Example relating trig function to side ratios

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:03.370
Sort the expressions
according to their values.

00:00:03.370 --> 00:00:05.640
You can put any number
of cards in a category

00:00:05.640 --> 00:00:07.717
or leave a category empty.

00:00:07.717 --> 00:00:09.550
And so we have this
diagram right over here,

00:00:09.550 --> 00:00:12.310
then we have these cards
that have these expressions.

00:00:12.310 --> 00:00:15.416
And we're supposed to sort
these into different buckets.

00:00:15.416 --> 00:00:16.790
So we're trying
to say, well what

00:00:16.790 --> 00:00:20.000
is the length of segment AC
over the length of segment

00:00:20.000 --> 00:00:21.030
BC equal to?

00:00:21.030 --> 00:00:23.130
Which of these expressions
is it equal to?

00:00:23.130 --> 00:00:27.109
And then we should drag it
into the appropriate buckets.

00:00:27.109 --> 00:00:28.650
So to figure these
out, I've actually

00:00:28.650 --> 00:00:32.750
already redrawn this
problem on my little

00:00:32.750 --> 00:00:35.050
I guess you call it scratch
pad or blackboard, whatever

00:00:35.050 --> 00:00:36.220
you want to call it.

00:00:36.220 --> 00:00:37.960
This right over here
is that same diagram

00:00:37.960 --> 00:00:39.110
blown up a little bit.

00:00:39.110 --> 00:00:41.799
Here are the expressions that
we need to drag into things.

00:00:41.799 --> 00:00:43.340
And here are the
buckets that we need

00:00:43.340 --> 00:00:45.506
to see which of these
expressions are equal to which

00:00:45.506 --> 00:00:46.930
of these expressions.

00:00:46.930 --> 00:00:48.510
So let's first look at this.

00:00:48.510 --> 00:00:52.530
The length of segment AC over
the length of segment BC.

00:00:52.530 --> 00:00:54.089
So let's think about what AC is.

00:00:54.089 --> 00:00:55.130
The length of segment AC.

00:00:55.130 --> 00:00:57.460
AC is this right here.

00:00:57.460 --> 00:01:02.480
So it's this length right
over here in purple over

00:01:02.480 --> 00:01:07.280
the length of segment BC, over
this length right over here.

00:01:07.280 --> 00:01:11.040
So it's the ratio of
the lengths of two

00:01:11.040 --> 00:01:12.560
sides of a right triangle.

00:01:12.560 --> 00:01:17.525
This is clearly a right
triangle, triangle ABC.

00:01:17.525 --> 00:01:18.900
And I could color
that in just so

00:01:18.900 --> 00:01:20.608
you know what triangle
I'm talking about.

00:01:20.608 --> 00:01:24.730
Triangle ABC is
this entire triangle

00:01:24.730 --> 00:01:26.112
that we could focus on.

00:01:26.112 --> 00:01:27.820
So you could imagine
that it's reasonable

00:01:27.820 --> 00:01:29.820
that the ratio of two
sides of a right triangle

00:01:29.820 --> 00:01:32.440
are going to be the sine
of one of its angles.

00:01:32.440 --> 00:01:35.040
And they give us one of
the angles right over here.

00:01:35.040 --> 00:01:38.684
They give us this
angle right over here.

00:01:38.684 --> 00:01:40.600
You say, well all they
did is mark that angle.

00:01:40.600 --> 00:01:43.360
But notice, one arc is
here, one arc is here.

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So anywhere we see only one arc,
that's going to be 30 degrees.

00:01:46.880 --> 00:01:48.900
So this is 30 degrees as well.

00:01:48.900 --> 00:01:51.052
You have two arcs here,
that's 41 degrees.

00:01:51.052 --> 00:01:53.260
Two arcs here, this is going
to be congruent to that.

00:01:53.260 --> 00:01:55.070
This over here is
going to be 41 degrees.

00:01:55.070 --> 00:01:55.940
This is three arcs.

00:01:55.940 --> 00:01:58.360
They don't tell us how
many degrees that is,

00:01:58.360 --> 00:02:00.160
but this angle
with the three arcs

00:02:00.160 --> 00:02:02.380
is congruent to this
angle with the three arcs

00:02:02.380 --> 00:02:03.290
right over there.

00:02:03.290 --> 00:02:06.990
So anyway, this yellow
triangle, triangle ABC,

00:02:06.990 --> 00:02:10.620
we know the measure of
this angle is 30 degrees.

00:02:10.620 --> 00:02:12.590
And then they give
us these two sides.

00:02:12.590 --> 00:02:15.860
So how do these sides relate
to this 30 degree angle?

00:02:15.860 --> 00:02:18.610
Well, side AC is adjacent to it.

00:02:18.610 --> 00:02:22.270
It's literally, it's one of the
sides of the angle that is not

00:02:22.270 --> 00:02:25.190
the hypotenuse, so let
me write that down.

00:02:25.190 --> 00:02:26.606
This is adjacent.

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And what is BC?

00:02:30.510 --> 00:02:33.510
Well, BC is the hypotenuse
of this right triangle.

00:02:33.510 --> 00:02:36.030
It's the side opposite
the 90 degrees.

00:02:36.030 --> 00:02:38.860
So this is the hypotenuse.

00:02:38.860 --> 00:02:44.050
So what trig function,
when applied to 30 degrees,

00:02:44.050 --> 00:02:48.220
is equal to the adjacent
side over the hypotenuse?

00:02:48.220 --> 00:02:50.790
Let's write down sohcahtoa
just to remind ourselves.

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So soh cah toa.

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Sine of an angle is
opposite over hypotenuse,

00:02:57.840 --> 00:03:01.700
cosine of an angle is
adjacent over hypotenuse.

00:03:01.700 --> 00:03:06.950
So cosine, let's write this
down, cosine of 30 degrees

00:03:06.950 --> 00:03:11.580
is going to be equal to the
length of the adjacent side.

00:03:11.580 --> 00:03:17.080
So that is AC, over the length
of the hypotenuse, which

00:03:17.080 --> 00:03:18.840
is equal to BC.

00:03:18.840 --> 00:03:22.120
So this right over
here is the same thing

00:03:22.120 --> 00:03:25.300
as the cosine of 30 degrees.

00:03:25.300 --> 00:03:27.690
So let's drag it in there.

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This is equal to the
cosine of 30 degrees.

00:03:33.070 --> 00:03:35.700
Now let's look at the next one.

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Cosine of angle DEC.

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Where is DEC?

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So DEC-- D, E, C. So that's
this angle right over here.

00:03:46.680 --> 00:03:49.040
I'll put four arcs here so
we don't get it confused.

00:03:49.040 --> 00:03:50.790
So this is angle DEC.

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So what is the cosine of DEC?

00:03:53.640 --> 00:03:57.300
Well once again, cosine is
adjacent over hypotenuse.

00:03:57.300 --> 00:04:00.740
So cosine of angle DEC,
the adjacent side to this,

00:04:00.740 --> 00:04:02.986
well that's this
right over here.

00:04:02.986 --> 00:04:04.860
You might say, well
isn't this side adjacent?

00:04:04.860 --> 00:04:07.527
Well that side, side DE, that
is the actual hypotenuse.

00:04:07.527 --> 00:04:09.360
So that's not going to
be the adjacent side.

00:04:09.360 --> 00:04:14.060
So the adjacent side
is, I could call it EC.

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It's the length of segment EC.

00:04:16.190 --> 00:04:21.800
And then the hypotenuse
is this right over here.

00:04:21.800 --> 00:04:24.770
The length of the hypotenuse--
the hypotenuse is side DE,

00:04:24.770 --> 00:04:27.050
or ED, however you
want to call it.

00:04:27.050 --> 00:04:31.650
And so the length of it is,
we could just write it as DE.

00:04:31.650 --> 00:04:33.132
Now what is this also equal to?

00:04:33.132 --> 00:04:34.590
We don't see this
choice over here.

00:04:34.590 --> 00:04:37.970
We don't have the ratio EC
over DE as one of these choices

00:04:37.970 --> 00:04:39.170
here.

00:04:39.170 --> 00:04:42.050
But what we do have is one of--
we do get one of the angles

00:04:42.050 --> 00:04:42.550
here.

00:04:42.550 --> 00:04:44.880
They give us this 41 degrees.

00:04:44.880 --> 00:04:47.670
And the ratio of this green
side over-- the length

00:04:47.670 --> 00:04:50.380
of this green side over this
orange side, what would that

00:04:50.380 --> 00:04:52.010
be in terms of, if
we wanted to apply

00:04:52.010 --> 00:04:54.110
a trig function to this angle?

00:04:54.110 --> 00:04:56.840
Well, relative to this
angle, the green side

00:04:56.840 --> 00:04:59.240
is the opposite side,
and the orange side

00:04:59.240 --> 00:05:00.800
is still the hypotenuse.

00:05:00.800 --> 00:05:03.120
So relative to 41
degrees-- so let's

00:05:03.120 --> 00:05:06.190
write this down--
relative to 41 degrees,

00:05:06.190 --> 00:05:09.310
this ratio is the opposite
over the hypotenuse.

00:05:09.310 --> 00:05:11.470
It's the cosine of
this angle, but it's

00:05:11.470 --> 00:05:13.260
the sine of this
angle right over here.

00:05:13.260 --> 00:05:15.930
Sine is opposite
over hypotenuse.

00:05:15.930 --> 00:05:18.700
So this is equal to the sine
of this angle right over here.

00:05:18.700 --> 00:05:22.200
It's equal to the
sine of 41 degrees.

00:05:22.200 --> 00:05:24.790
So that is this one
right over here.

00:05:24.790 --> 00:05:29.430
The sine of 41 degrees.

00:05:29.430 --> 00:05:33.000
So let's drag that in to
the appropriate bucket.

00:05:35.850 --> 00:05:39.280
Sine of 41 degrees
is the same thing

00:05:39.280 --> 00:05:42.830
as the cosine of angle DEC.

00:05:42.830 --> 00:05:44.780
Only have two left.

00:05:44.780 --> 00:05:51.060
So now we have to figure out
what the sine of angle CDA is.

00:05:51.060 --> 00:05:53.890
So let's see, where is CDA?

00:05:53.890 --> 00:05:57.380
CDA is this entire angle.

00:05:57.380 --> 00:05:59.987
It's this entire
angle right over here.

00:05:59.987 --> 00:06:02.070
So I could put a bunch of
arcs here if I want just

00:06:02.070 --> 00:06:04.510
to show that it's different
than all the other ones.

00:06:04.510 --> 00:06:07.090
So that's that angle
right over there.

00:06:07.090 --> 00:06:11.950
So now we're really dealing
with this larger right triangle.

00:06:11.950 --> 00:06:13.850
Let me highlight it
in this pink color.

00:06:13.850 --> 00:06:18.340
So we're now dealing with
this larger right triangle

00:06:18.340 --> 00:06:19.289
right over here.

00:06:19.289 --> 00:06:21.080
We care about the sine
of this whole thing.

00:06:21.080 --> 00:06:23.003
Remember, sine is
opposite over hypotenuse.

00:06:26.250 --> 00:06:35.800
So the opposite side
is going to be side CA.

00:06:35.800 --> 00:06:40.700
So this is going to be
equal to the length of CA

00:06:40.700 --> 00:06:42.840
over the hypotenuse,
which is AD.

00:06:47.970 --> 00:06:52.080
So that is going to be over AD.

00:06:52.080 --> 00:06:55.520
And once again, we don't
see that as a choice here.

00:06:55.520 --> 00:06:59.260
But maybe we can express
this ratio-- maybe

00:06:59.260 --> 00:07:01.370
this ratio is a trig
function applied

00:07:01.370 --> 00:07:02.562
to one of the other angles.

00:07:02.562 --> 00:07:04.020
And they give us
one of the angles.

00:07:04.020 --> 00:07:05.686
They give us this
angle right over here.

00:07:05.686 --> 00:07:08.080
I guess we could
call this angle DAC.

00:07:08.080 --> 00:07:09.990
This is 30 degrees.

00:07:09.990 --> 00:07:12.145
So relative to this
angle, what two sides

00:07:12.145 --> 00:07:13.270
are we taking the ratio of?

00:07:13.270 --> 00:07:15.780
We're taking, now, the ratio
of, relative to this angle,

00:07:15.780 --> 00:07:19.850
the adjacent side
over the hypotenuse.

00:07:19.850 --> 00:07:22.440
So this is the adjacent
side over the hypotenuse.

00:07:22.440 --> 00:07:24.260
What deals with adjacent
over hypotenuse?

00:07:24.260 --> 00:07:25.840
Well, cosine.

00:07:25.840 --> 00:07:28.670
So this is equal to the
cosine of this angle.

00:07:28.670 --> 00:07:32.960
So this is equal to
cosine of 30 degrees.

00:07:32.960 --> 00:07:36.770
Sine of CDA is
equal to the cosine

00:07:36.770 --> 00:07:39.180
of this angle right over here.

00:07:39.180 --> 00:07:44.490
So this one is equal to
this right over here.

00:07:44.490 --> 00:07:48.730
So let me drag that in.

00:07:48.730 --> 00:07:52.182
So this one is equal to--
so you can see that I just

00:07:52.182 --> 00:07:53.510
dragged it in-- equal to that.

00:07:53.510 --> 00:07:55.320
Now we have one left.

00:07:55.320 --> 00:07:56.180
We have one left.

00:07:56.180 --> 00:07:58.740
Home stretch, we should
be getting excited.

00:07:58.740 --> 00:08:00.770
AE over EB.

00:08:00.770 --> 00:08:05.150
AE, let me use this color.

00:08:05.150 --> 00:08:06.450
Length of segment AE.

00:08:06.450 --> 00:08:08.820
That's this length
right over here.

00:08:08.820 --> 00:08:10.540
Let me make that stand out more.

00:08:10.540 --> 00:08:12.501
Let me do it in this red.

00:08:12.501 --> 00:08:16.240
This color right over here,
that's length of segment AE,

00:08:16.240 --> 00:08:18.734
over length of segment EB.

00:08:22.250 --> 00:08:25.180
This is EB right over here.

