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PROFESSOR: Hi.
00:00:32.110 --> 00:00:36.460
Today we will discuss the
inverse hyperbolic functions
00:00:36.460 --> 00:00:40.840
and, with this lecture, finish
our block on the logarithmic
00:00:40.840 --> 00:00:43.640
exponential and hyperbolic
functions.
00:00:43.640 --> 00:00:47.380
And what we're going to find is
that much of what we have
00:00:47.380 --> 00:00:52.020
to say today is simply a
specific application to a
00:00:52.020 --> 00:00:55.470
special function of the same
theory that we were talking
00:00:55.470 --> 00:00:57.200
about in general before.
00:00:57.200 --> 00:01:00.530
Recall that we can always talk
about an inverse function if
00:01:00.530 --> 00:01:03.690
the original function is
a one-to-one function.
00:01:03.690 --> 00:01:07.880
For example, to introduce
today's topic, suppose we take
00:01:07.880 --> 00:01:11.385
the function 'y' equals
hyperbolic sine 'x'.
00:01:11.385 --> 00:01:13.080
'y' equals 'sinh x'.
00:01:13.080 --> 00:01:17.500
Well, as we saw last time,
the graph of 'y'
00:01:17.500 --> 00:01:20.480
equals 'sinh x' is this.
00:01:20.480 --> 00:01:23.390
And we can see in a glance
that this clearly is a
00:01:23.390 --> 00:01:25.160
one-to-one function.
00:01:25.160 --> 00:01:27.850
In fact, the derivative
of sinh is cosh.
00:01:27.850 --> 00:01:30.480
And as we've seen also
last time, the cosh
00:01:30.480 --> 00:01:31.970
can never be negative.
00:01:31.970 --> 00:01:35.170
In fact, the cosh can't
be less than one.
00:01:35.170 --> 00:01:38.050
So here is a curve that's
always rising--
00:01:38.050 --> 00:01:41.380
and, in fact, in general
rising quite steeply.
00:01:41.380 --> 00:01:45.770
In any event, we can therefore
talk about the inverse
00:01:45.770 --> 00:01:47.320
hyperbolic sine.
00:01:47.320 --> 00:01:47.990
All right?
00:01:47.990 --> 00:01:50.420
And how do we get the inverse
function in general?
00:01:50.420 --> 00:01:53.820
Well we've always seen that to
invert a function, all we have
00:01:53.820 --> 00:01:57.350
to do is to reflect the original
graph with respect to
00:01:57.350 --> 00:01:59.090
the line, 'y' equals 'x'.
00:01:59.090 --> 00:02:03.400
Or, again, in more slow motion,
we rotate through 90
00:02:03.400 --> 00:02:06.750
degrees and then flip
the graph over.
00:02:06.750 --> 00:02:09.310
And, in any event, whichever way
you want to look at this
00:02:09.310 --> 00:02:13.240
thing, we obtain that the
graph 'y' equals inverse
00:02:13.240 --> 00:02:17.300
hyperbolic sine 'x' is this
particular curve over here.
00:02:17.300 --> 00:02:21.140
This is the graph of the inverse
hyperbolic sine.
00:02:21.140 --> 00:02:25.850
Now, again, what was the main
issue or the main property of
00:02:25.850 --> 00:02:27.120
inverse functions?
00:02:27.120 --> 00:02:30.870
The idea was that once we knew
the original function, we
00:02:30.870 --> 00:02:33.000
could, by a change in emphasis,
00:02:33.000 --> 00:02:34.990
study the inverse function.
00:02:34.990 --> 00:02:39.620
Well, by way of illustration,
let's suppose we take the
00:02:39.620 --> 00:02:43.020
functional relationship 'y'
equals inverse hyperbolic sine
00:02:43.020 --> 00:02:46.970
'x' and we say, let's
find the 'ydx'.
00:02:46.970 --> 00:02:50.870
Recall that what we know how to
do, is how to differentiate
00:02:50.870 --> 00:02:52.370
the hyperbolic sine.
00:02:52.370 --> 00:02:55.820
Consequently, given that y
equals the inverse hyperbolic
00:02:55.820 --> 00:03:00.010
sine 'x', we simply switch the
emphasis and say, this is the
00:03:00.010 --> 00:03:03.820
same as saying that 'x'
equals 'sinh y'.
00:03:03.820 --> 00:03:07.240
And if 'x' equal 'sinh y',
we already know that the
00:03:07.240 --> 00:03:11.190
derivative of 'x' with respect
to 'y' is 'cosh y'.
00:03:11.190 --> 00:03:15.160
And since the derivative of 'y'
with respect to 'x' is the
00:03:15.160 --> 00:03:18.950
reciprocal of the derivative of
'x' with respect to 'y', we
00:03:18.950 --> 00:03:24.060
can conclude from this that the
'ydx' is '1 over cosh y'.
00:03:24.060 --> 00:03:26.460
In a certain manner
of speaking, we're
00:03:26.460 --> 00:03:27.520
all finished now.
00:03:27.520 --> 00:03:31.810
We have found that the
derivative off inverse 'sinh
00:03:31.810 --> 00:03:35.070
x' is '1 over cosh y'.
00:03:35.070 --> 00:03:38.000
The only problem is, is as we've
said many times also
00:03:38.000 --> 00:03:41.670
before that if 'y' is given as
a function of 'x', we would
00:03:41.670 --> 00:03:44.760
like the 'ydx' expressed
in terms of 'x'.
00:03:44.760 --> 00:03:47.370
Or, to put it in still different
words, usually when
00:03:47.370 --> 00:03:50.510
you're given an expression like
the 'ydx', you're asked
00:03:50.510 --> 00:03:55.210
to evaluate this when 'x' is
some particular value, not
00:03:55.210 --> 00:03:57.340
when 'y' is some particular
value.
00:03:57.340 --> 00:04:00.390
At any rate, we do much the same
here as we did with the
00:04:00.390 --> 00:04:02.910
inverse circular
trig functions.
00:04:02.910 --> 00:04:08.670
Namely, having arrived at '1
over cosh y' and remembering
00:04:08.670 --> 00:04:12.810
that the relationship that ties
in 'x' and 'y' is at 'x'
00:04:12.810 --> 00:04:17.769
equals 'sinh y', we invoke the
fundamental identity that
00:04:17.769 --> 00:04:21.810
'cosh squared y' minus 'sinh
squared y' is 1.
