Polynomial end behavior exercise example

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:04.560
The equations below all define
the same polynomial function,

00:00:04.560 --> 00:00:05.520
f of x.

00:00:05.520 --> 00:00:07.880
So they've just written
f of x in different ways.

00:00:07.880 --> 00:00:10.640
Here they've fully factored
f of x out as negative 4,

00:00:10.640 --> 00:00:12.500
times x minus 3,
times x minus 2, times

00:00:12.500 --> 00:00:14.470
x plus 1, times x plus 5.

00:00:14.470 --> 00:00:17.440
Then here, it looks like
they factored it once,

00:00:17.440 --> 00:00:19.504
and then they factored
it into two quadratics,

00:00:19.504 --> 00:00:20.920
and then they
completed the square

00:00:20.920 --> 00:00:22.920
for each of those quadratics.

00:00:22.920 --> 00:00:24.900
And then here, this
is just factoring it

00:00:24.900 --> 00:00:27.450
into two quadratic expressions.

00:00:27.450 --> 00:00:30.620
And then here, this is the
thing just fully expanded out,

00:00:30.620 --> 00:00:32.465
multiplied out as a polynomial.

00:00:35.650 --> 00:00:38.310
Which form of f of
x is most useful

00:00:38.310 --> 00:00:40.621
for determining end behavior?

00:00:40.621 --> 00:00:42.120
So when we think
about end behavior,

00:00:42.120 --> 00:00:44.992
we really want to focus on
the highest degree term.

00:00:44.992 --> 00:00:47.200
We really want to focus on
this term right over here,

00:00:47.200 --> 00:00:49.920
because as x becomes
very, very, very positive,

00:00:49.920 --> 00:00:51.490
this term is going to dominate.

00:00:51.490 --> 00:00:53.460
And as x becomes very,
very, very negative,

00:00:53.460 --> 00:00:55.270
this term is going to dominate.

00:00:55.270 --> 00:00:57.420
And so this term is
really going to describe

00:00:57.420 --> 00:00:58.660
what the end behavior is.

00:00:58.660 --> 00:01:00.930
We can even think about
the end behavior here.

00:01:00.930 --> 00:01:03.690
We have an even
exponent, so it's

00:01:03.690 --> 00:01:06.570
going to have behavior
like x squared,

00:01:06.570 --> 00:01:09.420
which is just another way of
saying that as x gets positive,

00:01:09.420 --> 00:01:12.130
the function is going to
get positive, very positive.

00:01:12.130 --> 00:01:13.990
And as x becomes very
negative, the function

00:01:13.990 --> 00:01:15.364
is going to become
very positive,

00:01:15.364 --> 00:01:16.890
because it's an even exponent.

00:01:16.890 --> 00:01:19.550
And the fact that there's a
negative coefficient in front

00:01:19.550 --> 00:01:23.830
of this highest
degree x term, this

00:01:23.830 --> 00:01:25.990
tells us it's going to
be similar to the shape

00:01:25.990 --> 00:01:27.370
of negative x squared.

00:01:27.370 --> 00:01:29.357
It's going to be
downward opening

00:01:29.357 --> 00:01:31.190
if you were to think
about its end behavior.

00:01:31.190 --> 00:01:33.080
So which form of the
function is most useful?

00:01:33.080 --> 00:01:35.550
Definitely D, the
expanded out version.

00:01:35.550 --> 00:01:37.290
So I'll go with D.

00:01:37.290 --> 00:01:40.330
So like the equation below, it's
similar end behavior f of x.

00:01:40.330 --> 00:01:41.830
We already talked about it.

00:01:41.830 --> 00:01:46.790
It's an even exponent, or it's
an even degree, we could say.

00:01:46.790 --> 00:01:49.220
And then we have a negative
out in front of it.

00:01:49.220 --> 00:01:52.040
So it's going to have
similar end behavior to y

00:01:52.040 --> 00:01:54.045
is equal to negative x squared.

00:01:54.045 --> 00:01:55.670
And then they say,
which form of f of x

00:01:55.670 --> 00:01:57.845
is most useful for
finding zeroes?

00:01:57.845 --> 00:01:59.470
When we're finding
zeroes, we're trying

00:01:59.470 --> 00:02:02.940
to find the x values that would
make the function equal to 0.

00:02:02.940 --> 00:02:04.452
And this first form
right over here

00:02:04.452 --> 00:02:06.660
where we're all factored
out-- that's the most useful

00:02:06.660 --> 00:02:08.169
for finding 0s.

00:02:08.169 --> 00:02:11.160
We know that if we wanted to
set this whole thing to be

00:02:11.160 --> 00:02:14.219
equal to 0-- we said, hey,
when does this equal 0?

00:02:14.219 --> 00:02:16.510
That means that one of these
expressions-- at least one

00:02:16.510 --> 00:02:18.190
of these expressions
need to be equal 0.

00:02:18.190 --> 00:02:20.500
We can independently
set them equal to 0

00:02:20.500 --> 00:02:21.580
and then solve for x.

00:02:21.580 --> 00:02:24.890
So, this would be 0 when x is
equal to 3, x is equal to 2,

00:02:24.890 --> 00:02:26.890
x is negative 1,
and x is negative 5.

00:02:26.890 --> 00:02:33.007
So A, Is definitely the
most useful for finding 0s.


Another polynomial end behavior example

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:03.320
So we've got three functions
graphed right over here.

00:00:03.320 --> 00:00:05.320
And then we have three
equations right over here

00:00:05.320 --> 00:00:07.999
that could describe
these functions.

00:00:07.999 --> 00:00:09.540
And once again, we
need to figure out

00:00:09.540 --> 00:00:11.862
which ones will match
with which other ones.

00:00:11.862 --> 00:00:14.320
And I encourage you to pause
this video right now and think

00:00:14.320 --> 00:00:15.890
about it.

00:00:15.890 --> 00:00:17.664
So let's go equation
by equation.

00:00:17.664 --> 00:00:20.080
And at least let's try to write
them in a form or at least

00:00:20.080 --> 00:00:23.740
look at parts of them that will
make it a little easier for us

00:00:23.740 --> 00:00:25.880
to figure out which
ones match up.

00:00:25.880 --> 00:00:29.740
So let's look at this
first one right over here.

00:00:29.740 --> 00:00:32.670
So this first one,
instead of just taking

00:00:32.670 --> 00:00:34.640
the trouble of multiplying
all of this out,

00:00:34.640 --> 00:00:36.776
I really care about
the highest degree term

00:00:36.776 --> 00:00:38.150
and maybe the
lowest degree term.

00:00:38.150 --> 00:00:40.130
So let me just try
to figure those out.

00:00:40.130 --> 00:00:42.550
So I could rewrite
that one as y is

00:00:42.550 --> 00:00:45.024
equal to-- the highest
degree term is essentially

00:00:45.024 --> 00:00:47.440
going to be, we're going to
take the product of these four

00:00:47.440 --> 00:00:50.290
expressions: negative 1/11
times x minus 5 times this times

00:00:50.290 --> 00:00:50.960
this.

00:00:50.960 --> 00:00:53.310
So if we take the highest
degree part of each of them,

00:00:53.310 --> 00:00:55.351
that's going to give us
our highest degree terms.

00:00:55.351 --> 00:00:59.610
So it's going to be
negative 1/11 times x times

00:00:59.610 --> 00:01:01.827
x squared times x squared.

00:01:01.827 --> 00:01:03.660
So that's going to be
x to the fifth, right?

00:01:03.660 --> 00:01:05.826
x squared times x squared
is x to the fourth times x

00:01:05.826 --> 00:01:06.940
is x to the fifth.

00:01:06.940 --> 00:01:09.551
Negative 1/11 x to the fifth.

00:01:09.551 --> 00:01:11.550
And then it's going to
be plus a bunch of stuff.

00:01:11.550 --> 00:01:14.008
And then our lowest degree term
is going to be, let's see--

00:01:14.008 --> 00:01:17.520
It's going to be negative
1/11 times negative

00:01:17.520 --> 00:01:19.485
5, which is positive 5/11.

00:01:19.485 --> 00:01:21.110
Actually let me just
write it this way.

00:01:21.110 --> 00:01:27.580
Plus negative 1/11 times
negative 5 times positive 10

00:01:27.580 --> 00:01:29.922
times 2x.

00:01:29.922 --> 00:01:31.213
I guess we could work this out.

00:01:31.213 --> 00:01:36.334
This is what, positive 100/11.

00:01:36.334 --> 00:01:37.750
But let's just
leave it like this.

00:01:37.750 --> 00:01:39.950
That's going to be our
lowest degree term.

00:01:39.950 --> 00:01:42.720
I just took the lowest degree
from each of these expressions

00:01:42.720 --> 00:01:46.230
that we're going to be
multiplying together.

00:01:46.230 --> 00:01:47.980
And so, this is another
way of viewing it.

00:01:47.980 --> 00:01:50.410
And what's useful about this
one is we could immediately

00:01:50.410 --> 00:01:52.810
think about the
end behavior here.

00:01:52.810 --> 00:01:54.780
So what do I mean
by end behavior?

00:01:54.780 --> 00:01:56.330
Well we've covered this already.

00:01:56.330 --> 00:01:58.962
Let's look at the coefficient
on the highest degree term.

00:01:58.962 --> 00:02:01.170
And let's also think about
whether the highest degree

00:02:01.170 --> 00:02:03.240
term is odd or even.

00:02:03.240 --> 00:02:07.150
Since we're dealing
with an odd degree

00:02:07.150 --> 00:02:09.210
polynomial-- this
is x to the fifth--

00:02:09.210 --> 00:02:12.120
and this coefficient on
the highest degree term

00:02:12.120 --> 00:02:13.860
is negative, this
is going to have

00:02:13.860 --> 00:02:23.299
end behavior like
negative x to the third.

00:02:23.299 --> 00:02:25.340
And what would negative
x to the third look like?

00:02:25.340 --> 00:02:27.514
Well when x is
really negative, it's

00:02:27.514 --> 00:02:28.680
going to be really positive.

00:02:28.680 --> 00:02:30.390
Because x to the third is going
to be really negative, but then

00:02:30.390 --> 00:02:32.530
you're multiplying
it times a negative.

00:02:32.530 --> 00:02:34.874
And then when x is really
positive, x to the 3rd

00:02:34.874 --> 00:02:36.290
is going to be
even more positive,

00:02:36.290 --> 00:02:37.530
but we have a
negative out front,

00:02:37.530 --> 00:02:39.030
so it's going to
be really negative.

00:02:41.610 --> 00:02:44.670
So this function is going
to look something like this,

00:02:44.670 --> 00:02:48.200
and it's going to do some
crazy things in between.

00:02:48.200 --> 00:02:49.830
So which of these
look like that?

00:02:49.830 --> 00:02:53.700
Well let's see... f starts high,
then it goes high up again.

00:02:53.700 --> 00:02:55.090
So it's not that one.

00:02:55.090 --> 00:02:58.035
g starts low and ends up high.

00:02:58.035 --> 00:02:59.920
This starts high, ends up low.

00:02:59.920 --> 00:03:03.250
So the only one that's left--
this actually kind of plays

00:03:03.250 --> 00:03:06.390
tricks with the eye--
the only one that's left

00:03:06.390 --> 00:03:09.660
is the dotted purple
one which is-- actually

00:03:09.660 --> 00:03:11.110
do they even have
a label on this?

00:03:11.110 --> 00:03:13.360
Well we'll call that one h.

00:03:13.360 --> 00:03:15.100
That one is h.

00:03:15.100 --> 00:03:18.470
And let me highlight it,
because I know it's hard to see.

00:03:18.470 --> 00:03:19.830
So this one.

00:03:19.830 --> 00:03:20.940
This one right over here.

00:03:20.940 --> 00:03:22.670
Notice: end behavior.

00:03:22.670 --> 00:03:25.000
It starts for very
negative values

00:03:25.000 --> 00:03:27.627
and has a very high
value for our function.

00:03:27.627 --> 00:03:29.710
Then it does all sorts of
crazy things in between.

00:03:33.414 --> 00:03:35.080
And you can actually
see that it clearly

00:03:35.080 --> 00:03:38.440
has one, two, three,
four, five roots.

00:03:38.440 --> 00:03:41.130
But that's not what we're really
concerned with for this sake.

00:03:41.130 --> 00:03:43.290
And then after it does
all that crazy stuff,

00:03:43.290 --> 00:03:45.480
as x gets really positive,
the value of the function

00:03:45.480 --> 00:03:46.870
becomes really negative.

