WEBVTT
Kind: captions
Language: en
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hi.
00:00:01.060 --> 00:00:02.900
okay so you have an integral
00:00:02.900 --> 00:00:05.620
and inside the integral youll have a function
00:00:05.800 --> 00:00:07.480
derivative of a function
00:00:09.700 --> 00:00:10.800
okay and
00:00:11.560 --> 00:00:12.760
its the chain rule
00:00:13.040 --> 00:00:14.080
repeated the function
00:00:18.300 --> 00:00:18.800
no.
00:00:19.900 --> 00:00:21.980
dress yeah youre in the middle.
00:00:22.060 --> 00:00:23.900
youre usually in the middle
00:00:24.260 --> 00:00:25.940
its just throwing me off.
00:00:32.480 --> 00:00:36.160
the chain rule created a function inside of a function.
00:00:36.200 --> 00:00:37.640
remember the example
00:00:37.880 --> 00:00:40.240
i had something like sin(x^3)
00:00:40.740 --> 00:00:42.260
the derivative of that
00:00:43.600 --> 00:00:45.260
we got cos(x^3)
00:00:46.780 --> 00:00:48.100
times 3x^2
00:00:48.820 --> 00:00:49.320
right?
00:00:50.020 --> 00:00:51.620
so if you had an integral
00:00:52.280 --> 00:00:53.640
of 3x^2
00:00:54.340 --> 00:00:56.300
cos(x^3)
00:00:57.540 --> 00:00:58.040
dx
00:00:59.120 --> 00:01:01.520
its not obvious when you look at that
00:01:02.000 --> 00:01:02.640
that this
00:01:03.160 --> 00:01:04.920
just turns into sin(x^3).
00:01:05.700 --> 00:01:08.720
you guys certainly arent at that level of sophistication
00:01:08.720 --> 00:01:10.880
where you just sort of look and go oh cool
00:01:11.280 --> 00:01:12.400
thats sin(x^3)
00:01:13.960 --> 00:01:15.060
+c right?
00:01:16.340 --> 00:01:17.540
what you want to do
00:01:17.540 --> 00:01:19.960
is you want to be able to look at some integral
00:01:20.400 --> 00:01:22.920
and figure out what the original function was
00:01:22.920 --> 00:01:25.840
if the integral contains both the function and the derivative.
00:01:26.220 --> 00:01:29.660
to do that you use substitution and that helps us..a lot easier.
00:01:30.360 --> 00:01:32.760
so if for example i had an integral of
00:01:40.780 --> 00:01:44.920
thats going to be one of these types of functions cause you have a function and its derivative.
00:01:44.940 --> 00:01:48.620
its not exactly obvious what the original function is.
00:01:48.620 --> 00:01:50.540
you could probably play around and get there.
00:01:51.660 --> 00:01:53.760
but in the beginning it might be a little tricky
00:01:53.760 --> 00:01:54.500
so what you do
00:01:55.080 --> 00:01:58.520
we say well whats the simplified function in there?
00:01:58.520 --> 00:01:59.780
and its derivative.
00:01:59.880 --> 00:02:02.040
well the derivative of x^2 is 2x.
00:02:02.760 --> 00:02:04.200
i got x so thats close.
00:02:05.320 --> 00:02:05.960
so if i let
00:02:06.440 --> 00:02:07.240
the letter u
00:02:08.600 --> 00:02:10.440
stand for 5+x^2
00:02:11.460 --> 00:02:12.800
can you hear me back there?
00:02:15.000 --> 00:02:16.040
can you hear me?
00:02:16.300 --> 00:02:17.260
how about now?
00:02:17.760 --> 00:02:18.400
alright.
00:02:18.720 --> 00:02:21.200
so if we let u=5+x^2
00:02:21.980 --> 00:02:22.480
du
00:02:23.360 --> 00:02:23.920
would be
00:02:25.300 --> 00:02:26.260
well du/dx
00:02:28.320 --> 00:02:29.120
would be 2x.
00:02:30.000 --> 00:02:30.500
ok?
00:02:31.380 --> 00:02:33.020
so if i cross multiply
00:02:34.540 --> 00:02:38.500
i get du is 2x dx.
00:02:39.740 --> 00:02:43.740
so i look inside the integral and i say well i almost have 2xdx.
00:02:44.300 --> 00:02:47.300
i have x dx right? i have x
00:02:47.780 --> 00:02:48.280
dx
00:02:48.860 --> 00:02:49.680
so if i
00:02:51.360 --> 00:02:52.640
if i multiply by 1/2
00:02:53.180 --> 00:02:54.220
or divide it by 2
00:02:55.420 --> 00:02:58.320
i get 1/2du=x dx.
00:02:59.920 --> 00:03:01.900
so now i can move over to the integral
00:03:01.900 --> 00:03:05.800
and i can substitute so instead of the integral in terms of x
00:03:06.180 --> 00:03:08.460
its going to be an integral in terms of u.
00:03:09.320 --> 00:03:09.820
but
00:03:10.500 --> 00:03:12.100
i cant have any x's left.
00:03:12.100 --> 00:03:13.880
it has to all be in terms of u.
00:03:15.080 --> 00:03:16.200
so i say well this
00:03:16.720 --> 00:03:18.480
is now the square root of u.
00:03:19.100 --> 00:03:20.220
thats easy to do.
00:03:21.260 --> 00:03:23.780
easy integral. and x dx
00:03:24.400 --> 00:03:25.060
is 1/2
00:03:26.100 --> 00:03:28.100
du. i put the 1/2 on the outside
00:03:28.100 --> 00:03:30.160
cause we're integrating not differentiating.
00:03:30.160 --> 00:03:32.880
if you multiply by a constant the constant just stays there.
00:03:32.880 --> 00:03:35.820
so you pull it out of the u make it easier to integrate.
00:03:36.620 --> 00:03:39.660
alright this is the same thing as saying a half
00:03:40.600 --> 00:03:42.120
u^1/2 right?
00:03:45.380 --> 00:03:46.680
so now we integrate that.
00:03:47.800 --> 00:03:48.840
the integral is
00:03:49.560 --> 00:03:50.060
1/2
00:03:52.300 --> 00:03:52.800
u^3/2
00:03:53.760 --> 00:03:54.600
over 3/2
00:03:56.280 --> 00:03:56.780
+c.