00:04:21.810 --> 00:04:25.050
From which we can conclude
that 'cosh y'--
00:04:25.050 --> 00:04:27.270
and here we have to be a
little bit careful--
00:04:27.270 --> 00:04:31.700
if we solve this algebraically
we find that 'cosh y' is plus
00:04:31.700 --> 00:04:36.180
or minus the square root of
'1 plus 'sinh squared y''.
00:04:36.180 --> 00:04:39.670
But recalling the cosh could
never be negative, it means
00:04:39.670 --> 00:04:42.030
that for this particular
problem, the minus
00:04:42.030 --> 00:04:43.740
sign does not apply.
00:04:43.740 --> 00:04:48.770
In other words, since cosh has
to be at least as big one, you
00:04:48.770 --> 00:04:53.180
see, that 'cosh y' is plus the
square root of '1 plus 'sinh
00:04:53.180 --> 00:04:54.370
squared y''.
00:04:54.370 --> 00:04:57.150
Now keep in mind why we
went to this identity
00:04:57.150 --> 00:04:58.220
in the first place.
00:04:58.220 --> 00:05:00.840
The reason we went to this
identity in the first place,
00:05:00.840 --> 00:05:03.390
is if we come back to the
beginning of our problem we
00:05:03.390 --> 00:05:08.020
see that we had that 'x'
is equal to 'sinh y'.
00:05:08.020 --> 00:05:12.460
In other words, down here now,
all we do is replace 'sinh y'
00:05:12.460 --> 00:05:14.490
by a synonym, namely 'x'.
00:05:14.490 --> 00:05:18.820
And we arrive at the fact that
'cosh y' is the square root of
00:05:18.820 --> 00:05:21.340
'1 plus 'x squared''.
00:05:21.340 --> 00:05:26.200
Therefore, since 'dy/dx' is the
reciprocal of 'cosh y',
00:05:26.200 --> 00:05:29.550
the derivative of 'y' with
respect to 'x' is '1 over the
00:05:29.550 --> 00:05:32.000
'square root of '1 plus
'x squared'''.
00:05:32.000 --> 00:05:34.900
And this is summarized in
our last step over here.
00:05:34.900 --> 00:05:38.270
In other words, the derivative
of the inverse hyperbolic sine
00:05:38.270 --> 00:05:41.730
of 'x' with respect to 'x' is '1
over the 'square root of '1
00:05:41.730 --> 00:05:43.160
plus 'x squared'''.
00:05:43.160 --> 00:05:46.290
Now, you see, the first thing I
want to point out here is to
00:05:46.290 --> 00:05:48.890
observe that without worrying
about what's important about
00:05:48.890 --> 00:05:53.100
this result, is to observe
again that we obtain this
00:05:53.100 --> 00:05:57.540
information virtually free of
charge by knowing the calculus
00:05:57.540 --> 00:05:59.680
of the regular hyperbolic
functions.
00:05:59.680 --> 00:06:01.980
You see, this result
was obtained
00:06:01.980 --> 00:06:04.610
without any new knowledge.
00:06:04.610 --> 00:06:07.640
The other thing I'd like to
point out here is somewhat
00:06:07.640 --> 00:06:08.780
more subtle.
00:06:08.780 --> 00:06:11.990
And we also mentioned this in
the same context, but from a
00:06:11.990 --> 00:06:14.550
different point of view, when
we dealt with the inverse
00:06:14.550 --> 00:06:15.920
circular functions.
00:06:15.920 --> 00:06:18.970
Many people will say things
like, who needs the inverse
00:06:18.970 --> 00:06:20.230
hyperbolic functions?
00:06:20.230 --> 00:06:23.120
How many times am I going to be
confronted with having to
00:06:23.120 --> 00:06:25.750
work with inverse hyperbolic
functions?
00:06:25.750 --> 00:06:29.930
And the interesting point that's
typified by this result
00:06:29.930 --> 00:06:32.360
is that if we now invert
this emphasis--
00:06:32.360 --> 00:06:35.090
in other words, if we now
read this equation
00:06:35.090 --> 00:06:36.420
from right to left--
00:06:36.420 --> 00:06:40.000
observe that if you start with
the function '1 over the
00:06:40.000 --> 00:06:41.970
'square root of '1 plus
'x squared''''--
00:06:41.970 --> 00:06:45.790
which is hardly a hyperbolic
function, this is a fairly
00:06:45.790 --> 00:06:48.070
straightforward algebraic
function--
00:06:48.070 --> 00:06:52.000
notice that the inverse
derivative leads to an inverse
00:06:52.000 --> 00:06:53.260
hyperbolic sine.
00:06:53.260 --> 00:06:57.370
In other words, stated from a
different perspective and
00:06:57.370 --> 00:07:00.500
using our language of the
indefinite integral, notice
00:07:00.500 --> 00:07:04.320
that what we have here is that
the indefinite integral 'dx'
00:07:04.320 --> 00:07:07.760
over the 'square root of '1 plus
'x squared''' is inverse
00:07:07.760 --> 00:07:10.860
hyperbolic sine 'x'
plus a constant.
00:07:10.860 --> 00:07:13.970
Now what does this mean,
say, geometrically?
00:07:13.970 --> 00:07:17.270
Suppose we take the curve 'y'
equals '1 over the 'square
00:07:17.270 --> 00:07:19.060
root of '1 plus 'x squared'''.
00:07:19.060 --> 00:07:21.590
Without beating this thing to
death, it should be fairly
00:07:21.590 --> 00:07:26.240
straightforward at this stage of
the game that the graph of
00:07:26.240 --> 00:07:28.510
this function can be obtained,
and it looks
00:07:28.510 --> 00:07:29.810
something like this.
00:07:29.810 --> 00:07:33.590
In fact, intuitively notice that
'y' will be maximum when
00:07:33.590 --> 00:07:35.330
my denominator is smallest.
00:07:35.330 --> 00:07:38.000
My denominator is smallest
when 'x' is zero.
00:07:38.000 --> 00:07:43.190
So the maximum value of 'y'
occurs when 'x' is 0, at which
00:07:43.190 --> 00:07:44.740
case, 'y' is 1.
00:07:44.740 --> 00:07:47.340
Also, if I replace 'x'
by minus 'x', I
00:07:47.340 --> 00:07:49.710
don't change the function.
