WEBVTT

f5cbf3eb-917b-4908-9073-a35fd4281bec-0
00:00:05.200 --> 00:00:06.160
Hi, I'm Richard.

ee93ce1e-b462-4822-92cb-7aea55ebe5d0-0
00:00:06.160 --> 00:00:10.360
We're going to be looking at TMUA 2021
paper one question 10 here it is.

745c8bc6-077c-43c6-9746-460635bed3b8-0
00:00:10.360 --> 00:00:14.960
It says use the trapezium rule with three
strips to estimate this integral.

79fbe970-7120-48db-a0b9-abc44850bb2e-0
00:00:15.240 --> 00:00:20.297
It's the integral from 1/2 to two of this
function which is 2 times log base 10 of

79fbe970-7120-48db-a0b9-abc44850bb2e-1
00:00:20.297 --> 00:00:21.760
X all with respect to X.

9cfa9d0c-05d9-4d34-919f-e0c5f2034085-0
00:00:22.160 --> 00:00:24.228
So when we're setting up the trapezium
rule,

9cfa9d0c-05d9-4d34-919f-e0c5f2034085-1
00:00:24.228 --> 00:00:28.088
it's it's really important to think about
this interval that we're integrating over

9cfa9d0c-05d9-4d34-919f-e0c5f2034085-2
00:00:28.088 --> 00:00:30.800
from 1/2 to two and the number of strips
that we're using.

fe7e8e75-fcf6-4ab0-bf84-6dd4c4a8899a-0
00:00:30.800 --> 00:00:32.520
So I'm going to draw a little diagram
here.

8c0c5410-ed6a-4ff0-973f-d44e08e61212-0
00:00:33.240 --> 00:00:36.468
The left hand end is at a half,
the right hand end is at 2,

8c0c5410-ed6a-4ff0-973f-d44e08e61212-1
00:00:36.468 --> 00:00:39.320
and there are going to be 3 strips with
equal width.

c9a1080a-6b26-4c50-b41b-635bf9f7c2bd-0
00:00:39.680 --> 00:00:43.780
And I can see that means the first
trapezium is going to run from 1/2 to one,

c9a1080a-6b26-4c50-b41b-635bf9f7c2bd-1
00:00:43.780 --> 00:00:46.723
the second trapezium is going to run from
one to 1 1/2,

c9a1080a-6b26-4c50-b41b-635bf9f7c2bd-2
00:00:46.723 --> 00:00:49.720
and the third trapezium is going to run
from 1/2 to two.

733035c4-2e7c-48f8-bf60-292730f8aa1c-0
00:00:50.240 --> 00:00:55.240
So in order to introduce some notation,
I'm going to call my function F of X.

dcf30b04-ce26-4ba4-84df-e93b391c1661-0
00:00:55.240 --> 00:01:01.147
So F of X is going to be two log base 10
of X and I'm going to use the regular

dcf30b04-ce26-4ba4-84df-e93b391c1661-1
00:01:01.147 --> 00:01:03.840
trapezium rule to get this estimate.

5079710c-2355-4717-a012-cfd70d257bba-0
00:01:04.200 --> 00:01:08.600
So in the trapezium rule I have H which
is the width of my trapezia.

afc01af4-c039-4955-acfb-c9c21c6a8a9d-0
00:01:09.000 --> 00:01:11.800
I can see that this is going to be 1/2 in
my case.

80a0e2e5-a1f5-4433-9024-5e3f369fe670-0
00:01:12.160 --> 00:01:16.480
So if I write out the general formula,
it will be in this case H / 2.

20847b15-dee9-4ac3-a3c4-5ce1571b1392-0
00:01:16.480 --> 00:01:22.322
I know H is going to be 1/2 and then I'm
going to have the function at a half the

20847b15-dee9-4ac3-a3c4-5ce1571b1392-1
00:01:22.322 --> 00:01:23.320
left hand end.

