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HERBERT GROSS: Hi, our lecture
today is about curve plotting.
00:00:34.770 --> 00:00:38.710
And actually, I call it 'Curve
Plotting with and without
00:00:38.710 --> 00:00:41.600
Calculus' to emphasize the
fact that what we're
00:00:41.600 --> 00:00:45.340
interested in is curve plotting
and that calculus, in
00:00:45.340 --> 00:00:49.460
particular, differentiation,
gives us a powerful tool that
00:00:49.460 --> 00:00:52.960
is not available to us, at least
in an accessible form
00:00:52.960 --> 00:00:54.820
without the calculus.
00:00:54.820 --> 00:00:58.550
Let's see what I mean by this
by going back to a typical
00:00:58.550 --> 00:01:01.660
high school type analytical
geometry problem.
00:01:01.660 --> 00:01:06.980
For example, sketch the curve
'y' equals 'x squared'.
00:01:06.980 --> 00:01:10.380
Now we all know that the graph
of 'y' equals 'x squared' is
00:01:10.380 --> 00:01:12.080
something like this.
00:01:12.080 --> 00:01:13.660
And how'd we find that?
00:01:13.660 --> 00:01:16.480
You may recall that in the truer
sense of the word, the
00:01:16.480 --> 00:01:21.420
pre-calculus approach really is
curve plotting as opposed
00:01:21.420 --> 00:01:22.635
to curve sketching.
00:01:22.635 --> 00:01:23.950
And I hope I make that
a little bit
00:01:23.950 --> 00:01:25.660
clearer as we go along.
00:01:25.660 --> 00:01:29.740
Namely, you look to see for
certain values of 'x' what
00:01:29.740 --> 00:01:33.700
value of 'y' corresponds
to that.
00:01:33.700 --> 00:01:36.960
And then we locate the
corresponding point 'x' comma
00:01:36.960 --> 00:01:40.320
'y' in the plane of
the blackboard.
00:01:40.320 --> 00:01:43.930
And what we then do is somehow
or other, take a French curve,
00:01:43.930 --> 00:01:47.780
or whatever it happens to be,
and we sketch a smooth curve
00:01:47.780 --> 00:01:49.310
through the given points.
00:01:49.310 --> 00:01:51.450
This is the usual technique.
00:01:51.450 --> 00:01:54.210
The question that comes up is,
that as long as there are
00:01:54.210 --> 00:01:57.350
spaces between the points that
you've sketched, how can you
00:01:57.350 --> 00:02:01.890
be sure that the curve that
you've drawn isn't inaccurate?
00:02:01.890 --> 00:02:05.120
In other words, starting in
reverse here, let's suppose
00:02:05.120 --> 00:02:08.080
that these are the points I've
happened to sketch for the
00:02:08.080 --> 00:02:09.800
curve 'y' equals 'x squared'.
00:02:09.800 --> 00:02:12.910
And by the way, as is often the
case in our course, don't
00:02:12.910 --> 00:02:15.210
be misled that we pick something
as simple as 'y'
00:02:15.210 --> 00:02:16.400
equals 'x squared'.
00:02:16.400 --> 00:02:18.890
I simply, again, wanted to
pick something that was
00:02:18.890 --> 00:02:22.350
straightforward enough so that
we could concentrate our
00:02:22.350 --> 00:02:26.680
attention on what the
mathematical implications were
00:02:26.680 --> 00:02:29.430
rather than to be bogged
down by computation.
00:02:29.430 --> 00:02:30.510
But the idea is this.
00:02:30.510 --> 00:02:33.340
Going back to the problem 'y'
equals 'x squared', suppose we
00:02:33.340 --> 00:02:36.490
had located these points and
now we said, let's sketch a
00:02:36.490 --> 00:02:37.930
smooth curve through these.
00:02:37.930 --> 00:02:41.350
What would have been wrong
with say, doing
00:02:41.350 --> 00:02:42.600
something like this?
00:02:45.080 --> 00:02:49.260
Why, for example, couldn't
this have been the curve?
00:02:49.260 --> 00:02:54.290
Now again, notice that in
terms of mathematical
00:02:54.290 --> 00:02:57.220
analysis, not necessarily
calculus but in terms of
00:02:57.220 --> 00:03:00.210
mathematical analysis, we could
immediately strike out
00:03:00.210 --> 00:03:01.410
something like this.
00:03:01.410 --> 00:03:05.590
For example, notice in terms
of our input versus output,
00:03:05.590 --> 00:03:08.360
this says that for any real
input since the square of a
00:03:08.360 --> 00:03:11.240
real number cannot be negative,
the output can never
00:03:11.240 --> 00:03:12.120
be negative.
00:03:12.120 --> 00:03:15.910
And pictorially, this means that
no portion of our diagram
00:03:15.910 --> 00:03:18.350
can be below the x-axis.
00:03:18.350 --> 00:03:20.980
In other words, without
knowing anything about
00:03:20.980 --> 00:03:23.940
calculus but knowing a little
bit about arithmetic, we can
00:03:23.940 --> 00:03:26.790
supplement our knowledge
of how the points go by
00:03:26.790 --> 00:03:27.640
saying look-it .
00:03:27.640 --> 00:03:30.870
Something like this can't happen
because in this region
00:03:30.870 --> 00:03:33.310
over here 'y' would
be negative.
00:03:33.310 --> 00:03:35.430
And we know from the
relationship that 'y' equals
00:03:35.430 --> 00:03:38.060
'x squared' that 'y'
cannot be negative.
00:03:38.060 --> 00:03:42.820
Well you see, armed with this
information, we could say, OK,
00:03:42.820 --> 00:03:45.750
given the same points here,
why couldn't the curve go
00:03:45.750 --> 00:03:48.780
through something like this?
00:03:48.780 --> 00:03:50.240
That's what the question
mark is in here for.
00:03:50.240 --> 00:03:52.530
Obviously, this is
not the graph of
00:03:52.530 --> 00:03:53.470
'y' equals 'x squared'.
00:03:53.470 --> 00:03:57.280
But the question is, noticing
that this curve doesn't go
00:03:57.280 --> 00:04:00.440
below the x-axis, what's
wrong with this?
00:04:00.440 --> 00:04:04.710
And again, we can get bailed
out by a pre-calculus
00:04:04.710 --> 00:04:06.710
knowledge of mathematics.
00:04:06.710 --> 00:04:09.980
Among other things, notice the
rather interesting property
00:04:09.980 --> 00:04:13.710
that if we replace 'x' by 'minus
x' here, we get the
00:04:13.710 --> 00:04:15.890
same curve as before.
00:04:15.890 --> 00:04:19.790
To generalize this result,
notice that in this particular
00:04:19.790 --> 00:04:23.580
case, 'f of x' gives us
the same result as
00:04:23.580 --> 00:04:25.470
'f of 'minus x''.
00:04:25.470 --> 00:04:27.330
Now this will not happen
in general.
