WEBVTT
00:00:00.000 --> 00:00:07.420
PROFESSOR: Hi.
00:00:07.420 --> 00:00:08.970
Welcome back to recitation.
00:00:08.970 --> 00:00:12.680
I have here a little bit of
a strange problem for you.
00:00:12.680 --> 00:00:15.640
So let me just tell it to
you, and then I'll give you
00:00:15.640 --> 00:00:16.640
some time to work on it.
00:00:16.640 --> 00:00:19.010
So I want to define
a function, g of x,
00:00:19.010 --> 00:00:21.390
and I want to
define it piecewise.
00:00:21.390 --> 00:00:26.020
So when x is positive, I just
want g of x to be 1 over x.
00:00:26.020 --> 00:00:31.320
But when x is negative, I want
g of x to be 1 of x plus 2.
00:00:31.320 --> 00:00:33.510
So I've got a little graph
here of the function.
00:00:33.510 --> 00:00:35.610
So you've got, you know,
when x is positive,
00:00:35.610 --> 00:00:37.800
it's just your usual
y equals 1 over x.
00:00:37.800 --> 00:00:40.969
But when x is negative, I've
taken, I've shifted it up by 2.
00:00:40.969 --> 00:00:42.510
So this is a perfectly
good function.
00:00:42.510 --> 00:00:43.627
It's not defined at 0.
00:00:43.627 --> 00:00:45.130
OK?
00:00:45.130 --> 00:00:46.840
So what I would
like you to do is
00:00:46.840 --> 00:00:49.010
to compute the derivative
of this function, wherever
00:00:49.010 --> 00:00:50.550
it's defined.
00:00:50.550 --> 00:00:54.331
And you'll notice, when you get
there, that you'll have some..
00:00:54.331 --> 00:00:58.440
you'll get some answer, and
maybe you'll notice something
00:00:58.440 --> 00:01:00.730
a little weird
about that answer.
00:01:00.730 --> 00:01:03.670
So if you notice something weird
about it, what I want you to do
00:01:03.670 --> 00:01:05.434
is try and explain
why this is true.
00:01:05.434 --> 00:01:07.100
And if you don't
notice something weird,
00:01:07.100 --> 00:01:11.000
then, you know, come back and
we'll talk about it together.
00:01:11.000 --> 00:01:14.350
So why don't you pause the
video, go do that computation,
00:01:14.350 --> 00:01:17.400
and think about what, if there's
something strange going on
00:01:17.400 --> 00:01:18.245
here.
00:01:18.245 --> 00:01:20.453
And then come back and we
can talk about it together.
00:01:28.510 --> 00:01:29.180
Welcome back.
00:01:29.180 --> 00:01:31.284
Hopefully you had some fun
working on this problem
00:01:31.284 --> 00:01:32.200
and thinking about it.
00:01:32.200 --> 00:01:34.100
So let's do the
first part, which
00:01:34.100 --> 00:01:36.020
is just the computational part.
00:01:36.020 --> 00:01:39.002
Let's have a go at it.
00:01:39.002 --> 00:01:40.960
So, because this function
is defined piecewise,
00:01:40.960 --> 00:01:43.630
when we compute a derivative, we
can just compute the derivative
00:01:43.630 --> 00:01:44.630
on the different pieces.
00:01:44.630 --> 00:01:46.860
So the function
isn't defined at 0,
00:01:46.860 --> 00:01:49.260
so of course, it doesn't
have a derivative at 0.
00:01:49.260 --> 00:01:51.990
But then we can compute a
derivative when x is positive,
00:01:51.990 --> 00:01:54.580
and we can compute a
derivative when x is negative.
00:01:57.320 --> 00:02:04.020
So when x is bigger than
0, g prime of x, well,
00:02:04.020 --> 00:02:10.020
that's just d over
dx of 1 over x.
00:02:10.020 --> 00:02:11.730
So that's something
we're familiar with.
00:02:11.730 --> 00:02:14.355
Its minus 1 over x squared.
00:02:14.355 --> 00:02:15.875
So that's for x positive.
00:02:19.870 --> 00:02:28.530
When x is less than 0, g
prime of x is d over dx of 1
00:02:28.530 --> 00:02:32.760
over x plus 2, because
that's what g of x is.
00:02:32.760 --> 00:02:35.360
And, OK, and so this is,
well, the plus 2 gets killed,
00:02:35.360 --> 00:02:37.910
and so then we have the
derivative of 1 over x.
00:02:37.910 --> 00:02:41.350
That's minus 1 over x squared.
00:02:41.350 --> 00:02:43.704
So one thing you've noticed
is that this is minus 1
00:02:43.704 --> 00:02:46.120
over x squared here, and it's
minus 1 over x squared here.
00:02:46.120 --> 00:02:48.620
So although we defined
this piecewise,
00:02:48.620 --> 00:02:50.900
we could, we can
summarize this by saying,
00:02:50.900 --> 00:02:58.360
so the derivative is minus
1 over x squared always,
00:02:58.360 --> 00:03:02.639
so for all x not
equal to-- you know,
00:03:02.639 --> 00:03:04.430
it doesn't have a
derivative at x equals 0.
