the two outer
rays that form the angle,

00:03:15.560 --> 00:03:17.780
and we think about this
angle right over here,

00:03:17.780 --> 00:03:22.220
what's this measure of this
wide angle right over there?

00:03:22.220 --> 00:03:24.712
Well, it's going to be x plus z.

00:03:27.300 --> 00:03:34.140
And that angle is supplementary
to this angle right over here

00:03:34.140 --> 00:03:36.380
that has measure y.

00:03:36.380 --> 00:03:39.460
So the measure of
x-- the measure

00:03:39.460 --> 00:03:41.530
of this wide angle,
which is x plus z,

00:03:41.530 --> 00:03:44.570
plus the measure of this
magenta angle, which is y,

00:03:44.570 --> 00:03:47.110
must be equal to 180
degrees because these two

00:03:47.110 --> 00:03:49.080
angles are supplementary.

00:03:49.080 --> 00:03:55.360
So x-- so the measure of
the wide angle, x plus z,

00:03:55.360 --> 00:03:58.580
plus the measure of the
magenta angle, which

00:03:58.580 --> 00:04:00.600
is supplementary
to the wide angle,

00:04:00.600 --> 00:04:05.229
it must be equal to 180 degrees
because they are supplementary.

00:04:05.229 --> 00:04:06.770
Well we could just
reorder this if we

00:04:06.770 --> 00:04:08.050
want to put in
alphabetical order.

00:04:08.050 --> 00:04:09.310
But we've just
completed our proof.

00:04:09.310 --> 00:04:10.860
The measure of the
interior angles

00:04:10.860 --> 00:04:13.260
of the triangle,
x plus z plus y.

00:04:13.260 --> 00:04:16.800
We could write this
as x plus y plus z

00:04:16.800 --> 00:04:18.750
if the lack of
alphabetical order

00:04:18.750 --> 00:04:20.079
is making you uncomfortable.

00:04:20.079 --> 00:04:22.980
We could just rewrite
this as x plus y plus z

00:04:22.980 --> 00:04:26.360
is equal to 180 degrees.

00:04:26.360 --> 00:04:28.350
And we are done.


How to find the average rate of change from a formula

WEBVTT
Kind: captions
Language: en

00:00:00.810 --> 00:00:04.500
y equals 1/8 x to the
third minus x squared.

00:00:04.500 --> 00:00:07.140
Over which interval does y
of x have an average rate

00:00:07.140 --> 00:00:09.190
of change of 1/2?

00:00:09.190 --> 00:00:11.420
So let's go interval by
interval and calculate

00:00:11.420 --> 00:00:12.660
the average rate of change.

00:00:12.660 --> 00:00:16.680
So first let's think about
this interval right over here.

00:00:16.680 --> 00:00:19.020
x is between negative 2 and 2.

00:00:19.020 --> 00:00:23.202
So negative 2 is less than
x, which is less than 2.

00:00:23.202 --> 00:00:25.660
So let's just think about what
is the value of our function

00:00:25.660 --> 00:00:28.030
when x is equal to negative 2?

00:00:28.030 --> 00:00:32.880
So y of negative 2
is equal to 1/8 times

00:00:32.880 --> 00:00:38.690
negative 2 to the third power
minus negative 2 squared, which

00:00:38.690 --> 00:00:41.550
is equal to--
let's see, negative

00:00:41.550 --> 00:00:44.040
2 the third power is negative 8.

00:00:44.040 --> 00:00:47.100
Negative 8 divided
by 8 is negative 1.

00:00:47.100 --> 00:00:49.300
Negative 2 squared
is positive 4,

00:00:49.300 --> 00:00:52.690
but then you're going to have to
subtract that, so it's minus 4.

00:00:52.690 --> 00:00:55.370
So this is equal to negative 5.

00:00:55.370 --> 00:01:01.360
And y of 2 is equal
to 1/8 times 2

00:01:01.360 --> 00:01:06.590
to the third power
minus 2 squared.

00:01:06.590 --> 00:01:12.840
And that's going to be equal
to 1/8 times 8 is 1 minus 4,

00:01:12.840 --> 00:01:15.266
which is equal to negative 3.

00:01:15.266 --> 00:01:17.390
So if you want to find your
average rate of change,

00:01:17.390 --> 00:01:20.200
you want to figure out how much
does the value of your function

00:01:20.200 --> 00:01:25.950
change, and divide that by
how much your x has changed.

00:01:25.950 --> 00:01:29.790
So we could make a table here.

00:01:29.790 --> 00:01:30.950
x, y.

00:01:30.950 --> 00:01:34.800
When x is negative
2, y is negative 5.

00:01:34.800 --> 00:01:41.350
When x is positive
2, y is negative 3.

00:01:41.350 --> 00:01:43.290
So how much did your y change?

00:01:43.290 --> 00:01:55.030
Well, your y increased by 2,
and your x increased by 4.

00:01:57.600 --> 00:01:59.600
And you could get these
numbers-- you could just

00:01:59.600 --> 00:02:02.820
look at it that y increased
from that point to that point,

00:02:02.820 --> 00:02:04.830
x increased from that
point to that point.

00:02:04.830 --> 00:02:09.220
Or you could say hey, negative 3
minus negative 5 is positive 2.

00:02:09.220 --> 00:02:12.240
That's the difference between
negative 3 and negative 5.

00:02:12.240 --> 00:02:15.800
If you said 2 minus
negative 4, well

00:02:15.800 --> 00:02:18.572
that gives you, once again, the
distance, or the difference.

00:02:18.572 --> 00:02:19.780
It would give you positive 4.

00:02:19.780 --> 00:02:21.405
But if you look at
it here, it's clear.

00:02:21.405 --> 00:02:23.320
When y increased
by 2-- Or we could

00:02:23.320 --> 00:02:27.170
say when x increased
by 4, y increased by 2.

00:02:27.170 --> 00:02:30.070
So our average rate of
change over this interval

00:02:30.070 --> 00:02:38.810
is going to be
average rate of change

00:02:38.810 --> 00:02:48.560
of y with respect to x is going
to be equal to, well, when

00:02:48.560 --> 00:02:53.980
x changed by 4, by positive
4, y changed by positive 2.

00:02:53.980 --> 00:02:57.042
So it's equal to 1/2.

00:02:57.042 --> 00:02:59.000
So it does look like the
average rate of change

00:02:59.000 --> 00:03:04.180
over this interval
right over here is 1/2.

00:03:04.180 --> 00:03:06.280
So we got lucky
in this situation,

00:03:06.280 --> 00:03:09.300
our first choice-- and
this is a multiple choice,

00:03:09.300 --> 00:03:11.890
so it's not a multiselect here.

