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HERBERT GROSS: Hi, our lecture
today is about curve plotting.
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And actually, I call it 'Curve
Plotting with and without
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00:00:38,710 --> 00:00:41,600
Calculus' to emphasize the
fact that what we're
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00:00:41,600 --> 00:00:45,340
interested in is curve plotting
and that calculus, in
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00:00:45,340 --> 00:00:49,460
particular, differentiation,
gives us a powerful tool that
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00:00:49,460 --> 00:00:52,960
is not available to us, at least
in an accessible form
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00:00:52,960 --> 00:00:54,820
without the calculus.
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00:00:54,820 --> 00:00:58,550
Let's see what I mean by this
by going back to a typical
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00:00:58,550 --> 00:01:01,660
high school type analytical
geometry problem.
19
00:01:01,660 --> 00:01:06,980
For example, sketch the curve
'y' equals 'x squared'.
20
00:01:06,980 --> 00:01:10,380
Now we all know that the graph
of 'y' equals 'x squared' is
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00:01:10,380 --> 00:01:12,080
something like this.
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00:01:12,080 --> 00:01:13,660
And how'd we find that?
23
00:01:13,660 --> 00:01:16,480
You may recall that in the truer
sense of the word, the
24
00:01:16,480 --> 00:01:21,420
pre-calculus approach really is
curve plotting as opposed
25
00:01:21,420 --> 00:01:22,635
to curve sketching.
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00:01:22,635 --> 00:01:23,950
And I hope I make that
a little bit
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00:01:23,950 --> 00:01:25,660
clearer as we go along.
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00:01:25,660 --> 00:01:29,740
Namely, you look to see for
certain values of 'x' what
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00:01:29,740 --> 00:01:33,700
value of 'y' corresponds
to that.
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00:01:33,700 --> 00:01:36,960
And then we locate the
corresponding point 'x' comma
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00:01:36,960 --> 00:01:40,320
'y' in the plane of
the blackboard.
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00:01:40,320 --> 00:01:43,930
And what we then do is somehow
or other, take a French curve,
33
00:01:43,930 --> 00:01:47,780
or whatever it happens to be,
and we sketch a smooth curve
34
00:01:47,780 --> 00:01:49,310
through the given points.
35
00:01:49,310 --> 00:01:51,450
This is the usual technique.
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00:01:51,450 --> 00:01:54,210
The question that comes up is,
that as long as there are
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00:01:54,210 --> 00:01:57,350
spaces between the points that
you've sketched, how can you
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00:01:57,350 --> 00:02:01,890
be sure that the curve that
you've drawn isn't inaccurate?
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00:02:01,890 --> 00:02:05,120
In other words, starting in
reverse here, let's suppose
40
00:02:05,120 --> 00:02:08,080
that these are the points I've
happened to sketch for the
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00:02:08,080 --> 00:02:09,800
curve 'y' equals 'x squared'.
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00:02:09,800 --> 00:02:12,910
And by the way, as is often the
case in our course, don't
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00:02:12,910 --> 00:02:15,210
be misled that we pick something
as simple as 'y'
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00:02:15,210 --> 00:02:16,400
equals 'x squared'.
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00:02:16,400 --> 00:02:18,890
I simply, again, wanted to
pick something that was
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00:02:18,890 --> 00:02:22,350
straightforward enough so that
we could concentrate our
47
00:02:22,350 --> 00:02:26,680
attention on what the
mathematical implications were
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00:02:26,680 --> 00:02:29,430
rather than to be bogged
down by computation.
49
00:02:29,430 --> 00:02:30,510
But the idea is this.
50
00:02:30,510 --> 00:02:33,340
Going back to the problem 'y'
equals 'x squared', suppose we
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00:02:33,340 --> 00:02:36,490
had located these points and
now we said, let's sketch a
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00:02:36,490 --> 00:02:37,930
smooth curve through these.
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00:02:37,930 --> 00:02:41,350
What would have been wrong
with say, doing
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00:02:41,350 --> 00:02:42,600
something like this?
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00:02:42,600 --> 00:02:45,080
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00:02:45,080 --> 00:02:49,260
Why, for example, couldn't
this have been the curve?
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00:02:49,260 --> 00:02:54,290
Now again, notice that in
terms of mathematical
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00:02:54,290 --> 00:02:57,220
analysis, not necessarily
calculus but in terms of
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00:02:57,220 --> 00:03:00,210
mathematical analysis, we could
immediately strike out
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00:03:00,210 --> 00:03:01,410
something like this.
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00:03:01,410 --> 00:03:05,590
For example, notice in terms
of our input versus output,
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00:03:05,590 --> 00:03:08,360
this says that for any real
input since the square of a
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00:03:08,360 --> 00:03:11,240
real number cannot be negative,
the output can never
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00:03:11,240 --> 00:03:12,120
be negative.
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And pictorially, this means that
no portion of our diagram
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00:03:15,910 --> 00:03:18,350
can be below the x-axis.
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In other words, without
knowing anything about
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00:03:20,980 --> 00:03:23,940
calculus but knowing a little
bit about arithmetic, we can
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00:03:23,940 --> 00:03:26,790
supplement our knowledge
of how the points go by
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saying look-it .
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00:03:27,640 --> 00:03:30,870
Something like this can't happen
because in this region
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00:03:30,870 --> 00:03:33,310
over here 'y' would
be negative.
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00:03:33,310 --> 00:03:35,430
And we know from the
relationship that 'y' equals
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00:03:35,430 --> 00:03:38,060
'x squared' that 'y'
cannot be negative.
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00:03:38,060 --> 00:03:42,820
Well you see, armed with this
information, we could say, OK,
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00:03:42,820 --> 00:03:45,750
given the same points here,
why couldn't the curve go
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through something like this?
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That's what the question
mark is in here for.
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00:03:50,240 --> 00:03:52,530
Obviously, this is
not the graph of
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00:03:52,530 --> 00:03:53,470
'y' equals 'x squared'.
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00:03:53,470 --> 00:03:57,280
But the question is, noticing
that this curve doesn't go
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below the x-axis, what's
wrong with this?
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00:04:00,440 --> 00:04:04,710
And again, we can get bailed
out by a pre-calculus
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00:04:04,710 --> 00:04:06,710
knowledge of mathematics.
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00:04:06,710 --> 00:04:09,980
Among other things, notice the
rather interesting property
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that if we replace 'x' by 'minus
x' here, we get the
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00:04:13,710 --> 00:04:15,890
same curve as before.
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00:04:15,890 --> 00:04:19,790
To generalize this result,
notice that in this particular
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00:04:19,790 --> 00:04:23,580
case, 'f of x' gives us
the same result as
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00:04:23,580 --> 00:04:25,470
'f of 'minus x''.
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00:04:25,470 --> 00:04:27,330
Now this will not happen
in general.
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00:04:27,330 --> 00:04:30,760
What does that mean if we know
that 'f of x' equals 'f of
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00:04:30,760 --> 00:04:31,560
'minus x''?
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00:04:31,560 --> 00:04:33,370
Well, look it.
95
00:04:33,370 --> 00:04:36,950
The relationship between 'x' and
'minus x' is clear, they
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00:04:36,950 --> 00:04:42,580
are located symmetrically with
respect to the y-axis.
