WEBVTT

3879bcf9-2068-4a79-b12c-d81fc60424ef-0
00:00:05.360 --> 00:00:09.425
Hi,
I'm Fiona and this is Tamila 2021 Paper 2

3879bcf9-2068-4a79-b12c-d81fc60424ef-1
00:00:09.425 --> 00:00:10.840
and question 12.

f690d7ff-b5b2-4adc-af7d-4aef6597cf3f-0
00:00:11.600 --> 00:00:16.227
We are asked which of the following
statements about polynomials F&

f690d7ff-b5b2-4adc-af7d-4aef6597cf3f-1
00:00:16.227 --> 00:00:17.320
G is or are true.

bf5194b8-d8c0-4740-b0a5-f5078af1d974-0
00:00:17.640 --> 00:00:23.479
And then we're given three statements to
investigate to see whether they are true

bf5194b8-d8c0-4740-b0a5-f5078af1d974-1
00:00:23.479 --> 00:00:24.120
or false.

cd4ffe70-3e0f-49c7-a3c1-95705f4ff6e5-0
00:00:24.400 --> 00:00:28.240
So let's go through each statement in
turn and investigate them.

857a01f7-91b5-4914-9cd1-a9426b8ca3e2-0
00:00:28.560 --> 00:00:34.560
Statement one starts with the condition
that the function F needs to be greater

857a01f7-91b5-4914-9cd1-a9426b8ca3e2-1
00:00:34.560 --> 00:00:36.960
than or equal to the function G.

fe7101f6-6848-4f2e-ad5e-f477d0fd75f9-0
00:00:36.960 --> 00:00:40.280
So I'm going to draw a scenario in which
that is true.

42b431f7-ac1e-4189-bae6-2b502dab7360-0
00:00:47.560 --> 00:00:53.013
So here we have function F of X which
I've drawn in orange and G of X which

42b431f7-ac1e-4189-bae6-2b502dab7360-1
00:00:53.013 --> 00:00:54.520
I've drawn in purple.

588a6a59-e0b0-420b-96e5-0afc8276effb-0
00:00:54.760 --> 00:01:02.374
And let's say that I have drawn those two
functions between the values zero and X

588a6a59-e0b0-420b-96e5-0afc8276effb-1
00:01:02.374 --> 00:01:10.174
because the next part of the statement is
all about the integral of these functions

588a6a59-e0b0-420b-96e5-0afc8276effb-2
00:01:10.174 --> 00:01:12.960
between the values zero and X.

9ba2a03a-3db0-4c11-ab1e-eaa5e82596d5-0
00:01:13.600 --> 00:01:17.497
Now when we think about the integral of
of a function,

9ba2a03a-3db0-4c11-ab1e-eaa5e82596d5-1
00:01:17.497 --> 00:01:23.165
we can think of that as being the area
enclosed between that function and the X

9ba2a03a-3db0-4c11-ab1e-eaa5e82596d5-2
00:01:23.165 --> 00:01:23.520
axis.

07973e61-8a26-40aa-9099-31fa7b2448d8-0
00:01:23.720 --> 00:01:27.353
If the function is positive and above the
X axis,

07973e61-8a26-40aa-9099-31fa7b2448d8-1
00:01:27.353 --> 00:01:32.585
then the area value will contribute
positively towards the value of the

07973e61-8a26-40aa-9099-31fa7b2448d8-2
00:01:32.585 --> 00:01:33.240
integral.

a6bd7731-e7e8-4a48-8337-3ef7aab3e196-0
00:01:33.600 --> 00:01:38.739
If the function is below the X axis and
therefore has negative values,

a6bd7731-e7e8-4a48-8337-3ef7aab3e196-1
00:01:38.739 --> 00:01:44.458
then the area value for that the value of
that area will contribute negatively

a6bd7731-e7e8-4a48-8337-3ef7aab3e196-2
00:01:44.458 --> 00:01:46.919
towards the value of the integral.

f8d12dfc-2b7e-4211-afee-8695a915d9ca-0
00:01:47.240 --> 00:01:54.976
So thinking about the integral as areas,
we can see that the integral of F of T

f8d12dfc-2b7e-4211-afee-8695a915d9ca-1
00:01:54.976 --> 00:02:03.098
with respect to T would give us the value
and the positive values of this area here

f8d12dfc-2b7e-4211-afee-8695a915d9ca-2
00:02:03.098 --> 00:02:06.000
that I'm shading in in orange.