00:03:46.870 --> 00:03:57.030
So its end behavior is like
negative x to the third.

00:03:57.030 --> 00:04:02.380
So we can feel confident that
this function right over here

00:04:02.380 --> 00:04:05.006
is h of x.

00:04:05.006 --> 00:04:07.130
Now, in other videos I've
looked at the intercepts.

00:04:07.130 --> 00:04:09.220
But what is the
y-intercept for this one?

00:04:09.220 --> 00:04:11.831
Well, when x equals 0, this
whole thing is equal to 0.

00:04:11.831 --> 00:04:13.830
Because the lowest degree
term isn't a constant.

00:04:13.830 --> 00:04:15.010
It has an x in it.

00:04:15.010 --> 00:04:16.610
So when x is 0, this
whole thing is 0.

00:04:16.610 --> 00:04:17.649
And that's consistent.

00:04:17.649 --> 00:04:19.876
Actually all three of these
have a y-intercept of 0.

00:04:19.876 --> 00:04:21.709
And you see that's
because all three of them

00:04:21.709 --> 00:04:24.620
have a lowest degree
term that involves an x.

00:04:24.620 --> 00:04:27.180
So we can really just
concern ourselves

00:04:27.180 --> 00:04:29.370
with the highest degree terms.

00:04:29.370 --> 00:04:31.260
So now let's think
about the next function.

00:04:31.260 --> 00:04:32.850
And I'm gonna give myself
some real estate here.

00:04:32.850 --> 00:04:34.600
I don't want to scroll
down just because I

00:04:34.600 --> 00:04:37.650
want to be able to
look at the graph.

00:04:37.650 --> 00:04:43.390
So let's look at-- oh
my phone's ringing.

00:04:43.390 --> 00:04:44.360
This is more important.

00:04:44.360 --> 00:04:48.182
Let me work on this video-- So
this next one right over here.

00:04:48.182 --> 00:04:50.140
Let's just think about
what it might look like.

00:04:50.140 --> 00:04:52.556
It would look like y is equal
to-- its highest degree term

00:04:52.556 --> 00:04:54.960
is 110 times x squared
times x squared.

00:04:54.960 --> 00:04:57.450
So positive 110 times x
squared times x squared,

00:04:57.450 --> 00:04:58.730
which is x to the fourth.

00:04:58.730 --> 00:05:00.530
And then plus a
bunch of other stuff.

00:05:00.530 --> 00:05:02.550
So what's its end
behavior going to be?

00:05:02.550 --> 00:05:07.080
Well it has an even exponent
on its highest degree.

00:05:07.080 --> 00:05:10.240
Or it's an even
degree polynomial.

00:05:10.240 --> 00:05:12.520
And the coefficient on
that highest degree term

00:05:12.520 --> 00:05:13.620
is positive.

00:05:13.620 --> 00:05:19.670
So it's going to
have end behavior

00:05:19.670 --> 00:05:24.390
like positive x squared.

00:05:24.390 --> 00:05:27.290
So when x is really negative,
the value of the function

00:05:27.290 --> 00:05:28.782
is going to be really positive.

00:05:28.782 --> 00:05:30.240
And then when x is
really positive,

00:05:30.240 --> 00:05:32.580
the value of the function is
going to be really positive.

00:05:32.580 --> 00:05:34.288
And it might do some
kind of crazy things

00:05:34.288 --> 00:05:36.470
in between, because this
obviously isn't x squared.

00:05:36.470 --> 00:05:38.480
It has all sorts of
crazy terms there.

00:05:38.480 --> 00:05:41.200
So which of these
functions looks like that?

00:05:41.200 --> 00:05:42.150
Well let's see.

00:05:42.150 --> 00:05:45.840
Our f has that,
when x gets really

00:05:45.840 --> 00:05:47.980
negative it gets
really positive.

00:05:47.980 --> 00:05:50.620
And when x gets really positive,
it gets really positive.

00:05:50.620 --> 00:05:53.650
And it's the only one that
has that type of behavior.

00:05:53.650 --> 00:05:58.410
So we can say fairly confidently
that f of x is this function.

00:05:58.410 --> 00:06:00.087
Or this function is f of x.

00:06:00.087 --> 00:06:01.920
And then once again,
deductive reasoning, we

00:06:01.920 --> 00:06:05.390
can think that this
last one is g of x.

00:06:05.390 --> 00:06:07.710
That this last one right
over here is g of x.

00:06:07.710 --> 00:06:08.666
But let's verify it.

00:06:08.666 --> 00:06:11.165
So if we were trying to think
about its highest degree term,

00:06:11.165 --> 00:06:15.150
it would be y is
equal to 1/11 times

00:06:15.150 --> 00:06:17.260
x times x squared
times x squared.

00:06:17.260 --> 00:06:19.220
So 1/11 x to the fifth.

00:06:19.220 --> 00:06:21.390
And then plus all sorts
of other business.

00:06:21.390 --> 00:06:25.430
So here you have an odd
degree-- it's a fifth degree--

00:06:25.430 --> 00:06:27.050
and you have a
positive coefficient

00:06:27.050 --> 00:06:28.710
on that fifth degree term.

00:06:28.710 --> 00:06:32.020
So it's going to
have end behavior

00:06:32.020 --> 00:06:34.370
like positive x to the third.

00:06:34.370 --> 00:06:37.200
So when x is really negative,
the value of the function

00:06:37.200 --> 00:06:38.589
is going to be really negative.

00:06:38.589 --> 00:06:40.630
And when x is positive,
the value of the function

00:06:40.630 --> 00:06:42.250
is going to be really positive.

00:06:42.250 --> 00:06:44.732
And it's going to do all sorts
of crazy things in between.

00:06:44.732 --> 00:06:45.940
So which one looks like that?

00:06:45.940 --> 00:06:47.320
Well g looks like that.

00:06:47.320 --> 00:06:50.110
When x gets negative,
the value of our function

00:06:50.110 --> 00:06:51.270
gets really negative.

00:06:51.270 --> 00:06:53.550
As x gets really positive,
the value of our function

00:06:53.550 --> 00:06:55.040
becomes really positive.

00:06:55.040 --> 00:06:59.790
And it does all sorts
of craziness in between.

00:06:59.790 --> 00:07:04.410
So we can feel good
that that one is g of x.


Polynomial end behavior example

WEBVTT
Kind: captions
Language: en

00:00:00.779 --> 00:00:02.820
So we have three functions
that are graphed here.

00:00:02.820 --> 00:00:05.680
We have f of x graphed
in this dotted magenta.

00:00:05.680 --> 00:00:07.660
We have g of x in this green.

00:00:07.660 --> 00:00:10.240
And then we have h of x
in this dotted purple.

00:00:10.240 --> 00:00:13.310
And then we have three
potential equations

00:00:13.310 --> 00:00:15.730
that could also be used
to define functions.

00:00:15.730 --> 00:00:18.890
And what I want you to
do is think about which

00:00:18.890 --> 00:00:24.300
of these equations match up to
which of these function graphs.

00:00:24.300 --> 00:00:26.050
So let's go through
them one by one.

00:00:26.050 --> 00:00:28.980
So this first one is
reasonably straightforward.

00:00:28.980 --> 00:00:31.280
This is just a
straight-up quadratic.

00:00:31.280 --> 00:00:35.514
Right over here, y is equal
to 1/2 x squared minus 9/2.

00:00:35.514 --> 00:00:37.930
And we see that the coefficient
on the second degree term,

00:00:37.930 --> 00:00:40.263
or the highest degree term
right over, here is positive.

00:00:40.263 --> 00:00:41.320
It is 1/2.

00:00:41.320 --> 00:00:43.710
So this is going to be an
upward-opening parabola.

00:00:43.710 --> 00:00:47.745
It's going to look
something like that.

00:00:47.745 --> 00:00:49.870
And when we look at all of
these options over here,

00:00:49.870 --> 00:00:52.340
there's only one upward
opening parabola.

00:00:52.340 --> 00:00:54.250
And that is g of x.

00:00:54.250 --> 00:00:58.080
So we know that if this has
got to be one of these three,

00:00:58.080 --> 00:01:00.580
this right over here is g of x.

00:01:00.580 --> 00:01:05.030
So we could say that g of
x is equal to 1/2 x squared

00:01:05.030 --> 00:01:06.710
minus 9/2.

00:01:06.710 --> 00:01:11.540
Another giveaway that g of x, or
that g is being described by y

00:01:11.540 --> 00:01:14.510
is equal to 1/2 x
squared, minus 9/2,

00:01:14.510 --> 00:01:16.650
is you see when x
is equal to 0, we

00:01:16.650 --> 00:01:18.840
have our y-intercept
at negative 9/2,

00:01:18.840 --> 00:01:20.930
which is the same thing
as negative 4 and 1/2.

00:01:20.930 --> 00:01:23.540
And you see that is indeed the
case for g right over there.

00:01:23.540 --> 00:01:26.830
It's when x is 0, y
is negative 4 and 1/2.

00:01:26.830 --> 00:01:28.970
So we could feel pretty
confident about that.

00:01:28.970 --> 00:01:30.530
Now let's look at
these next two.

00:01:30.530 --> 00:01:32.270
And what I'm going
to do, let me try

00:01:32.270 --> 00:01:34.300
to rewrite them in
a neutral color.

00:01:34.300 --> 00:01:36.260
So we don't use blue yet.

00:01:36.260 --> 00:01:40.140
So let's think about this one. y
is equal to negative 1/10 times

00:01:40.140 --> 00:01:44.686
x plus 3 times x
squared minus 9.

00:01:44.686 --> 00:01:46.560
Now, we could take the
trouble of multiplying

00:01:46.560 --> 00:01:47.561
all of this stuff out.

00:01:47.561 --> 00:01:50.060
Or we could just think about
what is the highest-degree term

00:01:50.060 --> 00:01:50.630
going to look like?

00:01:50.630 --> 00:01:52.490
And we could also think about
what does the lowest degree

00:01:52.490 --> 00:01:53.114
term look like?

00:01:53.114 --> 00:01:54.460
What is the constant term?

00:01:54.460 --> 00:01:57.620
So the highest-degree term is
going to be negative 1/10 times

00:01:57.620 --> 00:02:00.140
x times x squared.

00:02:00.140 --> 00:02:04.480
So this is going to be y
is equal to negative 1/10

00:02:04.480 --> 00:02:06.084
x to the third power.

00:02:06.084 --> 00:02:08.500
Then you're going to have a
bunch of other terms in there.

00:02:11.050 --> 00:02:13.300
And then you're going to
have your lowest-- maybe I'll

00:02:13.300 --> 00:02:15.327
write a plus although
it might be a minus,

00:02:15.327 --> 00:02:17.660
but then you're going have a
bunch of other terms there.

00:02:17.660 --> 00:02:20.160
And then you're going to have
your lowest-degree term, which

00:02:20.160 --> 00:02:25.630
is going to be negative 9
times 3, which is negative 27,

00:02:25.630 --> 00:02:27.547
times negative 1/10.

00:02:27.547 --> 00:02:29.130
Well, that's just
going to be positive

00:02:29.130 --> 00:02:35.426
27/10, which is the
same thing as 2.7.

00:02:35.426 --> 00:02:36.800
Now, just from
this, without even

00:02:36.800 --> 00:02:38.650
knowing all of this
business in the middle,

00:02:38.650 --> 00:02:43.050
we can start to think about
what this graph is actually

00:02:43.050 --> 00:02:44.500
going to look like.

00:02:44.500 --> 00:02:50.114
First of all, we see it is
a third-degree polynomial,

00:02:50.114 --> 00:02:51.530
or it's a third-degree
expression,

00:02:51.530 --> 00:02:52.660
I guess we could say.

00:02:52.660 --> 00:02:57.020
And its coefficient on the
third-degree term is negative.

00:02:57.020 --> 00:03:00.220
So that means it's going to have
end behavior like negative x

00:03:00.220 --> 00:03:00.790
to the third.

00:03:00.790 --> 00:03:03.730
x to the third looks like this.

00:03:03.730 --> 00:03:09.450
Negative x to the third will
look something like this.

00:03:09.450 --> 00:03:12.150
So its end behavior is
going to look like that.

00:03:12.150 --> 00:03:14.400
We could think about what
it does in the middle.

00:03:14.400 --> 00:03:16.299
It might jump up and
do something like that.

00:03:16.299 --> 00:03:18.340
But that's what it's end
behavior is going to be.