00:03:11.890 --> 00:03:13.630
So our first one
met our criteria.

00:03:13.630 --> 00:03:15.240
So we know that
that's the answer.

00:03:15.240 --> 00:03:17.100
But let me do one more
of these other ones

00:03:17.100 --> 00:03:19.500
to show you why that
is not the answer.

00:03:19.500 --> 00:03:21.170
So let's find the
average rate of change

00:03:21.170 --> 00:03:23.190
between that point
and that point.

00:03:23.190 --> 00:03:25.750
So let's do another-- let
me do it in another color.

00:03:25.750 --> 00:03:27.350
So I'll do this one in purple.

00:03:27.350 --> 00:03:31.190
So 0 is less than x,
which is less than 4.

00:03:31.190 --> 00:03:34.210
And I'll just do the
table right over here.

00:03:34.210 --> 00:03:35.750
x and y.

00:03:35.750 --> 00:03:38.600
So when x is 0, what is y?

00:03:38.600 --> 00:03:40.440
Well, it's going to be
1/8 times 0 minus 0,

00:03:40.440 --> 00:03:42.410
and y is just going to be 0.

00:03:42.410 --> 00:03:45.314
When x is 4, what is y?

00:03:45.314 --> 00:03:46.730
Well y is going
to be-- let's see,

00:03:46.730 --> 00:03:49.030
it's going to be 1/8--
trying to do this in my head.

00:03:49.030 --> 00:03:51.020
1/8 times 4 to the third.

00:03:51.020 --> 00:03:53.400
4 to the third is 64.

00:03:53.400 --> 00:03:55.016
1/8 of 64 is 8.

00:03:55.016 --> 00:03:59.500
It's going to be 8 minus
4 squared, which is 16.

00:03:59.500 --> 00:04:03.390
8 minus 16 is negative 8.

00:04:03.390 --> 00:04:08.223
So in this one, x
is increasing by 4,

00:04:08.223 --> 00:04:12.426
x has increased by 4,
what happened to y?

00:04:12.426 --> 00:04:15.600
y has decreased by 8.

00:04:15.600 --> 00:04:18.910
So the average rate of change
of y with respect to x here

00:04:18.910 --> 00:04:22.570
is y changed-- so I could
write my change in y--

00:04:22.570 --> 00:04:25.260
this Greek letter delta
just literally is shorthand

00:04:25.260 --> 00:04:26.420
for change in.

00:04:26.420 --> 00:04:34.250
My change in y is negative
8 when my change in x is 4.

00:04:34.250 --> 00:04:38.350
So the average rate of
change here is negative 2.

00:04:38.350 --> 00:04:42.420
It's negative because as x
increased, your y decreased.

00:04:42.420 --> 00:04:47.750
And on average, for every 1 that
x increased, y decreased by 2.

00:04:47.750 --> 00:04:49.680
That's where you get
your negative 2 here.

00:04:49.680 --> 00:04:52.810
So this clearly, the average
rate of change is not 1/2.

00:04:52.810 --> 00:04:54.886
So that confirms that
that is not the answer.

00:04:54.886 --> 00:04:56.510
And we know that
these other two, and I

00:04:56.510 --> 00:04:58.551
encourage you to try them
out, will also give you

00:04:58.551 --> 00:05:00.230
an average rate of
change of something

00:05:00.230 --> 00:05:02.589
other than positive 1/2.


Adding, subtracting numbers in different formats

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:05.646
79% minus 79.1
minus 58 and 1/10.

00:00:05.646 --> 00:00:07.270
And I encourage you
to pause this video

00:00:07.270 --> 00:00:10.809
and try to compute this
expression on your own.

00:00:10.809 --> 00:00:12.350
Well, the thing that
jumps out at you

00:00:12.350 --> 00:00:14.750
is that these are in
different formats.

00:00:14.750 --> 00:00:15.764
This is a percentage.

00:00:15.764 --> 00:00:17.090
These are different
representations.

00:00:17.090 --> 00:00:17.964
There's a percentage.

00:00:17.964 --> 00:00:18.770
This is a decimal.

00:00:18.770 --> 00:00:19.882
This is a mixed number.

00:00:19.882 --> 00:00:22.090
And so to make sense of it,
it's probably a good idea

00:00:22.090 --> 00:00:23.790
to get them all in
the same format.

00:00:23.790 --> 00:00:25.206
And it seems like
we could get all

00:00:25.206 --> 00:00:27.450
of these into a decimal
format pretty easily.

00:00:27.450 --> 00:00:28.920
So let's go that way.

00:00:28.920 --> 00:00:33.510
So 79%, that literally
means 79 per 100.

00:00:33.510 --> 00:00:36.099
If you wanted to write it as a
fraction, it would be 79/100.

00:00:36.099 --> 00:00:37.890
But if you wanted to
write it as a decimal,

00:00:37.890 --> 00:00:44.450
it's 0.79, which could be 0-- or
we would write it down a 0.79.

00:00:44.450 --> 00:00:49.682
Now, 79.1 is already
written as a decimal,

00:00:49.682 --> 00:00:50.890
so we'll just write it again.

00:00:50.890 --> 00:00:53.850
So minus 79.1.

00:00:53.850 --> 00:00:56.140
And then, 58 and 1/10.

00:00:56.140 --> 00:00:58.510
Well, 1/10 is the
same thing as 0.1.

00:00:58.510 --> 00:01:02.400
So you could view this as 58--
well, and literally as 1/10.

00:01:02.400 --> 00:01:05.890
So it's minus 58 and 1/10.

00:01:05.890 --> 00:01:08.387
Or, you could view this as 58.1.

00:01:08.387 --> 00:01:09.970
So now they're all
in the same format,

00:01:09.970 --> 00:01:12.101
let's actually do
the computation.

00:01:12.101 --> 00:01:13.850
Now, the first thing
that jumps out at you

00:01:13.850 --> 00:01:15.474
is you have a fairly
small number here.

00:01:15.474 --> 00:01:16.690
Small positive number.

00:01:16.690 --> 00:01:18.760
It's less than 1.

00:01:18.760 --> 00:01:21.510
And you're subtracting fairly
large numbers over here.

00:01:21.510 --> 00:01:23.805
So your whole answer is
going to be negative.

00:01:23.805 --> 00:01:25.430
And to make sense of
this a little bit,

00:01:25.430 --> 00:01:26.805
what I'm going to
do is I'm going

00:01:26.805 --> 00:01:28.884
to factor out a negative sign.

00:01:28.884 --> 00:01:30.300
And that'll make
the computation--

00:01:30.300 --> 00:01:33.270
at least in my brain, it's going
to make it a little bit easier.

