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PROFESSOR: Hi.
00:00:33.500 --> 00:00:35.960
Today we're going to study
something called the
00:00:35.960 --> 00:00:37.760
hyperbolic functions.
00:00:37.760 --> 00:00:43.220
And in a sense, we're going to
show that in a very major way,
00:00:43.220 --> 00:00:47.170
the study of the hyperbolic
functions mimics the study of
00:00:47.170 --> 00:00:49.350
the circular functions.
00:00:49.350 --> 00:00:53.160
And so in a sense, what we will
do is parrot much of what
00:00:53.160 --> 00:00:55.640
we did for circular functions
to wind up with
00:00:55.640 --> 00:00:57.650
our hyperbolic functions.
00:00:57.650 --> 00:01:00.800
Since the basic difference
between a hyperbola and a
00:01:00.800 --> 00:01:05.010
circle is a sign, namely 'x
squared' plus 'y squared'
00:01:05.010 --> 00:01:08.420
equals 1 versus, say, 'x
squared' minus 'y squared'
00:01:08.420 --> 00:01:11.140
equals 1, I've entitled today's
lesson, 'What a
00:01:11.140 --> 00:01:13.130
difference a sign makes'.
00:01:13.130 --> 00:01:16.980
Now, by way of brief review,
recall that the circular
00:01:16.980 --> 00:01:20.470
functions originated as follows:
we said consider the
00:01:20.470 --> 00:01:23.050
circle whose equation is
'x squared' plus 'y
00:01:23.050 --> 00:01:24.510
squared' equals 1.
00:01:24.510 --> 00:01:27.690
That's a circle over here.
00:01:27.690 --> 00:01:30.940
Then what we did was picking
this as a starting point, and
00:01:30.940 --> 00:01:36.390
taking any length, 't', we wrap
't' around the circle.
00:01:36.390 --> 00:01:40.890
When the length, 't', terminated
at point, 'p', we
00:01:40.890 --> 00:01:42.340
defined what?
00:01:42.340 --> 00:01:48.420
Cosine 't' to equal 'x' and
sine 't' to equal 'y'.
00:01:48.420 --> 00:01:51.170
That was basically the geometric
interpretation of
00:01:51.170 --> 00:01:55.010
the circular functions, after
which we showed that there was
00:01:55.010 --> 00:01:57.480
a physical meaning
to this as well.
00:01:57.480 --> 00:02:00.460
Now you see what we'd like to
do is the same thing only
00:02:00.460 --> 00:02:02.430
which regard now to
what I would call
00:02:02.430 --> 00:02:03.890
the hyperbolic functions.
00:02:03.890 --> 00:02:07.140
And you may recall that in our
lecture on circular functions,
00:02:07.140 --> 00:02:09.630
I mentioned the hyperbola
very briefly.
00:02:09.630 --> 00:02:12.870
Now I'd like to go back to the
hyperbola in somewhat more
00:02:12.870 --> 00:02:18.020
detail, namely we now look at
the equation 'x squared' minus
00:02:18.020 --> 00:02:19.110
'y squared' equals 1.
00:02:19.110 --> 00:02:21.070
See just to change of
sign over here.
00:02:21.070 --> 00:02:23.180
The graph of 'x squared'
minus 'y squared'
00:02:23.180 --> 00:02:26.130
equals 1 is the hyperbola.
00:02:26.130 --> 00:02:31.920
And by the way, notice already
one significant change when we
00:02:31.920 --> 00:02:33.360
change the sign here.
00:02:33.360 --> 00:02:35.790
Notice that in the case of 'x
squared' plus 'y squared'
00:02:35.790 --> 00:02:40.560
equals 1, our curve essentially
came in one piece.
00:02:40.560 --> 00:02:44.910
Now you see our curve has
two different branches.
00:02:44.910 --> 00:02:48.430
And for the sake of uniformity,
for the sake of
00:02:48.430 --> 00:02:51.390
well-definedness, let's pick the
branch that we will deal
00:02:51.390 --> 00:02:54.650
with to be the positive branch,
meaning the branch
00:02:54.650 --> 00:02:57.010
that's to the right
of the y-axis.
00:02:57.010 --> 00:03:00.150
To mimic precisely what we did
in the case of the circular
00:03:00.150 --> 00:03:02.530
functions, what we
do now is what?
00:03:02.530 --> 00:03:08.100
Given any length, 't', we start
at 's', mark 't' off
00:03:08.100 --> 00:03:12.310
along the upper branch
of this double-valued
00:03:12.310 --> 00:03:14.170
curve if 't' is positive.
00:03:14.170 --> 00:03:17.240
Mark it off along the lower
branch if 't' is negative.
00:03:17.240 --> 00:03:20.730
Wherever 't' terminates,
we call that point 'p'.
00:03:20.730 --> 00:03:23.710
And then to complete our analogy
with the circular
00:03:23.710 --> 00:03:27.900
functions, we define the
x-coordinate of 'p' to be the
00:03:27.900 --> 00:03:33.260
hyperbolic cosine written
C-O-S-H and pronounced cosh.
00:03:33.260 --> 00:03:35.220
'Cosh t' is 'x'.
00:03:35.220 --> 00:03:39.120
And the hyperbolic sine is
the y-coordinate of 'p'.
00:03:39.120 --> 00:03:42.440
And that's written S-I-N with
an h, hyperbolic sine
00:03:42.440 --> 00:03:44.030
pronounced cinch.
00:03:44.030 --> 00:03:47.030
In other words, 'cosh t' is
the x-coordinate of 'p'.
00:03:47.030 --> 00:03:50.380
'Sinh t' is the y-coordinate
of 'p'.
00:03:50.380 --> 00:03:53.250
And the same way that 'x
squared' plus 'y squared'
00:03:53.250 --> 00:03:57.350
equals 1 gave us the circular
identity that 'sine squared t'
00:03:57.350 --> 00:04:02.480
plus 'cosine squared t' is 1,
the fact that 'x squared'
00:04:02.480 --> 00:04:06.560
minus 'y squared' is 1 gives
us the hyperbolic identity
00:04:06.560 --> 00:04:11.410
that 'cosh squared t' minus
'sinh squared t' is 1.