58fb8dc6-37f4-404b-8347-98a9677bf9a3-0
00:02:06.320 --> 00:02:14.083
And the integral of G of T with respect
to T would give us this purple area that

58fb8dc6-37f4-404b-8347-98a9677bf9a3-1
00:02:14.083 --> 00:02:16.000
I'm shading in here.

e3498eae-81b8-49bf-a12e-6c5ecebffa0f-0
00:02:16.440 --> 00:02:23.799
Now I can see that in this example that
the value given with the orange shaded

e3498eae-81b8-49bf-a12e-6c5ecebffa0f-1
00:02:23.799 --> 00:02:30.320
region is greater than the value given
with the purple shaded region.

b7e02eef-a90c-45e2-9389-57b8eb7cca3d-0
00:02:30.560 --> 00:02:36.649
And with a bit of thought I can see that
that would actually be true no matter

b7e02eef-a90c-45e2-9389-57b8eb7cca3d-1
00:02:36.649 --> 00:02:42.200
what I chose as my functions F&
G as long as this condition is met.

f1fc86f5-1944-4a14-a398-cb2bb0a0fd7b-0
00:02:42.400 --> 00:02:46.333
So if F of X is greater than or equal to
G of X,

f1fc86f5-1944-4a14-a398-cb2bb0a0fd7b-1
00:02:46.333 --> 00:02:52.755
that means F lies above or in line with
the function G for all values of X that

f1fc86f5-1944-4a14-a398-cb2bb0a0fd7b-2
00:02:52.755 --> 00:02:54.200
we're considering.

0ecfde58-97df-45c4-a2e2-a89d9200ee47-0
00:02:54.920 --> 00:03:00.378
So we can see that that does force the
integral of F of T with respect to T

0ecfde58-97df-45c4-a2e2-a89d9200ee47-1
00:03:00.378 --> 00:03:05.981
between zero and X to be greater than or
equal to the integral of G of T with

0ecfde58-97df-45c4-a2e2-a89d9200ee47-2
00:03:05.981 --> 00:03:08.279
respect to T between zero and X.

1f5f4700-f30e-4860-ab99-3710cc898b0f-0
00:03:08.640 --> 00:03:11.920
Let's just think of one more example to
further convince ourselves.

745149bf-591e-43d3-8233-acbd98cb34a5-0
00:03:11.920 --> 00:03:14.440
Because what if these functions were
negative?

353799a4-525d-4a09-9525-684e58a74a17-0
00:03:14.600 --> 00:03:17.960
So let's say that they lay below the X
axis.

0a6840f9-e483-4611-90e2-97c4f0ae6bd7-0
00:03:17.960 --> 00:03:19.440
So here's my X axis.

02ee33ec-6575-4a3b-82e2-fb567ea26339-0
00:03:19.680 --> 00:03:21.000
Here's my Y axis.

a532d81f-6a93-468f-8ef4-41ee98ea31f6-0
00:03:21.560 --> 00:03:29.063
Then let's say that F is somewhere here,
so that's F of X&G is somewhere here,

a532d81f-6a93-468f-8ef4-41ee98ea31f6-1
00:03:29.063 --> 00:03:30.600
so that's G of X.

f9cade00-ab80-447c-89a2-84bde2c825ad-0
00:03:30.600 --> 00:03:35.456
Just keep continuing to consider the
example of these two functions,

f9cade00-ab80-447c-89a2-84bde2c825ad-1
00:03:35.456 --> 00:03:41.088
but now drawing them so that they have
negative values and they lie below the X

f9cade00-ab80-447c-89a2-84bde2c825ad-2
00:03:41.088 --> 00:03:41.440
axis.

fb15b20f-4566-4eaa-9282-092fdad1e1af-0
00:03:42.440 --> 00:03:43.880
Well, let's look at these areas.

e32a383c-4187-4b6f-920a-e65dfa202f84-0
00:03:44.120 --> 00:03:51.547
So the area represented by the integral
of F of T with respect to T would be this

e32a383c-4187-4b6f-920a-e65dfa202f84-1
00:03:51.547 --> 00:03:58.250
area and the area represented by the
integral of G of T with respect to T

e32a383c-4187-4b6f-920a-e65dfa202f84-2
00:03:58.250 --> 00:04:01.240
would be this purple shaded area.