00:08:25.180 --> 00:08:27.320
This is EB.

00:08:27.320 --> 00:08:33.590
So now we are focused on this
right triangle right over here.

00:08:33.590 --> 00:08:35.890
Well, we know the measure
of this angle over here.

00:08:35.890 --> 00:08:42.270
We have double arcs
right over here,

00:08:42.270 --> 00:08:44.500
and they say this is 41 degrees.

00:08:44.500 --> 00:08:47.320
So we have double
marks over here,

00:08:47.320 --> 00:08:49.960
and this is also going
to be 41 degrees.

00:08:49.960 --> 00:08:52.620
So relative to this
angle, what ratio is this?

00:08:52.620 --> 00:08:56.860
This is the opposite
over the hypotenuse.

00:08:56.860 --> 00:08:59.900
Opposite over the hypotenuse,
this right over here

00:08:59.900 --> 00:09:04.090
is going to be sine of that
angle, sine of 41 degrees.

00:09:04.090 --> 00:09:07.460
So it's equal to this
first one right over there.

00:09:07.460 --> 00:09:10.950
So let's drag it.

00:09:10.950 --> 00:09:13.190
So this is going to be
equal to sine of 41 degrees.

00:09:13.190 --> 00:09:14.890
So none of the ones
actually ended up

00:09:14.890 --> 00:09:17.960
being equal to the
tangent of 41 degrees.

00:09:17.960 --> 00:09:20.440
Now let's see if we
actually got this right.

00:09:20.440 --> 00:09:22.100
I hope I did.

00:09:22.100 --> 00:09:23.830
We did.


Pythagorean theorem and radii of circles

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:03.344
The center of this
circle is O. And I

00:00:03.344 --> 00:00:05.010
apologize if I'm a
little out of breath,

00:00:05.010 --> 00:00:06.550
I actually just
did some pull-ups.

00:00:06.550 --> 00:00:08.720
Anyway, the center
of this circle

00:00:08.720 --> 00:00:19.430
is O. Find the exact
length of OA, CD, and OF.

00:00:19.430 --> 00:00:21.380
So let's look at each of these.

00:00:21.380 --> 00:00:24.780
So CD, it's part of
a right triangle.

00:00:24.780 --> 00:00:27.290
It's one of the two shorter
sides of a right triangle.

00:00:27.290 --> 00:00:28.990
But we don't know
the hypotenuse,

00:00:28.990 --> 00:00:30.906
so we're not going to
be able to figure out CD

00:00:30.906 --> 00:00:32.090
out right, just yet.

00:00:32.090 --> 00:00:33.282
Same thing with OF.

00:00:33.282 --> 00:00:37.160
OF is one of the two shorter
sides of a right triangle.

00:00:37.160 --> 00:00:39.650
But we don't know its
hypotenuse either.

00:00:39.650 --> 00:00:41.094
Now let's look at OA.

00:00:41.094 --> 00:00:44.530
OA is the hypotenuse
of a right triangle,

00:00:44.530 --> 00:00:46.270
and they've given us
the two other sides.

00:00:46.270 --> 00:00:48.190
So we can use the
Pythagorean theorem

00:00:48.190 --> 00:00:50.250
to figure out the hypotenuse.

00:00:50.250 --> 00:00:54.680
So we know that 7 squared,
let's call this x.

00:00:54.680 --> 00:01:03.790
We know that 7 squared
plus 24 squared

00:01:03.790 --> 00:01:06.580
is going to be equal to
the length of OA squared.

00:01:06.580 --> 00:01:09.260
It's going to be
equal to x squared.

00:01:09.260 --> 00:01:13.532
7 squared is 49,
and 24 squared, well

00:01:13.532 --> 00:01:15.240
let's do a multiplication
right over here

00:01:15.240 --> 00:01:18.560
to figure out 24 times 24.

00:01:18.560 --> 00:01:20.660
4 times 4 is 16.

00:01:20.660 --> 00:01:24.570
4 times 2 is 8
plus-- so that's 96.

00:01:24.570 --> 00:01:28.970
And then two times 24 is 48.

00:01:28.970 --> 00:01:31.380
Add them together, we get 6.

00:01:31.380 --> 00:01:36.590
9 plus 8 is 17, 576.

00:01:36.590 --> 00:01:42.160
So 49 plus 576 is
equal to x squared.

00:01:42.160 --> 00:01:45.660
And so let's think about
what this is going to be.

00:01:45.660 --> 00:01:51.680
And this is going to be the
same thing as 50 plus 575.

00:01:51.680 --> 00:01:54.120
I just took one away from
this and added one here.

00:01:54.120 --> 00:01:58.220
So 50 plus 575 is 625.

00:01:58.220 --> 00:02:04.560
So 625 is equal to x squared.

00:02:04.560 --> 00:02:11.610
And you might recognize
that 25 times 25 is 625.

00:02:11.610 --> 00:02:16.890
So x is equal to 25.

00:02:16.890 --> 00:02:20.110
And if you don't believe me
you could multiply that out

00:02:20.110 --> 00:02:21.750
on your own.

00:02:21.750 --> 00:02:24.260
So x is equal to 25.

00:02:24.260 --> 00:02:27.810
Or another way of thinking
about it, the exact length of OA

00:02:27.810 --> 00:02:32.570
is equal to 25.

00:02:32.570 --> 00:02:35.480
Now, how can we somehow
use that information

00:02:35.480 --> 00:02:37.460
to figure out this other stuff?

00:02:37.460 --> 00:02:39.900
Well all of these
other right triangles,

00:02:39.900 --> 00:02:45.310
all of their hypotenuses
are a radius of the circle,

00:02:45.310 --> 00:02:46.085
and so is OA.

00:02:46.085 --> 00:02:47.790
OA is a radius of the circle.

00:02:47.790 --> 00:02:49.520
OG is a radius of the circle.

00:02:49.520 --> 00:02:51.760
OC is a radius of the circle.

00:02:51.760 --> 00:02:54.600
Well, we just figured out the
radius of the circle is 25.

00:02:54.600 --> 00:03:01.560
So OG is going to be 25, and
OC is going to be 25 as well.

00:03:01.560 --> 00:03:04.820
So now we just have to apply the
Pythagorean theorem a few more

00:03:04.820 --> 00:03:06.340
times.

00:03:06.340 --> 00:03:08.780
So right over
here, if I call OF,

00:03:08.780 --> 00:03:11.280
let's just call that, I don't
know for the sake of argument,

00:03:11.280 --> 00:03:14.150
let's call that
length equal to a.

00:03:14.150 --> 00:03:17.040
So here, for this
triangle, we see

00:03:17.040 --> 00:03:21.510
that a squared plus
the square root of 141

00:03:21.510 --> 00:03:28.270
squared-- I'll just write that
as 141-- so plus 141 is going

00:03:28.270 --> 00:03:35.950
to be equal to 25 squared,
which we already know to be 625.

00:03:35.950 --> 00:03:39.590
If we subtract 141
from both sides

00:03:39.590 --> 00:03:40.840
let's see where do we get.

00:03:40.840 --> 00:03:48.600
So let's do 625 minus 141
we get 5 minus 1 is 4.

00:03:48.600 --> 00:03:52.610
And then 12 here, and
we can put a 5 there.

00:03:52.610 --> 00:03:54.744
12 minus 4 is 8.

00:03:54.744 --> 00:03:57.120
5 minus 1 is 4.

00:03:57.120 --> 00:03:58.770
So we get 484.

00:03:58.770 --> 00:04:07.810
So we get a squared
is equal to 484.

00:04:07.810 --> 00:04:13.127
So what squared is equal to 484?

00:04:13.127 --> 00:04:15.460
Actually, I'll just try to
do a prime factorization here

00:04:15.460 --> 00:04:16.251
to figure this out.

00:04:16.251 --> 00:04:30.260
So 484 is 2 times 242,
which is 2 times 121, which

00:04:30.260 --> 00:04:34.579
is the same thing
as 11 times 11.

00:04:34.579 --> 00:04:36.370
So another way of
thinking about it is this

00:04:36.370 --> 00:04:39.860
is 2 squared-- so 484,
I'll write it over here.

00:04:39.860 --> 00:04:43.960
484 is equal to 2
squared times 11 squared.

00:04:43.960 --> 00:04:46.120
Or it's the same
thing as 2 times 11

00:04:46.120 --> 00:04:50.190
squared, which is 22 squared.

00:04:50.190 --> 00:04:53.620
So in this case a is equal to--
let me just clean all that up

00:04:53.620 --> 00:04:58.260
so I have some space to work
with-- a is equal to 22.

00:04:58.260 --> 00:04:59.910
Let me write that down.

00:04:59.910 --> 00:05:04.360
a is equal to 22, so that's
equal to 22 right here.

00:05:04.360 --> 00:05:06.880
And that's the
length of segment OF.

00:05:06.880 --> 00:05:08.630
So this is 22.

00:05:08.630 --> 00:05:10.830
And then finally CD,
once again we just

00:05:10.830 --> 00:05:12.810
apply the Pythagorean theorem.

00:05:12.810 --> 00:05:15.190
Let's just call this, I don't
know, I've already used a.

00:05:15.190 --> 00:05:16.160
I've already used x.

00:05:16.160 --> 00:05:20.055
I don't know, I'll call this b.

00:05:20.055 --> 00:05:24.760
So we see that b squared
plus 15 squared, which

00:05:24.760 --> 00:05:28.760
is the same thing as
225-- 15 squared is 225--

00:05:28.760 --> 00:05:30.930
is going to be
equal to 25 squared,

00:05:30.930 --> 00:05:33.450
is going to be equal to 625.

00:05:33.450 --> 00:05:38.970
Subtract 225 from both sides you
get b squared is equal to 400,

00:05:38.970 --> 00:05:41.620
and the square root of 400
is pretty easy to calculate.

00:05:41.620 --> 00:05:43.200
B is equal to 20.

00:05:45.870 --> 00:05:51.861
So segment OA is 25,
CD is 20, and OF is 22.


Equation for a circle using the Pythagorean Theorem

WEBVTT
Kind: captions
Language: en

00:00:00.970 --> 00:00:04.030
So I've drawn a circle
here with radius r.

00:00:04.030 --> 00:00:07.940
And its center is at
the point, h comma k.

00:00:07.940 --> 00:00:13.900
So its x-coordinate is
h, and its y-coordinate

00:00:13.900 --> 00:00:16.200
is k right over here.

00:00:16.200 --> 00:00:20.610
What I want to do is figure
out a general formula

00:00:20.610 --> 00:00:23.555
for the equation of this circle.

00:00:23.555 --> 00:00:24.930
In order to do
that, we just have

00:00:24.930 --> 00:00:27.550
to remind ourselves that
a circle, all the points

00:00:27.550 --> 00:00:30.410
on the circle, are the
ones that are exactly

00:00:30.410 --> 00:00:32.619
the radius away from the center.

00:00:32.619 --> 00:00:37.086
Their distance from the center
is exactly going to be r.

00:00:37.086 --> 00:00:38.710
So let's think about
this a little bit.

00:00:38.710 --> 00:00:42.720
What constraints do we have to
put on our x's and y's for them

00:00:42.720 --> 00:00:47.180
to be exactly r away
from the center?

00:00:47.180 --> 00:00:51.940
So let's say that this right
over here is one of the points,

00:00:51.940 --> 00:00:54.190
so this is x comma y.

00:00:54.190 --> 00:00:57.070
And this is one of the
points that is r away,

00:00:57.070 --> 00:00:59.830
exactly r away from the center.

00:00:59.830 --> 00:01:01.620
So what's a
relationship going to be

00:01:01.620 --> 00:01:04.661
between x, y; h, k; and r?

00:01:04.661 --> 00:01:07.160
And I could have put this point
x, y anywhere on the circle.

00:01:07.160 --> 00:01:08.618
I could have done
it here, as well.

00:01:08.618 --> 00:01:09.750
This would also be r away.

00:01:09.750 --> 00:01:14.182
This is also going to be
r away from the center.

00:01:14.182 --> 00:01:16.140
Well, it might not jump
out at you immediately,

00:01:16.140 --> 00:01:19.200
but we can actually use
the Pythagorean Theorem.

00:01:19.200 --> 00:01:22.970
All we need to do, let's draw
one going from the center.

00:01:22.970 --> 00:01:26.910
Let's draw one segment that
is parallel to the x-axis,

00:01:26.910 --> 00:01:28.310
going like this.

00:01:28.310 --> 00:01:30.520
And let's draw another
segment starting at x, y,

00:01:30.520 --> 00:01:34.530
that is parallel to the y-axis,
that's completely vertical.

00:01:34.530 --> 00:01:36.800
So this one is
completely vertical.

00:01:36.800 --> 00:01:38.650
This one is
completely horizontal.

00:01:38.650 --> 00:01:42.690
So this is going to be a
right angle right over here.

00:01:42.690 --> 00:01:44.490
So if we can figure
out the-- if we

00:01:44.490 --> 00:01:49.370
can express the height of this
line in terms of the variables

00:01:49.370 --> 00:01:51.310
we've just put out,
and if we can express

00:01:51.310 --> 00:01:53.640
the height or the width
of this line right

00:01:53.640 --> 00:01:55.780
over here in terms of
these variables somehow,

00:01:55.780 --> 00:01:58.360
then we can relate
those between that

00:01:58.360 --> 00:02:00.780
and r using the
Pythagorean Theorem. r

00:02:00.780 --> 00:02:03.690
is the hypotenuse of
this right triangle.

00:02:03.690 --> 00:02:06.190
Well, what's this
height going to be?

00:02:06.190 --> 00:02:08.570
Well, we know this
point right over here

00:02:08.570 --> 00:02:10.619
has a y-coordinate of k.

00:02:10.619 --> 00:02:12.160
This point over
here, well, that just

00:02:12.160 --> 00:02:14.500
has a y-coordinate, I
guess we could just say,

00:02:14.500 --> 00:02:17.440
that has a y-coordinate of y.

00:02:17.440 --> 00:02:21.190
So this distance
right over here is

00:02:21.190 --> 00:02:24.060
going to be the same exact
thing as this distance

00:02:24.060 --> 00:02:25.200
right over here.