00:04:11.410 --> 00:04:14.130
So again, notice great
similarities, great
00:04:14.130 --> 00:04:15.460
differences.
00:04:15.460 --> 00:04:19.519
I'd like to make just a brief
aside to point out how in one
00:04:19.519 --> 00:04:23.980
sense this difference isn't
too significant.
00:04:23.980 --> 00:04:27.710
Yet in another sense, it's
very significant.
00:04:27.710 --> 00:04:30.370
Notice the following: if you
look at the equation, 'x
00:04:30.370 --> 00:04:34.830
squared' minus 'y squared'
equals 1, and if I can draw on
00:04:34.830 --> 00:04:38.420
your previous knowledge of the
square root of minus 1 just
00:04:38.420 --> 00:04:40.530
for a moment for sake
of illustration.
00:04:40.530 --> 00:04:43.070
If we think of 'i' as being a
number whose square is minus
00:04:43.070 --> 00:04:46.210
1, noticed another way of
writing 'x squared' minus 'y
00:04:46.210 --> 00:04:49.430
squared' is to write
'x squared' plus
00:04:49.430 --> 00:04:51.120
the number 'iy squared'.
00:04:51.120 --> 00:04:54.290
You see 'i squared' is minus 1.
'y squared' is 'y squared'.
00:04:54.290 --> 00:04:56.600
This is just another way of
saying 'x squared' minus 'y
00:04:56.600 --> 00:04:57.980
squared' is 1.
00:04:57.980 --> 00:04:59.760
Again, it's not that crucial.
00:04:59.760 --> 00:05:03.710
But because much later in the
course when we study complex
00:05:03.710 --> 00:05:07.520
numbers, there is going to be
a rather nice identification
00:05:07.520 --> 00:05:11.580
between hyperbolic functions
and circular functions as a
00:05:11.580 --> 00:05:13.680
prelude of things to come.
00:05:13.680 --> 00:05:16.740
Notice that what we're saying
is if you plot 'x squared'
00:05:16.740 --> 00:05:21.280
minus 'y squared' equals 1, if
you plot that in the xy-
00:05:21.280 --> 00:05:26.010
plane, you get the hyperbola
that we talked about.
00:05:26.010 --> 00:05:30.280
On the other hand, if you change
the name of the y-axis
00:05:30.280 --> 00:05:34.300
to the iy-axis, notice that with
respect to this type of
00:05:34.300 --> 00:05:38.520
coordinate system, the graph
would be a circle.
00:05:38.520 --> 00:05:42.120
Again, just an aside to show you
structurally a connection
00:05:42.120 --> 00:05:46.510
between circles and hyperbolas
in terms of complex numbers.
00:05:46.510 --> 00:05:49.660
In that way, these things
out fairly similar.
00:05:49.660 --> 00:05:52.600
Now to show you how different
a sign change is, I thought
00:05:52.600 --> 00:05:55.660
you might be interested in the
following little device.
00:05:55.660 --> 00:05:59.580
Take any two numbers, 'a' and
'b', no matter how different,
00:05:59.580 --> 00:06:02.520
and form two numbers, one
of which is half the
00:06:02.520 --> 00:06:04.110
sum of 'a' and 'b'.
00:06:04.110 --> 00:06:07.750
And the other is half the
difference of 'a' and 'b'.
00:06:07.750 --> 00:06:11.790
It's a trivial verification to
show that the sum of 'x' and
00:06:11.790 --> 00:06:13.340
'y' is 'a'.
00:06:13.340 --> 00:06:16.210
And the difference
'x - y' is 'b'.
00:06:16.210 --> 00:06:19.200
In other words, no matter how
different 'a' and 'b' are, we
00:06:19.200 --> 00:06:23.330
can always find two numbers, 'x'
and 'y', such that just by
00:06:23.330 --> 00:06:26.910
changing the sign, namely if we
add the pair of numbers, we
00:06:26.910 --> 00:06:28.040
get one of the numbers.
00:06:28.040 --> 00:06:30.070
If we subtract them,
we get the other.
00:06:30.070 --> 00:06:32.430
Just to make this a little bit
more vivid, suppose, for
00:06:32.430 --> 00:06:35.320
example, 'a' happens
to be 1,000.
00:06:35.320 --> 00:06:37.750
And suppose 'b' happens
to be 4.
00:06:37.750 --> 00:06:40.080
Now there's no danger of
confusing these two numbers.
00:06:40.080 --> 00:06:41.030
They're quite different.
00:06:41.030 --> 00:06:47.860
On the other hand, notice that
half the sum is 502, and half
00:06:47.860 --> 00:06:52.160
the difference is 498.
00:06:52.160 --> 00:06:55.910
You see if I add these two
numbers I get 1,000.
00:06:55.910 --> 00:06:57.800
If I subtract them I get 4.
00:06:57.800 --> 00:07:00.570
And therefore, notice that
just by changing the sign
00:07:00.570 --> 00:07:05.960
here, I can effectively create
a change from 4 to 1,000.
00:07:05.960 --> 00:07:08.200
I just mentioned that to show
why these people who say
00:07:08.200 --> 00:07:11.690
things like, all I was, was off
by a sign, that making a
00:07:11.690 --> 00:07:15.000
sign mistake is very, very
crucial, especially if the
00:07:15.000 --> 00:07:18.550
answer later on has to be added
on to something else.
00:07:18.550 --> 00:07:21.820
But any rate, I didn't even
have to give this example.
00:07:21.820 --> 00:07:23.460
Except that I thought
it was a cute little
00:07:23.460 --> 00:07:24.800
aside for you to notice.
00:07:24.800 --> 00:07:28.030
From a more practical point of
view, if we just go back to
00:07:28.030 --> 00:07:30.660
these two graphs that we've
drawn, notice that just
00:07:30.660 --> 00:07:35.030
changing the plus sign to a
minus sign radically changed
00:07:35.030 --> 00:07:38.680
the shape of the curve that
we were talking about.
00:07:38.680 --> 00:07:40.130
So at any rate then,
this gives us
00:07:40.130 --> 00:07:41.530
the hyperbolic functions.