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00:04:42,580 --> 00:04:46,555
In other words, if this is 'x1',
this will be 'minus x1'.
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00:04:46,555 --> 00:04:48,250
Now, here's the point.
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00:04:48,250 --> 00:04:50,140
Let's suppose our curve
happens to be 'y'
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00:04:50,140 --> 00:04:51,190
equals 'f of x'.
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00:04:51,190 --> 00:04:55,660
When the input is 'x1', the
output will be 'f of x1'.
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00:04:55,660 --> 00:04:59,390
When the input is 'minus x1',
the output will be 'f of
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00:04:59,390 --> 00:05:00,750
'minus x1''.
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00:05:00,750 --> 00:05:05,560
To say that 'f of 'minus x1''
equals 'f of x1' is the same
105
00:05:05,560 --> 00:05:10,710
as saying that not only are
these two coordinates, 'x1'
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00:05:10,710 --> 00:05:14,490
and 'minus x1', symmetric with
respect to the y-axis, but the
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00:05:14,490 --> 00:05:16,010
outputs also are.
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00:05:16,010 --> 00:05:18,830
In other words, to say that 'f
of x1' equals 'f of 'minus
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00:05:18,830 --> 00:05:25,140
x1'', says that not only are
these two points the same left
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00:05:25,140 --> 00:05:28,370
and right displacement for the
y-axis, but they're also the
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00:05:28,370 --> 00:05:30,380
same height above the x-axis.
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00:05:30,380 --> 00:05:34,340
In short, this point is the
mirror image of this point.
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00:05:34,340 --> 00:05:38,090
In terms of a graph, if 'f of x'
equals 'f of 'minus x'' for
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00:05:38,090 --> 00:05:41,990
all 'x', the graph of 'y' equals
'f of x' is symmetric
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00:05:41,990 --> 00:05:44,220
with respect to the y-axis.
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00:05:44,220 --> 00:05:49,110
By the way, that's why we could
rule out this type of
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00:05:49,110 --> 00:05:50,330
possibility here.
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00:05:50,330 --> 00:05:54,760
For example, here's
'x1' over here.
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00:05:54,760 --> 00:05:56,240
Here's 'minus x1'.
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00:05:56,240 --> 00:06:00,350
And notice that for this choice
of 'x', the points on
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00:06:00,350 --> 00:06:03,730
the curve are not mirror images
of one another with
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00:06:03,730 --> 00:06:05,380
respect to the y-axis.
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00:06:05,380 --> 00:06:07,460
In short, even though we may
not know what this curve is
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00:06:07,460 --> 00:06:10,840
supposed to look like, the fact
that 'f of x' equals 'f
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00:06:10,840 --> 00:06:14,180
of 'minus x'' tells us that
whatever the graph is, it
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00:06:14,180 --> 00:06:16,670
should be symmetric with
respect to the y-axis.
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00:06:16,670 --> 00:06:19,860
This leads us to a rather
interesting aside.
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00:06:19,860 --> 00:06:22,720
It's something called 'even
and odd functions' that we
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00:06:22,720 --> 00:06:25,410
could technically leave out
here, but which I think is a
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00:06:25,410 --> 00:06:26,690
good place to bring in.
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00:06:26,690 --> 00:06:29,660
And the fact that there are
many, many places in our
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00:06:29,660 --> 00:06:32,060
advanced treatment that will
come up later where it's
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00:06:32,060 --> 00:06:35,060
important to understand what
these things mean, that
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00:06:35,060 --> 00:06:39,300
perhaps this is a nice place
to bring it into play.
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00:06:39,300 --> 00:06:41,080
So I call this an aside
and it's about
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00:06:41,080 --> 00:06:43,440
even and odd functions.
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00:06:43,440 --> 00:06:47,350
When we have the case that 'f of
x' equals 'f of 'minus x'',
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00:06:47,350 --> 00:06:50,040
that's the case where in terms
of the graph you have symmetry
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00:06:50,040 --> 00:06:53,880
with respect to the y-axis such
a function is called an
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00:06:53,880 --> 00:06:55,370
'even function'.
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00:06:55,370 --> 00:06:57,710
And I'll come back in a moment
and show you why it's called
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00:06:57,710 --> 00:07:00,740
even, though it's not
really important.
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00:07:00,740 --> 00:07:03,520
The counterpart to an even
function as you may guess
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00:07:03,520 --> 00:07:05,870
almost from the association
of ideas is
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00:07:05,870 --> 00:07:08,290
called an 'odd function'.
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00:07:08,290 --> 00:07:13,280
Now an odd function is one for
which 'f of x' is the negative
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00:07:13,280 --> 00:07:15,200
of 'f of 'minus x''.
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00:07:15,200 --> 00:07:19,310
See for 'f' to be odd, 'f of x'
has to be the negative of
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00:07:19,310 --> 00:07:20,760
'f of 'minus x''.
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00:07:20,760 --> 00:07:22,720
Now what does this
mean pictorially?
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00:07:22,720 --> 00:07:27,860
Again, 'x1' and 'minus x1' are
symmetrically located with
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00:07:27,860 --> 00:07:32,920
respect to the y-axis.
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00:07:32,920 --> 00:07:37,610
This height would be called 'f
of x1' and this height here
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00:07:37,610 --> 00:07:42,000
would be called 'f
of 'minus x1''.
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00:07:42,000 --> 00:07:47,340
And to say that these two
heights are equal in magnitude
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00:07:47,340 --> 00:07:49,530
but opposite in sign
means what?
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00:07:49,530 --> 00:07:53,130
That these two lengths are equal
but on opposite sides of
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00:07:53,130 --> 00:07:54,050
the x-axis.
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00:07:54,050 --> 00:07:57,960
In other words, here's
'f of x1'.
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00:07:57,960 --> 00:07:59,770
'f of 'minus x1'' is the height
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00:07:59,770 --> 00:08:01,330
corresponding to this point.
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00:08:01,330 --> 00:08:04,720
And the fact that these must
be equal in magnitude but
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00:08:04,720 --> 00:08:08,370
opposite in sign says that one
of the heights must be above
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00:08:08,370 --> 00:08:11,630
the axis, the other one must
be below the axis.
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00:08:11,630 --> 00:08:15,530
And if that's the case from some
very elementary geometry,
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00:08:15,530 --> 00:08:18,360
if 'f' is odd, what it
means with respect to
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00:08:18,360 --> 00:08:19,780
its graph is this.
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00:08:19,780 --> 00:08:25,420
If you put a ruler connecting
the origin with any point on
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00:08:25,420 --> 00:08:31,820
the curve, if you extend that
line that line will meet the
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00:08:31,820 --> 00:08:35,919
curve again, such that these
distances here will be equal.
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00:08:35,919 --> 00:08:38,830
This is called symmetry with
respect to the origin.
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00:08:38,830 --> 00:08:41,750
By the way, an example of
this kind of a curve is
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00:08:41,750 --> 00:08:43,480
'y' equals 'x cubed'.
174
00:08:43,480 --> 00:08:48,360
You see, if I replace 'x' by
'minus x', this becomes what?
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00:08:48,360 --> 00:08:50,550
'Minus x cubed'.
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00:08:50,550 --> 00:08:55,490
And 'minus 'minus x cubed''
is 'x cubed'.