00:04:01.060 --> 00:04:02.980
which, with a little algebra
00:04:04.900 --> 00:04:06.320
is the same as 1/3
00:04:07.660 --> 00:04:08.760
u^3/2 + c
00:04:09.540 --> 00:04:11.780
so now i just substitute back for u
00:04:12.580 --> 00:04:15.180
the handwriting gets harder as i move down the board.
00:04:16.580 --> 00:04:17.920
thats 1/3 of
00:04:18.380 --> 00:04:19.820
5+x^2
00:04:21.220 --> 00:04:21.860
to the 3/2
00:04:23.440 --> 00:04:23.940
+c.
00:04:24.740 --> 00:04:25.240
ok?
00:04:35.520 --> 00:04:36.160
alright.
00:04:36.160 --> 00:04:39.420
lets do another one and then you guys get to do a couple.
00:05:03.300 --> 00:05:04.180
so again i look
00:05:04.480 --> 00:05:06.580
now i could take sinx+3
00:05:07.120 --> 00:05:08.000
expand it out
00:05:08.000 --> 00:05:09.600
its to the 10 so it just wouldnt work
00:05:09.880 --> 00:05:11.640
i could distribute cosine
00:05:12.320 --> 00:05:14.880
and probably get a whole bunch of messy
00:05:14.880 --> 00:05:16.860
integrals so why would i do that?
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i look and i say
00:05:19.460 --> 00:05:21.140
the derivative of cosine
00:05:21.920 --> 00:05:22.420
is -sin
00:05:22.420 --> 00:05:23.960
but that wouldnt be very useful
00:05:24.060 --> 00:05:26.380
cause then id have a- this would be u
00:05:26.380 --> 00:05:28.280
and this would be du to the tenth
00:05:28.280 --> 00:05:29.700
which is not helpful.
00:05:30.060 --> 00:05:31.800
but if i let u be sinx
00:05:32.660 --> 00:05:33.160
+3
00:05:33.460 --> 00:05:35.620
the derivative of sinx+3 is cosx
00:05:36.400 --> 00:05:38.480
and this would now become u^10.
00:05:38.480 --> 00:05:39.220
which is easy.
00:05:39.220 --> 00:05:42.160
and this would become du so lets do the substitution.
00:05:42.660 --> 00:05:46.960
i let u=sinx+3
00:05:47.580 --> 00:05:51.500
du is just cosxdx.
00:05:52.640 --> 00:05:53.140
ok?
00:05:53.580 --> 00:05:55.760
so if i have du is cosxdx
00:05:56.560 --> 00:05:58.640
and u is sinx+3 now i substitute
00:06:00.340 --> 00:06:00.840
this
00:06:02.600 --> 00:06:03.320
thats u^10
00:06:04.280 --> 00:06:04.780
this
00:06:06.500 --> 00:06:07.000
is du.
00:06:08.240 --> 00:06:10.880
so usually when you do the substitution
00:06:10.880 --> 00:06:13.560
with these problems youll end up with a power rule
00:06:14.220 --> 00:06:16.700
or something simple like a sin or a cos
00:06:17.120 --> 00:06:17.620
ok?
00:06:18.240 --> 00:06:20.560
pretty much all we can give you right now.
00:06:20.920 --> 00:06:21.720
maybe a log.
00:06:22.140 --> 00:06:24.300
you can integrate a log which is sort of power rule.
00:06:25.480 --> 00:06:26.440
okay so this is
00:06:28.320 --> 00:06:29.620
(u^11)/11
00:06:31.280 --> 00:06:31.780
+c
00:06:31.780 --> 00:06:34.320
now you subtract -you substitute back
00:06:34.980 --> 00:06:36.280
you get sinx+3
00:06:38.000 --> 00:06:38.720
to the 11th
00:06:39.620 --> 00:06:40.120
over 11
00:06:42.020 --> 00:06:42.520
+c.
00:06:44.160 --> 00:06:46.800
how we doing so far? questions?
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no?
00:06:49.080 --> 00:06:50.960
i dont know what youre doing questions?
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im totally confused?
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ill just sit here and do instagram? yes?
00:07:05.860 --> 00:07:08.180
alright so first is you have to figure out
00:07:08.320 --> 00:07:11.120
whats going to be u and whats going to be du.
00:07:11.360 --> 00:07:13.760
you could sort of do trial and error.
00:07:15.500 --> 00:07:16.460
but the idea is
00:07:16.460 --> 00:07:18.940
you want to find 1 function and its derivative.
00:07:19.120 --> 00:07:20.560
so i look here and i say
00:07:20.960 --> 00:07:22.600
if i let u be cosine
00:07:22.800 --> 00:07:26.160
the derivative of cosine is -sin thats kind of sin.
00:07:26.440 --> 00:07:28.360
but this would have been the u
00:07:28.760 --> 00:07:30.460
and this would be du^10
00:07:30.820 --> 00:07:31.860
which is weird.
00:07:32.660 --> 00:07:37.740
so i say lets try the other way around what if i let u be sinx well then itd be u+3 which isnt very useful.
00:07:38.320 --> 00:07:40.520
but if i let u=sinx+3
00:07:41.600 --> 00:07:43.760
du is just cosxdx.
00:07:44.560 --> 00:07:45.060
this
00:07:45.700 --> 00:07:46.660
becomes u^10.
00:07:47.020 --> 00:07:49.320
and the remainder is just du.
00:07:51.860 --> 00:07:53.940
ive substituted all the parts.
00:07:54.280 --> 00:07:57.020
(sinx+3)^10 becomes u^10
00:07:57.820 --> 00:07:59.540
and cosxdx becomes du.
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notice
00:08:00.780 --> 00:08:03.580
i dont let u=(sinx+3)^10
00:08:04.220 --> 00:08:04.720
okay?
00:08:04.960 --> 00:08:06.240
thats not helpful.
00:08:06.240 --> 00:08:08.040
because the derivative of something to the tenth
00:08:08.040 --> 00:08:10.360
would be 10 times the thing to the 9th.
00:08:10.680 --> 00:08:12.520
i just let it be whats inside
00:08:12.660 --> 00:08:13.620
the function.
00:08:13.620 --> 00:08:16.060
because having stuff to the tenth is a problem
00:08:16.060 --> 00:08:16.860
for integrating.
00:08:16.860 --> 00:08:19.300
right cause you make it to the 11th divided by 11.
00:08:19.300 --> 00:08:21.040
its whats inside that i have to fix.