00:07:41.530 --> 00:07:45.410
And a very natural question to
raise at this time, is why the
00:07:45.410 --> 00:07:47.030
hyperbolic functions now?
00:07:47.030 --> 00:07:48.420
Why study them now?
00:07:48.420 --> 00:07:50.950
Why didn't we do it before?
00:07:50.950 --> 00:07:52.850
Why did we wait until
we were dealing
00:07:52.850 --> 00:07:55.000
with exponential functions?
00:07:55.000 --> 00:07:57.800
The answer is that the
exponential functions give us
00:07:57.800 --> 00:08:01.310
a very vivid way of
reconstructing
00:08:01.310 --> 00:08:02.820
the hyperbolic functions.
00:08:02.820 --> 00:08:05.620
And the key lies in the
following fact.
00:08:05.620 --> 00:08:10.290
It lies in the fact that the
derivative of 'e' to the 'u'
00:08:10.290 --> 00:08:14.360
with respect to 'x' is 'e'
to the 'u' times 'du/dx'.
00:08:14.360 --> 00:08:19.790
In particular, if I take the
derivative of 'e' to the 't'
00:08:19.790 --> 00:08:22.550
with respect to 't', I
get 'e' to the 't'.
00:08:22.550 --> 00:08:25.520
If I take the derivative of
'e' to the minus 't' with
00:08:25.520 --> 00:08:27.460
respect to 't', I get what?
00:08:27.460 --> 00:08:30.710
'e' to the minus 't' times the
derivative of minus 't' with
00:08:30.710 --> 00:08:31.780
respect to 't'.
00:08:31.780 --> 00:08:33.159
That's minus 1.
00:08:33.159 --> 00:08:36.080
So the derivative of 'e' to the
minus 't' is minus 'e' to
00:08:36.080 --> 00:08:37.340
the minus 't'.
00:08:37.340 --> 00:08:39.990
Well what does this have
to do with anything?
00:08:39.990 --> 00:08:44.610
Notice that if I now construct a
new function, 'e' to the 't'
00:08:44.610 --> 00:08:48.490
plus 'e' to the minus 't', since
the derivative of a sum
00:08:48.490 --> 00:08:51.360
is the sum of derivatives,
notice that this result here
00:08:51.360 --> 00:08:54.160
tells me that the derivative of
'e' to the 't' plus 'e' to
00:08:54.160 --> 00:08:58.410
the minus 't' with respect to
't' is 'e' to the 't' minus
00:08:58.410 --> 00:09:01.260
'e' to the minus 't'.
00:09:01.260 --> 00:09:03.740
Now let me do this operation
once again.
00:09:03.740 --> 00:09:07.550
Suppose I now take 'e' to the
't' minus 'e' to the minus 't'
00:09:07.550 --> 00:09:09.890
and I differentiate that
with respect to 't'.
00:09:09.890 --> 00:09:12.400
Well, the derivative 'e'
to the 't' gives me
00:09:12.400 --> 00:09:13.760
'e' to the 't' again.
00:09:13.760 --> 00:09:16.540
The derivative of 'e' to the
minus 't' is minus 'e'
00:09:16.540 --> 00:09:17.750
to the minus 't'.
00:09:17.750 --> 00:09:19.820
Minus times minus is plus.
00:09:19.820 --> 00:09:23.040
So observe that if I
differentiate 'e' to the 't'
00:09:23.040 --> 00:09:27.700
minus 'e' to the minus 't', I
come back to 'e' to the 't'
00:09:27.700 --> 00:09:29.770
plus 'e' to the minus 't'.
00:09:29.770 --> 00:09:33.640
Notice already how this starts
to behave, at least a little
00:09:33.640 --> 00:09:37.670
bit, like our friends sine and
cosine, except again, for a
00:09:37.670 --> 00:09:42.430
little change in sign, namely if
I give this function here a
00:09:42.430 --> 00:09:43.560
special name.
00:09:43.560 --> 00:09:47.780
And to make it suggestive, I'm
going to call it 'c of t'.
00:09:47.780 --> 00:09:49.750
'c' sort of to indicate
cosine.
00:09:49.750 --> 00:09:51.090
But I don't want
to write cosine
00:09:51.090 --> 00:09:52.330
because it's not a cosine.
00:09:52.330 --> 00:09:55.390
Let me just call this function
here 'c of t' and its
00:09:55.390 --> 00:10:00.150
companion, as you may guess,
I will call 's of t'.
00:10:00.150 --> 00:10:03.720
'c of t' is 'e' to the 't' plus
'e' to the minus 't'.
00:10:03.720 --> 00:10:08.610
's of t' is 'e' to the 't' minus
'e' to the minus 't'.
00:10:08.610 --> 00:10:11.400
This pair of functions has
the property that what?
00:10:11.400 --> 00:10:14.840
The derivative of one of them is
always the other, which is
00:10:14.840 --> 00:10:17.260
not quite what sine
and cosine did.
00:10:17.260 --> 00:10:20.850
Remember if you differentiated
the sine, you got the cosine.
00:10:20.850 --> 00:10:23.490
But if you differentiated
the cosine, you
00:10:23.490 --> 00:10:24.630
didn't get the sine.
00:10:24.630 --> 00:10:26.640
You got minus the sine.
00:10:26.640 --> 00:10:29.710
So you see, again, a similarity,
but not exactly
00:10:29.710 --> 00:10:30.950
the same thing.
00:10:30.950 --> 00:10:34.820
Nonetheless we get a little
bit suspicious about this.
00:10:34.820 --> 00:10:37.170
We see that the derivative of
'c' is 's', the derivative of
00:10:37.170 --> 00:10:37.950
's' is 'c'.
00:10:37.950 --> 00:10:41.450
We remember that there was a
basic identity involving sine
00:10:41.450 --> 00:10:43.430
squared plus cosine squared.
00:10:43.430 --> 00:10:46.350
We might get the premonition
that maybe there's some
00:10:46.350 --> 00:10:50.200
relationship between 's squared'
and 'c squared'.
00:10:50.200 --> 00:10:52.820
And one of the best ways of
checking out a premonition is
00:10:52.820 --> 00:10:53.660
to just check it out.
