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PROFESSOR: Hi.
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Today we're going to study
something called the
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hyperbolic functions.
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00:00:37,760 --> 00:00:43,220
And in a sense, we're going to
show that in a very major way,
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the study of the hyperbolic
functions mimics the study of
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the circular functions.
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And so in a sense, what we will
do is parrot much of what
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we did for circular functions
to wind up with
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00:00:55,640 --> 00:00:57,650
our hyperbolic functions.
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00:00:57,650 --> 00:01:00,800
Since the basic difference
between a hyperbola and a
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circle is a sign, namely 'x
squared' plus 'y squared'
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00:01:05,010 --> 00:01:08,420
equals 1 versus, say, 'x
squared' minus 'y squared'
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00:01:08,420 --> 00:01:11,140
equals 1, I've entitled today's
lesson, 'What a
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difference a sign makes'.
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00:01:13,130 --> 00:01:16,980
Now, by way of brief review,
recall that the circular
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00:01:16,980 --> 00:01:20,470
functions originated as follows:
we said consider the
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00:01:20,470 --> 00:01:23,050
circle whose equation is
'x squared' plus 'y
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00:01:23,050 --> 00:01:24,510
squared' equals 1.
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00:01:24,510 --> 00:01:27,690
That's a circle over here.
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00:01:27,690 --> 00:01:30,940
Then what we did was picking
this as a starting point, and
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00:01:30,940 --> 00:01:36,390
taking any length, 't', we wrap
't' around the circle.
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00:01:36,390 --> 00:01:40,890
When the length, 't', terminated
at point, 'p', we
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00:01:40,890 --> 00:01:42,340
defined what?
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00:01:42,340 --> 00:01:48,420
Cosine 't' to equal 'x' and
sine 't' to equal 'y'.
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00:01:48,420 --> 00:01:51,170
That was basically the geometric
interpretation of
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00:01:51,170 --> 00:01:55,010
the circular functions, after
which we showed that there was
34
00:01:55,010 --> 00:01:57,480
a physical meaning
to this as well.
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00:01:57,480 --> 00:02:00,460
Now you see what we'd like to
do is the same thing only
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00:02:00,460 --> 00:02:02,430
which regard now to
what I would call
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00:02:02,430 --> 00:02:03,890
the hyperbolic functions.
38
00:02:03,890 --> 00:02:07,140
And you may recall that in our
lecture on circular functions,
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00:02:07,140 --> 00:02:09,630
I mentioned the hyperbola
very briefly.
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00:02:09,630 --> 00:02:12,870
Now I'd like to go back to the
hyperbola in somewhat more
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00:02:12,870 --> 00:02:18,020
detail, namely we now look at
the equation 'x squared' minus
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00:02:18,020 --> 00:02:19,110
'y squared' equals 1.
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00:02:19,110 --> 00:02:21,070
See just to change of
sign over here.
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00:02:21,070 --> 00:02:23,180
The graph of 'x squared'
minus 'y squared'
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00:02:23,180 --> 00:02:26,130
equals 1 is the hyperbola.
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00:02:26,130 --> 00:02:31,920
And by the way, notice already
one significant change when we
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00:02:31,920 --> 00:02:33,360
change the sign here.
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00:02:33,360 --> 00:02:35,790
Notice that in the case of 'x
squared' plus 'y squared'
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00:02:35,790 --> 00:02:40,560
equals 1, our curve essentially
came in one piece.
50
00:02:40,560 --> 00:02:44,910
Now you see our curve has
two different branches.
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And for the sake of uniformity,
for the sake of
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well-definedness, let's pick the
branch that we will deal
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with to be the positive branch,
meaning the branch
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00:02:54,650 --> 00:02:57,010
that's to the right
of the y-axis.
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00:02:57,010 --> 00:03:00,150
To mimic precisely what we did
in the case of the circular
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00:03:00,150 --> 00:03:02,530
functions, what we
do now is what?
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00:03:02,530 --> 00:03:08,100
Given any length, 't', we start
at 's', mark 't' off
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00:03:08,100 --> 00:03:12,310
along the upper branch
of this double-valued
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00:03:12,310 --> 00:03:14,170
curve if 't' is positive.
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00:03:14,170 --> 00:03:17,240
Mark it off along the lower
branch if 't' is negative.
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00:03:17,240 --> 00:03:20,730
Wherever 't' terminates,
we call that point 'p'.
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And then to complete our analogy
with the circular
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functions, we define the
x-coordinate of 'p' to be the
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00:03:27,900 --> 00:03:33,260
hyperbolic cosine written
C-O-S-H and pronounced cosh.
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'Cosh t' is 'x'.
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00:03:35,220 --> 00:03:39,120
And the hyperbolic sine is
the y-coordinate of 'p'.
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00:03:39,120 --> 00:03:42,440
And that's written S-I-N with
an h, hyperbolic sine
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00:03:42,440 --> 00:03:44,030
pronounced cinch.
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00:03:44,030 --> 00:03:47,030
In other words, 'cosh t' is
the x-coordinate of 'p'.
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00:03:47,030 --> 00:03:50,380
'Sinh t' is the y-coordinate
of 'p'.
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00:03:50,380 --> 00:03:53,250
And the same way that 'x
squared' plus 'y squared'
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00:03:53,250 --> 00:03:57,350
equals 1 gave us the circular
identity that 'sine squared t'
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00:03:57,350 --> 00:04:02,480
plus 'cosine squared t' is 1,
the fact that 'x squared'
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00:04:02,480 --> 00:04:06,560
minus 'y squared' is 1 gives
us the hyperbolic identity
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00:04:06,560 --> 00:04:11,410
that 'cosh squared t' minus
'sinh squared t' is 1.
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00:04:11,410 --> 00:04:14,130
So again, notice great
similarities, great
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differences.
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00:04:15,460 --> 00:04:19,519
I'd like to make just a brief
aside to point out how in one
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00:04:19,519 --> 00:04:23,980
sense this difference isn't
too significant.
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00:04:23,980 --> 00:04:27,710
Yet in another sense, it's
very significant.
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00:04:27,710 --> 00:04:30,370
Notice the following: if you
look at the equation, 'x
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00:04:30,370 --> 00:04:34,830
squared' minus 'y squared'
equals 1, and if I can draw on
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00:04:34,830 --> 00:04:38,420
your previous knowledge of the
square root of minus 1 just
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00:04:38,420 --> 00:04:40,530
for a moment for sake
of illustration.