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00:08:55,490 --> 00:08:57,940
By the way, all I'm saying
now is what?
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00:08:57,940 --> 00:09:02,060
If I were to take a ruler and
place it here and let this
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00:09:02,060 --> 00:09:05,110
line go from curve to curve.
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00:09:05,110 --> 00:09:07,370
all I'm saying is that
this portion
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00:09:07,370 --> 00:09:08,890
would equal this portion.
182
00:09:08,890 --> 00:09:12,300
And by the way, maybe you can
already guess where the words
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00:09:12,300 --> 00:09:14,410
even and odd come from.
184
00:09:14,410 --> 00:09:17,420
Notice that when I dealt with
'y' equals 'x squared' I had
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00:09:17,420 --> 00:09:18,990
an even function.
186
00:09:18,990 --> 00:09:20,470
The exponent was even.
187
00:09:20,470 --> 00:09:23,170
When I dealt with 'y'
equals 'x cubed',
188
00:09:23,170 --> 00:09:24,740
I had an odd function.
189
00:09:24,740 --> 00:09:26,230
The exponent was odd.
190
00:09:26,230 --> 00:09:28,140
And by the way, there
are other examples.
191
00:09:28,140 --> 00:09:32,710
But for the time being, notice
something like say, 'y' equals
192
00:09:32,710 --> 00:09:35,420
'x to the fourth' plus
'x squared' say.
193
00:09:35,420 --> 00:09:46,030
194
00:09:46,030 --> 00:09:49,100
If I replace 'x' by 'minus x',
I get back the same thing.
195
00:09:49,100 --> 00:09:52,190
In other words, here's a case
where if 'y' equals 'f of x',
196
00:09:52,190 --> 00:09:56,150
'f of x' is the same as
'f of 'minus x''.
197
00:09:56,150 --> 00:09:59,980
An example of an odd function
might be 'y' equals 'x cubed'
198
00:09:59,980 --> 00:10:02,220
say, plus 'x'.
199
00:10:02,220 --> 00:10:03,720
See first power over here.
200
00:10:03,720 --> 00:10:08,080
If I replace 'x' by 'minus
x', I have 'minus x
201
00:10:08,080 --> 00:10:10,840
cubed' plus 'minus x'.
202
00:10:10,840 --> 00:10:14,120
That of course, is just 'minus
x cubed' minus 'x'.
203
00:10:14,120 --> 00:10:18,040
And that's minus the quantity
'x cubed + x'.
204
00:10:18,040 --> 00:10:21,240
That when I replace 'x' by
'minus x' all I do is change
205
00:10:21,240 --> 00:10:22,300
the sign here.
206
00:10:22,300 --> 00:10:26,070
In other words, that would be an
example for which 'f of x'
207
00:10:26,070 --> 00:10:30,350
is minus 'f of 'minus x''.
208
00:10:30,350 --> 00:10:32,670
By the way, while we're dealing
with examples like
209
00:10:32,670 --> 00:10:36,120
this, unlike the case with whole
numbers where a whole
210
00:10:36,120 --> 00:10:41,000
number is either even or odd
but not both, it's rather
211
00:10:41,000 --> 00:10:45,590
important to notice that a
function need not be either
212
00:10:45,590 --> 00:10:47,550
even or odd.
213
00:10:47,550 --> 00:10:50,220
And by the way, if you think of
our geometric definition,
214
00:10:50,220 --> 00:10:52,560
that's not too hard to see.
215
00:10:52,560 --> 00:10:55,370
Namely, there's no reason why a
curve drawn at random should
216
00:10:55,370 --> 00:10:58,430
be symmetric either with respect
to the y-axis or with
217
00:10:58,430 --> 00:10:59,390
respect to the origin.
218
00:10:59,390 --> 00:11:03,940
In fact, maybe the quickest way
to see this is to put an
219
00:11:03,940 --> 00:11:07,060
odd power of 'x' in with
an even power of
220
00:11:07,060 --> 00:11:08,580
'x' in the same diagram.
221
00:11:08,580 --> 00:11:14,420
If I now replace 'x' by 'minus
x' you see 'minus x' cubed is
222
00:11:14,420 --> 00:11:15,990
of course, 'minus x cubed'.
223
00:11:15,990 --> 00:11:19,670
But 'minus x' quantity squared
is just 'x squared'.
224
00:11:19,670 --> 00:11:22,750
And now you'll notice that if
I compare these two, I don't
225
00:11:22,750 --> 00:11:25,040
get the same thing nor
do I get the same
226
00:11:25,040 --> 00:11:26,850
thing with a sign change.
227
00:11:26,850 --> 00:11:30,020
You see, in other words, this is
not either equal to this or
228
00:11:30,020 --> 00:11:32,020
to the negative of this.
229
00:11:32,020 --> 00:11:34,310
Obviously you understand that
to be the negative of this,
230
00:11:34,310 --> 00:11:37,770
there would also have to be
a minus sign over here.
231
00:11:37,770 --> 00:11:43,400
Or if we wanted to go into more
detail about this, it is
232
00:11:43,400 --> 00:11:46,390
perhaps exciting to know that
whereas it's not true that
233
00:11:46,390 --> 00:11:50,480
every function is either even or
odd, every function can be
234
00:11:50,480 --> 00:11:53,880
written as the sum of two
functions, one of which is
235
00:11:53,880 --> 00:11:56,750
even, one of which is odd.
236
00:11:56,750 --> 00:12:00,800
Just to give you an idea of what
that means let me just
237
00:12:00,800 --> 00:12:02,180
write down something here.
238
00:12:02,180 --> 00:12:05,220
This will come back to be more
important later on, but for
239
00:12:05,220 --> 00:12:09,060
the time being, just to show you
a connecting thread here
240
00:12:09,060 --> 00:12:11,970
as long as we're talking about
even and odd functions, all I
241
00:12:11,970 --> 00:12:14,480
want to see is the following
identity.
242
00:12:14,480 --> 00:12:20,400
If you write ''f of x' plus
'f of 'minus x'' over 2.
243
00:12:20,400 --> 00:12:22,840
And don't worry about what
motivates this, I just wanted
244
00:12:22,840 --> 00:12:25,820
to show you this to keep
this fairly complete.
245
00:12:25,820 --> 00:12:31,000
Now suppose I add on to that ''f
of x' minus 'f of 'minus
246
00:12:31,000 --> 00:12:33,690
x'' over 2.
247
00:12:33,690 --> 00:12:35,740
You see, notice that the
expression on the right-hand
248
00:12:35,740 --> 00:12:38,750
side is just writing 'f
of x' the hard way.
249
00:12:38,750 --> 00:12:41,720
See, here's half of 'f of x',
here's half of 'f of x'.
250
00:12:41,720 --> 00:12:43,190
The sum is 'f of x'.
251
00:12:43,190 --> 00:12:45,470
And here's half of
'f of 'minus x''.
252
00:12:45,470 --> 00:12:48,250
And I'm subtracting half
of 'f of 'minus x''.
253
00:12:48,250 --> 00:12:49,490
That drops out.
254
00:12:49,490 --> 00:12:55,080
The point is that this is always
an even function and
255
00:12:55,080 --> 00:12:57,550
this is always an
odd function.