00:07:49.710 --> 00:07:51.270
And therefore the graph--
00:07:51.270 --> 00:07:52.520
it's an even function--
00:07:52.520 --> 00:07:54.180
the graph is symmetric
with respect to
00:07:54.180 --> 00:07:55.910
the y-axis, et cetera.
00:07:55.910 --> 00:07:58.300
At any rate, I have a
picture like this.
00:07:58.300 --> 00:08:02.330
And now suppose I want to find
the area of the region 'R',
00:08:02.330 --> 00:08:05.970
where 'R' is bounded above by
this curve, below by the
00:08:05.970 --> 00:08:10.230
x-axis, on the left by the
y-axis, and on the right by
00:08:10.230 --> 00:08:11.970
the line 'x' equals 't'.
00:08:11.970 --> 00:08:15.810
The area of the region 'R',
which is a function of 't', is
00:08:15.810 --> 00:08:17.290
given by what?
00:08:17.290 --> 00:08:21.710
The definite integral from 0 to
't', 'dx' over the 'square
00:08:21.710 --> 00:08:23.940
root of '1 plus 'x squared'''.
00:08:23.940 --> 00:08:27.730
The point is I could, as we
talked about in the previous
00:08:27.730 --> 00:08:32.100
block, try to evaluate this as
the limit of a sum-- in other
00:08:32.100 --> 00:08:33.590
words, an infinite sum--
00:08:33.590 --> 00:08:36.620
and go through all sorts of work
to try to do this thing.
00:08:36.620 --> 00:08:39.799
But the first fundamental
theorem tells us in this
00:08:39.799 --> 00:08:43.490
particular case that this
particular area just turns out
00:08:43.490 --> 00:08:47.110
to be the inverse hyperbolic
sine of 't'.
00:08:47.110 --> 00:08:51.170
Notice that a non-trigonometric
region has
00:08:51.170 --> 00:08:55.060
as its answer an inverse
hyperbolic
00:08:55.060 --> 00:08:56.380
trigonometric function.
00:08:56.380 --> 00:08:58.520
Or, if you want this thing
more specifically, for
00:08:58.520 --> 00:09:02.010
example, notice that if you want
the area of this region
00:09:02.010 --> 00:09:07.096
from 0 to 1, the answer to this
problem would have just
00:09:07.096 --> 00:09:10.470
been the inverse hyperbolic
sine of 1.
00:09:10.470 --> 00:09:12.750
In other words, 'e to the
1' minus 'e to the
00:09:12.750 --> 00:09:14.870
minus 1' over 2.
00:09:14.870 --> 00:09:18.490
Notice how 'e' sneaks into
a problem which basically
00:09:18.490 --> 00:09:21.240
doesn't seem to have any
relationship to 'e'.
00:09:21.240 --> 00:09:23.590
By the way-- as a very
brief aside--
00:09:23.590 --> 00:09:26.930
for what it's worth, it's rather
interesting to observe
00:09:26.930 --> 00:09:31.830
that this last equation that
we've written down gives a
00:09:31.830 --> 00:09:37.150
rather interesting definition of
the inverse hyperbolic sine
00:09:37.150 --> 00:09:40.420
without having to refer to
a hyperbola or anything
00:09:40.420 --> 00:09:41.420
trigonometric.
00:09:41.420 --> 00:09:44.320
In other words, notice that the
inverse hyperbolic sine
00:09:44.320 --> 00:09:48.540
can be defined as an integral,
which is what we've really
00:09:48.540 --> 00:09:49.330
done over here.
00:09:49.330 --> 00:09:50.990
But again, that's
just an aside.
00:09:50.990 --> 00:09:54.440
The main point that I wanted us
to get a hold of over here
00:09:54.440 --> 00:09:58.850
was the fact that you solve
non-hyperbolic functions
00:09:58.850 --> 00:10:01.140
conveniently if we
have mastered
00:10:01.140 --> 00:10:02.830
the hyperbolic functions.
00:10:02.830 --> 00:10:05.380
Well, at any rate, here's
another interesting question
00:10:05.380 --> 00:10:06.270
that comes up.
00:10:06.270 --> 00:10:08.770
And I thought that we should
mention this, also.
00:10:08.770 --> 00:10:12.420
Notice that we arrived at
this result by doing
00:10:12.420 --> 00:10:13.510
the thing in reverse.
00:10:13.510 --> 00:10:16.750
You'll recall that we started
with 'y' equals inverse
00:10:16.750 --> 00:10:19.840
hyperbolic sine 'x' and show
the derivative of that
00:10:19.840 --> 00:10:22.930
function was '1 over the 'square
root of '1 plus 'x
00:10:22.930 --> 00:10:23.920
squared'''.
00:10:23.920 --> 00:10:26.340
An interesting question might
have been, what if we had
00:10:26.340 --> 00:10:31.910
started with the integral being
given and we hadn't have
00:10:31.910 --> 00:10:34.890
differentiated the inverse
hyperbolic sine?
00:10:34.890 --> 00:10:37.910
How could we have got
from here to the
00:10:37.910 --> 00:10:39.630
inverse hyperbolic sine?
00:10:39.630 --> 00:10:41.940
And I thought I would mention
this because there may be some
00:10:41.940 --> 00:10:45.370
confusion, especially if you've
taken to heart certain
00:10:45.370 --> 00:10:49.290
advice that I gave you earlier
when we dealt with the inverse
00:10:49.290 --> 00:10:51.070
circular functions.
00:10:51.070 --> 00:10:53.490
Remember I told you that
whenever you see something
00:10:53.490 --> 00:10:57.810
like the sum of two squares, to
think of a right triangle?
00:10:57.810 --> 00:11:01.750
In other words, if you have the
square root of '1 plus 'x
00:11:01.750 --> 00:11:04.920
squared'', it seems to me that
the triangle that suggests
00:11:04.920 --> 00:11:06.610
itself is something like this.
00:11:06.610 --> 00:11:10.010
In other words, if I call this
side 'x', I call this side 1,
00:11:10.010 --> 00:11:12.740
and this the square root of '1
plus 'x squared'', it would
00:11:12.740 --> 00:11:15.800
seem to me that I could make
a circular trigonometric
00:11:15.800 --> 00:11:17.160
substitution over here.