00:10:53.660 --> 00:10:55.350
I mean let's just see
what happens.
00:10:55.350 --> 00:10:57.830
In other words, starting with
this definition of 'c'--
00:10:57.830 --> 00:10:59.080
this is a binomial--
00:10:59.080 --> 00:11:00.350
let's square it.
00:11:00.350 --> 00:11:01.690
You see if we square
it, it's what?
00:11:01.690 --> 00:11:05.120
It's 'e' to the '2t', the first
term squared, plus twice
00:11:05.120 --> 00:11:06.820
the product of these
two terms.
00:11:06.820 --> 00:11:09.710
But 'e' to the 't' times 'e' to
the minus 't' is 'e' to the
00:11:09.710 --> 00:11:15.120
0, which is 1, plus 'e' to the
minus 't squared', which is
00:11:15.120 --> 00:11:16.890
'e' to the minus '2t'.
00:11:16.890 --> 00:11:19.570
In other words, putting this
all together, the square of
00:11:19.570 --> 00:11:23.990
'c' is 'e' to the '2t' plus
twice 1, which is 2, plus 'e'
00:11:23.990 --> 00:11:26.070
to the minus '2t'.
00:11:26.070 --> 00:11:30.450
Similarly, when we square 's',
we get exactly the same thing
00:11:30.450 --> 00:11:32.930
only with a minus 2
as the middle term
00:11:32.930 --> 00:11:36.150
rather than a plus 2.
00:11:36.150 --> 00:11:38.570
Well now we look at 'c squared'
and 's squared'.
00:11:38.570 --> 00:11:41.990
And just by looking at this
expression, any hope that 'c
00:11:41.990 --> 00:11:44.220
squared' plus 's squared'
will give us a nice
00:11:44.220 --> 00:11:45.660
identity should vanish.
00:11:45.660 --> 00:11:48.780
Because you see, if you add
these, you're going to get an
00:11:48.780 --> 00:11:50.610
'e' to the '2t' term
in here, in fact,
00:11:50.610 --> 00:11:51.980
twice 'e' to the '2t'.
00:11:51.980 --> 00:11:54.450
Here you'll get twice 'e'
to the minus '2t'.
00:11:54.450 --> 00:11:57.090
And only the 2 and the minus
2 will drop out.
00:11:57.090 --> 00:12:01.440
On the other hand, should you
elect to subtract 's squared'
00:12:01.440 --> 00:12:05.100
from 'c squared', the 'e' to the
'2t' terms cancel, the 'e'
00:12:05.100 --> 00:12:07.800
to the minus '2t' term
cancels, and we
00:12:07.800 --> 00:12:09.360
wind up with what?
00:12:09.360 --> 00:12:14.020
'c squared' minus 's squared'
is identically 4.
00:12:14.020 --> 00:12:16.780
And by the way, notice
up to this point, no
00:12:16.780 --> 00:12:18.770
use of the word hyperbola.
00:12:18.770 --> 00:12:20.210
OK?
00:12:20.210 --> 00:12:23.220
However, if you think of the
so-called unit hyperbola, the
00:12:23.220 --> 00:12:26.570
one for which 'x squared' minus
'y squared' is 1, notice
00:12:26.570 --> 00:12:29.900
that 'c squared' minus
's squared' equals 4
00:12:29.900 --> 00:12:32.010
almost has that form.
00:12:32.010 --> 00:12:33.270
Well it has the form.
00:12:33.270 --> 00:12:36.590
All that's spoiling it
is the 4 on the side.
00:12:36.590 --> 00:12:38.135
Now you could pretend
the 4 was a 1.
00:12:38.135 --> 00:12:39.450
But that would be cheating.
00:12:39.450 --> 00:12:42.390
What we do instead is just
divide both sides of this
00:12:42.390 --> 00:12:48.570
equation by 4, observing that
'c squared t' over 4 is just
00:12:48.570 --> 00:12:51.660
'c of t' over 2 squared.
00:12:51.660 --> 00:12:55.710
's squared t' over 4 is just
's of t' over 2 squared.
00:12:55.710 --> 00:12:59.150
In other words, dividing through
by 4, this equation
00:12:59.150 --> 00:13:01.540
can be put into this form.
00:13:01.540 --> 00:13:06.780
And now you see we satisfy the
basic hyperbolic identity of
00:13:06.780 --> 00:13:10.190
this squared minus this
squared equals 1.
00:13:10.190 --> 00:13:13.200
By the way, since the c's and
the s's may not seem familiar
00:13:13.200 --> 00:13:15.570
to us, what we can do is what?
00:13:15.570 --> 00:13:18.180
Go back to the basic
definition.
00:13:18.180 --> 00:13:21.680
'c of t' meant 'e' to the 't'
plus 'e' to the minus 't'.
00:13:21.680 --> 00:13:25.480
Therefore, half of 'c of t' is
'e' to the 't' plus 'e' to the
00:13:25.480 --> 00:13:27.410
minus 't' over 2.
00:13:27.410 --> 00:13:31.540
's of t' was 'e' to the 't'
minus 'e' to the minus 't'.
00:13:31.540 --> 00:13:35.550
Therefore 's of t' over 2 is
'e' to the 't' minus 'e' to
00:13:35.550 --> 00:13:37.750
the minus 't' over 2.
00:13:37.750 --> 00:13:41.340
And again, the reason I bring
this out, is it in most
00:13:41.340 --> 00:13:44.170
textbooks they simply define
the hyperbolic
00:13:44.170 --> 00:13:45.560
functions this way.
00:13:45.560 --> 00:13:47.440
I find it a poor motivation.
00:13:47.440 --> 00:13:50.610
I don't find it easy to motivate
where are the clear
00:13:50.610 --> 00:13:53.630
blue sky, you bring in a
2 in the denominator.
00:13:53.630 --> 00:13:57.400
All I want you to see in terms
of motivating what's going on
00:13:57.400 --> 00:14:01.840
so far, is that if I just
defined a function to be 'e of
00:14:01.840 --> 00:14:05.200
t' plus 'e' to the minus 't' and
another one to be 'e of t'
00:14:05.200 --> 00:14:08.840
minus 'e' to the minus 't', I
could have shown that the
00:14:08.840 --> 00:14:11.920
first squared minus the
second squared was 4.