00:02:25.200 --> 00:02:27.420
So it's going to be y minus k.

00:02:33.310 --> 00:02:35.060
And what's this
distance going to be?

00:02:35.060 --> 00:02:36.310
Well, same exact argument.

00:02:36.310 --> 00:02:37.630
This is h.

00:02:37.630 --> 00:02:40.590
And this point right over
here, if we go all the way,

00:02:40.590 --> 00:02:44.890
or this point on the x-axis,
well, we'll just call that x.

00:02:44.890 --> 00:02:47.750
So what's this distance
right over here?

00:02:47.750 --> 00:02:53.452
Well, this distance is
going to be x minus h.

00:02:57.610 --> 00:03:01.380
So what is the relationship
between this side of length x

00:03:01.380 --> 00:03:04.280
minus h, this side of
length y minus k, and r?

00:03:04.280 --> 00:03:06.210
Well, it's the
Pythagorean Theorem.

00:03:06.210 --> 00:03:08.250
The sum of the squares
of the two shorter sides

00:03:08.250 --> 00:03:10.880
is going to be equal to the
square of the hypotenuse.

00:03:10.880 --> 00:03:14.180
So one of the shorter
sides has length this,

00:03:14.180 --> 00:03:15.630
has length x minus h.

00:03:15.630 --> 00:03:17.410
So I could write x minus h.

00:03:17.410 --> 00:03:20.710
So the length of
that side squared

00:03:20.710 --> 00:03:24.920
plus the length of this side--
let me do this in a color

00:03:24.920 --> 00:03:28.400
that I haven't used yet--
plus the length of this side

00:03:28.400 --> 00:03:33.460
squared, so plus
y minus k squared

00:03:33.460 --> 00:03:36.170
is going to be equal to
the hypotenuse squared,

00:03:36.170 --> 00:03:39.780
is going to be equal
to the radius squared.

00:03:39.780 --> 00:03:44.250
And just like that, we have
found the general equation

00:03:44.250 --> 00:03:51.290
of a circle, where h comma k
is the center of the circle

00:03:51.290 --> 00:03:53.060
and r is the radius.

00:03:53.060 --> 00:03:58.220
Any combination of x's and y's
that meet this, that for which

00:03:58.220 --> 00:04:01.180
this is true given your
center, given h and k

00:04:01.180 --> 00:04:07.450
and given the radius,
then any x's and y's that

00:04:07.450 --> 00:04:11.210
will meet these constraints
are going to be on the circle.

00:04:11.210 --> 00:04:13.780
So another way you could think
about it, if someone said,

00:04:13.780 --> 00:04:17.399
hey, look I have
a center-- let me

00:04:17.399 --> 00:04:22.164
write it this way-- the center
of my circle is at the point,

00:04:22.164 --> 00:04:25.920
I don't know, 5, negative 5.

00:04:25.920 --> 00:04:32.200
And if they said that
the radius is equal to 4,

00:04:32.200 --> 00:04:34.930
then, we could immediately say
what the equation of the circle

00:04:34.930 --> 00:04:35.990
is going to be.

00:04:35.990 --> 00:04:39.460
It's going to be x minus
this right over here-- let me

00:04:39.460 --> 00:04:45.006
color code this a little bit--
this right over here is our h.

00:04:45.006 --> 00:04:48.380
This right over here is our k.

00:04:48.380 --> 00:04:50.240
And this is our radius.

00:04:50.240 --> 00:04:52.530
This right over
here is our radius.

00:04:52.530 --> 00:04:57.160
So the equation of the
circle would be x minus h,

00:04:57.160 --> 00:05:10.030
so x minus 5 squared plus
y minus k. k is negative 5.

00:05:10.030 --> 00:05:12.670
So if you subtract
negative 5, that

00:05:12.670 --> 00:05:15.770
would be the same thing
as adding positive 5.

00:05:15.770 --> 00:05:20.970
So y plus 5 squared
is equal to r squared.

00:05:20.970 --> 00:05:25.690
Well, r in the example I
just made up is equal to 16.

00:05:25.690 --> 00:05:28.140
So someone told you that the
center is at 5, negative 5,

00:05:28.140 --> 00:05:29.786
the radius is
four, well, this is

00:05:29.786 --> 00:05:31.410
going to be the
equation of the circle.

00:05:31.410 --> 00:05:33.530
On the other hand,
if someone gave you

00:05:33.530 --> 00:05:35.860
this as the equation
of the circle,

00:05:35.860 --> 00:05:38.100
you know that the
center is going

00:05:38.100 --> 00:05:44.390
to be, well, it's going
to be at x equals 5.

00:05:44.390 --> 00:05:47.280
And y equals, and
since this is a plus,

00:05:47.280 --> 00:05:51.310
you know that this is going
to have to be, they're k.

00:05:51.310 --> 00:05:54.690
The center is going to
be at 5, negative 5.

00:05:54.690 --> 00:05:56.120
Another way to
think about is what

00:05:56.120 --> 00:05:59.410
are the x and y's that will
make each of these terms

00:05:59.410 --> 00:06:00.200
equal to zero?

00:06:00.200 --> 00:06:02.890
So if you have x equals
5 here, then this thing's

00:06:02.890 --> 00:06:04.100
going to be equal to 0.

00:06:04.100 --> 00:06:06.500
If you have y equals
negative 5, then

00:06:06.500 --> 00:06:09.370
this thing is going
to be equal to 0.

00:06:09.370 --> 00:06:12.824
If you had a circle centered
at the origin at zero,

00:06:12.824 --> 00:06:15.490
then you're just going to have x
squared plus y squared is going

00:06:15.490 --> 00:06:17.543
to be equal to the
radius squared.


Hypotenuse of right triangle inscribed in circle

WEBVTT
Kind: captions
Language: en

00:00:00.684 --> 00:00:02.100
What I want to do
in this video is

00:00:02.100 --> 00:00:05.570
attempt to find the diameter
of this circle right over here.

00:00:05.570 --> 00:00:07.910
And I encourage you
to pause the video

00:00:07.910 --> 00:00:10.990
and try this out on your own.

00:00:10.990 --> 00:00:13.920
Well, let's think about what's
going on right over here.

00:00:13.920 --> 00:00:17.545
AB is definitely a
diameter of the circle.

00:00:17.545 --> 00:00:18.420
It's a straight line.

00:00:18.420 --> 00:00:20.290
It's going through the
center of the circle.

00:00:20.290 --> 00:00:23.250
O is the center of the
circle right over here.

00:00:23.250 --> 00:00:25.210
And so what do we know?

00:00:25.210 --> 00:00:28.630
Well, we could look at this
angle right over here, angle C,

00:00:28.630 --> 00:00:32.090
and think about it is
an inscribed angle.

00:00:32.090 --> 00:00:34.900
And think about the
arc that it intercepts.

00:00:34.900 --> 00:00:38.800
It intercepts this
arc right over here.

00:00:38.800 --> 00:00:41.525
This arc is exactly
half of the circle.

00:00:45.170 --> 00:00:46.660
Angle C is inscribed.

00:00:46.660 --> 00:00:50.330
If you take these two sides
or the two sides of the angle,

00:00:50.330 --> 00:00:54.690
it intercepts at A and B,
and so it intercepts an arc,

00:00:54.690 --> 00:00:56.630
this green arc right over here.

00:00:56.630 --> 00:01:00.694
So the central angle right
over here is 180 degrees,

00:01:00.694 --> 00:01:02.860
and the inscribed angle is
going to be half of that.

00:01:02.860 --> 00:01:04.680
It's going to be 90 degrees.

00:01:04.680 --> 00:01:06.230
Or another way of
thinking about it,

00:01:06.230 --> 00:01:08.460
it's going to be a right angle.

00:01:08.460 --> 00:01:11.050
And what that does
for us is it tells us

00:01:11.050 --> 00:01:19.770
that triangle ACB
is a right triangle.

00:01:19.770 --> 00:01:24.110
This is a right triangle, and
the diameter is its hypotenuse.

00:01:24.110 --> 00:01:26.560
So we can just apply the
Pythagorean theorem here.

00:01:26.560 --> 00:01:37.800
15 squared plus 8 squared--
let me do this in magenta--

00:01:37.800 --> 00:01:41.890
is going to be the length
of side AB squared.

00:01:41.890 --> 00:01:47.030
So this side right over here,
let me just call that x.

00:01:47.030 --> 00:01:51.630
That's going to be
equal to x squared.

00:01:51.630 --> 00:01:55.700
So 15 squared, that's 225.

00:01:55.700 --> 00:02:01.850
8 squared is 64, plus
64-- I want to do

00:02:01.850 --> 00:02:06.650
that in green-- is
equal to x squared.

00:02:06.650 --> 00:02:15.110
225 plus 64 is 289 is
equal to x squared.

00:02:15.110 --> 00:02:19.572
And then 289 is 17 squared.

00:02:19.572 --> 00:02:21.030
And you could try
out a few numbers

00:02:21.030 --> 00:02:22.700
if you're unsure about that.

00:02:22.700 --> 00:02:26.430
So x is equal to 17.

00:02:26.430 --> 00:02:32.130
So the diameter of this
circle right over here is 17.


Example with tangent and radius

WEBVTT
Kind: captions
Language: en

00:00:00.960 --> 00:00:04.980
Line AC is tangent to
circle O at point C.

00:00:04.980 --> 00:00:09.950
So this is line AC, tangent
to circle O at point

00:00:09.950 --> 00:00:13.640
C. What is the
length of segment AC?

00:00:13.640 --> 00:00:16.850
What is this distance right over
here, between point A and point

00:00:16.850 --> 00:00:17.350
C?

00:00:17.350 --> 00:00:19.550
And I encourage you
now to pause this video

00:00:19.550 --> 00:00:22.290
and try this out on your own.

00:00:22.290 --> 00:00:24.500
So I'm assuming you've
given a go at it.

00:00:24.500 --> 00:00:26.580
So the key thing
to realize here,

00:00:26.580 --> 00:00:31.280
since AC is tangent to the
circle at point C, that

00:00:31.280 --> 00:00:33.770
means it's going to be
perpendicular to the radius

00:00:33.770 --> 00:00:36.710
between the center of
the circle and point C.

00:00:36.710 --> 00:00:40.480
So this right over
here is a right angle.

00:00:40.480 --> 00:00:42.700
And the reason
why that is useful

00:00:42.700 --> 00:00:47.660
is now we know that triangle
AOC is a right triangle.

00:00:47.660 --> 00:00:49.576
So if we know two
of its sides, we

00:00:49.576 --> 00:00:50.950
could use the
Pythagorean theorem

00:00:50.950 --> 00:00:52.660
to figure out the third.

00:00:52.660 --> 00:00:55.260
Now, we clearly know OC.

00:00:55.260 --> 00:00:57.700
Now OA, we don't
know the entire side.

00:00:57.700 --> 00:01:00.760
They only give us
that AB is equal to 2.

00:01:00.760 --> 00:01:03.410
But the thing that might
jump out in your mind

00:01:03.410 --> 00:01:06.820
is OB is a radius.

00:01:06.820 --> 00:01:09.190
It's going to be the same
length as any radius.

00:01:09.190 --> 00:01:10.967
So this is going
to be 3 as well.

00:01:10.967 --> 00:01:13.050
It's the distance between
the center of the circle

00:01:13.050 --> 00:01:15.920
and a point on the circle, just
like the distance between O

00:01:15.920 --> 00:01:18.300
and C. So this is
going to be 3 as well.

00:01:18.300 --> 00:01:20.220
And so now we are
able to figure out

00:01:20.220 --> 00:01:26.760
that the hypotenuse of
this triangle has length 5.

00:01:26.760 --> 00:01:31.620
And so we need to figure out
what the length of segment AC

00:01:31.620 --> 00:01:32.190
is.

00:01:32.190 --> 00:01:33.773
So let's just call
that, I don't know.

00:01:33.773 --> 00:01:38.470
I'll call that x.

00:01:38.470 --> 00:01:42.430
And so we know that x
squared plus 3 squared--

00:01:42.430 --> 00:01:46.340
I'm just applying the
Pythagorean theorem here--

00:01:46.340 --> 00:01:48.800
is going to be equal to the
length of the hypotenuse

00:01:48.800 --> 00:01:52.594
squared, is going to
be equal to 5 squared.

00:01:52.594 --> 00:01:54.010
And I know this
is the hypotenuse.

00:01:54.010 --> 00:01:56.090
It's the side opposite
the 90-degree angle.

00:01:56.090 --> 00:01:59.530
It's the longest side
of the right triangle.

00:01:59.530 --> 00:02:09.650
So x squared plus
9 is equal to 25.

00:02:09.650 --> 00:02:13.900
Subtract 9 from
both sides, and you

00:02:13.900 --> 00:02:18.020
get x squared is equal to 16.

00:02:18.020 --> 00:02:19.940
And so it should jump
out at you that x

00:02:19.940 --> 00:02:23.810
is going to be equal to 4.

00:02:23.810 --> 00:02:25.440
So x is equal to 4.

00:02:25.440 --> 00:02:29.860
x is the same thing as
the length of segment AC,

00:02:29.860 --> 00:02:32.657
so the length of
segment AC is 4.


Measure of circumscribed angle

WEBVTT
Kind: captions
Language: en

00:00:00.860 --> 00:00:05.470
Angle A is a circumscribed
angle on circle O.

00:00:05.470 --> 00:00:07.950
So this is angle
A right over here.

00:00:07.950 --> 00:00:11.360
Then when they say it's
a circumscribed angle,

00:00:11.360 --> 00:00:13.550
that means that the
two sides of the angle

00:00:13.550 --> 00:00:15.820
are tangent to the circle.

00:00:15.820 --> 00:00:18.530
So AC is tangent to
the circle at point

00:00:18.530 --> 00:00:23.090
C. AB is tangent to
the circle at point B.

00:00:23.090 --> 00:00:25.440
What is the measure of angle A?

00:00:25.440 --> 00:00:28.730
Now, I encourage you
to pause the video now

00:00:28.730 --> 00:00:30.840
and to try this out on your own.