00:11:17.160 --> 00:11:20.510
In fact, what seems to dictate
itself over here, is to make
00:11:20.510 --> 00:11:24.750
the substitution, let
tan theta equal 'x'.
00:11:24.750 --> 00:11:28.420
Now if I let tan theta equal
'x', watch what happens here.
00:11:28.420 --> 00:11:32.080
I get 'secant squared theta 'd
theta'' equals 'dx', taking
00:11:32.080 --> 00:11:33.850
the differential
of both sides.
00:11:33.850 --> 00:11:37.020
I also get, looking at my
reference triangle, that the
00:11:37.020 --> 00:11:41.120
square root of '1 plus 'x
squared'' is secant theta.
00:11:41.120 --> 00:11:44.910
See, this over this
is secant theta.
00:11:44.910 --> 00:11:48.830
At any rate, then, making the
substitution in here,
00:11:48.830 --> 00:11:52.800
replacing 'dx' by 'secant
squared theta 'd theta'', and
00:11:52.800 --> 00:11:56.350
the square root of '1 plus 'x
squared'' by secant theta, I
00:11:56.350 --> 00:11:59.970
wind up with integral of 'secant
theta 'd theta''.
00:11:59.970 --> 00:12:02.970
Now you see what I've done here,
is I have successfully
00:12:02.970 --> 00:12:06.130
transformed an integral
in terms of 'x' into
00:12:06.130 --> 00:12:07.640
one in terms of theta.
00:12:07.640 --> 00:12:11.190
But without belaboring this
point, it turns out that at
00:12:11.190 --> 00:12:16.270
this stage of the game, we do
not know how to exhibit a
00:12:16.270 --> 00:12:18.480
function whose derivative
with respect to
00:12:18.480 --> 00:12:20.430
theta is secant theta.
00:12:20.430 --> 00:12:23.150
In other words, we've made the
substitution but we wind up
00:12:23.150 --> 00:12:26.570
with an integral that's just as
tough to handle as the one
00:12:26.570 --> 00:12:27.580
that we started with.
00:12:27.580 --> 00:12:30.580
You see, in this case, trying
to make a circular
00:12:30.580 --> 00:12:32.590
trigonometric substitution
wouldn't have
00:12:32.590 --> 00:12:34.200
helped us very much.
00:12:34.200 --> 00:12:36.400
What I'd like to show you is,
again, an interesting
00:12:36.400 --> 00:12:39.270
connection between the circular
functions and the
00:12:39.270 --> 00:12:40.740
hyperbolic functions.
00:12:40.740 --> 00:12:45.380
Namely, when we did this
particular thing over here
00:12:45.380 --> 00:12:48.350
using our reference triangle,
what was the reference
00:12:48.350 --> 00:12:51.100
triangle really taking
the place of?
00:12:51.100 --> 00:12:55.190
Notice that if we let 'x' equal
tan theta, certainly '1
00:12:55.190 --> 00:12:58.910
plus 'x squared'' is 1 plus
tan squared theta.
00:12:58.910 --> 00:13:02.740
But there is a trigonometric
identity that says 1 plus tan
00:13:02.740 --> 00:13:05.390
squared theta is secant
squared theta.
00:13:05.390 --> 00:13:08.400
I think the usual way that's
given is that secant squared
00:13:08.400 --> 00:13:11.810
theta minus tan squared
theta is 1.
00:13:11.810 --> 00:13:13.290
This is the result
that we used.
00:13:13.290 --> 00:13:16.090
We didn't really use the
triangle other than to get
00:13:16.090 --> 00:13:18.380
this result more visually.
00:13:18.380 --> 00:13:22.860
The point is, is there a
hyperbolic function that has
00:13:22.860 --> 00:13:24.370
the same format?
00:13:24.370 --> 00:13:27.500
Is there a hyperbolic identity
which says that the difference
00:13:27.500 --> 00:13:29.340
of two squares is one?
00:13:29.340 --> 00:13:33.260
The answer is, well remember our
basic hyperbolic identity
00:13:33.260 --> 00:13:38.280
is the cosh squared theta minus
sinh squared theta is 1.
00:13:38.280 --> 00:13:43.130
Structurally, notice that sinh
theta does for the hyperbolic
00:13:43.130 --> 00:13:47.620
functions what tan theta does
for the circular functions.
00:13:47.620 --> 00:13:49.150
See the common structure,
here?
00:13:49.150 --> 00:13:52.380
We're going to reinforce this
in the next block by doing
00:13:52.380 --> 00:13:53.520
problems like this again.
00:13:53.520 --> 00:13:56.060
But for the time being, I
thought I would like to point
00:13:56.060 --> 00:13:57.920
this thing out to you.
00:13:57.920 --> 00:14:01.060
What the approach is, is that
when you try a circular
00:14:01.060 --> 00:14:04.460
function substitution and it
doesn't give you the answer
00:14:04.460 --> 00:14:07.490
that you want-- meaning that you
wind up with an integral
00:14:07.490 --> 00:14:09.930
that's just as tough to handle
as the original one--
00:14:09.930 --> 00:14:13.820
you look for the corresponding
hyperbolic function.
00:14:13.820 --> 00:14:17.940
What hyperbolic function plays
to the hyperbolic identity the
00:14:17.940 --> 00:14:20.900
same role that this
trigonometric function play to
00:14:20.900 --> 00:14:22.400
the circular identity?
00:14:22.400 --> 00:14:26.770
In this case, we replace tan
theta by sinh theta.
00:14:26.770 --> 00:14:29.010
Instead of making the
substitution 'x' equals
00:14:29.010 --> 00:14:32.130
tangent theta, we make
the substitution
00:14:32.130 --> 00:14:34.300
'x' equals sinh theta.
00:14:34.300 --> 00:14:37.150
And now watch what happens as
we work this thing quite
00:14:37.150 --> 00:14:38.200
mechanically.
00:14:38.200 --> 00:14:42.520
The differential of sinh theta
is 'cosh theta 'd theta''.
00:14:42.520 --> 00:14:45.780
And the square root of '1 plus
'x squared'' is the square
00:14:45.780 --> 00:14:48.440
root of 1 plus sin
squared theta.