00:14:11.920 --> 00:14:14.930
And the motivation of putting
the 2 in the denominator is
00:14:14.930 --> 00:14:18.350
for no greater reason than to
help us satisfy the equation
00:14:18.350 --> 00:14:20.540
'x squared' minus 'y
squared' equals 1.
00:14:20.540 --> 00:14:23.910
In fact, this is now the
connection between what we
00:14:23.910 --> 00:14:26.910
were doing a few months ago
and what we're doing now.
00:14:26.910 --> 00:14:32.680
Namely if I now call this 'cosh
t' and I call this 'sinh
00:14:32.680 --> 00:14:36.930
t', do I get my basic
hyperbola?
00:14:36.930 --> 00:14:41.070
Namely, suppose I take a curve
whose equation is given
00:14:41.070 --> 00:14:45.575
parametrically by 'x' equal
'cosh t' and 'y' equal 'sinh
00:14:45.575 --> 00:14:52.930
t', where cosh and sinh are
now defined as over here
00:14:52.930 --> 00:14:55.970
without any reference to the
hyperbola, in the same way
00:14:55.970 --> 00:15:00.700
that we could define the
circular functions in terms of
00:15:00.700 --> 00:15:03.130
calculus without having to
appeal to the circle.
00:15:03.130 --> 00:15:06.090
What I'm saying is, suppose we
had never mentioned the word
00:15:06.090 --> 00:15:08.110
hyperbolic function up to now.
00:15:08.110 --> 00:15:11.250
We've developed a function
called 'cosh t', a function
00:15:11.250 --> 00:15:14.980
called 'sinh t' just from these
two relationships here.
00:15:14.980 --> 00:15:19.530
If I eliminate the parameter 't'
here, namely if I square
00:15:19.530 --> 00:15:24.840
both sides of each equation
and subtract, I get 'x
00:15:24.840 --> 00:15:28.520
squared' minus 'y squared' is
'cosh squared t' minus 'sinh
00:15:28.520 --> 00:15:29.530
squared t'.
00:15:29.530 --> 00:15:31.400
But that I already say is 1.
00:15:31.400 --> 00:15:35.200
In other words, this
parametrically gives us the
00:15:35.200 --> 00:15:38.790
hyperbola 'x squared' minus
'y squared' equals 1.
00:15:38.790 --> 00:15:41.520
By the way, let's make
another observation.
00:15:41.520 --> 00:15:44.380
Notice that the exponential
could never be negative.
00:15:44.380 --> 00:15:47.930
See 'e' to the minus 't' is 1
over 'e' to the 't', which is
00:15:47.930 --> 00:15:49.060
still positive.
00:15:49.060 --> 00:15:51.630
Notice that the cosh can
never be negative.
00:15:51.630 --> 00:15:53.670
See, 'e' to the 't' is
a positive number.
00:15:53.670 --> 00:15:56.390
'e' to the minus 't' is
a positive number.
00:15:56.390 --> 00:16:00.040
Therefore, notice that in this
particular relationship, not
00:16:00.040 --> 00:16:03.030
only is the curve given by 'x
squared' minus 'y squared'
00:16:03.030 --> 00:16:07.240
equals 1, but since the cosh
can never be negative, it's
00:16:07.240 --> 00:16:11.540
also characterized by
'x' is positive.
00:16:11.540 --> 00:16:14.160
And you see that
tells us what?
00:16:14.160 --> 00:16:16.630
That what we're getting this
way is not the entire
00:16:16.630 --> 00:16:20.830
hyperbola, but just the branch
for which 'x' is positive,
00:16:20.830 --> 00:16:23.130
which is exactly what we
were talking about
00:16:23.130 --> 00:16:24.390
earlier in the lesson.
00:16:27.090 --> 00:16:30.220
Again, I leave further details
of this to the exercises.
00:16:30.220 --> 00:16:33.230
All I want to do in today's
lesson is to get the flavor of
00:16:33.230 --> 00:16:35.640
what's going on, how we can
invent the hyperbolic
00:16:35.640 --> 00:16:38.390
functions without reference
to a hyperbola
00:16:38.390 --> 00:16:39.990
until we're all done.
00:16:39.990 --> 00:16:41.830
See, I hope you see this
juxtaposition.
00:16:41.830 --> 00:16:42.930
We've done two things here.
00:16:42.930 --> 00:16:45.910
One is we've started with the
hyperbola and showed how we
00:16:45.910 --> 00:16:47.530
could invent this
construction.
00:16:47.530 --> 00:16:50.630
The other is we started with the
exponential and showed how
00:16:50.630 --> 00:16:53.970
we could construct two functions
of the exponential
00:16:53.970 --> 00:16:57.270
which had the property that they
could be identified with
00:16:57.270 --> 00:16:59.750
the thing that was called the
hyperbolic cosine and
00:16:59.750 --> 00:17:00.680
hyperbolic sine.
00:17:00.680 --> 00:17:03.260
In other words, we could take
either of the two approaches
00:17:03.260 --> 00:17:04.450
and derive the other.
00:17:04.450 --> 00:17:07.670
But I don't want to put in all
of those details here.
00:17:07.670 --> 00:17:08.510
OK?
00:17:08.510 --> 00:17:11.700
At any rate, to show you where
calculus comes in, in terms of
00:17:11.700 --> 00:17:15.040
what I meant by saying that
as far as this course is
00:17:15.040 --> 00:17:17.800
concerned, we have now finished
the groundwork both
00:17:17.800 --> 00:17:20.339
with respect to differential
and integral calculus.
00:17:20.339 --> 00:17:23.140
Suppose, for example,
I now want to define
00:17:23.140 --> 00:17:24.810
the hyperbolic tangent.
00:17:24.810 --> 00:17:27.119
See y equals hyperbolic
tangent 'x'.
00:17:27.119 --> 00:17:28.190
How could I do that?
00:17:28.190 --> 00:17:32.250
Well, I mimic the circular
function definition.