00:03:18.340 --> 00:03:22.480
It has end behavior like
negative x to the third.

00:03:22.480 --> 00:03:25.040
Now, which of these graphs
have that type of end behavior?

00:03:25.040 --> 00:03:29.550
Well, it's pretty clear that h
of x, when x is very negative,

00:03:29.550 --> 00:03:31.570
h of x is positive,
just like this.

00:03:31.570 --> 00:03:34.000
When x is very positive,
h of x is very negative.

00:03:34.000 --> 00:03:36.880
It is the only of these three
graphs that have that behavior.

00:03:36.880 --> 00:03:40.749
So we can feel pretty
confident that this is h of x.

00:03:40.749 --> 00:03:42.290
Now, if we're not
sure, we could also

00:03:42.290 --> 00:03:44.940
look at the y-intercept
right over here.

00:03:44.940 --> 00:03:48.210
When x is equal to 0, all
these other terms are 0.

00:03:48.210 --> 00:03:50.566
And so y is going to be 2.7.

00:03:50.566 --> 00:03:52.690
And we see that is indeed
the case right over here.

00:03:52.690 --> 00:03:55.080
When x is 0, y is 2.7.

00:03:55.080 --> 00:03:57.250
So that gives us some
confidence as well.

00:03:57.250 --> 00:04:00.040
Now, let's look at
this third expression.

00:04:00.040 --> 00:04:02.310
And obviously we
could just deduce

00:04:02.310 --> 00:04:05.840
that there's only one equation
left and one function left.

00:04:05.840 --> 00:04:07.590
So this has got to be f of x.

00:04:07.590 --> 00:04:09.130
But let's say we
didn't know that.

00:04:09.130 --> 00:04:11.340
Let's see how we
could reason it out.

00:04:11.340 --> 00:04:16.990
So once again, let's think
about multiplying-- figure

00:04:16.990 --> 00:04:18.490
out what the highest
degree term is.

00:04:18.490 --> 00:04:20.073
Well, the highest-degree
term is going

00:04:20.073 --> 00:04:22.050
to be 1/10 times x
squared times x squared.

00:04:22.050 --> 00:04:26.180
So we can rewrite this
one as 1/10-- x squared

00:04:26.180 --> 00:04:28.921
times x squared is x
to the fourth power.

00:04:28.921 --> 00:04:30.670
Then we're going to
have a bunch of terms.

00:04:30.670 --> 00:04:34.550
And then our lowest-degree term
is going to be 9 times 9 times

00:04:34.550 --> 00:04:35.550
1/10.

00:04:35.550 --> 00:04:38.170
That's 81 divided by 10.

00:04:38.170 --> 00:04:41.290
So that is 8.1.

00:04:41.290 --> 00:04:43.966
So what type of end to
behavior is this going to have?

00:04:43.966 --> 00:04:47.800
You have an even-degree
polynomial right over here,

00:04:47.800 --> 00:04:50.830
and it has a
positive coefficient.

00:04:50.830 --> 00:04:52.860
So it's going to
have end behavior

00:04:52.860 --> 00:04:55.600
like a second-degree polynomial
with a positive coefficient.

00:04:55.600 --> 00:04:58.180
Or it's going to have end
behavior like x squared.

00:04:58.180 --> 00:05:00.570
So it's going to look--
when x is a really negative,

00:05:00.570 --> 00:05:02.840
it's going to be
really positive.

00:05:02.840 --> 00:05:04.390
And then when x is
really positive,

00:05:04.390 --> 00:05:05.990
it's going to be
really positive.

00:05:05.990 --> 00:05:08.140
So it's end behavior is
going to look like that,

00:05:08.140 --> 00:05:10.710
and it might do some
business in between.

00:05:10.710 --> 00:05:12.890
Well, which of these
graphs do that?

00:05:12.890 --> 00:05:15.270
Well, it's pretty
clear that f definitely

00:05:15.270 --> 00:05:19.240
has that type of end behavior.

00:05:19.240 --> 00:05:21.529
It does something a little
interesting in the middle.

00:05:21.529 --> 00:05:23.070
And if we wanted to
be sure, we could

00:05:23.070 --> 00:05:24.840
look at this intercept
right over here.

00:05:24.840 --> 00:05:27.510
When x is 0, all these other
terms are going to be 0,

00:05:27.510 --> 00:05:30.250
and you're going to be left just
with this y-intercept, which

00:05:30.250 --> 00:05:31.780
is 8.1.

00:05:31.780 --> 00:05:34.380
And so when x is 0
we do see, indeed,

00:05:34.380 --> 00:05:37.240
that f is sitting here at 8.1.

00:05:37.240 --> 00:05:40.040
So we can feel pretty confident
that that right over there

00:05:40.040 --> 00:05:42.360
is f of x.


Polynomial end behavior

WEBVTT
Kind: captions
Language: en

00:00:00.024 --> 00:00:02.190
What I want to do in this
video is talk a little bit

00:00:02.190 --> 00:00:04.560
about polynomial end behavior.

00:00:04.560 --> 00:00:06.310
And this is really
just talking about what

00:00:06.310 --> 00:00:11.230
happens to a polynomial if
as x becomes really large

00:00:11.230 --> 00:00:13.250
or really, really,
really negative.

00:00:13.250 --> 00:00:17.700
For example, we're familiar
with quadratic polynomials where

00:00:17.700 --> 00:00:23.440
y is equal to ax
squared plus bx plus c.

00:00:23.440 --> 00:00:26.160
We know that if a
is greater than 0,

00:00:26.160 --> 00:00:28.811
this is going to be an upward
opening parabola of some kind.

00:00:28.811 --> 00:00:30.560
So it's going to look
something like that,

00:00:30.560 --> 00:00:34.240
the graph of this equation,
or of this function,

00:00:34.240 --> 00:00:35.200
you could say.

00:00:35.200 --> 00:00:38.389
And if a is less
than 0, it's going

00:00:38.389 --> 00:00:39.805
to be a downward
opening parabola.

00:00:43.270 --> 00:00:45.570
We've spent less time with
third degree polynomials,

00:00:45.570 --> 00:00:47.310
but we've also seen
those a little bit.

00:00:47.310 --> 00:00:51.020
So for example, if you have
the third degree polynomial,

00:00:51.020 --> 00:00:59.330
y is equal to ax to the third
plus bx squared plus cx plus d,

00:00:59.330 --> 00:01:05.500
if a is greater than 0-- I don't
want to use that brown color.

00:01:05.500 --> 00:01:11.710
If a is greater than 0,
when x is really, really,

00:01:11.710 --> 00:01:14.090
really negative, this whole
thing is going to be really,

00:01:14.090 --> 00:01:15.360
really, really negative.

00:01:15.360 --> 00:01:19.730
And then it's going to increase
as x becomes less negative.

00:01:19.730 --> 00:01:21.540
It's going do
something-- it might

00:01:21.540 --> 00:01:23.630
do a little bit of
funky stuff in between.

00:01:23.630 --> 00:01:26.400
But then as x becomes more
and more and more positive,

00:01:26.400 --> 00:01:29.930
it will become more and more
and more positive as well.

00:01:29.930 --> 00:01:35.900
So it might look something like
this when a is greater than 0.

00:01:35.900 --> 00:01:37.800
But what about when
a is less than 0?

00:01:37.800 --> 00:01:40.040
Well then, just like
here, we would flip it.

00:01:40.040 --> 00:01:46.180
We would flip it so that
if a is less than 0,

00:01:46.180 --> 00:01:48.290
when x is really
negative, you're

00:01:48.290 --> 00:01:50.282
going to multiply that
times a negative a

00:01:50.282 --> 00:01:51.990
and you're going to
get a positive value.

00:01:51.990 --> 00:01:54.100
So it's going to look
something like this.

00:01:54.100 --> 00:01:55.980
And then it's going
to go like this.

00:01:55.980 --> 00:01:58.740
It might do a little bit of this
type of business in between.

00:01:58.740 --> 00:02:01.950
But then its end behavior,
it starts decreasing again.

00:02:01.950 --> 00:02:03.540
It starts decreasing.

00:02:03.540 --> 00:02:05.660
So when we talk
about end behavior,

00:02:05.660 --> 00:02:08.330
we're talking about the idea
of what is this function?

00:02:08.330 --> 00:02:11.190
What does this polynomial do
as x becomes really, really,

00:02:11.190 --> 00:02:13.300
really, really positive
and as x becomes really,

00:02:13.300 --> 00:02:14.680
really, really, really negative?

00:02:14.680 --> 00:02:18.065
And kind of fully recognizing
that some weird things might

00:02:18.065 --> 00:02:19.190
be happening in the middle.

00:02:19.190 --> 00:02:20.689
But we just want
to think about what

00:02:20.689 --> 00:02:23.440
happens at extreme values of x.

00:02:23.440 --> 00:02:25.320
Now obviously for
the second degree

00:02:25.320 --> 00:02:28.630
polynomial nothing really
weird happens in the middle.

00:02:28.630 --> 00:02:30.380
But for a third
degree polynomial,

00:02:30.380 --> 00:02:33.600
we sort of see that
some interesting things

00:02:33.600 --> 00:02:35.320
can start happening
in the middle.

00:02:35.320 --> 00:02:38.650
But the end behavior for
third degree polynomial

00:02:38.650 --> 00:02:41.300
is that if a is
greater than 0-- we're

00:02:41.300 --> 00:02:43.540
starting really small,
really low values--

00:02:43.540 --> 00:02:46.870
and as a becomes positive,
we get to really high values.

00:02:46.870 --> 00:02:49.130
If a is less than 0
we have the opposite.

00:02:49.130 --> 00:02:51.620
And these are kind of the two
prototypes for polynomials.

00:02:51.620 --> 00:02:54.160
Because from there we can
start thinking about any degree

00:02:54.160 --> 00:02:55.380
polynomial.

00:02:55.380 --> 00:02:59.990
So let's just think about the
situation of a fourth degree

00:02:59.990 --> 00:03:01.280
polynomial.

00:03:01.280 --> 00:03:06.600
So let's say y is equal
to ax to the fourth power

00:03:06.600 --> 00:03:13.360
plus bx to the third plus cx
squared plus dx plus-- I don't

00:03:13.360 --> 00:03:17.080
want to write e because e has
other meanings in mathematics.

00:03:17.080 --> 00:03:21.704
I'll say plus-- I'm really
running out of letters here.

00:03:21.704 --> 00:03:23.870
I'll just use f, although
this isn't the function f.

00:03:23.870 --> 00:03:27.230
This is just a constant
f right over here.

00:03:27.230 --> 00:03:29.860
So let's just think about
what this might look like.

00:03:29.860 --> 00:03:31.409
Let's think about
its end behavior,

00:03:31.409 --> 00:03:33.700
and we could think about it
relative to a second degree

00:03:33.700 --> 00:03:35.040
polynomial.

00:03:35.040 --> 00:03:38.690
So its end behavior, if x is
really, really, really, really

00:03:38.690 --> 00:03:41.400
negative, x to the fourth is
still going to be positive.

00:03:41.400 --> 00:03:46.345
And if a is greater than 0
when x is really, really,

00:03:46.345 --> 00:03:48.470
really negative, we're
going to have really, really

00:03:48.470 --> 00:03:52.920
positive values, just
like a second degree.

00:03:52.920 --> 00:03:55.274
And when x is really
positive, same thing.

00:03:55.274 --> 00:03:56.940
x to the fourth is
going to be positive,

00:03:56.940 --> 00:03:58.870
times a is still
going to be positive.

00:03:58.870 --> 00:04:02.050
So its end behavior
is going to look

00:04:02.050 --> 00:04:05.280
very similar to a second
degree polynomial.

00:04:05.280 --> 00:04:07.280
Now, it might do--
in fact it probably

00:04:07.280 --> 00:04:09.060
will do some funky
stuff in between.

00:04:09.060 --> 00:04:15.570
It might do something that looks
kind of like that in between.

00:04:15.570 --> 00:04:18.220
But we care about
the end behavior.

00:04:18.220 --> 00:04:20.430
I guess you could call
the stuff that I've dotted

00:04:20.430 --> 00:04:23.050
lined in the middle, this is
called the non-end behavior,

00:04:23.050 --> 00:04:24.390
the middle behavior.

00:04:24.390 --> 00:04:26.680
This will obviously be
different than a second degree

00:04:26.680 --> 00:04:27.540
polynomial.

00:04:27.540 --> 00:04:30.560
But what happens at the
ends will be the same.

