WEBVTT

96861570-3988-429f-b695-21751f9611a7-0
00:00:05.760 --> 00:00:06.920
Hi my name is Richard.

cbe403d0-a318-46d3-85c7-b379e7b6a21c-0
00:00:06.920 --> 00:00:10.600
We're going to be looking at TAMUA 2021
paper one question 7.

981a9626-2b6b-4e67-bc25-8627998dc4f3-0
00:00:10.720 --> 00:00:11.840
Here is the question.

e9e8c0bf-6ae1-47ce-82e9-aa64d65d1b78-0
00:00:12.200 --> 00:00:19.427
So it says the function F is such that F
of 0 = 0 and also X * F of X is greater

e9e8c0bf-6ae1-47ce-82e9-aa64d65d1b78-1
00:00:19.427 --> 00:00:22.640
than 0 for all non zero values of X.

62c4fa29-2532-44d6-8b7b-4b66b240794a-0
00:00:23.440 --> 00:00:28.117
We're told that the integral between -2
and two of F of X with respect to X is 4,

62c4fa29-2532-44d6-8b7b-4b66b240794a-1
00:00:28.117 --> 00:00:32.909
and also that the integral between -2 and
two of the modulus of F of X with respect

62c4fa29-2532-44d6-8b7b-4b66b240794a-2
00:00:32.909 --> 00:00:33.480
to X is 8.

4210930b-a11b-4a3c-9124-d6ce6f3639db-0
00:00:33.840 --> 00:00:38.666
And we're asked to work out the value of
the integral from -2 to 0 of F of the

4210930b-a11b-4a3c-9124-d6ce6f3639db-1
00:00:38.666 --> 00:00:40.560
modulus of X with respect to X.

a8b5e80c-e010-49de-9191-ce752ad417db-0
00:00:41.480 --> 00:00:43.598
OK,
I think the place to start with this

a8b5e80c-e010-49de-9191-ce752ad417db-1
00:00:43.598 --> 00:00:46.440
question is this mysterious condition on
the function.

c4c51705-cc11-4cdc-a363-b63f5594ab0d-0
00:00:46.440 --> 00:00:49.240
Here X * F of X is greater than 0.

adc43f8f-4afc-4019-863e-ff5a079e1025-0
00:00:49.520 --> 00:00:52.936
It looks like it could be quite a
complicated thing to think about,

adc43f8f-4afc-4019-863e-ff5a079e1025-1
00:00:52.936 --> 00:00:56.855
but actually if we break this down,
we'll see that it's a just a very compact

adc43f8f-4afc-4019-863e-ff5a079e1025-2
00:00:56.855 --> 00:00:59.720
way of saying something quite simple
about the function.

1d4826f5-433f-4e14-b8de-78735c129a1e-0
00:01:00.760 --> 00:01:04.621
So what we're being told is that whenever
we multiply X by F of X,

1d4826f5-433f-4e14-b8de-78735c129a1e-1
00:01:04.621 --> 00:01:06.120
we get a positive outcome.

1c8a191f-01d3-437b-9c1a-bee482cb8b25-0
00:01:07.080 --> 00:01:11.240
And what that must mean is that the sign
of X&F of X agree.

5b5f927b-31df-4be1-812c-1cd1abbfc9f0-0
00:01:11.400 --> 00:01:15.440
So if X is positive,
then F of X must also be positive.

b29f6ae1-ae97-405e-ae91-02765ed2c9f5-0
00:01:16.440 --> 00:01:19.650
And if X is negative,
then F of X must also be negative in

b29f6ae1-ae97-405e-ae91-02765ed2c9f5-1
00:01:19.650 --> 00:01:23.840
order that when we multiply them together,
we always get a positive outcome.

f4140125-909c-49e4-9900-fa3c635dcc9d-0
00:01:24.520 --> 00:01:29.592
So actually this condition is just
telling us that when X is greater than 0,

f4140125-909c-49e4-9900-fa3c635dcc9d-1
00:01:29.592 --> 00:01:31.240
F of X is greater than 0.

d1139e3b-e44f-4037-8ab1-b3d97290205a-0
00:01:31.680 --> 00:01:35.600
And that tells us something about where
the graph is for positive X.

fe421ebb-94ce-4145-aeaa-aba83e3f87fa-0
00:01:36.040 --> 00:01:38.680
And when X is less than 0,
F of X is less than 0,

fe421ebb-94ce-4145-aeaa-aba83e3f87fa-1
00:01:38.680 --> 00:01:41.480
and that tells us where the graph is for
negative X.

