WEBVTT
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PROFESSOR: Welcome
back to recitation.
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In this video, I want us to
compute the following limit.
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It's the limit is n goes to
infinity of the sum for i
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equals 0 to n minus 1 of the
following, 2 over n times
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the quantity 2i over n
quantity squared minus 1.
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Now this might look
a little intimidating
00:00:25.630 --> 00:00:28.600
to try and take a
limit of this, but what
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I'd like you to do,
as a hint to you,
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is that you should think about
this as being potentially
00:00:34.800 --> 00:00:38.680
a Riemann sum of a
certain function.
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So if you can figure
out the function,
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and you can figure out
the appropriate interval
00:00:43.370 --> 00:00:46.090
that you're taking a Riemann
sum over, as n goes to infinity,
00:00:46.090 --> 00:00:49.220
you should be able to
write this as an integral.
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We know how to use the
fundamental theorem of calculus
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to determine that a definite
integral in many cases.
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Hopefully this is a function for
which we know a way to do that.
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So that's my hint to you.
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Think about it, it's a Riemann
sum approximating an integral,
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and I'll give you a
while to work on it,
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and then I'll be back.
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OK, welcome back.
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Well hopefully it's been fun
for you to look at this problem
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so far.
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Let me just remind you
what we were doing.
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We were trying to
compute a limit
00:01:23.714 --> 00:01:27.540
as n goes to infinity of the
sum from i equals 0 to n minus 1
00:01:27.540 --> 00:01:33.320
of 2 over n times 2i
over n squared minus 1.
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So I gave you the big hint
that this is probably going
00:01:36.860 --> 00:01:38.220
to be written as an integral.
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So let me show you some
pieces of this sum that
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should help us see what the
integral is, and then I'll
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make a guess about
what this is, and then
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I'll try to give an educated
way to check my guess.
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So the first thing we noticed
is that there is one thing,
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this is a product of two
functions, and one of them--
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well, of n, I guess.
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But this is a product
of two things.
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One thing appears
in every single term
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that you have for i.
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So the sum has n
terms, and they're all
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going to be 2 over n times this,
and the i is going to change.
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But this does not change, this
2 over n does not change, right?
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In fact, I could even
pull that out if I wanted.
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But, I don't want to pull
it out of the sum right now.
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I want us to look
at what's actually
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going on in this product.
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So if this thing is appearing
over and over again,
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and we know this is
probably a Riemann sum,
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then chances are
this is our delta x.
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So delta x being
equal to 2 over n,
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we know delta x equals b minus
a over n, where b and a are
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our left endpoint-- oh,
sorry, our right endpoint--
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and our left endpoint.
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Right?
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We integrate from a to b.
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So b minus a is the
length of the interval.
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So this is really
dividing up whatever
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interval we're integrating
over, into n equal subintervals.
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So that's my first thought,
is that b minus a over n
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is equal to 2 over n.
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And now we want to try
and figure out, well,
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what the heck is this.
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Well, when we take
a Riemann sum,
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remember when we take a
Riemann sum what we get.
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We get the sum of delta
x times f of x sub i,
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and i is what's varying
from 0 to n minus 1.
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Let me put a little
curve in here
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so we see those are
two different things.
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So this is i equals
0 to n minus 1.
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I have this delta x here.
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I'm anticipating this
is some f of x sub i.
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And so the question
is, what f is it?
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Right?
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If I know what f it is, than
I know that this sum will
00:03:34.900 --> 00:03:37.690
be equal to something,
the integral from a
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to b of f of x dx, and a
and b will differ by 2.
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So that's where I'm heading.
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So now the question is,
what is this a function of?
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What function is
this, I should say.
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Now first guess
would be something
00:03:53.036 --> 00:03:57.270
like, well, I'm taking some
quantity, I'm squaring it,
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and subtracting 1.
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So my first guess for this
function is x squared minus 1.
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I mean, that seems easy to me.
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Let's see if this would
actually even make sense just
00:04:07.690 --> 00:04:12.180
by looking at the subscripts,
or sorry, the index, the indices
00:04:12.180 --> 00:04:14.170
I have here.
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So what do I have?
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Well, when I put in
i equals 0-- let's
00:04:17.990 --> 00:04:20.720
put down some of these values--
when I put in i equals 0,
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I get 2 times 0 over
n squared minus 1.
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When I put in i equals
1, I get 2 times 1
00:04:29.030 --> 00:04:32.470
over n squared minus 1.
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And I go all the
way up, to 2 times n
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minus 1 over n squared minus 1.
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So it's kind of a long
sum there, but these are,
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this is what our
sum of these things
00:04:46.220 --> 00:04:49.230
looks like if I pull
out the 2 over n.