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00:04:40,530 --> 00:04:43,070
If we think of 'i' as being a
number whose square is minus
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00:04:43,070 --> 00:04:46,210
1, noticed another way of
writing 'x squared' minus 'y
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00:04:46,210 --> 00:04:49,430
squared' is to write
'x squared' plus
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00:04:49,430 --> 00:04:51,120
the number 'iy squared'.
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00:04:51,120 --> 00:04:54,290
You see 'i squared' is minus 1.
'y squared' is 'y squared'.
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00:04:54,290 --> 00:04:56,600
This is just another way of
saying 'x squared' minus 'y
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00:04:56,600 --> 00:04:57,980
squared' is 1.
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00:04:57,980 --> 00:04:59,760
Again, it's not that crucial.
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00:04:59,760 --> 00:05:03,710
But because much later in the
course when we study complex
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00:05:03,710 --> 00:05:07,520
numbers, there is going to be
a rather nice identification
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00:05:07,520 --> 00:05:11,580
between hyperbolic functions
and circular functions as a
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prelude of things to come.
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00:05:13,680 --> 00:05:16,740
Notice that what we're saying
is if you plot 'x squared'
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00:05:16,740 --> 00:05:21,280
minus 'y squared' equals 1, if
you plot that in the xy-
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00:05:21,280 --> 00:05:26,010
plane, you get the hyperbola
that we talked about.
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00:05:26,010 --> 00:05:30,280
On the other hand, if you change
the name of the y-axis
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00:05:30,280 --> 00:05:34,300
to the iy-axis, notice that with
respect to this type of
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00:05:34,300 --> 00:05:38,520
coordinate system, the graph
would be a circle.
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00:05:38,520 --> 00:05:42,120
Again, just an aside to show you
structurally a connection
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between circles and hyperbolas
in terms of complex numbers.
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00:05:46,510 --> 00:05:49,660
In that way, these things
out fairly similar.
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00:05:49,660 --> 00:05:52,600
Now to show you how different
a sign change is, I thought
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you might be interested in the
following little device.
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Take any two numbers, 'a' and
'b', no matter how different,
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00:05:59,580 --> 00:06:02,520
and form two numbers, one
of which is half the
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00:06:02,520 --> 00:06:04,110
sum of 'a' and 'b'.
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00:06:04,110 --> 00:06:07,750
And the other is half the
difference of 'a' and 'b'.
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00:06:07,750 --> 00:06:11,790
It's a trivial verification to
show that the sum of 'x' and
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00:06:11,790 --> 00:06:13,340
'y' is 'a'.
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00:06:13,340 --> 00:06:16,210
And the difference
'x - y' is 'b'.
115
00:06:16,210 --> 00:06:19,200
In other words, no matter how
different 'a' and 'b' are, we
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00:06:19,200 --> 00:06:23,330
can always find two numbers, 'x'
and 'y', such that just by
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00:06:23,330 --> 00:06:26,910
changing the sign, namely if we
add the pair of numbers, we
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00:06:26,910 --> 00:06:28,040
get one of the numbers.
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00:06:28,040 --> 00:06:30,070
If we subtract them,
we get the other.
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00:06:30,070 --> 00:06:32,430
Just to make this a little bit
more vivid, suppose, for
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00:06:32,430 --> 00:06:35,320
example, 'a' happens
to be 1,000.
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00:06:35,320 --> 00:06:37,750
And suppose 'b' happens
to be 4.
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00:06:37,750 --> 00:06:40,080
Now there's no danger of
confusing these two numbers.
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00:06:40,080 --> 00:06:41,030
They're quite different.
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00:06:41,030 --> 00:06:47,860
On the other hand, notice that
half the sum is 502, and half
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00:06:47,860 --> 00:06:52,160
the difference is 498.
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00:06:52,160 --> 00:06:55,910
You see if I add these two
numbers I get 1,000.
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00:06:55,910 --> 00:06:57,800
If I subtract them I get 4.
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00:06:57,800 --> 00:07:00,570
And therefore, notice that
just by changing the sign
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00:07:00,570 --> 00:07:05,960
here, I can effectively create
a change from 4 to 1,000.
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00:07:05,960 --> 00:07:08,200
I just mentioned that to show
why these people who say
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00:07:08,200 --> 00:07:11,690
things like, all I was, was off
by a sign, that making a
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00:07:11,690 --> 00:07:15,000
sign mistake is very, very
crucial, especially if the
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00:07:15,000 --> 00:07:18,550
answer later on has to be added
on to something else.
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00:07:18,550 --> 00:07:21,820
But any rate, I didn't even
have to give this example.
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00:07:21,820 --> 00:07:23,460
Except that I thought
it was a cute little
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00:07:23,460 --> 00:07:24,800
aside for you to notice.
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00:07:24,800 --> 00:07:28,030
From a more practical point of
view, if we just go back to
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00:07:28,030 --> 00:07:30,660
these two graphs that we've
drawn, notice that just
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00:07:30,660 --> 00:07:35,030
changing the plus sign to a
minus sign radically changed
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00:07:35,030 --> 00:07:38,680
the shape of the curve that
we were talking about.
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00:07:38,680 --> 00:07:40,130
So at any rate then,
this gives us
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00:07:40,130 --> 00:07:41,530
the hyperbolic functions.
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00:07:41,530 --> 00:07:45,410
And a very natural question to
raise at this time, is why the
145
00:07:45,410 --> 00:07:47,030
hyperbolic functions now?
146
00:07:47,030 --> 00:07:48,420
Why study them now?
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00:07:48,420 --> 00:07:50,950
Why didn't we do it before?
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00:07:50,950 --> 00:07:52,850
Why did we wait until
we were dealing
149
00:07:52,850 --> 00:07:55,000
with exponential functions?
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00:07:55,000 --> 00:07:57,800
The answer is that the
exponential functions give us
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00:07:57,800 --> 00:08:01,310
a very vivid way of
reconstructing
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00:08:01,310 --> 00:08:02,820
the hyperbolic functions.
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00:08:02,820 --> 00:08:05,620
And the key lies in the
following fact.
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00:08:05,620 --> 00:08:10,290
It lies in the fact that the
derivative of 'e' to the 'u'
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00:08:10,290 --> 00:08:14,360
with respect to 'x' is 'e'
to the 'u' times 'du/dx'.