00:04:30.560 --> 00:04:32.530
And the reason why, when
you square something,

00:04:32.530 --> 00:04:34.321
or you raise something
to the fourth power,

00:04:34.321 --> 00:04:37.470
you raise anything to any even
power for a very large-- as

00:04:37.470 --> 00:04:40.540
long as a is greater than 0,
for very large positive values,

00:04:40.540 --> 00:04:42.040
you're going to get
positive values.

00:04:42.040 --> 00:04:44.182
And for very large
negative values,

00:04:44.182 --> 00:04:46.140
you're going to get very
large positive values.

00:04:46.140 --> 00:04:48.665
You take a negative number,
raise it to the fourth power,

00:04:48.665 --> 00:04:52.020
or the second power, you're
going to get a positive value.

00:04:52.020 --> 00:04:54.300
Likewise, if a is
less than 0, you're

00:04:54.300 --> 00:04:57.070
going to have very similar
end behavior to this case.

00:04:57.070 --> 00:05:01.520
For a polynomial where
the highest degree

00:05:01.520 --> 00:05:06.980
term is even-- so this is
a is less than 0-- your end

00:05:06.980 --> 00:05:09.259
behavior when a is really,
really, really, really

00:05:09.259 --> 00:05:11.050
negative, this thing
is going to be really,

00:05:11.050 --> 00:05:12.150
really, really positive.

00:05:12.150 --> 00:05:13.850
We're going to be multiplying
it times a negative,

00:05:13.850 --> 00:05:14.750
so it's going to
be really, really,

00:05:14.750 --> 00:05:16.040
really, really negative.

00:05:16.040 --> 00:05:17.350
So it'll look like this.

00:05:17.350 --> 00:05:20.992
And likewise, when x is really,
really, really positive,

00:05:20.992 --> 00:05:21.950
you get the same thing.

00:05:21.950 --> 00:05:24.366
Because you're going to be
multiplying a positive times a,

00:05:24.366 --> 00:05:26.010
which is negative,
and in between it

00:05:26.010 --> 00:05:30.140
might be doing
something like that.

00:05:30.140 --> 00:05:32.090
But its end
behavior, you see, is

00:05:32.090 --> 00:05:35.530
very similar to a second
degree polynomial.

00:05:35.530 --> 00:05:40.320
So if you ignore this, its
end behavior is very similar.

00:05:40.320 --> 00:05:43.330
Now the same is true
for a fifth degree

00:05:43.330 --> 00:05:45.530
if you were to compare
it to a third degree.

00:05:45.530 --> 00:05:51.040
And the overall idea here is
what happens to this value when

00:05:51.040 --> 00:05:54.000
we get really large x's
or really small x's?

00:05:54.000 --> 00:05:55.590
Are we taking it
to an even power?

00:05:55.590 --> 00:05:58.240
In which case for either really
negative values or really

00:05:58.240 --> 00:06:00.760
positive values we're going
to get positive values.

00:06:00.760 --> 00:06:02.740
And then it depends what
our coefficient a is.

00:06:02.740 --> 00:06:07.420
Or are we taking
it to an odd power?

00:06:07.420 --> 00:06:09.380
So the general
idea-- and actually

00:06:09.380 --> 00:06:12.460
let me just do a fifth degree,
just to make the clear.

00:06:12.460 --> 00:06:14.100
So if I had something
of the form y

00:06:14.100 --> 00:06:19.209
is equal to ax to the fifth
plus bx to the fourth plus--

00:06:19.209 --> 00:06:21.750
and it just went all the way--
I won't even have to write it.

00:06:21.750 --> 00:06:27.139
This thing, if a
is greater than 0,

00:06:27.139 --> 00:06:28.430
would look something like this.

00:06:28.430 --> 00:06:34.420
Its end behavior is very similar
to a third degree polynomial

00:06:34.420 --> 00:06:35.990
where a is greater than 0.

00:06:35.990 --> 00:06:37.310
At the end it would do this.

00:06:37.310 --> 00:06:45.470
Now, it might do kind of
some craziness like this.

00:06:45.470 --> 00:06:46.810
I have to get this right.

00:06:46.810 --> 00:06:51.580
So 1, 2, 3-- it might do some
craziness like this in between.

00:06:51.580 --> 00:06:56.080
But then for really large x's
it will look the same as ax

00:06:56.080 --> 00:06:59.050
to the third when a
is greater than 0.

00:06:59.050 --> 00:07:02.830
So once again, very,
very similar end behavior

00:07:02.830 --> 00:07:05.060
when a is greater than
0, and very similar

00:07:05.060 --> 00:07:07.140
end behavior when
a is less than 0.

00:07:07.140 --> 00:07:08.140
It would look like this.

00:07:08.140 --> 00:07:10.960
At the ends at a
negative value it

00:07:10.960 --> 00:07:12.699
will be positive
because this part

00:07:12.699 --> 00:07:13.990
is going to be really negative.

00:07:13.990 --> 00:07:16.073
But then it's going to be
multiplied by a negative

00:07:16.073 --> 00:07:16.830
to get a positive.

00:07:16.830 --> 00:07:20.170
And for really positive values
of x, it will be negative.

00:07:20.170 --> 00:07:23.270
Because once again, this a
term is going to be negative.

00:07:23.270 --> 00:07:25.500
And then what it does
in between-- at least

00:07:25.500 --> 00:07:27.030
for the sake of
this video-- we're

00:07:27.030 --> 00:07:28.590
not really thinking about.

00:07:28.590 --> 00:07:30.687
So the big takeaway
here-- and this

00:07:30.687 --> 00:07:32.270
is kind of a little
bit of a drum roll

00:07:32.270 --> 00:07:34.560
here, we're talking
about end behavior--

00:07:34.560 --> 00:07:40.050
if you're looking at an
even degree polynomial

00:07:40.050 --> 00:07:43.120
it's going to have end
behavior like a second degree

00:07:43.120 --> 00:07:43.680
polynomial.

00:07:43.680 --> 00:07:45.150
If you ignore what happens
in the middle, what

00:07:45.150 --> 00:07:47.760
happens at really negative
values of x and really positive

00:07:47.760 --> 00:07:50.240
values of x is going to be
very similar to a second degree

00:07:50.240 --> 00:07:51.360
polynomial.

00:07:51.360 --> 00:07:54.460
And if your degree
is odd, you're

00:07:54.460 --> 00:07:57.950
going to have very similar
end behavior to a third degree

00:07:57.950 --> 00:07:58.737
polynomial.

00:07:58.737 --> 00:08:00.820
You might do all sorts of
craziness in the middle,

00:08:00.820 --> 00:08:03.695
but given for a given a,
whether it's greater than 0

00:08:03.695 --> 00:08:06.920
or less than 0, you will
have end behavior like this,

00:08:06.920 --> 00:08:09.800
or end behavior like that.


Using the Pythagorean trig identity

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:03.670
Let's say that we're told
that some angle theta, which

00:00:03.670 --> 00:00:05.300
is going to be
expressed in radians,

00:00:05.300 --> 00:00:09.580
is between negative 3 pi
over 2 and negative pi.

00:00:09.580 --> 00:00:11.235
It's greater than
negative 3 pi over 2

00:00:11.235 --> 00:00:13.840
and it's less than negative pi.

00:00:13.840 --> 00:00:18.220
And we're also told that sine
of theta is equal to 1/2.

00:00:18.220 --> 00:00:20.080
Just from this
information can we

00:00:20.080 --> 00:00:24.080
figure out what the tangent of
theta is going to be equal to?

00:00:24.080 --> 00:00:28.430
And I encourage you to pause the
video and try this on your own.

00:00:28.430 --> 00:00:30.590
In case you're stumped,
I will give you a hint.

00:00:30.590 --> 00:00:34.100
You should use the Pythagorean
identity, the fact that sine

00:00:34.100 --> 00:00:39.450
squared theta, plus cosine
squared theta is equal to 1.

00:00:39.450 --> 00:00:40.940
So let's do it.

00:00:40.940 --> 00:00:46.430
So we know the Pythagorean
identity, sine squared theta,

00:00:46.430 --> 00:00:52.830
plus cosine squared
theta is equal to 1.

00:00:52.830 --> 00:00:55.230
We know what sine
squared theta is.

00:00:55.230 --> 00:00:56.570
Sine theta is 1/2.

00:00:56.570 --> 00:01:00.250
So this could be
rewritten as 1/2 squared,

00:01:00.250 --> 00:01:06.860
plus cosine squared
theta, is equal to 1.

00:01:06.860 --> 00:01:13.450
Or we could write this as
1/4 plus cosine squared

00:01:13.450 --> 00:01:17.110
theta is equal to 1.

00:01:17.110 --> 00:01:19.890
Or we could subtract
1/4 from both sides,

00:01:19.890 --> 00:01:24.210
and we get cosine squared
theta is equal to-- let's see.

00:01:24.210 --> 00:01:25.990
You subtract 1/4 from
the left hand side,

00:01:25.990 --> 00:01:27.010
then this 1/4 goes away.

00:01:27.010 --> 00:01:28.051
That was the whole point.

00:01:28.051 --> 00:01:30.510
1 minus 1/4 is 3/4.

00:01:34.100 --> 00:01:36.870
So what could
cosine of theta be?

00:01:36.870 --> 00:01:40.280
Well, when I square
it, I get positive 3/4.

00:01:40.280 --> 00:01:44.680
So it could be the positive or
negative square root of 3/4.

00:01:44.680 --> 00:01:50.200
So cosine of theta
could be equal

00:01:50.200 --> 00:01:57.310
to the positive or negative
square root of 3 over 4,

00:01:57.310 --> 00:02:01.370
which is the same thing as
the positive or negative

00:02:01.370 --> 00:02:08.500
square root of 3, over the
square root of 4, which is 2.

00:02:08.500 --> 00:02:11.830
So it's the positive or negative
square root of 3 over 2.

00:02:11.830 --> 00:02:15.440
But how do we know which
one of these it actually is?

00:02:15.440 --> 00:02:18.200
Well, that's where this
information becomes useful.

00:02:22.130 --> 00:02:23.630
Let's draw our unit circle.

00:02:23.630 --> 00:02:24.580
If you're saying,
well, why am I even

00:02:24.580 --> 00:02:25.980
worried about cosine of theta?

00:02:25.980 --> 00:02:28.680
Well, if you know sine of
theta you know cosine of theta.

00:02:28.680 --> 00:02:31.750
Tangent of theta is just sine
of theta over cosine theta.

00:02:31.750 --> 00:02:33.760
So then you will know
the tangent of theta.

00:02:33.760 --> 00:02:35.760
But let's look at
the unit circle

00:02:35.760 --> 00:02:40.000
to figure out which value
of cosine we should use.

00:02:40.000 --> 00:02:43.890
So let me draw it,
the unit circle.

00:02:43.890 --> 00:02:45.930
That's my y-axis.

00:02:45.930 --> 00:02:49.040
That is my x-axis.

00:02:49.040 --> 00:02:52.280
And I will draw the
unit circle in pink.

00:02:52.280 --> 00:02:56.050
So that's my best attempt
at drawing a circle.

00:02:56.050 --> 00:03:00.440
Please forgive me for its
lack of perfect roundness.

00:03:00.440 --> 00:03:07.960
And it says theta is greater
than negative 3 pi over 2.

00:03:07.960 --> 00:03:09.869
So where is negative
3 pi over 2?

00:03:09.869 --> 00:03:10.410
So let's see.

00:03:10.410 --> 00:03:12.986
This is negative pi over 2.

00:03:12.986 --> 00:03:14.360
So this is one
side of the angle.

00:03:14.360 --> 00:03:15.690
Let me do this in a color.

00:03:15.690 --> 00:03:17.550
So this one side of
the angle is going

00:03:17.550 --> 00:03:18.960
to be along the positive x-axis.

00:03:18.960 --> 00:03:21.090
And we want to figure out
where the other side is.

00:03:21.090 --> 00:03:25.020
So this right over here
that's negative pi over 2.

00:03:25.020 --> 00:03:28.150
This is negative pi.

00:03:28.150 --> 00:03:31.260
So it's between negative pi,
which is right over here.

00:03:31.260 --> 00:03:32.740
So let me make that clear.

00:03:32.740 --> 00:03:36.320
Negative pi is right over here.

00:03:36.320 --> 00:03:40.860
It's between negative pi
and negative 3 pi over 2.

00:03:40.860 --> 00:03:46.430
Negative 3 pi over 2
is right over here.

00:03:46.430 --> 00:03:53.790
So our angle theta is going
to put us someplace over here.