00:01:33.270 --> 00:01:35.240
So if we factor out
a negative sign,

00:01:35.240 --> 00:01:37.310
this becomes-- so we're
going to factor it out.

00:01:37.310 --> 00:01:38.970
Actually, let me
just do it this way.

00:01:38.970 --> 00:01:41.980
So if we factor out
a negative sign,

00:01:41.980 --> 00:01:43.310
then this will become negative.

00:01:43.310 --> 00:01:44.268
This would be positive.

00:01:44.268 --> 00:01:46.230
And this would be positive.

00:01:46.230 --> 00:01:48.660
And just to verify this,
imagine distributing

00:01:48.660 --> 00:01:50.900
this negative sign, or
if this was a negative 1.

00:01:50.900 --> 00:01:53.170
Negative 1 times
this is positive.

00:01:53.170 --> 00:01:54.850
Negative 1 times
this is negative.

00:01:54.850 --> 00:01:56.790
Negative 1 times
this is negative.

00:01:56.790 --> 00:02:00.150
So these two expressions
are the exact same thing.

00:02:00.150 --> 00:02:01.670
And the reason why
I did that is now

00:02:01.670 --> 00:02:03.590
we'll do the more
natural thing of we

00:02:03.590 --> 00:02:04.970
will add these two numbers.

00:02:04.970 --> 00:02:08.422
We'll get a positive number,
a larger positive number

00:02:08.422 --> 00:02:10.380
than what we're going to
subtract from it right

00:02:10.380 --> 00:02:10.960
over here.

00:02:10.960 --> 00:02:12.501
So we can use our
traditional method.

00:02:12.501 --> 00:02:15.110
Although, we can't forget
about this negative out here.

00:02:15.110 --> 00:02:16.390
So let's first do that.

00:02:16.390 --> 00:02:20.310
Let's add 79.1 plus 58.1.

00:02:20.310 --> 00:02:25.220
So 79.1 plus 58.1.

00:02:25.220 --> 00:02:28.160
So 0.1 plus a 0.1 is 0.2.

00:02:28.160 --> 00:02:30.420
9 plus 8 is 17.

00:02:30.420 --> 00:02:33.590
So that's seven 1's and one 10.

00:02:33.590 --> 00:02:37.610
So one 10 plus seven 10's
is going to get us to 0.8.

00:02:37.610 --> 00:02:42.470
Plus 0.5 gets us to
thirteen 10's, or 130.

00:02:42.470 --> 00:02:46.740
So we have 137.2 is this
part right over here.

00:02:46.740 --> 00:02:48.082
So 100.

00:02:48.082 --> 00:02:49.040
Let me write this down.

00:02:49.040 --> 00:02:53.510
So we have 137.2.

00:02:53.510 --> 00:02:57.500
And then from that,
adding a negative 0.79

00:02:57.500 --> 00:02:59.790
is equivalent of
subtracting 0.79.

00:02:59.790 --> 00:03:01.010
So let's do that.

00:03:01.010 --> 00:03:06.400
Let's subtract
0.79, making a point

00:03:06.400 --> 00:03:09.740
to align our decimal points
so that we're subtracting

00:03:09.740 --> 00:03:12.250
the right place from
the right place.

00:03:12.250 --> 00:03:14.240
And now let's do
our subtraction.

00:03:14.240 --> 00:03:16.432
So right now we're
subtracting 9 from nothing.

00:03:16.432 --> 00:03:17.890
We could write a
0 right over here,

00:03:17.890 --> 00:03:20.260
but we still face an issue
in the hundredths place.

00:03:20.260 --> 00:03:22.880
We're also subtracting
a 0.7 from 0.2.

00:03:22.880 --> 00:03:25.520
So we're going to have
to regroup a little bit

00:03:25.520 --> 00:03:27.340
in the numerator in
order to subtract.

00:03:27.340 --> 00:03:29.110
Or at least, in
order to subtract

00:03:29.110 --> 00:03:32.170
using the most
traditional technique.

00:03:32.170 --> 00:03:35.930
So let's take a tenth from the
2, so it's only one tenth now,

00:03:35.930 --> 00:03:37.340
and give it to the hundredths.

00:03:37.340 --> 00:03:40.190
So one tenth is ten hundredths.

00:03:40.190 --> 00:03:42.565
So we could subtract that
ten hundredths minus nine

00:03:42.565 --> 00:03:44.550
hundredths is one hundredth.

00:03:44.550 --> 00:03:46.040
Now in the tenths place.

00:03:46.040 --> 00:03:49.220
We don't have enough up
here, so let's take 1

00:03:49.220 --> 00:03:50.170
from the one's place.

00:03:50.170 --> 00:03:52.360
So that becomes a 6.

00:03:52.360 --> 00:03:53.870
1 is ten tenths.

00:03:53.870 --> 00:03:55.932
So now we have 11 tenths.

00:03:59.080 --> 00:04:01.970
11 minus 7 is 4.

00:04:01.970 --> 00:04:02.920
Add our decimal place.

00:04:02.920 --> 00:04:04.660
6 minus 0 is 6.

00:04:04.660 --> 00:04:07.190
And then we got our
13 just like that.

00:04:10.252 --> 00:04:12.710
So outside the parentheses, I
still have the negative sign.

00:04:12.710 --> 00:04:15.220
When I computed all of this
inside the parentheses,

00:04:15.220 --> 00:04:18.983
I got 136.41.

00:04:18.983 --> 00:04:21.399
And then we can't forget about
the negative sign out here.

00:04:21.399 --> 00:04:25.250
So this whole thing
computes to negative 136.41.


Adding, subtracting fractions, decimals, percentages

WEBVTT
Kind: captions
Language: en

00:00:00.910 --> 00:00:08.460
So we have 0.79 minus
4/3 minus 1/2 plus 150%.

00:00:08.460 --> 00:00:10.460
So we have four
different numbers

00:00:10.460 --> 00:00:11.720
written in different formats.

00:00:11.720 --> 00:00:14.250
Here it's a decimal, here
we have two fractions,

00:00:14.250 --> 00:00:16.230
and then here we
have a percentage.

00:00:16.230 --> 00:00:17.980
So the easiest
thing to do would be

00:00:17.980 --> 00:00:20.350
to write all of these
in the same format.

00:00:20.350 --> 00:00:23.950
And for me, the easiest format
to do this computation in

00:00:23.950 --> 00:00:25.920
would be to write
them all as fractions.

00:00:25.920 --> 00:00:28.211
And the reason why I want to
do that, in particular, is

00:00:28.211 --> 00:00:31.670
because 4/3, when you divide
by 3, when you divide 1/3, 2/3,

00:00:31.670 --> 00:00:34.850
4/3, you're going to
have a repeating decimal.