156
00:08:14,360 --> 00:08:19,790
In particular, if I take the
derivative of 'e' to the 't'
157
00:08:19,790 --> 00:08:22,550
with respect to 't', I
get 'e' to the 't'.
158
00:08:22,550 --> 00:08:25,520
If I take the derivative of
'e' to the minus 't' with
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00:08:25,520 --> 00:08:27,460
respect to 't', I get what?
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00:08:27,460 --> 00:08:30,710
'e' to the minus 't' times the
derivative of minus 't' with
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00:08:30,710 --> 00:08:31,780
respect to 't'.
162
00:08:31,780 --> 00:08:33,159
That's minus 1.
163
00:08:33,159 --> 00:08:36,080
So the derivative of 'e' to the
minus 't' is minus 'e' to
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00:08:36,080 --> 00:08:37,340
the minus 't'.
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00:08:37,340 --> 00:08:39,990
Well what does this have
to do with anything?
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00:08:39,990 --> 00:08:44,610
Notice that if I now construct a
new function, 'e' to the 't'
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00:08:44,610 --> 00:08:48,490
plus 'e' to the minus 't', since
the derivative of a sum
168
00:08:48,490 --> 00:08:51,360
is the sum of derivatives,
notice that this result here
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00:08:51,360 --> 00:08:54,160
tells me that the derivative of
'e' to the 't' plus 'e' to
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00:08:54,160 --> 00:08:58,410
the minus 't' with respect to
't' is 'e' to the 't' minus
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00:08:58,410 --> 00:09:01,260
'e' to the minus 't'.
172
00:09:01,260 --> 00:09:03,740
Now let me do this operation
once again.
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00:09:03,740 --> 00:09:07,550
Suppose I now take 'e' to the
't' minus 'e' to the minus 't'
174
00:09:07,550 --> 00:09:09,890
and I differentiate that
with respect to 't'.
175
00:09:09,890 --> 00:09:12,400
Well, the derivative 'e'
to the 't' gives me
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00:09:12,400 --> 00:09:13,760
'e' to the 't' again.
177
00:09:13,760 --> 00:09:16,540
The derivative of 'e' to the
minus 't' is minus 'e'
178
00:09:16,540 --> 00:09:17,750
to the minus 't'.
179
00:09:17,750 --> 00:09:19,820
Minus times minus is plus.
180
00:09:19,820 --> 00:09:23,040
So observe that if I
differentiate 'e' to the 't'
181
00:09:23,040 --> 00:09:27,700
minus 'e' to the minus 't', I
come back to 'e' to the 't'
182
00:09:27,700 --> 00:09:29,770
plus 'e' to the minus 't'.
183
00:09:29,770 --> 00:09:33,640
Notice already how this starts
to behave, at least a little
184
00:09:33,640 --> 00:09:37,670
bit, like our friends sine and
cosine, except again, for a
185
00:09:37,670 --> 00:09:42,430
little change in sign, namely if
I give this function here a
186
00:09:42,430 --> 00:09:43,560
special name.
187
00:09:43,560 --> 00:09:47,780
And to make it suggestive, I'm
going to call it 'c of t'.
188
00:09:47,780 --> 00:09:49,750
'c' sort of to indicate
cosine.
189
00:09:49,750 --> 00:09:51,090
But I don't want
to write cosine
190
00:09:51,090 --> 00:09:52,330
because it's not a cosine.
191
00:09:52,330 --> 00:09:55,390
Let me just call this function
here 'c of t' and its
192
00:09:55,390 --> 00:10:00,150
companion, as you may guess,
I will call 's of t'.
193
00:10:00,150 --> 00:10:03,720
'c of t' is 'e' to the 't' plus
'e' to the minus 't'.
194
00:10:03,720 --> 00:10:08,610
's of t' is 'e' to the 't' minus
'e' to the minus 't'.
195
00:10:08,610 --> 00:10:11,400
This pair of functions has
the property that what?
196
00:10:11,400 --> 00:10:14,840
The derivative of one of them is
always the other, which is
197
00:10:14,840 --> 00:10:17,260
not quite what sine
and cosine did.
198
00:10:17,260 --> 00:10:20,850
Remember if you differentiated
the sine, you got the cosine.
199
00:10:20,850 --> 00:10:23,490
But if you differentiated
the cosine, you
200
00:10:23,490 --> 00:10:24,630
didn't get the sine.
201
00:10:24,630 --> 00:10:26,640
You got minus the sine.
202
00:10:26,640 --> 00:10:29,710
So you see, again, a similarity,
but not exactly
203
00:10:29,710 --> 00:10:30,950
the same thing.
204
00:10:30,950 --> 00:10:34,820
Nonetheless we get a little
bit suspicious about this.
205
00:10:34,820 --> 00:10:37,170
We see that the derivative of
'c' is 's', the derivative of
206
00:10:37,170 --> 00:10:37,950
's' is 'c'.
207
00:10:37,950 --> 00:10:41,450
We remember that there was a
basic identity involving sine
208
00:10:41,450 --> 00:10:43,430
squared plus cosine squared.
209
00:10:43,430 --> 00:10:46,350
We might get the premonition
that maybe there's some
210
00:10:46,350 --> 00:10:50,200
relationship between 's squared'
and 'c squared'.
211
00:10:50,200 --> 00:10:52,820
And one of the best ways of
checking out a premonition is
212
00:10:52,820 --> 00:10:53,660
to just check it out.
213
00:10:53,660 --> 00:10:55,350
I mean let's just see
what happens.
214
00:10:55,350 --> 00:10:57,830
In other words, starting with
this definition of 'c'--
215
00:10:57,830 --> 00:10:59,080
this is a binomial--
216
00:10:59,080 --> 00:11:00,350
let's square it.
217
00:11:00,350 --> 00:11:01,690
You see if we square
it, it's what?
218
00:11:01,690 --> 00:11:05,120
It's 'e' to the '2t', the first
term squared, plus twice
219
00:11:05,120 --> 00:11:06,820
the product of these
two terms.
220
00:11:06,820 --> 00:11:09,710
But 'e' to the 't' times 'e' to
the minus 't' is 'e' to the
221
00:11:09,710 --> 00:11:15,120
0, which is 1, plus 'e' to the
minus 't squared', which is
222
00:11:15,120 --> 00:11:16,890
'e' to the minus '2t'.