5c314923-8dd3-4158-aae4-0d139d37fec1-0
00:01:23.760 --> 00:01:28.720
I'm going to have twice the function of
one added, so two times the function of 1.

d160ac82-82e5-436d-b64a-0dce10f1689b-0
00:01:29.320 --> 00:01:31.720
I'm going to have two times the function
at 1 1/2.

a8986d31-b099-4d00-97aa-a7ef53b4ce06-0
00:01:32.320 --> 00:01:34.800
So F of I'll write that as 3 / 2.

97af8a7c-3f9a-4cc7-b6d2-c7b6e0dd72e6-0
00:01:35.320 --> 00:01:37.640
And then I'm going to have the function
at 2:00.

3329f197-4b64-4d3b-ad53-c4a13a523d82-0
00:01:38.520 --> 00:01:43.433
So this is my expression for the
trapezium rule estimate to this integral

3329f197-4b64-4d3b-ad53-c4a13a523d82-1
00:01:43.433 --> 00:01:45.160
using 3 trapezia 3 strips.

606392b5-f378-4709-857f-7f2cec84eb20-0
00:01:45.520 --> 00:01:47.320
So I just need to plug in all my values.

17363dfd-a495-4693-a9bb-cc30f89c2518-0
00:01:47.320 --> 00:01:50.000
Now I know that this H is 1/2.

81d32b14-f31b-474c-aeb6-1a1ad0aac0b5-0
00:01:50.000 --> 00:01:54.440
So what I've actually got at the front
here is 1/2 of a half 1/2 * 1/2.

52acf86f-ecc5-4bd9-a6f1-f1553702c351-0
00:01:55.320 --> 00:01:56.440
I'll open up my brackets.

b76eec53-3195-4b12-9236-863033a33407-0
00:01:56.440 --> 00:02:01.560
I have F over half,
so that's two log base 10 over half.

de7121b5-83c0-4ad4-a75b-f6aaba28a4bb-0
00:02:02.440 --> 00:02:06.760
I have two F of 1,
so that's two log 2 * 2.

1b242709-5f41-4f3e-9537-fe21715685c6-0
00:02:06.760 --> 00:02:12.000
So this will be 4 log base 12:50.

e0189f1a-c22d-4d7b-b44b-c97afc306a3f-0
00:02:13.040 --> 00:02:15.840
I've got 2 * F of 3 / 2, so another 4.

4f584e4f-e20d-41bf-b351-27016ba72793-0
00:02:16.400 --> 00:02:22.211
Multiplying this by two, four,
log base 10 of 3 / 2,

4f584e4f-e20d-41bf-b351-27016ba72793-1
00:02:22.211 --> 00:02:28.680
and then finally F of two,
which is 2 log base ten of two.

c8e02289-c80c-450c-a0a3-4fe9a01d98fc-0
00:02:29.680 --> 00:02:33.120
The first thing I notice is that we have
log base 12:50 here.

b4d374bf-aed3-4330-a9e4-46d5ef3e6b25-0
00:02:33.720 --> 00:02:38.578
Now log base anything of 1 is always 0,
so this term is just zero and I can

b4d374bf-aed3-4330-a9e4-46d5ef3e6b25-1
00:02:38.578 --> 00:02:40.880
effectively just cross that one out.

bd5d9b97-340a-4943-b176-16dc57070b9d-0
00:02:40.880 --> 00:02:45.512
The other thing I notice is that I've got
all these even numbers in my bracket,

bd5d9b97-340a-4943-b176-16dc57070b9d-1
00:02:45.512 --> 00:02:48.465
24 and 2,
so I could cancel that with one of these

bd5d9b97-340a-4943-b176-16dc57070b9d-2
00:02:48.465 --> 00:02:49.160
halves here.

d256c76f-61e1-4e63-a4a9-44c1ab652832-0
00:02:49.160 --> 00:02:50.520
So I think I'll do that first of all.

e417eb6e-672f-43c4-bd92-bac6bca68878-0
00:02:51.240 --> 00:02:52.240
So we'll have 1/2.

e5bf1e8e-cb25-454c-afd2-b39d1ec7d6c6-0
00:02:52.400 --> 00:02:55.720
I'm cancelling this half with the 2,
the four, and the 2.

eda22f67-1bc9-418e-9d8a-5181bf836696-0
00:02:56.120 --> 00:03:04.811
So what I'll have in my bracket is now
just log base 10 over half +2,

eda22f67-1bc9-418e-9d8a-5181bf836696-1
00:03:04.811 --> 00:03:11.640
log base 10 of 3 / 2,
and then just one log base 1:50.