00:14:48.440 --> 00:14:51.010
But notice that because
of the relationship
00:14:51.010 --> 00:14:52.470
between sinh and cosh--
00:14:52.470 --> 00:14:55.540
that's how we rig this thing,
that's why we chose 'x' to be
00:14:55.540 --> 00:14:56.510
sinh theta--
00:14:56.510 --> 00:15:00.260
notice that the square root of
1 plus sinh squared theta is
00:15:00.260 --> 00:15:02.510
just cosh theta.
00:15:02.510 --> 00:15:06.410
And therefore, when we now
substitute for the 'dx' over
00:15:06.410 --> 00:15:09.790
the square root of '1 plus 'x
squared'', we get what?
00:15:09.790 --> 00:15:12.830
For 'dx' we have 'cosh
theta 'd theta''.
00:15:12.830 --> 00:15:14.970
For the square root of
'1 plus 'x squared''
00:15:14.970 --> 00:15:16.460
we have cosh theta.
00:15:16.460 --> 00:15:20.240
'Cosh theta 'd theta'' over cosh
theta is just 'd theta'.
00:15:20.240 --> 00:15:24.350
And now we see the answer is,
quite simply, theta plus 'c'.
00:15:24.350 --> 00:15:26.180
But what was theta?
00:15:26.180 --> 00:15:30.970
Since sinh theta was 'x',
theta was inverse
00:15:30.970 --> 00:15:33.220
hyperbolic sine 'x'.
00:15:33.220 --> 00:15:37.080
And, you see, this is a
technique whereby starting
00:15:37.080 --> 00:15:42.000
with the integral 'dx' over the
'square root of '1 plus 'x
00:15:42.000 --> 00:15:45.010
squared''', we can show that
we must have started with
00:15:45.010 --> 00:15:46.070
inverse sinh.
00:15:46.070 --> 00:15:48.810
At any rate, this will be
reinforced in homework
00:15:48.810 --> 00:15:51.400
problems, it will be reinforced
in the next block
00:15:51.400 --> 00:15:53.410
when we talk about techniques
of integration.
00:15:53.410 --> 00:15:57.050
But I just wanted to again
show the similarity, the
00:15:57.050 --> 00:16:00.790
things in common, between the
hyperbolic functions and the
00:16:00.790 --> 00:16:04.000
circular functions and how
they're intertwined.
00:16:04.000 --> 00:16:07.150
Let's make a few more comments
while we're at it.
00:16:07.150 --> 00:16:10.600
You know, we mentioned that the
hyperbolic functions were
00:16:10.600 --> 00:16:13.040
really combinations of
exponential functions.
00:16:13.040 --> 00:16:16.690
Remember, 'cosh x' was ''e to
the x' plus 'e to the minus
00:16:16.690 --> 00:16:19.010
x'' over 2, et cetera.
00:16:19.010 --> 00:16:23.390
So somehow or other, if the
hyperbolic functions can be
00:16:23.390 --> 00:16:26.800
expressed in terms of
exponentials, it would seem
00:16:26.800 --> 00:16:29.460
that the inverse hyperbolic
functions should be
00:16:29.460 --> 00:16:33.570
expressible in terms of the
inverse of exponentials--
00:16:33.570 --> 00:16:36.590
namely, in terms
of logarithms.
00:16:36.590 --> 00:16:39.430
And so I thought that I would
try to go through some of
00:16:39.430 --> 00:16:41.010
these finer points with you.
00:16:41.010 --> 00:16:43.890
And, for example, ask the
following question.
00:16:43.890 --> 00:16:49.460
Given that 'y' equals inverse
'sinh x', is there a way of
00:16:49.460 --> 00:16:54.200
writing this in terms of
something that uses our
00:16:54.200 --> 00:16:57.360
natural logarithms?
00:16:57.360 --> 00:16:58.450
Another reason being, what?
00:16:58.450 --> 00:17:01.990
That if we've already learned
natural logs and exponentials,
00:17:01.990 --> 00:17:06.490
it would seem that whenever we
can reduce unfamiliar names to
00:17:06.490 --> 00:17:09.510
more familiar ones,
psychologically we feel much
00:17:09.510 --> 00:17:12.069
more at home in dealing
with the concepts.
00:17:12.069 --> 00:17:15.010
In other words, one might feel
strange working with inverse
00:17:15.010 --> 00:17:18.030
hyperbolic sine because he
hasn't seen that very much.
00:17:18.030 --> 00:17:20.730
But if he's used to seeing
logarithms, that wouldn't seem
00:17:20.730 --> 00:17:21.560
quite as strange.
00:17:21.560 --> 00:17:25.260
At any rate, let's see how
one could proceed here.
00:17:25.260 --> 00:17:28.790
Starting out with 'y' equals
inverse 'sinh x', notice that
00:17:28.790 --> 00:17:31.100
by the property, the basic
definition of inverse
00:17:31.100 --> 00:17:35.020
functions, I can now write
that 'x' equals 'sinh y'.
00:17:35.020 --> 00:17:37.420
Now, for obvious reasons,
since I want to get the
00:17:37.420 --> 00:17:40.440
inverse of exponentials in here,
it would seem to me that
00:17:40.440 --> 00:17:43.710
I should express 'sinh y' in
terms of exponentials.
00:17:43.710 --> 00:17:48.510
And going again back to basic
definitions, 'sinh y' is ''e
00:17:48.510 --> 00:17:52.080
to the y' minus 'e to the
minus y'' over 2.
00:17:52.080 --> 00:17:55.430
In other words, in terms of
exponentials, 'x' is equal to
00:17:55.430 --> 00:17:59.570
''e to the y' minus 'e to
the minus y'' over 2.
00:17:59.570 --> 00:18:04.420
If I now cross multiply, I get
that '2x' is equal to 'e to
00:18:04.420 --> 00:18:06.360
the y' minus--
00:18:06.360 --> 00:18:09.770
now notice that 'e to the minus
y' is just '1 over 'e to
00:18:09.770 --> 00:18:13.840
the y'', so I wind up now with
this particular equation.
00:18:13.840 --> 00:18:17.300
And multiplying through by 'e to
the y', to clear fractions
00:18:17.300 --> 00:18:20.190
in my denominator, to clear
my denominators, I
00:18:20.190 --> 00:18:21.570
wind up with what?
00:18:21.570 --> 00:18:28.950
'e to the 2y' minus ''2x 'e
to the y'' - 1' equals 0.