00:04:27.330 --> 00:04:30.760
What does that mean if we know
that 'f of x' equals 'f of
00:04:30.760 --> 00:04:31.560
'minus x''?
00:04:31.560 --> 00:04:33.370
Well, look it.
00:04:33.370 --> 00:04:36.950
The relationship between 'x' and
'minus x' is clear, they
00:04:36.950 --> 00:04:42.580
are located symmetrically with
respect to the y-axis.
00:04:42.580 --> 00:04:46.555
In other words, if this is 'x1',
this will be 'minus x1'.
00:04:46.555 --> 00:04:48.250
Now, here's the point.
00:04:48.250 --> 00:04:50.140
Let's suppose our curve
happens to be 'y'
00:04:50.140 --> 00:04:51.190
equals 'f of x'.
00:04:51.190 --> 00:04:55.660
When the input is 'x1', the
output will be 'f of x1'.
00:04:55.660 --> 00:04:59.390
When the input is 'minus x1',
the output will be 'f of
00:04:59.390 --> 00:05:00.750
'minus x1''.
00:05:00.750 --> 00:05:05.560
To say that 'f of 'minus x1''
equals 'f of x1' is the same
00:05:05.560 --> 00:05:10.710
as saying that not only are
these two coordinates, 'x1'
00:05:10.710 --> 00:05:14.490
and 'minus x1', symmetric with
respect to the y-axis, but the
00:05:14.490 --> 00:05:16.010
outputs also are.
00:05:16.010 --> 00:05:18.830
In other words, to say that 'f
of x1' equals 'f of 'minus
00:05:18.830 --> 00:05:25.140
x1'', says that not only are
these two points the same left
00:05:25.140 --> 00:05:28.370
and right displacement for the
y-axis, but they're also the
00:05:28.370 --> 00:05:30.380
same height above the x-axis.
00:05:30.380 --> 00:05:34.340
In short, this point is the
mirror image of this point.
00:05:34.340 --> 00:05:38.090
In terms of a graph, if 'f of x'
equals 'f of 'minus x'' for
00:05:38.090 --> 00:05:41.990
all 'x', the graph of 'y' equals
'f of x' is symmetric
00:05:41.990 --> 00:05:44.220
with respect to the y-axis.
00:05:44.220 --> 00:05:49.110
By the way, that's why we could
rule out this type of
00:05:49.110 --> 00:05:50.330
possibility here.
00:05:50.330 --> 00:05:54.760
For example, here's
'x1' over here.
00:05:54.760 --> 00:05:56.240
Here's 'minus x1'.
00:05:56.240 --> 00:06:00.350
And notice that for this choice
of 'x', the points on
00:06:00.350 --> 00:06:03.730
the curve are not mirror images
of one another with
00:06:03.730 --> 00:06:05.380
respect to the y-axis.
00:06:05.380 --> 00:06:07.460
In short, even though we may
not know what this curve is
00:06:07.460 --> 00:06:10.840
supposed to look like, the fact
that 'f of x' equals 'f
00:06:10.840 --> 00:06:14.180
of 'minus x'' tells us that
whatever the graph is, it
00:06:14.180 --> 00:06:16.670
should be symmetric with
respect to the y-axis.
00:06:16.670 --> 00:06:19.860
This leads us to a rather
interesting aside.
00:06:19.860 --> 00:06:22.720
It's something called 'even
and odd functions' that we
00:06:22.720 --> 00:06:25.410
could technically leave out
here, but which I think is a
00:06:25.410 --> 00:06:26.690
good place to bring in.
00:06:26.690 --> 00:06:29.660
And the fact that there are
many, many places in our
00:06:29.660 --> 00:06:32.060
advanced treatment that will
come up later where it's
00:06:32.060 --> 00:06:35.060
important to understand what
these things mean, that
00:06:35.060 --> 00:06:39.300
perhaps this is a nice place
to bring it into play.
00:06:39.300 --> 00:06:41.080
So I call this an aside
and it's about
00:06:41.080 --> 00:06:43.440
even and odd functions.
00:06:43.440 --> 00:06:47.350
When we have the case that 'f of
x' equals 'f of 'minus x'',
00:06:47.350 --> 00:06:50.040
that's the case where in terms
of the graph you have symmetry
00:06:50.040 --> 00:06:53.880
with respect to the y-axis such
a function is called an
00:06:53.880 --> 00:06:55.370
'even function'.
00:06:55.370 --> 00:06:57.710
And I'll come back in a moment
and show you why it's called
00:06:57.710 --> 00:07:00.740
even, though it's not
really important.
00:07:00.740 --> 00:07:03.520
The counterpart to an even
function as you may guess
00:07:03.520 --> 00:07:05.870
almost from the association
of ideas is
00:07:05.870 --> 00:07:08.290
called an 'odd function'.
00:07:08.290 --> 00:07:13.280
Now an odd function is one for
which 'f of x' is the negative
00:07:13.280 --> 00:07:15.200
of 'f of 'minus x''.
00:07:15.200 --> 00:07:19.310
See for 'f' to be odd, 'f of x'
has to be the negative of
00:07:19.310 --> 00:07:20.760
'f of 'minus x''.
00:07:20.760 --> 00:07:22.720
Now what does this
mean pictorially?
00:07:22.720 --> 00:07:27.860
Again, 'x1' and 'minus x1' are
symmetrically located with
00:07:27.860 --> 00:07:32.920
respect to the y-axis.
00:07:32.920 --> 00:07:37.610
This height would be called 'f
of x1' and this height here
00:07:37.610 --> 00:07:42.000
would be called 'f
of 'minus x1''.
00:07:42.000 --> 00:07:47.340
And to say that these two
heights are equal in magnitude
00:07:47.340 --> 00:07:49.530
but opposite in sign
means what?
00:07:49.530 --> 00:07:53.130
That these two lengths are equal
but on opposite sides of
00:07:53.130 --> 00:07:54.050
the x-axis.
00:07:54.050 --> 00:07:57.960
In other words, here's
'f of x1'.
00:07:57.960 --> 00:07:59.770
'f of 'minus x1'' is the height
00:07:59.770 --> 00:08:01.330
corresponding to this point.
00:08:01.330 --> 00:08:04.720
And the fact that these must
be equal in magnitude but
00:08:04.720 --> 00:08:08.370
opposite in sign says that one
of the heights must be above
00:08:08.370 --> 00:08:11.630
the axis, the other one must
be below the axis.
00:08:11.630 --> 00:08:15.530
And if that's the case from some
very elementary geometry,
00:08:15.530 --> 00:08:18.360
if 'f' is odd, what it
means with respect to
00:08:18.360 --> 00:08:19.780
its graph is this.
00:08:19.780 --> 00:08:25.420
If you put a ruler connecting
the origin with any point on
00:08:25.420 --> 00:08:31.820
the curve, if you extend that
line that line will meet the
00:08:31.820 --> 00:08:35.919
curve again, such that these
distances here will be equal.