256
00:12:57,550 --> 00:12:59,830
And just to review the
definition so that you see
257
00:12:59,830 --> 00:13:03,890
what happens here, all I'm
saying is if I replace 'x' by
258
00:13:03,890 --> 00:13:06,720
'minus x' here, what happens?
259
00:13:06,720 --> 00:13:11,340
If I replace 'x' by 'minus x'
this becomes 'f of 'minus x''.
260
00:13:11,340 --> 00:13:15,390
And if I replace 'x' by 'minus
x' here since minus minus is
261
00:13:15,390 --> 00:13:17,970
plus, this becomes 'f of x'.
262
00:13:17,970 --> 00:13:20,800
Now notice that when you add
two terms, the sum is
263
00:13:20,800 --> 00:13:23,140
independent of the order
in which you add them.
264
00:13:23,140 --> 00:13:27,170
'f of x' plus 'f of 'minus x'',
therefore, is the same as
265
00:13:27,170 --> 00:13:29,680
'f of 'minus x''
plus 'f of x'.
266
00:13:29,680 --> 00:13:33,000
In other words, when I replace
'x' by 'minus x' in this
267
00:13:33,000 --> 00:13:36,970
bracketed function, I do not
change the value of what's in
268
00:13:36,970 --> 00:13:38,070
the brackets.
269
00:13:38,070 --> 00:13:43,230
On the other hand, if I
interchange 'x' with 'minus x'
270
00:13:43,230 --> 00:13:46,740
here, notice that since I'm
subtracting look what happens.
271
00:13:46,740 --> 00:13:48,790
I replace 'x' by 'minus x'.
272
00:13:48,790 --> 00:13:50,770
This gives me an 'f
of 'minus x''.
273
00:13:50,770 --> 00:13:52,810
Now I make the same
replacement here.
274
00:13:52,810 --> 00:13:54,840
Minus minus is positive.
275
00:13:54,840 --> 00:13:58,130
But now notice that if I look at
this expression here, I've
276
00:13:58,130 --> 00:13:59,440
changed the order.
277
00:13:59,440 --> 00:14:01,960
See 'f of x' minus
'f of 'minus x''.
278
00:14:01,960 --> 00:14:04,990
Here, 'f of 'minus x''
minus 'f of x'.
279
00:14:04,990 --> 00:14:08,650
And when you change the order
you change the sign.
280
00:14:08,650 --> 00:14:11,750
In other words then, all I'm
saying is that when we talk
281
00:14:11,750 --> 00:14:14,230
about even and odd functions,
they play a very important
282
00:14:14,230 --> 00:14:19,130
role in calculus and in other
mathematical analysis topics.
283
00:14:19,130 --> 00:14:22,590
That not every function is
either even or odd, but every
284
00:14:22,590 --> 00:14:26,340
function that's defined on the
appropriate domain is the sum
285
00:14:26,340 --> 00:14:29,020
of both an even and
an odd function.
286
00:14:29,020 --> 00:14:31,730
But in terms of curve plotting,
the main point is
287
00:14:31,730 --> 00:14:35,120
not so much these extra remarks
as much as what?
288
00:14:35,120 --> 00:14:38,930
An even function is symmetric
with respect to the y-axis and
289
00:14:38,930 --> 00:14:42,110
an odd function is symmetric
with respect to the origin.
290
00:14:42,110 --> 00:14:46,000
At any rate, if we now go back
to our curve 'y' equals 'x
291
00:14:46,000 --> 00:14:54,290
squared' the fact that 'f of x'
equals 'x squared' is even
292
00:14:54,290 --> 00:14:57,670
means that whatever our graph
looks like, it must be
293
00:14:57,670 --> 00:15:00,200
symmetric with respect
to the y-axis.
294
00:15:00,200 --> 00:15:02,780
Now give or take how I've drawn
this, this should be
295
00:15:02,780 --> 00:15:05,100
symmetric with respect
to the y-axis.
296
00:15:05,100 --> 00:15:09,210
If it doesn't look that way,
imagine that it is that way.
297
00:15:09,210 --> 00:15:10,570
And so the idea is what?
298
00:15:10,570 --> 00:15:13,650
Well, any knowledge of calculus
whatsoever, what I
299
00:15:13,650 --> 00:15:16,880
was able to do here is show that
whatever the graph of 'y'
300
00:15:16,880 --> 00:15:20,960
equals 'x squared' is, it must
never dip below the x-axis.
301
00:15:20,960 --> 00:15:23,230
And whatever the curve looks
like to the right of the
302
00:15:23,230 --> 00:15:26,900
y-axis, it must be the mirror
image of that to the left of
303
00:15:26,900 --> 00:15:28,000
the y-axis.
304
00:15:28,000 --> 00:15:31,010
Again, this is how much one
can do without calculus.
305
00:15:31,010 --> 00:15:34,100
And most of you who are
practicing engineers, I'm sure
306
00:15:34,100 --> 00:15:37,140
not only understand this type
of technique as far as the
307
00:15:37,140 --> 00:15:40,440
pre-calculus is concerned, but
can probably draw curves much
308
00:15:40,440 --> 00:15:41,530
better than I can.
309
00:15:41,530 --> 00:15:44,260
In fact, even if you're not
practicing engineers you can
310
00:15:44,260 --> 00:15:46,800
probably draw curves much
better than I can.
311
00:15:46,800 --> 00:15:48,270
But that part is irrelevant.
312
00:15:48,270 --> 00:15:50,180
What I wanted to show
up to now--
313
00:15:50,180 --> 00:15:51,930
and this is what's important
to stress--
314
00:15:51,930 --> 00:15:55,360
is that to get as far as I've
gotten so far, I did not have
315
00:15:55,360 --> 00:15:58,090
to have any knowledge
of calculus.
316
00:15:58,090 --> 00:16:02,130
The way calculus comes in, as
I say again, is a supplement
317
00:16:02,130 --> 00:16:04,000
to our previous techniques.
318
00:16:04,000 --> 00:16:06,630
For example, let's suppose
we did have the
319
00:16:06,630 --> 00:16:09,270
curve drawn this way.
320
00:16:09,270 --> 00:16:13,150
From this, we certainly aren't
contradicting the fact that
321
00:16:13,150 --> 00:16:15,140
'x' can't be negative.
322
00:16:15,140 --> 00:16:16,750
We're not contradicting
the fact that
323
00:16:16,750 --> 00:16:18,110
'f' is an even function.
324
00:16:18,110 --> 00:16:20,120
But how do we know that this
is the wrong picture?
325
00:16:20,120 --> 00:16:24,710
Well, given that 'y' equals 'x
squared', we can easily verify
326
00:16:24,710 --> 00:16:29,400
that 'dy dx' is '2x.' Knowing
that 'dy dx' is '2x', that
327
00:16:29,400 --> 00:16:32,400
tells us among other things
that 'dy dx' and 'x'
328
00:16:32,400 --> 00:16:33,740
have the same sign.
329
00:16:33,740 --> 00:16:37,310
In other words, 'dy dx' is
positive if 'x' is positive,
330
00:16:37,310 --> 00:16:40,690
'dy dx' is negative if
'x' is negative.
331
00:16:40,690 --> 00:16:43,760
In terms of geometry,
that means what?