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So here I get 0 squared minus 1.
00:04:53.380 --> 00:04:54.760
That looks pretty good.
00:04:54.760 --> 00:04:58.170
Here I get 2 times 1
over n squared minus 1.
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So it does look like I'm doing
something, taking something,
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squaring it, subtracting 1.
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Does it make sense
that these are
00:05:03.950 --> 00:05:05.780
the kind of x values
I would expect
00:05:05.780 --> 00:05:10.000
to get if this were the Riemann
sum of x squared minus 1?
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It does, and let's
think about why.
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I'm starting at x equals 0
here, it sure looks like.
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Let's look at what happens when
I go all the way over here.
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What happens when n gets
really, really big, is it
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this ratio approaches 2.
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So it's 2 times n minus 1
over n. n minus 1 over n,
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as n gets arbitrarily
large, as n gets really big,
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then this approaches 2.
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So this is approaching
2 squared minus 1.
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So it's giving me more
evidence that this is probably
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the function x squared minus 1.
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And now I'm starting to
believe the interval is 0 to 2.
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I know it's a length 2
interval, and it's looking
00:05:49.480 --> 00:05:51.250
like the interval is 0 to 2.
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Let's come back and talk
about one more thing.
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The one other thing
that you should notice
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is that how does
this value differ
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from this value, and the next,
and the next, and the next?
00:06:01.310 --> 00:06:04.050
They differ by 2 over n.
00:06:04.050 --> 00:06:06.410
So each time whatever
input I had previously,
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I'm now adding 2 over
n to the next input.
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And that should
make sense of what
00:06:10.970 --> 00:06:12.940
we know about Riemann
sums, because what I do,
00:06:12.940 --> 00:06:16.620
is I divide my interval
into these subintervals
00:06:16.620 --> 00:06:18.440
of length 2 over n.
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I'm evaluating it
at one x-value,
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that-- I'm starting,
in this case, at 0.
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Then the next interval
is 2 over n more.
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Then I evaluate at that x-value.
00:06:27.225 --> 00:06:31.260
The next one is 2 over n more,
and I evaluate at that x-value.
00:06:31.260 --> 00:06:33.640
So this is looking like--
I'm going to write it here,
00:06:33.640 --> 00:06:41.950
this is my guess-- integral from
0 to 2 of x squared minus 1 dx.
00:06:41.950 --> 00:06:45.050
And now to make myself
feel good about this--
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I'm pretty sure it's that.
00:06:46.510 --> 00:06:48.410
To make you feel
good about this,
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I'm going to divide this
into four subintervals,
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and I'm going to show you
what the Riemann sum with four
00:06:54.140 --> 00:06:55.880
intervals looks
like, and then we
00:06:55.880 --> 00:07:00.380
can talk about how it relates
to this one over here.
00:07:00.380 --> 00:07:02.510
OK, so let me draw a graph.
00:07:02.510 --> 00:07:04.260
Actually, I'll use
just white chalk again.
00:07:04.260 --> 00:07:10.490
Let me draw a graph of x
squared minus 1 from 0 to 2.
00:07:10.490 --> 00:07:18.080
So 0, 1, 2, minus 1.
00:07:18.080 --> 00:07:22.490
OK, so at 0, x squared
minus 1 is negative 1.
00:07:22.490 --> 00:07:26.820
At x equals 1, x
squared minus 1 is 0.
00:07:26.820 --> 00:07:31.460
And at 2, x squared
minus 1 is 3.
00:07:31.460 --> 00:07:36.770
So hopefully, this is all
going to go into the video,
00:07:36.770 --> 00:07:39.370
and-- in the video
screen, I mean.
00:07:39.370 --> 00:07:41.120
And there we go,
something like that.
00:07:41.120 --> 00:07:43.410
So this is, you know,
it continues over here,
00:07:43.410 --> 00:07:45.870
but I'm really only
interested in this part.
00:07:45.870 --> 00:07:47.840
So now let's look at what
the subintervals are.
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And now I'm going to
get some colored chalk.
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So what are the subintervals?
00:07:51.460 --> 00:07:52.798
I'm taking 1 over 4, OK?
00:07:56.020 --> 00:08:01.204
And so delta x, in
this case, is 2 over 4,
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which is equal to 1/2.
00:08:02.120 --> 00:08:04.730
Right?
00:08:04.730 --> 00:08:07.122
And so what are my,
what are-- so what
00:08:07.122 --> 00:08:08.330
is my sum going to look like?
00:08:08.330 --> 00:08:10.329
Well, I am going to tell
you that I'm also going
00:08:10.329 --> 00:08:11.680
to use left-handed endpoints.