00:08:35.919 --> 00:08:38.830
This is called symmetry with
respect to the origin.
00:08:38.830 --> 00:08:41.750
By the way, an example of
this kind of a curve is
00:08:41.750 --> 00:08:43.480
'y' equals 'x cubed'.
00:08:43.480 --> 00:08:48.360
You see, if I replace 'x' by
'minus x', this becomes what?
00:08:48.360 --> 00:08:50.550
'Minus x cubed'.
00:08:50.550 --> 00:08:55.490
And 'minus 'minus x cubed''
is 'x cubed'.
00:08:55.490 --> 00:08:57.940
By the way, all I'm saying
now is what?
00:08:57.940 --> 00:09:02.060
If I were to take a ruler and
place it here and let this
00:09:02.060 --> 00:09:05.110
line go from curve to curve.
00:09:05.110 --> 00:09:07.370
all I'm saying is that
this portion
00:09:07.370 --> 00:09:08.890
would equal this portion.
00:09:08.890 --> 00:09:12.300
And by the way, maybe you can
already guess where the words
00:09:12.300 --> 00:09:14.410
even and odd come from.
00:09:14.410 --> 00:09:17.420
Notice that when I dealt with
'y' equals 'x squared' I had
00:09:17.420 --> 00:09:18.990
an even function.
00:09:18.990 --> 00:09:20.470
The exponent was even.
00:09:20.470 --> 00:09:23.170
When I dealt with 'y'
equals 'x cubed',
00:09:23.170 --> 00:09:24.740
I had an odd function.
00:09:24.740 --> 00:09:26.230
The exponent was odd.
00:09:26.230 --> 00:09:28.140
And by the way, there
are other examples.
00:09:28.140 --> 00:09:32.710
But for the time being, notice
something like say, 'y' equals
00:09:32.710 --> 00:09:35.420
'x to the fourth' plus
'x squared' say.
00:09:46.030 --> 00:09:49.100
If I replace 'x' by 'minus x',
I get back the same thing.
00:09:49.100 --> 00:09:52.190
In other words, here's a case
where if 'y' equals 'f of x',
00:09:52.190 --> 00:09:56.150
'f of x' is the same as
'f of 'minus x''.
00:09:56.150 --> 00:09:59.980
An example of an odd function
might be 'y' equals 'x cubed'
00:09:59.980 --> 00:10:02.220
say, plus 'x'.
00:10:02.220 --> 00:10:03.720
See first power over here.
00:10:03.720 --> 00:10:08.080
If I replace 'x' by 'minus
x', I have 'minus x
00:10:08.080 --> 00:10:10.840
cubed' plus 'minus x'.
00:10:10.840 --> 00:10:14.120
That of course, is just 'minus
x cubed' minus 'x'.
00:10:14.120 --> 00:10:18.040
And that's minus the quantity
'x cubed + x'.
00:10:18.040 --> 00:10:21.240
That when I replace 'x' by
'minus x' all I do is change
00:10:21.240 --> 00:10:22.300
the sign here.
00:10:22.300 --> 00:10:26.070
In other words, that would be an
example for which 'f of x'
00:10:26.070 --> 00:10:30.350
is minus 'f of 'minus x''.
00:10:30.350 --> 00:10:32.670
By the way, while we're dealing
with examples like
00:10:32.670 --> 00:10:36.120
this, unlike the case with whole
numbers where a whole
00:10:36.120 --> 00:10:41.000
number is either even or odd
but not both, it's rather
00:10:41.000 --> 00:10:45.590
important to notice that a
function need not be either
00:10:45.590 --> 00:10:47.550
even or odd.
00:10:47.550 --> 00:10:50.220
And by the way, if you think of
our geometric definition,
00:10:50.220 --> 00:10:52.560
that's not too hard to see.
00:10:52.560 --> 00:10:55.370
Namely, there's no reason why a
curve drawn at random should
00:10:55.370 --> 00:10:58.430
be symmetric either with respect
to the y-axis or with
00:10:58.430 --> 00:10:59.390
respect to the origin.
00:10:59.390 --> 00:11:03.940
In fact, maybe the quickest way
to see this is to put an
00:11:03.940 --> 00:11:07.060
odd power of 'x' in with
an even power of
00:11:07.060 --> 00:11:08.580
'x' in the same diagram.
00:11:08.580 --> 00:11:14.420
If I now replace 'x' by 'minus
x' you see 'minus x' cubed is
00:11:14.420 --> 00:11:15.990
of course, 'minus x cubed'.
00:11:15.990 --> 00:11:19.670
But 'minus x' quantity squared
is just 'x squared'.
00:11:19.670 --> 00:11:22.750
And now you'll notice that if
I compare these two, I don't
00:11:22.750 --> 00:11:25.040
get the same thing nor
do I get the same
00:11:25.040 --> 00:11:26.850
thing with a sign change.
00:11:26.850 --> 00:11:30.020
You see, in other words, this is
not either equal to this or
00:11:30.020 --> 00:11:32.020
to the negative of this.
00:11:32.020 --> 00:11:34.310
Obviously you understand that
to be the negative of this,
00:11:34.310 --> 00:11:37.770
there would also have to be
a minus sign over here.
00:11:37.770 --> 00:11:43.400
Or if we wanted to go into more
detail about this, it is
00:11:43.400 --> 00:11:46.390
perhaps exciting to know that
whereas it's not true that
00:11:46.390 --> 00:11:50.480
every function is either even or
odd, every function can be
00:11:50.480 --> 00:11:53.880
written as the sum of two
functions, one of which is
00:11:53.880 --> 00:11:56.750
even, one of which is odd.
00:11:56.750 --> 00:12:00.800
Just to give you an idea of what
that means let me just
00:12:00.800 --> 00:12:02.180
write down something here.
00:12:02.180 --> 00:12:05.220
This will come back to be more
important later on, but for
00:12:05.220 --> 00:12:09.060
the time being, just to show you
a connecting thread here
00:12:09.060 --> 00:12:11.970
as long as we're talking about
even and odd functions, all I
00:12:11.970 --> 00:12:14.480
want to see is the following
identity.
00:12:14.480 --> 00:12:20.400
If you write ''f of x' plus
'f of 'minus x'' over 2.
00:12:20.400 --> 00:12:22.840
And don't worry about what
motivates this, I just wanted
00:12:22.840 --> 00:12:25.820
to show you this to keep
this fairly complete.
00:12:25.820 --> 00:12:31.000
Now suppose I add on to that ''f
of x' minus 'f of 'minus
00:12:31.000 --> 00:12:33.690
x'' over 2.
00:12:33.690 --> 00:12:35.740
You see, notice that the
expression on the right-hand
00:12:35.740 --> 00:12:38.750
side is just writing 'f
of x' the hard way.
00:12:38.750 --> 00:12:41.720
See, here's half of 'f of x',
here's half of 'f of x'.
