WEBVTT

d39447b9-e688-40cd-be93-50a12e0bbdbf-0
00:00:05.800 --> 00:00:06.680
Hi I'm Fiona.

26ff4bb8-b537-4766-bad6-7b2b3cb8f199-0
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Let's have a look at Tamura 2021,
paper 2 and question 20.

bfb043bf-f16e-430b-874c-ea3913cc701c-0
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We are told that the sequence of
functions F1F2F3 and so on is defined by

bfb043bf-f16e-430b-874c-ea3913cc701c-1
00:00:21.332 --> 00:00:27.880
F1 of X equals modulus of XFN plus one of
X equals the modulus of FN of X + X.

408f842e-e351-488f-8f7b-c40ab7e7557e-0
00:00:28.440 --> 00:00:32.111
We're told N is greater than or equal to
1,

408f842e-e351-488f-8f7b-c40ab7e7557e-1
00:00:32.111 --> 00:00:38.869
and we're asked to find the value of the
integral between -1 and one of F99 of X

408f842e-e351-488f-8f7b-c40ab7e7557e-2
00:00:38.869 --> 00:00:39.120
DX.

da3b74cc-00a0-42cd-88b0-c94793c413f8-0
00:00:39.440 --> 00:00:43.600
And then we're given options A to H to
choose from as our correct answer.

6ce92b66-fd85-42e2-9274-8a98ff8f27d6-0
00:00:44.720 --> 00:00:47.880
Let's start by getting a feel for these F
functions.

66043ba1-1bea-4551-96b7-d3bc4b52825c-0
00:00:48.040 --> 00:00:52.655
F1 of X is the modulus of X,
so I'm just going to draw a graph of that

66043ba1-1bea-4551-96b7-d3bc4b52825c-1
00:00:52.655 --> 00:00:53.240
function.

a1569663-5f51-4d85-91f2-1e0c1602d2f1-0
00:00:59.040 --> 00:01:05.305
So we can see for this function that
where for values of X on the right hand

a1569663-5f51-4d85-91f2-1e0c1602d2f1-1
00:01:05.305 --> 00:01:09.942
side of the Y axis that is X greater than
or equal to 0,

a1569663-5f51-4d85-91f2-1e0c1602d2f1-2
00:01:09.942 --> 00:01:13.360
this function behaves like the line y = X.

cf647d24-32c5-4aa3-b997-d122e8c2535b-0
00:01:13.840 --> 00:01:21.280
And for values of X to the left of the Y
axis that is X less than 0,

cf647d24-32c5-4aa3-b997-d122e8c2535b-1
00:01:21.280 --> 00:01:26.240
this function behaves like the line y = -,
X.

00e2e2aa-c978-4e36-8fee-9a067f2810eb-0
00:01:27.040 --> 00:01:32.748
So this is important because we're going
to need to think about these F functions

00e2e2aa-c978-4e36-8fee-9a067f2810eb-1
00:01:32.748 --> 00:01:37.343
for two different cases,
the first being values of X greater than

00e2e2aa-c978-4e36-8fee-9a067f2810eb-2
00:01:37.343 --> 00:01:41.520
or equal to 0 and the second being values
of X less than 0.

98642747-83cd-47a9-a9a1-09f2e95cfbdd-0
00:01:41.880 --> 00:01:45.440
So now let's think about F2 of X.

6fbeae7f-4dd6-4003-863b-e9dfc7c45e3a-0
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F2 of X would be the modulus of the
modulus of X + X.

919a5ed3-cc11-4e33-a215-147b73e83663-0
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So let's think about the graph of that
function.

466efd4a-aafe-4079-9f5d-8fbf41cfffd4-0
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For values of X greater than or equal to
0,

466efd4a-aafe-4079-9f5d-8fbf41cfffd4-1
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what we would have in the function rule
or the function equation would be X + X

466efd4a-aafe-4079-9f5d-8fbf41cfffd4-2
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and that would be two X.

ce6184ed-1fbb-4d0d-8af0-f1493be08baf-0
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So we would have the line 2X.

c9781949-8d5d-491a-892d-13db986ca604-0
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Let me draw it in orange here just to
distinguish it.

cac8c38f-6820-40ee-9f79-c018563a4d68-0
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So this function would behave like the
line y = 2,

cac8c38f-6820-40ee-9f79-c018563a4d68-1
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X for X greater than or equal to 0.