0618c305-0f46-45aa-b371-39065ba21326-0
00:01:41.880 --> 00:01:45.188
So I think that's a good place to start
with this question,

0618c305-0f46-45aa-b371-39065ba21326-1
00:01:45.188 --> 00:01:48.000
to imagine a function which has got this
property.

af5fcfb1-7a6f-48a0-9eaa-a1ac6180a732-0
00:01:48.560 --> 00:01:50.280
So let's have axes.

23213341-cfb4-455f-8d72-d8b518027185-0
00:01:50.680 --> 00:01:54.723
Let's put in the origin because we know
that the graph of the function goes

23213341-cfb4-455f-8d72-d8b518027185-1
00:01:54.723 --> 00:01:56.320
through the origin F of 0 = 0.

e191130b-680f-4e54-b814-291dc78c59ff-0
00:01:57.040 --> 00:01:59.454
So we know that this means that for
positive X,

e191130b-680f-4e54-b814-291dc78c59ff-1
00:01:59.454 --> 00:02:02.120
we're going to get positive values for
our function.

cee18572-3619-468f-9085-2e1cbb51ff66-0
00:02:02.440 --> 00:02:05.920
So I'm just going to Draw Something which
kind of looks like that.

1bf3aeb6-4f6f-405a-bf92-d8e2fae90b21-0
00:02:06.880 --> 00:02:09.156
And actually,
I'm going to draw up as far as two

1bf3aeb6-4f6f-405a-bf92-d8e2fae90b21-1
00:02:09.156 --> 00:02:12.920
because I know that in this question,
everything is happening between -2 and +2.

36de62f8-bbfa-45c8-a41c-79e94b3108af-0
00:02:13.600 --> 00:02:18.042
So positive values of the function for
positive values of X and negative values

36de62f8-bbfa-45c8-a41c-79e94b3108af-1
00:02:18.042 --> 00:02:20.320
of the function for negative values of X.

b16878a8-fa25-4055-a689-71c3e0f0b7f4-0
00:02:20.320 --> 00:02:22.720
So I'm just going to draw anything that
kind of looks like that.

5aea60be-d97f-4bcc-b2ba-eb4c26340375-0
00:02:22.720 --> 00:02:27.806
So this is my graph for negative X and
this is -2 and I'm just going to box that

5aea60be-d97f-4bcc-b2ba-eb4c26340375-1
00:02:27.806 --> 00:02:28.560
off as well.

c23f12bf-c769-4782-b17f-d7c04b592b49-0
00:02:29.320 --> 00:02:32.389
OK,
So what I've drawn here is an example of

c23f12bf-c769-4782-b17f-d7c04b592b49-1
00:02:32.389 --> 00:02:37.776
the graph of a function which has got
these two properties F of 0 = 0 and also

c23f12bf-c769-4782-b17f-d7c04b592b49-2
00:02:37.776 --> 00:02:38.800
this condition.

c05f4a64-b6dc-40dd-bd70-93f46bf7e8b1-0
00:02:39.800 --> 00:02:41.657
OK,
let's let's interpret the other

c05f4a64-b6dc-40dd-bd70-93f46bf7e8b1-1
00:02:41.657 --> 00:02:44.960
information about the integrals in the
context of this picture.

2c8908f5-a77e-4f8a-8597-ded352ed8320-0
00:02:45.800 --> 00:02:51.543
So the first thing we have is the
integral from -2 to 2 of F of X with

2c8908f5-a77e-4f8a-8597-ded352ed8320-1
00:02:51.543 --> 00:02:53.000
respect to X is 4.

9488c458-a58c-4286-8033-ce4d9ad7fc7c-0
00:02:53.640 --> 00:02:56.501
OK,
now I know if I were to integrate this

9488c458-a58c-4286-8033-ce4d9ad7fc7c-1
00:02:56.501 --> 00:03:00.960
function between -2 and +2,
the area of this would be significant.