00:00:30.840 --> 00:00:32.600
And I'll give you a hint.

00:00:32.600 --> 00:00:35.960
It will leverage the fact that
this is a circumscribed angle

00:00:35.960 --> 00:00:38.580
as you could imagine.

00:00:38.580 --> 00:00:41.140
So I'm assuming you've
given a go at it.

00:00:41.140 --> 00:00:42.560
So the other piece
of information

00:00:42.560 --> 00:00:45.490
they give us is that angle D,
which is an inscribed angle,

00:00:45.490 --> 00:00:50.960
is 48 degrees and it intercepts
the same arc-- so this

00:00:50.960 --> 00:00:53.940
is the arc that it intercepts,
arc CB I guess you could call

00:00:53.940 --> 00:00:56.640
it-- it intercepts this
arc right over here.

00:00:56.640 --> 00:00:57.740
It's the inscribed angle.

00:00:57.740 --> 00:01:02.320
The central angle that
intersects that same arc

00:01:02.320 --> 00:01:04.840
is going to be twice
the inscribed angle.

00:01:04.840 --> 00:01:07.235
So this is going
to be 96 degrees.

00:01:07.235 --> 00:01:09.610
I could put three markers here
just because we've already

00:01:09.610 --> 00:01:11.370
used the double marker.

00:01:11.370 --> 00:01:15.855
Notice, they both intercept
arc CB so some people would

00:01:15.855 --> 00:01:18.530
say the measure of
arc CB is 96 degrees,

00:01:18.530 --> 00:01:21.180
the central angle is 96
degrees, the inscribed angle

00:01:21.180 --> 00:01:23.570
is going to be half
of that, 48 degrees.

00:01:23.570 --> 00:01:25.850
So how does this help us?

00:01:25.850 --> 00:01:29.560
Well, a key clue is that angle
is a circumscribed angle.

00:01:29.560 --> 00:01:34.410
So that means AC and AB are
each tangent to the circle.

00:01:34.410 --> 00:01:37.450
Well, a line that is
tangent to the circle

00:01:37.450 --> 00:01:40.830
is going to be perpendicular to
the radius of the circle that

00:01:40.830 --> 00:01:44.570
intersects the circle
at the same point.

00:01:44.570 --> 00:01:49.930
So this right over here is
going to be a 90-degree angle,

00:01:49.930 --> 00:01:53.730
and this right over here is
going to be a 90-degree angle.

00:01:53.730 --> 00:01:56.380
OC is perpendicular to CA.

00:01:56.380 --> 00:02:00.330
OB, which is a radius,
is perpendicular to BA,

00:02:00.330 --> 00:02:03.160
which is a tangent line, and
they both intersect right

00:02:03.160 --> 00:02:06.520
over here at B. Now, this
might jump out at you.

00:02:06.520 --> 00:02:08.509
We have a quadrilateral
going on here.

00:02:08.509 --> 00:02:13.480
ABOC is a quadrilateral,
so its sides

00:02:13.480 --> 00:02:20.310
are going to add
up to 360 degrees.

00:02:20.310 --> 00:02:23.170
So we could know, we
could write it this way.

00:02:23.170 --> 00:02:26.380
We could write the
measure of angle A

00:02:26.380 --> 00:02:37.930
plus 90 degrees plus another
90 degrees plus 96 degrees

00:02:37.930 --> 00:02:40.880
is going to be equal
to 360 degrees.

00:02:46.620 --> 00:02:49.720
Or another way of thinking
about it, if we subtract 180

00:02:49.720 --> 00:02:52.670
from both sides, if we
subtract that from both sides,

00:02:52.670 --> 00:02:59.870
we get the measure of
angle A plus 96 degrees

00:02:59.870 --> 00:03:05.142
is going to be equal
to 180 degrees.

00:03:05.142 --> 00:03:06.600
Or another way of
thinking about it

00:03:06.600 --> 00:03:09.950
is the measure of angle A
or that angle A and angle O

00:03:09.950 --> 00:03:12.860
right over here-- you
could call it angle COB--

00:03:12.860 --> 00:03:15.620
that these are going to be
supplementary angles if they

00:03:15.620 --> 00:03:18.990
add up to 180 degrees.

00:03:18.990 --> 00:03:22.130
So if we subtract 96
degrees from both sides,

00:03:22.130 --> 00:03:27.561
we get the measure
of angle A is equal

00:03:27.561 --> 00:03:30.060
to-- I don't want to make that
look like a less than symbol,

00:03:30.060 --> 00:03:32.380
let make it-- measure of
angle-- this one actually

00:03:32.380 --> 00:03:35.010
looks more like a--
measure of angle A

00:03:35.010 --> 00:03:37.980
is equal to 180 minus 96.

00:03:37.980 --> 00:03:39.840
Let's see, 180 minus
90 would be 90,

00:03:39.840 --> 00:03:46.190
and then we subtract another
6 gets us to 84 degrees.


Help support Khan Academy

WEBVTT
Kind: captions
Language: en

00:00:01.821 --> 00:00:04.443
Sal: Hi, this is Sal
Khan of the Khan Academy.

00:00:04.443 --> 00:00:06.363
I just wanted to thank everyone out there

00:00:06.363 --> 00:00:08.043
for another great year.

00:00:08.043 --> 00:00:10.675
We're now reaching over 10
million students per month

00:00:10.675 --> 00:00:12.056
and then many of you all have noticed

00:00:12.056 --> 00:00:13.781
amazing partnerships and the content

00:00:13.781 --> 00:00:16.836
that we've been able to work
together to put out there.

00:00:16.836 --> 00:00:19.589
One thing I do like remind
people at this time of year

00:00:19.589 --> 00:00:23.250
is that we are a 501(c)(3)
not-for-profit organization.

00:00:23.250 --> 00:00:24.864
The only reason why we can exist,

00:00:24.864 --> 00:00:27.027
why we can continue to
put out this content,

00:00:27.027 --> 00:00:30.919
is with the help of
supporters like yourself.

00:00:30.919 --> 00:00:32.094
I'll just throw that out there.

00:00:32.094 --> 00:00:33.607
If you're in the position,

00:00:33.607 --> 00:00:37.192
whether it's $5, $10,
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every penny is very much appreciated by us

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Solving a consecutive integer problem algebraically

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:06.070
The sum of four consecutive
odd integers is 136.

00:00:06.070 --> 00:00:08.390
What are the four integers?

00:00:08.390 --> 00:00:10.020
So before even
attempting to tackle it,

00:00:10.020 --> 00:00:11.970
let's think about
what it means to be

00:00:11.970 --> 00:00:15.460
a consecutive odd integer, what
four consecutive odd integers

00:00:15.460 --> 00:00:16.880
could be referring to.

00:00:16.880 --> 00:00:18.900
So we could start,
let's say, we had 3.

00:00:18.900 --> 00:00:21.580
The next odd integer-- so we
want consecutive odd integers.

00:00:21.580 --> 00:00:24.600
So we want the next odd
integer-- would be 5.

00:00:24.600 --> 00:00:26.820
Then the one after that
would be 7, and the one

00:00:26.820 --> 00:00:28.090
after that would be 9.

00:00:28.090 --> 00:00:30.310
These are consecutive
odd integers.

00:00:30.310 --> 00:00:32.270
Another example-- we
could start at 11.

00:00:32.270 --> 00:00:35.390
Then the next odd integer is 13.

00:00:35.390 --> 00:00:37.010
The next one is 15.

00:00:37.010 --> 00:00:38.650
The next one is 17.

00:00:38.650 --> 00:00:41.510
The example of
non-consecutive odd integers,

00:00:41.510 --> 00:00:44.870
if someone went from
3 straight to 7,

00:00:44.870 --> 00:00:46.280
these are not consecutive.

00:00:46.280 --> 00:00:51.340
The next odd integer
after 3 is 5, not 7.

00:00:51.340 --> 00:00:54.010
So these are examples of
consecutive odd integers.

00:00:54.010 --> 00:00:56.940
These are not
consecutive odd integers.

00:00:56.940 --> 00:00:58.750
So with that out of
the way, let's actually

00:00:58.750 --> 00:01:00.410
try to tackle this question.

00:01:00.410 --> 00:01:03.330
And I encourage you to pause
this video right now and try

00:01:03.330 --> 00:01:08.150
to tackle this on your own
before I take a go at it.

00:01:08.150 --> 00:01:10.400
Well as you can imagine,
a little bit of algebra

00:01:10.400 --> 00:01:11.860
might be handy here.

00:01:11.860 --> 00:01:20.460
So let's let x be equal to
the smallest of the four.

00:01:24.910 --> 00:01:32.440
So if x is the smallest of the
four consecutive odd integers,

00:01:32.440 --> 00:01:36.850
how can we express the
other three in terms of x?

00:01:36.850 --> 00:01:39.740
We'll just use these
examples right over here.

00:01:39.740 --> 00:01:43.650
If that is x, then how would
we express this in terms of x?

00:01:43.650 --> 00:01:46.810
Well, this would be x plus 2.

00:01:46.810 --> 00:01:49.020
The next odd integer
is going to be

00:01:49.020 --> 00:01:51.530
the previous odd integer plus 2.

00:01:51.530 --> 00:01:53.707
You're essentially
skipping the next number,

00:01:53.707 --> 00:01:54.790
which is going to be even.

00:01:54.790 --> 00:01:56.650
So if you added 1, you'd
just get to an even number,

00:01:56.650 --> 00:01:58.770
so you have to add 2 to
get to the next odd one.

00:01:58.770 --> 00:02:01.880
Well if you add 2 again,
if you add 2 to x plus 2,

00:02:01.880 --> 00:02:03.790
well now you get to x plus 4.

00:02:03.790 --> 00:02:05.080
You add 2 to that.

00:02:05.080 --> 00:02:08.240
You get to x plus 6.

00:02:08.240 --> 00:02:10.550
And you see that
again right over here.

00:02:10.550 --> 00:02:14.490
If this was x, then
this is x plus 2.

00:02:14.490 --> 00:02:19.230
This is x plus 4,
and this is x plus 6.

00:02:19.230 --> 00:02:22.520
So in general, if x is the
smallest of the integers,

00:02:22.520 --> 00:02:26.140
we can define the other
three as x plus 2, x plus 4,

00:02:26.140 --> 00:02:27.410
and x plus 6.

00:02:27.410 --> 00:02:31.080
And let's take the sum of
them and set that equal to 136

00:02:31.080 --> 00:02:32.560
and then solve for x.

00:02:32.560 --> 00:02:34.690
So the smallest one is x.

00:02:34.690 --> 00:02:38.500
Now the next smallest one
is going to be x plus 2.

00:02:38.500 --> 00:02:42.160
Now the one after that's
going to be x plus 4,

00:02:42.160 --> 00:02:45.090
and then the one after that
is going to be x plus 6.

00:02:45.090 --> 00:02:48.880
So this is the sum of the
four consecutive odd integers,

00:02:48.880 --> 00:02:53.480
and they tell us that that
is going to be equal to 136.

00:02:53.480 --> 00:02:56.050
This is equal to 136.

00:02:56.050 --> 00:02:58.210
And now we can just
literally solve for x.

00:02:58.210 --> 00:03:00.500
We have this one unknown here.

00:03:00.500 --> 00:03:03.710
So let's add these
x terms together.

00:03:03.710 --> 00:03:08.030
So we have one x, two
x's, three x's, four x's.

00:03:08.030 --> 00:03:10.940
So we can rewrite
those literally as 4x.

00:03:10.940 --> 00:03:17.080
And then we have 2 plus 4, which
is 6, plus another 6 is 12.

00:03:17.080 --> 00:03:22.720
4x plus 12 is equal to 136.

00:03:22.720 --> 00:03:24.610
So to solve for x, a
good starting point

00:03:24.610 --> 00:03:26.620
would be to just to
isolate the x terms

00:03:26.620 --> 00:03:29.269
on one side of the equation
or try to get rid of this 12.

00:03:29.269 --> 00:03:30.810
Well to get rid of
that 12, we'd want

00:03:30.810 --> 00:03:32.490
to subtract 12 from
the left-hand side.

00:03:32.490 --> 00:03:33.980
But we can't just do it
from the left-hand side.

00:03:33.980 --> 00:03:35.760
Then this equality
wouldn't hold anymore.

00:03:35.760 --> 00:03:38.980
If these two things were equal
before subtracting the 12,

00:03:38.980 --> 00:03:40.800
well then if we want
to keep them equal,

00:03:40.800 --> 00:03:42.800
if we want the left and
the right to stay equal,

00:03:42.800 --> 00:03:45.720
we've got to subtract
12 from both sides.

00:03:45.720 --> 00:03:47.770
So subtracting 12
from both sides

00:03:47.770 --> 00:03:52.210
gives us, well on the left-hand
side, we're just left with 4x.

00:03:52.210 --> 00:03:56.840
And on the right-hand side,
we are left with 136 minus 12

00:03:56.840 --> 00:04:03.587
is 124.

00:04:03.587 --> 00:04:04.420
Did I do that right?

00:04:04.420 --> 00:04:05.280
Yeah, 124.

00:04:05.280 --> 00:04:05.960
So what is x?

00:04:05.960 --> 00:04:13.750
Well, we just divide both
sides by 4 to solve for x.

00:04:13.750 --> 00:04:19.560
And we get-- do that in
the same, original color--

00:04:19.560 --> 00:04:24.210
x is equal to 124 divided by 4.

00:04:24.210 --> 00:04:24.900
So let's see.

00:04:24.900 --> 00:04:27.720
100 divided by 4 is 25.

00:04:27.720 --> 00:04:30.730
24 divided by 4 is 6.

00:04:30.730 --> 00:04:32.386
25 plus 6 is 31.

00:04:32.386 --> 00:04:34.510
And if you don't feel like
doing that in your head,

00:04:34.510 --> 00:04:36.843
you could also, of course,
do traditional long division.

00:04:36.843 --> 00:04:40.390
Goes into 124-- 4
doesn't go into 1.

00:04:40.390 --> 00:04:42.780
4 goes into 12 three times.

00:04:42.780 --> 00:04:44.140
3 times 4 is 12.