00:00:34.850 --> 00:00:36.480
So to avoid that,
I want to put all

00:00:36.480 --> 00:00:38.940
of these-- I want to rewrite
all of these as fractions.

00:00:38.940 --> 00:00:40.670
So let's do them one at a time.

00:00:40.670 --> 00:00:44.879
So 0.79, this is the
same thing as 79/100,

00:00:44.879 --> 00:00:46.170
so I'll just write it that way.

00:00:46.170 --> 00:00:50.160
So this is the same
thing as 79 over 100.

00:00:50.160 --> 00:00:52.350
Then of course,
we have minus 4/3.

00:00:55.690 --> 00:00:59.950
Then we have minus 1/2.

00:00:59.950 --> 00:01:02.795
And then finally,
we have-- I don't

00:01:02.795 --> 00:01:04.760
want to run out of colors here.

00:01:04.760 --> 00:01:07.050
Finally we have 150%.

00:01:07.050 --> 00:01:11.570
Well, 150%, percent literally
means per cent, per hundred.

00:01:11.570 --> 00:01:17.760
So this is plus 150 per 100.

00:01:17.760 --> 00:01:19.734
So now we've written
them all as fractions.

00:01:19.734 --> 00:01:21.900
And in order to do all the
subtraction and addition,

00:01:21.900 --> 00:01:24.290
we have to find a
common denominator.

00:01:24.290 --> 00:01:30.160
So what's the least common
multiple of 100, 3, 2, and 100?

00:01:30.160 --> 00:01:32.690
Well, 100 is divisible
by 2, so 100 is actually

00:01:32.690 --> 00:01:34.287
the least common
multiple of 102.

00:01:34.287 --> 00:01:36.620
So we really have to just
find the least common multiple

00:01:36.620 --> 00:01:39.200
between 100 and 300.

00:01:39.200 --> 00:01:41.140
And that's just going to be 300.

00:01:41.140 --> 00:01:44.680
There's no other common
factors between 100 and 3.

00:01:44.680 --> 00:01:49.830
So let's write all of them with
300 as the common denominator.

00:01:49.830 --> 00:01:52.890
So let me do this in
this reddish color.

00:01:52.890 --> 00:01:55.300
So 79 over 100 is
the same thing.

00:01:55.300 --> 00:01:58.480
If I were to write it over
300, to go from 100 to 300

00:01:58.480 --> 00:02:00.710
in the denominator,
I'm multiplying by 3,

00:02:00.710 --> 00:02:04.330
so I have to multiply the
numerator by 3 as well.

00:02:04.330 --> 00:02:06.370
So I'm going to multiply
it by 3 as well.

00:02:06.370 --> 00:02:10.100
Let's see, 80 times
3 would be 240.

00:02:10.100 --> 00:02:12.740
So it's going to be
3 less than that.

00:02:12.740 --> 00:02:19.640
So 240 minus 3 is 237.

00:02:19.640 --> 00:02:21.490
Now 4/3.

00:02:21.490 --> 00:02:24.360
Well, to get the
denominator to be 300,

00:02:24.360 --> 00:02:26.570
we have to multiply
the denominator by 100,

00:02:26.570 --> 00:02:29.980
so we have to multiply the
numerator by 100 as well.

00:02:29.980 --> 00:02:38.920
1/2, if our denominator is 300,
we multiplied the denominator

00:02:38.920 --> 00:02:41.083
by 150 to go from
200 to 300, so we

00:02:41.083 --> 00:02:44.310
have to multiply the
numerator by 150.

00:02:44.310 --> 00:02:50.350
And then finally, 150
over 100, well, we're

00:02:50.350 --> 00:02:53.630
multiplying the
denominator by 3 to get

00:02:53.630 --> 00:02:56.049
to 300, to go from 100 to 300.

00:02:56.049 --> 00:02:58.090
So we have to do the same
thing in the numerator.

00:02:58.090 --> 00:03:01.881
So 3 times 150 is 450.

00:03:01.881 --> 00:03:03.380
So now we have the
same denominator,

00:03:03.380 --> 00:03:05.420
and we can now add
our numerators.

00:03:05.420 --> 00:03:09.360
So this is going to be
equal to-- actually,

00:03:09.360 --> 00:03:12.050
I could just do it right over
here on the right-hand side.

00:03:12.050 --> 00:03:15.970
This is going to be equal
to some stuff over 300.

00:03:15.970 --> 00:03:29.142
So it's going to be 237
minus 400 and minus 150

00:03:29.142 --> 00:03:31.350
and-- this actually should
be a plus right over here.

00:03:31.350 --> 00:03:39.050
This should be plus 450.

00:03:39.050 --> 00:03:41.890
And so let's see if we could
simplify this a little bit.

00:03:41.890 --> 00:03:45.780
We're subtracting 400,
and we're subtracting 150.

00:03:45.780 --> 00:03:50.770
So these two would be the
same thing as subtracting 550.

00:03:50.770 --> 00:03:55.630
And then we have a
positive 237, and we're

00:03:55.630 --> 00:03:59.980
adding it to a positive 450.

00:03:59.980 --> 00:04:01.540
Or actually, maybe
another easier way

00:04:01.540 --> 00:04:05.730
to think about this is
negative 550 plus 450

00:04:05.730 --> 00:04:08.770
is going to get us negative 100.

00:04:08.770 --> 00:04:10.680
And so this simplifies
things a good bit.

00:04:10.680 --> 00:04:16.370
Now we have 237 minus
100 is going to be 137.

00:04:16.370 --> 00:04:24.120
So it equals 137 in
the numerator over 300.

00:04:24.120 --> 00:04:27.890
And this is about as simplified
as I can think of making it.

00:04:27.890 --> 00:04:31.126
And so this is our
final answer, 137/300.


Adding, subtracting fractions

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:05.320
We have negative 3/4
minus 7/6 minus 3/6.

00:00:05.320 --> 00:00:06.790
And there's many
ways to do this.

00:00:06.790 --> 00:00:08.520
But it immediately
jumps out at me

00:00:08.520 --> 00:00:11.400
that these last two numbers
have a 6 in the denominator.

00:00:11.400 --> 00:00:13.270
So I'm going to worry
about these first.

00:00:13.270 --> 00:00:17.390
I'm going to view this as
negative 7/6 minus 3/6.

00:00:17.390 --> 00:00:21.540
So if we have negative
7/6 minus 3/6,

00:00:21.540 --> 00:00:24.930
that's going to be the same
thing as negative 7 minus 3

00:00:24.930 --> 00:00:26.620
over 6.