332
00:16:43,760 --> 00:16:47,020
That the curve is always rising
for positive values of
333
00:16:47,020 --> 00:16:50,730
'x' and always falling for
negative values of 'x'.
334
00:16:50,730 --> 00:16:54,060
Well, you see with that as
a hint, I say ah-ha.
335
00:16:54,060 --> 00:16:55,480
This can't happen.
336
00:16:55,480 --> 00:16:57,340
Because look what's happening
over here.
337
00:16:57,340 --> 00:16:58,840
Or for that matter, over here.
338
00:16:58,840 --> 00:17:03,740
Here the curve is falling for
positive values of 'x'.
339
00:17:03,740 --> 00:17:06,280
And that contradicts the fact
that the curve must always be
340
00:17:06,280 --> 00:17:08,240
rising when 'x' is positive.
341
00:17:08,240 --> 00:17:11,660
In a similar way, we know that
the curve can't be rising when
342
00:17:11,660 --> 00:17:16,319
'x' is negative, yet get over
here and here too, we've drawn
343
00:17:16,319 --> 00:17:17,730
the curve to be rising.
344
00:17:17,730 --> 00:17:22,200
That again is contradicted
by this diagram.
345
00:17:22,200 --> 00:17:24,099
So you see the knowledge
of the first
346
00:17:24,099 --> 00:17:26,410
derivative does what?
347
00:17:26,410 --> 00:17:29,420
It tells us where the curve is
rising or falling, and that
348
00:17:29,420 --> 00:17:31,590
gives us another way of checking
whether the graph
349
00:17:31,590 --> 00:17:33,610
we've drawn is accurate
or not.
350
00:17:33,610 --> 00:17:35,690
By the way, it's not all
quite that simple.
351
00:17:35,690 --> 00:17:39,570
And notice again, subtlety how
step by step we strengthen our
352
00:17:39,570 --> 00:17:41,290
procedures each time.
353
00:17:41,290 --> 00:17:44,690
For example, now knowing that
the curve must always be
354
00:17:44,690 --> 00:17:48,390
rising when 'x' is positive and
always falling when 'x' is
355
00:17:48,390 --> 00:17:51,810
negative, how about this
possibility for the graph of
356
00:17:51,810 --> 00:17:53,770
'y' equals 'x squared'.
357
00:17:53,770 --> 00:17:56,260
You see this curve is always
fallen for negative values of
358
00:17:56,260 --> 00:17:59,630
'x', it's always rising for
positive values of 'x'.
359
00:17:59,630 --> 00:18:02,380
360
00:18:02,380 --> 00:18:04,720
Let's take a look now at what
the second derivative means.
361
00:18:04,720 --> 00:18:08,470
If 'y' equals 'x squared',
obviously as we saw before,
362
00:18:08,470 --> 00:18:14,150
'dy dx' is '2x' and the second
derivative of 'y' with respect
363
00:18:14,150 --> 00:18:16,450
to 'x' is 2.
364
00:18:16,450 --> 00:18:19,280
And 2 is a constant, which
is always positive.
365
00:18:19,280 --> 00:18:20,210
This says what?
366
00:18:20,210 --> 00:18:23,110
That the second derivative
is always positive.
367
00:18:23,110 --> 00:18:24,740
Now what is the second
derivative?
368
00:18:24,740 --> 00:18:27,230
The second derivative is the
first derivative of the first
369
00:18:27,230 --> 00:18:28,270
derivative.
370
00:18:28,270 --> 00:18:31,780
That means the rate of change
of the rate of change.
371
00:18:31,780 --> 00:18:34,250
Well, the rate of change of the
rate of change is called
372
00:18:34,250 --> 00:18:36,850
acceleration.
373
00:18:36,850 --> 00:18:39,600
So if the rate of change of the
rate of change is positive
374
00:18:39,600 --> 00:18:42,010
that means that the curve must
be accelerating, or the
375
00:18:42,010 --> 00:18:43,470
function is accelerating.
376
00:18:43,470 --> 00:18:45,810
And if it's negative, function
is decelerating.
377
00:18:45,810 --> 00:18:48,290
What does that mean in
terms of a picture?
378
00:18:48,290 --> 00:18:50,530
And the author of the
text uses a very
379
00:18:50,530 --> 00:18:51,980
descriptive phrase here.
380
00:18:51,980 --> 00:18:56,010
He talks about 'holding water'
and 'spilling water'.
381
00:18:56,010 --> 00:19:00,040
Notice, for example, here the
curve would tend to collect
382
00:19:00,040 --> 00:19:03,180
water, whereas here if water
were poured on it, the curve
383
00:19:03,180 --> 00:19:05,070
would tend to spill water.
384
00:19:05,070 --> 00:19:08,140
Holding water represents
acceleration.
385
00:19:08,140 --> 00:19:10,680
You see that not only is the
curve rising here, but it's
386
00:19:10,680 --> 00:19:13,710
rising at a faster
and faster rate.
387
00:19:13,710 --> 00:19:15,860
Again, more primitively,
in terms of
388
00:19:15,860 --> 00:19:17,780
slopes, notice that what?
389
00:19:17,780 --> 00:19:20,330
Not only is the slope positive,
but as you move
390
00:19:20,330 --> 00:19:24,350
along this portion of the curve,
the slope increases as
391
00:19:24,350 --> 00:19:26,020
you move along.
392
00:19:26,020 --> 00:19:28,900
And what typifies
this portion?
393
00:19:28,900 --> 00:19:32,500
That even though the slope is
always positive, it decreases
394
00:19:32,500 --> 00:19:34,010
as you move along the curve.
395
00:19:34,010 --> 00:19:36,480
In other words, 'holding water'
corresponds to the
396
00:19:36,480 --> 00:19:39,320
second derivative being
positive, 'spilling water'
397
00:19:39,320 --> 00:19:42,410
corresponds to the second
derivative being negative.
398
00:19:42,410 --> 00:19:45,540
Returning then to our original
problem, the fact that this
399
00:19:45,540 --> 00:19:52,160
thing here is greater than 0
for all 'x' says that this
400
00:19:52,160 --> 00:19:54,080
curve could never spill water.
401
00:19:54,080 --> 00:19:57,650
And that rules out this
portion in here.
402
00:19:57,650 --> 00:19:58,990
In other words, now
we put everything
403
00:19:58,990 --> 00:20:00,670
together, we know what?
404
00:20:00,670 --> 00:20:03,660
The curve can never go
below the x-axis.
405
00:20:03,660 --> 00:20:06,960
It's symmetric with respect
to the y-axis.
406
00:20:06,960 --> 00:20:09,350
It's always rising when
'x' is positive and
407
00:20:09,350 --> 00:20:11,700
always holding water.
408
00:20:11,700 --> 00:20:15,160
Now you see this is what I call
curve sketching versus
409
00:20:15,160 --> 00:20:16,080
curve plotting.
410
00:20:16,080 --> 00:20:18,900
With the information that I
have from calculus, I know
411
00:20:18,900 --> 00:20:22,010
what's going on for each
point, not just for the
412
00:20:22,010 --> 00:20:25,590
isolated points that I happened
to have plotted in
413
00:20:25,590 --> 00:20:26,840
the data for.