828d7462-f80b-4f95-b749-e1350519d6ee-0
00:03:00.960 --> 00:03:04.040
So a is representing the area in here.

fe33ebfc-8b43-49bd-9b1d-eefab0f54fba-0
00:03:04.240 --> 00:03:05.360
It's a positive value.

5cae9554-d1bc-415d-b61d-a70c2fc5d516-0
00:03:05.360 --> 00:03:08.491
It's an area,
and also the area of this bit would be

5cae9554-d1bc-415d-b61d-a70c2fc5d516-1
00:03:08.491 --> 00:03:09.200
significant.

d82c7021-a599-40ec-86ca-4de762f75505-0
00:03:09.200 --> 00:03:11.925
So B is representing the area inside this
shape,

d82c7021-a599-40ec-86ca-4de762f75505-1
00:03:11.925 --> 00:03:15.040
and that is also a positive value because
it's an area.

600109c2-d9ac-4f44-a2ad-2653b8bf62c5-0
00:03:16.120 --> 00:03:18.766
OK,
So if I were to integrate F of X between

600109c2-d9ac-4f44-a2ad-2653b8bf62c5-1
00:03:18.766 --> 00:03:21.353
-2 and two,
this part would make a positive

600109c2-d9ac-4f44-a2ad-2653b8bf62c5-2
00:03:21.353 --> 00:03:25.293
contribution to that integral,
and this part would make a negative

600109c2-d9ac-4f44-a2ad-2653b8bf62c5-3
00:03:25.293 --> 00:03:28.879
contribution to that integral because
it's below the X axis.

3b0bbd69-a3f6-4c8b-9489-c36b1099ea15-0
00:03:29.480 --> 00:03:34.040
So this integral would actually turn out
to be a -, b.

2ee705d9-0539-4604-84f2-805f8144da9c-0
00:03:35.960 --> 00:03:39.280
And in the question we are told that that
value is 4.

cf9054cd-9733-4810-a9b7-39cba2e8bc66-0
00:03:39.280 --> 00:03:42.720
So we are being told that A-B in my
picture would be 4.

5fba2b32-2f05-4688-bdfd-1003020fe802-0
00:03:44.080 --> 00:03:45.658
OK,
so we've used the first bit of

5fba2b32-2f05-4688-bdfd-1003020fe802-1
00:03:45.658 --> 00:03:46.200
information.

42ee40d4-ee93-4877-b9fd-31ede120e262-0
00:03:46.200 --> 00:03:48.560
We've got this relationship between the
two areas.

ad0644b3-9019-48a6-a4f3-8ea864bf711d-0
00:03:49.080 --> 00:03:51.520
Now let's have a look at the second piece
of information.

52fffd72-74e8-4c94-9b1c-91608df3af76-0
00:03:51.520 --> 00:03:54.600
This is about the integral of the modulus
of F of X.

f3288496-d011-434b-8b46-1f1150115346-0
00:03:55.000 --> 00:03:58.505
So the first thing I want to do is think
about what the graph of the modulus of F

f3288496-d011-434b-8b46-1f1150115346-1
00:03:58.505 --> 00:04:00.600
of X would look like in this particular
example.

130e753b-e21a-493e-b9d8-460a7b8222b9-0
00:04:01.280 --> 00:04:05.617
So to get the graph of the modulus of F
of XI need to reflect any parts of the

130e753b-e21a-493e-b9d8-460a7b8222b9-1
00:04:05.617 --> 00:04:08.802
graph of F of X which are below the X
axis in the X axis,

130e753b-e21a-493e-b9d8-460a7b8222b9-2
00:04:08.802 --> 00:04:10.560
so they appear above the X axis.

0174f795-1767-49a3-be72-82d158b1622e-0
00:04:11.640 --> 00:04:15.880
Any parts which are already above the X
axis will remain in the same place.

9a6bb8c3-19da-4153-95a2-124da776c854-0
00:04:15.880 --> 00:04:19.600
So this is the graph of the modulus of F
of X.