223
00:11:16,890 --> 00:11:19,570
In other words, putting this
all together, the square of
224
00:11:19,570 --> 00:11:23,990
'c' is 'e' to the '2t' plus
twice 1, which is 2, plus 'e'
225
00:11:23,990 --> 00:11:26,070
to the minus '2t'.
226
00:11:26,070 --> 00:11:30,450
Similarly, when we square 's',
we get exactly the same thing
227
00:11:30,450 --> 00:11:32,930
only with a minus 2
as the middle term
228
00:11:32,930 --> 00:11:36,150
rather than a plus 2.
229
00:11:36,150 --> 00:11:38,570
Well now we look at 'c squared'
and 's squared'.
230
00:11:38,570 --> 00:11:41,990
And just by looking at this
expression, any hope that 'c
231
00:11:41,990 --> 00:11:44,220
squared' plus 's squared'
will give us a nice
232
00:11:44,220 --> 00:11:45,660
identity should vanish.
233
00:11:45,660 --> 00:11:48,780
Because you see, if you add
these, you're going to get an
234
00:11:48,780 --> 00:11:50,610
'e' to the '2t' term
in here, in fact,
235
00:11:50,610 --> 00:11:51,980
twice 'e' to the '2t'.
236
00:11:51,980 --> 00:11:54,450
Here you'll get twice 'e'
to the minus '2t'.
237
00:11:54,450 --> 00:11:57,090
And only the 2 and the minus
2 will drop out.
238
00:11:57,090 --> 00:12:01,440
On the other hand, should you
elect to subtract 's squared'
239
00:12:01,440 --> 00:12:05,100
from 'c squared', the 'e' to the
'2t' terms cancel, the 'e'
240
00:12:05,100 --> 00:12:07,800
to the minus '2t' term
cancels, and we
241
00:12:07,800 --> 00:12:09,360
wind up with what?
242
00:12:09,360 --> 00:12:14,020
'c squared' minus 's squared'
is identically 4.
243
00:12:14,020 --> 00:12:16,780
And by the way, notice
up to this point, no
244
00:12:16,780 --> 00:12:18,770
use of the word hyperbola.
245
00:12:18,770 --> 00:12:20,210
OK?
246
00:12:20,210 --> 00:12:23,220
However, if you think of the
so-called unit hyperbola, the
247
00:12:23,220 --> 00:12:26,570
one for which 'x squared' minus
'y squared' is 1, notice
248
00:12:26,570 --> 00:12:29,900
that 'c squared' minus
's squared' equals 4
249
00:12:29,900 --> 00:12:32,010
almost has that form.
250
00:12:32,010 --> 00:12:33,270
Well it has the form.
251
00:12:33,270 --> 00:12:36,590
All that's spoiling it
is the 4 on the side.
252
00:12:36,590 --> 00:12:38,135
Now you could pretend
the 4 was a 1.
253
00:12:38,135 --> 00:12:39,450
But that would be cheating.
254
00:12:39,450 --> 00:12:42,390
What we do instead is just
divide both sides of this
255
00:12:42,390 --> 00:12:48,570
equation by 4, observing that
'c squared t' over 4 is just
256
00:12:48,570 --> 00:12:51,660
'c of t' over 2 squared.
257
00:12:51,660 --> 00:12:55,710
's squared t' over 4 is just
's of t' over 2 squared.
258
00:12:55,710 --> 00:12:59,150
In other words, dividing through
by 4, this equation
259
00:12:59,150 --> 00:13:01,540
can be put into this form.
260
00:13:01,540 --> 00:13:06,780
And now you see we satisfy the
basic hyperbolic identity of
261
00:13:06,780 --> 00:13:10,190
this squared minus this
squared equals 1.
262
00:13:10,190 --> 00:13:13,200
By the way, since the c's and
the s's may not seem familiar
263
00:13:13,200 --> 00:13:15,570
to us, what we can do is what?
264
00:13:15,570 --> 00:13:18,180
Go back to the basic
definition.
265
00:13:18,180 --> 00:13:21,680
'c of t' meant 'e' to the 't'
plus 'e' to the minus 't'.
266
00:13:21,680 --> 00:13:25,480
Therefore, half of 'c of t' is
'e' to the 't' plus 'e' to the
267
00:13:25,480 --> 00:13:27,410
minus 't' over 2.
268
00:13:27,410 --> 00:13:31,540
's of t' was 'e' to the 't'
minus 'e' to the minus 't'.
269
00:13:31,540 --> 00:13:35,550
Therefore 's of t' over 2 is
'e' to the 't' minus 'e' to
270
00:13:35,550 --> 00:13:37,750
the minus 't' over 2.
271
00:13:37,750 --> 00:13:41,340
And again, the reason I bring
this out, is it in most
272
00:13:41,340 --> 00:13:44,170
textbooks they simply define
the hyperbolic
273
00:13:44,170 --> 00:13:45,560
functions this way.
274
00:13:45,560 --> 00:13:47,440
I find it a poor motivation.
275
00:13:47,440 --> 00:13:50,610
I don't find it easy to motivate
where are the clear
276
00:13:50,610 --> 00:13:53,630
blue sky, you bring in a
2 in the denominator.
277
00:13:53,630 --> 00:13:57,400
All I want you to see in terms
of motivating what's going on
278
00:13:57,400 --> 00:14:01,840
so far, is that if I just
defined a function to be 'e of
279
00:14:01,840 --> 00:14:05,200
t' plus 'e' to the minus 't' and
another one to be 'e of t'
280
00:14:05,200 --> 00:14:08,840
minus 'e' to the minus 't', I
could have shown that the
281
00:14:08,840 --> 00:14:11,920
first squared minus the
second squared was 4.
282
00:14:11,920 --> 00:14:14,930
And the motivation of putting
the 2 in the denominator is
283
00:14:14,930 --> 00:14:18,350
for no greater reason than to
help us satisfy the equation
284
00:14:18,350 --> 00:14:20,540
'x squared' minus 'y
squared' equals 1.
285
00:14:20,540 --> 00:14:23,910
In fact, this is now the
connection between what we
286
00:14:23,910 --> 00:14:26,910
were doing a few months ago
and what we're doing now.
287
00:14:26,910 --> 00:14:32,680
Namely if I now call this 'cosh
t' and I call this 'sinh
288
00:14:32,680 --> 00:14:36,930
t', do I get my basic
hyperbola?