00:00:26.620 --> 00:00:28.790
And of course, we
have this negative 3/4

00:00:28.790 --> 00:00:30.900
out front that
we're going to add

00:00:30.900 --> 00:00:32.110
to whatever we get over here.

00:00:32.110 --> 00:00:35.980
So this is these two terms
that I'm adding together.

00:00:35.980 --> 00:00:39.770
Negative 7 minus
3 is negative 10.

00:00:39.770 --> 00:00:42.610
So it's negative 10 over 6.

00:00:42.610 --> 00:00:45.230
And then I'm going to have
to add that to negative 3/4.

00:00:52.550 --> 00:00:56.620
And now I have to worry about
finding a common denominator.

00:00:56.620 --> 00:01:02.800
Let me write that so
they have a similar size.

00:01:02.800 --> 00:01:05.390
So now I have to worry about
finding a common denominator.

00:01:05.390 --> 00:01:09.690
What is the smallest number that
is a multiple of both 4 and 6?

00:01:09.690 --> 00:01:11.620
Well, it might jump out
at you that it's 12.

00:01:11.620 --> 00:01:14.040
You can literally just go
through the multiples of 4.

00:01:14.040 --> 00:01:15.890
Or you could look at
the prime factorization

00:01:15.890 --> 00:01:16.850
of both of these numbers.

00:01:16.850 --> 00:01:18.308
And what's the
smallest number that

00:01:18.308 --> 00:01:20.950
has all of the prime
factors of both of these?

00:01:20.950 --> 00:01:25.380
So you need two 2s, and
you need a 2 and a 3.

00:01:25.380 --> 00:01:29.570
So if you have two 2s and a
3, that's 4 times 3 is 12.

00:01:29.570 --> 00:01:34.520
So let's rewrite this
as something over 12

00:01:34.520 --> 00:01:37.500
plus something over 12.

00:01:40.760 --> 00:01:43.000
Well, to get your
denominator from 4 to 12,

00:01:43.000 --> 00:01:44.860
you have to multiply by 3.

00:01:44.860 --> 00:01:47.410
So let's multiply our
numerator by 3 as well.

00:01:47.410 --> 00:01:50.120
So if we multiply
negative 3 times 3,

00:01:50.120 --> 00:01:52.130
you're going to have negative 9.

00:01:52.130 --> 00:01:54.260
And to get your
denominator from 6 to 12,

00:01:54.260 --> 00:01:56.270
you have to multiply by 2.

00:01:56.270 --> 00:01:58.322
So let's multiply our
numerator by 2 as well so

00:01:58.322 --> 00:02:00.280
that we don't change the
value of the fraction.

00:02:00.280 --> 00:02:02.890
So that's going
to be negative 20.

00:02:02.890 --> 00:02:04.490
And now we're ready to add.

00:02:04.490 --> 00:02:07.380
Our common denominator is 12.

00:02:07.380 --> 00:02:12.690
And so this is going to be
negative 9 plus negative 20,

00:02:12.690 --> 00:02:15.531
or we could even write
that as minus 20,

00:02:15.531 --> 00:02:18.780
over 12, which is equal to--
and we deserve a drum roll now.

00:02:18.780 --> 00:02:23.980
This is negative 29 over 12.

00:02:23.980 --> 00:02:26.980
And 29 is a prime
number, so it's not

00:02:26.980 --> 00:02:30.090
going to share any common
factors other than 1 with 12.

00:02:30.090 --> 00:02:34.794
So we also have this in
the most simplified form.


Where a function is not differentiable

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:03.320
Consider f which is defined
for all real numbers.

00:00:03.320 --> 00:00:07.881
At what arguments x is f
of x not differentiable?

00:00:07.881 --> 00:00:09.380
So to think about
that, I'm actually

00:00:09.380 --> 00:00:12.350
going to try to visualize what
f prime of x must look like.

00:00:12.350 --> 00:00:16.960
So I'm going to do f prime
of x in this purple color.

00:00:16.960 --> 00:00:18.940
So if we look at f
of x right over here,

00:00:18.940 --> 00:00:22.530
it looks like its slope is
pretty much consistently

00:00:22.530 --> 00:00:26.900
negative 2 over this interval
between x equals, I guess,

00:00:26.900 --> 00:00:29.380
it's like negative 8
and 1/2 all the way

00:00:29.380 --> 00:00:32.280
up to x equals negative 2.

00:00:32.280 --> 00:00:35.780
It looks like the slope
is a constant negative 2.

00:00:35.780 --> 00:00:40.170
So if I were to
draw its derivative,

00:00:40.170 --> 00:00:43.476
its derivative would
look something like this.

00:00:43.476 --> 00:00:46.540
Its derivative looks
something like this.

00:00:46.540 --> 00:00:49.570
But then something interesting
happens at x equals negative 2.

00:00:49.570 --> 00:00:54.666
Right as we cross x
equals negative 2,

00:00:54.666 --> 00:00:56.290
it looks like the
slope goes from being

00:00:56.290 --> 00:00:58.320
negative to being positive.

00:00:58.320 --> 00:01:00.087
And it looks like
right out the get

00:01:00.087 --> 00:01:02.420
go, if I were to estimate the
slope of its tangent line,

00:01:02.420 --> 00:01:03.410
it starts changing.

00:01:03.410 --> 00:01:04.500
It's not a line anymore.

00:01:04.500 --> 00:01:05.670
It's a curve.

00:01:05.670 --> 00:01:07.870
The slope of the tangent
line right at this point

00:01:07.870 --> 00:01:09.770
looks like it's
around-- I don't know--

00:01:09.770 --> 00:01:12.830
it looks like it's
around 3 and 1/2.

00:01:12.830 --> 00:01:18.440
Because if I were to draw a
tangent line right over here,

00:01:18.440 --> 00:01:20.810
it looks like if I move
1 in the x direction,

00:01:20.810 --> 00:01:23.560
I move up about 3 and
1/2 in the y direction.

00:01:23.560 --> 00:01:25.650
So I'm just trying to,
obviously, estimate it.

00:01:25.650 --> 00:01:30.210
So it looks like the
slope goes up to 3 and 1/2

00:01:30.210 --> 00:01:31.710
right when I cross that point.

00:01:31.710 --> 00:01:37.240
And then the slope becomes
lower and lower and lower

00:01:37.240 --> 00:01:40.800
all the way until I get to
this point right over here,

00:01:40.800 --> 00:01:43.770
all the way until I
get to x equals 2.

00:01:43.770 --> 00:01:46.850
And it looks like it continues
to get lower all the way

00:01:46.850 --> 00:01:49.190
until you get to x equals 3.