82e0a035-3e48-4cfd-b9ad-dfbd705a9255-0
00:04:19.600 --> 00:04:24.777
It's the same for these values of X where
F of X is already positive and where F of

82e0a035-3e48-4cfd-b9ad-dfbd705a9255-1
00:04:24.777 --> 00:04:25.640
X is negative.

795a1775-1038-45d4-a064-801a42e61dde-0
00:04:25.880 --> 00:04:30.826
In these negative values of X,
we're going to get the reflection of the

795a1775-1038-45d4-a064-801a42e61dde-1
00:04:30.826 --> 00:04:32.200
graph in the X axis.

c83c14e0-7f17-4f9c-8c70-062adfa7a3d2-0
00:04:32.520 --> 00:04:34.200
So that's supposed to be the reflection
of this.

5505abf1-6f55-4aa2-b5b4-9a417fa14edc-0
00:04:34.640 --> 00:04:39.011
And as such,
this area here is identical to this area,

5505abf1-6f55-4aa2-b5b4-9a417fa14edc-1
00:04:39.011 --> 00:04:40.840
and it would also be B.

aefa8513-8f20-4b21-8685-c8858f6373da-0
00:04:40.840 --> 00:04:42.760
Remember,
B is representing the area there.

6088758d-9308-4255-90c3-501b897f61e7-0
00:04:42.760 --> 00:04:43.960
It's a positive value.

a69d91fc-1a8b-4c51-a43c-c3ad552e60c2-0
00:04:45.000 --> 00:04:49.491
OK, so if I integrate the orange function,
let's label it,

a69d91fc-1a8b-4c51-a43c-c3ad552e60c2-1
00:04:49.491 --> 00:04:54.440
it's Y equals the modulus of F of X
between -2 and two, I get 8.

c75c5ec6-37f2-48b9-a65d-06d08c6a8b73-0
00:04:55.000 --> 00:04:58.760
I can see that in terms of my A&B,
that would just be A+B.

97e0d2d3-295c-4497-9def-545ef0235aa2-0
00:04:58.760 --> 00:05:00.000
It's just a positive function.

963bd3ea-2e17-43bd-abaa-979a3b123b08-0
00:05:00.000 --> 00:05:01.720
I would get this area plus this area.

686aee9c-2005-44d5-b914-8b57ddedb6b0-0
00:05:02.080 --> 00:05:08.866
So what I'm being told here is that A+B
as the integral of the modulus of F of X

686aee9c-2005-44d5-b914-8b57ddedb6b0-1
00:05:08.866 --> 00:05:10.040
is equal to 8.

02f9097f-6aa7-4e6c-92e0-b700df9d7be1-0
00:05:11.240 --> 00:05:13.800
So I have a pair of simultaneous
equations in A&B.

7f0a9b39-d129-4e2f-a92d-bb460b805b5a-0
00:05:14.040 --> 00:05:15.600
Fortunately,
they're fairly straightforward.

f3692420-9449-45f7-9334-bf6f52af762f-0
00:05:15.600 --> 00:05:19.600
If I just add these two together,
I'll have two a = 12.

e4a9299d-c46c-4798-bb3c-9dba452bb885-0
00:05:20.320 --> 00:05:22.720
So I think I can get that a is 6 from
that.

4dcc8d98-7581-42ee-aea7-e72d00c80eee-0
00:05:23.840 --> 00:05:27.066
And then if I say consider the second one
with a being 6,

4dcc8d98-7581-42ee-aea7-e72d00c80eee-1
00:05:27.066 --> 00:05:28.680
I can see that B must be two.

d92e9624-0683-4dff-a958-f346fdb184e7-0
00:05:29.920 --> 00:05:34.236
So this these two bits of information
have enabled me to work out both of these

d92e9624-0683-4dff-a958-f346fdb184e7-1
00:05:34.236 --> 00:05:34.560
areas.

808085d2-31de-44d2-8701-69689681330f-0
00:05:34.760 --> 00:05:35.600
A&B.

fa8df7f9-6d32-4921-b4ad-e937b9a46416-0
00:05:36.320 --> 00:05:42.310
Hopefully that will be helpful in terms
of working out what this is the integral

fa8df7f9-6d32-4921-b4ad-e937b9a46416-1
00:05:42.310 --> 00:05:46.600
from -2 to 0 of F of the modulus of X
with respect to XI.