00:04:44.140 --> 00:04:47.216
You subtract, bring
down the next 4.

00:04:47.216 --> 00:04:49.230
4 goes into 4 one time.

00:04:49.230 --> 00:04:51.460
You get no remainder.

00:04:51.460 --> 00:04:52.690
So x is equal to 31.

00:04:52.690 --> 00:04:54.890
So x is the smallest
of the four integers.

00:04:54.890 --> 00:04:58.290
So this right over
here, x is 31.

00:04:58.290 --> 00:05:01.210
x plus 2 is going to be 33.

00:05:01.210 --> 00:05:04.560
x plus 4 is going to be 35.

00:05:04.560 --> 00:05:09.240
And x plus 6 is going to be 37.

00:05:09.240 --> 00:05:12.340
So our four consecutive
odd integers

00:05:12.340 --> 00:05:17.500
are 31, 33, 35, and 37.


Power series intro

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:04.182
We've already seen many
examples of infinite series.

00:00:04.182 --> 00:00:06.765
But what's exciting about what
we're about to do in this video

00:00:06.765 --> 00:00:09.879
is we're going to use infinite
series to define a function.

00:00:09.879 --> 00:00:11.420
And the most common
one that you will

00:00:11.420 --> 00:00:17.120
see in your mathematical
careers is the power series.

00:00:17.120 --> 00:00:20.010
And I'm about to write a general
case of the power series.

00:00:20.010 --> 00:00:22.250
So I could imagine
a function, f of x,

00:00:22.250 --> 00:00:26.100
being defined as
the infinite sum.

00:00:26.100 --> 00:00:30.900
So going from n equals 0
to infinity of a sub n--

00:00:30.900 --> 00:00:33.760
so a sub n is just going to be
the coefficient on each term--

00:00:33.760 --> 00:00:37.320
times our variable x
minus some constant c.

00:00:37.320 --> 00:00:39.160
You could almost
imagine this is shifting

00:00:39.160 --> 00:00:42.260
our function to the n-th power.

00:00:42.260 --> 00:00:43.910
So if I were to
expand this out, I

00:00:43.910 --> 00:00:46.390
have my first term's
coefficient, a sub 0,

00:00:46.390 --> 00:00:50.630
times x minus c
to the 0-th power,

00:00:50.630 --> 00:00:54.400
plus a sub 1 times x minus
c to the first power.

00:00:54.400 --> 00:00:57.180
This one, of course, will
simplify just a sub 0.

00:00:57.180 --> 00:01:01.590
This would simplify to a sub
1 times x minus c plus a sub

00:01:01.590 --> 00:01:04.700
2 times x minus c squared.

00:01:04.700 --> 00:01:07.587
And I could just keep
going on and on and on.

00:01:07.587 --> 00:01:09.170
Now, when you see
this, you might say,

00:01:09.170 --> 00:01:10.760
aren't our geometric
series, don't

00:01:10.760 --> 00:01:13.440
those look like a special
case of a power series

00:01:13.440 --> 00:01:17.260
if our common ratio was an x
instead of an r in that case,

00:01:17.260 --> 00:01:21.920
or if our common ratio was a
variable, I guess I could say?

00:01:21.920 --> 00:01:23.060
And you would be right.

00:01:23.060 --> 00:01:25.490
That absolutely
would be the case.

00:01:25.490 --> 00:01:28.800
So a geometric series.

00:01:28.800 --> 00:01:31.050
So let's just think about
defining a function in terms

00:01:31.050 --> 00:01:32.482
of a geometric series.

00:01:32.482 --> 00:01:33.940
And of course, we
don't have to use

00:01:33.940 --> 00:01:36.650
x all the time as the
independent variable,

00:01:36.650 --> 00:01:38.650
but this is kind of the
most typical convention.

00:01:38.650 --> 00:01:40.900
I guess we could also use r
as an independent variable

00:01:40.900 --> 00:01:42.240
if we wanted as well.

00:01:42.240 --> 00:01:44.067
But let's imagine
a function g of x.

00:01:44.067 --> 00:01:45.900
We could have g of r
if we wanted, but let's

00:01:45.900 --> 00:01:50.200
do g of x is equal
to the sum from n

00:01:50.200 --> 00:01:57.060
equals 0 to infinity
of a times x to the n.

00:01:57.060 --> 00:02:01.220
So this is kind of a typical
geometric series here.

00:02:01.220 --> 00:02:04.020
And what's the difference
between this and this?

00:02:04.020 --> 00:02:06.259
Well, the difference is is
here, for every term we're

00:02:06.259 --> 00:02:07.800
going to have the
same coefficient a,

00:02:07.800 --> 00:02:09.864
while over here we have a sub n.

00:02:09.864 --> 00:02:12.280
We're multiplying by a different
thing every time up here.

00:02:12.280 --> 00:02:14.610
We're multiplying by the
same thing over here.

00:02:14.610 --> 00:02:16.930
And in this case, this
particular geometric series

00:02:16.930 --> 00:02:19.420
I just made, instead of
having x minus c to the n,

00:02:19.420 --> 00:02:20.580
we have just x to the n.

00:02:20.580 --> 00:02:22.705
So you could say, well,
this is a special case when

00:02:22.705 --> 00:02:24.270
c is equal to 0.

00:02:24.270 --> 00:02:25.440
And we can expand it out.

00:02:25.440 --> 00:02:28.040
This is a times
x to the 0, which

00:02:28.040 --> 00:02:31.540
is just going to be a, plus
a times x to the first,

00:02:31.540 --> 00:02:33.330
plus a times x squared.

00:02:33.330 --> 00:02:36.210
And we just go on and
on and on forever.

00:02:36.210 --> 00:02:38.540
Now, what's exciting
about this is

00:02:38.540 --> 00:02:43.510
we know that this, under
certain conditions,

00:02:43.510 --> 00:02:45.950
will actually give
us a finite value.

00:02:45.950 --> 00:02:47.640
This will actually converge.

00:02:47.640 --> 00:02:51.350
This will actually, I guess,
give us a sensical answer.

00:02:51.350 --> 00:02:53.520
So under what conditions
does that happen?

00:02:53.520 --> 00:02:56.279
Well, this converges
if each of these terms

00:02:56.279 --> 00:02:57.820
gets smaller and
smaller and smaller.

00:02:57.820 --> 00:03:00.390
And each of these terms gets
smaller and smaller and smaller

00:03:00.390 --> 00:03:06.470
if the absolute value of our
common ratio is less than 1.

00:03:06.470 --> 00:03:08.100
So let me write that down.

00:03:08.100 --> 00:03:14.410
So this converges if
the absolute value

00:03:14.410 --> 00:03:18.181
of our common ratio
is less than 1.

00:03:18.181 --> 00:03:19.680
Or another way of
thinking about it,

00:03:19.680 --> 00:03:23.020
this is another way
of saying that x

00:03:23.020 --> 00:03:26.120
is in the interval
between-- it's less than 1

00:03:26.120 --> 00:03:29.280
and it is greater
than negative 1.

00:03:29.280 --> 00:03:31.640
And this term right over
here, now x is a variable.

00:03:31.640 --> 00:03:33.300
x can vary between those values.

00:03:33.300 --> 00:03:35.640
We're defining a
function in terms of x.

00:03:35.640 --> 00:03:37.740
We call this the
interval of convergence.

00:03:46.090 --> 00:03:48.487
And so we know that if
x is in this interval,

00:03:48.487 --> 00:03:50.070
this is going to
give us a finite sum.

00:03:50.070 --> 00:03:52.380
And we know what
that finite sum is.

00:03:52.380 --> 00:03:56.620
It's going to be equal
to-- if it converges.

00:03:56.620 --> 00:03:58.950
So if it converges,
this is going

00:03:58.950 --> 00:04:01.950
to be equal to our first
term, which is just

00:04:01.950 --> 00:04:05.490
a-- this simplifies to a
right over here-- over 1

00:04:05.490 --> 00:04:07.580
minus our common ratio.

00:04:07.580 --> 00:04:08.820
What's our common ratio?

00:04:08.820 --> 00:04:11.360
Our common ratio in
this example is x.

00:04:11.360 --> 00:04:15.120
Going from one term to the next,
we're just multiplying by x.

00:04:15.120 --> 00:04:19.640
We're just multiplying
by x right over there.

00:04:19.640 --> 00:04:21.510
Now, this is pretty
neat, because we're

00:04:21.510 --> 00:04:23.390
going to be able
to use this fact

00:04:23.390 --> 00:04:26.030
to put more
traditionally-defined functions

00:04:26.030 --> 00:04:28.750
into this form, and
then try to expand them

00:04:28.750 --> 00:04:31.660
out using a geometric series.

00:04:31.660 --> 00:04:33.722
And this whole idea
of using power series,

00:04:33.722 --> 00:04:35.430
or in this special
case, geometric series

00:04:35.430 --> 00:04:37.280
to represent functions,
has all sorts

00:04:37.280 --> 00:04:39.750
of applications in
engineering and finance.

00:04:39.750 --> 00:04:43.980
Using a finite number of
terms of these series,

00:04:43.980 --> 00:04:46.130
you could kind of
approximate the functions

00:04:46.130 --> 00:04:48.020
in a way that's simpler
for the human brain

00:04:48.020 --> 00:04:49.470
to understand, or
maybe a simpler

00:04:49.470 --> 00:04:51.690
way to manipulate in some way.

00:04:51.690 --> 00:04:54.610
But what's interesting
here is instead of just

00:04:54.610 --> 00:04:57.540
going from the sum
to-- instead of going

00:04:57.540 --> 00:05:01.922
from this expanded-out version
to this kind of finite value,

00:05:01.922 --> 00:05:04.130
we're now going to start
being able to take something

00:05:04.130 --> 00:05:06.930
in this form and expand it
out into a geometric series.

00:05:06.930 --> 00:05:08.734
But we have to be
careful to make sure

00:05:08.734 --> 00:05:11.150
that we're only doing it over
the interval of convergence.

00:05:11.150 --> 00:05:13.790
This is only going to be
true over the interval

00:05:13.790 --> 00:05:15.010
of convergence.

00:05:15.010 --> 00:05:18.550
Now, one other term you might
see in your mathematical career

00:05:18.550 --> 00:05:20.220
is a radius.

00:05:20.220 --> 00:05:21.945
Radius of convergence.

00:05:24.970 --> 00:05:28.770
And this is how far--
up to what value,

00:05:28.770 --> 00:05:30.180
but not including this value.

00:05:30.180 --> 00:05:36.760
So as long as our x value stays
less than a certain amount

00:05:36.760 --> 00:05:40.680
from our c value, then
this thing will converge.

00:05:40.680 --> 00:05:44.470
Now in this case,
our c value is 0.

00:05:44.470 --> 00:05:46.720
So we could ask
ourselves a question.

00:05:49.290 --> 00:05:53.120
As long as x stays
within some value of 0,

00:05:53.120 --> 00:05:54.540
this thing is going to converge.

00:05:54.540 --> 00:05:55.914
Well, you see it
right over here.

00:05:55.914 --> 00:06:01.302
As long as x stays
within one of 0.

00:06:01.302 --> 00:06:03.010
It can't go all the
way to 1, but as long

00:06:03.010 --> 00:06:05.000
as it stays less
than 1, or as long

00:06:05.000 --> 00:06:06.500
as it stays greater
than negative 1.

00:06:10.300 --> 00:06:12.490
It can stray anything
less than one

00:06:12.490 --> 00:06:14.380
away from 0, either in
the positive direction

00:06:14.380 --> 00:06:16.460
or the negative direction.

00:06:16.460 --> 00:06:18.560
Then this thing
will still converge.

00:06:18.560 --> 00:06:23.640
So we could say that our radius
of convergence is equal to 1.

00:06:23.640 --> 00:06:26.550
Another way to think about it,
our interval of convergence--

00:06:26.550 --> 00:06:28.490
we're going from
negative 1 to 1,

00:06:28.490 --> 00:06:29.980
not including those
two boundaries,

00:06:29.980 --> 00:06:31.690
so our interval is 2.

00:06:31.690 --> 00:06:34.450
So our radius of
convergence is half of that.

00:06:39.550 --> 00:06:41.920
As long as x stays
within one of 0,

00:06:41.920 --> 00:06:44.440
and that's the same thing as
saying this right over here,

00:06:44.440 --> 00:06:48.310
this series is
going to converge.


Geometric series interval of convergence

WEBVTT
Kind: captions
Language: en

00:00:00.644 --> 00:00:02.310
So as we talked about
in the last video,

00:00:02.310 --> 00:00:04.070
we've seen many
examples of starting

00:00:04.070 --> 00:00:06.340
with a geometric
series expanded out,

00:00:06.340 --> 00:00:09.540
and then assuming
that its common ratio,

00:00:09.540 --> 00:00:13.220
that the absolute value of the
common ratio is less than 1,

00:00:13.220 --> 00:00:15.140
finding what the sum
of that might be.

00:00:15.140 --> 00:00:17.667
We've proven with this
formula in previous videos.

00:00:17.667 --> 00:00:19.250
But now let's go the
other way around.

00:00:19.250 --> 00:00:21.170
Let's try to take
some function-- let's

00:00:21.170 --> 00:00:26.560
say h of x being equal to
1 over 3 plus x squared--

00:00:26.560 --> 00:00:28.409
and let's try to
put it in this form.

00:00:28.409 --> 00:00:29.950
And then once we
put it in that form,

00:00:29.950 --> 00:00:32.970
we can think about what a
and our common ratio is.

00:00:32.970 --> 00:00:38.473
And then try to represent it
as an actual geometric series.

00:00:38.473 --> 00:00:40.080
So I encourage you
to pause the video

00:00:40.080 --> 00:00:42.405
and try to do that right now.

00:00:42.405 --> 00:00:44.530
So let's see, the first
thing that you might notice

00:00:44.530 --> 00:00:46.620
is we have a 1 here
instead of a 3.

00:00:46.620 --> 00:00:48.500
So let's try to factor out a 3.

00:00:48.500 --> 00:00:56.320
So this is equal to 1 over 3
times 1 plus x squared over 3.