289
00:14:36,930 --> 00:14:41,070
Namely, suppose I take a curve
whose equation is given
290
00:14:41,070 --> 00:14:45,575
parametrically by 'x' equal
'cosh t' and 'y' equal 'sinh
291
00:14:45,575 --> 00:14:52,930
t', where cosh and sinh are
now defined as over here
292
00:14:52,930 --> 00:14:55,970
without any reference to the
hyperbola, in the same way
293
00:14:55,970 --> 00:15:00,700
that we could define the
circular functions in terms of
294
00:15:00,700 --> 00:15:03,130
calculus without having to
appeal to the circle.
295
00:15:03,130 --> 00:15:06,090
What I'm saying is, suppose we
had never mentioned the word
296
00:15:06,090 --> 00:15:08,110
hyperbolic function up to now.
297
00:15:08,110 --> 00:15:11,250
We've developed a function
called 'cosh t', a function
298
00:15:11,250 --> 00:15:14,980
called 'sinh t' just from these
two relationships here.
299
00:15:14,980 --> 00:15:19,530
If I eliminate the parameter 't'
here, namely if I square
300
00:15:19,530 --> 00:15:24,840
both sides of each equation
and subtract, I get 'x
301
00:15:24,840 --> 00:15:28,520
squared' minus 'y squared' is
'cosh squared t' minus 'sinh
302
00:15:28,520 --> 00:15:29,530
squared t'.
303
00:15:29,530 --> 00:15:31,400
But that I already say is 1.
304
00:15:31,400 --> 00:15:35,200
In other words, this
parametrically gives us the
305
00:15:35,200 --> 00:15:38,790
hyperbola 'x squared' minus
'y squared' equals 1.
306
00:15:38,790 --> 00:15:41,520
By the way, let's make
another observation.
307
00:15:41,520 --> 00:15:44,380
Notice that the exponential
could never be negative.
308
00:15:44,380 --> 00:15:47,930
See 'e' to the minus 't' is 1
over 'e' to the 't', which is
309
00:15:47,930 --> 00:15:49,060
still positive.
310
00:15:49,060 --> 00:15:51,630
Notice that the cosh can
never be negative.
311
00:15:51,630 --> 00:15:53,670
See, 'e' to the 't' is
a positive number.
312
00:15:53,670 --> 00:15:56,390
'e' to the minus 't' is
a positive number.
313
00:15:56,390 --> 00:16:00,040
Therefore, notice that in this
particular relationship, not
314
00:16:00,040 --> 00:16:03,030
only is the curve given by 'x
squared' minus 'y squared'
315
00:16:03,030 --> 00:16:07,240
equals 1, but since the cosh
can never be negative, it's
316
00:16:07,240 --> 00:16:11,540
also characterized by
'x' is positive.
317
00:16:11,540 --> 00:16:14,160
And you see that
tells us what?
318
00:16:14,160 --> 00:16:16,630
That what we're getting this
way is not the entire
319
00:16:16,630 --> 00:16:20,830
hyperbola, but just the branch
for which 'x' is positive,
320
00:16:20,830 --> 00:16:23,130
which is exactly what we
were talking about
321
00:16:23,130 --> 00:16:24,390
earlier in the lesson.
322
00:16:24,390 --> 00:16:27,090
323
00:16:27,090 --> 00:16:30,220
Again, I leave further details
of this to the exercises.
324
00:16:30,220 --> 00:16:33,230
All I want to do in today's
lesson is to get the flavor of
325
00:16:33,230 --> 00:16:35,640
what's going on, how we can
invent the hyperbolic
326
00:16:35,640 --> 00:16:38,390
functions without reference
to a hyperbola
327
00:16:38,390 --> 00:16:39,990
until we're all done.
328
00:16:39,990 --> 00:16:41,830
See, I hope you see this
juxtaposition.
329
00:16:41,830 --> 00:16:42,930
We've done two things here.
330
00:16:42,930 --> 00:16:45,910
One is we've started with the
hyperbola and showed how we
331
00:16:45,910 --> 00:16:47,530
could invent this
construction.
332
00:16:47,530 --> 00:16:50,630
The other is we started with the
exponential and showed how
333
00:16:50,630 --> 00:16:53,970
we could construct two functions
of the exponential
334
00:16:53,970 --> 00:16:57,270
which had the property that they
could be identified with
335
00:16:57,270 --> 00:16:59,750
the thing that was called the
hyperbolic cosine and
336
00:16:59,750 --> 00:17:00,680
hyperbolic sine.
337
00:17:00,680 --> 00:17:03,260
In other words, we could take
either of the two approaches
338
00:17:03,260 --> 00:17:04,450
and derive the other.
339
00:17:04,450 --> 00:17:07,670
But I don't want to put in all
of those details here.
340
00:17:07,670 --> 00:17:08,510
OK?
341
00:17:08,510 --> 00:17:11,700
At any rate, to show you where
calculus comes in, in terms of
342
00:17:11,700 --> 00:17:15,040
what I meant by saying that
as far as this course is
343
00:17:15,040 --> 00:17:17,800
concerned, we have now finished
the groundwork both
344
00:17:17,800 --> 00:17:20,339
with respect to differential
and integral calculus.
345
00:17:20,339 --> 00:17:23,140
Suppose, for example,
I now want to define
346
00:17:23,140 --> 00:17:24,810
the hyperbolic tangent.
347
00:17:24,810 --> 00:17:27,119
See y equals hyperbolic
tangent 'x'.
348
00:17:27,119 --> 00:17:28,190
How could I do that?
349
00:17:28,190 --> 00:17:32,250
Well, I mimic the circular
function definition.
350
00:17:32,250 --> 00:17:35,970
I say, OK, I will define the
hyperbolic tangent to be the
351
00:17:35,970 --> 00:17:38,740
hyperbolic sine over the
hyperbolic cosine.
352
00:17:38,740 --> 00:17:43,450
By the way, in the same way that
one can show in terms of
353
00:17:43,450 --> 00:17:47,440
geometric constructions on the
unit circle how one constructs
354
00:17:47,440 --> 00:17:51,080
the tangent and the cotangent
and the cosecant and the
355
00:17:51,080 --> 00:17:54,770
secant, et cetera, once sine
and cosine and given.