00:08:11.680 --> 00:08:14.130
And I mentioned earlier
why that is, I believe.
00:08:14.130 --> 00:08:15.580
Maybe I didn't.
00:08:15.580 --> 00:08:18.300
But, I started
off at i equals 0,
00:08:18.300 --> 00:08:20.350
and my first input value was 0.
00:08:20.350 --> 00:08:24.510
My last input value had an n
minus 1 in it instead of an n.
00:08:24.510 --> 00:08:28.380
So I was doing, somehow,
the second-to-last place
00:08:28.380 --> 00:08:29.630
that I was interested in here.
00:08:29.630 --> 00:08:32.700
So it's definitely more of
a left-hand endpoint thing.
00:08:32.700 --> 00:08:35.020
So I'm going to do this
with left-hand endpoints.
00:08:35.020 --> 00:08:37.300
And I'm not going
to simplify as I go.
00:08:37.300 --> 00:08:40.740
I'm going to write it out
in sort of an expanded form.
00:08:40.740 --> 00:08:42.559
OK, so let's write it
out in expanded form.
00:08:42.559 --> 00:08:46.700
So the Riemann sum-- this is
y equals x squared minus 1.
00:08:46.700 --> 00:08:51.470
The Riemann sum is, the
first term is 1/2 times what?
00:08:51.470 --> 00:08:54.520
It's the value,
this x-value, which
00:08:54.520 --> 00:09:01.240
is 0, evaluated on this
curve, so 0 squared minus 1.
00:09:01.240 --> 00:09:04.010
The next term-- I'll just write
them right below each other--
00:09:04.010 --> 00:09:04.905
is 1/2.
00:09:04.905 --> 00:09:08.170
'Cause again, let's
draw a picture of what
00:09:08.170 --> 00:09:09.170
the first one is, sorry.
00:09:09.170 --> 00:09:12.570
It's this rectangle.
00:09:12.570 --> 00:09:13.460
Right?
00:09:13.460 --> 00:09:17.810
It's evaluated--
It's length 1/2,
00:09:17.810 --> 00:09:20.410
and it's the function
evaluated at 0.
00:09:20.410 --> 00:09:21.920
The next one is
length 1/2, and it's
00:09:21.920 --> 00:09:23.570
going to be the
function evaluated
00:09:23.570 --> 00:09:26.280
at whatever this left-hand
endpoint is here.
00:09:26.280 --> 00:09:29.780
So it's going to be this area.
00:09:29.780 --> 00:09:33.080
So it's going to be
length 1/2, and then
00:09:33.080 --> 00:09:36.520
the height is going
to be at x equals 1/2,
00:09:36.520 --> 00:09:41.790
so 1/2 quantity squared minus 1.
00:09:41.790 --> 00:09:45.420
The next one is going
to be this interval.
00:09:45.420 --> 00:09:49.190
Well, there's no rectangle
to draw because it's just,
00:09:49.190 --> 00:09:51.960
the output is zero at
the left endpoint here.
00:09:51.960 --> 00:09:54.770
So it's just going
to be-- it's going
00:09:54.770 --> 00:09:58.750
to have output equal to 0,
at length 1/2 and height 0.
00:09:58.750 --> 00:10:00.020
But we'll write it out anyway.
00:10:00.020 --> 00:10:03.102
It's going to be 1/2
times the quantity-- now,
00:10:03.102 --> 00:10:09.370
I went up 1/2 more, so
it's going to be two 1/2's,
00:10:09.370 --> 00:10:11.590
two times 1/2 squared minus 1.
00:10:11.590 --> 00:10:14.710
Let me just show
you why I did this.
00:10:14.710 --> 00:10:16.610
OK, if we look at
the picture, here I'd
00:10:16.610 --> 00:10:18.900
gone up 1/2 from
my initial value.
00:10:18.900 --> 00:10:21.710
Here I'd gone up another
1/2 from my initial value.
00:10:21.710 --> 00:10:24.496
So that's two 1/2's from
my initial value of 0.
00:10:24.496 --> 00:10:26.496
The next one is going to
be three 1/2's, so this
00:10:26.496 --> 00:10:29.501
is three 1/2's away, or
commonly known as 3/2.
00:10:29.501 --> 00:10:30.000
OK?
00:10:32.710 --> 00:10:37.970
So that one is going to be--
1/2 is the base length again,
00:10:37.970 --> 00:10:44.000
times the quantity 3
times 1/2 squared minus 1.
00:10:44.000 --> 00:10:47.265
And that is in the
picture, this rectangle.
00:10:51.951 --> 00:10:52.450
Great.
00:10:52.450 --> 00:10:53.780
So what are we see here?