00:01:49.190 --> 00:01:51.120
So it looks like the
slope of the line

00:01:51.120 --> 00:01:53.520
is-- it looks like it's getting
lower at a constant rate,

00:01:53.520 --> 00:01:54.780
I guess I could say.

00:01:54.780 --> 00:01:56.321
So it looks like
it's doing something

00:01:56.321 --> 00:02:04.130
like this over this interval.

00:02:04.130 --> 00:02:08.350
But then right as x crosses
3, this becomes a flat line.

00:02:08.350 --> 00:02:10.130
The slope is 0 here.

00:02:10.130 --> 00:02:15.030
So right as x crosses
3, the slope becomes 0.

00:02:15.030 --> 00:02:16.900
So we immediately
see there are points

00:02:16.900 --> 00:02:21.560
where it looks like
the slope jumps.

00:02:21.560 --> 00:02:23.540
And at these points
we really don't

00:02:23.540 --> 00:02:25.520
have a defined derivative.

00:02:25.520 --> 00:02:27.980
The slope jumps there as well.

00:02:27.980 --> 00:02:31.580
And so at what arguments
is f not differentiable?

00:02:31.580 --> 00:02:35.180
Well, it's not differentiable
when x is equal to negative 2.

00:02:35.180 --> 00:02:36.880
When x is equal to
negative 2, we really

00:02:36.880 --> 00:02:39.320
don't have a slope there.

00:02:39.320 --> 00:02:41.090
Remember, when
we're trying to find

00:02:41.090 --> 00:02:42.910
the slope of the
tangent line, we

00:02:42.910 --> 00:02:45.290
take the limit of the
slope of the secant line

00:02:45.290 --> 00:02:48.070
between that point and some
other point on the curve.

00:02:48.070 --> 00:02:50.060
If we did that as we
approached from the left,

00:02:50.060 --> 00:02:52.710
it looks like the
derivative is negative 2.

00:02:52.710 --> 00:02:55.260
If we do that from the right,
it looks like the derivative

00:02:55.260 --> 00:02:58.460
is something like
positive 3 and 1/2.

00:02:58.460 --> 00:03:01.490
And so we're not getting the
same limit of the secant line

00:03:01.490 --> 00:03:05.170
as we approach from the left and
as we approach from the right.

00:03:05.170 --> 00:03:08.275
And the same thing is
happening at x is equal to 3.

00:03:08.275 --> 00:03:11.460
At x equals 3, as we
approach from the left,

00:03:11.460 --> 00:03:15.400
the slope looks like
it is decreasing.

00:03:15.400 --> 00:03:19.596
It is approaching-- I don't
know-- maybe around negative 1.

00:03:19.596 --> 00:03:20.970
But as we approach
from the right

00:03:20.970 --> 00:03:22.660
it looks like the slope is 0.

00:03:22.660 --> 00:03:25.590
So we do not have the same
limit of the secant slope

00:03:25.590 --> 00:03:27.940
as we approach from the
left- and right-hand sides.

00:03:27.940 --> 00:03:30.530
So at both of these points
we see the derivative jump,

00:03:30.530 --> 00:03:34.005
and it looks like f of
x is not differentiable.


Graphs of functions and their derivatives example 1

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:04.580
We're asked, at which points
on the graph is f of x times

00:00:04.580 --> 00:00:08.200
f prime of x equal to 0?

00:00:08.200 --> 00:00:11.130
So if I have the product of
two things and it's equal to 0,

00:00:11.130 --> 00:00:13.430
that tells us that at least
one of these two things

00:00:13.430 --> 00:00:15.240
need to be equal to 0.

00:00:15.240 --> 00:00:19.210
So first of all, let's see are
there any points when f of x

00:00:19.210 --> 00:00:21.520
is equal to 0?

00:00:21.520 --> 00:00:23.900
So we're plotting f of
x on the vertical axis.

00:00:23.900 --> 00:00:25.950
We could call this
graph right over here,

00:00:25.950 --> 00:00:28.780
we could say this is
y is equal to f of x.

00:00:28.780 --> 00:00:33.440
So at any point, does the y
value of this curve equal 0?

00:00:33.440 --> 00:00:36.360
So it's positive, positive,
positive, positive, positive,

00:00:36.360 --> 00:00:37.320
positive.

00:00:37.320 --> 00:00:38.995
But it is decreasing
right over here.

00:00:38.995 --> 00:00:40.120
Well, it's decreasing here.

00:00:40.120 --> 00:00:41.040
Then it's increasing.

00:00:41.040 --> 00:00:42.220
Then it's decreasing.

00:00:42.220 --> 00:00:44.740
And it does get to
0 right over here,

00:00:44.740 --> 00:00:47.457
but that's not one of
the labeled points.

00:00:47.457 --> 00:00:49.540
And they want us to pick
one of the labeled points

00:00:49.540 --> 00:00:52.070
or maybe even more than one
of these labeled points.

00:00:52.070 --> 00:00:56.480
So we're going to focus on where
f prime of x is equal to 0.

00:00:56.480 --> 00:00:58.070
And we just have
to remind ourselves

00:00:58.070 --> 00:01:00.650
what f prime of x
even represents.

00:01:00.650 --> 00:01:04.360
f prime of x represents the
slope of the tangent line

00:01:04.360 --> 00:01:06.590
at that value of x.

00:01:06.590 --> 00:01:10.810
So for example, f
prime of 0-- which

00:01:10.810 --> 00:01:13.340
is the x value for this
point right over here--

00:01:13.340 --> 00:01:15.310
is going to be some
negative value.

00:01:15.310 --> 00:01:18.360
It's the slope of
the tangent line.

00:01:18.360 --> 00:01:21.900
Similarly, f prime
of x, when x is

00:01:21.900 --> 00:01:24.289
equal to 4-- that's what's
going on right over here--

00:01:24.289 --> 00:01:26.330
that's going to be the
slope of the tangent line.

00:01:26.330 --> 00:01:28.469
That's going to be
a positive value.

00:01:28.469 --> 00:01:30.010
So if you look at
all of these, where

00:01:30.010 --> 00:01:32.510
is the slope of
the tangent line 0?

00:01:32.510 --> 00:01:34.050
And what does a 0
slope look like?

00:01:34.050 --> 00:01:36.230
Well, it looks like
a horizontal line.

00:01:36.230 --> 00:01:38.875
So where is the slope of the
tangent line here horizontal?

00:01:38.875 --> 00:01:40.500
Well, the only one
that jumps out at me

00:01:40.500 --> 00:01:42.400
is point B right over here.

00:01:42.400 --> 00:01:44.630
It looks like the slope
of the tangent line

00:01:44.630 --> 00:01:51.280
would indeed be horizontal
right over here.