7ff68e4b-e209-4128-bdf3-ef689ba1b185-0
00:03:13.040 --> 00:03:15.317
OK,
I want to push all this together as a

7ff68e4b-e209-4128-bdf3-ef689ba1b185-1
00:03:15.317 --> 00:03:16.240
single logarithm.

993630ce-3975-465c-80c6-d94f68312d4a-0
00:03:16.240 --> 00:03:21.752
I want to use the fact that log of A plus
log of B is log of A * B I've got this 2

993630ce-3975-465c-80c6-d94f68312d4a-1
00:03:21.752 --> 00:03:24.874
here,
but I can push that up as a power on 3 /

993630ce-3975-465c-80c6-d94f68312d4a-2
00:03:24.874 --> 00:03:27.000
2, so I'm going to do that next.

bb970956-fbf1-44b8-92a5-cf36a073dd0d-0
00:03:27.480 --> 00:03:37.120
And we have 1/2 log base 10 over half
plus log base 10.

1425980c-a047-45bd-bce2-b2dde9024d03-0
00:03:37.200 --> 00:03:39.800
And I'm saying that twice.

4b9b4b9d-4bff-4df8-9f35-ef7959d6f354-0
00:03:39.800 --> 00:03:44.600
Log base 10 of 3 / 2 is the same as log
base 10 of 3 / 2 ^2.

47d6ce22-9be3-4560-8d47-912c34f4a443-0
00:03:45.240 --> 00:03:52.829
So this will be log base 10 of 9 / 4,
squaring the three and squaring the two,

47d6ce22-9be3-4560-8d47-912c34f4a443-1
00:03:52.829 --> 00:03:55.520
and then plus log base 1:50.

9c322451-bfd8-4f5d-9713-84c5ec2a8ed2-0
00:03:56.480 --> 00:03:58.706
OK,
so now I can push all this together as a

9c322451-bfd8-4f5d-9713-84c5ec2a8ed2-1
00:03:58.706 --> 00:04:01.280
single logarithm moving to the bottom of
the board.

3347b9f8-ceeb-42fd-bf5e-52cff4fb2efe-0
00:04:01.280 --> 00:04:02.640
I'll keep my half here.

d6675f52-823b-4503-8eff-69d295971d43-0
00:04:06.000 --> 00:04:12.434
And then I have log base 10 and I can
just multiply these values together using

d6675f52-823b-4503-8eff-69d295971d43-1
00:04:12.434 --> 00:04:13.560
that property.

82b86709-5537-453f-9d44-6c0d08714794-0
00:04:13.560 --> 00:04:22.958
Log of A plus log of B plus log of C is
log of A * B * C So in this case log base

82b86709-5537-453f-9d44-6c0d08714794-1
00:04:22.958 --> 00:04:25.480
10 of 1/2 * 9 / 4 * 2.

797d4216-8d07-464b-9170-4009656a3d7c-0
00:04:25.840 --> 00:04:27.080
It's a nice cancellation.