414
00:20:26,840 --> 00:20:28,970
415
00:20:28,970 --> 00:20:31,910
See, the calculus fills
in the missing
416
00:20:31,910 --> 00:20:33,600
data very, very nicely.
417
00:20:33,600 --> 00:20:38,230
Now you see this does not mean
I'm going to replace this by--
418
00:20:38,230 --> 00:20:39,980
my previous analysis
by calculus.
419
00:20:39,980 --> 00:20:42,950
It means I'm going to add
calculus as one of my bags of
420
00:20:42,950 --> 00:20:43,890
tools here.
421
00:20:43,890 --> 00:20:47,970
Notice again that there is a
very nice relationship between
422
00:20:47,970 --> 00:20:49,810
pictures and analysis.
423
00:20:49,810 --> 00:20:51,450
And I'm not going to
belabor that point.
424
00:20:51,450 --> 00:20:53,820
All I'm saying is that if
you add to our previous
425
00:20:53,820 --> 00:20:54,810
identifications--
426
00:20:54,810 --> 00:20:56,430
what identifications?
427
00:20:56,430 --> 00:20:58,780
Like increase means rising.
428
00:20:58,780 --> 00:21:01,060
I mean this, if a derivative
is positive,
429
00:21:01,060 --> 00:21:02,240
the curve is rising.
430
00:21:02,240 --> 00:21:04,970
If a derivative is negative,
the curve is falling.
431
00:21:04,970 --> 00:21:07,450
If the second derivative
is positive, the
432
00:21:07,450 --> 00:21:08,770
curve is holding water.
433
00:21:08,770 --> 00:21:10,920
If the second derivative is
negative, the curve is
434
00:21:10,920 --> 00:21:11,820
spilling water.
435
00:21:11,820 --> 00:21:15,360
And again, we have ample
exercises and portions of this
436
00:21:15,360 --> 00:21:17,630
in the reading material
to illustrate the
437
00:21:17,630 --> 00:21:19,030
computational aspects.
438
00:21:19,030 --> 00:21:22,090
But again, all I want you to
get from this lecture is
439
00:21:22,090 --> 00:21:24,740
what's happening here
conceptually.
440
00:21:24,740 --> 00:21:28,640
Let's look at this, a few
more applications.
441
00:21:28,640 --> 00:21:31,990
We're going to find in
subsequent lectures as well as
442
00:21:31,990 --> 00:21:34,280
other portions of the course,
we're going to be interested,
443
00:21:34,280 --> 00:21:37,810
for example, in things called
stationary points.
444
00:21:37,810 --> 00:21:40,540
A 'stationary point' is a point
at which the curve is
445
00:21:40,540 --> 00:21:42,230
neither rising nor falling.
446
00:21:42,230 --> 00:21:44,830
In other words, if the curve
happens to be smooth, it's
447
00:21:44,830 --> 00:21:48,520
characterized by the fact that
'dy dx' is 0 it such a point.
448
00:21:48,520 --> 00:21:51,840
In terms of the language of
functions to say that the
449
00:21:51,840 --> 00:21:55,100
curve is smooth means that the
function is differentiable and
450
00:21:55,100 --> 00:21:58,910
what we're saying is to be
stationary at 'x' equals 'x1',
451
00:21:58,910 --> 00:22:03,460
it must be that 'f prime
of x1' equals 0.
452
00:22:03,460 --> 00:22:06,370
And the importance of stationary
points can be seen
453
00:22:06,370 --> 00:22:09,020
in terms of a physical
interpretation.
454
00:22:09,020 --> 00:22:11,970
We haven't used our freely
falling body for quite a
455
00:22:11,970 --> 00:22:14,610
while, let's go back
to such an example.
456
00:22:14,610 --> 00:22:20,750
Suppose a particle is projected
vertically upward in
457
00:22:20,750 --> 00:22:23,880
the absence of air resistance,
et cetera, with an initial
458
00:22:23,880 --> 00:22:26,690
speed of 160 feet per second.
459
00:22:26,690 --> 00:22:30,560
It can then be shown that the
height 's' to which the ball
460
00:22:30,560 --> 00:22:33,520
rises in feet at time
't' is given by
461
00:22:33,520 --> 00:22:37,100
'160t - 16't squared''.
462
00:22:37,100 --> 00:22:43,460
A very natural question to ask
is, when will the ball be at
463
00:22:43,460 --> 00:22:45,450
its maximum height?
464
00:22:45,450 --> 00:22:47,360
And I'm sure you can see in
terms of this physical
465
00:22:47,360 --> 00:22:50,870
example, the ball will be at
its maximum height when the
466
00:22:50,870 --> 00:22:53,020
velocity is 0.
467
00:22:53,020 --> 00:22:57,840
In other words, since this is
a smooth type of motion, if
468
00:22:57,840 --> 00:23:02,910
the velocity is not 0, the
ball is still rising.
469
00:23:02,910 --> 00:23:06,010
If the velocity is positive,
the ball is still rising.
470
00:23:06,010 --> 00:23:10,340
If the velocity is negative, the
ball is already falling.
471
00:23:10,340 --> 00:23:13,060
Consequently, if the speed is
smooth, which it is in this
472
00:23:13,060 --> 00:23:16,430
case, the only way it can go
from rising to falling is to
473
00:23:16,430 --> 00:23:20,550
first level off and the
velocity must be 0.
474
00:23:20,550 --> 00:23:22,760
But you see, what
is the velocity?
475
00:23:22,760 --> 00:23:26,440
The velocity is the derivative
of the displacement.
476
00:23:26,440 --> 00:23:28,500
In other words, without solving
this problem, which is
477
00:23:28,500 --> 00:23:32,220
not important here, to find the
time at which you have the
478
00:23:32,220 --> 00:23:34,960
maximum height, you
simply do what?
479
00:23:34,960 --> 00:23:37,410
Set the derivative equal to 0.
480
00:23:37,410 --> 00:23:40,870
In other words, stationary
points tell us where we have
481
00:23:40,870 --> 00:23:42,740
high and low points
for functions.
482
00:23:42,740 --> 00:23:45,480
And knowing where we have high
and low points is a very
483
00:23:45,480 --> 00:23:47,440
important portion in
curve plotting.
484
00:23:47,440 --> 00:23:50,580
We'll talk about this in a
future lecture very shortly in
485
00:23:50,580 --> 00:23:51,460
more detail.
486
00:23:51,460 --> 00:23:55,670
But coming back here to what we
were talking about before,
487
00:23:55,670 --> 00:23:59,980
what we're saying in terms of
curve plotting is that where
488
00:23:59,980 --> 00:24:05,600
the derivative is 0 gives us a
good candidate to have either
489
00:24:05,600 --> 00:24:09,300
a low point on the curve
or to have a high
490
00:24:09,300 --> 00:24:11,940
point on the curve.
491
00:24:11,940 --> 00:24:14,990
However, we should not read
more into this than what's
492
00:24:14,990 --> 00:24:16,340
already there.
493
00:24:16,340 --> 00:24:20,960
Namely, it's possible
that you have what?
494
00:24:20,960 --> 00:24:25,950
A situation like this, in which
the derivative is 0
495
00:24:25,950 --> 00:24:28,930
here, but the curve is
rising every place.