00:18:28.950 --> 00:18:33.590
And if I now recall that 'e to
the 2y' is the square of 'e to
00:18:33.590 --> 00:18:37.810
the y', observe that what I
now have is a quadratic
00:18:37.810 --> 00:18:40.300
equation in 'e to the y'.
00:18:40.300 --> 00:18:43.880
I have a quadratic equation
in 'e to the y'.
00:18:43.880 --> 00:18:47.210
Now, since I have a quadratic
equation in 'e to the y', I
00:18:47.210 --> 00:18:51.710
can use the quadratic formula
to solve for 'e to the y'.
00:18:51.710 --> 00:18:53.640
If I do this I get what?
00:18:53.640 --> 00:18:54.500
Remember how this thing works.
00:18:54.500 --> 00:18:59.230
I take the coefficient of this
term minus that, that's '2x'
00:18:59.230 --> 00:19:05.220
plus or minus the square root of
this squared minus 4 times
00:19:05.220 --> 00:19:09.050
this times the coefficient
of 'e to the y' squared.
00:19:09.050 --> 00:19:12.910
Leaving the details as being
fairly obvious, 'e to the y'
00:19:12.910 --> 00:19:17.840
is '2x' plus or minus the square
root of ''4x' squared'
00:19:17.840 --> 00:19:23.180
plus 4 all over 2.
00:19:23.180 --> 00:19:26.190
And noticing now that the 4 can
be factored out here as a
00:19:26.190 --> 00:19:29.980
2, and that I can cancel a 2,
then from both the numerator
00:19:29.980 --> 00:19:35.680
and denominator, I wind up with
'e to the y' is 'x' plus
00:19:35.680 --> 00:19:40.350
or minus the square root
of 'x squared' plus 1.
00:19:40.350 --> 00:19:43.960
The point to keep in mind, now,
is remember that in terms
00:19:43.960 --> 00:19:47.850
of exponentials, 'e to the
y' can never be negative.
00:19:47.850 --> 00:19:51.960
Observe that the square root
of 'x squared plus 1' is
00:19:51.960 --> 00:19:56.020
bigger than 'x' in magnitude,
you see.
00:19:56.020 --> 00:19:57.620
See, 'x' would be just
the positive
00:19:57.620 --> 00:19:59.010
square root of 'x squared'.
00:19:59.010 --> 00:20:02.470
So the positive square root of
'x squared plus 1' is bigger
00:20:02.470 --> 00:20:04.450
than 'x' in magnitude.
00:20:04.450 --> 00:20:07.770
Consequently, if I use the minus
sign here, I'd be taking
00:20:07.770 --> 00:20:10.000
away more than what I had.
00:20:10.000 --> 00:20:12.490
That would make my answer
negative, which would be a
00:20:12.490 --> 00:20:16.200
contradiction, since 'e to
the y' can't be negative.
00:20:16.200 --> 00:20:20.110
Again, in terms of this
particular problem, the minus
00:20:20.110 --> 00:20:23.870
root, the minus sign
here is extraneous.
00:20:23.870 --> 00:20:25.920
And we therefore wind
up with what?
00:20:25.920 --> 00:20:29.050
'e to the y' is 'x' plus
the square root of 'x
00:20:29.050 --> 00:20:30.290
squared plus 1'.
00:20:30.290 --> 00:20:34.870
Therefore 'y' itself is the
'log of x' plus the square
00:20:34.870 --> 00:20:38.030
root of 'x squared plus 1' to
the base 'e', which we've
00:20:38.030 --> 00:20:42.890
already seen is called
the natural log.
00:20:42.890 --> 00:20:46.860
Going back now, say, from the
first step to the last, I
00:20:46.860 --> 00:20:52.040
guess we can now fill in what's
really happened here.
00:20:52.040 --> 00:20:56.930
A synonym for the inverse 'sinh
x' is the natural log of
00:20:56.930 --> 00:21:01.020
'x' plus the square root
of 'x squared plus 1'.
00:21:01.020 --> 00:21:05.980
So notice that we can study the
inverse sinh, for example,
00:21:05.980 --> 00:21:09.040
in terms of a suitably
chosen natural log.
00:21:09.040 --> 00:21:11.260
And of course there are many
more examples that we could
00:21:11.260 --> 00:21:12.670
use along these lines.
00:21:12.670 --> 00:21:17.390
But again, I think that with
the previous explanation,
00:21:17.390 --> 00:21:21.150
coupled with the fact that there
will be ample exercises
00:21:21.150 --> 00:21:23.790
in the like, I think the message
has become clear as
00:21:23.790 --> 00:21:26.410
far as the inverse hyperbolic
sine is concerned.
00:21:26.410 --> 00:21:29.580
What I would like to do now is
to turn to another facet of
00:21:29.580 --> 00:21:33.050
inverse functions, something
that involves principal values
00:21:33.050 --> 00:21:35.100
the same as it did with the
circular functions.
00:21:35.100 --> 00:21:39.320
We wind up with the same problem
as before when we come
00:21:39.320 --> 00:21:42.780
to the idea that, technically
speaking, you cannot talk
00:21:42.780 --> 00:21:45.620
about an inverse function unless
the original function
00:21:45.620 --> 00:21:47.140
is one-to-one.
00:21:47.140 --> 00:21:49.700
And so therefore, when one
talks about the inverse
00:21:49.700 --> 00:21:54.300
hyperbolic cosine, one is in a
way looking for trouble if one
00:21:54.300 --> 00:21:56.370
doesn't keep his eye
on exactly what's
00:21:56.370 --> 00:21:57.690
going on around here.
00:21:57.690 --> 00:22:00.950
Namely, if we look at the graph
'y' equals hyperbolic
00:22:00.950 --> 00:22:05.750
cosine 'x', observe that whereas
the function is single
00:22:05.750 --> 00:22:08.310
valued, it is not one-to-one.
00:22:08.310 --> 00:22:12.420
In fact, there is a zero
derivative at this point here,
00:22:12.420 --> 00:22:15.720
which leads us to believe that
maybe what we should have done
00:22:15.720 --> 00:22:20.280
was to have broken this curve
down into the union of two
00:22:20.280 --> 00:22:21.830
one-to-one functions.
00:22:21.830 --> 00:22:26.510
Let me call this curve 'y'
equals 'c1 of x' and let me
00:22:26.510 --> 00:22:31.430
call this branch here 'y'
equals 'c2 of x'.