62466ae4-55e1-4e08-9ac9-6d8acfe75893-0
00:05:46.600 --> 00:05:51.249
Want to think about what it's graph will
look like in between -2 and 0 where we

62466ae4-55e1-4e08-9ac9-6d8acfe75893-1
00:05:51.249 --> 00:05:52.760
are doing the integration.

14bb64bf-7b19-49dc-9b2d-ee94d1120970-0
00:05:53.440 --> 00:05:56.709
So I know this is like one of those
standard topics that you may be able to

14bb64bf-7b19-49dc-9b2d-ee94d1120970-1
00:05:56.709 --> 00:05:59.720
sort of see this straight away,
but I just want to think this through

14bb64bf-7b19-49dc-9b2d-ee94d1120970-2
00:05:59.720 --> 00:06:00.280
from scratch.

5fe0b836-5ee7-4745-ba5e-28782eab31bf-0
00:06:00.880 --> 00:06:04.630
So I'm going to start by imagining that X
is -2 and it's going to travel to the

5fe0b836-5ee7-4745-ba5e-28782eab31bf-1
00:06:04.630 --> 00:06:07.443
right and reach 0,
and I want to think about what the graph

5fe0b836-5ee7-4745-ba5e-28782eab31bf-2
00:06:07.443 --> 00:06:08.240
looks like there.

81fbac2c-c77c-455c-8f56-670c277cf1bf-0
00:06:08.920 --> 00:06:12.138
So when X is -2,
the modulus of X would be two,

81fbac2c-c77c-455c-8f56-670c277cf1bf-1
00:06:12.138 --> 00:06:15.960
and we would apply after that and we
would get F of two.

961395c3-9fd6-4996-abef-19893a311d00-0
00:06:16.440 --> 00:06:21.352
So when X is -2 in this function,
we are getting the same as F of two over

961395c3-9fd6-4996-abef-19893a311d00-1
00:06:21.352 --> 00:06:21.680
here.

d2cb9ab9-a5e4-4fec-8e3f-aec93a13359e-0
00:06:21.680 --> 00:06:28.121
So it will be this value here,
which I'll do in blue as we move to the

d2cb9ab9-a5e4-4fec-8e3f-aec93a13359e-1
00:06:28.121 --> 00:06:29.120
right here.

c6cf1940-80fd-46f5-a84c-3d24a7c295c2-0
00:06:29.600 --> 00:06:35.320
Say we move up to -1.9, we'll get F of 1.
9 in this function.

91dc40a4-2df3-4f36-8c3d-a9f403e39bdc-0
00:06:35.800 --> 00:06:39.818
So hopefully you can see from that,
continuing to use that idea,

91dc40a4-2df3-4f36-8c3d-a9f403e39bdc-1
00:06:39.818 --> 00:06:44.640
what we're going to end up with is a
reflection of this part of y = F of X in

91dc40a4-2df3-4f36-8c3d-a9f403e39bdc-2
00:06:44.640 --> 00:06:45.320
the Y axis.

b27ae761-ef7f-4e58-9251-d940d6e4e2e0-0
00:06:46.040 --> 00:06:47.600
So I'll now attempt to draw that.

2d30e832-1b93-47f1-a673-d3674c871977-0
00:06:47.640 --> 00:06:50.760
I'm reflecting this in blue.

78c88c83-972a-46c9-a10d-af8dda7d06c9-0
00:06:50.960 --> 00:06:57.400
So this blue curve is the reflection of
this part of y = F of X in the Y axis.

b8e07098-85ed-4658-95e8-3730988ddd23-0
00:06:58.320 --> 00:07:04.538
And now we can see straight away that
given that this is the graph of y = F of

b8e07098-85ed-4658-95e8-3730988ddd23-1
00:07:04.538 --> 00:07:09.104
the modulus of X,
if we were to integrate that between -2

b8e07098-85ed-4658-95e8-3730988ddd23-2
00:07:09.104 --> 00:07:12.646
and 0,
that is exactly the same as this area

b8e07098-85ed-4658-95e8-3730988ddd23-3
00:07:12.646 --> 00:07:13.040
here.

a780e229-03e1-4f4d-b5bf-d137785abaa6-0
00:07:13.120 --> 00:07:18.160
And we can see that this is going to be A
and we now know that a is 6.