00:08:22.260 --> 00:08:25.620
but then i substitute and now i just get u^11/11
00:08:27.240 --> 00:08:28.760
and i substitute back.
00:08:29.060 --> 00:08:29.560
ok?
00:08:30.260 --> 00:08:31.960
lets have you guys try a couple.
00:08:36.880 --> 00:08:39.520
there you go, theres 3 to start off with.
00:08:41.820 --> 00:08:43.580
so you look at the first one
00:08:43.820 --> 00:08:45.340
and you say to yourself
00:08:45.640 --> 00:08:47.160
what do i do with that 7?
00:08:47.260 --> 00:08:49.720
cause thats annoying to have that 7 there.
00:08:49.780 --> 00:08:51.220
just take the 7 out.
00:09:01.200 --> 00:09:01.840
alright.
00:09:02.660 --> 00:09:03.300
remember
00:09:03.300 --> 00:09:05.720
you move constants in and out of the integrand
00:09:05.720 --> 00:09:07.480
youre supposed to multiply ok?
00:09:11.600 --> 00:09:13.860
so let u=x^2+8.
00:09:18.260 --> 00:09:18.760
du
00:09:22.280 --> 00:09:23.560
du is 2xdx.
00:09:25.640 --> 00:09:27.400
now i look and i say alright
00:09:27.500 --> 00:09:28.460
this would be u
00:09:28.820 --> 00:09:32.580
and i have x left so i just have to play with du for a second.
00:09:33.040 --> 00:09:34.400
make that 1/2du
00:09:37.960 --> 00:09:38.800
is xdx.
00:09:41.520 --> 00:09:42.480
so far so good?
00:09:43.020 --> 00:09:44.300
so now i substitute
00:09:44.980 --> 00:09:46.100
and i have 7
00:09:46.720 --> 00:09:47.800
times 1/2
00:09:49.200 --> 00:09:50.200
times u^5
00:09:52.900 --> 00:09:53.400
du.
00:09:54.200 --> 00:09:55.300
thats 7/2
00:09:55.680 --> 00:09:57.840
or 3.5 if youre feeling decimal.
00:09:59.040 --> 00:09:59.540
ok?
00:10:01.200 --> 00:10:02.240
so that becomes
00:10:03.300 --> 00:10:04.180
7/2..
00:10:04.880 --> 00:10:08.220
times u^6/6
00:10:08.480 --> 00:10:09.600
plus a constant.
00:10:14.200 --> 00:10:15.960
and then substitute back.
00:10:16.980 --> 00:10:18.020
so you would get
00:10:19.260 --> 00:10:20.600
7/12..
00:10:21.680 --> 00:10:24.760
(x^2+8)^6
00:10:25.200 --> 00:10:27.520
+c. to anticipate your question..
00:10:27.740 --> 00:10:30.220
no you dont have to turn this into 7/12
00:10:30.220 --> 00:10:33.860
you could leave it 7 times 1/2 with the thing over 6, thats fine.
00:10:34.240 --> 00:10:34.740
ok?
00:10:34.740 --> 00:10:37.820
simplifying is not important to a question like this.
00:10:37.820 --> 00:10:40.580
demonstrate that you know what youre doing.
00:10:52.320 --> 00:10:54.160
alright how about this one?
00:10:55.280 --> 00:10:58.240
well the 5, yeah you could just take the 5 out.
00:11:04.580 --> 00:11:07.300
then you could think of that if you wanted
00:11:13.880 --> 00:11:14.380
ok?
00:11:14.940 --> 00:11:15.980
cause lnx/x
00:11:16.120 --> 00:11:18.260
is the same as lnx -1/x
00:11:18.260 --> 00:11:19.740
you dont have to do this.
00:11:20.060 --> 00:11:22.140
but if it makes it easier for you
00:11:22.640 --> 00:11:23.140
so then
00:11:23.380 --> 00:11:24.420
u would be
00:11:25.220 --> 00:11:25.900
lnx
00:11:28.060 --> 00:11:29.020
du would be 1/x
00:11:29.960 --> 00:11:30.460
dx.
00:11:34.380 --> 00:11:35.680
when you write on the board
00:11:36.000 --> 00:11:37.040
step to the side
00:11:37.040 --> 00:11:38.520
and see what youre doing
00:11:38.620 --> 00:11:40.140
rather than doing this
00:11:41.560 --> 00:11:42.940
students find very annoying.
00:11:44.200 --> 00:11:46.760
thought id share that teachable moment with you.
00:11:46.760 --> 00:11:48.320
in case some of you out of desperation
00:11:48.320 --> 00:11:49.820
rather than going to med school
00:11:49.820 --> 00:11:51.760
end up with something terrible like this.
00:11:59.960 --> 00:12:00.940
the u=lnx
00:12:01.480 --> 00:12:02.940
du is 1/xdx
00:12:03.520 --> 00:12:05.760
so now this is just the integral of
00:12:06.060 --> 00:12:06.560
du.
00:12:22.940 --> 00:12:24.940
so u is lnx
00:12:25.640 --> 00:12:27.200
du is 1/xdx
00:12:27.200 --> 00:12:28.220
so this is just u
00:12:28.260 --> 00:12:30.260
and the integral of u is u^2/2.
00:12:36.800 --> 00:12:38.440
and now you just substitute back.
00:12:39.460 --> 00:12:40.500
so that would be
00:12:43.740 --> 00:12:44.860
5/2
00:12:45.920 --> 00:12:46.640
lnx
00:12:48.120 --> 00:12:48.840
squared..
00:12:49.420 --> 00:12:50.540
plus a constant.
00:12:51.760 --> 00:12:54.060
which you could write in more than 1 way ok?
00:12:56.580 --> 00:12:57.540
so far so good?
00:12:58.920 --> 00:12:59.560
alright.
00:13:00.840 --> 00:13:03.460
what about tan6xsc^2xdx?
00:13:05.540 --> 00:13:07.700
well if i let u=tanx
00:13:09.780 --> 00:13:10.280
du
00:13:11.620 --> 00:13:12.840
is sec^2x
00:13:14.920 --> 00:13:15.420
dx.
00:13:27.160 --> 00:13:28.600
so now this is just
00:13:30.200 --> 00:13:32.580
u^6du
00:13:34.940 --> 00:13:36.720
which is u^7/7
00:13:38.460 --> 00:13:39.580
plus a constant.