00:17:32.250 --> 00:17:35.970
I say, OK, I will define the
hyperbolic tangent to be the
00:17:35.970 --> 00:17:38.740
hyperbolic sine over the
hyperbolic cosine.
00:17:38.740 --> 00:17:43.450
By the way, in the same way that
one can show in terms of
00:17:43.450 --> 00:17:47.440
geometric constructions on the
unit circle how one constructs
00:17:47.440 --> 00:17:51.080
the tangent and the cotangent
and the cosecant and the
00:17:51.080 --> 00:17:54.770
secant, et cetera, once sine
and cosine and given.
00:17:54.770 --> 00:17:58.560
It can also be shown that we can
construct these things in
00:17:58.560 --> 00:18:01.070
terms of geometry from
the hyperbola.
00:18:01.070 --> 00:18:03.680
But because that gets
computationally involved, I've
00:18:03.680 --> 00:18:07.320
also saved those aspects
for the exercises.
00:18:07.320 --> 00:18:10.440
I'm not going to worry about
how you construct the
00:18:10.440 --> 00:18:13.020
hyperbolic tangent given the
hyperbolic sine and the
00:18:13.020 --> 00:18:14.650
hyperbolic cosine right now.
00:18:14.650 --> 00:18:16.560
All I'm saying is suppose
you've made up this
00:18:16.560 --> 00:18:17.460
definition.
00:18:17.460 --> 00:18:19.960
And now if somebody says gee,
I would like to find the
00:18:19.960 --> 00:18:22.950
derivative of the hyperbolic
tangent of 'x'.
00:18:22.950 --> 00:18:26.120
What I want to point out is that
the recipes that we need
00:18:26.120 --> 00:18:28.540
are no different from anything
we've had before.
00:18:28.540 --> 00:18:31.980
Namely, since the hyperbolic
tangent of 'x' is 'sinh x'
00:18:31.980 --> 00:18:34.480
over 'cosh x', I use
the same quotient
00:18:34.480 --> 00:18:36.380
rule as I used before.
00:18:36.380 --> 00:18:39.140
Namely, to differentiate the
hyperbolic tangent, I
00:18:39.140 --> 00:18:40.650
differentiate this quotient.
00:18:40.650 --> 00:18:42.520
How do we differentiate
a quotient?
00:18:42.520 --> 00:18:47.480
It's the denominator times the
derivative of the numerator.
00:18:47.480 --> 00:18:49.430
Well the numerator
is 'sinh x'.
00:18:49.430 --> 00:18:52.010
The derivative 'sinh
x' is 'cosh x'.
00:18:52.010 --> 00:18:56.090
That gives me a 'cosh squared
x' over here minus the
00:18:56.090 --> 00:19:00.990
numerator, which is 'sinh x'
times the derivative of the
00:19:00.990 --> 00:19:01.830
denominator.
00:19:01.830 --> 00:19:05.580
Well the derivative of 'cosh
x' is 'sinh x'--
00:19:05.580 --> 00:19:07.280
that give me another factor--
00:19:07.280 --> 00:19:08.725
over the square of
the denominator.
00:19:12.900 --> 00:19:16.000
Now my basic identity is that
cosh squared minus sinh
00:19:16.000 --> 00:19:17.200
squared is 1.
00:19:17.200 --> 00:19:22.430
So this is just 1 over
'cosh squared x'.
00:19:22.430 --> 00:19:25.060
And if I want to write this
all on one line, I could
00:19:25.060 --> 00:19:31.690
invent the notation that the
hyperbolic secant is 1 over
00:19:31.690 --> 00:19:33.890
the hyperbolic cosine.
00:19:33.890 --> 00:19:37.680
In other words, I could invent
a notation like this.
00:19:37.680 --> 00:19:40.110
But again, the important point
is not so much what this
00:19:40.110 --> 00:19:43.300
answer is, but how I can
derive it from first
00:19:43.300 --> 00:19:43.960
principles.
00:19:43.960 --> 00:19:47.800
And by the way, in terms of the
inverse derivative, once I
00:19:47.800 --> 00:19:51.380
know that the derivative of
hyperbolic tangent is a square
00:19:51.380 --> 00:19:55.140
of the hyperbolic secant, I can
immediately add to my list
00:19:55.140 --> 00:19:57.130
of inverse derivatives.
00:19:57.130 --> 00:20:00.280
Namely I just have
to write it down.
00:20:04.780 --> 00:20:07.640
See there's an integral formula
that I now have.
00:20:07.640 --> 00:20:11.480
But again, all I want to
emphasize here is how the
00:20:11.480 --> 00:20:14.820
details take care of themselves
in the same way as
00:20:14.820 --> 00:20:17.990
always once we have our
basic definitions.
00:20:17.990 --> 00:20:21.370
It might be interesting just
in terms of cementing down
00:20:21.370 --> 00:20:24.740
what sinh and cosh look like,
in the same way that we have
00:20:24.740 --> 00:20:27.470
talked about sine and cosine,
it might be nice to
00:20:27.470 --> 00:20:29.600
graph sinh and cosh.
00:20:29.600 --> 00:20:32.900
And again notice, in terms of
curve plotting, how do we plot
00:20:32.900 --> 00:20:34.280
'y' equals 'f of x'?
00:20:34.280 --> 00:20:36.200
The general procedure
was what?
00:20:36.200 --> 00:20:38.860
Always take the first and second
derivatives so that you
00:20:38.860 --> 00:20:40.500
can see what the slope
is, what the
00:20:40.500 --> 00:20:42.630
concavity is, et cetera.
00:20:42.630 --> 00:20:44.510
The derivative of
cosh is sinh.
00:20:44.510 --> 00:20:46.440
The derivative of
sinh is cosh.
00:20:46.440 --> 00:20:50.380
By the way, again, if these
terms seem alien to you, you
00:20:50.380 --> 00:20:54.950
can always rewrite them in terms
of the basic definition,
00:20:54.950 --> 00:20:57.740
in terms of 'e' to the 'x' and
'e' to the minus 'x', and
00:20:57.740 --> 00:21:01.020
carry out the differentiation
in a straightforward way.