08406b25-154e-4815-8c97-da8650da3df4-0
00:07:18.280 --> 00:07:20.840
So we can say that this is equal to 6.

bcd8a4cf-6a06-4e20-a96f-7228f3d43960-0
00:07:20.840 --> 00:07:26.200
And we look over up to the left to see
our answers and we can see their six as G.

bb9600c0-bf7e-4ad0-ad0e-de2addd3dcd4-0
00:07:28.040 --> 00:07:30.160
OK, this is a very interesting question.

635c2599-5739-4add-b47e-fad0a6ce38c4-0
00:07:30.240 --> 00:07:33.098
For me,
the key thing here is how to use this

635c2599-5739-4add-b47e-fad0a6ce38c4-1
00:07:33.098 --> 00:07:33.720
condition.

3a82cb6c-be05-4c18-aae7-aed6929226b9-0
00:07:34.000 --> 00:07:39.049
So we were told that in this question,
our function F had the property that X *

3a82cb6c-be05-4c18-aae7-aed6929226b9-1
00:07:39.049 --> 00:07:43.973
F of X was greater than 0 for all non
zero values of X and we had to use that

3a82cb6c-be05-4c18-aae7-aed6929226b9-2
00:07:43.973 --> 00:07:44.920
to reduce that.

f1c0243d-114e-413d-a956-d897bbbaf471-0
00:07:44.920 --> 00:07:47.649
When X was positive,
our function was positive,

f1c0243d-114e-413d-a956-d897bbbaf471-1
00:07:47.649 --> 00:07:49.640
so it's graph was in this quadrant.

4d033e4e-c09c-4eac-86db-b75a4e2855f7-0
00:07:49.920 --> 00:07:52.560
And when X was negative,
our function was negative.

56f1c5b6-7ec7-49c8-a32b-4cf5fd4cb644-0
00:07:52.560 --> 00:07:53.880
So it's graph was in this quadrant.

8ecda0db-f46e-45b9-9300-d307bb504b04-0
00:07:53.880 --> 00:07:57.760
That was kind of the key to unlocking how
to use these integrals.

b97cc335-1759-42f5-9efa-8662930b660e-0
00:07:58.160 --> 00:08:03.838
So just going back to this condition,
imagine we were told that we have two

b97cc335-1759-42f5-9efa-8662930b660e-1
00:08:03.838 --> 00:08:07.200
values A&
B and a * b is greater than 0.

ae736269-d869-4a9d-8f78-9e53511ffeef-0
00:08:08.040 --> 00:08:13.085
That's telling us that if A is positive,
then B is positive, and if A is negative,

ae736269-d869-4a9d-8f78-9e53511ffeef-1
00:08:13.085 --> 00:08:14.240
then B is negative.

d60eae62-1ddc-4432-afa2-826908875d4d-0
00:08:14.720 --> 00:08:19.011
I guess we might imagine a variance on
that where we were told that A * b is

d60eae62-1ddc-4432-afa2-826908875d4d-1
00:08:19.011 --> 00:08:19.680
less than 0.

2c306c07-e45f-41c2-8917-e7ef2a5a95d0-0
00:08:20.240 --> 00:08:23.347
So here, if we knew that A was negative,
B must be positive,

2c306c07-e45f-41c2-8917-e7ef2a5a95d0-1
00:08:23.347 --> 00:08:26.200
and if we knew that B was negative,
A must be positive.

c6d202fe-94d2-4812-a5b9-8a3cb17c19c8-0
00:08:26.600 --> 00:08:30.560
So that's the kind of way that you can
use information presented like that.

b4a05be9-c32f-47f2-b9da-f7a531e7311f-0
00:08:30.600 --> 00:08:35.749
It's really just a very compact way of
telling you about the possibilities of

b4a05be9-c32f-47f2-b9da-f7a531e7311f-1
00:08:35.749 --> 00:08:38.720
the signs of these two values X&
F of X.