00:13:40.540 --> 00:13:41.100
so thats
00:13:43.120 --> 00:13:44.760
n^7x/7
00:13:45.920 --> 00:13:47.040
plus a constant.
00:14:05.200 --> 00:14:06.080
suppose i had
00:14:14.140 --> 00:14:16.380
suppose i had something like that.
00:14:21.840 --> 00:14:22.960
thats annoying.
00:14:24.060 --> 00:14:26.480
cause what am i going to use u substitution on?
00:14:28.880 --> 00:14:29.600
any ideas?
00:14:30.240 --> 00:14:32.060
not going to do u substitution.
00:14:32.720 --> 00:14:33.220
first
00:14:33.660 --> 00:14:35.800
whats 1-cos^2 the same as?
00:14:37.100 --> 00:14:38.940
really? you guys know that?
00:14:40.260 --> 00:14:41.300
thats amazing.
00:14:44.420 --> 00:14:47.060
that would just be the integral of sinx.
00:14:48.940 --> 00:14:51.340
dont forget your trig substitutions.
00:14:52.160 --> 00:14:52.660
ok?
00:14:53.100 --> 00:14:56.740
because we can make things trickier by first making you do a lot of trig
00:14:56.740 --> 00:14:59.080
so remember how we did the integral of tangent?
00:14:59.560 --> 00:15:03.880
you rewrite that first as sin/cos and its a much easier integral.
00:15:04.880 --> 00:15:06.320
this is just -cos.
00:15:11.360 --> 00:15:13.560
now i could do these as definite integrals.
00:15:19.380 --> 00:15:20.740
lets do one of those.
00:15:48.020 --> 00:15:49.060
i like that one.
00:16:05.280 --> 00:16:07.120
by the way when in doubt just write pi.
00:16:09.120 --> 00:16:11.100
pi..0..1..
00:16:12.120 --> 00:16:12.620
100..
00:16:12.620 --> 00:16:14.740
heck maybe one of those will turn up
00:16:16.980 --> 00:16:18.700
pi is not going to be the right answer to this.
00:16:26.000 --> 00:16:28.640
alright thats probably long enough.
00:16:31.400 --> 00:16:33.920
so lets let u=2-e^x
00:16:39.760 --> 00:16:40.260
du
00:16:42.060 --> 00:16:45.540
is -e^x dx
00:16:47.360 --> 00:16:49.280
and thats almost what i have.
00:16:49.740 --> 00:16:51.300
except for the negative thing.
00:16:52.120 --> 00:16:52.780
-du
00:16:54.520 --> 00:16:55.560
is e^xdx.
00:16:56.260 --> 00:16:57.940
so leaving out the limits
00:16:59.400 --> 00:17:04.040
this would now be negative the integral of du/u.
00:17:04.040 --> 00:17:05.920
these are called the limits of integration.
00:17:07.240 --> 00:17:09.400
now you could change the limits.
00:17:09.840 --> 00:17:11.660
ok which we're not going to do today.
00:17:12.020 --> 00:17:13.220
change the limits
00:17:13.220 --> 00:17:17.680
and then you do the whole integral in terms of u and then you dont have to change the
00:17:18.380 --> 00:17:19.980
ok but one step at a time.
00:17:20.040 --> 00:17:21.800
learn how to do this first.
00:17:23.800 --> 00:17:24.440
so this is
00:17:26.520 --> 00:17:27.900
-ln(u)
00:17:32.300 --> 00:17:34.300
which is -ln of..
00:17:34.760 --> 00:17:36.340
2-e^x
00:17:36.700 --> 00:17:39.500
now we're going to do that from
00:17:39.980 --> 00:17:42.920
0 to ln4.
00:17:50.240 --> 00:17:50.740
ok.
00:17:51.140 --> 00:17:52.780
what is e^ln4?
00:17:54.200 --> 00:17:54.900
4.
00:17:55.060 --> 00:17:56.820
ok make sure you know that.
00:17:57.440 --> 00:17:59.740
e^lnx is x.
00:18:00.480 --> 00:18:01.680
so this becomes..
00:18:03.920 --> 00:18:04.720
negative..
00:18:05.360 --> 00:18:06.560
ln of..
00:18:07.180 --> 00:18:08.220
2-4.
00:18:09.180 --> 00:18:09.680
plus
00:18:10.860 --> 00:18:12.900
ln(2-1).
00:18:15.800 --> 00:18:18.200
you can tell why its plus..minus minus.
00:18:26.380 --> 00:18:27.340
so far so good?
00:18:29.040 --> 00:18:30.200
okay whats ln1?
00:18:31.800 --> 00:18:32.800
ln1 is 0.
00:18:34.940 --> 00:18:37.260
and the ln-2 but its absolute value
00:18:38.060 --> 00:18:39.460
so its ln2 which is
00:18:41.880 --> 00:18:43.520
-ln2.
00:18:45.600 --> 00:18:47.580
thats why you have the absolute values.
00:18:53.000 --> 00:18:55.960
so i look at this and i say i need to integrate.
00:18:56.400 --> 00:18:57.900
we need to do a substitution
00:18:58.300 --> 00:19:00.260
and i look at the denominator and numerator
00:19:00.260 --> 00:19:02.460
and i have to make u whats in the denominator.
00:19:03.060 --> 00:19:04.740
so i let u=2-e^x.
00:19:05.080 --> 00:19:06.420
the derivative of that
00:19:06.700 --> 00:19:07.760
is -e^x
00:19:07.760 --> 00:19:08.960
the derivative of 2 is 0.
00:19:10.120 --> 00:19:12.040
and then its a minus sign here
00:19:12.440 --> 00:19:14.280
so -du is e^x so i go over here
00:19:14.700 --> 00:19:16.140
and now i say okay this
00:19:17.140 --> 00:19:17.860
becomes u.
00:19:18.920 --> 00:19:19.420
this
00:19:20.240 --> 00:19:21.120
becomes -du.
00:19:22.980 --> 00:19:24.340
the integral of du/u
00:19:24.340 --> 00:19:27.100
is the ln of u not the ln of the absolute value of u.
00:19:27.940 --> 00:19:30.440
so then i substitute back and i get the negative log
00:19:30.480 --> 00:19:32.620
of absolute value 2-e^x.