00:03:57.770 --> 00:04:00.070
And the whole reason I did
this-- so this whole arc

00:04:00.070 --> 00:04:02.400
right here-- you
could think of this

00:04:02.400 --> 00:04:06.174
as the measure of angle
theta right over there.

00:04:06.174 --> 00:04:07.590
And the whole
reason I did that is

00:04:07.590 --> 00:04:09.340
to think about whether
the cosine of theta

00:04:09.340 --> 00:04:11.230
is going to be
positive or negative.

00:04:11.230 --> 00:04:14.120
We clearly see it's in
the second quadrant.

00:04:14.120 --> 00:04:16.079
The cosine of theta
is the x-coordinate

00:04:16.079 --> 00:04:20.160
of this point where our angle
intersects the unit circle.

00:04:20.160 --> 00:04:23.636
So this point right
over here-- actually

00:04:23.636 --> 00:04:25.010
let me do it in
that orange color

00:04:25.010 --> 00:04:29.290
again-- this right over here,
that is the cosine of theta.

00:04:29.290 --> 00:04:31.270
Now is that a positive
or negative value?

00:04:31.270 --> 00:04:33.580
Well it's clearly
a negative value.

00:04:33.580 --> 00:04:38.010
So for the sake of this
example, our cosine theta

00:04:38.010 --> 00:04:39.220
is not a positive 1.

00:04:39.220 --> 00:04:40.720
It is a negative 1
So we could write

00:04:40.720 --> 00:04:45.240
the cosine theta is equal to
the negative square root of 3

00:04:45.240 --> 00:04:46.530
over 2.

00:04:46.530 --> 00:04:48.410
So we figured out cosine
theta, but we still

00:04:48.410 --> 00:04:49.910
have to figure out
tangent of theta.

00:04:49.910 --> 00:04:51.440
And we just have
to remind ourselves

00:04:51.440 --> 00:04:53.050
that the tangent
of theta is going

00:04:53.050 --> 00:05:00.745
to be equal to the sine of
theta over the cosine of theta.

00:05:03.360 --> 00:05:05.560
Well they told us the
sine of theta is 1/2.

00:05:05.560 --> 00:05:08.520
So it's going to be 1/2
over cosine of theta, which

00:05:08.520 --> 00:05:10.260
is negative square
root of 3 over 2.

00:05:13.050 --> 00:05:14.250
And what does that equal?

00:05:14.250 --> 00:05:18.920
Well that's the same thing
as 1/2 times the reciprocal

00:05:18.920 --> 00:05:19.510
of this.

00:05:19.510 --> 00:05:27.810
So times negative 2 over
the square root of 3.

00:05:27.810 --> 00:05:31.960
These twos will cancel out and
we are left with negative 1

00:05:31.960 --> 00:05:34.160
over the square root of 3.

00:05:34.160 --> 00:05:36.850
Now some people don't like a
radical in the denominator,

00:05:36.850 --> 00:05:37.350
like this.

00:05:37.350 --> 00:05:40.120
They don't like an
irrational denominator.

00:05:40.120 --> 00:05:42.060
So we could rationalize
the denominator here

00:05:42.060 --> 00:05:44.500
by multiplying by square root
of 3 over square root of 3.

00:05:48.230 --> 00:05:52.980
And so this will be equal
to negative square root of 3

00:05:52.980 --> 00:06:04.747
over 3 is the tangent of
this angle right over here.

00:06:04.747 --> 00:06:06.830
And that actually makes
sense, because the tangent

00:06:06.830 --> 00:06:09.490
of the angle is the
slope of this line.

00:06:09.490 --> 00:06:14.199
And we see that it is
indeed a negative slope.


Pythagorean trig identity from unit circle

WEBVTT
Kind: captions
Language: en

00:00:00.532 --> 00:00:02.875
Voiceover: Let's review the unit circle definition

00:00:02.875 --> 00:00:04.535
of trig functions a little bit.

00:00:04.815 --> 00:00:07.516
Right over here I've draw a unit circle, and when we say

00:00:07.516 --> 00:00:11.367
a unit circle we're talking about a circle with radius one.

00:00:11.367 --> 00:00:13.286
For example this point right over

00:00:13.286 --> 00:00:16.359
here is the point one comma zero.

00:00:16.359 --> 00:00:17.970
X is equal to one, Y is zero.

00:00:17.970 --> 00:00:20.690
This point is the point zero comma one.

00:00:20.690 --> 00:00:24.412
This is the point negative one comma zero and

00:00:24.412 --> 00:00:27.783
this is the point zero comma negative one.

00:00:28.223 --> 00:00:30.313
The radius over here, the distance from the center

00:00:30.313 --> 00:00:33.014
of the circle which is at the origin to any

00:00:33.014 --> 00:00:35.564
point in the circle or any point on the circle,

00:00:35.564 --> 00:00:39.625
I should say, this radius is equal to one.

00:00:40.375 --> 00:00:43.156
The unit circle definition of trig functions

00:00:43.156 --> 00:00:45.867
leverages this unit circle, that's why

00:00:45.867 --> 00:00:47.677
it's called the unit circle definition,

00:00:47.677 --> 00:00:50.937
and we saw that if we define an angle as the

00:00:50.937 --> 00:00:53.258
bottom side of the angle being along the positive

00:00:53.258 --> 00:00:56.035
x-axis and then the other side of that angle,

00:00:56.945 --> 00:00:59.767
thinking about where it intersects the unit circle.

00:00:59.987 --> 00:01:02.271
Let's say this is the angle theta.

00:01:02.541 --> 00:01:07.302
We define sine of theta and cosine of theta,

00:01:07.302 --> 00:01:11.133
or cosine theta and sine of theta as the x and y coordinates

00:01:11.133 --> 00:01:14.274
of this point at which this side of the angle,

00:01:14.274 --> 00:01:17.144
the side that is not the positive x-axis,

00:01:17.144 --> 00:01:19.835
where that intersects the unit circle.

00:01:20.035 --> 00:01:24.096
For example this point right over here. We would call this,

00:01:24.096 --> 00:01:25.676
the x coordinate of this point,

00:01:25.676 --> 00:01:27.736
so this value right over here,

00:01:28.206 --> 00:01:30.486
we would call that cosine of theta.

00:01:33.435 --> 00:01:36.137
The y coordinate of that point, which is this point

00:01:36.137 --> 00:01:41.588
right over here, we would call that sine of theta.

00:01:41.588 --> 00:01:43.118
In the previous videos on the unit circle we

00:01:43.118 --> 00:01:44.968
talked about why this is really just a natural

00:01:44.968 --> 00:01:48.119
extension of the so cah toa definition.

00:01:48.119 --> 00:01:50.309
What's useful is it starts to work for negative angles,

00:01:50.309 --> 00:01:53.510
it even works for 90 degree angles, it works for angles more

00:01:53.510 --> 00:01:55.961
than 90 degrees, it works for angles less than 90 degrees,

00:01:55.961 --> 00:01:57.671
so it's really, really, really useful.

00:01:57.901 --> 00:02:00.841
What I want to do is leverage what we already

00:02:00.841 --> 00:02:02.882
know about the unit circle definition of trig

00:02:02.882 --> 00:02:06.713
functions to help prove the Pythagorean identity.

00:02:06.713 --> 00:02:09.654
It almost falls out of the fact that this point right

00:02:09.654 --> 00:02:14.124
over here is on a circle, a circle of radius one.

00:02:15.105 --> 00:02:16.965
What is the equation of a circle with

00:02:16.965 --> 00:02:19.656
radius one centered at the origin?

00:02:20.476 --> 00:02:25.697
Well the equation of that is x squared, x squared ...

00:02:27.008 --> 00:02:28.858
We have other videos where we really prove this

00:02:28.858 --> 00:02:30.569
using the distance formula, which is really

00:02:30.569 --> 00:02:32.789
just an application of the Pythagorean theorem.

00:02:33.089 --> 00:02:36.930
The equation of a unit circle centered at the origin

00:02:36.930 --> 00:02:45.662
is x squared plus y squared is equal to one,

00:02:45.662 --> 00:02:48.303
is equal to the radius squared.

00:02:48.543 --> 00:02:51.014
This distance right over here is equal to one.

00:02:51.604 --> 00:02:54.354
We've already said that we're defining cosine of

00:02:54.354 --> 00:02:58.135
theta as the x coordinate of this point and we're

00:02:58.135 --> 00:03:00.356
defining sine of theta at the y coordinate of this

00:03:00.356 --> 00:03:02.526
point and this point is sitting on the circle.

00:03:02.526 --> 00:03:06.447
It has to satisfy this relationship right over here.

00:03:06.717 --> 00:03:10.338
That means if we're defining cosine of theta to be the x,

00:03:10.338 --> 00:03:14.058
to be this x value, sine of theta to be the y value and

00:03:14.058 --> 00:03:16.478
it has to satisfy this relationship that means that

00:03:16.781 --> 00:03:24.747
cosine of theta squared plus sine squared of theta

00:03:25.187 --> 00:03:29.389
needs to be equal to one.

00:03:29.330 --> 00:03:31.180
Or sine square theta plus cosine squared

00:03:31.180 --> 00:03:33.181
of theta needs to be equal to one.

00:03:33.100 --> 00:03:34.040
That's just from the point.

00:03:34.040 --> 00:03:36.461
This is the x, cosine theta is the x coordinate,

00:03:36.461 --> 00:03:38.212
sine theta is the y coordinate.

00:03:38.212 --> 00:03:40.523
They have to satisfy this relationship

00:03:40.523 --> 00:03:44.024
which defines a circle so cosine squared

00:03:44.024 --> 00:03:46.446
theta plus sine squared theta is one.

00:03:47.253 --> 00:03:49.803
This is called, as we've seen in other videos,

00:03:49.803 --> 00:03:53.174
this is called the Pythagorean identity.

00:03:53.174 --> 00:03:54.594
You say, "Why is that useful?"

00:03:54.594 --> 00:03:57.704
Well using this, if you know sine of theta you can figure

00:03:57.704 --> 00:04:01.415
out what cosine of theta is going to be, or vice versa.

00:04:01.415 --> 00:04:04.726
And if you know one of cosine of theta and then you could --

00:04:04.726 --> 00:04:06.696
Say you know cosine theta then you use this to figure out

00:04:06.696 --> 00:04:08.556
sine of theta, then you can figure out tangent of theta

00:04:08.556 --> 00:04:11.347
because tangent of theta is just sine over cosine.

00:04:11.787 --> 00:04:14.627
If you're a little bit confused as to why this is called

00:04:14.627 --> 00:04:18.427
the Pythagorean identity, well it really just falls

00:04:18.427 --> 00:04:21.518
out of where the equation of a circle even came from.

00:04:21.868 --> 00:04:23.688
If we look at this point right over here,

00:04:23.688 --> 00:04:26.239
we look at this point right over here,

00:04:26.239 --> 00:04:28.409
which we're saying is the x coordinate is cosine theta

00:04:28.409 --> 00:04:30.039
and the y coordinate is sine of theta,

00:04:30.039 --> 00:04:34.240
what is the distance between that point and the origin?

00:04:35.108 --> 00:04:38.079
Well to think about that we can construct a right triangle.

00:04:38.829 --> 00:04:41.390
This distance right over here.

00:04:43.520 --> 00:04:45.301
So that we could deal with any quadrant I'll

00:04:45.301 --> 00:04:49.112
make it the absolute value of the cosine theta

00:04:49.112 --> 00:04:51.032
is this distance right over here.

00:04:51.193 --> 00:04:54.083
And this distance right over here is

00:04:54.083 --> 00:04:57.206
the absolute value of the sine of theta.

00:05:02.032 --> 00:05:03.593
I obviously don't have to take the absolute value

00:05:03.593 --> 00:05:05.703
for this first quadrant here but if I went into the

00:05:05.703 --> 00:05:07.423
other quadrants and I were to setup a similar right

00:05:07.423 --> 00:05:10.144
triangle then the absolute value is at play.

00:05:10.784 --> 00:05:12.975
What do we know from the Pythagorean theorem?

00:05:12.975 --> 00:05:14.295
This is a right triangle here,

00:05:14.295 --> 00:05:16.006
the hypotenuse has length one,

00:05:16.006 --> 00:05:18.106
so we know that this expression squared,

00:05:18.106 --> 00:05:22.987
the absolute value of cosine of theta squared,

00:05:22.987 --> 00:05:26.188
plus this expression squared, which is this length,

00:05:26.188 --> 00:05:30.630
plus the absolute value of the sine of theta squared needs

00:05:30.630 --> 00:05:33.203
to be equal to the length of the hypotenuse squared, which

00:05:33.203 --> 00:05:36.174
is the same thing which is going to be equal to one squared.