00:00:56.320 --> 00:00:59.160
And now, since we don't want
that 3 in the denominator,

00:00:59.160 --> 00:01:02.450
we can think about
this as 1 over 3.

00:01:02.450 --> 00:01:07.630
So we could say this
is 1/3 over-- let

00:01:07.630 --> 00:01:08.880
me do it in that purple color.

00:01:08.880 --> 00:01:12.200
1/3 over 1.

00:01:12.200 --> 00:01:13.866
And we don't want to
just add something,

00:01:13.866 --> 00:01:15.620
we want to subtract
our common ratio.

00:01:15.620 --> 00:01:19.401
So 1 minus-- and let me
write our common ratio here

00:01:19.401 --> 00:01:19.900
in yellow.

00:01:19.900 --> 00:01:25.000
1 minus negative
x squared over 3.

00:01:25.000 --> 00:01:28.720
So now we've written
this in that form.

00:01:28.720 --> 00:01:34.720
And so now we could say that
the sum-- let me write it here

00:01:34.720 --> 00:01:36.740
in-- let me do it
in a new color.

00:01:36.740 --> 00:01:39.020
So let me do it in blue.

00:01:39.020 --> 00:01:43.040
So now we could say
that the sum from n

00:01:43.040 --> 00:01:48.015
equals 0 to infinity of-- let's
see, our first term is 1/3.

00:01:48.015 --> 00:01:55.320
1/3 times our common
ratio to the n-th power.

00:01:55.320 --> 00:02:01.360
Common ratio is negative
x squared over 3.

00:02:01.360 --> 00:02:03.440
And if we wanted
to expand this out,

00:02:03.440 --> 00:02:07.730
this would be equal to-- so
the first term is 1/3 times

00:02:07.730 --> 00:02:09.630
all of this to the 0-th power.

00:02:09.630 --> 00:02:11.820
So it's just going to be 1/3.

00:02:11.820 --> 00:02:13.700
And so each successive
term is just

00:02:13.700 --> 00:02:16.170
going to be the previous
term times our common ratio.

00:02:16.170 --> 00:02:19.710
So 1/3 times negative
x squared over 3

00:02:19.710 --> 00:02:28.036
is going to be
negative 1/9 x squared.

00:02:28.036 --> 00:02:29.410
To go from that
to that, you have

00:02:29.410 --> 00:02:32.140
to multiply by-- let's
see, 1/3 to negative 1/3,

00:02:32.140 --> 00:02:33.810
you have to multiply
it by negative 1/3.

00:02:33.810 --> 00:02:36.300
And we multiplied by
x squared as well.

00:02:36.300 --> 00:02:38.060
Now in our next
term, we're going

00:02:38.060 --> 00:02:40.420
to multiply by negative
x squared over 3 again.

00:02:40.420 --> 00:02:44.440
So it's going to be plus-- a
negative times a negative is

00:02:44.440 --> 00:02:50.675
a positive-- plus
1/27 x to the fourth.

00:02:50.675 --> 00:02:55.940
x squared times x squared,
x to the fourth power.

00:02:55.940 --> 00:03:00.060
And we just keep going
on and on and on.

00:03:00.060 --> 00:03:09.070
And when this converges, so over
the interval of convergence,

00:03:09.070 --> 00:03:12.290
this is going to
converge to h of x.

00:03:12.290 --> 00:03:14.290
Now, what is the interval
of convergence here?

00:03:14.290 --> 00:03:16.826
And I encourage you to pause
the video and think about it.

00:03:16.826 --> 00:03:18.200
Well, the interval
of convergence

00:03:18.200 --> 00:03:21.360
is the interval over
which your common ratio,

00:03:21.360 --> 00:03:25.980
the absolute value of your
common ratio, is less than 1.

00:03:25.980 --> 00:03:28.070
So let me write this
right over here.

00:03:28.070 --> 00:03:37.540
So our absolute value of
negative x squared over 3

00:03:37.540 --> 00:03:40.030
has to be less than 1.

00:03:40.030 --> 00:03:41.850
Well, the absolute
value, this is

00:03:41.850 --> 00:03:43.120
going to be a negative number.

00:03:43.120 --> 00:03:44.536
This is the same
thing as saying--

00:03:44.536 --> 00:03:46.510
let me scroll down a little bit.

00:03:46.510 --> 00:03:50.270
This is the same thing as saying
that the absolute value of x

00:03:50.270 --> 00:03:58.670
squared over 3 has
to be less than 1.

00:03:58.670 --> 00:04:01.070
And this is another
way of saying--

00:04:01.070 --> 00:04:05.040
well, one thing that might jump
out at you is that x squared,

00:04:05.040 --> 00:04:06.947
this is going to be
positive no matter what.

00:04:06.947 --> 00:04:08.530
Or I guess I should
say, this is going

00:04:08.530 --> 00:04:11.410
to be non-negative
no matter what.

00:04:11.410 --> 00:04:18.000
So this is another way of
saying that x squared over 3

00:04:18.000 --> 00:04:21.780
has to be less than 1.

00:04:21.780 --> 00:04:22.280
Right?

00:04:22.280 --> 00:04:24.280
I don't want to confuse you
in this step right over here.

00:04:24.280 --> 00:04:26.050
But the absolute value
of x squared over 3

00:04:26.050 --> 00:04:28.508
is just going to be x squared
over 3, because this is never

00:04:28.508 --> 00:04:29.960
going to take on
a negative value.

00:04:29.960 --> 00:04:32.150
And so we can multiply
both sides by 3.

00:04:32.150 --> 00:04:34.490
I'll go up here now to do it.

00:04:34.490 --> 00:04:37.340
Multiply both sides by
3 to say that x squared

00:04:37.340 --> 00:04:40.510
needs to be less than 3.

00:04:40.510 --> 00:04:45.490
And so that means that
the absolute value of x

00:04:45.490 --> 00:04:48.170
needs to be less than
the square root of 3.

00:04:48.170 --> 00:04:50.610
Or we could say
that x is greater

00:04:50.610 --> 00:04:52.710
than the negative
square root of 3,

00:04:52.710 --> 00:04:55.960
and it is less than
the square root of 3.

00:04:55.960 --> 00:04:58.720
So this is the interval
of convergence.

00:04:58.720 --> 00:05:07.690
This is the interval of
convergence for this series,

00:05:07.690 --> 00:05:08.736
for this power series.

00:05:08.736 --> 00:05:10.110
It's a geometric
series, which is

00:05:10.110 --> 00:05:12.110
a special case of
a power series.

00:05:12.110 --> 00:05:15.880
And over the interval
of convergence,

00:05:15.880 --> 00:05:23.650
that is going to be equal
to 1 over 3 plus x squared.

00:05:23.650 --> 00:05:25.420
So as long as x is
in this interval,

00:05:25.420 --> 00:05:27.360
it's going to take
on the same values

00:05:27.360 --> 00:05:30.662
as our original function,
which is a pretty neat idea.


Geometric series as a function

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:02.320
We have a function
right over here

00:00:02.320 --> 00:00:05.070
defined as an infinite series.

00:00:05.070 --> 00:00:06.820
What I want to attempt
to do in this video

00:00:06.820 --> 00:00:10.439
is to see if we can express
it in a more traditional form.

00:00:10.439 --> 00:00:12.230
And so the thing that
might jump out at you

00:00:12.230 --> 00:00:15.130
is, well, look, if this
is a geometric series,

00:00:15.130 --> 00:00:18.620
we know how to take the sum of
an infinite geometric series,

00:00:18.620 --> 00:00:21.940
at least over the x values
where the thing will actually

00:00:21.940 --> 00:00:22.710
converge.

00:00:22.710 --> 00:00:26.080
And so let's first see if
this is a geometric series.

00:00:26.080 --> 00:00:29.252
Well, the telltale pattern
of a geometric series

00:00:29.252 --> 00:00:30.960
is when you go from
one term to the next,

00:00:30.960 --> 00:00:33.159
you're multiplying
by a common ratio.

00:00:33.159 --> 00:00:33.700
So let's see.

00:00:33.700 --> 00:00:36.300
To go from 2 to
negative 8x squared,

00:00:36.300 --> 00:00:37.900
what do you have to multiply by?

00:00:37.900 --> 00:00:42.890
We have to multiply by
negative 4x squared.

00:00:42.890 --> 00:00:45.910
So we've multiplied by
negative 4x squared.

00:00:45.910 --> 00:00:47.890
Now, do we multiply
by that same amount

00:00:47.890 --> 00:00:49.630
to get to 32x to the fourth?

00:00:49.630 --> 00:00:50.580
Well sure.

00:00:50.580 --> 00:00:52.950
Negative 4x squared
times negative 8x squared

00:00:52.950 --> 00:00:55.570
is going to be positive
32x to the fourth.

00:00:55.570 --> 00:00:57.670
Multiply by negative
4x squared again,

00:00:57.670 --> 00:01:00.650
you're going to get to
negative 128x to the sixth.

00:01:00.650 --> 00:01:02.350
So the common ratio
is negative 4x

00:01:02.350 --> 00:01:06.720
squared, our first term is
2, so we can rewrite this.

00:01:06.720 --> 00:01:12.690
We can rewrite f of x as
being equal to the sum from n

00:01:12.690 --> 00:01:16.650
equals 0 to infinity
of-- let's see,

00:01:16.650 --> 00:01:23.890
our first term here is
2-- 2 times negative 4x

00:01:23.890 --> 00:01:29.620
squared to the n-th power.

00:01:29.620 --> 00:01:33.420
This is a geometric series
where our common ratio

00:01:33.420 --> 00:01:35.630
is negative 4x squared
to the n-th power.

00:01:35.630 --> 00:01:39.350
Now, when will this thing
right over here converge?

00:01:39.350 --> 00:01:41.640
Well, we know that
a geometric series

00:01:41.640 --> 00:01:46.020
will converge if the absolute
value of its common ratio

00:01:46.020 --> 00:01:48.760
is less than 1.

00:01:48.760 --> 00:01:50.470
So let me write this down.

00:01:50.470 --> 00:01:57.730
So converge if the absolute
value of the common ratio,

00:01:57.730 --> 00:02:00.875
negative 4x squared,
is less than 1.

00:02:00.875 --> 00:02:02.750
Well, this-- the way
it's written right now--

00:02:02.750 --> 00:02:04.940
is going to be a negative value.

00:02:04.940 --> 00:02:11.920
So the absolute value of this
is just going to be 4x squared.

00:02:11.920 --> 00:02:12.420
Right?

00:02:17.290 --> 00:02:20.580
x squared is going
to be non-negative,

00:02:20.580 --> 00:02:22.890
so 4x squared is going
to be non-negative.

00:02:22.890 --> 00:02:26.570
Negative 4x squared is
going to be non-positive.

00:02:26.570 --> 00:02:29.960
So if you're taking the absolute
value of a non-positive thing,

00:02:29.960 --> 00:02:33.140
that's going to be the same
thing as the absolute value

00:02:33.140 --> 00:02:35.210
of the negative of it.

00:02:35.210 --> 00:02:38.202
So this just has
to be less than 1.

00:02:38.202 --> 00:02:39.660
And the absolute
value of something

00:02:39.660 --> 00:02:41.700
that is strictly
non-negative like this,

00:02:41.700 --> 00:02:44.550
well, that's just going
to be 4x squared--

00:02:44.550 --> 00:02:46.350
these two statements
are equivalent--

00:02:46.350 --> 00:02:49.290
and that has to be less than 1.

00:02:49.290 --> 00:02:55.060
Can divide both sides by 4, you
get x squared is less than 1/4.

00:02:55.060 --> 00:03:00.790
And so we can say that
the absolute value of x

00:03:00.790 --> 00:03:07.720
has to be less than 1/4, or
we could say that negative 1/4

00:03:07.720 --> 00:03:12.460
has to be less than x, which has
to be less than positive 1/4.

00:03:12.460 --> 00:03:14.050
So expressed this
way, we're giving

00:03:14.050 --> 00:03:16.150
the interval of convergence.

00:03:16.150 --> 00:03:18.920
This thing will converge as
long as x is in this interval.

00:03:18.920 --> 00:03:20.770
Expressed this way,
we're really saying

00:03:20.770 --> 00:03:22.110
the radius of convergence.

00:03:22.110 --> 00:03:24.510
This will converge
as long as x is

00:03:24.510 --> 00:03:27.740
less than our radius
of convergence,

00:03:27.740 --> 00:03:29.650
as long as the
absolute value of x

00:03:29.650 --> 00:03:34.480
is less than our
radius of convergence,

00:03:34.480 --> 00:03:38.621
as long as x stays less
than 1/4 away from 0.

00:03:38.621 --> 00:03:40.120
To make it a little
bit clearer, you

00:03:40.120 --> 00:03:43.160
could rewrite this as the
distance between x and 0,

00:03:43.160 --> 00:03:44.910
as long as this- this,
you could view this

00:03:44.910 --> 00:03:46.380
as the distance
between x and 0--

00:03:46.380 --> 00:03:48.820
as long as this
stays less than 1/4,

00:03:48.820 --> 00:03:50.400
this thing is going to converge.

00:03:50.400 --> 00:03:52.510
So this is the interval
of convergence, this,

00:03:52.510 --> 00:03:54.110
you could view,
1/4, you could view

00:03:54.110 --> 00:03:55.522
as the radius of convergence.

00:03:55.522 --> 00:03:56.980
Now with that out
of the way, we've

00:03:56.980 --> 00:03:58.688
thought about where
this thing converges,

00:03:58.688 --> 00:04:00.740
let's think about
what it converges to.

00:04:00.740 --> 00:04:02.740
Well, we've done
this multiple times.

00:04:02.740 --> 00:04:08.250
This is going to be equal
to our first term, 2 over 1

00:04:08.250 --> 00:04:09.560
minus our common ratio.

00:04:13.940 --> 00:04:17.810
Our common ratio is
negative 4x squared.

00:04:17.810 --> 00:04:19.950
So this is going
to give us-- and we

00:04:19.950 --> 00:04:26.820
deserve a drum roll-- 2
over 1 plus 4x squared.