00:19:33.540 --> 00:19:34.040
now
00:19:34.200 --> 00:19:36.200
i think i have to evaluate that
00:19:37.060 --> 00:19:37.560
0
00:19:37.940 --> 00:19:38.820
to ln4
00:19:39.680 --> 00:19:40.880
so i plug in ln4
00:19:41.660 --> 00:19:43.440
and e^ln4 is 4.
00:19:44.000 --> 00:19:46.140
and i plug in 0 and e^0 is 1.
00:19:48.020 --> 00:19:50.100
so this becomes ln1 and ln1 is 0.
00:19:50.100 --> 00:19:51.640
thats why i get that number.
00:19:52.980 --> 00:19:55.000
so this becomes the ln2.
00:19:55.920 --> 00:19:57.840
thats it my answers the -ln2.
00:19:58.460 --> 00:20:00.180
if you really wanna show off, the ln1/2.
00:20:03.900 --> 00:20:04.400
yes?
00:20:09.360 --> 00:20:11.440
why is it the denominator here?
00:20:12.380 --> 00:20:14.300
whats the derivative of ln(u)?
00:20:17.620 --> 00:20:18.420
right? 1/u.
00:20:19.000 --> 00:20:20.940
so the integral of du/u
00:20:21.400 --> 00:20:21.960
is ln(u)
00:20:21.960 --> 00:20:23.520
you should remember some of these integrals.
00:20:28.560 --> 00:20:29.060
so
00:20:37.300 --> 00:20:38.740
you should know a power.
00:20:46.120 --> 00:20:47.800
know the integral of e^x.
00:21:01.500 --> 00:21:05.980
you should know the integral of dx/x is ln of the absolute value of x.
00:21:24.780 --> 00:21:27.880
ill make a handy review sheet of this for you guys tomorrow.
00:21:27.880 --> 00:21:30.260
and ill post it somewhere in blackboard.
00:21:31.220 --> 00:21:34.420
tomorrow i take a 2 hour train trip into the city.
00:21:34.420 --> 00:21:37.420
and about an hour and a half trip home from the city.
00:21:37.420 --> 00:21:43.420
so i could sit through the entire 3 and a half hours and look at my phone and check out my tumblr but i have a life.
00:21:43.600 --> 00:21:46.560
so instead im going to write calculus stuff.
00:21:46.920 --> 00:21:48.440
that way thats not off.
00:21:48.680 --> 00:21:50.520
so because i dont have a life
00:21:50.520 --> 00:21:52.680
i will write you guys a review sheet tomorrow,
00:21:52.680 --> 00:21:55.820
and some practice problems we dont have a practice exam yet.
00:21:55.820 --> 00:21:57.320
do that part tomorrow.
00:21:57.460 --> 00:21:59.300
ill put up the practice exam
00:22:00.080 --> 00:22:01.840
probably friday morning.
00:22:01.840 --> 00:22:04.380
i will not put up the answers friday morning
00:22:04.380 --> 00:22:06.660
cause i will be tired and irritable.
00:22:07.840 --> 00:22:09.580
cause i wont be thursday night ladies night at the bench
00:22:09.960 --> 00:22:10.460
no
00:22:11.500 --> 00:22:13.180
i will be doing calculus.
00:22:13.580 --> 00:22:15.180
how do i know about that?
00:22:16.460 --> 00:22:19.820
alright so the answers will go up over the weekend.
00:22:27.620 --> 00:22:30.580
i dont know if any of you used to go to jake star
00:22:30.580 --> 00:22:32.680
across the train station. they just closed.
00:22:32.680 --> 00:22:36.200
really sad. they say its going to be a new italian restaurant.
00:22:36.200 --> 00:22:39.520
so we have a nice big like pizza place across from the train station.
00:22:40.260 --> 00:22:43.260
and a new chinese restaurant to order chinese food.
00:22:45.160 --> 00:22:53.660
(unrelated to math, inaudible)
00:23:02.420 --> 00:23:04.340
am i forgetting any? oh sure.
00:23:24.000 --> 00:23:25.360
forgot a minus sign.
00:23:26.820 --> 00:23:27.700
thats minus.
00:23:28.340 --> 00:23:30.980
thats what you get for having awful handwriting.
00:23:30.980 --> 00:23:32.780
ill erase that and replace it.
00:23:41.260 --> 00:23:41.760
okay.
00:23:42.940 --> 00:23:43.900
so far so good?
00:23:46.980 --> 00:23:47.620
alright.
00:23:47.620 --> 00:23:49.160
lets do some other stuff.
00:23:53.160 --> 00:23:55.860
the function f(x)=3x^2+x
00:23:57.340 --> 00:23:57.840
a
00:23:59.220 --> 00:24:01.300
find the area of the rectangle.
00:24:01.300 --> 00:24:02.860
find the area of the curve.
00:24:03.240 --> 00:24:06.560
from x=1 to x=9 using 4 rectangles.
00:24:06.980 --> 00:24:08.580
right hand rectangles.
00:24:08.840 --> 00:24:09.340
b
00:24:09.340 --> 00:24:12.780
write a riemann sum for the area using n rectangles.
00:24:13.420 --> 00:24:13.920
c
00:24:14.140 --> 00:24:15.900
evaluate the riemann sum.
00:24:23.340 --> 00:24:27.300
wanna find the area under f(x)=3x^2+x
00:24:27.500 --> 00:24:29.780
x=1 to x=9 using
00:24:30.340 --> 00:24:33.120
4 right hand rectangles.
00:24:40.040 --> 00:24:42.440
curve would do something like that.
00:24:42.700 --> 00:24:44.700
thats not a good way to draw it.
00:24:52.040 --> 00:24:53.800
you want to find this area.
00:25:01.640 --> 00:25:03.640
im going to make 4 rectangles.
00:25:04.180 --> 00:25:05.740
its going to be right hand rectangles.
00:25:06.240 --> 00:25:07.960
and at each interval
00:25:08.200 --> 00:25:10.740
1 to 3, 3 to 5, 5 to 7, 7 to 9
00:25:11.980 --> 00:25:12.780
goes right.
00:25:14.320 --> 00:25:16.560
find the height of the rectangle.
00:25:16.560 --> 00:25:18.360
its going to overestimate.
00:25:20.780 --> 00:25:21.820
too big a number
00:25:24.040 --> 00:25:26.360
okay the width of each of these rectangles
00:25:26.480 --> 00:25:26.980
is 2.