00:22:31.430 --> 00:22:35.300
Notice that both 'c1
of x' and 'c2 of x'
00:22:35.300 --> 00:22:38.050
are one-to-one functions.
00:22:38.050 --> 00:22:40.960
In fact, let's write this
more formally using the
00:22:40.960 --> 00:22:43.140
picture as a guide.
00:22:43.140 --> 00:22:45.180
Let's do the following
analytically.
00:22:45.180 --> 00:22:46.230
Let's say this.
00:22:46.230 --> 00:22:51.690
Define 'c sub 1 of x' to be
'cosh x', provided that 'x' is
00:22:51.690 --> 00:22:53.130
at least as big as 0.
00:22:53.130 --> 00:22:55.770
Again, I mentioned this with the
circular functions, let me
00:22:55.770 --> 00:22:57.290
reinforce this again.
00:22:57.290 --> 00:23:00.860
To define a function, it's not
enough to tell the rule.
00:23:00.860 --> 00:23:03.410
You must also tell the domain.
00:23:03.410 --> 00:23:07.410
Notice that 'c1' is not the
same as cosh, because the
00:23:07.410 --> 00:23:10.010
domain of cosh is all
real numbers.
00:23:10.010 --> 00:23:14.370
The domain of 'c1' is just
the non-negative reals.
00:23:14.370 --> 00:23:18.710
At any rate, I define 'c1' to
be 'cosh x', where 'x' is at
00:23:18.710 --> 00:23:20.420
least as big as 0.
00:23:20.420 --> 00:23:24.710
I define 'c2 of x' to be
'cosh x', where 'x' is
00:23:24.710 --> 00:23:26.460
no bigger than 0.
00:23:26.460 --> 00:23:28.530
In other words, these two
functions are different,
00:23:28.530 --> 00:23:31.510
because even though the
functional relations are the
00:23:31.510 --> 00:23:33.960
same, the domains
are different.
00:23:33.960 --> 00:23:35.610
The interesting point is what?
00:23:35.610 --> 00:23:40.710
That 'cosh x' is the union
of 'c1' and 'c2'.
00:23:40.710 --> 00:23:45.050
But the important point is that
both 'c1' and 'c2' are
00:23:45.050 --> 00:23:46.910
one-to-one.
00:23:46.910 --> 00:23:48.540
And because they
are one-to-one,
00:23:48.540 --> 00:23:49.780
their inverses exist.
00:23:49.780 --> 00:23:53.710
In other words, I can talk
meaningfully about 'c1
00:23:53.710 --> 00:23:56.060
inverse' and 'c2 inverse'.
00:23:56.060 --> 00:24:02.090
In fact, pictorially,
what I have is this.
00:24:02.090 --> 00:24:08.530
See, if I take the curve 'y'
equals 'cosh x' and reflect it
00:24:08.530 --> 00:24:11.680
about the 45 degree line, this
is the curve that I get.
00:24:11.680 --> 00:24:14.080
You see, it's a double
value curve.
00:24:14.080 --> 00:24:19.100
All I'm saying is if we look at
'y' equals 'c1x', which is
00:24:19.100 --> 00:24:23.500
a one-to-one function, its
inverse is 'c1 inverse x',
00:24:23.500 --> 00:24:25.630
which is this piece over here.
00:24:25.630 --> 00:24:30.730
And if, on the other hand, we
look at 'y' equals 'c2x',
00:24:30.730 --> 00:24:34.000
that's this branch over here,
its inverse is this.
00:24:34.000 --> 00:24:37.400
You see, notice that these two
pieces are symmetric with
00:24:37.400 --> 00:24:40.540
respect to the line 'y' equals
'x', and these two pieces are
00:24:40.540 --> 00:24:43.530
symmetric with respect to
the line 'y' equals 'x'.
00:24:43.530 --> 00:24:47.200
As long as we break this down
into the union of two pieces,
00:24:47.200 --> 00:24:49.710
we can talk about inverse
functions.
00:24:49.710 --> 00:24:52.740
Now you see, the interesting
point is that what most
00:24:52.740 --> 00:24:57.790
authors traditionally refer to
as the inverse hyperbolic
00:24:57.790 --> 00:25:03.210
cosine of 'x' is really what
we call 'c1 inverse of x'.
00:25:03.210 --> 00:25:07.610
In other words, the definition
'y' equals inverse hyperbolic
00:25:07.610 --> 00:25:11.750
cosine 'x' is 'x'
equals cosh 'y'.
00:25:11.750 --> 00:25:15.330
And this is very important,
and 'y' is at
00:25:15.330 --> 00:25:18.000
least as big as 0.
00:25:18.000 --> 00:25:23.750
Notice that the domain of 'cosh
inverse x' is really 'x'
00:25:23.750 --> 00:25:26.110
has to be at least
as big as one.
00:25:26.110 --> 00:25:28.920
But that's not the important
point here.
00:25:28.920 --> 00:25:32.140
What I do want to see over here
is that when you put this
00:25:32.140 --> 00:25:35.660
restriction on, instead if you
left this restriction out,
00:25:35.660 --> 00:25:38.350
there would be no inverse
function here.
00:25:38.350 --> 00:25:39.950
I'll come back to that
in a moment.
00:25:39.950 --> 00:25:42.770
Let me just reinforce what we've
talked about before, and
00:25:42.770 --> 00:25:45.420
let's find the derivative
of 'inverse cosh x'.
00:25:45.420 --> 00:25:48.640
In other words, let's find
'dy/dx', if 'y' equals
00:25:48.640 --> 00:25:49.910
'inverse cosh x'.
00:25:49.910 --> 00:25:53.000
Well again, what is the
definition, 'y' equals
00:25:53.000 --> 00:25:54.330
'inverse cosh x'?
00:25:54.330 --> 00:26:00.230
It means 'x' equals cosh y',
where 'y' is positive.
00:26:00.230 --> 00:26:01.350
OK.
00:26:01.350 --> 00:26:05.200
If 'x' equals 'cosh y',
'dx/dy' is 'sinh y'.
00:26:05.200 --> 00:26:07.450
And we'll keep track of the
fact that 'y' is positive.
00:26:07.450 --> 00:26:09.820
Actually, 'y' is non-negative.