356
00:17:54,770 --> 00:17:58,560
It can also be shown that we can
construct these things in
357
00:17:58,560 --> 00:18:01,070
terms of geometry from
the hyperbola.
358
00:18:01,070 --> 00:18:03,680
But because that gets
computationally involved, I've
359
00:18:03,680 --> 00:18:07,320
also saved those aspects
for the exercises.
360
00:18:07,320 --> 00:18:10,440
I'm not going to worry about
how you construct the
361
00:18:10,440 --> 00:18:13,020
hyperbolic tangent given the
hyperbolic sine and the
362
00:18:13,020 --> 00:18:14,650
hyperbolic cosine right now.
363
00:18:14,650 --> 00:18:16,560
All I'm saying is suppose
you've made up this
364
00:18:16,560 --> 00:18:17,460
definition.
365
00:18:17,460 --> 00:18:19,960
And now if somebody says gee,
I would like to find the
366
00:18:19,960 --> 00:18:22,950
derivative of the hyperbolic
tangent of 'x'.
367
00:18:22,950 --> 00:18:26,120
What I want to point out is that
the recipes that we need
368
00:18:26,120 --> 00:18:28,540
are no different from anything
we've had before.
369
00:18:28,540 --> 00:18:31,980
Namely, since the hyperbolic
tangent of 'x' is 'sinh x'
370
00:18:31,980 --> 00:18:34,480
over 'cosh x', I use
the same quotient
371
00:18:34,480 --> 00:18:36,380
rule as I used before.
372
00:18:36,380 --> 00:18:39,140
Namely, to differentiate the
hyperbolic tangent, I
373
00:18:39,140 --> 00:18:40,650
differentiate this quotient.
374
00:18:40,650 --> 00:18:42,520
How do we differentiate
a quotient?
375
00:18:42,520 --> 00:18:47,480
It's the denominator times the
derivative of the numerator.
376
00:18:47,480 --> 00:18:49,430
Well the numerator
is 'sinh x'.
377
00:18:49,430 --> 00:18:52,010
The derivative 'sinh
x' is 'cosh x'.
378
00:18:52,010 --> 00:18:56,090
That gives me a 'cosh squared
x' over here minus the
379
00:18:56,090 --> 00:19:00,990
numerator, which is 'sinh x'
times the derivative of the
380
00:19:00,990 --> 00:19:01,830
denominator.
381
00:19:01,830 --> 00:19:05,580
Well the derivative of 'cosh
x' is 'sinh x'--
382
00:19:05,580 --> 00:19:07,280
that give me another factor--
383
00:19:07,280 --> 00:19:08,725
over the square of
the denominator.
384
00:19:08,725 --> 00:19:12,900
385
00:19:12,900 --> 00:19:16,000
Now my basic identity is that
cosh squared minus sinh
386
00:19:16,000 --> 00:19:17,200
squared is 1.
387
00:19:17,200 --> 00:19:22,430
So this is just 1 over
'cosh squared x'.
388
00:19:22,430 --> 00:19:25,060
And if I want to write this
all on one line, I could
389
00:19:25,060 --> 00:19:31,690
invent the notation that the
hyperbolic secant is 1 over
390
00:19:31,690 --> 00:19:33,890
the hyperbolic cosine.
391
00:19:33,890 --> 00:19:37,680
In other words, I could invent
a notation like this.
392
00:19:37,680 --> 00:19:40,110
But again, the important point
is not so much what this
393
00:19:40,110 --> 00:19:43,300
answer is, but how I can
derive it from first
394
00:19:43,300 --> 00:19:43,960
principles.
395
00:19:43,960 --> 00:19:47,800
And by the way, in terms of the
inverse derivative, once I
396
00:19:47,800 --> 00:19:51,380
know that the derivative of
hyperbolic tangent is a square
397
00:19:51,380 --> 00:19:55,140
of the hyperbolic secant, I can
immediately add to my list
398
00:19:55,140 --> 00:19:57,130
of inverse derivatives.
399
00:19:57,130 --> 00:20:00,280
Namely I just have
to write it down.
400
00:20:00,280 --> 00:20:04,780
401
00:20:04,780 --> 00:20:07,640
See there's an integral formula
that I now have.
402
00:20:07,640 --> 00:20:11,480
But again, all I want to
emphasize here is how the
403
00:20:11,480 --> 00:20:14,820
details take care of themselves
in the same way as
404
00:20:14,820 --> 00:20:17,990
always once we have our
basic definitions.
405
00:20:17,990 --> 00:20:21,370
It might be interesting just
in terms of cementing down
406
00:20:21,370 --> 00:20:24,740
what sinh and cosh look like,
in the same way that we have
407
00:20:24,740 --> 00:20:27,470
talked about sine and cosine,
it might be nice to
408
00:20:27,470 --> 00:20:29,600
graph sinh and cosh.
409
00:20:29,600 --> 00:20:32,900
And again notice, in terms of
curve plotting, how do we plot
410
00:20:32,900 --> 00:20:34,280
'y' equals 'f of x'?
411
00:20:34,280 --> 00:20:36,200
The general procedure
was what?
412
00:20:36,200 --> 00:20:38,860
Always take the first and second
derivatives so that you
413
00:20:38,860 --> 00:20:40,500
can see what the slope
is, what the
414
00:20:40,500 --> 00:20:42,630
concavity is, et cetera.
415
00:20:42,630 --> 00:20:44,510
The derivative of
cosh is sinh.
416
00:20:44,510 --> 00:20:46,440
The derivative of
sinh is cosh.
417
00:20:46,440 --> 00:20:50,380
By the way, again, if these
terms seem alien to you, you
418
00:20:50,380 --> 00:20:54,950
can always rewrite them in terms
of the basic definition,
419
00:20:54,950 --> 00:20:57,740
in terms of 'e' to the 'x' and
'e' to the minus 'x', and
420
00:20:57,740 --> 00:21:01,020
carry out the differentiation
in a straightforward way.
421
00:21:01,020 --> 00:21:02,410
Well here's the interesting
point.
422
00:21:02,410 --> 00:21:04,880
Let's plot 'y' equals
'cosh x'.
423
00:21:04,880 --> 00:21:08,330
Notice first of all that 'cosh
x' is an even function.