00:12:41.720 --> 00:12:43.190
The sum is 'f of x'.
00:12:43.190 --> 00:12:45.470
And here's half of
'f of 'minus x''.
00:12:45.470 --> 00:12:48.250
And I'm subtracting half
of 'f of 'minus x''.
00:12:48.250 --> 00:12:49.490
That drops out.
00:12:49.490 --> 00:12:55.080
The point is that this is always
an even function and
00:12:55.080 --> 00:12:57.550
this is always an
odd function.
00:12:57.550 --> 00:12:59.830
And just to review the
definition so that you see
00:12:59.830 --> 00:13:03.890
what happens here, all I'm
saying is if I replace 'x' by
00:13:03.890 --> 00:13:06.720
'minus x' here, what happens?
00:13:06.720 --> 00:13:11.340
If I replace 'x' by 'minus x'
this becomes 'f of 'minus x''.
00:13:11.340 --> 00:13:15.390
And if I replace 'x' by 'minus
x' here since minus minus is
00:13:15.390 --> 00:13:17.970
plus, this becomes 'f of x'.
00:13:17.970 --> 00:13:20.800
Now notice that when you add
two terms, the sum is
00:13:20.800 --> 00:13:23.140
independent of the order
in which you add them.
00:13:23.140 --> 00:13:27.170
'f of x' plus 'f of 'minus x'',
therefore, is the same as
00:13:27.170 --> 00:13:29.680
'f of 'minus x''
plus 'f of x'.
00:13:29.680 --> 00:13:33.000
In other words, when I replace
'x' by 'minus x' in this
00:13:33.000 --> 00:13:36.970
bracketed function, I do not
change the value of what's in
00:13:36.970 --> 00:13:38.070
the brackets.
00:13:38.070 --> 00:13:43.230
On the other hand, if I
interchange 'x' with 'minus x'
00:13:43.230 --> 00:13:46.740
here, notice that since I'm
subtracting look what happens.
00:13:46.740 --> 00:13:48.790
I replace 'x' by 'minus x'.
00:13:48.790 --> 00:13:50.770
This gives me an 'f
of 'minus x''.
00:13:50.770 --> 00:13:52.810
Now I make the same
replacement here.
00:13:52.810 --> 00:13:54.840
Minus minus is positive.
00:13:54.840 --> 00:13:58.130
But now notice that if I look at
this expression here, I've
00:13:58.130 --> 00:13:59.440
changed the order.
00:13:59.440 --> 00:14:01.960
See 'f of x' minus
'f of 'minus x''.
00:14:01.960 --> 00:14:04.990
Here, 'f of 'minus x''
minus 'f of x'.
00:14:04.990 --> 00:14:08.650
And when you change the order
you change the sign.
00:14:08.650 --> 00:14:11.750
In other words then, all I'm
saying is that when we talk
00:14:11.750 --> 00:14:14.230
about even and odd functions,
they play a very important
00:14:14.230 --> 00:14:19.130
role in calculus and in other
mathematical analysis topics.
00:14:19.130 --> 00:14:22.590
That not every function is
either even or odd, but every
00:14:22.590 --> 00:14:26.340
function that's defined on the
appropriate domain is the sum
00:14:26.340 --> 00:14:29.020
of both an even and
an odd function.
00:14:29.020 --> 00:14:31.730
But in terms of curve plotting,
the main point is
00:14:31.730 --> 00:14:35.120
not so much these extra remarks
as much as what?
00:14:35.120 --> 00:14:38.930
An even function is symmetric
with respect to the y-axis and
00:14:38.930 --> 00:14:42.110
an odd function is symmetric
with respect to the origin.
00:14:42.110 --> 00:14:46.000
At any rate, if we now go back
to our curve 'y' equals 'x
00:14:46.000 --> 00:14:54.290
squared' the fact that 'f of x'
equals 'x squared' is even
00:14:54.290 --> 00:14:57.670
means that whatever our graph
looks like, it must be
00:14:57.670 --> 00:15:00.200
symmetric with respect
to the y-axis.
00:15:00.200 --> 00:15:02.780
Now give or take how I've drawn
this, this should be
00:15:02.780 --> 00:15:05.100
symmetric with respect
to the y-axis.
00:15:05.100 --> 00:15:09.210
If it doesn't look that way,
imagine that it is that way.
00:15:09.210 --> 00:15:10.570
And so the idea is what?
00:15:10.570 --> 00:15:13.650
Well, any knowledge of calculus
whatsoever, what I
00:15:13.650 --> 00:15:16.880
was able to do here is show that
whatever the graph of 'y'
00:15:16.880 --> 00:15:20.960
equals 'x squared' is, it must
never dip below the x-axis.
00:15:20.960 --> 00:15:23.230
And whatever the curve looks
like to the right of the
00:15:23.230 --> 00:15:26.900
y-axis, it must be the mirror
image of that to the left of
00:15:26.900 --> 00:15:28.000
the y-axis.
00:15:28.000 --> 00:15:31.010
Again, this is how much one
can do without calculus.
00:15:31.010 --> 00:15:34.100
And most of you who are
practicing engineers, I'm sure
00:15:34.100 --> 00:15:37.140
not only understand this type
of technique as far as the
00:15:37.140 --> 00:15:40.440
pre-calculus is concerned, but
can probably draw curves much
00:15:40.440 --> 00:15:41.530
better than I can.
00:15:41.530 --> 00:15:44.260
In fact, even if you're not
practicing engineers you can
00:15:44.260 --> 00:15:46.800
probably draw curves much
better than I can.
00:15:46.800 --> 00:15:48.270
But that part is irrelevant.
00:15:48.270 --> 00:15:50.180
What I wanted to show
up to now--
00:15:50.180 --> 00:15:51.930
and this is what's important
to stress--
00:15:51.930 --> 00:15:55.360
is that to get as far as I've
gotten so far, I did not have
00:15:55.360 --> 00:15:58.090
to have any knowledge
of calculus.
00:15:58.090 --> 00:16:02.130
The way calculus comes in, as
I say again, is a supplement
00:16:02.130 --> 00:16:04.000
to our previous techniques.
00:16:04.000 --> 00:16:06.630
For example, let's suppose
we did have the
00:16:06.630 --> 00:16:09.270
curve drawn this way.
00:16:09.270 --> 00:16:13.150
From this, we certainly aren't
contradicting the fact that
00:16:13.150 --> 00:16:15.140
'x' can't be negative.
00:16:15.140 --> 00:16:16.750
We're not contradicting
the fact that
00:16:16.750 --> 00:16:18.110
'f' is an even function.
00:16:18.110 --> 00:16:20.120
But how do we know that this
is the wrong picture?
00:16:20.120 --> 00:16:24.710
Well, given that 'y' equals 'x
squared', we can easily verify
00:16:24.710 --> 00:16:29.400
that 'dy dx' is '2x.' Knowing
that 'dy dx' is '2x', that
00:16:29.400 --> 00:16:32.400
tells us among other things
that 'dy dx' and 'x'
00:16:32.400 --> 00:16:33.740
have the same sign.