00:05:36.174 --> 00:05:39.685
Or we could say, this is the same thing.

00:05:39.685 --> 00:05:41.578
If you're going to square something the sign,

00:05:41.578 --> 00:05:43.358
if negative it's going to be negative times a

00:05:43.358 --> 00:05:45.098
negative so it's just going to be positive so this

00:05:45.098 --> 00:05:47.329
is going to be the same thing as saying that the

00:05:47.329 --> 00:05:56.170
cosine squared theta plus sine squared theta is equal to one.

00:05:58.514 --> 00:06:01.617
This is why it's called the Pythagorean identity.

00:06:01.617 --> 00:06:02.837
Actually that's even where the equation of

00:06:02.837 --> 00:06:05.528
a circle comes from, it comes straight out of the

00:06:05.528 --> 00:06:10.701
Pythagorean theorem where your hypotenuse has length one.


Pythagorean trig identity from soh cah toa

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:02.750
So we've got a right
triangle drawn over here

00:00:02.750 --> 00:00:06.720
where this base's length
is a, the height here is b,

00:00:06.720 --> 00:00:09.540
and the length of
the hypotenuse is c.

00:00:09.540 --> 00:00:11.800
And we already know when
we see something like this,

00:00:11.800 --> 00:00:13.341
we know from the
Pythagorean theorem,

00:00:13.341 --> 00:00:14.990
the relationship
between a, b, and c,

00:00:14.990 --> 00:00:19.240
we know there's a
squared plus b squared

00:00:19.240 --> 00:00:22.380
is going to be equal to
the hypotenuse squared,

00:00:22.380 --> 00:00:25.040
is going to be
equal to c squared.

00:00:25.040 --> 00:00:26.490
What I want to do
in this video is

00:00:26.490 --> 00:00:31.650
explore how we can relate trig
functions to, essentially,

00:00:31.650 --> 00:00:32.830
the Pythagorean theorem.

00:00:32.830 --> 00:00:35.800
And to do that, let's pick
one of these non-right angles.

00:00:35.800 --> 00:00:38.290
So let's pick this angle
right over here as theta,

00:00:38.290 --> 00:00:40.775
and let's just think about
this what the sine of theta is

00:00:40.775 --> 00:00:42.760
and what the cosine
of theta is, and see

00:00:42.760 --> 00:00:46.140
if we can mess with them a
little bit to somehow leverage

00:00:46.140 --> 00:00:48.190
the Pythagorean theorem.

00:00:48.190 --> 00:00:51.010
So before we do that, let's
just write down sohcahtoa just

00:00:51.010 --> 00:00:53.510
so we remember the definitions
of these trig functions.

00:00:53.510 --> 00:00:56.000
So sine is opposite
over hypotenuse.

00:00:56.000 --> 00:00:58.600
Cah, cosine is adjacent
over hypotenuse.

00:00:58.600 --> 00:01:01.030
And toa, tan is
opposite over adjacent,

00:01:01.030 --> 00:01:03.920
we won't be using tan,
at least in this video.

00:01:03.920 --> 00:01:08.490
So let's think
about sine of theta.

00:01:08.490 --> 00:01:11.430
I will do it, I'll do
it in this blue color.

00:01:11.430 --> 00:01:14.240
So sine of theta is what?

00:01:14.240 --> 00:01:16.880
It is opposite over
hypotenuse, so it

00:01:16.880 --> 00:01:18.915
is equal to the
length of b or it

00:01:18.915 --> 00:01:21.870
is equal to b-- b
is the length-- b

00:01:21.870 --> 00:01:25.850
over the length of the
hypotenuse, which is c.

00:01:25.850 --> 00:01:27.210
Now what is cosine of theta?

00:01:31.150 --> 00:01:33.900
Well, the adjacent
side, the side

00:01:33.900 --> 00:01:36.965
of this angle that is not the
hypotenuse, it has length a.

00:01:36.965 --> 00:01:38.590
So it's the length
of the adjacent side

00:01:38.590 --> 00:01:42.510
over the length
of the hypotenuse.

00:01:42.510 --> 00:01:44.740
Now how could I
relate these things?

00:01:44.740 --> 00:01:48.062
Well it seems like, if
I square sine of theta,

00:01:48.062 --> 00:01:49.520
then I'm going to
have sine squared

00:01:49.520 --> 00:01:51.560
theta is equal to b
squared over c squared,

00:01:51.560 --> 00:01:53.229
and cosine squared
theta is going

00:01:53.229 --> 00:01:54.520
to be a squared over c squared.

00:01:54.520 --> 00:01:56.290
Seems like I might be able
to add them to get something

00:01:56.290 --> 00:01:58.456
that's pretty close to the
Pythagorean theorem here.

00:01:58.456 --> 00:01:59.940
So let's try that out.

00:01:59.940 --> 00:02:06.530
So sine squared theta is equal
to b squared over c squared.

00:02:06.530 --> 00:02:07.835
I just squared both sides.

00:02:11.190 --> 00:02:19.990
Cosine squared theta is equal
to a squared over c squared.

00:02:19.990 --> 00:02:20.970
So what's this sum?

00:02:20.970 --> 00:02:23.150
What's sine squared theta
plus cosine squared theta?

00:02:32.690 --> 00:02:34.260
Is going to be equal to what?

00:02:34.260 --> 00:02:40.920
Sine squared theta is b
squared over c squared,

00:02:40.920 --> 00:02:49.210
plus a squared over
c squared, which

00:02:49.210 --> 00:02:51.250
is going to be equal
to-- Well we have

00:02:51.250 --> 00:02:53.894
a common denominator
of c squared.

00:02:53.894 --> 00:02:56.060
And the numerator, we have
b squared plus a squared.

00:03:00.590 --> 00:03:02.875
Now, what is b squared
plus a squared?

00:03:02.875 --> 00:03:04.250
Well, we have it
right over here,

00:03:04.250 --> 00:03:06.750
Pythagorean theorem tells
us, b squared plus a squared

00:03:06.750 --> 00:03:10.050
or a squared plus b squared is
going to be equal to c squared.

00:03:10.050 --> 00:03:12.572
So this numerator
simplifies to c squared.

00:03:12.572 --> 00:03:14.780
And the whole expression is
c squared over c squared,

00:03:14.780 --> 00:03:17.840
which is just equal to 1.

00:03:17.840 --> 00:03:20.470
So using the sohcahtoa
definition, in a future video,

00:03:20.470 --> 00:03:22.190
we'll use the unit
circle definition.

00:03:22.190 --> 00:03:23.900
But you see just
using the units,

00:03:23.900 --> 00:03:26.970
just even using the sohcahtoa
definition of our trig

00:03:26.970 --> 00:03:29.920
functions, we see probably
the most important

00:03:29.920 --> 00:03:31.780
of all the trig identities.

00:03:31.780 --> 00:03:36.690
That the sine squared theta,
sine squared of an angle,

00:03:36.690 --> 00:03:39.705
plus the cosine squared
of that same angle--

00:03:39.705 --> 00:03:45.500
I'm introducing
orange unnecessarily--

00:03:45.500 --> 00:03:47.182
is going to be equal to 1.

00:03:47.182 --> 00:03:48.890
Now you might probably
be saying, OK Sal,

00:03:48.890 --> 00:03:51.410
that's kind of cool, but
what's the big deal about this?

00:03:51.410 --> 00:03:53.650
Why should I care about this?

00:03:53.650 --> 00:03:56.670
Well, the big deal is now you
give me the sine of an angle

00:03:56.670 --> 00:04:01.030
and I can solve this equation
for the cosine of that angle,

00:04:01.030 --> 00:04:02.800
or vice versa.

00:04:02.800 --> 00:04:07.600
So this is actually a pretty
powerful, powerful thing.

00:04:07.600 --> 00:04:10.160
And this is also part
of the motivation

00:04:10.160 --> 00:04:15.336
even for the unit circle
definition of trig functions.


Decay of cesium 137 example

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.560
Caesium-137 is a
radioactive tracer element

00:00:03.560 --> 00:00:05.830
used to study
upslope soil erosion

00:00:05.830 --> 00:00:08.090
and downstream sedimentation.

00:00:08.090 --> 00:00:11.750
It has a half-life of
approximately 30 days.

00:00:11.750 --> 00:00:13.480
So this half-life
of 30 days, this

00:00:13.480 --> 00:00:18.370
means that, if you were
to start with 2 kilograms

00:00:18.370 --> 00:00:23.690
of caesium-137, that
30 days later you're

00:00:23.690 --> 00:00:28.270
going to have 1
kilogram of caesium-137.

00:00:28.270 --> 00:00:32.360
The other kilogram has
decayed into other things.

00:00:32.360 --> 00:00:33.840
And if you waited
another 30 days,

00:00:33.840 --> 00:00:37.080
you would have half a
kilogram of caesium-137.

00:00:37.080 --> 00:00:42.490
Assume that the amount A in
becquerels of caesium-137

00:00:42.490 --> 00:00:47.150
in a soil sample is given by
the exponential function A

00:00:47.150 --> 00:00:51.530
is equal to c times r to the t--
where t is the number of days

00:00:51.530 --> 00:00:55.990
since the release of caesium-137
into soil and c and r

00:00:55.990 --> 00:00:57.330
are unknown constants.

00:00:57.330 --> 00:00:58.700
So this bears some explaining.

00:00:58.700 --> 00:01:00.920
What is this becquerel business?

00:01:00.920 --> 00:01:03.330
So normally, if I were
to talk about the amount

00:01:03.330 --> 00:01:06.360
of some element, I'd probably
be thinking in terms of mass

00:01:06.360 --> 00:01:08.640
and I might talk in
terms of kilograms.

00:01:08.640 --> 00:01:10.720
But some people might
also be referring

00:01:10.720 --> 00:01:14.060
to the amount of this
radioactive substance

00:01:14.060 --> 00:01:17.190
in terms of the amount of
radioactivity it produces.

00:01:17.190 --> 00:01:20.020
And becquerels is the
international unit

00:01:20.020 --> 00:01:23.430
of radioactivity-- named
after Henri Becquerel, who

00:01:23.430 --> 00:01:28.110
codiscovered radioactivity
with Marie Curie.

00:01:28.110 --> 00:01:34.220
So you could consider this
the amount of caesium-137

00:01:34.220 --> 00:01:39.270
that causes A becquerels
of radioactivity.

00:01:39.270 --> 00:01:42.316
But either way, we can just
think of it as a quantity.

00:01:42.316 --> 00:01:43.690
But it's really
the quantity that

00:01:43.690 --> 00:01:46.680
causes A becquerels of activity.

00:01:46.680 --> 00:01:50.120
So just to be
clear, the amount is

00:01:50.120 --> 00:01:52.130
given by the
exponential function

00:01:52.130 --> 00:01:55.630
A is equal to-- let me just
rewrite it-- A is equal to c

00:01:55.630 --> 00:02:00.060
times r to the t power--
where t is the number of days

00:02:00.060 --> 00:02:02.800
since the release of the
caesium-137 in the soil and c

00:02:02.800 --> 00:02:04.400
and r are unknown constants.

00:02:04.400 --> 00:02:05.610
Fair enough?

00:02:05.610 --> 00:02:09.590
So let's just be clear,
this is days since release.

00:02:09.590 --> 00:02:12.430
In addition, assume that we
know that the initial amount

00:02:12.430 --> 00:02:17.210
of caesium-137 released in
the soil is 8 becquerels.

00:02:19.790 --> 00:02:23.766
Solve for the unknown
constants c and r.

00:02:23.766 --> 00:02:24.890
So the initial in the soil.

00:02:24.890 --> 00:02:26.140
That's when t is equal to 0.

00:02:26.140 --> 00:02:27.770
When no days have passed.

00:02:27.770 --> 00:02:32.320
So we could say that the amount
at times 0-- well, that's

00:02:32.320 --> 00:02:36.220
going to be equal to c
times r to the 0 power,

00:02:36.220 --> 00:02:38.480
which is just going to be
equal to c times 1, which

00:02:38.480 --> 00:02:39.710
is equal to c.

00:02:39.710 --> 00:02:41.860
And they tell us what A of 0 is.

00:02:41.860 --> 00:02:45.810
They say A of 0 is 8 becquerels.