00:01:51.280 --> 00:01:53.980
Or another way you could think
of it is the instantaneous rate

00:01:53.980 --> 00:01:59.120
of change of the function,
right at x equals 2,

00:01:59.120 --> 00:02:02.790
looks like it's pretty
close to-- if this

00:02:02.790 --> 00:02:06.010
is x equals 2-- looks like
it's pretty close to 0.

00:02:06.010 --> 00:02:07.690
So out of all of
the choices here,

00:02:07.690 --> 00:02:12.280
I would say only B looks like
the derivative at x equals 2.

00:02:12.280 --> 00:02:15.770
Or the slope of the tangent
line at B, it looks like it's 0.

00:02:15.770 --> 00:02:18.950
So I'll say B right over here.

00:02:18.950 --> 00:02:21.850
And then they had this kind of
crazy, wacky expression here. f

00:02:21.850 --> 00:02:24.270
of x minus 6 over x.

00:02:24.270 --> 00:02:26.861
What is that greatest in value?

00:02:26.861 --> 00:02:28.110
And we have to interpret this.

00:02:28.110 --> 00:02:31.830
We have to think about what does
f of x minus 6 over x actually

00:02:31.830 --> 00:02:32.770
mean?

00:02:32.770 --> 00:02:34.780
Whenever I see
expressions like this,

00:02:34.780 --> 00:02:37.700
especially if I'm taking a
differential calculus class,

00:02:37.700 --> 00:02:39.660
I would say well,
this looks kind of

00:02:39.660 --> 00:02:44.890
like finding the slope
of a secant line.

00:02:44.890 --> 00:02:47.260
In fact, all of what we
know about derivatives

00:02:47.260 --> 00:02:50.460
is finding the limiting value
of the slope of a secant line.

00:02:50.460 --> 00:02:51.940
And this looks
kind of like that,

00:02:51.940 --> 00:02:55.610
especially if at some point,
my y value is a 6 here.

00:02:55.610 --> 00:02:58.240
And this could be the
change in y value.

00:02:58.240 --> 00:03:00.350
And if the corresponding
x value is 0,

00:03:00.350 --> 00:03:03.680
then this would be f of
x minus 6 over x minus 0.

00:03:03.680 --> 00:03:06.390
So do I have 0, 6
on this curve here?

00:03:06.390 --> 00:03:07.070
Well, sure.

00:03:07.070 --> 00:03:15.230
When x is equal to 0, we see
that f of x is equal to 6.

00:03:15.230 --> 00:03:18.590
So what this is right over
here-- let me rewrite this.

00:03:18.590 --> 00:03:33.615
This we could rewrite as f
of x minus 6 over x minus 0.

00:03:36.300 --> 00:03:37.270
So what is this?

00:03:37.270 --> 00:03:38.380
What does this represent?

00:03:38.380 --> 00:03:41.982
Well, this is equal
to the slope--

00:03:41.982 --> 00:03:43.440
let me do some of
that color-- this

00:03:43.440 --> 00:03:57.760
is equal to the slope of the
secant line between the points,

00:03:57.760 --> 00:04:03.770
x, f of x, x, and whatever
the corresponding f of x is.

00:04:03.770 --> 00:04:11.130
And we could write it as 0,
f of 0 because we see f of 0

00:04:11.130 --> 00:04:12.260
is equal to 6.

00:04:12.260 --> 00:04:15.000
This right over here is f of 0.

00:04:15.000 --> 00:04:17.149
In fact, let me just
write that as 6.

00:04:17.149 --> 00:04:18.500
And the point, 0, 6.

00:04:21.620 --> 00:04:23.439
So let's go through
each of these points

00:04:23.439 --> 00:04:25.480
and think about what the
slope of the secant line

00:04:25.480 --> 00:04:28.760
between those points are and
point A. This is essentially

00:04:28.760 --> 00:04:31.095
the slope of the secant
line between some point x, f

00:04:31.095 --> 00:04:35.190
of x, and essentially point
A. So let's draw this out.

00:04:35.190 --> 00:04:40.030
So between A and B you have
a fairly negative slope.

00:04:40.030 --> 00:04:42.530
Remember we want to
find the largest slope.

00:04:42.530 --> 00:04:44.390
So here it's fairly negative.

00:04:44.390 --> 00:04:46.440
Between A and C,
it's less negative.

00:04:49.310 --> 00:04:51.780
Between A and D, it's
even less negative.

00:04:51.780 --> 00:04:55.840
It's still negative,
but it's less negative.

00:04:55.840 --> 00:05:01.040
And then between A and E, it
becomes more negative now.

00:05:01.040 --> 00:05:03.710
And then between A and F, it
becomes even more negative.

00:05:08.560 --> 00:05:10.810
So when is the slope of
the secant line between one

00:05:10.810 --> 00:05:14.219
of these points
and A the greatest?

00:05:14.219 --> 00:05:16.010
Or I guess we could
say the least negative?

00:05:16.010 --> 00:05:19.080
Because it seems like
they're always negative.

00:05:19.080 --> 00:05:22.530
It would be between
point D and A.

00:05:22.530 --> 00:05:25.790
So when is this
greatest in value?

00:05:25.790 --> 00:05:30.070
Well, when we're
looking at point D.

00:05:30.070 --> 00:05:34.830
At point D, x is
equal to 6 and it

00:05:34.830 --> 00:05:38.620
looks like f of x is like
5 and 1/2 or something.

00:05:38.620 --> 00:05:44.500
So this will turn into f of
6, which is 5 and 1/2 or maybe

00:05:44.500 --> 00:05:46.470
it's even less than
that-- 5 and 1/3

00:05:46.470 --> 00:05:50.360
or something, minus
6 over 6 minus 0.

00:05:50.360 --> 00:05:52.550
That's how we'll
maximize this value.

00:05:52.550 --> 00:05:56.343
This is the least negative
slope of the secant line.


Figuring out which function is the derivative

WEBVTT
Kind: captions
Language: en

00:00:00.840 --> 00:00:03.530
So what we have plotted here is
some function-- let's call it

00:00:03.530 --> 00:00:06.030
f of x-- and its
derivative, f prime of x.

00:00:06.030 --> 00:00:07.470
And what we need
to figure out is

00:00:07.470 --> 00:00:12.860
which one is f of x and
which one is f prime of x?

00:00:12.860 --> 00:00:15.620
So let's take a stab at it.

00:00:15.620 --> 00:00:18.240
Let's think about what
would be the situation

00:00:18.240 --> 00:00:21.000
if the green
function were f of x.

00:00:21.000 --> 00:00:22.450
So let's see if this works out.