7e9c4880-2757-49f3-8759-cae84143c05d-0
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What about for X less than 0?

20daa22f-1a81-4f51-bbf8-8fc8c584c993-0
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Well,
in this modulus sign we would have minus

20daa22f-1a81-4f51-bbf8-8fc8c584c993-1
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X + X which would be the modulus of 0,
and that would be this line here which is

20daa22f-1a81-4f51-bbf8-8fc8c584c993-2
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the X axis.

6dc75446-855a-4a4b-9112-fd8a04324f10-0
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So that would be the line y = 0.

7b1f00eb-ae2f-4786-9a8c-ddd1e3cc6b16-0
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I'm going to transfer the graph of F2 of
X up under the graph of F1 of X so we can

7b1f00eb-ae2f-4786-9a8c-ddd1e3cc6b16-1
00:02:48.956 --> 00:02:51.800
compare and keep that for reference.

400c44be-469c-4e39-b021-8a1c95431fd0-0
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Now let's think of F3 of X.

1223d3b3-277a-420e-9455-b64624b165ec-0
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To generate F3 of X,
I'm going to be taking F2 of X, adding X,

1223d3b3-277a-420e-9455-b64624b165ec-1
00:03:08.480 --> 00:03:11.040
and then taking the modulus.

8bf6a8df-4838-42e9-a295-04de4613fd49-0
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Let's think about our two cases.

9b966afa-c181-48b2-adfd-8d6f088c4988-0
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So first of all thinking about values of
X on the right hand side of the Y axis.

b33ba42e-4cfc-4aa5-b474-678f544849de-0
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So values of X greater than or equal to
zero.

0f856d26-5cc0-4d83-9c4b-7ad297f9f3d7-0
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Well,
F2 of X behaves like the line y = 2 X.

bb90802f-00f8-4aee-acea-4c3a3a1f2a94-0
00:03:25.640 --> 00:03:30.251
Adding an X to that,
I get 3X and then if I take the modulus,

bb90802f-00f8-4aee-acea-4c3a3a1f2a94-1
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it remains unchanged because it's already
positive.

decbd91c-00f2-425f-88ba-ebf604081c11-0
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So for F3 of X,
for values of X greater than or equal to

decbd91c-00f2-425f-88ba-ebf604081c11-1
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0 behaves like y = 3 X.

8b38ed48-d99a-4178-a62b-aa0eea119e4e-0
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Now let's think about the second case,
which is values of X less than 0.

1d312808-75e9-463e-91ac-5676b9aaa309-0
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F2 of X behaves like the line y = 0.

146b4499-c76a-4cc9-ae56-22c10b7ccec7-0
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Adding an X gives me X.

5a2d4f55-88b9-43d6-bbf1-6283756d3cdd-0
00:03:56.360 --> 00:04:03.239
If I were to just graph this so the line
y = X for X less than 0 would give me

5a2d4f55-88b9-43d6-bbf1-6283756d3cdd-1
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this dotted line.

85540d13-de1a-4048-844e-f5f2878b819a-0
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But of course I'm going to take the
modulus of that,

85540d13-de1a-4048-844e-f5f2878b819a-1
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which means that I get the line y = -,
X on this side of the Y axis.

38018bea-8786-4c73-a62c-343a29da6835-0
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Now that means that the graph of F3 of X
is what I have here in yellow.

b8f6d6ac-5b23-4bca-a2fc-90a89c89c3e1-0
00:04:20.320 --> 00:04:23.720
I'm going to transfer that over to the
right hand side for reference.

7b217e14-935e-4110-9f40-84fcf8b7307f-0
00:04:28.680 --> 00:04:32.910
We're starting to get a sense that there
might be a pattern here for values of X

7b217e14-935e-4110-9f40-84fcf8b7307f-1
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greater than or equal to 0.

0a9de9d3-00d8-45c2-a424-56e866c2e1f7-0
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We can see that as N increases,
the coefficient on X increases here and

0a9de9d3-00d8-45c2-a424-56e866c2e1f7-1
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we get lines Y equals XY equals 2,
XY equals 3X,

0a9de9d3-00d8-45c2-a424-56e866c2e1f7-2
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or really Y equals NX for each N on the
left hand side of the Y axis.