00:16:33.740 --> 00:16:37.310
In other words, 'dy dx' is
positive if 'x' is positive,
00:16:37.310 --> 00:16:40.690
'dy dx' is negative if
'x' is negative.
00:16:40.690 --> 00:16:43.760
In terms of geometry,
that means what?
00:16:43.760 --> 00:16:47.020
That the curve is always rising
for positive values of
00:16:47.020 --> 00:16:50.730
'x' and always falling for
negative values of 'x'.
00:16:50.730 --> 00:16:54.060
Well, you see with that as
a hint, I say ah-ha.
00:16:54.060 --> 00:16:55.480
This can't happen.
00:16:55.480 --> 00:16:57.340
Because look what's happening
over here.
00:16:57.340 --> 00:16:58.840
Or for that matter, over here.
00:16:58.840 --> 00:17:03.740
Here the curve is falling for
positive values of 'x'.
00:17:03.740 --> 00:17:06.280
And that contradicts the fact
that the curve must always be
00:17:06.280 --> 00:17:08.240
rising when 'x' is positive.
00:17:08.240 --> 00:17:11.660
In a similar way, we know that
the curve can't be rising when
00:17:11.660 --> 00:17:16.319
'x' is negative, yet get over
here and here too, we've drawn
00:17:16.319 --> 00:17:17.730
the curve to be rising.
00:17:17.730 --> 00:17:22.200
That again is contradicted
by this diagram.
00:17:22.200 --> 00:17:24.099
So you see the knowledge
of the first
00:17:24.099 --> 00:17:26.410
derivative does what?
00:17:26.410 --> 00:17:29.420
It tells us where the curve is
rising or falling, and that
00:17:29.420 --> 00:17:31.590
gives us another way of checking
whether the graph
00:17:31.590 --> 00:17:33.610
we've drawn is accurate
or not.
00:17:33.610 --> 00:17:35.690
By the way, it's not all
quite that simple.
00:17:35.690 --> 00:17:39.570
And notice again, subtlety how
step by step we strengthen our
00:17:39.570 --> 00:17:41.290
procedures each time.
00:17:41.290 --> 00:17:44.690
For example, now knowing that
the curve must always be
00:17:44.690 --> 00:17:48.390
rising when 'x' is positive and
always falling when 'x' is
00:17:48.390 --> 00:17:51.810
negative, how about this
possibility for the graph of
00:17:51.810 --> 00:17:53.770
'y' equals 'x squared'.
00:17:53.770 --> 00:17:56.260
You see this curve is always
fallen for negative values of
00:17:56.260 --> 00:17:59.630
'x', it's always rising for
positive values of 'x'.
00:18:02.380 --> 00:18:04.720
Let's take a look now at what
the second derivative means.
00:18:04.720 --> 00:18:08.470
If 'y' equals 'x squared',
obviously as we saw before,
00:18:08.470 --> 00:18:14.150
'dy dx' is '2x' and the second
derivative of 'y' with respect
00:18:14.150 --> 00:18:16.450
to 'x' is 2.
00:18:16.450 --> 00:18:19.280
And 2 is a constant, which
is always positive.
00:18:19.280 --> 00:18:20.210
This says what?
00:18:20.210 --> 00:18:23.110
That the second derivative
is always positive.
00:18:23.110 --> 00:18:24.740
Now what is the second
derivative?
00:18:24.740 --> 00:18:27.230
The second derivative is the
first derivative of the first
00:18:27.230 --> 00:18:28.270
derivative.
00:18:28.270 --> 00:18:31.780
That means the rate of change
of the rate of change.
00:18:31.780 --> 00:18:34.250
Well, the rate of change of the
rate of change is called
00:18:34.250 --> 00:18:36.850
acceleration.
00:18:36.850 --> 00:18:39.600
So if the rate of change of the
rate of change is positive
00:18:39.600 --> 00:18:42.010
that means that the curve must
be accelerating, or the
00:18:42.010 --> 00:18:43.470
function is accelerating.
00:18:43.470 --> 00:18:45.810
And if it's negative, function
is decelerating.
00:18:45.810 --> 00:18:48.290
What does that mean in
terms of a picture?
00:18:48.290 --> 00:18:50.530
And the author of the
text uses a very
00:18:50.530 --> 00:18:51.980
descriptive phrase here.
00:18:51.980 --> 00:18:56.010
He talks about 'holding water'
and 'spilling water'.
00:18:56.010 --> 00:19:00.040
Notice, for example, here the
curve would tend to collect
00:19:00.040 --> 00:19:03.180
water, whereas here if water
were poured on it, the curve
00:19:03.180 --> 00:19:05.070
would tend to spill water.
00:19:05.070 --> 00:19:08.140
Holding water represents
acceleration.
00:19:08.140 --> 00:19:10.680
You see that not only is the
curve rising here, but it's
00:19:10.680 --> 00:19:13.710
rising at a faster
and faster rate.
00:19:13.710 --> 00:19:15.860
Again, more primitively,
in terms of
00:19:15.860 --> 00:19:17.780
slopes, notice that what?
00:19:17.780 --> 00:19:20.330
Not only is the slope positive,
but as you move
00:19:20.330 --> 00:19:24.350
along this portion of the curve,
the slope increases as
00:19:24.350 --> 00:19:26.020
you move along.
00:19:26.020 --> 00:19:28.900
And what typifies
this portion?
00:19:28.900 --> 00:19:32.500
That even though the slope is
always positive, it decreases
00:19:32.500 --> 00:19:34.010
as you move along the curve.
00:19:34.010 --> 00:19:36.480
In other words, 'holding water'
corresponds to the
00:19:36.480 --> 00:19:39.320
second derivative being
positive, 'spilling water'
00:19:39.320 --> 00:19:42.410
corresponds to the second
derivative being negative.
00:19:42.410 --> 00:19:45.540
Returning then to our original
problem, the fact that this
00:19:45.540 --> 00:19:52.160
thing here is greater than 0
for all 'x' says that this
00:19:52.160 --> 00:19:54.080
curve could never spill water.
00:19:54.080 --> 00:19:57.650
And that rules out this
portion in here.
00:19:57.650 --> 00:19:58.990
In other words, now
we put everything
00:19:58.990 --> 00:20:00.670
together, we know what?
00:20:00.670 --> 00:20:03.660
The curve can never go
below the x-axis.
00:20:03.660 --> 00:20:06.960
It's symmetric with respect
to the y-axis.
00:20:06.960 --> 00:20:09.350
It's always rising when
'x' is positive and
00:20:09.350 --> 00:20:11.700
always holding water.
00:20:11.700 --> 00:20:15.160
Now you see this is what I call
curve sketching versus
00:20:15.160 --> 00:20:16.080
curve plotting.