00:00:22.450 --> 00:00:26.640
If the green function were f
of x, does the orange function

00:00:26.640 --> 00:00:30.229
here, or the yellow function,
could that be f prime of x?

00:00:30.229 --> 00:00:31.770
So let's think about
what's happening

00:00:31.770 --> 00:00:34.160
to this green function
at different points.

00:00:34.160 --> 00:00:36.337
So this green function
right over here,

00:00:36.337 --> 00:00:38.170
right at this point if
we start at the left,

00:00:38.170 --> 00:00:41.695
has a positive slope.

00:00:49.520 --> 00:00:51.160
If this orange
function were f prime

00:00:51.160 --> 00:00:54.590
of x, if it were the derivative
of the green function,

00:00:54.590 --> 00:00:55.999
then it would have
to be positive

00:00:55.999 --> 00:00:57.540
because the green
function's slope is

00:00:57.540 --> 00:00:58.820
positive at that point.

00:00:58.820 --> 00:01:00.670
But we see that
it's not positive.

00:01:00.670 --> 00:01:03.980
So it's pretty clear that the
green function cannot be f

00:01:03.980 --> 00:01:06.822
of x, and the yellow function
cannot be its derivative,

00:01:06.822 --> 00:01:08.280
because if this
was its derivative,

00:01:08.280 --> 00:01:09.450
it would be positive here.

00:01:09.450 --> 00:01:13.150
So that quickly, we found out
that that can't be the case.

00:01:13.150 --> 00:01:15.200
But let's see if it could
work out the other way.

00:01:15.200 --> 00:01:18.640
So it's starting to feel--
just ruling that situation

00:01:18.640 --> 00:01:22.490
out-- that maybe that this is
f of x and the green function

00:01:22.490 --> 00:01:25.900
is f prime of x.

00:01:25.900 --> 00:01:29.420
So let's see if this
holds up to scrutiny.

00:01:29.420 --> 00:01:33.170
So what we have when we
start off at the left,

00:01:33.170 --> 00:01:35.420
f of x, or what we
think is f of x,

00:01:35.420 --> 00:01:38.364
has a reasonably positive slope.

00:01:38.364 --> 00:01:39.155
Is that consistent?

00:01:39.155 --> 00:01:40.070
Well, yeah, sure.

00:01:40.070 --> 00:01:43.580
Our green function is positive.

00:01:43.580 --> 00:01:45.500
In fact, at the
point, it's telling us

00:01:45.500 --> 00:01:49.040
that the slope of the tangent
line is around 2 and 1/2.

00:01:49.040 --> 00:01:51.910
And it actually does look like
the slope of the tangent line

00:01:51.910 --> 00:01:55.405
is exactly 2 and 1/2 of
this function right here.

00:01:55.405 --> 00:01:56.780
Actually, let me
erase this, just

00:01:56.780 --> 00:01:58.196
so we don't look
like we're trying

00:01:58.196 --> 00:02:00.960
to take the slope of the
tangent line of the derivative.

00:02:00.960 --> 00:02:02.160
So it looks just like that.

00:02:02.160 --> 00:02:04.610
So we see the slope of the
tangent line right over here

00:02:04.610 --> 00:02:07.980
looks like about 2 and 1/2,
and the value of this function

00:02:07.980 --> 00:02:09.910
up here looks like
it's about 2 and a 1/2.

00:02:09.910 --> 00:02:11.990
So, so far this
green function looks

00:02:11.990 --> 00:02:15.320
like a pretty good
candidate for the derivative

00:02:15.320 --> 00:02:16.490
of this yellow function.

00:02:16.490 --> 00:02:19.086
But let's keep going here.

00:02:19.086 --> 00:02:20.460
So let's think
about what happens

00:02:20.460 --> 00:02:22.140
as we move to the right.

00:02:22.140 --> 00:02:23.100
So here, let's see.

00:02:23.100 --> 00:02:27.630
It looks like the slope
of this yellow function--

00:02:27.630 --> 00:02:32.560
let me just use a color we
can see-- it keeps going up.

00:02:32.560 --> 00:02:36.020
It keeps going up,
keeps going up.

00:02:36.020 --> 00:02:39.090
And then at some point, it
reaches some maximum slope,

00:02:39.090 --> 00:02:43.140
and then it starts
to go down again.

00:02:43.140 --> 00:02:46.030
The slope starts to go
down again, all the way

00:02:46.030 --> 00:02:49.980
to the slope going all the
way down to 0 right over here.

00:02:49.980 --> 00:02:52.000
Well, does this green
function describe that?

00:02:52.000 --> 00:02:52.666
Well, let's see.

00:02:52.666 --> 00:02:54.940
The slope is positive
and increasing up

00:02:54.940 --> 00:02:57.440
to this point, which seemed
pretty consistent with what

00:02:57.440 --> 00:02:58.760
we just experienced.

00:02:58.760 --> 00:03:00.780
Then the slope stays
positive, but it's

00:03:00.780 --> 00:03:02.324
positive and decreasing.

00:03:02.324 --> 00:03:03.490
And that's what we saw here.

00:03:03.490 --> 00:03:05.910
The slope is positive and
decreasing all the way

00:03:05.910 --> 00:03:08.460
to the slope getting to 0
at this maximum point here.

00:03:08.460 --> 00:03:10.950
And we see, indeed, that
on this green function,

00:03:10.950 --> 00:03:12.410
the green function hits 0.

00:03:12.410 --> 00:03:15.110
So it seems like it's doing
a pretty good job of plotting

00:03:15.110 --> 00:03:18.230
the slope of the tangent
line of the orange function.

00:03:18.230 --> 00:03:23.140
And then our slope becomes
more and more negative.

00:03:23.140 --> 00:03:26.420
And then it hits some
point, some minimum point

00:03:26.420 --> 00:03:27.750
right over here.

00:03:27.750 --> 00:03:30.490
The slope hits some minimum
point right over here.

00:03:30.490 --> 00:03:33.156
And then it becomes
less and less negative.

00:03:35.556 --> 00:03:36.930
Let's see how well
I can draw it.

00:03:36.930 --> 00:03:39.480
The slope becomes
less and less negative

00:03:39.480 --> 00:03:43.160
until it hits a 0 slope again.

00:03:43.160 --> 00:03:47.860
And then it starts
becoming positive

00:03:47.860 --> 00:03:49.860
until it hits some
maximum slope.

00:03:49.860 --> 00:03:53.070
But then it stays positive,
but it becomes less positive.

00:03:53.070 --> 00:03:55.850
It becomes less
and less positive.

00:03:55.850 --> 00:03:59.070
So it looks pretty clear that
the orange function is f of x,

00:03:59.070 --> 00:04:02.710
and the green function
is f prime of x.