0c980551-fb86-4185-836c-0f0570f665a0-0
00:04:51.400 --> 00:04:57.218
So for X less than 0,
we can see that when we got to F3,

0c980551-fb86-4185-836c-0f0570f665a0-1
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we were back at the same result for F1.

024feaf7-972c-4069-baa3-7440d66a959f-0
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And so we can see that that pattern would
continue.

d01c3e07-3217-490c-ab0e-08be12928c44-0
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Now, if you're in an exam,
you probably don't want to give time to

d01c3e07-3217-490c-ab0e-08be12928c44-1
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drawing lots of these F functions,
especially not all the way up to F99.

225b2be4-66d4-4baf-b756-40774b19383c-0
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But we just want to get confirmation that
this pattern is in fact going to emerge.

480dd186-c8e8-4807-aeda-31481dacc7e9-0
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So I'm not in an exam,
so I'm going to draw more of these F

480dd186-c8e8-4807-aeda-31481dacc7e9-1
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functions so that we can see this pattern
that does emerge.

c746960f-517f-40fd-a30e-14ebf4a239c7-0
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Here's F4 of X.

2691feec-34b8-44fe-97a9-60c5c369dd28-0
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Here's F5 of X.

dde7a028-e925-4ab4-9fe7-eea2687777a8-0
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Here's F6 of X.

2e62d126-642b-46b1-8a8e-b45a68980b98-0
00:06:02.160 --> 00:06:11.433
Here's F7 of X and so on,
so we can see that this pattern does in

2e62d126-642b-46b1-8a8e-b45a68980b98-1
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fact emerge.

8069e49b-71b7-457d-b56a-2edfb0a632b0-0
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And now we want to just think about what
happens when N is odd.

9bc9d130-f2f3-429c-9824-3a1e5835d6e0-0
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Because 99 is odd,
we're asked for the integral between -1

9bc9d130-f2f3-429c-9824-3a1e5835d6e0-1
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and one of F99 of X with respect to X.

66264737-8544-49a1-ba9e-29a61e9367d0-0
00:06:27.880 --> 00:06:32.480
And so that will be the graph under the
function F 99.

64992da9-38d9-445b-84a8-9001b8e9e884-0
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And so when we graph F 99,
because of this pattern,

64992da9-38d9-445b-84a8-9001b8e9e884-1
00:06:37.758 --> 00:06:44.625
what we expect to see on the right hand
side for values of X greater than or

64992da9-38d9-445b-84a8-9001b8e9e884-2
00:06:44.625 --> 00:06:47.480
equal to 0 is the line y = 99 X.

578675ad-213a-434c-a565-706fff2afc81-0
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What we expect to see on the left hand
side for values of X less than 0 is the

578675ad-213a-434c-a565-706fff2afc81-1
00:06:55.265 --> 00:06:56.560
line y = -, X.

b185b3d8-b713-4a6e-9505-68ce8894fb66-0
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Now let's mark where -1 and one is.

cee2312f-df26-483d-9cf5-65281153fb7c-0
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And we know that the integral of a
function between the limits of

cee2312f-df26-483d-9cf5-65281153fb7c-1
00:07:10.474 --> 00:07:15.880
integration is the area between that
function and the X axis.

2854a43e-fa9e-4e63-95f5-33f69aee5228-0
00:07:16.160 --> 00:07:22.186
And so for the value of this integral,
we're looking for this shaded area of

2854a43e-fa9e-4e63-95f5-33f69aee5228-1
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here, here plus this shaded area here.

9c03e004-158b-4a88-af84-8566e518798f-0
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Well, on the left hand side,
if I imagine a unit circle,

9c03e004-158b-4a88-af84-8566e518798f-1
00:07:29.563 --> 00:07:34.713
so let's imagine that that's one there,
then this blue shaded region would be

9c03e004-158b-4a88-af84-8566e518798f-2
00:07:34.713 --> 00:07:36.760
half the area of a unit circle.