00:21:01.020 --> 00:21:02.410
Well here's the interesting
point.
00:21:02.410 --> 00:21:04.880
Let's plot 'y' equals
'cosh x'.
00:21:04.880 --> 00:21:08.330
Notice first of all that 'cosh
x' is an even function.
00:21:08.330 --> 00:21:12.000
If I replace 'x' by minus 'x',
all I get is what? 'e' to the
00:21:12.000 --> 00:21:15.420
minus 'x' plus 'e' to the 'x'
over 2, which is the same
00:21:15.420 --> 00:21:17.020
thing as I have over here.
00:21:17.020 --> 00:21:19.320
It's going to be a symmetric
function with a curve with
00:21:19.320 --> 00:21:20.680
respect to the y-axis.
00:21:20.680 --> 00:21:22.280
But again, what do I do?
00:21:22.280 --> 00:21:23.850
I compute the derivative.
00:21:23.850 --> 00:21:26.410
I can find out where the
derivative is 0.
00:21:26.410 --> 00:21:29.720
It's 0 when 'e' to the 'x'
equals 'e' to the minus 'x'.
00:21:29.720 --> 00:21:31.960
That happens only
when 'x' is 0.
00:21:31.960 --> 00:21:36.730
When 'x' is 0, this is 'e' to
the 0 plus 'e' to the 0.
00:21:36.730 --> 00:21:41.270
That's 1 plus 1, which is
2 over 2, which is one.
00:21:41.270 --> 00:21:45.040
So in other words, the
derivative is 0 when 'x' is 0
00:21:45.040 --> 00:21:47.010
and 'y' is 1.
00:21:47.010 --> 00:21:48.740
The second derivative is cosh.
00:21:48.740 --> 00:21:52.310
We've already seen that this
can never be negative.
00:21:52.310 --> 00:21:54.470
So the second derivative
is always positive.
00:21:54.470 --> 00:21:57.150
That means that the curve
is always holding water.
00:21:57.150 --> 00:22:00.680
Putting all this together, the
curve 'y' equals 'cosh x'
00:22:00.680 --> 00:22:03.450
looks something like this.
00:22:03.450 --> 00:22:05.100
It's a dangerous thing
to say it sort
00:22:05.100 --> 00:22:06.450
of resembles a parabola.
00:22:06.450 --> 00:22:10.500
It's nothing like a parabola
except what I mean is it has
00:22:10.500 --> 00:22:11.730
this type of shape.
00:22:11.730 --> 00:22:13.940
Notice it does not oscillate.
00:22:13.940 --> 00:22:16.680
It is not an oscillating
function like the cosine.
00:22:16.680 --> 00:22:18.380
It doesn't act periodically.
00:22:18.380 --> 00:22:21.110
This thing just keeps
going like this.
00:22:21.110 --> 00:22:21.630
All right?
00:22:21.630 --> 00:22:25.710
On the other hand, how can we
plot 'y' equals 'sinh x'?
00:22:25.710 --> 00:22:28.500
Well, I guess we come
right back to here
00:22:28.500 --> 00:22:29.620
and work from here.
00:22:29.620 --> 00:22:31.580
The derivative of
sinh is cosh.
00:22:31.580 --> 00:22:34.700
But cosh is always at least
as big as 1 we found out.
00:22:34.700 --> 00:22:37.230
Therefore, the slope of
'sinh x' is always at
00:22:37.230 --> 00:22:38.450
least as big as 1.
00:22:38.450 --> 00:22:42.550
And it's equal to
1 when 'x' is 0.
00:22:42.550 --> 00:22:44.490
Putting all these details
together,
00:22:44.490 --> 00:22:45.770
what we find is what?
00:22:45.770 --> 00:22:48.830
First of all, the sinh is an odd
function, meaning that if
00:22:48.830 --> 00:22:52.800
we replace 'x' by minus 'x',
we change the sign.
00:22:52.800 --> 00:22:56.300
So this is 'e to the minus x'
minus 'e to the x', which is a
00:22:56.300 --> 00:22:58.070
negative of what we
have over here.
00:22:58.070 --> 00:23:00.440
But those, again, are details
which are easy
00:23:00.440 --> 00:23:01.650
for you to fill in.
00:23:01.650 --> 00:23:04.940
The graph 'y' equals 'sinh x'
looks something like this.
00:23:04.940 --> 00:23:07.890
In other words, it's
a very steep curve.
00:23:07.890 --> 00:23:12.790
It's spilling water and
rising here, holding
00:23:12.790 --> 00:23:14.960
water and rising here.
00:23:14.960 --> 00:23:16.270
The curve is always rising.
00:23:16.270 --> 00:23:20.000
Again, another interesting thing
to observe here is that
00:23:20.000 --> 00:23:24.010
notice that when 'x' is very,
very large, 'e' to the minus
00:23:24.010 --> 00:23:25.870
'x' becomes negligible.
00:23:25.870 --> 00:23:29.060
And if 'e' to the minus 'x' is
negligible, notice that both
00:23:29.060 --> 00:23:32.660
sinh and cosh behave like
1/2 'e to the x'.
00:23:32.660 --> 00:23:34.970
In other words, this term
tends to drop out.
00:23:34.970 --> 00:23:37.540
And just to show you the
contrast here, I've sketched
00:23:37.540 --> 00:23:40.710
in the curve 'y' equals '1/2 'e
to the x'' to show you how
00:23:40.710 --> 00:23:43.950
it splits the difference for
large values of 'x' between
00:23:43.950 --> 00:23:45.140
these two in a way.
00:23:45.140 --> 00:23:49.870
And as 'x' gets larger, both
of these curves converge on
00:23:49.870 --> 00:23:52.100
'y' equals '1/2 'e to the x''.
00:23:52.100 --> 00:23:55.410
Again, I simply want to mention
enough here so that
00:23:55.410 --> 00:23:59.140
you get the idea of how we can
apply the same old calculus to
00:23:59.140 --> 00:24:00.860
our new function.