00:20:16.080 --> 00:20:18.900
With the information that I
have from calculus, I know
00:20:18.900 --> 00:20:22.010
what's going on for each
point, not just for the
00:20:22.010 --> 00:20:25.590
isolated points that I happened
to have plotted in
00:20:25.590 --> 00:20:26.840
the data for.
00:20:28.970 --> 00:20:31.910
See, the calculus fills
in the missing
00:20:31.910 --> 00:20:33.600
data very, very nicely.
00:20:33.600 --> 00:20:38.230
Now you see this does not mean
I'm going to replace this by--
00:20:38.230 --> 00:20:39.980
my previous analysis
by calculus.
00:20:39.980 --> 00:20:42.950
It means I'm going to add
calculus as one of my bags of
00:20:42.950 --> 00:20:43.890
tools here.
00:20:43.890 --> 00:20:47.970
Notice again that there is a
very nice relationship between
00:20:47.970 --> 00:20:49.810
pictures and analysis.
00:20:49.810 --> 00:20:51.450
And I'm not going to
belabor that point.
00:20:51.450 --> 00:20:53.820
All I'm saying is that if
you add to our previous
00:20:53.820 --> 00:20:54.810
identifications--
00:20:54.810 --> 00:20:56.430
what identifications?
00:20:56.430 --> 00:20:58.780
Like increase means rising.
00:20:58.780 --> 00:21:01.060
I mean this, if a derivative
is positive,
00:21:01.060 --> 00:21:02.240
the curve is rising.
00:21:02.240 --> 00:21:04.970
If a derivative is negative,
the curve is falling.
00:21:04.970 --> 00:21:07.450
If the second derivative
is positive, the
00:21:07.450 --> 00:21:08.770
curve is holding water.
00:21:08.770 --> 00:21:10.920
If the second derivative is
negative, the curve is
00:21:10.920 --> 00:21:11.820
spilling water.
00:21:11.820 --> 00:21:15.360
And again, we have ample
exercises and portions of this
00:21:15.360 --> 00:21:17.630
in the reading material
to illustrate the
00:21:17.630 --> 00:21:19.030
computational aspects.
00:21:19.030 --> 00:21:22.090
But again, all I want you to
get from this lecture is
00:21:22.090 --> 00:21:24.740
what's happening here
conceptually.
00:21:24.740 --> 00:21:28.640
Let's look at this, a few
more applications.
00:21:28.640 --> 00:21:31.990
We're going to find in
subsequent lectures as well as
00:21:31.990 --> 00:21:34.280
other portions of the course,
we're going to be interested,
00:21:34.280 --> 00:21:37.810
for example, in things called
stationary points.
00:21:37.810 --> 00:21:40.540
A 'stationary point' is a point
at which the curve is
00:21:40.540 --> 00:21:42.230
neither rising nor falling.
00:21:42.230 --> 00:21:44.830
In other words, if the curve
happens to be smooth, it's
00:21:44.830 --> 00:21:48.520
characterized by the fact that
'dy dx' is 0 it such a point.
00:21:48.520 --> 00:21:51.840
In terms of the language of
functions to say that the
00:21:51.840 --> 00:21:55.100
curve is smooth means that the
function is differentiable and
00:21:55.100 --> 00:21:58.910
what we're saying is to be
stationary at 'x' equals 'x1',
00:21:58.910 --> 00:22:03.460
it must be that 'f prime
of x1' equals 0.
00:22:03.460 --> 00:22:06.370
And the importance of stationary
points can be seen
00:22:06.370 --> 00:22:09.020
in terms of a physical
interpretation.
00:22:09.020 --> 00:22:11.970
We haven't used our freely
falling body for quite a
00:22:11.970 --> 00:22:14.610
while, let's go back
to such an example.
00:22:14.610 --> 00:22:20.750
Suppose a particle is projected
vertically upward in
00:22:20.750 --> 00:22:23.880
the absence of air resistance,
et cetera, with an initial
00:22:23.880 --> 00:22:26.690
speed of 160 feet per second.
00:22:26.690 --> 00:22:30.560
It can then be shown that the
height 's' to which the ball
00:22:30.560 --> 00:22:33.520
rises in feet at time
't' is given by
00:22:33.520 --> 00:22:37.100
'160t - 16't squared''.
00:22:37.100 --> 00:22:43.460
A very natural question to ask
is, when will the ball be at
00:22:43.460 --> 00:22:45.450
its maximum height?
00:22:45.450 --> 00:22:47.360
And I'm sure you can see in
terms of this physical
00:22:47.360 --> 00:22:50.870
example, the ball will be at
its maximum height when the
00:22:50.870 --> 00:22:53.020
velocity is 0.
00:22:53.020 --> 00:22:57.840
In other words, since this is
a smooth type of motion, if
00:22:57.840 --> 00:23:02.910
the velocity is not 0, the
ball is still rising.
00:23:02.910 --> 00:23:06.010
If the velocity is positive,
the ball is still rising.
00:23:06.010 --> 00:23:10.340
If the velocity is negative, the
ball is already falling.
00:23:10.340 --> 00:23:13.060
Consequently, if the speed is
smooth, which it is in this
00:23:13.060 --> 00:23:16.430
case, the only way it can go
from rising to falling is to
00:23:16.430 --> 00:23:20.550
first level off and the
velocity must be 0.
00:23:20.550 --> 00:23:22.760
But you see, what
is the velocity?
00:23:22.760 --> 00:23:26.440
The velocity is the derivative
of the displacement.
00:23:26.440 --> 00:23:28.500
In other words, without solving
this problem, which is
00:23:28.500 --> 00:23:32.220
not important here, to find the
time at which you have the
00:23:32.220 --> 00:23:34.960
maximum height, you
simply do what?
00:23:34.960 --> 00:23:37.410
Set the derivative equal to 0.
00:23:37.410 --> 00:23:40.870
In other words, stationary
points tell us where we have
00:23:40.870 --> 00:23:42.740
high and low points
for functions.
00:23:42.740 --> 00:23:45.480
And knowing where we have high
and low points is a very
00:23:45.480 --> 00:23:47.440
important portion in
curve plotting.
00:23:47.440 --> 00:23:50.580
We'll talk about this in a
future lecture very shortly in
00:23:50.580 --> 00:23:51.460
more detail.
00:23:51.460 --> 00:23:55.670
But coming back here to what we
were talking about before,
00:23:55.670 --> 00:23:59.980
what we're saying in terms of
curve plotting is that where
00:23:59.980 --> 00:24:05.600
the derivative is 0 gives us a
good candidate to have either
00:24:05.600 --> 00:24:09.300
a low point on the curve
or to have a high
00:24:09.300 --> 00:24:11.940
point on the curve.
00:24:11.940 --> 00:24:14.990
However, we should not read
more into this than what's
00:24:14.990 --> 00:24:16.340
already there.
00:24:16.340 --> 00:24:20.960
Namely, it's possible
that you have what?