00:02:45.810 --> 00:02:47.990
So this is going
to be equal to 8.

00:02:47.990 --> 00:02:51.920
So our constant here, the c is
just going to be equal to 8.

00:02:51.920 --> 00:02:53.410
What is the value
of the constant?

00:02:53.410 --> 00:02:55.650
We could just write
8 right over there.

00:02:57.930 --> 00:02:59.430
So the value of the
constant c is 8.

00:02:59.430 --> 00:03:01.560
What is the value
of the constant r?

00:03:01.560 --> 00:03:05.020
Round to the nearest thousandth.

00:03:05.020 --> 00:03:06.630
So we're starting with 8.

00:03:06.630 --> 00:03:07.780
So A of 0 is 8.

00:03:07.780 --> 00:03:10.457
How much are we going
to have after 30 days?

00:03:10.457 --> 00:03:12.040
And the reason why
I'm picking 30 days

00:03:12.040 --> 00:03:15.840
is-- that is the
half-life of caesium-137.

00:03:15.840 --> 00:03:19.760
So A of 30, remember our
t is in-- let me just

00:03:19.760 --> 00:03:21.660
switch colors just for fun.

00:03:21.660 --> 00:03:23.250
Remember, t is in days.

00:03:23.250 --> 00:03:25.290
So A of 30.

00:03:25.290 --> 00:03:28.300
So after 30 days, I
am going to-- if I

00:03:28.300 --> 00:03:32.400
want to use this
formula right over here,

00:03:32.400 --> 00:03:34.290
if I wanted to use
the description

00:03:34.290 --> 00:03:36.730
of this exponential function,
we already know that c is 8.

00:03:36.730 --> 00:03:41.490
It's going to be 8 times
r to the 30th power, which

00:03:41.490 --> 00:03:43.680
is going to be equal to what?

00:03:43.680 --> 00:03:45.805
Well, if we started
with 8, 30 days later,

00:03:45.805 --> 00:03:47.180
we're going to
have half as much.

00:03:47.180 --> 00:03:51.380
We're going to
have 4 becquerels.

00:03:51.380 --> 00:03:55.270
And now, we can use
this to solve for r.

00:03:55.270 --> 00:03:59.120
So you have 8 times r to the
30th power is equal to 4.

00:03:59.120 --> 00:04:01.000
Divide both sides by 8.

00:04:01.000 --> 00:04:04.500
You get r to the 30th power
is equal to 4 over 8--

00:04:04.500 --> 00:04:06.660
which is the same thing as 1/2.

00:04:06.660 --> 00:04:13.730
And then, we can raise both
sides to the 1/30th power.

00:04:13.730 --> 00:04:16.079
r to the 30th--
but then, you could

00:04:16.079 --> 00:04:18.016
think of the 30th root
of that or raising that

00:04:18.016 --> 00:04:19.390
to the 1/30th
power-- that's just

00:04:19.390 --> 00:04:27.060
going to give us r is equal to
one half to the 1/30th power.

00:04:27.060 --> 00:04:29.390
And that is something
that's very hard

00:04:29.390 --> 00:04:30.730
to compute in your head.

00:04:30.730 --> 00:04:32.696
So I suggest you use
a calculator for that.

00:04:32.696 --> 00:04:34.070
And they hint
because we're going

00:04:34.070 --> 00:04:37.440
to round to the
nearest thousandth.

00:04:37.440 --> 00:04:39.490
So let's get a
calculator right out.

00:04:39.490 --> 00:04:45.150
And so we're talking about
one half to the 1/30 power.

00:04:47.940 --> 00:04:54.980
So we get 0.9771599--
it keeps going.

00:04:54.980 --> 00:05:04.370
But they tell us to round to
the nearest thousandth, so 0.977

00:05:04.370 --> 00:05:07.030
rounded to the
nearest thousandth.

00:05:07.030 --> 00:05:10.710
And then, they finally say-- how
many becquerels of caesium-137

00:05:10.710 --> 00:05:17.920
remain in our sample 150 days
after its release in the soil?

00:05:17.920 --> 00:05:21.440
Use the rounded value of
r, and round this number

00:05:21.440 --> 00:05:23.620
to the nearest hundredth.

00:05:23.620 --> 00:05:27.070
So just to be clear, we
already know c and r.

00:05:27.070 --> 00:05:32.180
We know that the amount of
caesium-137 in becquerels--

00:05:32.180 --> 00:05:34.700
as a function of
time in days-- is

00:05:34.700 --> 00:05:44.460
going to be equal to 8
times 0.977 to the t power--

00:05:44.460 --> 00:05:46.419
where t is the number of
days that have passed.

00:05:46.419 --> 00:05:47.959
And they're essentially
saying, well,

00:05:47.959 --> 00:05:49.660
how much do we have
left after 150 days?

00:05:49.660 --> 00:05:53.480
So they want us to
calculate what is A of 150?

00:05:53.480 --> 00:06:04.950
Well, that's going to be 8
times 0.977 to the 150th power.

00:06:04.950 --> 00:06:07.860
And so clearly, we need
a calculator for this.

00:06:07.860 --> 00:06:09.670
So let's calculate that.

00:06:09.670 --> 00:06:11.590
So it's going to be
8 times-- and they

00:06:11.590 --> 00:06:15.710
tell us to use our rounded value
of r, not the exact value of r.

00:06:15.710 --> 00:06:26.985
So it's going to be 8 times
0.977 to the 150th power.

00:06:26.985 --> 00:06:29.110
And they want us to round
to the nearest hundredth.

00:06:29.110 --> 00:06:36.240
0.24.

00:06:36.240 --> 00:06:39.370
0.24 becquerels is kind
of the radioactivity level

00:06:39.370 --> 00:06:43.664
of the caesium-137
that we have left over.

00:06:43.664 --> 00:06:45.080
Now, one interesting
thing is they

00:06:45.080 --> 00:06:46.840
asked us to use the
rounded value of r.

00:06:46.840 --> 00:06:49.290
So we used the
rounded value of r.

00:06:49.290 --> 00:06:52.620
Because this right over
here is a multiple of 30,

00:06:52.620 --> 00:06:55.950
you could actually-- in not
too difficult of a way--

00:06:55.950 --> 00:06:57.923
find out the exact
value that's left over.

00:06:57.923 --> 00:07:00.173
And actually, you don't even
need a calculator for it.

00:07:00.173 --> 00:07:01.105
I encourage you to
pause your video

00:07:01.105 --> 00:07:02.271
and try to think about that.

00:07:02.271 --> 00:07:04.150
Find the exact value.

00:07:04.150 --> 00:07:09.810
Well, instead of writing
0.977, let's write A of t

00:07:09.810 --> 00:07:14.240
as being equal to 8 times our r.

00:07:14.240 --> 00:07:15.726
This is an approximate
value for r.

00:07:15.726 --> 00:07:17.350
If we wanted to be
a little more exact,

00:07:17.350 --> 00:07:23.070
we can say that our r is one
half to the 1/30th power.

00:07:23.070 --> 00:07:27.710
And we're going to raise
that to the t power.

00:07:27.710 --> 00:07:33.060
Or we could say A of
t is equal to 8 times

00:07:33.060 --> 00:07:36.001
one half to the t/30 power.

00:07:36.001 --> 00:07:37.500
If we raise something
to an exponent

00:07:37.500 --> 00:07:38.958
and then raise that
to an exponent,

00:07:38.958 --> 00:07:41.290
we can take the product
of those exponents.

00:07:41.290 --> 00:07:43.342
So that's one half
to the t/30 power.

00:07:43.342 --> 00:07:45.050
Let me actually do
that in another color.

00:07:45.050 --> 00:07:46.091
Let me do that in yellow.

00:07:46.091 --> 00:07:52.475
So that's 8 times one
half to the t/30 power.

00:07:55.550 --> 00:08:00.510
And actually, I don't need this
parentheses right over here.

00:08:00.510 --> 00:08:03.230
This is another way
to describe A of t.

00:08:03.230 --> 00:08:04.925
So what is A of 150?

00:08:04.925 --> 00:08:14.430
So A of 150 is going to be
equal to 8 times one half

00:08:14.430 --> 00:08:16.280
to the 150/30.

00:08:16.280 --> 00:08:17.330
Well, that's just 5.

00:08:17.330 --> 00:08:20.870
One half to the fifth power.

00:08:20.870 --> 00:08:22.880
Well, what's one half
to the fifth power?

00:08:22.880 --> 00:08:27.920
That is 1 to the fifth over
2 to the fifth or 1/32.

00:08:27.920 --> 00:08:36.799
So this right over here is 1/32,
which is equal to 8 over 32,

00:08:36.799 --> 00:08:39.630
which is equal to
1 over 4, which

00:08:39.630 --> 00:08:45.520
is equal to 1 over 4 or 0.25.

00:08:45.520 --> 00:08:48.320
So using our
approximation for r,

00:08:48.320 --> 00:08:51.400
we got 0.24 when we
rounded to the 150th power.

00:08:51.400 --> 00:08:54.150
So that's using our
approximation a lot.

00:08:54.150 --> 00:08:58.370
We're taking 150 of these and
multiplying them together,

00:08:58.370 --> 00:09:01.869
but it's not too far off
of what the real value is.

00:09:01.869 --> 00:09:03.660
And they asked us to
use the rounded value.

00:09:03.660 --> 00:09:08.780
But if we'd used the precise
value, the actual value,

00:09:08.780 --> 00:09:12.960
0.25 becquerels
would be left over.


Constructing linear and exponential functions from data

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:04.080
Consider the following table
of values for a linear function

00:00:04.080 --> 00:00:08.240
f of x is equal to mx plus b
and an exponential function g

00:00:08.240 --> 00:00:11.340
of x is equal to a
times r to the x.

00:00:11.340 --> 00:00:14.000
Write the equation
for each function.

00:00:14.000 --> 00:00:16.640
And so they give us,
for each x-value,

00:00:16.640 --> 00:00:19.657
what f of x is and
what g of x is.

00:00:19.657 --> 00:00:21.990
And we need to figure out the
equation for each function

00:00:21.990 --> 00:00:23.600
and type them in over here.

00:00:23.600 --> 00:00:25.190
So I copy and
pasted this problem

00:00:25.190 --> 00:00:26.910
on my little scratchpad.

00:00:26.910 --> 00:00:30.080
So let's first think
about the linear function.

00:00:30.080 --> 00:00:33.900
And to figure out the
equation of a line

00:00:33.900 --> 00:00:35.740
or a linear function
right over here,

00:00:35.740 --> 00:00:37.520
you really just need two points.

00:00:37.520 --> 00:00:40.760
And I always like to use the
situation when x equals 0

00:00:40.760 --> 00:00:43.840
because that makes
it very clear what

00:00:43.840 --> 00:00:47.130
the y-intercept is going to be.

00:00:47.130 --> 00:00:53.110
So, for example, we
can say that f of 0

00:00:53.110 --> 00:00:58.630
is going to be equal
to m times 0 plus b.

00:00:58.630 --> 00:01:01.330
Or this is just going
to be equal to b.

00:01:01.330 --> 00:01:04.590
And they tell us that
f of 0 is equal to 5.

00:01:04.590 --> 00:01:06.160
b is equal to 5.

00:01:06.160 --> 00:01:09.190
So we immediately know
that this b right over here

00:01:09.190 --> 00:01:10.187
is equal to 5.

00:01:10.187 --> 00:01:11.770
Now, we just have
to figure out the m.

00:01:11.770 --> 00:01:14.610
We have to figure out
the slope of this line.

00:01:14.610 --> 00:01:18.290
So just as a little bit
of a refresher on slope,

00:01:18.290 --> 00:01:21.350
the slope of this line is
going to be our change in y--

00:01:21.350 --> 00:01:23.970
or our change in our function
I guess we could say,

00:01:23.970 --> 00:01:25.560
if we say that this
y is equal to f

00:01:25.560 --> 00:01:27.835
of x-- over our change in x.

00:01:27.835 --> 00:01:29.460
And actually, let me
write it that way.

00:01:29.460 --> 00:01:31.080
We could write
this as our change

00:01:31.080 --> 00:01:34.140
in our function
over our change in x

00:01:34.140 --> 00:01:36.190
if you want to look
at it that way.

00:01:36.190 --> 00:01:38.080
So let's look at
this first change

00:01:38.080 --> 00:01:41.430
in x when x goes from 0 to 1.

00:01:41.430 --> 00:01:42.590
So we finish at 1.

00:01:42.590 --> 00:01:44.330
We started at 0.