00:03:04.430 --> 00:03:07.360
It's not defined at 0, it
can't have a derivative there.
00:03:07.360 --> 00:03:11.720
So, but we don't need the
piecewise definition, anymore.
00:03:11.720 --> 00:03:14.660
So that was kind of
interesting, that we
00:03:14.660 --> 00:03:18.100
can summarize the derivative
of this piecewise function
00:03:18.100 --> 00:03:19.685
in a non-piecewise way.
00:03:24.610 --> 00:03:28.810
Now, the thing is, we've
learned what the anti-derivative
00:03:28.810 --> 00:03:30.240
of this function is.
00:03:30.240 --> 00:03:40.820
So we know that the
anti-derivative of minus 1
00:03:40.820 --> 00:03:48.290
over x squared dx is 1
over x plus a constant.
00:03:48.290 --> 00:03:51.710
So we know that the functions
whose derivative is minus 1
00:03:51.710 --> 00:03:55.740
over x squared are of the
form, 1 over x plus a constant.
00:03:55.740 --> 00:03:59.480
The thing is, this function
g that we just talked about,
00:03:59.480 --> 00:04:01.125
this function g
isn't of that form.
00:04:01.125 --> 00:04:02.020
Right?
00:04:02.020 --> 00:04:05.370
You don't get this function
by taking the function
00:04:05.370 --> 00:04:08.010
1 over x and just
shifting it up or down.
00:04:08.010 --> 00:04:09.825
You-- something weird happens.
00:04:09.825 --> 00:04:11.350
You've shifted it
up on one piece
00:04:11.350 --> 00:04:12.970
and not on the other piece.
00:04:12.970 --> 00:04:16.815
And yet, it's still true
that the derivative of g
00:04:16.815 --> 00:04:19.170
is equal to minus
1 over x squared.
00:04:19.170 --> 00:04:21.749
So this is a little bit
of a head-scratcher.
00:04:21.749 --> 00:04:25.264
And I wanted to talk
about why this happens.
00:04:25.264 --> 00:04:27.680
And the thing is that there's
a sort of theoretical reason
00:04:27.680 --> 00:04:29.510
for this, which is
that you remember
00:04:29.510 --> 00:04:32.900
that the reason that we know
that anti-derivatives have
00:04:32.900 --> 00:04:36.130
this form, a function
plus a constant,
00:04:36.130 --> 00:04:38.280
is because we know
that constants
00:04:38.280 --> 00:04:40.640
are the functions
with derivative 0.
00:04:40.640 --> 00:04:45.860
And so we were able to apply
the mean value theorem in order
00:04:45.860 --> 00:04:49.540
to show that if two functions
have the same derivative, then
00:04:49.540 --> 00:04:52.480
they differ by each other,
differ from each other
00:04:52.480 --> 00:04:53.500
by a constant.
00:04:53.500 --> 00:04:55.560
If two functions have
the same derivative,
00:04:55.560 --> 00:04:57.870
they differ by a constant.
00:04:57.870 --> 00:05:01.870
And we used, as a really
crucial step in that proof,
00:05:01.870 --> 00:05:03.744
the mean value theorem.
00:05:03.744 --> 00:05:05.410
Now the thing is, the
mean value theorem
00:05:05.410 --> 00:05:09.090
has, as one of its assumptions,
as one of its hypotheses,
00:05:09.090 --> 00:05:10.940
that the functions that
you're working with
00:05:10.940 --> 00:05:12.669
are continuous
and differentiable
00:05:12.669 --> 00:05:13.377
in some interval.
00:05:13.377 --> 00:05:15.620
OK?
00:05:15.620 --> 00:05:18.830
So what's happened here is
that the functions that we're
00:05:18.830 --> 00:05:23.260
talking about, the function 1
over x and the function minus 1
00:05:23.260 --> 00:05:25.570
over x squared,
those functions are
00:05:25.570 --> 00:05:27.985
continuous and differentiable
on certain intervals.
00:05:27.985 --> 00:05:29.860
So if we look-- if we
go back to this picture
00:05:29.860 --> 00:05:35.370
here we see that this function
g of x, just like the function 1
00:05:35.370 --> 00:05:39.570
over x, it's continuous and
differentiable for positive x,
00:05:39.570 --> 00:05:42.040
it's continuous and
differentiable for negative x,
00:05:42.040 --> 00:05:44.960
but at 0, there's
a discontinuity.
00:05:44.960 --> 00:05:46.890
So there's no
interval that crosses
00:05:46.890 --> 00:05:51.260
0 on which this function is
continuous or differentiable.
00:05:51.260 --> 00:05:53.610
As a result, the
mean value theorem
00:05:53.610 --> 00:05:58.080
can't tell us anything about
intervals that cross 0.
00:05:58.080 --> 00:06:00.010
So if the mean value
theorem doesn't tell us
00:06:00.010 --> 00:06:03.300
anything in that case, it
means the conclusion isn't true
00:06:03.300 --> 00:06:05.710
and we get a situation-- sorry.