00:25:29.660 --> 00:25:32.460
the height of the first one f evaluated at 3
00:25:33.580 --> 00:25:34.220
then f(5)
00:25:35.420 --> 00:25:36.500
then f(7)
00:25:37.660 --> 00:25:38.960
and f(9).
00:25:42.120 --> 00:25:42.620
k?
00:25:44.860 --> 00:25:46.380
you take 3 and put it in.
00:25:50.200 --> 00:25:50.700
so 3
00:25:51.440 --> 00:25:53.120
times 3^2 is 27.
00:25:54.240 --> 00:25:55.120
+3 this is 30.
00:25:57.660 --> 00:26:00.580
3 times 5^2 is 75 +5 is 80.
00:26:02.760 --> 00:26:06.860
3 times 7^2 is 147 +7 is 154.
00:26:08.720 --> 00:26:13.360
3 times 9^2 is 243 +9 is 252.
00:26:17.040 --> 00:26:17.840
so thats uhh
00:26:18.820 --> 00:26:19.900
1,032?
00:26:21.040 --> 00:26:22.400
did i get that right?
00:26:23.140 --> 00:26:23.640
ok.
00:26:29.500 --> 00:26:31.020
how about that, right?
00:26:31.240 --> 00:26:32.600
ok it could be wrong.
00:26:34.120 --> 00:26:36.320
now lets write a riemann sum.
00:26:36.320 --> 00:26:42.520
riemann sum you have to just sort of think about whats going on but you want to do it in terms of i.
00:26:42.980 --> 00:26:45.880
you want to think about how wide each rectangle is.
00:26:46.220 --> 00:26:48.400
each rectangle is over 2y
00:26:48.900 --> 00:26:50.740
each rectangle is now, well
00:26:51.220 --> 00:26:53.840
i dont know let me take 9-1
00:26:54.840 --> 00:26:56.440
divide it by n you get
00:26:57.660 --> 00:26:58.360
8/n.
00:26:58.900 --> 00:27:01.580
so thats how wide each rectangle could be.
00:27:09.560 --> 00:27:11.880
the first one is going to be at f(3).
00:27:12.880 --> 00:27:18.460
well no its going to be at f(1) +8/n.
00:27:19.440 --> 00:27:25.180
the next one is going to be at f(1)+2(8/n).
00:27:26.240 --> 00:27:27.760
and 1 of f
00:27:31.140 --> 00:27:33.940
1+3( 8/n)'s
00:27:36.800 --> 00:27:37.440
et cetera
00:27:39.000 --> 00:27:39.800
the last one
00:27:40.540 --> 00:27:45.260
will be f(1)+n(8/n)
00:27:45.260 --> 00:27:46.960
well n over n cancel.
00:27:47.180 --> 00:27:48.480
that is 1+8=9
00:27:48.480 --> 00:27:49.600
which is what you want.
00:27:50.260 --> 00:27:50.820
so again
00:27:50.820 --> 00:27:52.560
how do we come up with this?
00:27:52.880 --> 00:27:55.360
i say well i start at the left end point
00:27:56.440 --> 00:27:59.000
and the width of each interval
00:27:59.360 --> 00:28:01.120
is 8/n. why is it 8/n?
00:28:01.120 --> 00:28:02.240
well cause its 8 wide
00:28:02.240 --> 00:28:04.560
and im dividing 8 into n rectangles
00:28:05.200 --> 00:28:06.640
so i take the first
00:28:07.380 --> 00:28:09.760
pint 1 and i move 8/n to the right.
00:28:10.140 --> 00:28:12.860
and thats the right end point of the first interval.
00:28:13.280 --> 00:28:16.560
and from there i would do another 8/n to the right so thats why its 2
00:28:17.260 --> 00:28:17.760
8/n's.
00:28:18.140 --> 00:28:20.260
then i move another 8/n to the right
00:28:20.260 --> 00:28:22.280
and thats 3 and i keep doing this n times.
00:28:22.620 --> 00:28:25.440
and the last one you should check the arithmetic should come out
00:28:25.440 --> 00:28:27.180
for right hand numbers it should come out 9
00:28:37.380 --> 00:28:39.380
what im writing here is identical to whats over there.
00:28:40.040 --> 00:28:40.840
8/n
00:28:41.980 --> 00:28:44.820
f(1+8/n)
00:28:45.400 --> 00:28:50.380
+ f(1+2(8/n))
00:28:51.420 --> 00:28:56.000
+ f(1+3(8/n))
00:28:58.260 --> 00:28:59.340
dot dot dot
00:29:01.860 --> 00:29:08.100
f(1+n(8/n)).
00:29:09.660 --> 00:29:11.180
so thats my riemann sum
00:29:11.500 --> 00:29:12.000
now
00:29:12.000 --> 00:29:14.640
you just sort of put it in sigma notation
00:29:14.640 --> 00:29:15.640
its the fun part.
00:29:16.660 --> 00:29:18.180
k well thats 8/n
00:29:22.160 --> 00:29:25.180
so youre always going to have the same thing in your sigma notation.
00:29:25.360 --> 00:29:27.460
this will be the width.
00:29:28.240 --> 00:29:30.900
that will be (b-a)/n.
00:29:31.820 --> 00:29:34.280
this will always be i=1 to n.
00:29:35.640 --> 00:29:36.580
inside
00:29:36.940 --> 00:29:38.620
youre just going to have f
00:29:39.060 --> 00:29:41.060
at the left end point
00:29:42.060 --> 00:29:43.760
plus the width
00:29:45.720 --> 00:29:46.420
times i.
00:29:46.800 --> 00:29:49.440
k thats what your riemann sum looks like
00:29:49.440 --> 00:29:51.860
then you take f and replace it with
00:29:52.300 --> 00:29:53.800
3x^2+x.
00:30:04.520 --> 00:30:05.020
ok?
00:30:09.120 --> 00:30:10.660
this okay so this
00:30:10.780 --> 00:30:12.380
is the same as this so 8/n
00:30:12.380 --> 00:30:14.200
thats the width of each of these.
00:30:14.940 --> 00:30:17.340
and i have f and im going to start at 1.
00:30:17.520 --> 00:30:20.200
and im going to keep adding 8/n's
00:30:20.200 --> 00:30:22.720
so im going to have 8/n thats the width each time
00:30:24.040 --> 00:30:24.840
times i.
00:30:26.180 --> 00:30:26.680
ok?
00:30:33.800 --> 00:30:35.400
cause im doing right hand rectangles.