00:24:00.860 --> 00:24:04.900
What I'd like to do in closing
is to try again from our point
00:24:04.900 --> 00:24:08.430
of view of being engineeringly
oriented to show why the
00:24:08.430 --> 00:24:12.070
hyperbolic functions might have
occurred in nature, in
00:24:12.070 --> 00:24:14.540
the same way that we tried
to show that the circular
00:24:14.540 --> 00:24:18.370
functions had a natural
outgrowth in terms of nature.
00:24:18.370 --> 00:24:20.820
You recall that the circular
functions were motivated in
00:24:20.820 --> 00:24:22.700
terms of simple harmonic
motion.
00:24:22.700 --> 00:24:25.650
The circular functions would
have been invented when we
00:24:25.650 --> 00:24:28.940
talked about motion, in which
the acceleration was
00:24:28.940 --> 00:24:31.440
proportional to the displacement
but in the
00:24:31.440 --> 00:24:33.040
opposite direction.
00:24:33.040 --> 00:24:34.870
Let's see what happens
when we work with
00:24:34.870 --> 00:24:36.270
the hyperbolic functions.
00:24:40.060 --> 00:24:46.000
Suppose now I have a particle
moving along the x-axis.
00:24:46.000 --> 00:24:46.880
OK?
00:24:46.880 --> 00:24:51.390
And I know this position at any
time, 't', is sinh of 'kt'
00:24:51.390 --> 00:24:54.690
where 'k' happens to
be a constant.
00:24:54.690 --> 00:24:56.480
What is 'dx/dt'?
00:24:56.480 --> 00:24:59.480
The derivative of 'sinh
u' with respect to
00:24:59.480 --> 00:25:01.360
'u' is 'cosh u'.
00:25:01.360 --> 00:25:05.610
So the derivative of 'sinh kt'
with respect to 't' would be
00:25:05.610 --> 00:25:09.950
'cosh kt' times the derivative
of 'kt' with respect to 't',
00:25:09.950 --> 00:25:11.230
which is just 'k'.
00:25:11.230 --> 00:25:14.350
In other words, the chain
rule again, OK?
00:25:14.350 --> 00:25:19.860
Now knowing that 'dx/dt' is
'k cosh kt', ''d 2 x' 'dt
00:25:19.860 --> 00:25:21.220
squared'' would be what?
00:25:21.220 --> 00:25:25.930
The derivative of 'cosh kt' is
'sinh kt' times the derivative
00:25:25.930 --> 00:25:28.000
of 'kt' with respect to 't'.
00:25:28.000 --> 00:25:30.190
That gives me another
factor of 'k'.
00:25:30.190 --> 00:25:32.460
And therefore the second
derivative of 'x' with respect
00:25:32.460 --> 00:25:36.100
to 't' is ''k squared'
sinh kt'.
00:25:36.100 --> 00:25:38.210
Now remember what
'sinh kt' is.
00:25:38.210 --> 00:25:43.300
We started with the knowledge
that 'sinh kt' is 'x'.
00:25:43.300 --> 00:25:46.740
Therefore replacing 'sinh kt'
by what's it's equal to, we
00:25:46.740 --> 00:25:48.430
wind up with what?
00:25:48.430 --> 00:25:51.470
The second derivative of 'x'
with respect to 't' is equal
00:25:51.470 --> 00:25:54.020
to 'k squared x'.
00:25:54.020 --> 00:25:55.490
See, no minus sign here.
00:25:55.490 --> 00:25:57.570
What happens when we
start with a sinh?
00:25:57.570 --> 00:26:00.520
If an object we moved in
according to the rule, 'x'
00:26:00.520 --> 00:26:04.310
equals 'sinh kt', its
acceleration would be
00:26:04.310 --> 00:26:06.310
proportional to the
displacement
00:26:06.310 --> 00:26:08.380
but in the same direction.
00:26:08.380 --> 00:26:09.500
You see what that means?
00:26:09.500 --> 00:26:12.500
That means the bigger the
displacement, the bigger the
00:26:12.500 --> 00:26:15.760
acceleration so the faster
the object keeps moving.
00:26:15.760 --> 00:26:18.010
In other words, this thing would
move at a faster and
00:26:18.010 --> 00:26:20.630
faster right away from the
origin, et cetera.
00:26:20.630 --> 00:26:22.350
That's not our main
concern here.
00:26:22.350 --> 00:26:25.560
What is interesting though from
a physical point of view,
00:26:25.560 --> 00:26:28.620
and we can use this as a summary
from a physical point
00:26:28.620 --> 00:26:31.780
of view, is that the hyperbolic
functions serve as
00:26:31.780 --> 00:26:37.270
a solution to the equation ''d
2 x' 'dt squared'' equals 'k
00:26:37.270 --> 00:26:39.770
squared x' where 'k'
is a constant.
00:26:39.770 --> 00:26:43.580
The circular functions, on the
other hand, solve the equation
00:26:43.580 --> 00:26:49.730
''d 2 x' 'dt squared'' is
minus 'k squared x'.
00:26:49.730 --> 00:26:55.330
In other words, from this point
of view, notice again,
00:26:55.330 --> 00:26:58.710
from a physical point of view
that the difference between
00:26:58.710 --> 00:27:02.240
the circular functions and the
hyperbolic functions are again
00:27:02.240 --> 00:27:04.140
just the difference in sign.
00:27:04.140 --> 00:27:05.510
As I say, this will be
00:27:05.510 --> 00:27:08.010
emphasized more in the exercises.
00:27:08.010 --> 00:27:10.580
The obvious next lesson, I think
you can see it coming by
00:27:10.580 --> 00:27:13.870
this stage in the game, will,
of course, be the inverse
00:27:13.870 --> 00:27:15.260
hyperbolic functions.
00:27:15.260 --> 00:27:17.800
And we'll see why those are
important next time.
00:27:17.800 --> 00:27:19.250
But until then, good bye.
00:27:22.270 --> 00:27:24.810
ANNOUNCER 2: Funding for the
publication of this video was
00:27:24.810 --> 00:27:29.520
provided by the Gabriella and
Paul Rosenbaum Foundation.
00:27:29.520 --> 00:27:33.700
Help OCW continue to provide
free and open access to MIT
00:27:33.700 --> 00:27:37.890
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at ocw.mit.edu/donate.