00:24:20.960 --> 00:24:25.950
A situation like this, in which
the derivative is 0
00:24:25.950 --> 00:24:28.930
here, but the curve is
rising every place.
00:24:28.930 --> 00:24:33.790
And secondly, there is the
possibility that if the
00:24:33.790 --> 00:24:35.975
derivative doesn't exist,
for example, if
00:24:35.975 --> 00:24:38.490
there's a sharp corner.
00:24:38.490 --> 00:24:41.170
Where you have a sharp corner
notice that-- see this is a
00:24:41.170 --> 00:24:43.410
straight line, this is
a straight line.
00:24:43.410 --> 00:24:45.490
In this special case, the
derivative is the slope of
00:24:45.490 --> 00:24:46.870
this straight line.
00:24:46.870 --> 00:24:48.940
Derivative here is the slope
of this straight line.
00:24:48.940 --> 00:24:51.700
Therefore, the derivative is
positive on this line,
00:24:51.700 --> 00:24:53.370
negative on this line.
00:24:53.370 --> 00:24:55.440
Yet the point is what?
00:24:55.440 --> 00:24:58.300
At their junction, there's
a discontinuity.
00:24:58.300 --> 00:25:01.050
The function is continuous,
but the derivative isn't.
00:25:01.050 --> 00:25:03.400
And at that particular point
notice that you have a high
00:25:03.400 --> 00:25:06.580
point even though the derivative
doesn't exist at
00:25:06.580 --> 00:25:07.840
that particular point.
00:25:07.840 --> 00:25:08.970
All we're saying is what?
00:25:08.970 --> 00:25:14.410
That for a smooth curve if there
is to be a high point or
00:25:14.410 --> 00:25:17.780
a low point, a maximum or a
minimum, and we'll talk about
00:25:17.780 --> 00:25:21.750
this as I say, in a future
lecture, it must be that at
00:25:21.750 --> 00:25:25.190
particular point the
derivative is 0.
00:25:25.190 --> 00:25:28.110
On the other hand, conversely,
if the derivative is 0, you
00:25:28.110 --> 00:25:29.960
may not have a high
or low point.
00:25:29.960 --> 00:25:32.280
It may be what we call a saddle
point, the curve just
00:25:32.280 --> 00:25:35.410
levels off after rising
and then rises again.
00:25:35.410 --> 00:25:38.440
And secondly, if the function
isn't differentiable, or the
00:25:38.440 --> 00:25:41.050
curve isn't smooth at that
particular point, you can have
00:25:41.050 --> 00:25:43.790
a high or a low point regardless
of what the
00:25:43.790 --> 00:25:46.740
knowledge about the
derivative is.
00:25:46.740 --> 00:25:49.130
Just a little buckshot here to
give you an idea of how we're
00:25:49.130 --> 00:25:51.250
going to use this material.
00:25:51.250 --> 00:25:56.030
A very related topic that's also
quite important here is
00:25:56.030 --> 00:25:58.740
something called points
of inflection.
00:25:58.740 --> 00:26:02.540
Points of inflection are, in a
way, to the second derivative
00:26:02.540 --> 00:26:05.930
what stationary points are
to the first derivative.
00:26:05.930 --> 00:26:09.020
In many cases, we are interested
in knowing, where
00:26:09.020 --> 00:26:11.090
does the curve change
its concavity?
00:26:11.090 --> 00:26:13.770
Where does it go, in other
words, from holding water to
00:26:13.770 --> 00:26:15.510
spelling water?
00:26:15.510 --> 00:26:19.780
And by the way, again, in
terms of a geometrical
00:26:19.780 --> 00:26:23.860
interpretation, there's a very
what I call exciting answer to
00:26:23.860 --> 00:26:24.780
this question.
00:26:24.780 --> 00:26:26.320
It almost results in what looks
00:26:26.320 --> 00:26:27.810
like an optical illusion.
00:26:27.810 --> 00:26:32.710
You see, if a curve is holding
water, the tangent line lies
00:26:32.710 --> 00:26:34.020
below the curve.
00:26:34.020 --> 00:26:37.680
If the curve is spilling water,
the tangent line to the
00:26:37.680 --> 00:26:40.210
curve at a point lies
above the curve.
00:26:40.210 --> 00:26:42.760
Consequently, at a 'point of
inflection', meaning where the
00:26:42.760 --> 00:26:47.280
curve changes concavity, the
tangent line on one side must
00:26:47.280 --> 00:26:50.410
be above the curve, on the other
side below the curve.
00:26:50.410 --> 00:26:53.350
And all I'm saying over here is
that you recognize a point
00:26:53.350 --> 00:26:57.680
of inflection by what?
00:26:57.680 --> 00:27:00.120
It's the situation in which
the tangent line
00:27:00.120 --> 00:27:01.970
to the curve appears--
00:27:01.970 --> 00:27:03.440
in fact, it actually does
in the manner of
00:27:03.440 --> 00:27:06.750
speaking, cross the curve.
00:27:06.750 --> 00:27:07.950
I think we talked about
this in the previous
00:27:07.950 --> 00:27:09.130
lecture, I'm not sure.
00:27:09.130 --> 00:27:11.110
But we talked about the idea
that a tangent line
00:27:11.110 --> 00:27:12.720
can cross the curve.
00:27:12.720 --> 00:27:15.250
And where can it cross the curve
and still be a tangent
00:27:15.250 --> 00:27:16.220
line at that point?
00:27:16.220 --> 00:27:18.160
At a point of inflection.
00:27:18.160 --> 00:27:21.660
By the way, if this is a
smooth type of thing in
00:27:21.660 --> 00:27:24.480
general, what we're saying is
that for a point of inflection
00:27:24.480 --> 00:27:29.890
to occur, the second derivative
must be 0.
00:27:29.890 --> 00:27:31.070
You see because that
means what?
00:27:31.070 --> 00:27:32.850
The curve is neither--
00:27:32.850 --> 00:27:34.810
it's going from holding to
spilling, so it goes through a
00:27:34.810 --> 00:27:37.980
transition where it's
doing neither.
00:27:37.980 --> 00:27:41.875
In the same as with first
derivatives, the mere fact
00:27:41.875 --> 00:27:44.500
that the second derivative
is 0 does not allow us to
00:27:44.500 --> 00:27:46.620
conclude that we have a
point of inflection.
00:27:46.620 --> 00:27:49.570
In fact, let me close with this
particular illustration.
00:27:49.570 --> 00:27:52.200
Let's take the curve 'y' equals
'x to the fourth'.
00:27:52.200 --> 00:27:54.280
The first derivative
is '4 x cubed'.
00:27:54.280 --> 00:27:56.230
The second derivative
is '12 x squared'.
00:27:56.230 --> 00:27:57.900
The curve is symmetric
with respect to
00:27:57.900 --> 00:28:00.280
the y-axis, et cetera.