00:04:26.820 --> 00:04:30.430
So this expression is
going to be equal to this,

00:04:30.430 --> 00:04:35.880
as long as x is within our
interval of convergence.


y-intercept of tangent line example

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.520
In terms of k, where
k does not equal 0,

00:00:03.520 --> 00:00:07.040
what is the y-intercept of
the line tangent to the curve

00:00:07.040 --> 00:00:10.170
f of x is equal to 1/x
at the point of the curve

00:00:10.170 --> 00:00:11.814
where x is equal to k?

00:00:11.814 --> 00:00:13.730
So let's just think about
what they're asking.

00:00:13.730 --> 00:00:15.120
So if I were to
draw myself-- let

00:00:15.120 --> 00:00:18.330
me draw some quick
axes right over here.

00:00:18.330 --> 00:00:20.670
So that's my y-axis.

00:00:20.670 --> 00:00:26.180
This is my x-axis
right over here.

00:00:26.180 --> 00:00:29.410
And the graph f of
x is equal to 1/x

00:00:29.410 --> 00:00:31.920
would look something like this.

00:00:31.920 --> 00:00:35.060
So it looks something like this.

00:00:35.060 --> 00:00:37.720
So it kind of spikes up
there, and then it comes down,

00:00:37.720 --> 00:00:38.930
and then it goes like this.

00:00:38.930 --> 00:00:40.888
And I'm just doing a
rough approximation of it.

00:00:40.888 --> 00:00:42.450
So it would look
something like that.

00:00:42.450 --> 00:00:48.120
And then on the negative side,
it looks something like this.

00:00:48.120 --> 00:00:50.830
So this is my hand-drawn
version of roughly

00:00:50.830 --> 00:00:52.170
what this graph looks like.

00:00:52.170 --> 00:00:56.890
So this right over here
is f of x is equal to 1/x.

00:00:56.890 --> 00:01:01.150
Now, we are concerning ourselves
with the point x equals k.

00:01:01.150 --> 00:01:04.010
So let's say-- I mean, it could
be anything that's non-zero,

00:01:04.010 --> 00:01:06.740
but let's just say that
this is k right over here.

00:01:06.740 --> 00:01:10.310
So that is the point k 1/k.

00:01:16.060 --> 00:01:19.590
We can visualize the line
tangent to the curve there.

00:01:19.590 --> 00:01:23.015
So it might look
something like this.

00:01:26.720 --> 00:01:29.130
And we need to figure
out its y-intercept.

00:01:29.130 --> 00:01:31.790
Where does it
intercept the y-axis?

00:01:31.790 --> 00:01:35.629
So we need to figure out
this point right over here.

00:01:35.629 --> 00:01:37.670
Well, the best way to do
it, if we can figure out

00:01:37.670 --> 00:01:39.440
the slope of the
tangent line here,

00:01:39.440 --> 00:01:41.730
the slope of the tangent
line is just the derivative

00:01:41.730 --> 00:01:42.982
of the line at that point.

00:01:42.982 --> 00:01:45.190
If we could figure out the
slope of the tangent line,

00:01:45.190 --> 00:01:47.820
we already know that
line contains the point.

00:01:50.560 --> 00:01:52.370
Let me do this in
a different color.

00:01:52.370 --> 00:01:56.940
We know it contains
the point k comma 1/k.

00:01:56.940 --> 00:01:59.470
So if we know its slope, we
know what point it contains,

00:01:59.470 --> 00:02:03.756
we can figure out what
its y-intercept is.

00:02:03.756 --> 00:02:05.380
So the first step is
just, well, what's

00:02:05.380 --> 00:02:07.477
the slope of the tangent line?

00:02:07.477 --> 00:02:09.560
Well, to figure out the
slope of the tangent line,

00:02:09.560 --> 00:02:11.460
let's take the derivative.

00:02:11.460 --> 00:02:16.000
So if we write f of x,
instead writing it as 1/x,

00:02:16.000 --> 00:02:18.466
I'll write it as x to
the negative 1 power.

00:02:18.466 --> 00:02:20.090
That makes it a little
bit more obvious

00:02:20.090 --> 00:02:22.460
that we're about to use
the power rule here.

00:02:22.460 --> 00:02:26.260
So the derivative
of f at any point x

00:02:26.260 --> 00:02:28.520
is going to be equal
to-- well, it's

00:02:28.520 --> 00:02:31.460
going to be the exponent
here is negative 1.

00:02:31.460 --> 00:02:34.920
So negative 1 times
x to the-- now

00:02:34.920 --> 00:02:38.550
we decrement the exponent
to the negative 2 power.

00:02:38.550 --> 00:02:42.680
Or I could say it's negative
x to the negative 2.

00:02:42.680 --> 00:02:46.130
Now, what we care about is
the slope when x equals k.

00:02:46.130 --> 00:02:57.090
So f prime of k is going
to be equal to negative k

00:02:57.090 --> 00:03:00.870
to the negative 2 power.

00:03:00.870 --> 00:03:05.320
Or another way of thinking about
it, this is equal to negative 1

00:03:05.320 --> 00:03:06.655
over k squared.

00:03:11.210 --> 00:03:12.710
So this right over
here is the slope

00:03:12.710 --> 00:03:16.330
of the tangent
line at that point.

00:03:16.330 --> 00:03:18.962
Now, let's just think
about what the equation

00:03:18.962 --> 00:03:19.920
of the tangent line is.

00:03:19.920 --> 00:03:22.890
And we could think about
it in slope-intercept form.

00:03:22.890 --> 00:03:25.650
So we know the equation of a
line in slope-intercept form

00:03:25.650 --> 00:03:31.360
is y is equal to mx plus
b, where m is the slope

00:03:31.360 --> 00:03:32.544
and b is the y-intercept.

00:03:32.544 --> 00:03:34.960
So if we can get it in this
form, then we know our answer.

00:03:34.960 --> 00:03:36.793
We know what the
y-intercept is going to be.

00:03:36.793 --> 00:03:37.610
It's going to be b.

00:03:37.610 --> 00:03:39.770
So let's think about
it a little bit.

00:03:39.770 --> 00:03:42.140
This equation, so
we could say y is

00:03:42.140 --> 00:03:45.540
equal to our m, our slope
of the tangent line, when

00:03:45.540 --> 00:03:48.820
x is equal to k, we just
figure out to be this business.

00:03:48.820 --> 00:03:51.192
It equals this thing
right over here.

00:03:51.192 --> 00:03:52.400
So let me write that in blue.

00:03:52.400 --> 00:04:01.050
Negative 1 over k
squared times x plus b.

00:04:01.050 --> 00:04:02.570
So how do we solve for b?

00:04:02.570 --> 00:04:06.440
Well, we know what y is
when x is equal to k.

00:04:06.440 --> 00:04:08.770
And so we can use
that to solve for b.

00:04:08.770 --> 00:04:16.610
We know that y is equal to
1/k when x is equal to k.

00:04:16.610 --> 00:04:23.740
So this is going to be
equal to negative 1--

00:04:23.740 --> 00:04:25.833
that's not the same color.

00:04:25.833 --> 00:04:34.655
Negative 1 over k
squared times k plus b.

00:04:41.500 --> 00:04:44.370
Now, what does this simplify to?

00:04:44.370 --> 00:04:47.340
See, k over k squared is
the same thing as 1/k,

00:04:47.340 --> 00:04:49.110
so this is going
to be negative 1/k.

00:04:49.110 --> 00:04:56.800
So this part, all of this
simplifies to negative 1/k.

00:04:56.800 --> 00:04:58.140
So how do we solve for b?

00:04:58.140 --> 00:05:03.410
Well, we could just
add 1/k to both sides

00:05:03.410 --> 00:05:06.987
and we are left with-- if
you add 1/k here-- actually,

00:05:06.987 --> 00:05:07.820
let me just do that.

00:05:07.820 --> 00:05:15.560
Plus 1/k, left-hand side,
you're left with 2/k,

00:05:15.560 --> 00:05:19.070
and on the right-hand side,
you're just left with b,

00:05:19.070 --> 00:05:21.250
is equal to b.

00:05:21.250 --> 00:05:22.300
So we're done.

00:05:22.300 --> 00:05:27.260
The y-intercept of
the line tangent

00:05:27.260 --> 00:05:32.887
to the curve when x equals
k is going to be 2/k.

00:05:32.887 --> 00:05:34.720
If we wanted the equation
of the line, well,

00:05:34.720 --> 00:05:35.430
we've done all the work.

00:05:35.430 --> 00:05:36.221
Let's write it out.

00:05:36.221 --> 00:05:37.300
It'll be satisfying.

00:05:37.300 --> 00:05:42.580
It's going to be y is equal
to negative 1/k squared

00:05:42.580 --> 00:05:49.400
x plus our
y-intercept, plus 2/k.

00:05:57.090 --> 00:05:58.940
And we're done.


Equation of normal line

WEBVTT
Kind: captions
Language: en

00:00:00.860 --> 00:00:02.500
So we have the
function f of x is

00:00:02.500 --> 00:00:05.290
equal to e to the
x over x squared.

00:00:05.290 --> 00:00:07.630
And what I want to
do in this video

00:00:07.630 --> 00:00:10.960
is find the equation,
not of the tangent line,

00:00:10.960 --> 00:00:17.220
but the equation of the normal
line, when x is equal to 1.

00:00:17.220 --> 00:00:22.970
So we care about the
equation of the normal line.

00:00:22.970 --> 00:00:26.250
So I encourage you to pause this
video and try this on your own.

00:00:26.250 --> 00:00:29.690
And if you need a little bit
of a hint, the hint I will give

00:00:29.690 --> 00:00:33.800
you is, is that the
slope of a normal line

00:00:33.800 --> 00:00:38.920
is going to be the
negative reciprocal

00:00:38.920 --> 00:00:41.020
of the slope of
the tangent line.

00:00:41.020 --> 00:00:46.640
If you imagine a
curve like this,

00:00:46.640 --> 00:00:49.686
and we want to find a
tangent line at a point,

00:00:49.686 --> 00:00:51.310
it's going to look
something like this.

00:00:51.310 --> 00:00:54.600
So the tangent line is
going to look like this.

00:00:54.600 --> 00:00:59.220
A normal line is perpendicular
to the tangent line.

00:00:59.220 --> 00:01:01.470
This is the tangent line.

00:01:01.470 --> 00:01:05.710
The normal line is going to
be perpendicular to that.

00:01:05.710 --> 00:01:08.310
It's going to go just like that.

00:01:08.310 --> 00:01:11.450
And if this has a
slope of m, then this

00:01:11.450 --> 00:01:14.100
has a slope of the
negative reciprocal of m.

00:01:14.100 --> 00:01:17.350
So negative 1/m.

00:01:17.350 --> 00:01:19.200
So with that as a
little bit of a hint,

00:01:19.200 --> 00:01:21.880
I encourage you to find the
equation of the normal line

00:01:21.880 --> 00:01:25.410
to this curve, when x equals 1.

00:01:25.410 --> 00:01:29.082
So let's find the slope
of the tangent line.

00:01:29.082 --> 00:01:30.790
And then we take the
negative reciprocal,

00:01:30.790 --> 00:01:33.070
we can find the slope
of the normal line.

00:01:33.070 --> 00:01:35.190
So to find the slope
of the tangent line,

00:01:35.190 --> 00:01:38.660
we just take the derivative here
and evaluate it at x equals 1.

00:01:38.660 --> 00:01:41.180
So f prime of x,
and actually, let me

00:01:41.180 --> 00:01:42.400
rewrite this a little bit.

00:01:42.400 --> 00:01:47.222
So f of x is equal to e to the
x times x to the negative 2.

00:01:47.222 --> 00:01:49.180
I like to rewrite it this
way, because I always

00:01:49.180 --> 00:01:50.721
forget the whole
quotient rule thing.

00:01:50.721 --> 00:01:52.770
I like the power
rule a lot more.

00:01:52.770 --> 00:01:54.610
And this allows me to
use the power rule.

00:01:54.610 --> 00:01:57.792
I'm sorry, not the power
rule, the product rule.

00:01:57.792 --> 00:01:59.250
So this allows me
to do the product

00:01:59.250 --> 00:02:01.150
rule instead of
the quotient rule.

00:02:01.150 --> 00:02:03.650
So the derivative
of this, f prime

00:02:03.650 --> 00:02:05.930
of x, is going to be the
derivative of e to the x.

00:02:05.930 --> 00:02:11.500
Which is just e to the x times
x to the negative 2, plus

00:02:11.500 --> 00:02:14.780
e to the x times the derivative
of x to the negative 2.

00:02:14.780 --> 00:02:18.650
Which is negative 2x to
the negative 3 power.

00:02:18.650 --> 00:02:21.200
I just used the power
rule right over here.

00:02:21.200 --> 00:02:24.710
So if I want to evaluate
when x is equal to 1,

00:02:24.710 --> 00:02:26.450
this is going to
be equal to-- let

00:02:26.450 --> 00:02:28.570
me do that in that
yellow color like.

00:02:28.570 --> 00:02:30.000
I like switching colors.

00:02:30.000 --> 00:02:32.950
This is going to be
equal to, let's see,

00:02:32.950 --> 00:02:35.580
this is going to be
e to the first power.

00:02:35.580 --> 00:02:39.100
Which is just e times
1 to the negative 2,

00:02:39.100 --> 00:02:46.290
which is just 1 plus e to the
first power, which is just e,

00:02:46.290 --> 00:02:48.410
times negative 2.

00:02:48.410 --> 00:02:50.210
1 to the negative 3 is just 1.

00:02:50.210 --> 00:02:51.644
So e times negative 2.

00:02:51.644 --> 00:02:52.810
So let me write it this way.

00:02:52.810 --> 00:02:57.390
So minus 2e.

00:02:57.390 --> 00:03:02.220
And e minus 2e is just going
to be equal to negative e.

00:03:02.220 --> 00:03:12.190
So this right over here, this is
the slope of the tangent line.

00:03:12.190 --> 00:03:14.390
And so if we want the
slope of the normal,

00:03:14.390 --> 00:03:16.100
we just take the
negative reciprocal.