9a231d31-11fc-4747-aa96-1e3f2466202f-0
00:07:36.960 --> 00:07:39.120
So the area would be 1/2.

e79e11e0-f930-40f2-81bf-527dd67266fa-0
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For this triangle,
the area would be 1/2 the base times the

e79e11e0-f930-40f2-81bf-527dd67266fa-1
00:07:44.124 --> 00:07:48.683
perpendicular height,
which is in this case 1/2 * 1 which is

e79e11e0-f930-40f2-81bf-527dd67266fa-2
00:07:48.683 --> 00:07:52.720
1/2 times the perpendicular height,
which will be 99.

72ebb07b-c90f-43e2-ae70-a21213cdafeb-0
00:07:53.040 --> 00:07:57.600
And if I add these two areas together,
I'll get the value of this integral.

9d1f8db8-4559-4a48-bad0-4afad2d0ee6d-0
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So this would be 1/2 + 99 halves,
so that would be 100 halves, which is 50.

fe5efe6b-9c23-433c-a035-556d2f171421-0
00:08:07.560 --> 00:08:11.362
And then I look at my options to choose
from and I can see that option E gives me

fe5efe6b-9c23-433c-a035-556d2f171421-1
00:08:11.362 --> 00:08:12.800
the value that I'm looking for.

390a0205-6784-4a98-8e67-6ebf77d47a60-0
00:08:13.000 --> 00:08:15.360
So option E is the answer to this
question.

2167f65d-4a4b-48b4-8e3c-1fdfb432839b-0
00:08:16.000 --> 00:08:17.960
Let's take a moment to reflect on this
question.

3c54595b-870e-4401-a358-02db399e5190-0
00:08:18.240 --> 00:08:24.242
It was important in this question for me
to be familiar with the modulus function,

3c54595b-870e-4401-a358-02db399e5190-1
00:08:24.242 --> 00:08:29.160
and in particular to be aware that there
are two cases to consider.

6bfe3947-5778-4fc6-a19c-177eac36063f-0
00:08:29.160 --> 00:08:33.596
So the case when X is greater than or
equal to 0 and the case when X is less

6bfe3947-5778-4fc6-a19c-177eac36063f-1
00:08:33.596 --> 00:08:34.000
than 0.

1d1916a2-808c-49df-9054-3f1004e42a6a-0
00:08:34.000 --> 00:08:37.000
That allowed me to generate these graphs
accurately.

8abf25a4-5ee5-4716-90c2-4be553d004e2-0
00:08:37.760 --> 00:08:40.000
Then I started to see a spot.

075a9f5a-70f9-4899-be66-e6a5d7706194-0
00:08:40.280 --> 00:08:47.243
I started to spot a pattern emerging and
that pattern was that for values of N

075a9f5a-70f9-4899-be66-e6a5d7706194-1
00:08:47.243 --> 00:08:52.268
that were even my,
the graph of my function behaved more

075a9f5a-70f9-4899-be66-e6a5d7706194-2
00:08:52.268 --> 00:08:54.560
like the graph of F2 of X.

27c79a94-058c-4e99-9ab7-be9c973cd48b-0
00:08:54.920 --> 00:08:59.380
And for values of N that were odd,
which is what I was interested in because

27c79a94-058c-4e99-9ab7-be9c973cd48b-1
00:08:59.380 --> 00:08:59.960
99 is odd.

866dfe99-7e86-40a6-8764-086acfc52aa7-0
00:09:00.360 --> 00:09:06.400
The graph of these F functions behave
most behave like the graph of F3 of X,

866dfe99-7e86-40a6-8764-086acfc52aa7-1
00:09:06.400 --> 00:09:12.284
where on the right hand side I the
function behaves like the line Y equals

866dfe99-7e86-40a6-8764-086acfc52aa7-2
00:09:12.284 --> 00:09:12.520
NX.

fd4f0d52-e7e0-4cf7-b4f4-95b904240add-0
00:09:12.760 --> 00:09:16.840
And on the left hand side the function is
always y = -, X.

1d8ab165-c1de-4692-a678-62b8248b9fbb-0
00:09:17.080 --> 00:09:22.201
And spotting that pattern was what
allowed me to be confident to be able to

1d8ab165-c1de-4692-a678-62b8248b9fbb-1
00:09:22.201 --> 00:09:25.571
graph F99 of X and to then think about
this area,

1d8ab165-c1de-4692-a678-62b8248b9fbb-2
00:09:25.571 --> 00:09:30.760
which was equivalent to the value of this
integral that I was asked to find.