00:28:00.280 --> 00:28:02.980
Using all of the given data you
know the second derivative
00:28:02.980 --> 00:28:05.250
is always positive,
what have you.
00:28:05.250 --> 00:28:06.940
We can sketch this curve.
00:28:06.940 --> 00:28:10.860
And again, in fact the
uninitiated say this curve
00:28:10.860 --> 00:28:12.190
looks like a parabola.
00:28:12.190 --> 00:28:13.890
What do you mean it looks
like a parabola?
00:28:13.890 --> 00:28:16.610
Well, he says, the parabola does
something like this too.
00:28:16.610 --> 00:28:19.340
Well, what do we mean by
something like this?
00:28:19.340 --> 00:28:21.370
I want to mention a
few points here.
00:28:21.370 --> 00:28:24.110
One is, of course, that
actually, the parabola 'y'
00:28:24.110 --> 00:28:25.360
equals 'x squared'.
00:28:30.780 --> 00:28:32.325
These are going to crisscross
very shortly here.
00:28:32.325 --> 00:28:33.520
It doesn't make any
difference.
00:28:33.520 --> 00:28:36.620
The parabola 'y' equals 'x
squared' has the same general
00:28:36.620 --> 00:28:38.760
shape but with a few different
properties, which we'll
00:28:38.760 --> 00:28:39.880
mention in a little while.
00:28:39.880 --> 00:28:42.640
But the point that I wanted to
mention here first of all is
00:28:42.640 --> 00:28:43.610
simply this.
00:28:43.610 --> 00:28:48.790
At the value 'x' equals 0,
'y double prime' is 0.
00:28:48.790 --> 00:28:51.850
So you notice that the second
derivative is 0 over here.
00:28:51.850 --> 00:28:54.930
Yet even though the second
derivative is 0, notice that
00:28:54.930 --> 00:28:57.530
the curve does not
change concavity.
00:28:57.530 --> 00:29:00.400
The curve here is always
holding water.
00:29:00.400 --> 00:29:03.450
The concluding remark that I
wanted to make is, what is the
00:29:03.450 --> 00:29:05.680
relationship between 'y' equals
'x squared' and 'y'
00:29:05.680 --> 00:29:06.670
equals 'x to the fourth'?
00:29:06.670 --> 00:29:08.580
Or how about 'y' equals
'x to the sixth'?
00:29:08.580 --> 00:29:10.720
Or 'y' equals 'x to the 12th'?
00:29:10.720 --> 00:29:13.240
Notice that any curve in
that family will look
00:29:13.240 --> 00:29:15.160
something like this.
00:29:15.160 --> 00:29:15.840
Only what?
00:29:15.840 --> 00:29:19.570
As the exponent goes up, the
curve becomes broader in the
00:29:19.570 --> 00:29:20.790
neighborhood of 0.
00:29:20.790 --> 00:29:25.010
And then once 'x' passes 1, the
curve rises more sharply.
00:29:25.010 --> 00:29:27.600
See what we're saying is, if the
magnitude of 'x' is less
00:29:27.600 --> 00:29:30.400
than 1, the higher a power
you raise it to,
00:29:30.400 --> 00:29:31.750
the smaller 'y' is.
00:29:31.750 --> 00:29:34.250
On the other hand, if the
absolute value of 'x' is
00:29:34.250 --> 00:29:40.020
greater than 1, the higher a
power you raise it to, the
00:29:40.020 --> 00:29:41.640
bigger the output becomes.
00:29:41.640 --> 00:29:43.160
But the idea is this.
00:29:43.160 --> 00:29:47.470
Notice that the exponent--
00:29:47.470 --> 00:29:50.540
in other words, how many
derivatives are 0, gives you a
00:29:50.540 --> 00:29:54.080
way of getting into a problem
that will become very, very
00:29:54.080 --> 00:29:55.960
crucial as this course
continues.
00:29:55.960 --> 00:29:59.905
And it's the idea of, can one
curve be more tangent to a
00:29:59.905 --> 00:30:01.380
line than another curve?
00:30:01.380 --> 00:30:06.720
You see, all of these curves are
tangent to the line what?
00:30:06.720 --> 00:30:09.840
The x-axis, 'y' equals
0 at 'x' equals 0.
00:30:09.840 --> 00:30:12.610
How do we distinguish between
these curves?
00:30:12.610 --> 00:30:17.370
Well, it seems that some of
these curves fit the x-axis
00:30:17.370 --> 00:30:20.390
better than others in a
neighborhood of the
00:30:20.390 --> 00:30:21.940
point 'x' equals 0.
00:30:21.940 --> 00:30:24.730
See the point that I want to
bring out as to 'y' curve
00:30:24.730 --> 00:30:27.510
plotting tells us things that we
don't learn in the ordinary
00:30:27.510 --> 00:30:28.970
physics class is this.
00:30:28.970 --> 00:30:32.290
If you study calculus the way
it comes up in most physics
00:30:32.290 --> 00:30:35.310
courses, we essentially don't go
past the second derivative.
00:30:35.310 --> 00:30:35.960
Why?
00:30:35.960 --> 00:30:38.880
Because in many cases, we're
studying distance.
00:30:38.880 --> 00:30:41.210
And the derivative of distance
is velocity.
00:30:41.210 --> 00:30:43.190
The second derivative of
distance, namely the
00:30:43.190 --> 00:30:47.380
derivative of velocity
is acceleration.
00:30:47.380 --> 00:30:49.690
And we don't usually talk
physically beyond
00:30:49.690 --> 00:30:50.780
acceleration.
00:30:50.780 --> 00:30:54.530
But notice that in terms of
curve plotting the third,
00:30:54.530 --> 00:30:59.440
fourth, fifth, sixth, seventh,
tenth derivatives all have a
00:30:59.440 --> 00:31:02.030
meaning that gives you
more information
00:31:02.030 --> 00:31:03.630
than what came before.
00:31:03.630 --> 00:31:06.970
Don't be deceived by the fact
that in other applications
00:31:06.970 --> 00:31:09.490
that you never go past the
second derivative means that
00:31:09.490 --> 00:31:13.610
there is no value to knowing how
higher order derivatives
00:31:13.610 --> 00:31:15.240
are related to plotting
curves.
00:31:15.240 --> 00:31:18.010
At any rate, I think this is
enough of an introduction to
00:31:18.010 --> 00:31:20.710
the topic of curve plotting
and curve sketching.
00:31:20.710 --> 00:31:23.530
We'll pursue these topics
further in our next lectures.
00:31:23.530 --> 00:31:25.020
So until next time, goodbye.
00:31:28.290 --> 00:31:30.820
ANNOUNCER: Funding for the
publication of this video was
00:31:30.820 --> 00:31:35.540
provided by the Gabriella and
Paul Rosenbaum Foundation.
00:31:35.540 --> 00:31:39.710
Help OCW continue to provide
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00:31:39.710 --> 00:31:43.910
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