424
00:21:08,330 --> 00:21:12,000
If I replace 'x' by minus 'x',
all I get is what? 'e' to the
425
00:21:12,000 --> 00:21:15,420
minus 'x' plus 'e' to the 'x'
over 2, which is the same
426
00:21:15,420 --> 00:21:17,020
thing as I have over here.
427
00:21:17,020 --> 00:21:19,320
It's going to be a symmetric
function with a curve with
428
00:21:19,320 --> 00:21:20,680
respect to the y-axis.
429
00:21:20,680 --> 00:21:22,280
But again, what do I do?
430
00:21:22,280 --> 00:21:23,850
I compute the derivative.
431
00:21:23,850 --> 00:21:26,410
I can find out where the
derivative is 0.
432
00:21:26,410 --> 00:21:29,720
It's 0 when 'e' to the 'x'
equals 'e' to the minus 'x'.
433
00:21:29,720 --> 00:21:31,960
That happens only
when 'x' is 0.
434
00:21:31,960 --> 00:21:36,730
When 'x' is 0, this is 'e' to
the 0 plus 'e' to the 0.
435
00:21:36,730 --> 00:21:41,270
That's 1 plus 1, which is
2 over 2, which is one.
436
00:21:41,270 --> 00:21:45,040
So in other words, the
derivative is 0 when 'x' is 0
437
00:21:45,040 --> 00:21:47,010
and 'y' is 1.
438
00:21:47,010 --> 00:21:48,740
The second derivative is cosh.
439
00:21:48,740 --> 00:21:52,310
We've already seen that this
can never be negative.
440
00:21:52,310 --> 00:21:54,470
So the second derivative
is always positive.
441
00:21:54,470 --> 00:21:57,150
That means that the curve
is always holding water.
442
00:21:57,150 --> 00:22:00,680
Putting all this together, the
curve 'y' equals 'cosh x'
443
00:22:00,680 --> 00:22:03,450
looks something like this.
444
00:22:03,450 --> 00:22:05,100
It's a dangerous thing
to say it sort
445
00:22:05,100 --> 00:22:06,450
of resembles a parabola.
446
00:22:06,450 --> 00:22:10,500
It's nothing like a parabola
except what I mean is it has
447
00:22:10,500 --> 00:22:11,730
this type of shape.
448
00:22:11,730 --> 00:22:13,940
Notice it does not oscillate.
449
00:22:13,940 --> 00:22:16,680
It is not an oscillating
function like the cosine.
450
00:22:16,680 --> 00:22:18,380
It doesn't act periodically.
451
00:22:18,380 --> 00:22:21,110
This thing just keeps
going like this.
452
00:22:21,110 --> 00:22:21,630
All right?
453
00:22:21,630 --> 00:22:25,710
On the other hand, how can we
plot 'y' equals 'sinh x'?
454
00:22:25,710 --> 00:22:28,500
Well, I guess we come
right back to here
455
00:22:28,500 --> 00:22:29,620
and work from here.
456
00:22:29,620 --> 00:22:31,580
The derivative of
sinh is cosh.
457
00:22:31,580 --> 00:22:34,700
But cosh is always at least
as big as 1 we found out.
458
00:22:34,700 --> 00:22:37,230
Therefore, the slope of
'sinh x' is always at
459
00:22:37,230 --> 00:22:38,450
least as big as 1.
460
00:22:38,450 --> 00:22:42,550
And it's equal to
1 when 'x' is 0.
461
00:22:42,550 --> 00:22:44,490
Putting all these details
together,
462
00:22:44,490 --> 00:22:45,770
what we find is what?
463
00:22:45,770 --> 00:22:48,830
First of all, the sinh is an odd
function, meaning that if
464
00:22:48,830 --> 00:22:52,800
we replace 'x' by minus 'x',
we change the sign.
465
00:22:52,800 --> 00:22:56,300
So this is 'e to the minus x'
minus 'e to the x', which is a
466
00:22:56,300 --> 00:22:58,070
negative of what we
have over here.
467
00:22:58,070 --> 00:23:00,440
But those, again, are details
which are easy
468
00:23:00,440 --> 00:23:01,650
for you to fill in.
469
00:23:01,650 --> 00:23:04,940
The graph 'y' equals 'sinh x'
looks something like this.
470
00:23:04,940 --> 00:23:07,890
In other words, it's
a very steep curve.
471
00:23:07,890 --> 00:23:12,790
It's spilling water and
rising here, holding
472
00:23:12,790 --> 00:23:14,960
water and rising here.
473
00:23:14,960 --> 00:23:16,270
The curve is always rising.
474
00:23:16,270 --> 00:23:20,000
Again, another interesting thing
to observe here is that
475
00:23:20,000 --> 00:23:24,010
notice that when 'x' is very,
very large, 'e' to the minus
476
00:23:24,010 --> 00:23:25,870
'x' becomes negligible.
477
00:23:25,870 --> 00:23:29,060
And if 'e' to the minus 'x' is
negligible, notice that both
478
00:23:29,060 --> 00:23:32,660
sinh and cosh behave like
1/2 'e to the x'.
479
00:23:32,660 --> 00:23:34,970
In other words, this term
tends to drop out.
480
00:23:34,970 --> 00:23:37,540
And just to show you the
contrast here, I've sketched
481
00:23:37,540 --> 00:23:40,710
in the curve 'y' equals '1/2 'e
to the x'' to show you how
482
00:23:40,710 --> 00:23:43,950
it splits the difference for
large values of 'x' between
483
00:23:43,950 --> 00:23:45,140
these two in a way.
484
00:23:45,140 --> 00:23:49,870
And as 'x' gets larger, both
of these curves converge on
485
00:23:49,870 --> 00:23:52,100
'y' equals '1/2 'e to the x''.
486
00:23:52,100 --> 00:23:55,410
Again, I simply want to mention
enough here so that
487
00:23:55,410 --> 00:23:59,140
you get the idea of how we can
apply the same old calculus to
488
00:23:59,140 --> 00:24:00,860
our new function.
489
00:24:00,860 --> 00:24:04,900
What I'd like to do in closing
is to try again from our point
490
00:24:04,900 --> 00:24:08,430
of view of being engineeringly
oriented to show why the
491
00:24:08,430 --> 00:24:12,070
hyperbolic functions might have
occurred in nature, in
492
00:24:12,070 --> 00:24:14,540
the same way that we tried
to show that the circular
493
00:24:14,540 --> 00:24:18,370
functions had a natural
outgrowth in terms of nature.