00:06:05.710 --> 00:06:07.070
I should rephrase that.
00:06:07.070 --> 00:06:10.120
It means the conclusion
doesn't have to be true.
00:06:10.120 --> 00:06:15.150
Our proof doesn't work in a case
where we have a discontinuity.
00:06:15.150 --> 00:06:17.270
And what happens,
in fact, is right
00:06:17.270 --> 00:06:20.850
what we have here, which is that
when you have a function that
00:06:20.850 --> 00:06:25.000
has a discontinuity and you look
at its anti-derivatives, what
00:06:25.000 --> 00:06:28.410
you can do is that, in addition
to shifting the whole thing up
00:06:28.410 --> 00:06:30.300
and down, you can
shift the pieces
00:06:30.300 --> 00:06:33.660
on either side of the
discontinuity separately.
00:06:33.660 --> 00:06:36.870
Just like in this case
we can shift the piece
00:06:36.870 --> 00:06:39.560
to the left of 0 separately
from the piece to the right of 0
00:06:39.560 --> 00:06:41.580
and get a function whose
derivative is still
00:06:41.580 --> 00:06:42.690
what we started with.
00:06:42.690 --> 00:06:46.650
So this function g of x, we
get by shifting part of 1
00:06:46.650 --> 00:06:49.730
over x up, and it gives us a
function whose derivative is
00:06:49.730 --> 00:06:53.020
still minus 1 over x squared.
00:06:53.020 --> 00:06:54.860
So this is true anytime
you have a function
00:06:54.860 --> 00:06:55.735
with a discontinuity.
00:06:55.735 --> 00:06:59.930
So one consequence of this--
I'm going to go back over here
00:06:59.930 --> 00:07:08.260
and just write down one special
case of this-- is that to say,
00:07:08.260 --> 00:07:12.710
we say that the
anti-derivative of 1 over x dx
00:07:12.710 --> 00:07:19.010
is equal to ln of the
absolute value of x plus c.
00:07:19.010 --> 00:07:23.900
What this really means is
that when x is positive,
00:07:23.900 --> 00:07:27.360
we have a single kind
of anti-derivative,
00:07:27.360 --> 00:07:30.470
and they're of the form,
ln x plus a constant.
00:07:30.470 --> 00:07:33.220
And when x is negative, we
have a single anti-derivative,
00:07:33.220 --> 00:07:36.800
that's-- or single family
of anti-derivatives,
00:07:36.800 --> 00:07:39.360
of the form ln of
minus x-- remember,
00:07:39.360 --> 00:07:42.740
absolutely value of x is minus
x when x is negative-- plus c.
00:07:42.740 --> 00:07:44.880
But if we consider x to
be positive and negative
00:07:44.880 --> 00:07:47.140
at the same time,
the two constants
00:07:47.140 --> 00:07:48.494
don't necessarily have to agree.
00:07:48.494 --> 00:07:50.160
You can have the same
situation that you
00:07:50.160 --> 00:07:53.760
had before where one side can
shift up and down independently
00:07:53.760 --> 00:07:57.490
of the other, because there's
that discontinuity at 0 there.
00:07:57.490 --> 00:07:59.710
So this is just something
to keep in mind.
00:07:59.710 --> 00:08:01.390
It also means you
have to be careful
00:08:01.390 --> 00:08:02.820
with certain substitutions.
00:08:02.820 --> 00:08:04.500
You don't want to
do substitutions
00:08:04.500 --> 00:08:06.290
that have discontinuities.
00:08:06.290 --> 00:08:08.630
If you do substitutions
that have discontinuities,
00:08:08.630 --> 00:08:12.150
you might accidentally
introduce a discontinuity
00:08:12.150 --> 00:08:16.020
and bad things can happen
that I won't go into now.
00:08:16.020 --> 00:08:18.490
You can make-- end
up with statements
00:08:18.490 --> 00:08:22.310
that don't make any sense by
making a substitution where
00:08:22.310 --> 00:08:24.130
the function that
you're substituting
00:08:24.130 --> 00:08:25.769
has a discontinuity in it.
00:08:25.769 --> 00:08:27.560
So you-- or another
way of saying it is you
00:08:27.560 --> 00:08:32.430
have to restrict to some
interval on which it really
00:08:32.430 --> 00:08:33.630
is continuous.
00:08:33.630 --> 00:08:35.890
And then on each of those
intervals it makes sense,
00:08:35.890 --> 00:08:37.620
but bad things could
happen when you
00:08:37.620 --> 00:08:39.850
cross those discontinuities.
00:08:39.850 --> 00:08:41.620
So this is a little
bit theoretical,
00:08:41.620 --> 00:08:43.650
but I think it's a nice
thing to be aware of,
00:08:43.650 --> 00:08:47.110
a nice thing to keep in
mind when you're working
00:08:47.110 --> 00:08:49.460
with some of these expressions.
00:08:49.460 --> 00:08:50.964
So I'll end there.