496
00:24:28,930 --> 00:24:33,790
And secondly, there is the
possibility that if the
497
00:24:33,790 --> 00:24:35,975
derivative doesn't exist,
for example, if
498
00:24:35,975 --> 00:24:38,490
there's a sharp corner.
499
00:24:38,490 --> 00:24:41,170
Where you have a sharp corner
notice that-- see this is a
500
00:24:41,170 --> 00:24:43,410
straight line, this is
a straight line.
501
00:24:43,410 --> 00:24:45,490
In this special case, the
derivative is the slope of
502
00:24:45,490 --> 00:24:46,870
this straight line.
503
00:24:46,870 --> 00:24:48,940
Derivative here is the slope
of this straight line.
504
00:24:48,940 --> 00:24:51,700
Therefore, the derivative is
positive on this line,
505
00:24:51,700 --> 00:24:53,370
negative on this line.
506
00:24:53,370 --> 00:24:55,440
Yet the point is what?
507
00:24:55,440 --> 00:24:58,300
At their junction, there's
a discontinuity.
508
00:24:58,300 --> 00:25:01,050
The function is continuous,
but the derivative isn't.
509
00:25:01,050 --> 00:25:03,400
And at that particular point
notice that you have a high
510
00:25:03,400 --> 00:25:06,580
point even though the derivative
doesn't exist at
511
00:25:06,580 --> 00:25:07,840
that particular point.
512
00:25:07,840 --> 00:25:08,970
All we're saying is what?
513
00:25:08,970 --> 00:25:14,410
That for a smooth curve if there
is to be a high point or
514
00:25:14,410 --> 00:25:17,780
a low point, a maximum or a
minimum, and we'll talk about
515
00:25:17,780 --> 00:25:21,750
this as I say, in a future
lecture, it must be that at
516
00:25:21,750 --> 00:25:25,190
particular point the
derivative is 0.
517
00:25:25,190 --> 00:25:28,110
On the other hand, conversely,
if the derivative is 0, you
518
00:25:28,110 --> 00:25:29,960
may not have a high
or low point.
519
00:25:29,960 --> 00:25:32,280
It may be what we call a saddle
point, the curve just
520
00:25:32,280 --> 00:25:35,410
levels off after rising
and then rises again.
521
00:25:35,410 --> 00:25:38,440
And secondly, if the function
isn't differentiable, or the
522
00:25:38,440 --> 00:25:41,050
curve isn't smooth at that
particular point, you can have
523
00:25:41,050 --> 00:25:43,790
a high or a low point regardless
of what the
524
00:25:43,790 --> 00:25:46,740
knowledge about the
derivative is.
525
00:25:46,740 --> 00:25:49,130
Just a little buckshot here to
give you an idea of how we're
526
00:25:49,130 --> 00:25:51,250
going to use this material.
527
00:25:51,250 --> 00:25:56,030
A very related topic that's also
quite important here is
528
00:25:56,030 --> 00:25:58,740
something called points
of inflection.
529
00:25:58,740 --> 00:26:02,540
Points of inflection are, in a
way, to the second derivative
530
00:26:02,540 --> 00:26:05,930
what stationary points are
to the first derivative.
531
00:26:05,930 --> 00:26:09,020
In many cases, we are interested
in knowing, where
532
00:26:09,020 --> 00:26:11,090
does the curve change
its concavity?
533
00:26:11,090 --> 00:26:13,770
Where does it go, in other
words, from holding water to
534
00:26:13,770 --> 00:26:15,510
spelling water?
535
00:26:15,510 --> 00:26:19,780
And by the way, again, in
terms of a geometrical
536
00:26:19,780 --> 00:26:23,860
interpretation, there's a very
what I call exciting answer to
537
00:26:23,860 --> 00:26:24,780
this question.
538
00:26:24,780 --> 00:26:26,320
It almost results in what looks
539
00:26:26,320 --> 00:26:27,810
like an optical illusion.
540
00:26:27,810 --> 00:26:32,710
You see, if a curve is holding
water, the tangent line lies
541
00:26:32,710 --> 00:26:34,020
below the curve.
542
00:26:34,020 --> 00:26:37,680
If the curve is spilling water,
the tangent line to the
543
00:26:37,680 --> 00:26:40,210
curve at a point lies
above the curve.
544
00:26:40,210 --> 00:26:42,760
Consequently, at a 'point of
inflection', meaning where the
545
00:26:42,760 --> 00:26:47,280
curve changes concavity, the
tangent line on one side must
546
00:26:47,280 --> 00:26:50,410
be above the curve, on the other
side below the curve.
547
00:26:50,410 --> 00:26:53,350
And all I'm saying over here is
that you recognize a point
548
00:26:53,350 --> 00:26:57,680
of inflection by what?
549
00:26:57,680 --> 00:27:00,120
It's the situation in which
the tangent line
550
00:27:00,120 --> 00:27:01,970
to the curve appears--
551
00:27:01,970 --> 00:27:03,440
in fact, it actually does
in the manner of
552
00:27:03,440 --> 00:27:06,750
speaking, cross the curve.
553
00:27:06,750 --> 00:27:07,950
I think we talked about
this in the previous
554
00:27:07,950 --> 00:27:09,130
lecture, I'm not sure.
555
00:27:09,130 --> 00:27:11,110
But we talked about the idea
that a tangent line
556
00:27:11,110 --> 00:27:12,720
can cross the curve.
557
00:27:12,720 --> 00:27:15,250
And where can it cross the curve
and still be a tangent
558
00:27:15,250 --> 00:27:16,220
line at that point?
559
00:27:16,220 --> 00:27:18,160
At a point of inflection.
560
00:27:18,160 --> 00:27:21,660
By the way, if this is a
smooth type of thing in
561
00:27:21,660 --> 00:27:24,480
general, what we're saying is
that for a point of inflection
562
00:27:24,480 --> 00:27:29,890
to occur, the second derivative
must be 0.
563
00:27:29,890 --> 00:27:31,070
You see because that
means what?
564
00:27:31,070 --> 00:27:32,850
The curve is neither--
565
00:27:32,850 --> 00:27:34,810
it's going from holding to
spilling, so it goes through a
566
00:27:34,810 --> 00:27:37,980
transition where it's
doing neither.
567
00:27:37,980 --> 00:27:41,875
In the same as with first
derivatives, the mere fact
568
00:27:41,875 --> 00:27:44,500
that the second derivative
is 0 does not allow us to
569
00:27:44,500 --> 00:27:46,620
conclude that we have a
point of inflection.
570
00:27:46,620 --> 00:27:49,570
In fact, let me close with this
particular illustration.
571
00:27:49,570 --> 00:27:52,200
Let's take the curve 'y' equals
'x to the fourth'.
572
00:27:52,200 --> 00:27:54,280
The first derivative
is '4 x cubed'.
573
00:27:54,280 --> 00:27:56,230
The second derivative
is '12 x squared'.
574
00:27:56,230 --> 00:27:57,900
The curve is symmetric
with respect to
575
00:27:57,900 --> 00:28:00,280
the y-axis, et cetera.
576
00:28:00,280 --> 00:28:02,980
Using all of the given data you
know the second derivative
577
00:28:02,980 --> 00:28:05,250
is always positive,
what have you.
578
00:28:05,250 --> 00:28:06,940
We can sketch this curve.