09db8e81-974d-4ab0-b0c4-b68f53cc18bd-0
00:04:27.080 --> 00:04:27.760
With the twos.

e2f02e63-c326-48cf-be86-a06b0f13a4c5-0
00:04:28.280 --> 00:04:31.800
I've got log base 10 of 9 / 4 but times
1/2.

b47c51d9-e574-4002-8e58-710f31fb6928-0
00:04:32.320 --> 00:04:41.960
So this is the same as log base 10 of 9 /
4 to the power of 1/2 or sqrt 9 / 4.

f01a69bc-1673-4c1e-8ef9-d26d704e080a-0
00:04:42.400 --> 00:04:43.840
I know that will be 3 / 2.

d0498a50-060f-4077-afee-659b0bc72a69-0
00:04:43.840 --> 00:04:49.400
So the answer is log base 10 of 3 / 2,
which I can see over here is answer B.

c4e3232d-f0eb-4814-a2e4-792985f659dc-0
00:04:50.360 --> 00:04:52.840
Let's just take a chance to reflect on
this question.

320b5ee6-9ace-4e71-8cee-5e6bd3178a7d-0
00:04:53.120 --> 00:04:54.000
Two things really.

8731b2d5-ebbb-4611-9d92-c32f69eb2985-0
00:04:54.000 --> 00:04:57.440
One is we're using the trapezium rule and
we're told with three strips.

d96f03f7-86ca-4de2-862c-1ad417f22ae6-0
00:04:57.440 --> 00:05:00.040
So it's important to get a feel for what
that will look like.

9968c154-e7c2-452c-9d4f-db1998e1a511-0
00:05:00.080 --> 00:05:02.750
In this case,
the integral was from 1/2 to two and we

9968c154-e7c2-452c-9d4f-db1998e1a511-1
00:05:02.750 --> 00:05:03.640
have three strips.

1ed6319a-1ec6-4fa6-b470-f03e9a5b38fa-0
00:05:03.640 --> 00:05:07.708
So this diagram was helpful to help us
establish what which function values

1ed6319a-1ec6-4fa6-b470-f03e9a5b38fa-1
00:05:07.708 --> 00:05:10.600
we'll need and what value of H we'll be
working with.

7c7676c8-5a2e-443d-90e2-8710a7e58905-0
00:05:11.200 --> 00:05:15.040
The second thing we could see would
happen here is because we're going to be

7c7676c8-5a2e-443d-90e2-8710a7e58905-1
00:05:15.040 --> 00:05:18.980
plugging in values into this function,
we're going to get a sum of lots of log

7c7676c8-5a2e-443d-90e2-8710a7e58905-2
00:05:18.980 --> 00:05:19.280
terms.

fcea712c-d632-45b2-af59-4dda71bba538-0
00:05:19.640 --> 00:05:23.446
So we could be expecting to use those
properties of logs,

fcea712c-d632-45b2-af59-4dda71bba538-1
00:05:23.446 --> 00:05:26.400
like log of A plus log of B is log of A *
b.

b8076e50-47b7-4860-9ad8-264297fdd916-0
00:05:26.800 --> 00:05:28.840
And of course we can extend that to three
values.

e660f8a3-60a4-4776-a645-33a61ebb6537-0
00:05:28.840 --> 00:05:32.688
Log of A plus log of B plus log of C is
log of A * b * C,

e660f8a3-60a4-4776-a645-33a61ebb6537-1
00:05:32.688 --> 00:05:34.480
which is what we used here.

a745869a-4d28-4c3c-9e24-e3cf78919ad0-0
00:05:34.800 --> 00:05:38.917
We also use the idea that if we have a
number in front of our log of expression,

a745869a-4d28-4c3c-9e24-e3cf78919ad0-1
00:05:38.917 --> 00:05:42.680
we can bring that up as a power of the
thing that we're taking logarithm.

66560f6e-d8a6-4130-b656-458aca31be62-0
00:05:43.000 --> 00:05:46.163
So for example,
we use that here where we had two log

66560f6e-d8a6-4130-b656-458aca31be62-1
00:05:46.163 --> 00:05:47.160
base 10 of 3 / 2.

24232d93-8d68-42e5-866c-636ab2e293ba-0
00:05:47.360 --> 00:05:52.625
We can bring that two in as a power and
say this is the same as log base 10 of 3

24232d93-8d68-42e5-866c-636ab2e293ba-1
00:05:52.625 --> 00:05:53.080
/ 2 ^2.