494
00:24:18,370 --> 00:24:20,820
You recall that the circular
functions were motivated in
495
00:24:20,820 --> 00:24:22,700
terms of simple harmonic
motion.
496
00:24:22,700 --> 00:24:25,650
The circular functions would
have been invented when we
497
00:24:25,650 --> 00:24:28,940
talked about motion, in which
the acceleration was
498
00:24:28,940 --> 00:24:31,440
proportional to the displacement
but in the
499
00:24:31,440 --> 00:24:33,040
opposite direction.
500
00:24:33,040 --> 00:24:34,870
Let's see what happens
when we work with
501
00:24:34,870 --> 00:24:36,270
the hyperbolic functions.
502
00:24:36,270 --> 00:24:40,060
503
00:24:40,060 --> 00:24:46,000
Suppose now I have a particle
moving along the x-axis.
504
00:24:46,000 --> 00:24:46,880
OK?
505
00:24:46,880 --> 00:24:51,390
And I know this position at any
time, 't', is sinh of 'kt'
506
00:24:51,390 --> 00:24:54,690
where 'k' happens to
be a constant.
507
00:24:54,690 --> 00:24:56,480
What is 'dx/dt'?
508
00:24:56,480 --> 00:24:59,480
The derivative of 'sinh
u' with respect to
509
00:24:59,480 --> 00:25:01,360
'u' is 'cosh u'.
510
00:25:01,360 --> 00:25:05,610
So the derivative of 'sinh kt'
with respect to 't' would be
511
00:25:05,610 --> 00:25:09,950
'cosh kt' times the derivative
of 'kt' with respect to 't',
512
00:25:09,950 --> 00:25:11,230
which is just 'k'.
513
00:25:11,230 --> 00:25:14,350
In other words, the chain
rule again, OK?
514
00:25:14,350 --> 00:25:19,860
Now knowing that 'dx/dt' is
'k cosh kt', ''d 2 x' 'dt
515
00:25:19,860 --> 00:25:21,220
squared'' would be what?
516
00:25:21,220 --> 00:25:25,930
The derivative of 'cosh kt' is
'sinh kt' times the derivative
517
00:25:25,930 --> 00:25:28,000
of 'kt' with respect to 't'.
518
00:25:28,000 --> 00:25:30,190
That gives me another
factor of 'k'.
519
00:25:30,190 --> 00:25:32,460
And therefore the second
derivative of 'x' with respect
520
00:25:32,460 --> 00:25:36,100
to 't' is ''k squared'
sinh kt'.
521
00:25:36,100 --> 00:25:38,210
Now remember what
'sinh kt' is.
522
00:25:38,210 --> 00:25:43,300
We started with the knowledge
that 'sinh kt' is 'x'.
523
00:25:43,300 --> 00:25:46,740
Therefore replacing 'sinh kt'
by what's it's equal to, we
524
00:25:46,740 --> 00:25:48,430
wind up with what?
525
00:25:48,430 --> 00:25:51,470
The second derivative of 'x'
with respect to 't' is equal
526
00:25:51,470 --> 00:25:54,020
to 'k squared x'.
527
00:25:54,020 --> 00:25:55,490
See, no minus sign here.
528
00:25:55,490 --> 00:25:57,570
What happens when we
start with a sinh?
529
00:25:57,570 --> 00:26:00,520
If an object we moved in
according to the rule, 'x'
530
00:26:00,520 --> 00:26:04,310
equals 'sinh kt', its
acceleration would be
531
00:26:04,310 --> 00:26:06,310
proportional to the
displacement
532
00:26:06,310 --> 00:26:08,380
but in the same direction.
533
00:26:08,380 --> 00:26:09,500
You see what that means?
534
00:26:09,500 --> 00:26:12,500
That means the bigger the
displacement, the bigger the
535
00:26:12,500 --> 00:26:15,760
acceleration so the faster
the object keeps moving.
536
00:26:15,760 --> 00:26:18,010
In other words, this thing would
move at a faster and
537
00:26:18,010 --> 00:26:20,630
faster right away from the
origin, et cetera.
538
00:26:20,630 --> 00:26:22,350
That's not our main
concern here.
539
00:26:22,350 --> 00:26:25,560
What is interesting though from
a physical point of view,
540
00:26:25,560 --> 00:26:28,620
and we can use this as a summary
from a physical point
541
00:26:28,620 --> 00:26:31,780
of view, is that the hyperbolic
functions serve as
542
00:26:31,780 --> 00:26:37,270
a solution to the equation ''d
2 x' 'dt squared'' equals 'k
543
00:26:37,270 --> 00:26:39,770
squared x' where 'k'
is a constant.
544
00:26:39,770 --> 00:26:43,580
The circular functions, on the
other hand, solve the equation
545
00:26:43,580 --> 00:26:49,730
''d 2 x' 'dt squared'' is
minus 'k squared x'.
546
00:26:49,730 --> 00:26:55,330
In other words, from this point
of view, notice again,
547
00:26:55,330 --> 00:26:58,710
from a physical point of view
that the difference between
548
00:26:58,710 --> 00:27:02,240
the circular functions and the
hyperbolic functions are again
549
00:27:02,240 --> 00:27:04,140
just the difference in sign.
550
00:27:04,140 --> 00:27:05,510
As I say, this will be
551
00:27:05,510 --> 00:27:08,010
emphasized more in the exercises.
552
00:27:08,010 --> 00:27:10,580
The obvious next lesson, I think
you can see it coming by
553
00:27:10,580 --> 00:27:13,870
this stage in the game, will,
of course, be the inverse
554
00:27:13,870 --> 00:27:15,260
hyperbolic functions.
555
00:27:15,260 --> 00:27:17,800
And we'll see why those are
important next time.
556
00:27:17,800 --> 00:27:19,250
But until then, good bye.
557
00:27:19,250 --> 00:27:22,270
558
00:27:22,270 --> 00:27:24,810
ANNOUNCER 2: Funding for the
publication of this video was
559
00:27:24,810 --> 00:27:29,520
provided by the Gabriella and
Paul Rosenbaum Foundation.
560
00:27:29,520 --> 00:27:33,700
Help OCW continue to provide
free and open access to MIT
561
00:27:33,700 --> 00:27:37,890
courses by making a donation
at ocw.mit.edu/donate.
562
00:27:37,890 --> 00:27:42,634