00:01:44.330 --> 00:01:49.100
And f of x finishes
at 7 and started at 5.

00:01:49.100 --> 00:01:52.010
So when x is 1, f of x is 7.

00:01:52.010 --> 00:01:55.900
When x is 0, f of x is 5.

00:01:55.900 --> 00:02:00.260
And we get a change
in our function of 2

00:02:00.260 --> 00:02:02.550
when x changes by 1.

00:02:02.550 --> 00:02:03.952
So our m is equal to 2.

00:02:03.952 --> 00:02:04.660
And you see that.

00:02:04.660 --> 00:02:08.660
When x increases by 1, our
function increases by 2.

00:02:08.660 --> 00:02:11.770
So now we know the
equation for f of x.

00:02:11.770 --> 00:02:23.260
f of x is going to be equal
to 2 times 2x plus b, or 5.

00:02:23.260 --> 00:02:25.100
So we figured out
what f of x is.

00:02:25.100 --> 00:02:27.790
Now we need to figure
out what g of x is.

00:02:27.790 --> 00:02:29.910
So g of x is an
exponential function.

00:02:29.910 --> 00:02:32.286
And there's really two things
that we need to figure out.

00:02:32.286 --> 00:02:33.910
We need to figure
out what a is, and we

00:02:33.910 --> 00:02:35.120
need to figure out what r is.

00:02:35.120 --> 00:02:36.360
And let me just rewrite that.

00:02:36.360 --> 00:02:40.700
So we know that g of x--
maybe I'll do it down here.

00:02:40.700 --> 00:02:47.780
g of x is equal to a
times r to the x power.

00:02:47.780 --> 00:02:50.820
And if we know what g of 0 is,
that's a pretty useful thing.

00:02:50.820 --> 00:02:53.300
Because r to the 0th
power, regardless

00:02:53.300 --> 00:02:55.270
of what r is-- or
I guess we could

00:02:55.270 --> 00:02:58.080
assume that r is not equal to 0.

00:02:58.080 --> 00:03:00.560
People can debate what
0 to the 0 power is.

00:03:00.560 --> 00:03:02.590
But if r is any
non-zero number, we

00:03:02.590 --> 00:03:04.900
know that if you raise it
to the 0 power, you get 1.

00:03:04.900 --> 00:03:06.970
And so that
essentially gives us a.

00:03:06.970 --> 00:03:10.050
So let's just write that down.

00:03:10.050 --> 00:03:15.420
g of 0 is a times r to
the 0 power, which is just

00:03:15.420 --> 00:03:18.540
going to be equal
to a times 1 or a.

00:03:18.540 --> 00:03:23.060
And they tell us what g of
0 is. g of 0 is equal to 3.

00:03:23.060 --> 00:03:25.780
So we know that a is equal to 3.

00:03:25.780 --> 00:03:32.190
So so far, we know that our g
of x can be written as 3 times r

00:03:32.190 --> 00:03:34.120
to the x power.

00:03:34.120 --> 00:03:36.270
So now we can just use any
one of the other values

00:03:36.270 --> 00:03:39.005
they gave us to solve for r.

00:03:39.005 --> 00:03:45.830
For example, they tell us
that g of 1 is equal to 2.

00:03:45.830 --> 00:03:49.680
So let's write
that down. g of 1,

00:03:49.680 --> 00:03:55.756
which would be 3 times r to
the first power, or just 3--

00:03:55.756 --> 00:03:56.630
let me just write it.

00:03:56.630 --> 00:03:58.380
It could be 3 times
r to the first power,

00:03:58.380 --> 00:04:00.600
or we could just write
that as 3 times r.

00:04:00.600 --> 00:04:05.500
They tell us that g
of 1 is equal to 2.

00:04:05.500 --> 00:04:08.080
So we get 3 times
r is equal to 2.

00:04:08.080 --> 00:04:11.810
Or we get that r
is equal to 2/3.

00:04:11.810 --> 00:04:15.630
Divide both sides of
this equation by 3.

00:04:15.630 --> 00:04:16.940
So r is 2/3.

00:04:16.940 --> 00:04:18.739
And we're done.

00:04:18.739 --> 00:04:24.120
g of x is equal to 3 times 2/3.

00:04:27.414 --> 00:04:29.080
Actually, let me just
write it this way.

00:04:29.080 --> 00:04:33.275
3 times 2/3 to the x power.

00:04:36.540 --> 00:04:39.220
You could write it that way
if you want, any which way.

00:04:39.220 --> 00:04:42.500
So 3 times 2/3 to the x power,
and f of x is 2x plus 5.

00:04:42.500 --> 00:04:44.300
So let's actually
just type that in.

00:04:44.300 --> 00:04:46.625
So f of x is 2x plus 5.

00:04:52.000 --> 00:04:55.190
And we can verify that that's
the expression that we want.

00:04:55.190 --> 00:05:07.569
And g of x is 3 times 2
over 3 to the x power.

00:05:07.569 --> 00:05:09.735
And let me just verify that
that's what I did there.

00:05:09.735 --> 00:05:11.460
I have a short memory.

00:05:11.460 --> 00:05:11.960
All right.

00:05:11.960 --> 00:05:13.701
Yeah, that looks right.

00:05:13.701 --> 00:05:14.200
All right.

00:05:14.200 --> 00:05:15.158
Let's check our answer.

00:05:15.158 --> 00:05:17.410
And we got it right.


Introduction to complex numbers

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:03.210
Now that we know a little bit
about the imaginary unit i,

00:00:03.210 --> 00:00:07.091
let's see if we can simplify
more involved expressions,

00:00:07.091 --> 00:00:08.340
like this one right over here.

00:00:08.340 --> 00:00:14.920
2 plus 3i plus 7i squared plus
5i to the third power plus 9i

00:00:14.920 --> 00:00:15.787
to the fourth power.

00:00:15.787 --> 00:00:18.120
And I encourage you to pause
the video right now and try

00:00:18.120 --> 00:00:21.680
to simplify this on your own.

00:00:21.680 --> 00:00:24.685
So as you can see here, we
have various powers of i.

00:00:24.685 --> 00:00:27.310
You could view this as
i to the first power.

00:00:27.310 --> 00:00:29.090
We have i squared here.

00:00:29.090 --> 00:00:32.060
And we already
know that i squared

00:00:32.060 --> 00:00:35.330
is defined to be negative 1.

00:00:35.330 --> 00:00:37.820
Then we have i to
the third power.

00:00:37.820 --> 00:00:41.971
I to the third power would just
be i times this, or negative i.

00:00:41.971 --> 00:00:43.720
And we already reviewed
this when we first

00:00:43.720 --> 00:00:47.140
introduced the imaginary
unit, i, but I'll do it again.

00:00:47.140 --> 00:00:50.080
i to the fourth power is
just going to be i times

00:00:50.080 --> 00:00:53.170
this, which is the same
thing as negative 1 times i.

00:00:53.170 --> 00:00:56.330
That's i to the third
power times i again.

00:00:56.330 --> 00:00:58.100
i times i is negative 1.

00:00:58.100 --> 00:01:01.690
So that's negative 1 times
negative 1, which is equal to 1

00:01:01.690 --> 00:01:02.710
again.

00:01:02.710 --> 00:01:11.220
So we can rewrite this
whole thing as 2 plus 3i.

00:01:11.220 --> 00:01:14.510
7i squared is going
to be the same thing,

00:01:14.510 --> 00:01:17.950
so i squared is negative 1.

00:01:17.950 --> 00:01:21.280
So this is the same thing
as 7 times negative 1.

00:01:21.280 --> 00:01:24.490
So that's just
going to be minus 7.

00:01:24.490 --> 00:01:26.660
And then we have 5i
to the third power.

00:01:26.660 --> 00:01:28.515
i to the third
power is negative i.

00:01:28.515 --> 00:01:31.475
So this could be
rewritten as negative i.

00:01:31.475 --> 00:01:37.320
So this term right over here
we could write as minus 5i,

00:01:37.320 --> 00:01:40.640
or negative 5i, depending on
how you want to think about it.

00:01:40.640 --> 00:01:45.080
And then finally, i to the
fourth power is just 1.

00:01:45.080 --> 00:01:46.660
So this is just equal to 1.

00:01:46.660 --> 00:01:50.960
So this whole term
just simplifies to 9.

00:01:50.960 --> 00:01:53.410
So how could we
simplify this more?

00:01:53.410 --> 00:01:57.580
Well we have several terms
that are not imaginary,

00:01:57.580 --> 00:01:59.670
that they are real numbers.

00:01:59.670 --> 00:02:04.620
For example, we have
this 2 is a real number.

00:02:04.620 --> 00:02:06.590
Negative 7 is a real number.

00:02:06.590 --> 00:02:08.699
And 9 is a real number.

00:02:08.699 --> 00:02:10.419
So we could just add those up.

00:02:10.419 --> 00:02:15.750
So 2 plus negative 7
would be negative 5.

00:02:15.750 --> 00:02:18.930
Negative 5 plus 9 would be 4.

00:02:18.930 --> 00:02:22.030
So the real numbers add up to 4.

00:02:22.030 --> 00:02:24.570
And now we have these
imaginary numbers.

00:02:24.570 --> 00:02:29.400
So 3 times i minus 5 times i.

00:02:29.400 --> 00:02:32.230
So if you have 3 of
something and then I

00:02:32.230 --> 00:02:35.960
were to subtract 5 of that
same something from it,

00:02:35.960 --> 00:02:38.587
now you're going to have
negative 2 of that something.

00:02:38.587 --> 00:02:40.920
Or another way of thinking
about it is the coefficients.

00:02:40.920 --> 00:02:44.180
3 minus 5 is negative 2.

00:02:44.180 --> 00:02:46.280
So three i's minus
five i's, that's

00:02:46.280 --> 00:02:50.000
going to give you negative 2i.

00:02:50.000 --> 00:02:52.830
Now you might say, well, can
we simplify this any further?

00:02:52.830 --> 00:02:54.290
Well no, you really can't.

00:02:54.290 --> 00:02:59.319
This right over here
is a real number.

00:02:59.319 --> 00:03:00.860
4 is a number that
we've been dealing

00:03:00.860 --> 00:03:03.540
with throughout our
mathematical careers.

00:03:03.540 --> 00:03:08.650
And negative 2i, that's
an imaginary number.

00:03:11.350 --> 00:03:16.170
And so what we really consider
this is this 4 minus 2i,

00:03:16.170 --> 00:03:18.150
we can now consider
this entire expression

00:03:18.150 --> 00:03:20.320
to really be a number.

00:03:20.320 --> 00:03:22.690
So this is a number
that has a real part

00:03:22.690 --> 00:03:24.850
and an imaginary part.

00:03:24.850 --> 00:03:30.920
And numbers like this
we call complex numbers.

00:03:30.920 --> 00:03:35.090
It is a complex number.

00:03:35.090 --> 00:03:36.210
Why is it complex?

00:03:36.210 --> 00:03:39.920
Well, it has a real part
and an imaginary part.

00:03:39.920 --> 00:03:43.610
And you might say, well,
gee, can't any real number

00:03:43.610 --> 00:03:45.200
be considered a complex number?

00:03:45.200 --> 00:03:49.490
For example, if I have
the real number 3,

00:03:49.490 --> 00:03:55.650
can't I just write the
real number 3 as 3 plus 0i?

00:03:55.650 --> 00:03:57.770
And you would be correct.

00:03:57.770 --> 00:03:59.920
Any real number is
a complex number.

00:03:59.920 --> 00:04:04.280
You could view this right
over here as a complex number.

00:04:04.280 --> 00:04:05.980
And actually, the
real numbers are

00:04:05.980 --> 00:04:09.140
a subset of the complex numbers.

00:04:09.140 --> 00:04:12.260
Likewise, imaginary
numbers are a subset

00:04:12.260 --> 00:04:14.040
of the complex numbers.

00:04:14.040 --> 00:04:18.195
For example, you could
rewrite i as a real part-- 0

00:04:18.195 --> 00:04:23.480
is a real number-- 0 plus i.

00:04:23.480 --> 00:04:26.430
So the imaginaries are a
subset of complex numbers.

00:04:26.430 --> 00:04:28.660
Real numbers are a subset
of complex numbers.

00:04:28.660 --> 00:04:30.700
And then complex
numbers also have

00:04:30.700 --> 00:04:34.140
all of the sums and differences,
or all of the numbers

00:04:34.140 --> 00:04:38.900
that have both real
and imaginary parts.