00:10:53.780 --> 00:10:57.590
If we look at this, these
four pieces, what do we have?
00:10:57.590 --> 00:10:59.890
We have a 1/2 in
front each time.
00:10:59.890 --> 00:11:01.100
Which, what was the 1/2?
00:11:01.100 --> 00:11:03.460
It was b minus a over n.
00:11:03.460 --> 00:11:07.230
So b minus a was 2, n was 4.
00:11:07.230 --> 00:11:10.127
So maybe I should have
kept that as 2 over 4.
00:11:10.127 --> 00:11:11.960
But, it's a little
easier to write it as 1/2
00:11:11.960 --> 00:11:13.350
because of what I'm doing next.
00:11:13.350 --> 00:11:15.470
I square something,
I subtract 1.
00:11:15.470 --> 00:11:19.590
I go up by the value that this
is from the initial one here.
00:11:19.590 --> 00:11:23.000
And so now I'm taking the
output of what was in here.
00:11:23.000 --> 00:11:27.410
I now take the output at
1/2 more than what was here.
00:11:27.410 --> 00:11:31.210
Now I take it at two 1/2's more
than what was here, or 1/2 more
00:11:31.210 --> 00:11:33.950
than what was there,
and then three 1/2's
00:11:33.950 --> 00:11:36.430
more than what was here, or
one more than what was there.
00:11:36.430 --> 00:11:38.846
That's kind of confusing, but
let's go back to the picture
00:11:38.846 --> 00:11:41.090
and see what it is.
00:11:41.090 --> 00:11:43.480
My delta x was 1/2, right?
00:11:43.480 --> 00:11:44.930
So I evaluate at
the first place,
00:11:44.930 --> 00:11:49.600
and I evaluate one more up, and
then I evaluate one more up,
00:11:49.600 --> 00:11:52.630
and I evaluate one more up,
which gives me outputs here,
00:11:52.630 --> 00:11:53.750
there, there, and there.
00:11:53.750 --> 00:11:55.370
Right?
00:11:55.370 --> 00:11:57.710
So really if you go
back and you look
00:11:57.710 --> 00:12:04.090
at the formulation of the
sum, this was 2 over n times
00:12:04.090 --> 00:12:09.250
quantity 2i over
n squared minus 1,
00:12:09.250 --> 00:12:13.210
you can see the 2
over n is my 1/2,
00:12:13.210 --> 00:12:15.850
and then this is maybe
the hardest part to see,
00:12:15.850 --> 00:12:18.200
but that's the 2 over
n is my 1/2 again,
00:12:18.200 --> 00:12:24.400
and the i is this thing
that's coming in as 1, 2, 3.
00:12:24.400 --> 00:12:27.810
And so that i was going
from 0 to n minus 1-- so I
00:12:27.810 --> 00:12:31.480
should have said 0, 1, 2, 3.
00:12:31.480 --> 00:12:31.980
Right?
00:12:31.980 --> 00:12:35.705
So that i is the 0 to n minus
1, and then I'm evaluating that,
00:12:35.705 --> 00:12:37.740
and then I add them all up.
00:12:37.740 --> 00:12:44.620
So when I take the sum, I get,
for n equals 4, I get this.
00:12:44.620 --> 00:12:46.512
So in fact, this
is just a guess,
00:12:46.512 --> 00:12:47.886
still maybe you
should, maybe you
00:12:47.886 --> 00:12:49.480
should convince yourself more.
00:12:49.480 --> 00:12:52.150
I'm actually convinced
at this point,
00:12:52.150 --> 00:12:54.280
based on not just this
evidence, but the evidence I
00:12:54.280 --> 00:12:57.030
understood before about
how the function works.
00:12:57.030 --> 00:13:00.140
Maybe you want to
compare it when n equals
00:13:00.140 --> 00:13:01.330
6, or something like that.
00:13:01.330 --> 00:13:02.788
You may need a
little more evidence
00:13:02.788 --> 00:13:05.797
to make you understand
this particular piece.
00:13:05.797 --> 00:13:07.380
But, hopefully that
makes sense to you
00:13:07.380 --> 00:13:10.150
that this is really just
i times delta x, and then
00:13:10.150 --> 00:13:11.780
evaluated somewhere.
00:13:11.780 --> 00:13:15.900
That's the main idea
of this component.
00:13:15.900 --> 00:13:17.989
OK, well hopefully this
is informative to you.
00:13:17.989 --> 00:13:19.780
If you want to know
the exact answer of how
00:13:19.780 --> 00:13:22.720
to compute the sum, obviously
you just take the integral.
00:13:22.720 --> 00:13:24.570
So I know you can do that.
00:13:24.570 --> 00:13:26.106
So that's where I'll stop.