00:03:16.100 --> 00:03:19.702
So the negative reciprocal
of this is going to be,

00:03:19.702 --> 00:03:21.410
well the reciprocal
is 1 over negative e,

00:03:21.410 --> 00:03:22.785
but we want the
negative of that.

00:03:22.785 --> 00:03:24.240
So it's going to be 1/e.

00:03:24.240 --> 00:03:27.260
This is going to be the
slope of the normal line.

00:03:34.590 --> 00:03:37.066
And then if we, and
our goal isn't just

00:03:37.066 --> 00:03:38.524
to the slope of
the normal line, we

00:03:38.524 --> 00:03:40.340
want the equation
of the normal line.

00:03:40.340 --> 00:03:42.560
And we know the
equation of a line

00:03:42.560 --> 00:03:45.860
can be represented
as y is equal to mx

00:03:45.860 --> 00:03:47.970
plus b, where m is the slope.

00:03:47.970 --> 00:03:52.050
So we can say it's going to be
y is equal to 1/e-- remember,

00:03:52.050 --> 00:03:56.660
we're doing the normal
line here-- times x plus b.

00:03:56.660 --> 00:03:59.250
And to solve for b, we
just have to recognize

00:03:59.250 --> 00:04:01.210
that we know a point
that this goes through.

00:04:01.210 --> 00:04:04.180
This goes through
the point x equals 1.

00:04:04.180 --> 00:04:09.970
And when x equals 1, what is y?

00:04:09.970 --> 00:04:13.430
Well, y is e to the 1st
over 1, which is just e.

00:04:13.430 --> 00:04:16.820
So this goes to the
point 1 comma e.

00:04:16.820 --> 00:04:22.800
So we know that when x is
equal to 1, y is equal to e.

00:04:22.800 --> 00:04:25.020
And now we can just solve for b.

00:04:25.020 --> 00:04:34.960
So we get e is
equal to 1/e plus b.

00:04:34.960 --> 00:04:38.630
Or we could just subtract
1 over e from both sides,

00:04:38.630 --> 00:04:45.160
and we would get b is
equal to e minus 1/e.

00:04:48.190 --> 00:04:53.355
And we could obviously right
this as e squared minus 1/e

00:04:53.355 --> 00:04:54.730
if we want to
write it like that.

00:04:54.730 --> 00:04:56.355
But could just leave
it just like this.

00:04:56.355 --> 00:04:58.170
So the equation of
the normal line--

00:04:58.170 --> 00:05:00.630
so we deserve our drum
roll right over here--

00:05:00.630 --> 00:05:10.020
is going to be y is equal
to 1/e times x, plus b.

00:05:10.020 --> 00:05:13.400
And b, plus b, is all of this.

00:05:13.400 --> 00:05:18.830
So plus e minus 1/e.

00:05:18.830 --> 00:05:25.530
So that right there is our
equation of the normal line.


Equation of tangent line example 1

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:03.100
We have the curve y is
equal to e to the x over 2

00:00:03.100 --> 00:00:05.160
plus x to the third power.

00:00:05.160 --> 00:00:08.660
And what we want to do
is find the equation

00:00:08.660 --> 00:00:13.910
of the tangent line to this
curve at the point x equals 1.

00:00:13.910 --> 00:00:17.840
And when x is equal to 1, y is
going to be equal to e over 3.

00:00:17.840 --> 00:00:21.230
It's going to be e over 3.

00:00:21.230 --> 00:00:24.630
So let's try to figure out the
equation of the tangent line

00:00:24.630 --> 00:00:27.106
to this curve at this point.

00:00:27.106 --> 00:00:28.730
And I encourage you
to pause this video

00:00:28.730 --> 00:00:31.730
and try this on your own first.

00:00:31.730 --> 00:00:35.230
Well, the slope of the
tangent line at this point

00:00:35.230 --> 00:00:39.160
is the same thing as the
derivative at this point.

00:00:39.160 --> 00:00:41.620
So let's try to find
the derivative of this

00:00:41.620 --> 00:00:45.410
or evaluate the derivative of
this function right over here

00:00:45.410 --> 00:00:47.180
at this point.

00:00:47.180 --> 00:00:49.440
So to do that, first
I'm going to rewrite it.

00:00:49.440 --> 00:00:51.537
You could use the
quotient rule if you like,

00:00:51.537 --> 00:00:53.120
but I always forget
the quotient rule.

00:00:53.120 --> 00:00:55.570
The product rule is much
easier for me to remember.

00:00:55.570 --> 00:00:58.400
So I can rewrite this as y
is equal to-- and I might

00:00:58.400 --> 00:01:06.080
as well color code it-- is equal
to e to the x times 2 plus x

00:01:06.080 --> 00:01:10.420
to the third to the
negative 1 power.

00:01:10.420 --> 00:01:14.000
And so the derivative of
this, so let me write it here.

00:01:14.000 --> 00:01:17.240
So y prime is going to be
equal to the derivative

00:01:17.240 --> 00:01:21.310
of this part of it, e to the x.

00:01:21.310 --> 00:01:24.272
So the derivative of e to
the x is just e to the x.

00:01:24.272 --> 00:01:25.230
Just let me write that.

00:01:25.230 --> 00:01:27.570
So we're going to take
the derivative of it.

00:01:27.570 --> 00:01:29.591
And that's what's
amazing about e to the x,

00:01:29.591 --> 00:01:31.090
is that the derivative
of e to the x

00:01:31.090 --> 00:01:33.920
is just e to the x
times this thing.

00:01:33.920 --> 00:01:40.170
So times 2 plus x to the
third to the negative 1.

00:01:40.170 --> 00:01:44.120
And then to that we're
going to add this thing.

00:01:44.120 --> 00:01:45.890
So not its derivative anymore.

00:01:45.890 --> 00:01:51.450
We're just going
to add e to the x

00:01:51.450 --> 00:01:54.500
times the derivative of
this thing right over here.

00:01:54.500 --> 00:01:58.010
So we're going to
take the derivative.

00:01:58.010 --> 00:02:00.330
So we can do the chain rule.

00:02:00.330 --> 00:02:03.050
It's going to be
the derivative of 2

00:02:03.050 --> 00:02:04.800
plus x to the third
to the negative 1

00:02:04.800 --> 00:02:08.410
power with respect to 2
plus x to the third times

00:02:08.410 --> 00:02:12.320
the derivative of 2 plus x to
the third with respect to x.

00:02:12.320 --> 00:02:15.455
So this is going to be
equal to negative-- I'll

00:02:15.455 --> 00:02:22.920
write it this way-- negative
2 plus x to the third

00:02:22.920 --> 00:02:25.700
to the negative 2 power.

00:02:25.700 --> 00:02:28.730
And then we're going to multiply
that times the derivative of 2

00:02:28.730 --> 00:02:31.160
plus x to the third
with respect to x.

00:02:31.160 --> 00:02:33.000
Well, derivative of
this with respect to x

00:02:33.000 --> 00:02:33.955
is just 3x squared.

00:02:39.280 --> 00:02:45.320
And of course, we could simplify
this a little bit if we like.

00:02:45.320 --> 00:02:47.510
But the whole point of
this is to actually find

00:02:47.510 --> 00:02:50.560
the value of the
derivative at this point.

00:02:50.560 --> 00:02:52.400
So let's evaluate.

00:02:52.400 --> 00:02:57.300
Let's evaluate y prime
when x is equal to 1.

00:02:57.300 --> 00:03:01.080
Y prime of 1 when
x is equal to 1.

00:03:01.080 --> 00:03:02.900
This thing will
simplify to-- let's see,

00:03:02.900 --> 00:03:10.960
this is going to be e times 2
plus 1 to the negative power.

00:03:10.960 --> 00:03:13.320
So that's just going
to be 1/3, right?

00:03:13.320 --> 00:03:15.000
2 plus 1 to the negative 1.

00:03:15.000 --> 00:03:16.545
So that's 3 to the negative 1.

00:03:16.545 --> 00:03:17.410
That's 1/3.

00:03:17.410 --> 00:03:24.820
So that's times 1/3 plus
e to the first power.

00:03:24.820 --> 00:03:26.360
Now let's see,
what does this do?

00:03:26.360 --> 00:03:28.560
This part right
over here, this is

00:03:28.560 --> 00:03:33.020
2 plus 1 to the
negative 2 power.

00:03:33.020 --> 00:03:35.040
So this-- actually,
let me-- I don't

00:03:35.040 --> 00:03:38.694
want to-- so this part right
over here is going to be,

00:03:38.694 --> 00:03:40.110
let's see, this
is going to be-- I

00:03:40.110 --> 00:03:41.943
don't want to make a
careless mistake here--

00:03:41.943 --> 00:03:44.500
is 3 to the negative 2 power.

00:03:44.500 --> 00:03:45.970
So 3 squared is 9.

00:03:45.970 --> 00:03:50.670
3 to the negative
second power is 1/9.

00:03:50.670 --> 00:03:52.815
And so it's going to be 1/9.

00:03:52.815 --> 00:03:54.940
Well, you're going to
multiply this negative there.

00:03:54.940 --> 00:03:57.150
So it's negative 1/9.

00:03:57.150 --> 00:04:00.930
And then we're going to
multiply that times 3 times 1.

00:04:00.930 --> 00:04:04.390
So it's negative 1/9 times 3.

00:04:04.390 --> 00:04:06.190
Times 3 right over here.

00:04:06.190 --> 00:04:13.350
So it's negative
3/9 or negative 1/3.

00:04:13.350 --> 00:04:18.529
So times negative 1/3.

00:04:18.529 --> 00:04:21.190
And all I did here
is I substituted 1

00:04:21.190 --> 00:04:23.320
for x and evaluated it.

00:04:23.320 --> 00:04:24.590
Now this is interesting.

00:04:24.590 --> 00:04:28.030
I have essentially--
let me rewrite this.

00:04:28.030 --> 00:04:37.250
This is equal to e over 3 minus
e over 3, which is equal to 0.

00:04:37.250 --> 00:04:42.280
So the slope of the derivative
when x is equal to 1 is 0,

00:04:42.280 --> 00:04:46.420
or the slope of the
tangent line is equal to 0.

00:04:49.380 --> 00:04:52.310
This simplified to a pretty
straightforward situation.

00:04:54.820 --> 00:04:57.480
If I wanted to write a line
in slope intercept form,

00:04:57.480 --> 00:05:03.170
I could write it like this. y
is equal to mx plus b, where

00:05:03.170 --> 00:05:06.250
m is the slope and b
is the y-intercept.

00:05:06.250 --> 00:05:08.830
Now we know that the slope of
the tangent line at this point,

00:05:08.830 --> 00:05:10.640
it has a slope, is 0.

00:05:10.640 --> 00:05:12.180
So this is going to be 0.

00:05:12.180 --> 00:05:14.394
So this whole term
is going to be 0.

00:05:14.394 --> 00:05:16.560
So it's just going to have
the form y is equal to b.

00:05:16.560 --> 00:05:19.330
This is just going to
be a horizontal line.

00:05:19.330 --> 00:05:20.970
So what is a
horizontal line that

00:05:20.970 --> 00:05:23.780
contains this point
right over here?

00:05:23.780 --> 00:05:28.292
Well, it contains the value
y is equal to e over 3.

00:05:28.292 --> 00:05:29.500
So this is a horizontal line.

00:05:29.500 --> 00:05:31.400
It has the same y
value the entire time.

00:05:31.400 --> 00:05:34.460
So if it has the
y value e over 3,

00:05:34.460 --> 00:05:37.260
then we know the equation
of the tangent line

00:05:37.260 --> 00:05:39.880
to this curve at
this point is going

00:05:39.880 --> 00:05:46.480
to be y is equal to e over 3.

00:05:46.480 --> 00:05:48.730
Another way you could think
about this right over here

00:05:48.730 --> 00:05:51.025
is, well, let's substitute
when x is equal to 1.

00:05:51.025 --> 00:05:52.400
Well, there's not
even an x here.

00:05:52.400 --> 00:05:55.850
But when x is any value,
y is equal to e over 3,

00:05:55.850 --> 00:05:58.740
you get b is equal
to e over 3, or you'd

00:05:58.740 --> 00:06:01.950
get y is equal to e over 3.

00:06:01.950 --> 00:06:03.340
So it's just a horizontal line.

00:06:03.340 --> 00:06:04.923
So let's actually
visualize this, just

00:06:04.923 --> 00:06:07.250
to make sure that this
actually makes sense.

00:06:07.250 --> 00:06:09.130
So let me get my
graphing calculator out.

00:06:11.890 --> 00:06:14.970
And so I'm in graphing mode.

00:06:14.970 --> 00:06:17.770
If you wanted to know
how to get there,

00:06:17.770 --> 00:06:21.682
you literally can just
go to graph, y equals,

00:06:21.682 --> 00:06:27.680
and I will do-- so e
to the x power divided

00:06:27.680 --> 00:06:33.175
by 2 plus x to the third power.

00:06:37.100 --> 00:06:38.280
That looks right.

00:06:38.280 --> 00:06:40.620
And I actually set the range
ahead of time to save time.

00:06:40.620 --> 00:06:42.500
So let me graph this.

00:06:42.500 --> 00:06:43.080
So let's see.

00:06:43.080 --> 00:06:44.840
Ooh, it does all sorts
of interesting things.

00:06:44.840 --> 00:06:45.380
All right.

00:06:45.380 --> 00:06:46.750
Oh, look at that.

00:06:46.750 --> 00:06:52.860
All right, so now we can trace
to get to when x is equal to 1.

00:06:52.860 --> 00:06:55.980
x equals 1.

00:06:55.980 --> 00:06:59.710
Right over there, you see
y is equal to e over 3,

00:06:59.710 --> 00:07:02.500
which this is kind of
its decimal expansion

00:07:02.500 --> 00:07:03.450
right over here.

00:07:03.450 --> 00:07:06.840
And it does look like
the slope right over here

00:07:06.840 --> 00:07:09.430
is 0, that the
tangent line is just

00:07:09.430 --> 00:07:12.780
going to be a horizontal
line at that point.

00:07:12.780 --> 00:07:15.593
So that makes me feel pretty
good about our answer.