00:30:36.120 --> 00:30:38.840
so im going to take the left most end point
00:30:39.660 --> 00:30:40.160
with 1
00:30:40.720 --> 00:30:43.300
but then each interval remember you use the right hand side of it
00:30:45.180 --> 00:30:46.900
each rectangle is 8/n wide
00:30:47.400 --> 00:30:49.940
so i have to start at 1 and move 8/n to the right
00:30:50.880 --> 00:30:53.120
and then another 8/n to the right.
00:30:54.340 --> 00:30:56.600
so if were doing left end point rectangles
00:30:56.640 --> 00:30:59.040
this would be 0 and that would be n-1.
00:30:59.320 --> 00:31:00.600
but all our formula
00:31:00.600 --> 00:31:04.000
the formula works better from 1 to n than from 0 to n-1.
00:31:08.080 --> 00:31:10.720
now the last step to get it exactly right
00:31:14.300 --> 00:31:15.260
is instead of f
00:31:15.720 --> 00:31:18.280
so i recommend if you do this on the exam
00:31:18.500 --> 00:31:19.700
if you have a question like this
00:31:19.700 --> 00:31:21.440
write out all these steps.
00:31:21.440 --> 00:31:22.880
youll get partial credit.
00:31:24.140 --> 00:31:25.600
so f is 3x^2+x
00:31:26.220 --> 00:31:27.420
so its going to be 3
00:31:28.120 --> 00:31:30.480
x is this. 1 +...
00:31:31.120 --> 00:31:33.460
(8i/n)^2
00:31:33.860 --> 00:31:35.080
plus x which is
00:31:35.460 --> 00:31:40.680
another 1+ 8i/n
00:31:56.360 --> 00:31:57.320
so far so good?
00:31:59.420 --> 00:32:02.780
now for the really fun part..evaluating this.
00:32:03.440 --> 00:32:05.120
i know you want to do this.
00:32:10.620 --> 00:32:12.940
i dont know if were going to have this on the exam
00:32:12.940 --> 00:32:15.760
i know if we had to put up a vote i know what the answer would be.
00:32:16.460 --> 00:32:19.540
if we do have it on the exam i suggest you do the problem last.
00:32:20.140 --> 00:32:20.640
ok?
00:32:21.240 --> 00:32:23.500
cause its time consuming work on other stuff first
00:32:23.500 --> 00:32:25.340
do the stuff that youre comfortable with.
00:32:26.380 --> 00:32:28.820
these are annoying, i have trouble with them.
00:32:28.880 --> 00:32:30.640
okay theyre very tedious.
00:32:30.640 --> 00:32:32.220
its easy to make mistakes.
00:32:32.220 --> 00:32:34.440
but you know when the answer has to come out right?
00:32:35.140 --> 00:32:36.300
this has to be
00:32:36.980 --> 00:32:39.000
the integral from 1 to 9
00:32:39.540 --> 00:32:40.860
3x^2+x
00:32:43.560 --> 00:32:47.040
its x^3+x^2/2
00:32:48.340 --> 00:32:49.120
1 to 9
00:32:49.300 --> 00:32:51.620
okay so you have to get that number.
00:32:54.840 --> 00:32:56.280
so what you do this way
00:32:56.280 --> 00:32:58.620
you get the same answer as what you get there.
00:33:22.100 --> 00:33:24.420
so now how do we evaluate this? well
00:33:24.880 --> 00:33:26.140
ive got 8/n
00:33:28.200 --> 00:33:28.980
sigma
00:33:29.420 --> 00:33:29.920
3
00:33:30.760 --> 00:33:31.320
times..
00:33:32.360 --> 00:33:36.740
multiply that out with 1+16i/n
00:33:37.340 --> 00:33:41.440
+64i^2/n^2
00:33:43.140 --> 00:33:47.080
+ 1+ 8i/n
00:33:47.300 --> 00:33:49.440
so all i did was foil out this.
00:33:53.720 --> 00:33:55.320
wheres the 64 come from?
00:33:55.840 --> 00:33:57.120
1 squared
00:33:57.360 --> 00:33:59.420
2 times 1 times 8/n
00:33:59.860 --> 00:34:00.640
16i/n
00:34:01.380 --> 00:34:03.320
+ 8i/n^2
00:34:03.580 --> 00:34:06.080
which is 64i^2/n^2.
00:34:09.680 --> 00:34:10.180
ok
00:34:11.600 --> 00:34:13.360
lets do some simplifying.
00:34:16.360 --> 00:34:19.300
im not going to write the i or the n cause i dont have time for that.
00:34:19.920 --> 00:34:22.860
so this 3 times 1 is 3+1 is 4.
00:34:24.040 --> 00:34:27.320
3 times 16 is 48 +8 is 65.
00:34:30.380 --> 00:34:32.680
3 times 64 is 192
00:34:33.540 --> 00:34:34.640
i^2
00:34:35.920 --> 00:34:36.840
over n^2.
00:34:43.560 --> 00:34:45.560
starting to get easier. okay?
00:34:47.340 --> 00:34:47.840
ok.
00:34:48.200 --> 00:34:49.540
3 times 1 is 3.
00:34:50.120 --> 00:34:51.020
+1 is 4.
00:34:52.120 --> 00:34:53.980
3 times 16 is 48
00:34:54.320 --> 00:34:55.780
+8 is 56.
00:34:56.860 --> 00:34:58.540
3 times 64 is 192
00:34:59.060 --> 00:35:00.660
plus nothing...so 192.
00:35:01.720 --> 00:35:02.220
ok?
00:35:02.940 --> 00:35:03.900
skipped a step
00:35:04.060 --> 00:35:06.380
i dont recommend you skip any steps
00:35:07.240 --> 00:35:07.740
ok
00:35:08.380 --> 00:35:11.480
i pity the TAs who have to grade questions like this.
00:35:15.980 --> 00:35:16.620
alright.
00:35:17.340 --> 00:35:19.620
lets make this 3 separate sums.