3dbb11a6-8d17-4ff1-aaf8-781176889951-0
00:04:01.600 --> 00:04:06.462
And we can see that because these would
be negative values,

3dbb11a6-8d17-4ff1-aaf8-781176889951-1
00:04:06.462 --> 00:04:11.000
the this this part of the statement is
still satisfied.

1d254c00-45d8-4e6e-ab1b-1600224c25d0-0
00:04:11.280 --> 00:04:13.080
So statement one is true.

57647685-f5bd-4258-997d-c539bbfa0ef7-0
00:04:13.080 --> 00:04:17.680
And I'm going to give that a tick just to
keep tracking as I go through.

f3dbab6d-5ac7-4db7-b48a-d770e7c7a30d-0
00:04:18.480 --> 00:04:20.560
Now let's look at statement two.

218caeba-8006-4a84-9ea0-0580ee80db28-0
00:04:20.920 --> 00:04:26.439
Statement two starts with the same
condition that F of X needs to be greater

218caeba-8006-4a84-9ea0-0580ee80db28-1
00:04:26.439 --> 00:04:28.160
than or equal to G of X.

5852ec39-9939-40cb-93b6-29a1f58f93e8-0
00:04:28.160 --> 00:04:34.573
So let's keep with this example,
and statement 2 is asking us to consider,

5852ec39-9939-40cb-93b6-29a1f58f93e8-1
00:04:34.573 --> 00:04:41.500
does the fact that F of X being greater
than G of X force the derivative of F of

5852ec39-9939-40cb-93b6-29a1f58f93e8-2
00:04:41.500 --> 00:04:45.520
X to be greater than the derivative of G
of X?

6e6c177c-4639-413c-a4d8-5248d8fe7769-0
00:04:45.840 --> 00:04:49.437
Well,
the derivative represents the gradient of

6e6c177c-4639-413c-a4d8-5248d8fe7769-1
00:04:49.437 --> 00:04:52.360
the function, and so let's have a look.

66c2be51-ed93-4e4e-9216-34e14f18ba0e-0
00:04:52.560 --> 00:04:56.680
I'm going to just section off this part
of my graph.

b2b59746-5a68-4138-9fd6-ef7346084217-0
00:04:57.880 --> 00:05:02.683
Let's look at the derivative of F and the
derivative of G within the box that I've

b2b59746-5a68-4138-9fd6-ef7346084217-1
00:05:02.683 --> 00:05:03.320
just drawn.

63e4b40d-f0ff-4774-a8a0-89f7125dabe8-0
00:05:03.880 --> 00:05:08.218
Well, within the box,
the derivative of X is either 0 or

63e4b40d-f0ff-4774-a8a0-89f7125dabe8-1
00:05:08.218 --> 00:05:11.719
negative,
and the derivative of G is actually

63e4b40d-f0ff-4774-a8a0-89f7125dabe8-2
00:05:11.719 --> 00:05:17.960
positive all the way through the portion
of the function that's lying in the box.

ddb936ac-6901-4355-888c-09011ab9d3cc-0
00:05:18.400 --> 00:05:26.118
And so this provides A counterexample and
shows a example for which G-X is actually

ddb936ac-6901-4355-888c-09011ab9d3cc-1
00:05:26.118 --> 00:05:27.680
greater than F-X.

16fa0082-a7a8-42b6-8955-c7550060cfca-0
00:05:27.920 --> 00:05:31.313
And so that shows us that statement 2 is
not true,

16fa0082-a7a8-42b6-8955-c7550060cfca-1
00:05:31.313 --> 00:05:36.236
because if we find a counterexample,
which is an example that disproves a

16fa0082-a7a8-42b6-8955-c7550060cfca-2
00:05:36.236 --> 00:05:39.297
claim,
then our claim is disproved and we can

16fa0082-a7a8-42b6-8955-c7550060cfca-3
00:05:39.297 --> 00:05:41.160
say this statement is false.