579
00:28:06,940 --> 00:28:10,860
And again, in fact the
uninitiated say this curve
580
00:28:10,860 --> 00:28:12,190
looks like a parabola.
581
00:28:12,190 --> 00:28:13,890
What do you mean it looks
like a parabola?
582
00:28:13,890 --> 00:28:16,610
Well, he says, the parabola does
something like this too.
583
00:28:16,610 --> 00:28:19,340
Well, what do we mean by
something like this?
584
00:28:19,340 --> 00:28:21,370
I want to mention a
few points here.
585
00:28:21,370 --> 00:28:24,110
One is, of course, that
actually, the parabola 'y'
586
00:28:24,110 --> 00:28:25,360
equals 'x squared'.
587
00:28:25,360 --> 00:28:30,780
588
00:28:30,780 --> 00:28:32,325
These are going to crisscross
very shortly here.
589
00:28:32,325 --> 00:28:33,520
It doesn't make any
difference.
590
00:28:33,520 --> 00:28:36,620
The parabola 'y' equals 'x
squared' has the same general
591
00:28:36,620 --> 00:28:38,760
shape but with a few different
properties, which we'll
592
00:28:38,760 --> 00:28:39,880
mention in a little while.
593
00:28:39,880 --> 00:28:42,640
But the point that I wanted to
mention here first of all is
594
00:28:42,640 --> 00:28:43,610
simply this.
595
00:28:43,610 --> 00:28:48,790
At the value 'x' equals 0,
'y double prime' is 0.
596
00:28:48,790 --> 00:28:51,850
So you notice that the second
derivative is 0 over here.
597
00:28:51,850 --> 00:28:54,930
Yet even though the second
derivative is 0, notice that
598
00:28:54,930 --> 00:28:57,530
the curve does not
change concavity.
599
00:28:57,530 --> 00:29:00,400
The curve here is always
holding water.
600
00:29:00,400 --> 00:29:03,450
The concluding remark that I
wanted to make is, what is the
601
00:29:03,450 --> 00:29:05,680
relationship between 'y' equals
'x squared' and 'y'
602
00:29:05,680 --> 00:29:06,670
equals 'x to the fourth'?
603
00:29:06,670 --> 00:29:08,580
Or how about 'y' equals
'x to the sixth'?
604
00:29:08,580 --> 00:29:10,720
Or 'y' equals 'x to the 12th'?
605
00:29:10,720 --> 00:29:13,240
Notice that any curve in
that family will look
606
00:29:13,240 --> 00:29:15,160
something like this.
607
00:29:15,160 --> 00:29:15,840
Only what?
608
00:29:15,840 --> 00:29:19,570
As the exponent goes up, the
curve becomes broader in the
609
00:29:19,570 --> 00:29:20,790
neighborhood of 0.
610
00:29:20,790 --> 00:29:25,010
And then once 'x' passes 1, the
curve rises more sharply.
611
00:29:25,010 --> 00:29:27,600
See what we're saying is, if the
magnitude of 'x' is less
612
00:29:27,600 --> 00:29:30,400
than 1, the higher a power
you raise it to,
613
00:29:30,400 --> 00:29:31,750
the smaller 'y' is.
614
00:29:31,750 --> 00:29:34,250
On the other hand, if the
absolute value of 'x' is
615
00:29:34,250 --> 00:29:40,020
greater than 1, the higher a
power you raise it to, the
616
00:29:40,020 --> 00:29:41,640
bigger the output becomes.
617
00:29:41,640 --> 00:29:43,160
But the idea is this.
618
00:29:43,160 --> 00:29:47,470
Notice that the exponent--
619
00:29:47,470 --> 00:29:50,540
in other words, how many
derivatives are 0, gives you a
620
00:29:50,540 --> 00:29:54,080
way of getting into a problem
that will become very, very
621
00:29:54,080 --> 00:29:55,960
crucial as this course
continues.
622
00:29:55,960 --> 00:29:59,905
And it's the idea of, can one
curve be more tangent to a
623
00:29:59,905 --> 00:30:01,380
line than another curve?
624
00:30:01,380 --> 00:30:06,720
You see, all of these curves are
tangent to the line what?
625
00:30:06,720 --> 00:30:09,840
The x-axis, 'y' equals
0 at 'x' equals 0.
626
00:30:09,840 --> 00:30:12,610
How do we distinguish between
these curves?
627
00:30:12,610 --> 00:30:17,370
Well, it seems that some of
these curves fit the x-axis
628
00:30:17,370 --> 00:30:20,390
better than others in a
neighborhood of the
629
00:30:20,390 --> 00:30:21,940
point 'x' equals 0.
630
00:30:21,940 --> 00:30:24,730
See the point that I want to
bring out as to 'y' curve
631
00:30:24,730 --> 00:30:27,510
plotting tells us things that we
don't learn in the ordinary
632
00:30:27,510 --> 00:30:28,970
physics class is this.
633
00:30:28,970 --> 00:30:32,290
If you study calculus the way
it comes up in most physics
634
00:30:32,290 --> 00:30:35,310
courses, we essentially don't go
past the second derivative.
635
00:30:35,310 --> 00:30:35,960
Why?
636
00:30:35,960 --> 00:30:38,880
Because in many cases, we're
studying distance.
637
00:30:38,880 --> 00:30:41,210
And the derivative of distance
is velocity.
638
00:30:41,210 --> 00:30:43,190
The second derivative of
distance, namely the
639
00:30:43,190 --> 00:30:47,380
derivative of velocity
is acceleration.
640
00:30:47,380 --> 00:30:49,690
And we don't usually talk
physically beyond
641
00:30:49,690 --> 00:30:50,780
acceleration.
642
00:30:50,780 --> 00:30:54,530
But notice that in terms of
curve plotting the third,
643
00:30:54,530 --> 00:30:59,440
fourth, fifth, sixth, seventh,
tenth derivatives all have a
644
00:30:59,440 --> 00:31:02,030
meaning that gives you
more information
645
00:31:02,030 --> 00:31:03,630
than what came before.
646
00:31:03,630 --> 00:31:06,970
Don't be deceived by the fact
that in other applications
647
00:31:06,970 --> 00:31:09,490
that you never go past the
second derivative means that
648
00:31:09,490 --> 00:31:13,610
there is no value to knowing how
higher order derivatives
649
00:31:13,610 --> 00:31:15,240
are related to plotting
curves.
650
00:31:15,240 --> 00:31:18,010
At any rate, I think this is
enough of an introduction to
651
00:31:18,010 --> 00:31:20,710
the topic of curve plotting
and curve sketching.
652
00:31:20,710 --> 00:31:23,530
We'll pursue these topics
further in our next lectures.
653
00:31:23,530 --> 00:31:25,020
So until next time, goodbye.
654
00:31:25,020 --> 00:31:28,290
655
00:31:28,290 --> 00:31:30,820
ANNOUNCER: Funding for the
publication of this video was
656
00:31:30,820 --> 00:31:35,540
provided by the Gabriella and
Paul Rosenbaum Foundation.
657
00:31:35,540 --> 00:31:39,710
Help OCW continue to provide
free and open access to MIT
658
00:31:39,710 --> 00:31:43,910
courses by making a donation
at ocw.mit.edu/donate.
659
00:31:43,910 --> 00:31:48,650