00:26:09.820 --> 00:26:14.970
Therefore the reciprocal of
'dx/dy' will be 'dy/dx'.
00:26:14.970 --> 00:26:19.370
In other words, 'dy/dx'
is '1 over sinh y'.
00:26:19.370 --> 00:26:23.010
And this would be a correct
answer, except, as usual, we
00:26:23.010 --> 00:26:26.360
would like to be able to express
'dy/dx' for a given
00:26:26.360 --> 00:26:27.350
value of 'x'.
00:26:27.350 --> 00:26:31.520
What we do now is, remembering
that 'x' is 'cosh y', we
00:26:31.520 --> 00:26:33.410
invoke the identity again.
00:26:33.410 --> 00:26:37.500
'Cosh squared y' minus 'sinh
squared y' is one.
00:26:37.500 --> 00:26:41.530
From which we can solve and find
that 'sinh y' is plus or
00:26:41.530 --> 00:26:44.550
minus the 'square root of
'x squared minus 1'.
00:26:44.550 --> 00:26:47.680
And by the way, I'm not
going to remove the
00:26:47.680 --> 00:26:49.270
extraneous sign here.
00:26:49.270 --> 00:26:52.130
Because in a certain manner
of speaking, it is only
00:26:52.130 --> 00:26:55.810
extraneous because we are
imposing the condition that
00:26:55.810 --> 00:26:59.340
'y' is positive.
00:26:59.340 --> 00:27:03.070
See, in other words, once we
assume that y' is positive--
00:27:03.070 --> 00:27:06.190
remember that 'sinh y' is
positive for positive values
00:27:06.190 --> 00:27:09.220
of 'y', and negative for
negative values of 'y'--
00:27:09.220 --> 00:27:12.850
consequently, the assumption
that 'y' is positive forces us
00:27:12.850 --> 00:27:15.520
to accept the fact that
'sinh y' is positive.
00:27:15.520 --> 00:27:19.030
And that's what forces us, in
terms of the restriction that
00:27:19.030 --> 00:27:23.530
we imposed the fact that y has
to be at least as big as 0,
00:27:23.530 --> 00:27:26.260
why we can get rid of
the minus sign here.
00:27:26.260 --> 00:27:31.300
And so we wind up with what?
'Sinh y' is positive 'square
00:27:31.300 --> 00:27:34.320
root of 'cosh squared
y' minus 1''.
00:27:34.320 --> 00:27:37.000
But 'cosh y' is 'x'
in this problem.
00:27:37.000 --> 00:27:40.130
In other words, 'sinh y', in
this problem, is the positive
00:27:40.130 --> 00:27:42.360
square root of 'x squared
minus 1'.
00:27:42.360 --> 00:27:44.560
By the way, don't
be nervous here.
00:27:44.560 --> 00:27:46.580
You might say, couldn't
this be imaginary?
00:27:46.580 --> 00:27:49.370
In other words, what happens if
'x squared' is less than 1?
00:27:49.370 --> 00:27:52.190
Remember, 'x' is at
least as big as 1.
00:27:52.190 --> 00:27:55.740
So this thing here in
the square root sign
00:27:55.740 --> 00:27:56.870
can never be negative.
00:27:56.870 --> 00:27:59.960
But at any rate, what we now
wind up with is that the
00:27:59.960 --> 00:28:04.120
derivative of 'inverse cosh x',
with respect to 'x', is '1
00:28:04.120 --> 00:28:07.520
over the 'square root of ''x
squared' minus 1'''.
00:28:07.520 --> 00:28:10.810
Now again, there's no law that
says that a person couldn't
00:28:10.810 --> 00:28:13.830
have been on the negative
branch of this curve.
00:28:13.830 --> 00:28:17.200
In other words, if all you mean
by inverse cosh is the
00:28:17.200 --> 00:28:21.070
inverse of cosh with no
restriction to branch, what
00:28:21.070 --> 00:28:22.590
we've really proven is this.
00:28:22.590 --> 00:28:24.720
And let me summarize on
this particular point.
00:28:24.720 --> 00:28:27.130
What we've really
proven is this.
00:28:27.130 --> 00:28:32.160
That the derivative of 'c1'
inverse is '1 over the 'square
00:28:32.160 --> 00:28:34.280
root of ''x squared'
minus 1'''.
00:28:34.280 --> 00:28:39.430
The derivative of 'c2' inverse
is '1 over minus the 'square
00:28:39.430 --> 00:28:41.360
root of ''x squared'
minus 1'''.
00:28:41.360 --> 00:28:44.090
I've taken the liberty of
putting the minus down with
00:28:44.090 --> 00:28:47.370
the square root sign, rather
than with the fraction itself,
00:28:47.370 --> 00:28:50.370
to emphasize the fact that
which of the two signs we
00:28:50.370 --> 00:28:53.860
choose depends on whether we're
looking at the branch
00:28:53.860 --> 00:28:57.030
for which 'y' is above the
x-axis, or the branch which
00:28:57.030 --> 00:28:58.830
'y' is below the x-axis.
00:28:58.830 --> 00:29:02.280
In other words, what we don't
want to happen here is for
00:29:02.280 --> 00:29:05.420
people to lose track of the fact
that all we have done is
00:29:05.420 --> 00:29:08.430
made a convention so we can
talk about one-to-one
00:29:08.430 --> 00:29:11.320
functions and inverse functions
more meaningfully.
00:29:11.320 --> 00:29:14.150
But you can be on either of
these particular branches.
00:29:14.150 --> 00:29:19.160
In any event, this does complete
our discussion of the
00:29:19.160 --> 00:29:21.010
hyperbolic functions.
00:29:21.010 --> 00:29:24.510
And we will now turn
our attention to
00:29:24.510 --> 00:29:26.240
utilizing these results.
00:29:26.240 --> 00:29:28.610
We will learn some techniques of
integration, and the like.
00:29:28.610 --> 00:29:31.150
But at any rate, until
next time, goodbye.
00:29:33.820 --> 00:29:36.840
MALE SPEAKER: Funding for the
publication of this video was
00:29:36.840 --> 00:29:41.560
provided by the Gabriella and
Paul Rosenbaum Foundation.
00:29:41.560 --> 00:29:45.730
Help OCW continue to provide
free and open access to MIT
00:29:45.730 --> 00:29:49.930
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