2b30b701-d148-4d68-a9af-b2d0f4f7674e-0
00:05:41.640 --> 00:05:47.209
I'm going to give an X there to just keep
that in mind as I track when I come to

2b30b701-d148-4d68-a9af-b2d0f4f7674e-1
00:05:47.209 --> 00:05:47.760
the end.

0aa4966a-4b5a-4950-8e5f-61f085d57385-0
00:05:48.400 --> 00:05:50.080
Now let's look at statement 3.

bc7f933e-1a61-4efa-8271-3320a361863c-0
00:05:50.320 --> 00:05:54.200
So statement three starts with a
different condition.

b1053cc9-1ced-4d61-8c62-807a1d893883-0
00:05:54.200 --> 00:06:01.568
Now we're looking at a condition for
which the gradient of F is greater than

b1053cc9-1ced-4d61-8c62-807a1d893883-1
00:06:01.568 --> 00:06:04.440
or equal to the gradient of G.

ff776375-67d3-49ce-bddc-f97ce601c2f6-0
00:06:04.440 --> 00:06:06.080
So let's draw that scenario.

3635432b-7b20-4642-a02e-9363190e4c4c-0
00:06:11.400 --> 00:06:15.092
OK,
here I have drawn an example in which the

3635432b-7b20-4642-a02e-9363190e4c4c-1
00:06:15.092 --> 00:06:21.515
gradient of the function G of X is either
0 or negative and the gradient of the

3635432b-7b20-4642-a02e-9363190e4c4c-2
00:06:21.515 --> 00:06:22.799
function F of X.

abfafdcb-bd27-4cee-bce3-924a6dacb451-0
00:06:22.800 --> 00:06:26.995
Here I've just chosen a linear function,
so the gradient is constant and it's

abfafdcb-bd27-4cee-bce3-924a6dacb451-1
00:06:26.995 --> 00:06:27.480
positive.

b5327a39-e0c9-463d-af4c-719b2a913df5-0
00:06:27.720 --> 00:06:31.560
And so I've drawn a scenario in which
this condition is satisfied.

66f2d8ad-d8c9-46e9-a43a-f72ae0947a1b-0
00:06:31.920 --> 00:06:38.720
Now let's look at whether this condition
forces the function F to be greater than

66f2d8ad-d8c9-46e9-a43a-f72ae0947a1b-1
00:06:38.720 --> 00:06:40.960
or equal to the function G.

650cfa26-51de-48c8-8d7d-93c8a4eb448b-0
00:06:41.160 --> 00:06:45.880
Well, just in the example I've drawn,
we can see that F is lying under G.

f37a3610-e9ad-48ed-870e-3507bfac605d-0
00:06:46.120 --> 00:06:51.558
That means that in this example,
G of X is actually less than or equal to

f37a3610-e9ad-48ed-870e-3507bfac605d-1
00:06:51.558 --> 00:06:55.600
F of X and so therefore giving another
counterexample.

a85e8395-7256-4c25-955f-da88139fac7c-0
00:06:55.840 --> 00:07:00.080
So that shows us that statement 3 is
actually false.

128113dd-4394-489a-b246-efebf8b83b7e-0
00:07:00.120 --> 00:07:05.396
So I'm going to put an X there and I can
see as I bring everything together that

128113dd-4394-489a-b246-efebf8b83b7e-1
00:07:05.396 --> 00:07:09.240
statement one is the only statement that
is actually true.

386bb7e3-57e4-41d1-bd49-3172930457ba-0
00:07:09.480 --> 00:07:15.799
So looking at my options to choose,
I can see that B is the option that one

386bb7e3-57e4-41d1-bd49-3172930457ba-1
00:07:15.799 --> 00:07:16.880
only is true.

82efdf91-1e20-41cd-b81c-3da642dbf5a7-0
00:07:17.080 --> 00:07:19.680
And so option B is the answer to this
question.