00:35:20.240 --> 00:35:20.740
i=
00:35:21.180 --> 00:35:22.600
1 to n of 4
00:35:23.680 --> 00:35:24.180
plus
00:35:24.760 --> 00:35:26.080
8/n
00:35:26.880 --> 00:35:28.360
times the sum
00:35:28.360 --> 00:35:29.800
i=1 to n
00:35:31.500 --> 00:35:32.840
56i/n
00:35:33.820 --> 00:35:35.280
+8/n
00:35:35.920 --> 00:35:38.880
times the sum of i=1 to n
00:35:39.260 --> 00:35:40.260
of 192
00:35:41.180 --> 00:35:42.360
i^2
00:35:43.560 --> 00:35:44.660
over n^2
00:36:02.340 --> 00:36:02.840
ok
00:36:08.400 --> 00:36:10.120
so 8/n
00:36:10.820 --> 00:36:13.320
the sum of i=1 to n
00:36:14.000 --> 00:36:14.840
of 4
00:36:15.900 --> 00:36:19.920
just take the four and multiply it by the n so thats just 8/n
00:36:20.580 --> 00:36:21.080
4n
00:36:21.680 --> 00:36:23.680
cause thats this formula.
00:36:24.640 --> 00:36:25.140
k?
00:36:25.460 --> 00:36:26.920
i=1 to n of a constant
00:36:27.480 --> 00:36:28.700
its the constant times n.
00:36:32.160 --> 00:36:33.580
the n's cancel
00:36:33.680 --> 00:36:35.040
this comes out to 32.
00:36:35.040 --> 00:36:37.360
we havent even done the limit yet, oh
00:36:38.040 --> 00:36:41.240
yeah when you write the riemann sum you have to write the limit as n goes to infinity.
00:36:41.460 --> 00:36:43.140
kind of forgot to put that in there.
00:36:43.140 --> 00:36:45.180
when were all done we'll take the limit.
00:36:46.860 --> 00:36:47.360
ok
00:36:48.160 --> 00:36:49.200
the middle term
00:36:49.360 --> 00:36:50.540
we have 8/n
00:36:52.260 --> 00:36:54.260
sigma i=1 to n.
00:36:55.660 --> 00:36:57.200
of 56
00:36:58.780 --> 00:36:59.900
i/n
00:37:03.020 --> 00:37:06.020
so 8 times 56 is 448.
00:37:09.640 --> 00:37:13.980
over n^2 so i can take the 8 times 56 and the n times n
00:37:14.580 --> 00:37:15.300
pull it out
00:37:15.940 --> 00:37:19.040
times the sum of i=1 to n of i.
00:37:21.120 --> 00:37:22.200
which is 448
00:37:23.640 --> 00:37:24.760
over n^2 times..
00:37:25.320 --> 00:37:26.040
so this one
00:37:27.160 --> 00:37:28.920
is (n times n + 1)2
00:37:35.840 --> 00:37:38.920
that simplifies to 224
00:37:41.280 --> 00:37:42.800
times (n+1)/n
00:37:47.520 --> 00:37:48.740
448/2
00:37:49.260 --> 00:37:50.100
is 224
00:37:50.380 --> 00:37:51.740
one of the n's cancel
00:37:52.240 --> 00:37:53.840
and im left with (n+1)/n
00:37:55.420 --> 00:37:56.460
we're not done.
00:37:58.120 --> 00:37:58.840
now ive got
00:38:00.020 --> 00:38:00.840
8/n
00:38:02.840 --> 00:38:05.500
times the sum of i=1 to n
00:38:06.500 --> 00:38:07.480
of 192
00:38:10.440 --> 00:38:12.120
i^2/n^2
00:38:13.660 --> 00:38:15.140
8 times 192
00:38:16.100 --> 00:38:18.180
is 1536
00:38:20.740 --> 00:38:23.380
i could just be making this stuff up and you could just write it down.
00:38:24.140 --> 00:38:24.640
right?
00:38:25.660 --> 00:38:27.420
lets see thats 1600
00:38:27.860 --> 00:38:29.700
minus...yeah 1536.
00:38:30.700 --> 00:38:31.780
i^2
00:38:33.500 --> 00:38:34.540
over n^3.
00:38:51.600 --> 00:38:52.100
ok
00:38:52.900 --> 00:38:53.400
and
00:38:58.540 --> 00:39:00.540
ive got the sigma there, sorry
00:39:06.240 --> 00:39:08.140
try to be less sloppy.
00:39:14.540 --> 00:39:16.700
and that is
00:39:17.460 --> 00:39:18.680
1536
00:39:19.820 --> 00:39:21.900
over n^3 times the last formula
00:39:22.260 --> 00:39:28.080
n(n+1)(2n+1)/6.
00:39:32.320 --> 00:39:36.920
which simplifies to 256 times (n+1)...
00:39:37.840 --> 00:39:39.200
times (2n+1)
00:39:40.400 --> 00:39:42.300
over n^2
00:39:46.460 --> 00:39:48.140
alright take the limits.
00:39:48.480 --> 00:39:50.960
the limit as n goes to infinity of 32 is 32.
00:39:51.760 --> 00:39:52.260
well
00:39:52.420 --> 00:39:55.460
the limit as n goes to infinity of n+(1/n) is 1.
00:39:55.460 --> 00:39:56.620
and this ones going to be
00:39:57.620 --> 00:39:58.840
24.
00:40:00.480 --> 00:40:01.280
and this one
00:40:02.820 --> 00:40:03.540
this limit
00:40:04.460 --> 00:40:05.680
comes out to 2.
00:40:13.240 --> 00:40:15.560
k so therefore thats the sum
00:40:16.200 --> 00:40:17.180
is 32
00:40:18.300 --> 00:40:19.520
+224
00:40:20.600 --> 00:40:22.800
+ 512.
00:40:23.720 --> 00:40:27.300
which is 768.
00:40:32.220 --> 00:40:34.580
now i wrote a whole bunch of this stuff out
00:40:34.740 --> 00:40:37.540
and put it on the website last week on blackboard.
00:40:37.760 --> 00:40:39.920
did any of you guys look at it yet?
00:40:40.520 --> 00:40:44.380
i recommend you look through it cause i basically work through one of these step by step.
00:40:44.380 --> 00:40:46.320
i write everything out nice and easy
00:40:50.100 --> 00:40:54.080
well 1536/6 = 256
00:40:54.360 --> 00:40:56.340
the limit as n goes to infinity of this
00:40:56.340 --> 00:40:57.300
is 2.
00:41:00.100 --> 00:41:01.900
ive got a 2 over 1 okay?
00:41:02.780 --> 00:41:04.960
so thats why i had 2 at the end.
00:41:06.240 --> 00:41:07.880
how do we feel about this stuff?