WEBVTT

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Hi,
I'm Fiona and this is Tamua 2021 paper

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one.

451f8740-2532-423a-b924-069df534c582-0
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In question 15,
we're given a diagram that shows the

451f8740-2532-423a-b924-069df534c582-1
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graph of y = a function F of X,
and we're told that the graph consists of

451f8740-2532-423a-b924-069df534c582-2
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alternating straight line segments of
gradient 1 and -1 and continues in this

451f8740-2532-423a-b924-069df534c582-3
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way for all values of X.

84b43743-3d01-4293-8077-b782e5cfbeb3-0
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We're given the function G,
which is defined as the sum from r = 1 to

84b43743-3d01-4293-8077-b782e5cfbeb3-1
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10 of F of 2 to the power of r - 1 X,
and then we're asked to find the value of

84b43743-3d01-4293-8077-b782e5cfbeb3-2
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the integral of G of X with respect to X
between zero and one.

bcab4f3d-58ac-4f36-a135-7d889e813c57-0
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So the first thing I want to do is to get
a feel of this function G of X by

bcab4f3d-58ac-4f36-a135-7d889e813c57-1
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considering each component,
each of these functions as our changes in

bcab4f3d-58ac-4f36-a135-7d889e813c57-2
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value.

0c6f1571-5a27-4eb4-b502-660975d9035a-0
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So I'm going to make a table.

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So here's my table.

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Let's consider what we get for each value
of R When r = 1,

33a8d013-bb7d-457f-8b67-99e7b5d13262-1
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we get F of 2 to the power of 0,
which is 1 * X, so just F of X.

46efa06a-1c36-4eda-a7e0-ce86d17a042d-0
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When R is 2, we get F of 2X.

ead4c41a-57c5-4502-9c2b-27ffa1b186b2-0
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When R is 3,
we get F of 2 to the power of 2,

ead4c41a-57c5-4502-9c2b-27ffa1b186b2-1
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so 2 ^2 we get F of 4X,
and so we're going up in powers of two.

e66f7446-e68e-4114-a61a-debf4ddee27b-0
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So when r = 4, we will get F of 8X.

b142c38d-3309-454c-9219-97e8b8b711fb-0
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Now let's think of the graph of each of
these functions.

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So we're going to need to think of these
functions between zero and one.

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We can see that F lies above the X axis
and is positive between X = 0 and X = 1.

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And so when we're thinking about
evaluating this integral,

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we're going to be thinking about the area
under each of these F functions.

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So in my table now I'm going to just
consider a little sketch of the graph of

d01f4df1-47cb-4939-9445-e348f6b28db1-1
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each of these components.

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So the graph of F of X between zero and
one is just what I'm given here.

ec14a92b-f07d-40bb-b63b-bc6a384c5398-0
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And if I think about this as being half
of a unit square,

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then the area represented here will be
will take a value of 1/2.

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Now let's think of F of 2X.

6069a8a6-456b-4bf7-8b9b-67c36f060e40-0
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Well, let me just input a value of 1.

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And when I input a value of 1,
I get F of two, which will take me here.

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And so the graph of F of 2X between zero
and one will look like this.

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And now if I think about what a unit
square would look like here,

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I can also divide that in half and see
that it's made-up of four triangles of

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equal area.

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And so the area here would also take a
value of 1/2.

4e4ba844-0d7b-47eb-a5ea-e650dc7c6466-0
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Now let's look at F of 4X between zero
and one.

9b8bc1c1-590e-442d-9751-e75436a8ff79-0
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Now if I put a value of X = 1 in here,
I get F of four.

5539ea6b-b6f4-4476-ad89-67e9b681cecc-0
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And so I can see what that looks like
here to apply my scaling to this graph.

afcb940b-3d42-40ab-8968-8e6aee05410d-0
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And I end up with a graph that looks
something like this.

e434920e-4535-4a42-b0fe-e9c15947a925-0
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And I can see that the area here also
gives a value of 1/2.

c3b18ff0-41fe-4719-aecd-09251965d9b9-0
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Now I could continue filling in this
table,

c3b18ff0-41fe-4719-aecd-09251965d9b9-1
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but what I can see here is that for each
component F of XF of two, XF of four,

c3b18ff0-41fe-4719-aecd-09251965d9b9-2
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XF of 8X and so on,
I'm getting a value for the area of 1/2.

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And so when I think about the integration
that I'm asked to perform,

adc25e30-1664-4931-925a-33404b5af8bb-1
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this is the integration of a sum of
functions which is the same as the sum of

adc25e30-1664-4931-925a-33404b5af8bb-2
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the integral of each of those functions.

51abbb76-bd3d-4332-9468-82678471d304-0
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And now I know that to get the value of
this integral, I just need to add 1/2.

ab0f7024-d411-4cf1-b7b1-73b80849e465-0
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Now because R goes from 1:00 to 10:00,
I need 10 of these halves and so my

ab0f7024-d411-4cf1-b7b1-73b80849e465-1
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calculation is 10 * 1/2 which equals 5.

9e3621fa-c4cb-45b1-a829-7af4b11defd0-0
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And looking at the options here,
I can see that that's option C,

9e3621fa-c4cb-45b1-a829-7af4b11defd0-1
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which is the answer to this question.